On partial fractional Sturm–Liouville equation and inclusion

Zohreh Zeinalabedini Charandabi; Hakimeh Mohammadi; Shahram Rezapour; Hashem Parvaneh Masiha

doi:10.1186/s13662-021-03478-7

On partial fractional Sturm–Liouville equation and inclusion

Charandabi, Zohreh Zeinalabedini; Mohammadi, Hakimeh; Rezapour, Shahram; Masiha, Hashem Parvaneh 2021-07-08 00:00:00 sh.rezapour@mail.cmuh.org.tw; The Sturm–Liouville diﬀerential equation is one of interesting problems which has rezapourshahram@yahoo.ca Department of Mathematics, been studied by researchers during recent decades. We study the existence of a Azarbaijan Shahid Madani solution for partial fractional Sturm–Liouville equation by using the α-ψ-contractive University, Tabriz, Iran 4 mappings. Also, we give an illustrative example. By using the α-ψ-multifunctions, we Department of Medical Research, China Medical University Hospital, prove the existence of solutions for inclusion version of the partial fractional China Medical University, Taichung, Sturm–Liouville problem. Finally by providing another example and some ﬁgures, we Taiwan try to illustrate the related inclusion result. Full list of author information is available at the end of the article MSC: Primary 34A08; secondary 34A12 Keywords: α-ψ-contractive map; Inclusion problem; The Caputo derivative; The partial fractional Sturm–Liouville equation; Two variables partial diﬀerential equation 1 Introduction It can be said that most physical or engineering phenomena can be modeled with some categories such time-dependent (or time-fractional), fractional diﬀerential and some vari- ables partial equations. One can ﬁnd many published papers on delayed time-fractional problems, fractional diﬀerential equations [1–30] and some variables partial fractional problems [31–36]. During the history of mathematics, physics and engineering, we can ﬁnd many equations which have a special role in progress of these sciences. One of the important frameworks of problems is the Sturm–Liouville diﬀerential equation (in brief SLDE) have been in the spotlight of the mathematicians of applied mathematics, engi- neering and scientists of physics, quantum mechanics, classical mechanics (see, [37, 38] and the references therein). In such a manner, it is important that mathematicians and researchers design complicated and more general abstract mathematical models of pro- cedures in the format of applicable fractional SLDE [33, 39–41]. One can ﬁnd a variety of recent work about this equation, but the aim of this work is studying partial version of the Sturm–Liouville diﬀerential equation. ˆ ˆ ˆ ˆ ˆ Let k =(k , k )where k , k >0 and J =[0, a ]and J =[0, b ]where a , b >0.For σ ∈ 1 2 1 2 a 0 b 0 0 0 0 0 1 1 L (J × J , R)= L (J × J ), the partial left-sided mixed Riemann–Liouville integral a b a b 0 0 0 0 © The Author(s) 2021. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Charandabi et al. Advances in Diﬀerence Equations (2021) 2021:323 Page 2 of 18 (of order k)isdeﬁnedby(see[42]) ˆ ˆ p q k –1 k –1 1 2 (p – s) (q – t) I σ p , q = σ (s, t) dt ds. ˆ ˆ 0 0 (k )(k ) 1 2 Also the partial derivative in the sense of Caputo (of order k)isdeﬁnedby ˆ ˆ k 1–k D σ p , q = I σ p , q c 0 ∂p ∂q ˆ ˆ p q k –1 k –1 2 1 2 (p – s) (q – t) ∂ = σ (s, t) dt ds. ˆ ˆ ∂s ∂t 0 0 (k )(k ) 1 2 Let (Z , d )be a metric space. P (Z )the set of all closed subsets of Z and 2 the Z cl set of all nonempty subsets of Z . It is well known that the Pompeiu–Hausdorﬀ metric PH : P (Z ) × P (Z ) → R ∪{∞} is deﬁned by d cl cl d d d d Z Z Z Z PH A , A = max sup d a , A , sup d A , a d Z Z 1 2 1 2 1 2 d d Z Z a ∈A a ∈A 1 1 2 2 d d d d Z Z Z Z for all A , A ∈ P(Z ), where d (a , A )= inf d (a , a )and d (A , Z Z Z 1 2 1 2 Z 1 2 1 a ∈A 1 1 a )= inf d d (a , a )[43]. We say that a set-valued mapping : Z → P (Z ) Z cl 2 1 2 a ∈A 2 2 is called Lipschitzian with Lipschitz constant k > 0 whenever PH ((σ ), (σ )) ≤ d 1 1 kd (σ , σ ) for all σ , σ ∈ Z .If 0 < k < 1, then we ay that is a contraction [43]. An op- Z 1 2 1 2 erator : [0, 1] → P (R) is called measurable whenever the function t → d (ω , (t)) = cl Z 0 inf{|ω – y| : y ∈ (t)} is measurable for all real constant ω [43, 44]. The following notions were introduced in 2012 [45]. • = {ψ | ψ (t)< ∞, ∀t >0} where ψ :[0, ∞) × [0, ∞) → [0, ∞). n=1 • Assume that α : Z × Z → [0, ∞) and T : Z → Z are two mappings. Now T is α-admissible whenever for each σ , σ ∈ Z with α(σ , σ ) ≥ 1,weget 1 2 1 2 α(T σ , T σ ) ≥ 1. 1 2 • T is α-ψ-contractive mapping whence α(σ , σ )d (T σ , T σ ) ≤ ψ(d (σ , σ )) for all 1 2 Z 1 2 Z 1 2 σ , σ ∈ Z . 1 2 Lemma 1 ([45]) Assume that the metric space (Z , d ) is complete, Tis α-admissible and α-ψ-contractive mapping and there exists σ ∈ Z such that α(σ , T σ ) ≥ 1. Further, 0 0 0 for every convergent sequence {σ } ⊆ Z with σ → σ and α(σ , σ ) ≥ 1 for all n ≥ 1, n n≥1 n n n+1 we have α(σ , σ ) ≥ 1 for all n ≥ 1. Then T has a ﬁxed point. After this, multifunction version of α-ψ-contractive maps introduced in 2013 as follows [46]. • A multifunction F : Z → CB(Z ) is α-admissible whenever for each σ ∈ Z and σ ∈ Fσ with α(σ , σ ) ≥ 1,wehave α(σ , w ) ≥ 1, for all w ∈ Fσ . 2 1 1 2 2 0 0 2 • The metric space Z possesses the C -property if for every convergent sequence {σ } ⊆ Z with σ → σ and α(σ , σ ) ≥ 1 for all n ≥ 1,there exists asubsequence n n≥1 n n n+1 {σ } of σ such that α(σ , σ ) ≥ 1 for all j ≥ 1. n j≥1 n n j j • F is α-ψ-contractive multifunction whenever α(σ , σ )PH (T σ , T σ ) ≤ ψ(d (σ , σ )) for all σ , σ ∈ Z . 1 2 d 1 2 Z 1 2 1 2 Z Charandabi et al. Advances in Diﬀerence Equations (2021) 2021:323 Page 3 of 18 Lemma 2 ([46]) Assume that the metric space (Z , d ) is complete, Fis α-admissible and α-ψ-contractive multifunction and there exist σ ∈ Z and σ ∈ Fσ such that α(σ , σ ) ≥ 0 1 0 0 1 1. If Z possesses the C -property, then F has a ﬁxed point. In this paper, ﬁrst we investigate the partial fractional Sturm–Liouville diﬀerential equa- tion ˆ ˆ D (l(p , q )D σ (p , q )) + o(p , q )σ (p , q )= h(p , q )f (σ (p , q )), c c 0 0 (p , q ) ∈ J × J (l(p , q )D σ (p , q )) = θ (p ), a b q =0 1 0 0 c (1) ⎪ (l(p , q )D σ (p , q )) = θ (q ), p =0 2 ⎪ c σ (p ,0) = κ(p )and σ (0, q )= ω(q ), ˆ ˆ where k, ∈ (0, 1] × (0, 1], D and D denote the Caputo partial fractional derivatives, c c 0 0 l, o, h belong to C(J × J )with l(p , q ) =0 for all (p , q ) ∈ J × J and f : R → R a b a b 0 0 0 0 is a function. Here, θ : J → R, θ : J → R, κ : J → R and ω : J → R are abso- 1 a 2 b a b 0 0 0 0 lutely continuous with θ (0) = θ (0) = κ(0) = ω(0). Also, we investigate the partial fractional 1 2 Sturm–Liouville diﬀerential inclusion problem ˆ ˆ D (l(p , q )D σ (p , q )) ∈ H(p q , σ (p , q )), (p , q ) ∈ J × J , ⎪ a b c c 0 0 0 0 (l(p , q )D σ (p , q )) = θ (p ), q =0 1 (2) ⎪ (l(p , q )D σ (p , q )) = θ (q ), p =0 2 ⎪ c σ (p ,0) = κ(p )and σ (0, q )= ω(q ), ˆ ˆ where k, ∈ (0, 1] × (0, 1], D and D denote the Caputo partial fractional derivatives, c c 0 0 l, o, h belong to C(J × J )with l(p , q ) =0 for all (p , q ) ∈ J × J and f : R → R a b a b 0 0 0 0 is a function. Here, θ : J → R, θ : J → R, κ : J → R and ω : J → R are abso- 1 a 2 b a b 0 0 0 0 lutely continuous with θ (0) = θ (0) = κ(0) = ω(0). Also, H : J × J × R → P (R)is an 1 2 a b cl 0 0 integrable bounded multifunction so that H(., ., σ ) is measurable for all σ ∈ R. 2 Main results Assume that Z = {σ |σ ∈ C(J × J )} and σ = sup |σ (p , q )|,where σ ∈ a b 0 0 (p ,q )∈J ×J 0 0 Z .Then (Z , .)is a Banach space. ˆ ˆ ˆ ˆ ˆ ˆ Lemma 3 Let k =(k , k ), =( ) ∈ (0, 1] × (0, 1] and g ∈ L (J × J ). Consider the 1 2 1 2 a b 0 0 problem ˆ ˆ D l p , q D σ p , q = g p , q,(3) c c 0 0 with boundary conditions (l(p , q )D σ (p , q )) = θ (p ), ⎪ y=0 1 (4) (l(p , q )D σ (p , q )) = θ (q ), x=0 2 σ (p ,0) = κ(p ) and σ (0, q )= ω(q ). Charandabi et al. Advances in Diﬀerence Equations (2021) 2021:323 Page 4 of 18 Then the function σ ∈ C(J × J ) is a solution of the problem (3)–(4) whenever a b 0 0 σ p , q = p , q p q s t 0 0 0 0 ˆ ˆ ˆ ˆ k –1 k –1 –1 –1 1 2 1 2 (p – s) (q – t) (s – ℘ ¯) (t – ζ ¯) g(℘ ¯, ζ ¯) 1 2 1 2 dt ds dζ d℘ , ¯ ¯ 2 1 ˆ ˆ ˆ ˆ (k )(k )( )( )l(s, t) 1 2 1 2 where θ (p )+ θ (q )– θ (0) ˆ 1 2 1 p , q = κ p + ω q – κ(0) + I . l(p , q ) Proof Note that Eq. (3)can be writtenas ˆ ˆ 1–k I l p , q D σ p , q = g p , q . 0 c ∂p ∂q Operating by I on both sides we get ˆ ˆ I l p , q D σ p , q = I g p , q . 0 c 0 ∂p ∂q Since I l p , q D σ p , q 0 c ∂p ∂q ˆ ˆ = l p , q D σ p , q – l p , q D σ p , q c c 0 0 q =0 ˆ ˆ – l p , q D σ p , q + l p , q D σ p , q c c 0 p =0 0 p =0,q =0 = l p , q D σ p , q – θ p – θ q + θ (0), 1 2 1 ˆ ˆ we get l(p , q )D σ (p , q )= θ (p )+ θ (q )– θ (0) + I g(p , q ). Hence, 1 2 1 c 0 θ (p )+ θ (q )– θ (0) 1 ˆ 1 2 1 ˆ D σ p , q = + I g p , q . c 0 l(p , q ) l(p , q ) This equation can be written as ∂ θ (p )+ θ (q )– θ (0) 1 ˆ 1 2 1 1– I σ p , q = + I g p , q . 0 0 ∂p ∂q l(p , q ) l(p , q ) Again operating by I on both sides, we obtain ∂ θ (p )+ θ (q )– θ (0) 1 ˆ ˆ ˆ 1 2 1 I σ p , q = I + I I g p , q . 0 0 0 0 ∂p ∂q l(p , q ) l(p , q ) Charandabi et al. Advances in Diﬀerence Equations (2021) 2021:323 Page 5 of 18 Since I σ p , q = σ p , q – σ p ,0 – σ 0, q + σ (0, 0) ∂p ∂q = σ p , q – κ p – ω q + κ(0), we get θ (p )+ θ (q )– θ (0) ˆ 1 2 1 σ p , q = κ p + ω q – κ(0) + I l(p , q ) 1 1 ˆ ˆ ˆ ˆ + I I g p , q = p , q + I I g p , q . 0 0 0 0 l(p , q ) l(p , q ) On the other hand, ˆ ˆ I I g p , q 0 0 l(p , q ) ˆ ˆ p q k –1 k –1 1 2 (p – s) (q – t) 1 = I g(s, t) dt ds ˆ ˆ l(s, t) 0 0 (k )(k ) 1 2 p q ˆ ˆ 1 1 k –1 k –1 1 2 = p – s q – t ˆ ˆ l(s, t) 0 0 (k )(k ) 1 2 ˆ ˆ s t –1 –1 1 2 (s – ℘ ) (t – ζ ) ¯ ¯ 1 2 × g(℘ , ζ ) d℘ dζ dt ds ¯ ¯ ¯ ¯ 1 2 1 2 ˆ ˆ )( 0 0 1 2 p q s t 0 0 0 0 ˆ ˆ ˆ ˆ k –1 k –1 –1 –1 1 2 1 2 (p – s) (q – t) (s – ℘ ) (t – ζ ) g(℘ , ζ ) ¯ ¯ ¯ ¯ 1 2 1 2 dt ds dζ d℘ ¯ ¯ 2 1 ˆ ˆ ˆ ˆ (k )(k )( )( )l(s, t) 1 2 1 2 and so σ p , q = p , q p q s t 0 0 0 0 ˆ ˆ ˆ ˆ k –1 k –1 –1 –1 1 2 1 2 (p – s) (q – t) (s – ℘ ¯) (t – ζ ¯) g(℘ ¯, ζ ¯) 1 2 1 2 dt ds dζ d℘ . ¯ ¯ 2 1 ˆ ˆ ˆ ˆ (k )(k )( )( )l(s, t) 1 2 1 2 This completes the proof. Now we establish and prove our ﬁrst main theorem. Theorem 4 Assume that υ : R × R → R is a function and : J × J → [0, ∞) is a a b 0 0 bounded function such that f σ p , q – f σ p , q ≤ p , q σ p , q – σ p , q 1 2 1 2 Charandabi et al. Advances in Diﬀerence Equations (2021) 2021:323 Page 6 of 18 for all σ , σ ∈ Z with υ(σ (p , q ), σ (p , q )) ≥ 0, where (p , q ) ∈ J ×J . Suppose that, 1 2 1 2 a b 0 0 for all σ , σ ∈ Z with υ(σ (p , q ), σ (p , q )) ≥ 0, we have υ(Q σ (p , q ), Q σ (p , q )) ≥ 1 2 1 2 1 2 ¯ ¯ 0 0 0, where Q σ p , q = p , q p q s t 0 0 0 0 ˆ ˆ ˆ ˆ k –1 k –1 –1 –1 1 2 1 2 (p – s) (q – t) (s – ℘ ) (t – ζ ) H(℘ , ζ , σ (℘ , ζ )) ¯ ¯ ¯ ¯ ¯ ¯ 1 2 1 2 1 2 dt ds dζ d℘ , ¯ ¯ 2 1 ˆ ˆ ˆ ˆ (k )(k )( )( )l(s, t) 1 2 1 2 θ (p )+θ (q )–θ (0) 1 2 1 (p , q )= κ(p )+ ω(q )– κ(0) + I ( ), l(p ,q ) H ℘ , ζ , σ (℘ , ζ ) = h(℘ , ζ )f σ (℘ , ζ ) – o(℘ , ζ )σ (℘ , ζ ) ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ 1 2 1 2 1 2 1 2 1 2 1 2 and there exists σ so that υ(σ (p , q ), Q σ (p , q )) ≥ 0 whenever (p , q ) ∈ J × J . 0 0 0 a b ¯ 0 0 Assume that, for every sequence {σ } ⊆ Z with σ → σ and υ(σ (p , q ), σ (p , q )) ≥ n n≥1 n n n+1 0 for all n ≥ 1 and (p , q ) ∈ J × J , we have υ(σ (p , q ), σ (p , q )) ≥ 0 for all n ≥ 1 a b n 0 0 ˆ ˆ ˆ ˆ k + k + 1 1 2 2 ∗ a b (h +o) 0 0 and (p , q ) ∈ J × J . If <1, then the fractional Sturm–Liouville a b 0 0 ˆ ˆ ˆ ˆ l(k + +1)(k + +1) 1 1 2 2 problem (1) has a solution, where = sup (p , q ) and (p ,q )∈J ×J l = inf l p , q . (p ,q )∈J ×J a b Proof By using Lemma 3, σ is a solution of the partial fractional Sturm–Liouville problem (1)ifand only if σ = Q σ .Let σ , σ ∈ Z and (p , q ) ∈ J × J with 0 0 1 2 a b ¯ 0 0 υ σ p , q , σ p , q ≥ 0. 1 2 Hence, we get Q σ p , q – Q σ p , q 1 2 ¯ ¯ 0 0 p q s t = p , q + 0 0 0 0 ˆ ˆ ˆ ˆ k –1 k –1 –1 –1 1 2 1 2 (p – s) (q – t) (s – ℘ ) (t – ζ ) H(℘ , ζ , σ (℘ , ζ )) ¯ ¯ ¯ ¯ ¯ ¯ 1 2 1 2 1 1 2 dt ds dζ d℘ ¯ ¯ 2 1 ˆ ˆ ˆ ˆ (k )(k )( )( )l(s, t) 1 2 1 2 p q s t – p , q – 0 0 0 0 ˆ ˆ ˆ ˆ k –1 k –1 –1 –1 1 2 1 2 (p – s) (q – t) (s – ℘ ) (t – ζ ) H(℘ , ζ , σ (℘ , ζ )) ¯ ¯ ¯ ¯ ¯ ¯ 1 2 1 2 1 2 dt ds dζ d℘ ¯ ¯ 2 1 ˆ ˆ ˆ ˆ (k )(k )( )( )l(s, t) 1 2 1 2 p q s t 0 0 0 0 ˆ ˆ ˆ ˆ k –1 k –1 –1 –1 1 2 1 2 (p – s) (q – t) (s – ℘ ) (t – ζ ) |N (℘ , ζ )| ¯ ¯ ¯ ¯ 1 2 1 2 dt ds dζ d℘ , ¯ ¯ 2 1 ˆ ˆ ˆ ˆ (k )(k )( )( )l(s, t) 1 2 1 2 Charandabi et al. Advances in Diﬀerence Equations (2021) 2021:323 Page 7 of 18 where N (℘ , ζ )= H(℘ , ζ , σ (℘ , ζ )) – H(℘ , ζ , σ (℘ , ζ )). Since ¯ ¯ ¯ ¯ 1 ¯ ¯ ¯ ¯ 2 ¯ ¯ 1 2 1 2 1 2 1 2 1 2 H ℘ , ζ , σ (℘ , ζ ) – H ℘ , ζ , σ (℘ , ζ ) ¯ ¯ 1 ¯ ¯ ¯ ¯ 2 ¯ ¯ 1 2 1 2 1 2 1 2 ≤ h(℘ , ζ ) f σ (℘ , ζ ) – f σ (℘ , ζ ) – o(℘ , ζ ) σ (℘ , ζ )– σ (℘ , ζ ) ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ 1 2 1 1 2 2 1 2 1 2 1 1 2 2 1 2 ≤ h(℘ , ζ ) f σ (℘ , ζ ) – f σ (℘ , ζ ) + o(℘ , ζ ) σ (℘ , ζ )– σ (℘ , ζ ) ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ 1 2 1 1 2 2 1 2 1 2 1 1 2 2 1 2 ≤ h(℘ , ζ ) (℘ , ζ ) σ (℘ , ζ )– σ (℘ , ζ ) + o(℘ , ζ ) σ (℘ , ζ )– σ (℘ , ζ ) ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 ≤ h + o σ – σ , 1 2 we have Q σ p , q – Q σ p , q 1 2 ¯ ¯ 0 0 (h + o)σ – σ 1 2 ˆ ˆ ˆ ˆ l(k )(k )( )( 1 2 1 2 (5) p q s t ˆ ˆ k –1 k –1 ˆ 1 2 –1 × p – s q – t (s – ℘ ) 0 0 0 0 –1 × (t – ζ ) dt ds dζ d℘ , ¯ ¯ ¯ 2 2 1 where p q s t ˆ ˆ k –1 k –1 ˆ ˆ 1 2 –1 –1 1 2 p – s q – t (s – ℘ ) (t – ζ ) dt ds dζ d℘ ¯ ¯ ¯ ¯ 1 2 2 1 0 0 0 0 ˆ ˆ ˆ ˆ p q k –1 k –1 1 1 2 2 s (p – s) t (q – t) = dζ d℘ ¯ ¯ 2 1 ˆ ˆ 0 0 1 2 ˆ ˆ ˆ ˆ k –1 q k –1 1 1 2 2 s (p – s) t (q – t) = d℘ × dζ ¯ ¯ 1 2 ˆ ˆ 1 2 a b 0 0 ˆ ˆ ˆ ˆ k –1 k –1 1 1 2 2 ≤ s (a – s) ds × t (b – t) dt. 0 0 ˆ ˆ 0 0 1 2 Put s = a u and t = b v.Thus, we obtain 0 0 p q s t ˆ ˆ k –1 k –1 ˆ ˆ 1 2 –1 –1 1 2 p – s q – t (s – ℘ ) (t – ζ ) dt ds dζ d℘ ¯ ¯ ¯ ¯ 1 2 2 1 0 0 0 0 ˆ ˆ ˆ ˆ k + k + 1 1 2 2 1 1 a b ˆ ˆ ˆ ˆ 0 0 k –1 k –1 1 1 2 2 ≤ u (1 – u) du × v (1 – v) dv. ˆ ˆ 0 0 1 2 On the other hand, ˆ ˆ +1)(k ) ˆ ˆ 1 1 k –1 1 1 ˆ ˆ B( +1, k )= u (1 – u) du = 1 1 ˆ ˆ 0 (k + +1) 1 1 and ˆ ˆ +1)(k ) ˆ ˆ 2 2 k –1 ˆ 2 2 B( +1, k )= v (1 – v) dv = . 2 2 ˆ ˆ 0 (k + +1) 2 2 Charandabi et al. Advances in Diﬀerence Equations (2021) 2021:323 Page 8 of 18 Hence, p q s t ˆ ˆ k –1 k –1 ˆ ˆ 1 2 –1 –1 1 2 p – s q – t (s – ℘ ) (t – ζ ) dt ds dζ d℘ ¯ ¯ ¯ ¯ 1 2 2 1 0 0 0 0 ˆ ˆ ˆ ˆ k + k + 1 1 2 2 ˆ ˆ ˆ ˆ a b ( +1)( +1)(k )(k ) 1 2 1 2 0 0 ≤ . ˆ ˆ ˆ ˆ ˆ ˆ (k + +1)(k + +1) 1 2 1 1 2 2 By using (5), we derive Q σ p , q – Q σ p , q 1 2 ¯ ¯ 0 0 (h + o)σ – σ 1 2 ˆ ˆ ˆ ˆ l(k )(k )( )( 1 2 1 2 ˆ ˆ ˆ ˆ k + k + 1 1 2 2 ˆ ˆ ˆ ˆ a b ( +1)( +1)(k )(k ) 1 2 1 2 0 0 ˆ ˆ ˆ ˆ ˆ ˆ (k + +1)(k + +1) 1 2 1 1 2 2 ˆ ˆ ˆ ˆ k + k + 1 1 2 2 ∗ a b (h + o) 0 0 = σ – σ , 1 2 ˆ ˆ ˆ ˆ l(k + +1)(k + +1) 1 1 2 2 ˆ ˆ ˆ ˆ k + k + 1 1 2 2 ∗ a b (h +o) 0 0 which means Q σ – Q σ ≤ σ – σ .Deﬁne ψ :[0, ∞) → [0, ∞) 1 2 1 2 ¯ ¯ ˆ ˆ 0 0 ˆ ˆ l(k + +1)(k + +1) 1 1 2 2 ˆ ˆ ˆ ˆ k + k + 1 1 2 2 ∗ a b (h +o) 0 0 and α : Z × Z → [0, ∞)by ψ(t)= t and ˆ ˆ ˆ ˆ l(k + +1)(k + +1) 1 1 2 2 1, υ(σ (p , q ), σ (p , q )) ≥ 0with(p , q ) ∈ J × J , 1 2 a b 0 0 α(σ , σ )= 1 2 0, otherwise. It is clear that ψ ∈ .If α(σ , σ ) ≥ 1, then υ(σ (p , q ), σ (p , q )) ≥ 0. From the hy- 1 2 1 2 potheses, υ(Q σ (p , q ), Q σ (p , q )) ≥ 0and so α(Q σ , Q σ ) ≥ 1. Thus, Q is an α- 1 2 1 2 ¯ ¯ ¯ ¯ ¯ 0 0 0 0 0 admissible mapping. Also, there exists σ ∈ Z such that α(σ , Q σ ) ≥ 1. For every se- 0 0 0 quence {σ } ⊆ Z with σ → σ and α(σ , σ ) ≥ 1 for all n ≥ 1, we have α(σ , σ ) ≥ 1 n n≥1 n n n+1 n for all n ≥ 1. Assume that α(σ , σ )=0. Then α(σ , σ )Q σ – Q σ =0 ≤ ψ(σ – σ ) 1 2 1 2 1 2 1 2 ¯ ¯ 0 0 and so α(σ , σ ) Q σ – Q σ ≤ ψ σ – σ 1 2 1 2 1 2 ¯ ¯ 0 0 for all σ , σ ∈ Z . Thus all conditions of Lemma 1 hold and so Q has a ﬁxed point which 1 2 is a solution for the partial fractional Sturm–Liouville problem (1). Example 1 Consider the partial fractional Sturm–Liouville equation √ 3 999 1999 89 79 ⎪ ( , ) 4 ( , ) –p –q – p e ⎪ 1000 2000 90 80 ⎪ D (100e cosh q D σ (p , q )) + σ (p , q ) c c 2 2 ⎪ 0 0 ⎪ 300(1+p +p ) ⎪ p ⎪ 2 1+p ⎨ = e σ (p , q ), (p , q ) ∈ [0, 1] × [0, 1], 89 79 4 (6) ( , ) 2 – p 90 80 1 (100e cosh q D σ (p , q )) = p , c q =0 0 100 89 79 ⎪ 4 ( , ) 3 ⎪ – p 90 80 1 (100e cosh q D σ (p , q )) = q , c q =0 ⎪ 0 ⎩ 1 1 σ (p ,0) = p and σ (0, q )= q . 500 350 Charandabi et al. Advances in Diﬀerence Equations (2021) 2021:323 Page 9 of 18 ∗ ∗ ∗ ∗ ∗ ∗ Figure 1 Plots of functions h(p , q ), l(p , q ), o(p , q ) in [0, 1] Figure 2 Plots of functions θ , θ , κ, ω in [0, 1] 1 2 999 1999 89 79 1 ˆ ˆ ˆ ˆ ˆ ˆ Put k =(k , k )=( , ), =( )=( , ), a =1, b =1, θ (p )= p , θ (q )= 1 2 1 2 0 0 1 2 1000 2000 99 80 100 3 2 –p –q 1 1 1 – p e q , κ(p )= p , ω(q )= q , l(p , q ) = 100e cosh q , o(p , q )= , 2 2 400 500 350 300(1+p +q ) 1+q h(p , q )= e . The diagrams are plotted in Figs. 1 and 2, and obviously they satisfy the conditions of the partial Sturm–Liouville diﬀerential problem. Put f (r)= r, υ(r , r )= 1 2 3whenever |r |≤ 1and |r |≤ 1and υ(r , r ) = –1 otherwise, and 1 2 1 2 Q σ p , q p q s t = p , q + 0 0 0 0 –1 –1 –1 –1 1000 2000 90 80 (p – s) (q – t) (s – ℘ ) (t – ζ ) H(℘ , ζ , σ (℘ , ζ )) ¯ ¯ ¯ ¯ ¯ ¯ 1 2 1 2 1 2 √ dt ds dζ d℘ , ¯ ¯ 2 1 – p cosh q 999 1999 89 79 ( )( )( )( )100e 1000 2000 99 80 50 where 1 1 p , q = p + p 500 350 1 1 p q – – 1 2 1 3 90 80 (p – s) (q – t) ( s + t ) 100 400 + √ dt ds. – p 89 79 0 0 100e cosh q ( )( ) 90 80 Charandabi et al. Advances in Diﬀerence Equations (2021) 2021:323 Page 10 of 18 Note that H(℘ , ζ , σ (℘ , ζ )) = h(℘ , ζ )f (σ (℘ , ζ )) – o(℘ , ζ )σ (℘ , ζ ). Now, assume that ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ 1 2 1 2 1 2 1 2 1 2 1 2 υ(σ (p , q ), σ (p , q )) ≥ 0. Then we have |σ (p , q )|≤ 1and |σ (p , q )|≤ 1. Suppose that 1 2 1 2 |σ (p , q )|≤ 1. Then we get H ℘ , ζ , σ (℘ , ζ ) ¯ ¯ ¯ ¯ 1 2 1 2 1 –℘ –ζ 1 2 e 1+ζ = e f σ (℘ , ζ ) – σ (℘ , ζ ) ¯ ¯ 1 ¯ ¯ 1 2 1 2 2 2 600 300(1 + ℘ + ζ ) ¯ ¯ 1 2 –p –p 1 e 1+ζ ≤ e σ (℘ , ζ ) + σ (℘ , ζ ) 1 ¯ ¯ 1 ¯ ¯ 1 2 2 2 1 2 600 300(1 + p + p ) e 1 e +2 ≤ + = 600 300 600 and so 1 1 s t 0 0 0 0 –1 –1 –1 –1 1000 2000 90 80 (p – s) (q – t) (s – ℘ ) (t – ζ ) |H(℘ , ζ , σ (℘ , ζ ))| ¯ ¯ ¯ ¯ ¯ ¯ 1 2 1 2 1 1 2 √ dt ds dζ d℘ ¯ ¯ 2 1 – p 999 1999 89 79 ( )( )( )( )100e cosh q 1000 2000 90 80 p q s t –1 –1 –1 1000 2000 ≤ 0.0000285064 p – s q – t (s – ℘ ) 0 0 0 0 –1 × (t – ζ ) dt ds dζ d℘ ¯ ¯ ¯ 2 2 1 89 79 999 1999 ( +1)( +1)( )( ) 90 80 1000 2000 ≤ 0.0000285064 × = 0.0000296489. 89 79 999 89 1999 79 ( + +1)( + +1) 90 80 1000 90 2000 80 Also, p , q 1 1 2 3 – 1 1 p q – 90 80 (p – s) (q – t) ( p + p ) 1 1 2 100 400 ≤ p + q + √ dt ds – p 89 79 500 350 0 0 100e cosh q ( )( ) 90 80 1 1 1 1 1 1 – – 90 80 ≤ + + 0.0000453519 (1 – s) (1 – t) dt ds 500 350 0 0 1 1 = + + 0.0000453519 × 1.0240364102 = 0.0053797753. 500 350 Therefore, Q σ p , q p q s t ≤ p , q + 0 0 0 0 –1 –1 –1 –1 1000 2000 90 80 (p – s) (q – t) (s – ℘ ) (t – ζ ) |H(℘ , ζ , σ (℘ , ζ ))| ¯ ¯ ¯ ¯ ¯ ¯ 1 2 1 2 1 2 √ dt ds dζ d℘ ¯ ¯ 2 1 cosh q 999 1999 89 79 – p ( )( )( )( )100e 1000 2000 99 80 50 ≤ 0.0053797753 + 0.0000296489 = 0.0054094242 ≤ 1. Charandabi et al. Advances in Diﬀerence Equations (2021) 2021:323 Page 11 of 18 Similarly, we obtain |Q σ (p , q )|≤ 1and υ(Q σ (p , q ), Q σ (p , q )) ≥ 0. Assume that 1 1 2 ¯ ¯ ¯ 0 0 0 {σ } ⊆ Z is a sequence such that σ → σ and n n≥1 n υ σ p , q , σ p , q ≥ 0 n n+1 for all n ≥ 1and (p , q ) ∈ J ×J .Thenwehave |σ (p , q )|≤ 1and |σ (p , q )|≤ 1for a b n n+1 0 0 all n ≥ 1and (p , q ) ∈ J × J .Since σ → σ , |σ (p , q )| = lim |σ (p , q )|≤ 1, we a b n n→∞ n 0 0 obtain υ(Q σ (p , q ), Q σ (p , q )) ≥ 0 for all n ≥ 1and (p , q ) ∈ J × J .Since |0|≤ 1 n a b ¯ ¯ 0 0 0 0 and |Q 0|≤ 1, we get υ(0, Q 0) ≥ 1. Note that =1, ¯ ¯ 0 0 – p l = inf l p , q = inf 100e cosh q = 100e, (p ,q )∈J ×J (p ,q )∈J ×J a b a b 0 0 0 0 –p –q e 1 o = sup o p , q = sup = , 2 2 300(1 + p + q ) 300 (p ,q )∈J ×J (p ,q )∈J ×J a a b b 0 0 0 0 1 e 1+q and h = sup e = .Hence, (p ,q )∈J ×J a 600 600 0 0 ˆ ˆ ˆ ˆ k + k + e 1 1 1 2 2 ∗ a b (h + o) 0 0 600 300 999 89 1999 79 ˆ ˆ ˆ ˆ 100e( + +1)( + +1) l(k + +1)(k + +1) 1 1 2 2 1000 90 2000 80 = 0.0000074014 ≤ 1. Now by using Theorem 4,the problem(6)has asolution. Deﬁnition 5 We say that a function σ ∈ C(J ×J ) is a solution for the partial fractional a b 0 0 Sturm–Liouville diﬀerential inclusion problem problem (2) whenever there is a function v in L (J × J , R)such that v(p , q ) ∈ H(p , q , σ (p , q )) for almost all (p , q ) ∈ J × a b a 0 0 0 J , (l(p , q )D σ (p , q )) = θ (p ), q =0 1 (l(p , q )D σ (p , q )) = θ (q ), p =0 2 σ (p ,0) = κ(p )and σ (0, q )= ω(q ), and σ p , q = p , q p q s t 0 0 0 0 ˆ ˆ ˆ ˆ k –1 k –1 –1 –1 1 2 1 2 (p – s) (q – t) (s – ℘ ) (t – ζ ) v(℘ , ζ ) ¯ ¯ ¯ ¯ 1 2 1 2 dt ds dζ d℘ , ¯ ¯ 2 1 ˆ ˆ ˆ ˆ (k )(k )( )( )l(s, t) 1 2 1 2 for all (p , q ) ∈ J × J . a b 0 0 Charandabi et al. Advances in Diﬀerence Equations (2021) 2021:323 Page 12 of 18 For given σ ∈ Z ,deﬁne theset S = v ∈ L (J × J )|v ∈ H p , q , σ p , q on J × J . H,σ a b a b 0 0 0 0 Assume that (H1) H : J × J × R → P (R) is an integrable bounded multifunction so that a b cl 0 0 H(·, ·, σ ) is measurable for all σ ∈ R. (H2) There exists ρ ∈ C(J × J , R ) so that a b 0 0 PH H p , q , r , H p , q , r ≤ ρ p , q ψ |r – r | d 1 2 1 2 ˆ ˆ ˆ ˆ k + k + 1 1 2 2 a b ρ 0 0 for all r , r ∈ R,where ψ ∈ , ≤ 1 and 1 2 ˆ ˆ ˆ ˆ l(k + +1)(k + +1) 1 1 2 2 l = inf l p , q . (p ,q )∈J ×J a b (H3) Deﬁne N : Z → 2 by N(σ)= h ∈ Z | there exists v ∈ S so that h p , q = w p , q H,σ for all p , q ∈ J × J , a b 0 0 where w p , q p q s t = p , q + 0 0 0 0 ˆ ˆ ˆ ˆ k –1 k –1 –1 –1 1 2 1 2 (p – s) (q – t) (s – ℘ ) (t – ζ ) v(℘ , ζ ) ¯ ¯ ¯ ¯ 1 2 1 2 dt ds dζ d℘ . ¯ ¯ 2 1 ˆ ˆ ˆ ˆ (k )(k )( )( )l(s, t) 1 2 1 2 (H4) Suppose that υ : R × R → R bearealvaluedfunctionand forevery conver- gent sequence {σ } ⊆ Z with σ → σ and υ(σ (p , q ), σ (p , q )) ≥ 0 for n n≥1 n n all n and (p , q ) ∈ J × J ,there exists asubsequence {σ } of σ so that a b n j≥1 n 0 0 j υ(σ (p , q ), σ (p , q )) ≥ 0 for all n and (p , q ) ∈ J × J . Assume that, for every n a b j 0 0 σ ∈ Z and h ∈ N(σ ) with υ(σ (p , q ), h(p , q )) ≥ 0 for each (p , q ) ∈ J × J , a b 0 0 there exists w ∈ N(σ ) so that υ(h(p , q ), w(p , q )) ≥ 0 for all (p , q ) ∈ J × J . a b 0 0 Suppose that there exists σ ∈ Z and h ∈ N(σ ) so that υ(σ (p , q ), h(p , q )) ≥ 0 0 0 0 for all (p , q ) ∈ J × J . a b 0 0 Theorem 6 Assume that (H1)–(H4) hold. Then the partial fractional Sturm–Liouville problem (2) has a solution. Proof We prove that the multifunction N : Z → 2 has a ﬁxed point which provides a solution for the partial fractional Sturm–Liouville problem (2). Note that the multifunc- tion (p , q ) → H(p , q , σ (p , q )) has a measurable selection. Since it has closed and has measurable values for aa σ ∈ Z , S is nonempty for every σ ∈ Z .Weprove that N(σ ) H,σ Charandabi et al. Advances in Diﬀerence Equations (2021) 2021:323 Page 13 of 18 is closed subset of Z . For this aim assume that σ ∈ Z and {h }⊂ N(σ)is a sequence with h → h.For each n,choose v ∈ S such that n n H,σ h p , q p q s t = p , q + 0 0 0 0 ˆ ˆ ˆ ˆ k –1 k –1 –1 –1 1 2 1 2 (p – s) (q – t) (s – ℘ ) (t – ζ ) v (℘ , ζ ) ¯ ¯ n ¯ ¯ 1 2 1 2 dt ds dζ d℘ , ¯ ¯ 2 1 ˆ ˆ ˆ ˆ (k )(k )( )( )l(s, t) 1 2 1 2 for all (p , q ) ∈ J ×J . On the other hand, H hascompactvalues.Thus, wemayassume a b 0 0 that {v } converges to some v ∈ L (J × J ). Hence, n a b 0 0 h p , q p q s t = p , q + 0 0 0 0 ˆ ˆ ˆ ˆ k –1 k –1 –1 –1 1 2 1 2 (p – s) (q – t) (s – ℘ ) (t – ζ ) v(℘ , ζ ) ¯ ¯ ¯ ¯ 1 2 1 2 dt ds dζ d℘ ¯ ¯ 2 1 ˆ ˆ ˆ ˆ (k )(k )( )( )l(s, t) 1 2 1 2 and so h ∈ N(σ ). Since H is a compact map, N(σ)is a bounded set for al σ ∈ Z .Now, deﬁne the function α : Z × Z → R by α(σ , σ ) ≥ 1whenever υ(σ (p , q ), σ (p , q )) ≥ 1 2 1 2 0 for almost all (p , q ) ∈ J × J and α(σ , σ ) = 0 otherwise. We show that N is α-ψ- a b 1 2 0 0 contractive. Let σ , σ ∈ Z and h ∈ N(σ ). Choose v ∈ S such that 1 2 1 2 1 H,σ h p , q p q s t = p , q + 0 0 0 0 ˆ ˆ ˆ ˆ k –1 k –1 –1 –1 1 2 1 2 (p – s) (q – t) (s – ℘ ) (t – ζ ) v (℘ , ζ ) ¯ ¯ 1 ¯ ¯ 1 2 1 2 dt ds dζ d℘ ¯ ¯ 2 1 ˆ ˆ ˆ ˆ (k )(k )( )( )l(s, t) 1 2 1 2 for almost all (p , q ) ∈ J × J .Thenweget a b 0 0 PH H p , q , σ p , q , H p , q , σ p , q d 1 2 ≤ ρ p , q ψ σ p , q – σ p , q , 1 2 for almost all (p , q ) ∈ J × J with υ(σ (p , q ), σ (p , q )) ≥ 0. Now, choose w ∈ a b 1 2 0 0 H(p , q , σ (p , q )) so that |v (t)– w|≤ ρ(p , q )ψ(|σ (p , q )– σ (p , q )|) for all (p , q ) ∈ 1 1 1 2 J × J .Now,deﬁne U : J × J → P(R)by a b a b 0 0 0 0 U p , q = w ∈ R| v p , q – w ≤ ρ p , q ψ σ p , q – σ p , q . 1 1 2 Since v and ω = ρ(p , q )ψ(|σ (p , q )– σ (p , q )|) are measurable, the multifunction 1 1 2 U(·, ·) ∩ H(·, ·, σ (·, ·)) is measurable. Choose v ∈ H(p , q , σ (p , q )) such that 2 1 v p , q – v p , q ≤ ρ p , q ψ σ p , q – σ p , q 1 2 1 2 (7) ≤ρ ψ σ – σ ∞ 1 2 Charandabi et al. Advances in Diﬀerence Equations (2021) 2021:323 Page 14 of 18 for all (p , q ) ∈ J × J . Now, choose the element h ∈ N(σ )deﬁned by a b 2 1 0 0 h p , q p q s t = p , q + 0 0 0 0 ˆ ˆ ˆ ˆ k –1 k –1 –1 –1 1 2 1 2 (p – s) (q – t) (s – ℘ ) (t – ζ ) v (℘ , ζ ) ¯ ¯ 2 ¯ ¯ 1 2 1 2 dt ds dζ d℘ ¯ ¯ 2 1 ˆ ˆ ˆ ˆ (k )(k )( )( )l(s, t) 1 2 1 2 for all (p , q ) ∈ J × J .Byusing (7), we get a b 0 0 h p , q – h p , q 2 1 p q s t 0 0 0 0 ˆ ˆ ˆ ˆ k –1 k –1 –1 –1 1 2 1 2 (p – s) (q – t) (s – ℘ ) (t – ζ ) |v (℘ , ζ )– v (℘ , ζ )| ¯ ¯ 2 ¯ ¯ 1 ¯ ¯ 1 2 1 2 1 2 dt ds dζ d℘ ¯ ¯ 2 1 ˆ ˆ ˆ ˆ (k )(k )( )( )l(s, t) 1 2 1 2 p q s t ≤ρ 0 0 0 0 ˆ ˆ ˆ ˆ k –1 k –1 –1 –1 1 2 1 2 (p – s) (q – t) (s – ℘ ) (t – ζ ) ¯ ¯ 1 2 dt ds dζ d℘ ψ σ – σ ¯ ¯ 1 2 2 1 ˆ ˆ ˆ ˆ (k )(k )( )( )l(s, t) 1 2 1 2 ˆ ˆ ˆ ˆ k + k + 1 1 2 2 a b ρ 0 0 ≤ ψ σ – σ ≤ ψ σ – σ 1 2 1 2 ˆ ˆ ˆ ˆ l(k + +1)(k + +1) 1 1 2 2 and so h – h ≤ ψ(σ – σ ). Note that 1 2 1 2 α(σ , σ )PH N(σ ), N(σ ) ≤ ψ σ – σ for all σ , σ ∈ Z . 1 2 d 1 2 1 2 1 2 Thus, N is α-ψ-contraction. Let σ ∈ Z and σ ∈ N(σ )be such that α(σ , σ ) ≥ 1. Then 1 2 1 1 2 υ(σ (p , q ), σ (p , q )) ≥ 0 for all (p , q ) ∈ J × J .Hence,there exists w ∈ N(σ )such 1 2 a b 2 0 0 that υ(σ (p , q ), w(p , q )) ≥ 0. This implies that α(σ , w) ≥ 1. Thus, N is α-admissible. 1 1 Now by using Lemma 2, N has a ﬁxed point which is a solution of the partial fractional Sturm–Liouville problem (2). Example 2 Consider the partial fractional Sturm–Liouville inclusion problem 2 2 1 1 1 1 –p –q ( , ) ( , ) e |σ (p ,q )| ⎪ 2 3 p q 7 8 D (13e D σ (p , q )) ∈ [σ (p , q ), σ (p , q )+ ], ⎪ c c 0 0 2 4(1+|σ (p ,p )|) ⎪ (p , q ) ∈ [0, 1] × [0, 1], 1 1 ( , ) 2 p q 7 8 3 (8) (13e D σ (p , q )) = p , c q =0 ⎪ 0 1 1 ⎪ ( , ) 3 ⎪ p q 7 8 1 ⎪ (13e D σ (p , q )) = q , c p =0 ⎪ 0 2 3 7 8 σ (p ,0) = p and σ (0, q )= q . 50 35 2 3 1 1 1 1 3 1 ˆ ˆ ˆ ˆ ˆ ˆ Put k =(k , k )=( , ), =( )=( , ), a =1, b =1, θ (p )= p , θ (q )= q , 1 2 1 2 1 2 2 3 7 8 40 25 2 3 7 8 p q κ(p )= p , ω(q )= q and l(p , q )=13e . The plotted diagrams in Figs. 3 and 50 35 Charandabi et al. Advances in Diﬀerence Equations (2021) 2021:323 Page 15 of 18 ∗ ∗ Figure 3 Plot of function l(p , q )in[0,1] Figure 4 Plots of functions θ , θ , κ, ω in [0, 1] 1 2 4 show that the conditions of the partial Sturm–Liouville diﬀerential inclusion problem 2 2 –p –q e |r| hold. Also, put H(p , q , r)=[r, r + ], υ(r , r )=1 whenever r ≥ 0and r ≥ 0 1 2 1 2 4(1+|r|) and υ(r , r )=–1 otherwise, 1 2 N(σ)= h ∈ Z | there exists v ∈ S so that h p , q = w p , q H,σ for all p , q ∈ J × J a b 0 0 where w p , q = p , q 1 2 6 7 p q s t – – – – 2 3 7 8 (p – s) (q – t) (s – ℘ ) (t – ζ ) v(℘ , ζ ) ¯ ¯ ¯ ¯ 1 2 1 2 + dt ds dζ d℘ ¯ ¯ 2 1 1 1 1 1 st 13e ( )( )( )( ) 0 0 0 0 2 3 7 8 1 1 – – 3 1 2 3 90 80 2 3 (p –s) (q –t) ( s + t ) p q 7 8 40 25 and (p , q )= p + p + dt ds. Assume that σ ∈ Z 89 79 50 35 0 0 st 13e ( )( ) 90 80 and h ∈ N(σ)with υ(σ (p , q ), h(p , q )) ≥ 0 for all (p , q ) ∈ [0, 1] × [0, 1]. Then we have Charandabi et al. Advances in Diﬀerence Equations (2021) 2021:323 Page 16 of 18 σ (p , q ) ≥ 0and h(p , q ) ≥ 0 for all (p , q ) ∈ [0, 1] × [0, 1]. Since σ (p , q ) ≥ 0, we get 2 2 –p –q e |σ (p , q )| H p , q , σ p , q = σ p , q , σ p , q + ⊆ [0, ∞). 4(1 + |σ (p , q )|) Choose v(p , q ) ∈ H(p , q , σ (p , q )) so that v(p , q ) ≥ 0 for all (p , q ) in [0, 1] × [0, 1]. Since (p , q ) ≥ 0, we get w p , q p q s t := p , q + 0 0 0 0 1 2 6 7 – – – – 2 3 7 8 (p – s) (q – t) (s – ℘ ) (t – ζ ) v(℘ , ζ ) ¯ ¯ ¯ ¯ 1 2 1 2 dt ds dζ d℘ ≥ 0 ¯ ¯ 2 1 1 1 1 1 st 13e ( )( )( )( ) 2 3 7 8 and so w(p , q ) ≥ 0. Thus, υ(h(p , q ), w(p , q )) ≥ 0. Note that υ(0, h(p , q )) ≥ 0for h ∈ N(σ ) and also for each convergent sequence {σ } ⊆ Z with σ → σ and n n≥1 n υ(σ (p , q ), σ (p , q )) ≥ 0 for all n and (p , q ) ∈ J × J ,there exists asubsequence n a b 0 0 {σ } of {σ } such that υ(σ (p , q ), σ (p , q )) ≥ 0. Thus, n j≥1 n n≥1 n j j PH H p , q , r , H p , q , r d 1 2 2 2 –p –q e |r | |r | 1 2 ≤ – 4 1+ |r | 1+ |r | 1 2 2 2 2 2 –p –q –p –q e e = |r | – |r | ≤ |r – r |. 1 2 1 2 4 4 2 2 –p –q 1 If ρ(p , q )= e and ψ(t)= t,then PH H p , q , r , H p , q , r ≤ p , q ψ |r – r | d 1 2 1 2 p q and ρ =1. Put l(p , q )=13e .Then l =13 and so ˆ ˆ ˆ ˆ k + k + 1 1 2 2 a b ρ 1 0 0 = = 0.0966114627 ≤ 1. 1 1 1 1 ˆ ˆ ˆ ˆ 13( + +1)( + +1) l(k + +1)(k + +1) 1 1 2 2 2 7 3 8 Now by using Theorem 6,the problem(8)has asolution. 3Conclusion In this work, we studied a partial fractional version of the Sturm–Liouville diﬀerential equation by using the Caputo derivative. Also, we reviewed inclusion version of the prob- lem. First, by using the technique of α-ψ-contractive mappings, we investigated the ex- istence of solutions for the partial fractional Sturm–Liouville equation. We presented an illustrated example to clear more the result. Secondly, we have investigated the partial fractional Sturm–Liouville inclusion problem by using the technique of α-ψ-contractive multifunctions. We provided an illustrated an example for explaining the second result. In this way, we provided some related ﬁgures for the examples. Charandabi et al. Advances in Diﬀerence Equations (2021) 2021:323 Page 17 of 18 Acknowledgements The ﬁrst author was supported by Tabriz Branch, Islamic Azad University. The second author was supported by Miandoab Branch, Islamic Azad University. The third author was supported by Azarbaijan Shahid Madani University. The fourth author was supported by K. N. Toosi University of Technology. The authors express their gratitude to the dear unknown referees for their helpful suggestions, which improved the ﬁnal version of this paper. Funding Not applicable. Availability of data and materials Data sharing not applicable to this article as no datasets were generated or analyzed during the current study. Ethics approval and consent to participate Not applicable. Competing interests The authors declare that they have no competing interests. Consent for publication Not applicable. Authors’ contributions The authors declare that the study was realized in collaboration with equal responsibility. All authors read and approved the ﬁnal manuscript. Author details 1 2 Department of Mathematics, Tabriz Branch, Islamic Azad University, Tabriz, Iran. Department of Mathematics, Miandoab Branch, Islamic Azad University, Miandoab, Iran. Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran. Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan. Department of Mathematics, K. N. Toosi University of Technology, P.O. Box 16315-1618, Tehran, 15418, Iran. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations. Received: 2 April 2021 Accepted: 23 June 2021 References 1. Afshari, H., Kalantari, S., Karapinar, E.: Solution of fractional diﬀerential equations via coupled ﬁxed point. Electron. J. Diﬀer. Equ. 15(286), 1 (2015). http://ejde.math.txstate.edu 2. Alqahtani, B., Aydi, H., Karapinar, E., Rakocevic, V.: A solution for Volterra fractional integral equations by hybrid contractions. Mathematics 7(8), 694 (2019). https://doi.org/10.3390/math7080694 3. Baleanu, D., Jajarmi, A., Mohammadi, H., Rezapour, S.: A new study on the mathematical modelling of human liver with Caputo–Fabrizio fractional derivative. Chaos Solitons Fractals 134, 109705 (2020). https://doi.org/10.1016/j.chaos.2020.109705 4. Rezapour, S., Ntouyas, S.K., Iqbal, M.Q., Hussain, A., Etemad, S., Tariboon, J.: An analytical survey on the solutions of the generalized double-order φ-integrodiﬀerential equation. J. Funct. Spaces 2021, Article ID 6667757 (2021). https://doi.org/10.1155/2021/6667757 5. Abbas, S., Benchohra, M.: Darboux problem for perturbed partial diﬀerential equations of fractional order with ﬁnite delay. Nonlinear Anal. Hybrid Syst. 3(4), 597–604 (2009). https://doi.org/10.1016/j.nahs.2009.05.001 6. Rezapour, S., Sakar, F.M., Aydogan, S.M., Ravash, E.: Some results on a system of multiterm fractional integro-diﬀerential equations. Turk. J. Math. 44(6), 2004–2020 (2021). https://doi.org/10.3906/mat-1903-51 7. Baleanu, D., Mousalou, A., Rezapour, S.: On the existence of solutions for some inﬁnite coeﬃcient-symmetric Caputo–Fabrizio fractional integro-diﬀerential equations. Bound. Value Probl. 2017(1), 145 (2017). https://doi.org/10.1186/s13661-017-0867-9 8. Baleanu, D., Rezapour, S., Etemad, S., Alsaedi, A.: On a time-fractional partial integro-diﬀerential equation via three-point boundary value conditions. Math. Probl. Eng. 2015, Article ID 896871 (2015). https://doi.org/10.1155/2015/785738 9. Baitiche, Z., Derbazi, C., Benchora, M.: ψ-Caputo fractional diﬀerential equations with multi-point boundary conditions by topological degree theory. Results Nonlinear Anal. 3(4), 167–178 (2020) 10. Karapinar, E., Fulga, A., Rashid, M., Shahid, L., Aydi, H.: Large contractions on quasi-metric spaces with an application to nonlinear fractional diﬀerential equations. Mathematics 7(5), 444 (2019). https://doi.org/10.3390/math7050444 11. Mohammadi, H., Kumar, S., Rezapour, S., Etemad, S.: A theoretical study of the Caputo–Fabrizio fractional modeling for hearing loss due to Mumps virus with optimal control. Chaos Solitons Fractals 144, 110668 (2021). https://doi.org/10.1016/j.chaos.2021.110668 12. Ali, A., Shah, K., Abdeljawad, T., Mahariq, I., Rashdan, M.: Mathematical analysis of nonlinear integral boundary value problem of proportional delay implicit fractional diﬀerential equations with impulsive conditions. Bound. Value Probl. 2021, 7 (2021). https://doi.org/10.1186/s13661-021-01484-y 13. Tuan, N.H., Mohammadi, H., Rezapour, S.: A mathematical model for COVID-19 transmission by using the Caputo fractional derivative. Chaos Solitons Fractals 140, 110107 (2020). https://doi.org/10.1016/j.chaos.2020.110107 14. Murugusundaramoorthy, G.: Application of Pascal distribution series to Ronning type star-like and convex functions. Adv. Theory Nonlinear Anal. Appl. 4(4), 243–251 (2020). https://doi.org/10.31197/atnaa.743436 Charandabi et al. Advances in Diﬀerence Equations (2021) 2021:323 Page 18 of 18 15. Patil, J., Chaudhari, A., Abdo, M., Hardan, B.: Upper and lower solution method for positive solution of generalized Caputo fractional diﬀerential equations. Adv. Theory Nonlinear Anal. Appl. 4(4), 279–291 (2020). https://doi.org/10.31197/atnaa.709442 16. Patil, J., Chaudhari, A., Abdo, M., Hardan, B.: Upper and lower solution method for positive solution of generalized Caputo fractional diﬀerential equations. Adv. Theory Nonlinear Anal. Appl. 4(4), 279–291 (2020). https://doi.org/10.31197/atnaa.709442 17. Muthaiah, S., Murugesan, M., Thangaraj, N.G.: Existence of solutions for nonlocal boundary value problem of Hadamard fractional diﬀerential equations. Adv. Theory Nonlinear Anal. Appl 3(3), 162–173 (2019). https://doi.org/10.31197/atnaa.579701 18. Aydogan, M.S., Baleanu, D., Mousalou, A., Rezapour, S.: On high order fractional integro-diﬀerential equations including the Caputo–Fabrizio derivative. Bound. Value Probl. 2018, 90 (2018). https://doi.org/10.1186/s13661-018-1008-9 19. Jarad, F., Abdeljawad, T.: A modiﬁed Laplace transform for certain generalized fractional operators. Results Nonlinear Anal. 1(2), 88–98 (2019) 20. Marino, G., Scardamglia, B., Karapinar, E.: Strong convergence theorem for strict pseudo-contractions in Hilbert spaces. J. Inequal. Appl. 2016, 134 (2016). https://doi.org/10.1186/s13660-016-1072-6 21. Baleanu, D., Mousalou, A., Rezapour, S.: A new method for investigating approximate solutions of some fractional integro-diﬀerential equations involving the Caputo-Fabrizio derivative. Adv. Diﬀer. Equ. 2017, 51 (2017). https://doi.org/10.1186/s13662-017-1088-3 22. Afshari, H., Marasi, H., Aydi, H.: Existence and uniqueness of positive solutions for boundary value problems of fractional diﬀerential equations. Filomat 31(9), 2675–2682 (2017). https://doi.org/10.2298/FIL1709675A 23. Marasi, H., Afshari, H., Zhai, C.B.: Some existence and uniqueness results for nonlinear fractional partial diﬀerential equations. Rocky Mt. J. Math. 47(2), 571–585 (2017). https://doi.org/10.1216/RMJ-2017-47-2-571 24. Bachir, F.S., Said, A., Benbachir, M., Benchohra, M.: Hilfer–Hadamard fractional diﬀerential equations; existence and attractivity. Adv. Theory Nonlinear Anal. Appl 5(1), 49–57 (2021). https://doi.org/10.31197/atnaa.848928 25. Abdeljawad, T., Al-Mdallal, Q.M., Hammouchc, Z., Jarad, F.: Existence and uniqueness of positive solutions for boundary value problems of fractional diﬀerential equations. Adv. Theory Nonlinear Anal. Appl 4(4), 214–215 (2020). https://doi.org/10.31197/atnaa.810371 26. Tuan, N.H., Huynh, L.N., Baleanu, D., Can, N.H.: On a terminal value problem for a generalization of the fractional diﬀusion equation with hyper-Bessel operator. Math. Methods Appl. Sci. 43(6), 2858–2882 (2020). https://doi.org/10.1002/mma.6087 27. Can, N.H., Nikan, O., Rasoulizadeh, M.N., Gasimov, Y.S.: Numerical computation of the time nonlinear fractional generalized equal width model arising in shallow water channel. Therm. Sci. 24(1), 49–58 (2020). https://doi.org/10.2298/TSCI20S1049C 28. Tuan, N.H., Thuch, T.N., Can, N.H., O’Regan, D.: Regularization of a multidimensional diﬀusion equation with conformable time derivative and discrete data. Math. Methods Appl. Sci. 44(4), 2879–2891 (2021). https://doi.org/10.1002/mma.6133 29. Baleanu, D., Mohammadi, H., Rezapour, S.: Analysis of the model of HIV-1 infection of CD4 T-cell with a new approach of fractional derivative. Adv. Diﬀer. Equ. 2020, Article ID 71 (2020) 30. Rezapour, S., Mohammadi, H., Jajarmi, A.: A new mathematical model for Zika virus transmission. Adv. Diﬀer. Equ. 2020, 589 (2020). https://doi.org/10.1186/s13662-020-03044-7 31. Abbas, S., Benchohra, M.: Fractional order partial hyperbolic diﬀerential equations involving Caputo derivative. Stud. Univ. Babes–Bolyai ¸ Math. 57(4), 469–479 (2012) 32. Abbas, S., Benchohra, M.: Partial hyperbolic diﬀerential equations with ﬁnite delay involving the Caputo fractional derivative. Commun. Math. Anal. 7(2), 62–72 (2009) 33. Abbas, S., Benchohra, M., N’Guerekata, G.M.: Darboux problem for perturbed partial diﬀerential equations of fractional order with ﬁnite delay. Nonlinear Anal. Hybrid Syst. 3(4), 597–604 (2009). https://doi.org/10.1016/j.nahs.2009.05.001 34. Ahmad, B., Ntouyas, S.K., Tariboon, J.: A study of mixed Hadamard and Riemann–Liouville fractional integro-diﬀerential inclusions via endpoint theory. Appl. Math. Lett. 52, 9–14 (2016). https://doi.org/10.1016/j.aml.2015.08.002 35. Benchohra, M., Henderson, J., Mostefai, F.Z.: Weak solutions for hyperbolic partial fractional diﬀerential inclusions in Banach spaces. Comput. Math. Appl. 64(10), 3101–3107 (2012). https://doi.org/10.1016/j.camwa.2011.12.055 36. Etemad, S., Rezapour, S.: On the existence of solution for three variables partial fractional-diﬀerential equation and inclusion. J. Adv. Math. Stud. 8(2), 224–233 (2015) 37. Joannopoulos, J.D., Johnson, S.G., Winnn, J.N., Meade, R.D.: Photonic Crystals: Molding the Flow of Light. Princeton University Press, Princeton (2008) 38. Teschl, G.: Mathematical Methods in Quantum Mechanics: With Applications to Schrodinger Operators. Am. Math. Soc., New York (2014) 39. Ashrafyan, Y.: A new kind of uniqueness theorems for inverse Sturm–Liouville problems. Bound. Value Probl. 2017,79 (2017). https://doi.org/10.1186/s13661-017-0813-x 40. Adiguzel, R. S., Aksoy, U., Karapinar, E., Erhan, I. M.: On the solution of a boundary value problem associated with a fractional diﬀerential equation. Math. Method. Appl. Sci. (2020). https://doi.org/10.1002/mma.6652 41. Liu, Y., He, T., Shi, H.: Three positive solutions of Sturm–Liouville boundary value problems for fractional diﬀerential equations. Diﬀer. Equ. Appl. 5(1), 127–152 (2013) 42. Podlubny, I.: Fractional Diﬀerential Equations. Academic Press, San Diego (1999) 43. Deimling, K.: Multi-Valued Diﬀerential Equations. de Gruyter, Berlin (1992) 44. Aubin, J., Cellina, A.: Diﬀerential Inclusions: Set-Valued Maps and Viability Theory. Springer, Berlin (1984). https://doi.org/10.1007/978-3-642-69512-4 45. Samet, B., Vetro, C., Vetro, P.: Fixed point theorem for α-ψ contractive type mappings. Nonlinear Anal. 75(4), 2154–2165 (2012). https://doi.org/10.1016/j.na.2011.10.014 46. Mohammadi, B., Rezapour, S., Shahzad, N.: Some results on ﬁxed points of α-ψ-Ciric generalized multifunctions. Fixed Point Theory Appl. 2013, 24 (2013). https://doi.org/10.1186/1687-1812-2013-24 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Difference Equations Springer Journals http://www.deepdyve.com/lp/springer-journals/on-partial-fractional-sturm-liouville-equation-and-inclusion-cAoFPqeKyK

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References (42)

D. Baleanu, A. Jajarmi, H. Mohammadi, S. Rezapour (2020)
A new study on the mathematical modelling of human liver with Caputo–Fabrizio fractional derivative
Chaos Solitons Fractals, 134
N.H. Can, O. Nikan, M.N. Rasoulizadeh, Y.S. Gasimov (2020)
Numerical computation of the time nonlinear fractional generalized equal width model arising in shallow water channel
Therm. Sci., 24
B. Ahmad, S.K. Ntouyas, J. Tariboon (2016)
A study of mixed Hadamard and Riemann–Liouville fractional integro-differential inclusions via endpoint theory
Appl. Math. Lett., 52
E. Karapinar, A. Fulga, M. Rashid, L. Shahid, H. Aydi (2019)
Large contractions on quasi-metric spaces with an application to nonlinear fractional differential equations
Mathematics, 7
S. Abbas, M. Benchohra (2009)
Partial hyperbolic differential equations with finite delay involving the Caputo fractional derivative
Commun. Math. Anal., 7
S. Etemad, S. Rezapour (2015)
On the existence of solution for three variables partial fractional-differential equation and inclusion
J. Adv. Math. Stud., 8
D. Baleanu, S. Rezapour, S. Etemad, A. Alsaedi (2015)
On a time-fractional partial integro-differential equation via three-point boundary value conditions
Math. Probl. Eng., 2015
D. Baleanu, H. Mohammadi, S. Rezapour (2020)
Analysis of the model of HIV-1 infection of CD4+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${CD}4^{+}$\end{document} T-cell with a new approach of fractional derivative
Adv. Differ. Equ., 2020
N.H. Tuan, T.N. Thuch, N.H. Can, D. O’Regan (2021)
Regularization of a multidimensional diffusion equation with conformable time derivative and discrete data
Math. Methods Appl. Sci., 44
A. Ali, K. Shah, T. Abdeljawad, I. Mahariq, M. Rashdan (2021)
Mathematical analysis of nonlinear integral boundary value problem of proportional delay implicit fractional differential equations with impulsive conditions
Bound. Value Probl., 2021
Y. Liu, T. He, H. Shi (2013)
Three positive solutions of Sturm–Liouville boundary value problems for fractional differential equations
Differ. Equ. Appl., 5
H. Mohammadi, S. Kumar, S. Rezapour, S. Etemad (2021)
A theoretical study of the Caputo–Fabrizio fractional modeling for hearing loss due to Mumps virus with optimal control
Chaos Solitons Fractals, 144
F.S. Bachir, A. Said, M. Benbachir, M. Benchohra (2021)
Hilfer–Hadamard fractional differential equations; existence and attractivity
Adv. Theory Nonlinear Anal. Appl, 5
S. Rezapour, H. Mohammadi, A. Jajarmi (2020)
A new mathematical model for Zika virus transmission
Adv. Differ. Equ., 2020
B. Mohammadi, S. Rezapour, N. Shahzad (2013)
Some results on fixed points of α-ψ-Ciric generalized multifunctions
Fixed Point Theory Appl., 2013
N.H. Tuan, H. Mohammadi, S. Rezapour (2020)
A mathematical model for COVID-19 transmission by using the Caputo fractional derivative
Chaos Solitons Fractals, 140
J. Aubin, A. Cellina (1984)
10.1007/978-3-642-69512-4
Differential Inclusions: Set-Valued Maps and Viability Theory
H. Marasi, H. Afshari, C.B. Zhai (2017)
Some existence and uniqueness results for nonlinear fractional partial differential equations
Rocky Mt. J. Math., 47
Z. Baitiche, C. Derbazi, M. Benchora (2020)
ψ-Caputo fractional differential equations with multi-point boundary conditions by topological degree theory
Results Nonlinear Anal., 3
S. Muthaiah, M. Murugesan, N.G. Thangaraj (2019)
Existence of solutions for nonlocal boundary value problem of Hadamard fractional differential equations
Adv. Theory Nonlinear Anal. Appl, 3
G. Marino, B. Scardamglia, E. Karapinar (2016)
Strong convergence theorem for strict pseudo-contractions in Hilbert spaces
J. Inequal. Appl., 2016
S. Rezapour, S.K. Ntouyas, M.Q. Iqbal, A. Hussain, S. Etemad, J. Tariboon (2021)
An analytical survey on the solutions of the generalized double-order ϕ-integrodifferential equation
J. Funct. Spaces, 2021
S. Abbas, M. Benchohra, G.M. N’Guerekata (2009)
Darboux problem for perturbed partial differential equations of fractional order with finite delay
Nonlinear Anal. Hybrid Syst., 3
G. Murugusundaramoorthy (2020)
Application of Pascal distribution series to Ronning type star-like and convex functions
Adv. Theory Nonlinear Anal. Appl., 4
K. Deimling (1992)
10.1515/9783110874228
Multi-Valued Differential Equations
H. Afshari, H. Marasi, H. Aydi (2017)
Existence and uniqueness of positive solutions for boundary value problems of fractional differential equations
Filomat, 31
S. Abbas, M. Benchohra (2012)
Fractional order partial hyperbolic differential equations involving Caputo derivative
Stud. Univ. Babeş–Bolyai Math., 57
B. Samet, C. Vetro, P. Vetro (2012)
Fixed point theorem for α-ψ contractive type mappings
Nonlinear Anal., 75
F. Jarad, T. Abdeljawad (2019)
A modified Laplace transform for certain generalized fractional operators
Results Nonlinear Anal., 1
J. Patil, A. Chaudhari, M. Abdo, B. Hardan (2020)
Upper and lower solution method for positive solution of generalized Caputo fractional differential equations
Adv. Theory Nonlinear Anal. Appl., 4
S. Rezapour, F.M. Sakar, S.M. Aydogan, E. Ravash (2021)
Some results on a system of multiterm fractional integro-differential equations
Turk. J. Math., 44
H. Afshari, S. Kalantari, E. Karapinar (2015)
Solution of fractional differential equations via coupled fixed point
Electron. J. Differ. Equ., 15
T. Abdeljawad, Q.M. Al-Mdallal, Z. Hammouchc, F. Jarad (2020)
Existence and uniqueness of positive solutions for boundary value problems of fractional differential equations
Adv. Theory Nonlinear Anal. Appl, 4
R. S. Adiguzel, U. Aksoy, E. Karapinar, I. M. Erhan (2020)
On the solution of a boundary value problem associated with a fractional differential equation
Math. Method. Appl. Sci.
M. Benchohra, J. Henderson, F.Z. Mostefai (2012)
Weak solutions for hyperbolic partial fractional differential inclusions in Banach spaces
Comput. Math. Appl., 64
D. Baleanu, A. Mousalou, S. Rezapour (2017)
On the existence of solutions for some infinite coefficient-symmetric Caputo–Fabrizio fractional integro-differential equations
Bound. Value Probl., 2017
D. Baleanu, A. Mousalou, S. Rezapour (2017)
A new method for investigating approximate solutions of some fractional integro-differential equations involving the Caputo-Fabrizio derivative
Adv. Differ. Equ., 2017
B. Alqahtani, H. Aydi, E. Karapinar, V. Rakocevic (2019)
A solution for Volterra fractional integral equations by hybrid contractions
Mathematics, 7
Y. Ashrafyan (2017)
A new kind of uniqueness theorems for inverse Sturm–Liouville problems
Bound. Value Probl., 2017
M.S. Aydogan, D. Baleanu, A. Mousalou, S. Rezapour (2018)
On high order fractional integro-differential equations including the Caputo–Fabrizio derivative
Bound. Value Probl., 2018
G. Teschl (2014)
10.1090/gsm/157
Mathematical Methods in Quantum Mechanics: With Applications to Schrodinger Operators
N.H. Tuan, L.N. Huynh, D. Baleanu, N.H. Can (2020)
On a terminal value problem for a generalization of the fractional diffusion equation with hyper-Bessel operator
Math. Methods Appl. Sci., 43

Publisher: Springer Journals
Copyright: Copyright © The Author(s) 2021
eISSN: 1687-1847
DOI: 10.1186/s13662-021-03478-7
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Abstract

sh.rezapour@mail.cmuh.org.tw; The Sturm–Liouville diﬀerential equation is one of interesting problems which has rezapourshahram@yahoo.ca Department of Mathematics, been studied by researchers during recent decades. We study the existence of a Azarbaijan Shahid Madani solution for partial fractional Sturm–Liouville equation by using the α-ψ-contractive University, Tabriz, Iran 4 mappings. Also, we give an illustrative example. By using the α-ψ-multifunctions, we Department of Medical Research, China Medical University Hospital, prove the existence of solutions for inclusion version of the partial fractional China Medical University, Taichung, Sturm–Liouville problem. Finally by providing another example and some ﬁgures, we Taiwan try to illustrate the related inclusion result. Full list of author information is available at the end of the article MSC: Primary 34A08; secondary 34A12 Keywords: α-ψ-contractive map; Inclusion problem; The Caputo derivative; The partial fractional Sturm–Liouville equation; Two variables partial diﬀerential equation 1 Introduction It can be said that most physical or engineering phenomena can be modeled with some categories such time-dependent (or time-fractional), fractional diﬀerential and some vari- ables partial equations. One can ﬁnd many published papers on delayed time-fractional problems, fractional diﬀerential equations [1–30] and some variables partial fractional problems [31–36]. During the history of mathematics, physics and engineering, we can ﬁnd many equations which have a special role in progress of these sciences. One of the important frameworks of problems is the Sturm–Liouville diﬀerential equation (in brief SLDE) have been in the spotlight of the mathematicians of applied mathematics, engi- neering and scientists of physics, quantum mechanics, classical mechanics (see, [37, 38] and the references therein). In such a manner, it is important that mathematicians and researchers design complicated and more general abstract mathematical models of pro- cedures in the format of applicable fractional SLDE [33, 39–41]. One can ﬁnd a variety of recent work about this equation, but the aim of this work is studying partial version of the Sturm–Liouville diﬀerential equation. ˆ ˆ ˆ ˆ ˆ Let k =(k , k )where k , k >0 and J =[0, a ]and J =[0, b ]where a , b >0.For σ ∈ 1 2 1 2 a 0 b 0 0 0 0 0 1 1 L (J × J , R)= L (J × J ), the partial left-sided mixed Riemann–Liouville integral a b a b 0 0 0 0 © The Author(s) 2021. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Charandabi et al. Advances in Diﬀerence Equations (2021) 2021:323 Page 2 of 18 (of order k)isdeﬁnedby(see[42]) ˆ ˆ p q k –1 k –1 1 2 (p – s) (q – t) I σ p , q = σ (s, t) dt ds. ˆ ˆ 0 0 (k )(k ) 1 2 Also the partial derivative in the sense of Caputo (of order k)isdeﬁnedby ˆ ˆ k 1–k D σ p , q = I σ p , q c 0 ∂p ∂q ˆ ˆ p q k –1 k –1 2 1 2 (p – s) (q – t) ∂ = σ (s, t) dt ds. ˆ ˆ ∂s ∂t 0 0 (k )(k ) 1 2 Let (Z , d )be a metric space. P (Z )the set of all closed subsets of Z and 2 the Z cl set of all nonempty subsets of Z . It is well known that the Pompeiu–Hausdorﬀ metric PH : P (Z ) × P (Z ) → R ∪{∞} is deﬁned by d cl cl d d d d Z Z Z Z PH A , A = max sup d a , A , sup d A , a d Z Z 1 2 1 2 1 2 d d Z Z a ∈A a ∈A 1 1 2 2 d d d d Z Z Z Z for all A , A ∈ P(Z ), where d (a , A )= inf d (a , a )and d (A , Z Z Z 1 2 1 2 Z 1 2 1 a ∈A 1 1 a )= inf d d (a , a )[43]. We say that a set-valued mapping : Z → P (Z ) Z cl 2 1 2 a ∈A 2 2 is called Lipschitzian with Lipschitz constant k > 0 whenever PH ((σ ), (σ )) ≤ d 1 1 kd (σ , σ ) for all σ , σ ∈ Z .If 0 < k < 1, then we ay that is a contraction [43]. An op- Z 1 2 1 2 erator : [0, 1] → P (R) is called measurable whenever the function t → d (ω , (t)) = cl Z 0 inf{|ω – y| : y ∈ (t)} is measurable for all real constant ω [43, 44]. The following notions were introduced in 2012 [45]. • = {ψ | ψ (t)< ∞, ∀t >0} where ψ :[0, ∞) × [0, ∞) → [0, ∞). n=1 • Assume that α : Z × Z → [0, ∞) and T : Z → Z are two mappings. Now T is α-admissible whenever for each σ , σ ∈ Z with α(σ , σ ) ≥ 1,weget 1 2 1 2 α(T σ , T σ ) ≥ 1. 1 2 • T is α-ψ-contractive mapping whence α(σ , σ )d (T σ , T σ ) ≤ ψ(d (σ , σ )) for all 1 2 Z 1 2 Z 1 2 σ , σ ∈ Z . 1 2 Lemma 1 ([45]) Assume that the metric space (Z , d ) is complete, Tis α-admissible and α-ψ-contractive mapping and there exists σ ∈ Z such that α(σ , T σ ) ≥ 1. Further, 0 0 0 for every convergent sequence {σ } ⊆ Z with σ → σ and α(σ , σ ) ≥ 1 for all n ≥ 1, n n≥1 n n n+1 we have α(σ , σ ) ≥ 1 for all n ≥ 1. Then T has a ﬁxed point. After this, multifunction version of α-ψ-contractive maps introduced in 2013 as follows [46]. • A multifunction F : Z → CB(Z ) is α-admissible whenever for each σ ∈ Z and σ ∈ Fσ with α(σ , σ ) ≥ 1,wehave α(σ , w ) ≥ 1, for all w ∈ Fσ . 2 1 1 2 2 0 0 2 • The metric space Z possesses the C -property if for every convergent sequence {σ } ⊆ Z with σ → σ and α(σ , σ ) ≥ 1 for all n ≥ 1,there exists asubsequence n n≥1 n n n+1 {σ } of σ such that α(σ , σ ) ≥ 1 for all j ≥ 1. n j≥1 n n j j • F is α-ψ-contractive multifunction whenever α(σ , σ )PH (T σ , T σ ) ≤ ψ(d (σ , σ )) for all σ , σ ∈ Z . 1 2 d 1 2 Z 1 2 1 2 Z Charandabi et al. Advances in Diﬀerence Equations (2021) 2021:323 Page 3 of 18 Lemma 2 ([46]) Assume that the metric space (Z , d ) is complete, Fis α-admissible and α-ψ-contractive multifunction and there exist σ ∈ Z and σ ∈ Fσ such that α(σ , σ ) ≥ 0 1 0 0 1 1. If Z possesses the C -property, then F has a ﬁxed point. In this paper, ﬁrst we investigate the partial fractional Sturm–Liouville diﬀerential equa- tion ˆ ˆ D (l(p , q )D σ (p , q )) + o(p , q )σ (p , q )= h(p , q )f (σ (p , q )), c c 0 0 (p , q ) ∈ J × J (l(p , q )D σ (p , q )) = θ (p ), a b q =0 1 0 0 c (1) ⎪ (l(p , q )D σ (p , q )) = θ (q ), p =0 2 ⎪ c σ (p ,0) = κ(p )and σ (0, q )= ω(q ), ˆ ˆ where k, ∈ (0, 1] × (0, 1], D and D denote the Caputo partial fractional derivatives, c c 0 0 l, o, h belong to C(J × J )with l(p , q ) =0 for all (p , q ) ∈ J × J and f : R → R a b a b 0 0 0 0 is a function. Here, θ : J → R, θ : J → R, κ : J → R and ω : J → R are abso- 1 a 2 b a b 0 0 0 0 lutely continuous with θ (0) = θ (0) = κ(0) = ω(0). Also, we investigate the partial fractional 1 2 Sturm–Liouville diﬀerential inclusion problem ˆ ˆ D (l(p , q )D σ (p , q )) ∈ H(p q , σ (p , q )), (p , q ) ∈ J × J , ⎪ a b c c 0 0 0 0 (l(p , q )D σ (p , q )) = θ (p ), q =0 1 (2) ⎪ (l(p , q )D σ (p , q )) = θ (q ), p =0 2 ⎪ c σ (p ,0) = κ(p )and σ (0, q )= ω(q ), ˆ ˆ where k, ∈ (0, 1] × (0, 1], D and D denote the Caputo partial fractional derivatives, c c 0 0 l, o, h belong to C(J × J )with l(p , q ) =0 for all (p , q ) ∈ J × J and f : R → R a b a b 0 0 0 0 is a function. Here, θ : J → R, θ : J → R, κ : J → R and ω : J → R are abso- 1 a 2 b a b 0 0 0 0 lutely continuous with θ (0) = θ (0) = κ(0) = ω(0). Also, H : J × J × R → P (R)is an 1 2 a b cl 0 0 integrable bounded multifunction so that H(., ., σ ) is measurable for all σ ∈ R. 2 Main results Assume that Z = {σ |σ ∈ C(J × J )} and σ = sup |σ (p , q )|,where σ ∈ a b 0 0 (p ,q )∈J ×J 0 0 Z .Then (Z , .)is a Banach space. ˆ ˆ ˆ ˆ ˆ ˆ Lemma 3 Let k =(k , k ), =( ) ∈ (0, 1] × (0, 1] and g ∈ L (J × J ). Consider the 1 2 1 2 a b 0 0 problem ˆ ˆ D l p , q D σ p , q = g p , q,(3) c c 0 0 with boundary conditions (l(p , q )D σ (p , q )) = θ (p ), ⎪ y=0 1 (4) (l(p , q )D σ (p , q )) = θ (q ), x=0 2 σ (p ,0) = κ(p ) and σ (0, q )= ω(q ). Charandabi et al. Advances in Diﬀerence Equations (2021) 2021:323 Page 4 of 18 Then the function σ ∈ C(J × J ) is a solution of the problem (3)–(4) whenever a b 0 0 σ p , q = p , q p q s t 0 0 0 0 ˆ ˆ ˆ ˆ k –1 k –1 –1 –1 1 2 1 2 (p – s) (q – t) (s – ℘ ¯) (t – ζ ¯) g(℘ ¯, ζ ¯) 1 2 1 2 dt ds dζ d℘ , ¯ ¯ 2 1 ˆ ˆ ˆ ˆ (k )(k )( )( )l(s, t) 1 2 1 2 where θ (p )+ θ (q )– θ (0) ˆ 1 2 1 p , q = κ p + ω q – κ(0) + I . l(p , q ) Proof Note that Eq. (3)can be writtenas ˆ ˆ 1–k I l p , q D σ p , q = g p , q . 0 c ∂p ∂q Operating by I on both sides we get ˆ ˆ I l p , q D σ p , q = I g p , q . 0 c 0 ∂p ∂q Since I l p , q D σ p , q 0 c ∂p ∂q ˆ ˆ = l p , q D σ p , q – l p , q D σ p , q c c 0 0 q =0 ˆ ˆ – l p , q D σ p , q + l p , q D σ p , q c c 0 p =0 0 p =0,q =0 = l p , q D σ p , q – θ p – θ q + θ (0), 1 2 1 ˆ ˆ we get l(p , q )D σ (p , q )= θ (p )+ θ (q )– θ (0) + I g(p , q ). Hence, 1 2 1 c 0 θ (p )+ θ (q )– θ (0) 1 ˆ 1 2 1 ˆ D σ p , q = + I g p , q . c 0 l(p , q ) l(p , q ) This equation can be written as ∂ θ (p )+ θ (q )– θ (0) 1 ˆ 1 2 1 1– I σ p , q = + I g p , q . 0 0 ∂p ∂q l(p , q ) l(p , q ) Again operating by I on both sides, we obtain ∂ θ (p )+ θ (q )– θ (0) 1 ˆ ˆ ˆ 1 2 1 I σ p , q = I + I I g p , q . 0 0 0 0 ∂p ∂q l(p , q ) l(p , q ) Charandabi et al. Advances in Diﬀerence Equations (2021) 2021:323 Page 5 of 18 Since I σ p , q = σ p , q – σ p ,0 – σ 0, q + σ (0, 0) ∂p ∂q = σ p , q – κ p – ω q + κ(0), we get θ (p )+ θ (q )– θ (0) ˆ 1 2 1 σ p , q = κ p + ω q – κ(0) + I l(p , q ) 1 1 ˆ ˆ ˆ ˆ + I I g p , q = p , q + I I g p , q . 0 0 0 0 l(p , q ) l(p , q ) On the other hand, ˆ ˆ I I g p , q 0 0 l(p , q ) ˆ ˆ p q k –1 k –1 1 2 (p – s) (q – t) 1 = I g(s, t) dt ds ˆ ˆ l(s, t) 0 0 (k )(k ) 1 2 p q ˆ ˆ 1 1 k –1 k –1 1 2 = p – s q – t ˆ ˆ l(s, t) 0 0 (k )(k ) 1 2 ˆ ˆ s t –1 –1 1 2 (s – ℘ ) (t – ζ ) ¯ ¯ 1 2 × g(℘ , ζ ) d℘ dζ dt ds ¯ ¯ ¯ ¯ 1 2 1 2 ˆ ˆ )( 0 0 1 2 p q s t 0 0 0 0 ˆ ˆ ˆ ˆ k –1 k –1 –1 –1 1 2 1 2 (p – s) (q – t) (s – ℘ ) (t – ζ ) g(℘ , ζ ) ¯ ¯ ¯ ¯ 1 2 1 2 dt ds dζ d℘ ¯ ¯ 2 1 ˆ ˆ ˆ ˆ (k )(k )( )( )l(s, t) 1 2 1 2 and so σ p , q = p , q p q s t 0 0 0 0 ˆ ˆ ˆ ˆ k –1 k –1 –1 –1 1 2 1 2 (p – s) (q – t) (s – ℘ ¯) (t – ζ ¯) g(℘ ¯, ζ ¯) 1 2 1 2 dt ds dζ d℘ . ¯ ¯ 2 1 ˆ ˆ ˆ ˆ (k )(k )( )( )l(s, t) 1 2 1 2 This completes the proof. Now we establish and prove our ﬁrst main theorem. Theorem 4 Assume that υ : R × R → R is a function and : J × J → [0, ∞) is a a b 0 0 bounded function such that f σ p , q – f σ p , q ≤ p , q σ p , q – σ p , q 1 2 1 2 Charandabi et al. Advances in Diﬀerence Equations (2021) 2021:323 Page 6 of 18 for all σ , σ ∈ Z with υ(σ (p , q ), σ (p , q )) ≥ 0, where (p , q ) ∈ J ×J . Suppose that, 1 2 1 2 a b 0 0 for all σ , σ ∈ Z with υ(σ (p , q ), σ (p , q )) ≥ 0, we have υ(Q σ (p , q ), Q σ (p , q )) ≥ 1 2 1 2 1 2 ¯ ¯ 0 0 0, where Q σ p , q = p , q p q s t 0 0 0 0 ˆ ˆ ˆ ˆ k –1 k –1 –1 –1 1 2 1 2 (p – s) (q – t) (s – ℘ ) (t – ζ ) H(℘ , ζ , σ (℘ , ζ )) ¯ ¯ ¯ ¯ ¯ ¯ 1 2 1 2 1 2 dt ds dζ d℘ , ¯ ¯ 2 1 ˆ ˆ ˆ ˆ (k )(k )( )( )l(s, t) 1 2 1 2 θ (p )+θ (q )–θ (0) 1 2 1 (p , q )= κ(p )+ ω(q )– κ(0) + I ( ), l(p ,q ) H ℘ , ζ , σ (℘ , ζ ) = h(℘ , ζ )f σ (℘ , ζ ) – o(℘ , ζ )σ (℘ , ζ ) ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ 1 2 1 2 1 2 1 2 1 2 1 2 and there exists σ so that υ(σ (p , q ), Q σ (p , q )) ≥ 0 whenever (p , q ) ∈ J × J . 0 0 0 a b ¯ 0 0 Assume that, for every sequence {σ } ⊆ Z with σ → σ and υ(σ (p , q ), σ (p , q )) ≥ n n≥1 n n n+1 0 for all n ≥ 1 and (p , q ) ∈ J × J , we have υ(σ (p , q ), σ (p , q )) ≥ 0 for all n ≥ 1 a b n 0 0 ˆ ˆ ˆ ˆ k + k + 1 1 2 2 ∗ a b (h +o) 0 0 and (p , q ) ∈ J × J . If <1, then the fractional Sturm–Liouville a b 0 0 ˆ ˆ ˆ ˆ l(k + +1)(k + +1) 1 1 2 2 problem (1) has a solution, where = sup (p , q ) and (p ,q )∈J ×J l = inf l p , q . (p ,q )∈J ×J a b Proof By using Lemma 3, σ is a solution of the partial fractional Sturm–Liouville problem (1)ifand only if σ = Q σ .Let σ , σ ∈ Z and (p , q ) ∈ J × J with 0 0 1 2 a b ¯ 0 0 υ σ p , q , σ p , q ≥ 0. 1 2 Hence, we get Q σ p , q – Q σ p , q 1 2 ¯ ¯ 0 0 p q s t = p , q + 0 0 0 0 ˆ ˆ ˆ ˆ k –1 k –1 –1 –1 1 2 1 2 (p – s) (q – t) (s – ℘ ) (t – ζ ) H(℘ , ζ , σ (℘ , ζ )) ¯ ¯ ¯ ¯ ¯ ¯ 1 2 1 2 1 1 2 dt ds dζ d℘ ¯ ¯ 2 1 ˆ ˆ ˆ ˆ (k )(k )( )( )l(s, t) 1 2 1 2 p q s t – p , q – 0 0 0 0 ˆ ˆ ˆ ˆ k –1 k –1 –1 –1 1 2 1 2 (p – s) (q – t) (s – ℘ ) (t – ζ ) H(℘ , ζ , σ (℘ , ζ )) ¯ ¯ ¯ ¯ ¯ ¯ 1 2 1 2 1 2 dt ds dζ d℘ ¯ ¯ 2 1 ˆ ˆ ˆ ˆ (k )(k )( )( )l(s, t) 1 2 1 2 p q s t 0 0 0 0 ˆ ˆ ˆ ˆ k –1 k –1 –1 –1 1 2 1 2 (p – s) (q – t) (s – ℘ ) (t – ζ ) |N (℘ , ζ )| ¯ ¯ ¯ ¯ 1 2 1 2 dt ds dζ d℘ , ¯ ¯ 2 1 ˆ ˆ ˆ ˆ (k )(k )( )( )l(s, t) 1 2 1 2 Charandabi et al. Advances in Diﬀerence Equations (2021) 2021:323 Page 7 of 18 where N (℘ , ζ )= H(℘ , ζ , σ (℘ , ζ )) – H(℘ , ζ , σ (℘ , ζ )). Since ¯ ¯ ¯ ¯ 1 ¯ ¯ ¯ ¯ 2 ¯ ¯ 1 2 1 2 1 2 1 2 1 2 H ℘ , ζ , σ (℘ , ζ ) – H ℘ , ζ , σ (℘ , ζ ) ¯ ¯ 1 ¯ ¯ ¯ ¯ 2 ¯ ¯ 1 2 1 2 1 2 1 2 ≤ h(℘ , ζ ) f σ (℘ , ζ ) – f σ (℘ , ζ ) – o(℘ , ζ ) σ (℘ , ζ )– σ (℘ , ζ ) ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ 1 2 1 1 2 2 1 2 1 2 1 1 2 2 1 2 ≤ h(℘ , ζ ) f σ (℘ , ζ ) – f σ (℘ , ζ ) + o(℘ , ζ ) σ (℘ , ζ )– σ (℘ , ζ ) ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ 1 2 1 1 2 2 1 2 1 2 1 1 2 2 1 2 ≤ h(℘ , ζ ) (℘ , ζ ) σ (℘ , ζ )– σ (℘ , ζ ) + o(℘ , ζ ) σ (℘ , ζ )– σ (℘ , ζ ) ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 ≤ h + o σ – σ , 1 2 we have Q σ p , q – Q σ p , q 1 2 ¯ ¯ 0 0 (h + o)σ – σ 1 2 ˆ ˆ ˆ ˆ l(k )(k )( )( 1 2 1 2 (5) p q s t ˆ ˆ k –1 k –1 ˆ 1 2 –1 × p – s q – t (s – ℘ ) 0 0 0 0 –1 × (t – ζ ) dt ds dζ d℘ , ¯ ¯ ¯ 2 2 1 where p q s t ˆ ˆ k –1 k –1 ˆ ˆ 1 2 –1 –1 1 2 p – s q – t (s – ℘ ) (t – ζ ) dt ds dζ d℘ ¯ ¯ ¯ ¯ 1 2 2 1 0 0 0 0 ˆ ˆ ˆ ˆ p q k –1 k –1 1 1 2 2 s (p – s) t (q – t) = dζ d℘ ¯ ¯ 2 1 ˆ ˆ 0 0 1 2 ˆ ˆ ˆ ˆ k –1 q k –1 1 1 2 2 s (p – s) t (q – t) = d℘ × dζ ¯ ¯ 1 2 ˆ ˆ 1 2 a b 0 0 ˆ ˆ ˆ ˆ k –1 k –1 1 1 2 2 ≤ s (a – s) ds × t (b – t) dt. 0 0 ˆ ˆ 0 0 1 2 Put s = a u and t = b v.Thus, we obtain 0 0 p q s t ˆ ˆ k –1 k –1 ˆ ˆ 1 2 –1 –1 1 2 p – s q – t (s – ℘ ) (t – ζ ) dt ds dζ d℘ ¯ ¯ ¯ ¯ 1 2 2 1 0 0 0 0 ˆ ˆ ˆ ˆ k + k + 1 1 2 2 1 1 a b ˆ ˆ ˆ ˆ 0 0 k –1 k –1 1 1 2 2 ≤ u (1 – u) du × v (1 – v) dv. ˆ ˆ 0 0 1 2 On the other hand, ˆ ˆ +1)(k ) ˆ ˆ 1 1 k –1 1 1 ˆ ˆ B( +1, k )= u (1 – u) du = 1 1 ˆ ˆ 0 (k + +1) 1 1 and ˆ ˆ +1)(k ) ˆ ˆ 2 2 k –1 ˆ 2 2 B( +1, k )= v (1 – v) dv = . 2 2 ˆ ˆ 0 (k + +1) 2 2 Charandabi et al. Advances in Diﬀerence Equations (2021) 2021:323 Page 8 of 18 Hence, p q s t ˆ ˆ k –1 k –1 ˆ ˆ 1 2 –1 –1 1 2 p – s q – t (s – ℘ ) (t – ζ ) dt ds dζ d℘ ¯ ¯ ¯ ¯ 1 2 2 1 0 0 0 0 ˆ ˆ ˆ ˆ k + k + 1 1 2 2 ˆ ˆ ˆ ˆ a b ( +1)( +1)(k )(k ) 1 2 1 2 0 0 ≤ . ˆ ˆ ˆ ˆ ˆ ˆ (k + +1)(k + +1) 1 2 1 1 2 2 By using (5), we derive Q σ p , q – Q σ p , q 1 2 ¯ ¯ 0 0 (h + o)σ – σ 1 2 ˆ ˆ ˆ ˆ l(k )(k )( )( 1 2 1 2 ˆ ˆ ˆ ˆ k + k + 1 1 2 2 ˆ ˆ ˆ ˆ a b ( +1)( +1)(k )(k ) 1 2 1 2 0 0 ˆ ˆ ˆ ˆ ˆ ˆ (k + +1)(k + +1) 1 2 1 1 2 2 ˆ ˆ ˆ ˆ k + k + 1 1 2 2 ∗ a b (h + o) 0 0 = σ – σ , 1 2 ˆ ˆ ˆ ˆ l(k + +1)(k + +1) 1 1 2 2 ˆ ˆ ˆ ˆ k + k + 1 1 2 2 ∗ a b (h +o) 0 0 which means Q σ – Q σ ≤ σ – σ .Deﬁne ψ :[0, ∞) → [0, ∞) 1 2 1 2 ¯ ¯ ˆ ˆ 0 0 ˆ ˆ l(k + +1)(k + +1) 1 1 2 2 ˆ ˆ ˆ ˆ k + k + 1 1 2 2 ∗ a b (h +o) 0 0 and α : Z × Z → [0, ∞)by ψ(t)= t and ˆ ˆ ˆ ˆ l(k + +1)(k + +1) 1 1 2 2 1, υ(σ (p , q ), σ (p , q )) ≥ 0with(p , q ) ∈ J × J , 1 2 a b 0 0 α(σ , σ )= 1 2 0, otherwise. It is clear that ψ ∈ .If α(σ , σ ) ≥ 1, then υ(σ (p , q ), σ (p , q )) ≥ 0. From the hy- 1 2 1 2 potheses, υ(Q σ (p , q ), Q σ (p , q )) ≥ 0and so α(Q σ , Q σ ) ≥ 1. Thus, Q is an α- 1 2 1 2 ¯ ¯ ¯ ¯ ¯ 0 0 0 0 0 admissible mapping. Also, there exists σ ∈ Z such that α(σ , Q σ ) ≥ 1. For every se- 0 0 0 quence {σ } ⊆ Z with σ → σ and α(σ , σ ) ≥ 1 for all n ≥ 1, we have α(σ , σ ) ≥ 1 n n≥1 n n n+1 n for all n ≥ 1. Assume that α(σ , σ )=0. Then α(σ , σ )Q σ – Q σ =0 ≤ ψ(σ – σ ) 1 2 1 2 1 2 1 2 ¯ ¯ 0 0 and so α(σ , σ ) Q σ – Q σ ≤ ψ σ – σ 1 2 1 2 1 2 ¯ ¯ 0 0 for all σ , σ ∈ Z . Thus all conditions of Lemma 1 hold and so Q has a ﬁxed point which 1 2 is a solution for the partial fractional Sturm–Liouville problem (1). Example 1 Consider the partial fractional Sturm–Liouville equation √ 3 999 1999 89 79 ⎪ ( , ) 4 ( , ) –p –q – p e ⎪ 1000 2000 90 80 ⎪ D (100e cosh q D σ (p , q )) + σ (p , q ) c c 2 2 ⎪ 0 0 ⎪ 300(1+p +p ) ⎪ p ⎪ 2 1+p ⎨ = e σ (p , q ), (p , q ) ∈ [0, 1] × [0, 1], 89 79 4 (6) ( , ) 2 – p 90 80 1 (100e cosh q D σ (p , q )) = p , c q =0 0 100 89 79 ⎪ 4 ( , ) 3 ⎪ – p 90 80 1 (100e cosh q D σ (p , q )) = q , c q =0 ⎪ 0 ⎩ 1 1 σ (p ,0) = p and σ (0, q )= q . 500 350 Charandabi et al. Advances in Diﬀerence Equations (2021) 2021:323 Page 9 of 18 ∗ ∗ ∗ ∗ ∗ ∗ Figure 1 Plots of functions h(p , q ), l(p , q ), o(p , q ) in [0, 1] Figure 2 Plots of functions θ , θ , κ, ω in [0, 1] 1 2 999 1999 89 79 1 ˆ ˆ ˆ ˆ ˆ ˆ Put k =(k , k )=( , ), =( )=( , ), a =1, b =1, θ (p )= p , θ (q )= 1 2 1 2 0 0 1 2 1000 2000 99 80 100 3 2 –p –q 1 1 1 – p e q , κ(p )= p , ω(q )= q , l(p , q ) = 100e cosh q , o(p , q )= , 2 2 400 500 350 300(1+p +q ) 1+q h(p , q )= e . The diagrams are plotted in Figs. 1 and 2, and obviously they satisfy the conditions of the partial Sturm–Liouville diﬀerential problem. Put f (r)= r, υ(r , r )= 1 2 3whenever |r |≤ 1and |r |≤ 1and υ(r , r ) = –1 otherwise, and 1 2 1 2 Q σ p , q p q s t = p , q + 0 0 0 0 –1 –1 –1 –1 1000 2000 90 80 (p – s) (q – t) (s – ℘ ) (t – ζ ) H(℘ , ζ , σ (℘ , ζ )) ¯ ¯ ¯ ¯ ¯ ¯ 1 2 1 2 1 2 √ dt ds dζ d℘ , ¯ ¯ 2 1 – p cosh q 999 1999 89 79 ( )( )( )( )100e 1000 2000 99 80 50 where 1 1 p , q = p + p 500 350 1 1 p q – – 1 2 1 3 90 80 (p – s) (q – t) ( s + t ) 100 400 + √ dt ds. – p 89 79 0 0 100e cosh q ( )( ) 90 80 Charandabi et al. Advances in Diﬀerence Equations (2021) 2021:323 Page 10 of 18 Note that H(℘ , ζ , σ (℘ , ζ )) = h(℘ , ζ )f (σ (℘ , ζ )) – o(℘ , ζ )σ (℘ , ζ ). Now, assume that ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ 1 2 1 2 1 2 1 2 1 2 1 2 υ(σ (p , q ), σ (p , q )) ≥ 0. Then we have |σ (p , q )|≤ 1and |σ (p , q )|≤ 1. Suppose that 1 2 1 2 |σ (p , q )|≤ 1. Then we get H ℘ , ζ , σ (℘ , ζ ) ¯ ¯ ¯ ¯ 1 2 1 2 1 –℘ –ζ 1 2 e 1+ζ = e f σ (℘ , ζ ) – σ (℘ , ζ ) ¯ ¯ 1 ¯ ¯ 1 2 1 2 2 2 600 300(1 + ℘ + ζ ) ¯ ¯ 1 2 –p –p 1 e 1+ζ ≤ e σ (℘ , ζ ) + σ (℘ , ζ ) 1 ¯ ¯ 1 ¯ ¯ 1 2 2 2 1 2 600 300(1 + p + p ) e 1 e +2 ≤ + = 600 300 600 and so 1 1 s t 0 0 0 0 –1 –1 –1 –1 1000 2000 90 80 (p – s) (q – t) (s – ℘ ) (t – ζ ) |H(℘ , ζ , σ (℘ , ζ ))| ¯ ¯ ¯ ¯ ¯ ¯ 1 2 1 2 1 1 2 √ dt ds dζ d℘ ¯ ¯ 2 1 – p 999 1999 89 79 ( )( )( )( )100e cosh q 1000 2000 90 80 p q s t –1 –1 –1 1000 2000 ≤ 0.0000285064 p – s q – t (s – ℘ ) 0 0 0 0 –1 × (t – ζ ) dt ds dζ d℘ ¯ ¯ ¯ 2 2 1 89 79 999 1999 ( +1)( +1)( )( ) 90 80 1000 2000 ≤ 0.0000285064 × = 0.0000296489. 89 79 999 89 1999 79 ( + +1)( + +1) 90 80 1000 90 2000 80 Also, p , q 1 1 2 3 – 1 1 p q – 90 80 (p – s) (q – t) ( p + p ) 1 1 2 100 400 ≤ p + q + √ dt ds – p 89 79 500 350 0 0 100e cosh q ( )( ) 90 80 1 1 1 1 1 1 – – 90 80 ≤ + + 0.0000453519 (1 – s) (1 – t) dt ds 500 350 0 0 1 1 = + + 0.0000453519 × 1.0240364102 = 0.0053797753. 500 350 Therefore, Q σ p , q p q s t ≤ p , q + 0 0 0 0 –1 –1 –1 –1 1000 2000 90 80 (p – s) (q – t) (s – ℘ ) (t – ζ ) |H(℘ , ζ , σ (℘ , ζ ))| ¯ ¯ ¯ ¯ ¯ ¯ 1 2 1 2 1 2 √ dt ds dζ d℘ ¯ ¯ 2 1 cosh q 999 1999 89 79 – p ( )( )( )( )100e 1000 2000 99 80 50 ≤ 0.0053797753 + 0.0000296489 = 0.0054094242 ≤ 1. Charandabi et al. Advances in Diﬀerence Equations (2021) 2021:323 Page 11 of 18 Similarly, we obtain |Q σ (p , q )|≤ 1and υ(Q σ (p , q ), Q σ (p , q )) ≥ 0. Assume that 1 1 2 ¯ ¯ ¯ 0 0 0 {σ } ⊆ Z is a sequence such that σ → σ and n n≥1 n υ σ p , q , σ p , q ≥ 0 n n+1 for all n ≥ 1and (p , q ) ∈ J ×J .Thenwehave |σ (p , q )|≤ 1and |σ (p , q )|≤ 1for a b n n+1 0 0 all n ≥ 1and (p , q ) ∈ J × J .Since σ → σ , |σ (p , q )| = lim |σ (p , q )|≤ 1, we a b n n→∞ n 0 0 obtain υ(Q σ (p , q ), Q σ (p , q )) ≥ 0 for all n ≥ 1and (p , q ) ∈ J × J .Since |0|≤ 1 n a b ¯ ¯ 0 0 0 0 and |Q 0|≤ 1, we get υ(0, Q 0) ≥ 1. Note that =1, ¯ ¯ 0 0 – p l = inf l p , q = inf 100e cosh q = 100e, (p ,q )∈J ×J (p ,q )∈J ×J a b a b 0 0 0 0 –p –q e 1 o = sup o p , q = sup = , 2 2 300(1 + p + q ) 300 (p ,q )∈J ×J (p ,q )∈J ×J a a b b 0 0 0 0 1 e 1+q and h = sup e = .Hence, (p ,q )∈J ×J a 600 600 0 0 ˆ ˆ ˆ ˆ k + k + e 1 1 1 2 2 ∗ a b (h + o) 0 0 600 300 999 89 1999 79 ˆ ˆ ˆ ˆ 100e( + +1)( + +1) l(k + +1)(k + +1) 1 1 2 2 1000 90 2000 80 = 0.0000074014 ≤ 1. Now by using Theorem 4,the problem(6)has asolution. Deﬁnition 5 We say that a function σ ∈ C(J ×J ) is a solution for the partial fractional a b 0 0 Sturm–Liouville diﬀerential inclusion problem problem (2) whenever there is a function v in L (J × J , R)such that v(p , q ) ∈ H(p , q , σ (p , q )) for almost all (p , q ) ∈ J × a b a 0 0 0 J , (l(p , q )D σ (p , q )) = θ (p ), q =0 1 (l(p , q )D σ (p , q )) = θ (q ), p =0 2 σ (p ,0) = κ(p )and σ (0, q )= ω(q ), and σ p , q = p , q p q s t 0 0 0 0 ˆ ˆ ˆ ˆ k –1 k –1 –1 –1 1 2 1 2 (p – s) (q – t) (s – ℘ ) (t – ζ ) v(℘ , ζ ) ¯ ¯ ¯ ¯ 1 2 1 2 dt ds dζ d℘ , ¯ ¯ 2 1 ˆ ˆ ˆ ˆ (k )(k )( )( )l(s, t) 1 2 1 2 for all (p , q ) ∈ J × J . a b 0 0 Charandabi et al. Advances in Diﬀerence Equations (2021) 2021:323 Page 12 of 18 For given σ ∈ Z ,deﬁne theset S = v ∈ L (J × J )|v ∈ H p , q , σ p , q on J × J . H,σ a b a b 0 0 0 0 Assume that (H1) H : J × J × R → P (R) is an integrable bounded multifunction so that a b cl 0 0 H(·, ·, σ ) is measurable for all σ ∈ R. (H2) There exists ρ ∈ C(J × J , R ) so that a b 0 0 PH H p , q , r , H p , q , r ≤ ρ p , q ψ |r – r | d 1 2 1 2 ˆ ˆ ˆ ˆ k + k + 1 1 2 2 a b ρ 0 0 for all r , r ∈ R,where ψ ∈ , ≤ 1 and 1 2 ˆ ˆ ˆ ˆ l(k + +1)(k + +1) 1 1 2 2 l = inf l p , q . (p ,q )∈J ×J a b (H3) Deﬁne N : Z → 2 by N(σ)= h ∈ Z | there exists v ∈ S so that h p , q = w p , q H,σ for all p , q ∈ J × J , a b 0 0 where w p , q p q s t = p , q + 0 0 0 0 ˆ ˆ ˆ ˆ k –1 k –1 –1 –1 1 2 1 2 (p – s) (q – t) (s – ℘ ) (t – ζ ) v(℘ , ζ ) ¯ ¯ ¯ ¯ 1 2 1 2 dt ds dζ d℘ . ¯ ¯ 2 1 ˆ ˆ ˆ ˆ (k )(k )( )( )l(s, t) 1 2 1 2 (H4) Suppose that υ : R × R → R bearealvaluedfunctionand forevery conver- gent sequence {σ } ⊆ Z with σ → σ and υ(σ (p , q ), σ (p , q )) ≥ 0 for n n≥1 n n all n and (p , q ) ∈ J × J ,there exists asubsequence {σ } of σ so that a b n j≥1 n 0 0 j υ(σ (p , q ), σ (p , q )) ≥ 0 for all n and (p , q ) ∈ J × J . Assume that, for every n a b j 0 0 σ ∈ Z and h ∈ N(σ ) with υ(σ (p , q ), h(p , q )) ≥ 0 for each (p , q ) ∈ J × J , a b 0 0 there exists w ∈ N(σ ) so that υ(h(p , q ), w(p , q )) ≥ 0 for all (p , q ) ∈ J × J . a b 0 0 Suppose that there exists σ ∈ Z and h ∈ N(σ ) so that υ(σ (p , q ), h(p , q )) ≥ 0 0 0 0 for all (p , q ) ∈ J × J . a b 0 0 Theorem 6 Assume that (H1)–(H4) hold. Then the partial fractional Sturm–Liouville problem (2) has a solution. Proof We prove that the multifunction N : Z → 2 has a ﬁxed point which provides a solution for the partial fractional Sturm–Liouville problem (2). Note that the multifunc- tion (p , q ) → H(p , q , σ (p , q )) has a measurable selection. Since it has closed and has measurable values for aa σ ∈ Z , S is nonempty for every σ ∈ Z .Weprove that N(σ ) H,σ Charandabi et al. Advances in Diﬀerence Equations (2021) 2021:323 Page 13 of 18 is closed subset of Z . For this aim assume that σ ∈ Z and {h }⊂ N(σ)is a sequence with h → h.For each n,choose v ∈ S such that n n H,σ h p , q p q s t = p , q + 0 0 0 0 ˆ ˆ ˆ ˆ k –1 k –1 –1 –1 1 2 1 2 (p – s) (q – t) (s – ℘ ) (t – ζ ) v (℘ , ζ ) ¯ ¯ n ¯ ¯ 1 2 1 2 dt ds dζ d℘ , ¯ ¯ 2 1 ˆ ˆ ˆ ˆ (k )(k )( )( )l(s, t) 1 2 1 2 for all (p , q ) ∈ J ×J . On the other hand, H hascompactvalues.Thus, wemayassume a b 0 0 that {v } converges to some v ∈ L (J × J ). Hence, n a b 0 0 h p , q p q s t = p , q + 0 0 0 0 ˆ ˆ ˆ ˆ k –1 k –1 –1 –1 1 2 1 2 (p – s) (q – t) (s – ℘ ) (t – ζ ) v(℘ , ζ ) ¯ ¯ ¯ ¯ 1 2 1 2 dt ds dζ d℘ ¯ ¯ 2 1 ˆ ˆ ˆ ˆ (k )(k )( )( )l(s, t) 1 2 1 2 and so h ∈ N(σ ). Since H is a compact map, N(σ)is a bounded set for al σ ∈ Z .Now, deﬁne the function α : Z × Z → R by α(σ , σ ) ≥ 1whenever υ(σ (p , q ), σ (p , q )) ≥ 1 2 1 2 0 for almost all (p , q ) ∈ J × J and α(σ , σ ) = 0 otherwise. We show that N is α-ψ- a b 1 2 0 0 contractive. Let σ , σ ∈ Z and h ∈ N(σ ). Choose v ∈ S such that 1 2 1 2 1 H,σ h p , q p q s t = p , q + 0 0 0 0 ˆ ˆ ˆ ˆ k –1 k –1 –1 –1 1 2 1 2 (p – s) (q – t) (s – ℘ ) (t – ζ ) v (℘ , ζ ) ¯ ¯ 1 ¯ ¯ 1 2 1 2 dt ds dζ d℘ ¯ ¯ 2 1 ˆ ˆ ˆ ˆ (k )(k )( )( )l(s, t) 1 2 1 2 for almost all (p , q ) ∈ J × J .Thenweget a b 0 0 PH H p , q , σ p , q , H p , q , σ p , q d 1 2 ≤ ρ p , q ψ σ p , q – σ p , q , 1 2 for almost all (p , q ) ∈ J × J with υ(σ (p , q ), σ (p , q )) ≥ 0. Now, choose w ∈ a b 1 2 0 0 H(p , q , σ (p , q )) so that |v (t)– w|≤ ρ(p , q )ψ(|σ (p , q )– σ (p , q )|) for all (p , q ) ∈ 1 1 1 2 J × J .Now,deﬁne U : J × J → P(R)by a b a b 0 0 0 0 U p , q = w ∈ R| v p , q – w ≤ ρ p , q ψ σ p , q – σ p , q . 1 1 2 Since v and ω = ρ(p , q )ψ(|σ (p , q )– σ (p , q )|) are measurable, the multifunction 1 1 2 U(·, ·) ∩ H(·, ·, σ (·, ·)) is measurable. Choose v ∈ H(p , q , σ (p , q )) such that 2 1 v p , q – v p , q ≤ ρ p , q ψ σ p , q – σ p , q 1 2 1 2 (7) ≤ρ ψ σ – σ ∞ 1 2 Charandabi et al. Advances in Diﬀerence Equations (2021) 2021:323 Page 14 of 18 for all (p , q ) ∈ J × J . Now, choose the element h ∈ N(σ )deﬁned by a b 2 1 0 0 h p , q p q s t = p , q + 0 0 0 0 ˆ ˆ ˆ ˆ k –1 k –1 –1 –1 1 2 1 2 (p – s) (q – t) (s – ℘ ) (t – ζ ) v (℘ , ζ ) ¯ ¯ 2 ¯ ¯ 1 2 1 2 dt ds dζ d℘ ¯ ¯ 2 1 ˆ ˆ ˆ ˆ (k )(k )( )( )l(s, t) 1 2 1 2 for all (p , q ) ∈ J × J .Byusing (7), we get a b 0 0 h p , q – h p , q 2 1 p q s t 0 0 0 0 ˆ ˆ ˆ ˆ k –1 k –1 –1 –1 1 2 1 2 (p – s) (q – t) (s – ℘ ) (t – ζ ) |v (℘ , ζ )– v (℘ , ζ )| ¯ ¯ 2 ¯ ¯ 1 ¯ ¯ 1 2 1 2 1 2 dt ds dζ d℘ ¯ ¯ 2 1 ˆ ˆ ˆ ˆ (k )(k )( )( )l(s, t) 1 2 1 2 p q s t ≤ρ 0 0 0 0 ˆ ˆ ˆ ˆ k –1 k –1 –1 –1 1 2 1 2 (p – s) (q – t) (s – ℘ ) (t – ζ ) ¯ ¯ 1 2 dt ds dζ d℘ ψ σ – σ ¯ ¯ 1 2 2 1 ˆ ˆ ˆ ˆ (k )(k )( )( )l(s, t) 1 2 1 2 ˆ ˆ ˆ ˆ k + k + 1 1 2 2 a b ρ 0 0 ≤ ψ σ – σ ≤ ψ σ – σ 1 2 1 2 ˆ ˆ ˆ ˆ l(k + +1)(k + +1) 1 1 2 2 and so h – h ≤ ψ(σ – σ ). Note that 1 2 1 2 α(σ , σ )PH N(σ ), N(σ ) ≤ ψ σ – σ for all σ , σ ∈ Z . 1 2 d 1 2 1 2 1 2 Thus, N is α-ψ-contraction. Let σ ∈ Z and σ ∈ N(σ )be such that α(σ , σ ) ≥ 1. Then 1 2 1 1 2 υ(σ (p , q ), σ (p , q )) ≥ 0 for all (p , q ) ∈ J × J .Hence,there exists w ∈ N(σ )such 1 2 a b 2 0 0 that υ(σ (p , q ), w(p , q )) ≥ 0. This implies that α(σ , w) ≥ 1. Thus, N is α-admissible. 1 1 Now by using Lemma 2, N has a ﬁxed point which is a solution of the partial fractional Sturm–Liouville problem (2). Example 2 Consider the partial fractional Sturm–Liouville inclusion problem 2 2 1 1 1 1 –p –q ( , ) ( , ) e |σ (p ,q )| ⎪ 2 3 p q 7 8 D (13e D σ (p , q )) ∈ [σ (p , q ), σ (p , q )+ ], ⎪ c c 0 0 2 4(1+|σ (p ,p )|) ⎪ (p , q ) ∈ [0, 1] × [0, 1], 1 1 ( , ) 2 p q 7 8 3 (8) (13e D σ (p , q )) = p , c q =0 ⎪ 0 1 1 ⎪ ( , ) 3 ⎪ p q 7 8 1 ⎪ (13e D σ (p , q )) = q , c p =0 ⎪ 0 2 3 7 8 σ (p ,0) = p and σ (0, q )= q . 50 35 2 3 1 1 1 1 3 1 ˆ ˆ ˆ ˆ ˆ ˆ Put k =(k , k )=( , ), =( )=( , ), a =1, b =1, θ (p )= p , θ (q )= q , 1 2 1 2 1 2 2 3 7 8 40 25 2 3 7 8 p q κ(p )= p , ω(q )= q and l(p , q )=13e . The plotted diagrams in Figs. 3 and 50 35 Charandabi et al. Advances in Diﬀerence Equations (2021) 2021:323 Page 15 of 18 ∗ ∗ Figure 3 Plot of function l(p , q )in[0,1] Figure 4 Plots of functions θ , θ , κ, ω in [0, 1] 1 2 4 show that the conditions of the partial Sturm–Liouville diﬀerential inclusion problem 2 2 –p –q e |r| hold. Also, put H(p , q , r)=[r, r + ], υ(r , r )=1 whenever r ≥ 0and r ≥ 0 1 2 1 2 4(1+|r|) and υ(r , r )=–1 otherwise, 1 2 N(σ)= h ∈ Z | there exists v ∈ S so that h p , q = w p , q H,σ for all p , q ∈ J × J a b 0 0 where w p , q = p , q 1 2 6 7 p q s t – – – – 2 3 7 8 (p – s) (q – t) (s – ℘ ) (t – ζ ) v(℘ , ζ ) ¯ ¯ ¯ ¯ 1 2 1 2 + dt ds dζ d℘ ¯ ¯ 2 1 1 1 1 1 st 13e ( )( )( )( ) 0 0 0 0 2 3 7 8 1 1 – – 3 1 2 3 90 80 2 3 (p –s) (q –t) ( s + t ) p q 7 8 40 25 and (p , q )= p + p + dt ds. Assume that σ ∈ Z 89 79 50 35 0 0 st 13e ( )( ) 90 80 and h ∈ N(σ)with υ(σ (p , q ), h(p , q )) ≥ 0 for all (p , q ) ∈ [0, 1] × [0, 1]. Then we have Charandabi et al. Advances in Diﬀerence Equations (2021) 2021:323 Page 16 of 18 σ (p , q ) ≥ 0and h(p , q ) ≥ 0 for all (p , q ) ∈ [0, 1] × [0, 1]. Since σ (p , q ) ≥ 0, we get 2 2 –p –q e |σ (p , q )| H p , q , σ p , q = σ p , q , σ p , q + ⊆ [0, ∞). 4(1 + |σ (p , q )|) Choose v(p , q ) ∈ H(p , q , σ (p , q )) so that v(p , q ) ≥ 0 for all (p , q ) in [0, 1] × [0, 1]. Since (p , q ) ≥ 0, we get w p , q p q s t := p , q + 0 0 0 0 1 2 6 7 – – – – 2 3 7 8 (p – s) (q – t) (s – ℘ ) (t – ζ ) v(℘ , ζ ) ¯ ¯ ¯ ¯ 1 2 1 2 dt ds dζ d℘ ≥ 0 ¯ ¯ 2 1 1 1 1 1 st 13e ( )( )( )( ) 2 3 7 8 and so w(p , q ) ≥ 0. Thus, υ(h(p , q ), w(p , q )) ≥ 0. Note that υ(0, h(p , q )) ≥ 0for h ∈ N(σ ) and also for each convergent sequence {σ } ⊆ Z with σ → σ and n n≥1 n υ(σ (p , q ), σ (p , q )) ≥ 0 for all n and (p , q ) ∈ J × J ,there exists asubsequence n a b 0 0 {σ } of {σ } such that υ(σ (p , q ), σ (p , q )) ≥ 0. Thus, n j≥1 n n≥1 n j j PH H p , q , r , H p , q , r d 1 2 2 2 –p –q e |r | |r | 1 2 ≤ – 4 1+ |r | 1+ |r | 1 2 2 2 2 2 –p –q –p –q e e = |r | – |r | ≤ |r – r |. 1 2 1 2 4 4 2 2 –p –q 1 If ρ(p , q )= e and ψ(t)= t,then PH H p , q , r , H p , q , r ≤ p , q ψ |r – r | d 1 2 1 2 p q and ρ =1. Put l(p , q )=13e .Then l =13 and so ˆ ˆ ˆ ˆ k + k + 1 1 2 2 a b ρ 1 0 0 = = 0.0966114627 ≤ 1. 1 1 1 1 ˆ ˆ ˆ ˆ 13( + +1)( + +1) l(k + +1)(k + +1) 1 1 2 2 2 7 3 8 Now by using Theorem 6,the problem(8)has asolution. 3Conclusion In this work, we studied a partial fractional version of the Sturm–Liouville diﬀerential equation by using the Caputo derivative. Also, we reviewed inclusion version of the prob- lem. First, by using the technique of α-ψ-contractive mappings, we investigated the ex- istence of solutions for the partial fractional Sturm–Liouville equation. We presented an illustrated example to clear more the result. Secondly, we have investigated the partial fractional Sturm–Liouville inclusion problem by using the technique of α-ψ-contractive multifunctions. We provided an illustrated an example for explaining the second result. In this way, we provided some related ﬁgures for the examples. Charandabi et al. Advances in Diﬀerence Equations (2021) 2021:323 Page 17 of 18 Acknowledgements The ﬁrst author was supported by Tabriz Branch, Islamic Azad University. The second author was supported by Miandoab Branch, Islamic Azad University. The third author was supported by Azarbaijan Shahid Madani University. The fourth author was supported by K. N. Toosi University of Technology. The authors express their gratitude to the dear unknown referees for their helpful suggestions, which improved the ﬁnal version of this paper. Funding Not applicable. Availability of data and materials Data sharing not applicable to this article as no datasets were generated or analyzed during the current study. Ethics approval and consent to participate Not applicable. Competing interests The authors declare that they have no competing interests. Consent for publication Not applicable. Authors’ contributions The authors declare that the study was realized in collaboration with equal responsibility. All authors read and approved the ﬁnal manuscript. Author details 1 2 Department of Mathematics, Tabriz Branch, Islamic Azad University, Tabriz, Iran. Department of Mathematics, Miandoab Branch, Islamic Azad University, Miandoab, Iran. Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran. Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan. Department of Mathematics, K. N. Toosi University of Technology, P.O. Box 16315-1618, Tehran, 15418, Iran. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations. Received: 2 April 2021 Accepted: 23 June 2021 References 1. Afshari, H., Kalantari, S., Karapinar, E.: Solution of fractional diﬀerential equations via coupled ﬁxed point. Electron. J. Diﬀer. Equ. 15(286), 1 (2015). http://ejde.math.txstate.edu 2. Alqahtani, B., Aydi, H., Karapinar, E., Rakocevic, V.: A solution for Volterra fractional integral equations by hybrid contractions. Mathematics 7(8), 694 (2019). https://doi.org/10.3390/math7080694 3. Baleanu, D., Jajarmi, A., Mohammadi, H., Rezapour, S.: A new study on the mathematical modelling of human liver with Caputo–Fabrizio fractional derivative. Chaos Solitons Fractals 134, 109705 (2020). https://doi.org/10.1016/j.chaos.2020.109705 4. Rezapour, S., Ntouyas, S.K., Iqbal, M.Q., Hussain, A., Etemad, S., Tariboon, J.: An analytical survey on the solutions of the generalized double-order φ-integrodiﬀerential equation. J. Funct. Spaces 2021, Article ID 6667757 (2021). https://doi.org/10.1155/2021/6667757 5. Abbas, S., Benchohra, M.: Darboux problem for perturbed partial diﬀerential equations of fractional order with ﬁnite delay. Nonlinear Anal. Hybrid Syst. 3(4), 597–604 (2009). https://doi.org/10.1016/j.nahs.2009.05.001 6. Rezapour, S., Sakar, F.M., Aydogan, S.M., Ravash, E.: Some results on a system of multiterm fractional integro-diﬀerential equations. Turk. J. Math. 44(6), 2004–2020 (2021). https://doi.org/10.3906/mat-1903-51 7. Baleanu, D., Mousalou, A., Rezapour, S.: On the existence of solutions for some inﬁnite coeﬃcient-symmetric Caputo–Fabrizio fractional integro-diﬀerential equations. Bound. Value Probl. 2017(1), 145 (2017). https://doi.org/10.1186/s13661-017-0867-9 8. Baleanu, D., Rezapour, S., Etemad, S., Alsaedi, A.: On a time-fractional partial integro-diﬀerential equation via three-point boundary value conditions. Math. Probl. Eng. 2015, Article ID 896871 (2015). https://doi.org/10.1155/2015/785738 9. Baitiche, Z., Derbazi, C., Benchora, M.: ψ-Caputo fractional diﬀerential equations with multi-point boundary conditions by topological degree theory. Results Nonlinear Anal. 3(4), 167–178 (2020) 10. Karapinar, E., Fulga, A., Rashid, M., Shahid, L., Aydi, H.: Large contractions on quasi-metric spaces with an application to nonlinear fractional diﬀerential equations. Mathematics 7(5), 444 (2019). https://doi.org/10.3390/math7050444 11. Mohammadi, H., Kumar, S., Rezapour, S., Etemad, S.: A theoretical study of the Caputo–Fabrizio fractional modeling for hearing loss due to Mumps virus with optimal control. Chaos Solitons Fractals 144, 110668 (2021). https://doi.org/10.1016/j.chaos.2021.110668 12. Ali, A., Shah, K., Abdeljawad, T., Mahariq, I., Rashdan, M.: Mathematical analysis of nonlinear integral boundary value problem of proportional delay implicit fractional diﬀerential equations with impulsive conditions. Bound. Value Probl. 2021, 7 (2021). https://doi.org/10.1186/s13661-021-01484-y 13. Tuan, N.H., Mohammadi, H., Rezapour, S.: A mathematical model for COVID-19 transmission by using the Caputo fractional derivative. Chaos Solitons Fractals 140, 110107 (2020). https://doi.org/10.1016/j.chaos.2020.110107 14. Murugusundaramoorthy, G.: Application of Pascal distribution series to Ronning type star-like and convex functions. Adv. Theory Nonlinear Anal. Appl. 4(4), 243–251 (2020). https://doi.org/10.31197/atnaa.743436 Charandabi et al. Advances in Diﬀerence Equations (2021) 2021:323 Page 18 of 18 15. Patil, J., Chaudhari, A., Abdo, M., Hardan, B.: Upper and lower solution method for positive solution of generalized Caputo fractional diﬀerential equations. Adv. Theory Nonlinear Anal. Appl. 4(4), 279–291 (2020). https://doi.org/10.31197/atnaa.709442 16. Patil, J., Chaudhari, A., Abdo, M., Hardan, B.: Upper and lower solution method for positive solution of generalized Caputo fractional diﬀerential equations. Adv. Theory Nonlinear Anal. Appl. 4(4), 279–291 (2020). https://doi.org/10.31197/atnaa.709442 17. Muthaiah, S., Murugesan, M., Thangaraj, N.G.: Existence of solutions for nonlocal boundary value problem of Hadamard fractional diﬀerential equations. Adv. Theory Nonlinear Anal. Appl 3(3), 162–173 (2019). https://doi.org/10.31197/atnaa.579701 18. Aydogan, M.S., Baleanu, D., Mousalou, A., Rezapour, S.: On high order fractional integro-diﬀerential equations including the Caputo–Fabrizio derivative. Bound. Value Probl. 2018, 90 (2018). https://doi.org/10.1186/s13661-018-1008-9 19. Jarad, F., Abdeljawad, T.: A modiﬁed Laplace transform for certain generalized fractional operators. Results Nonlinear Anal. 1(2), 88–98 (2019) 20. Marino, G., Scardamglia, B., Karapinar, E.: Strong convergence theorem for strict pseudo-contractions in Hilbert spaces. J. Inequal. Appl. 2016, 134 (2016). https://doi.org/10.1186/s13660-016-1072-6 21. Baleanu, D., Mousalou, A., Rezapour, S.: A new method for investigating approximate solutions of some fractional integro-diﬀerential equations involving the Caputo-Fabrizio derivative. Adv. Diﬀer. Equ. 2017, 51 (2017). https://doi.org/10.1186/s13662-017-1088-3 22. Afshari, H., Marasi, H., Aydi, H.: Existence and uniqueness of positive solutions for boundary value problems of fractional diﬀerential equations. Filomat 31(9), 2675–2682 (2017). https://doi.org/10.2298/FIL1709675A 23. Marasi, H., Afshari, H., Zhai, C.B.: Some existence and uniqueness results for nonlinear fractional partial diﬀerential equations. Rocky Mt. J. Math. 47(2), 571–585 (2017). https://doi.org/10.1216/RMJ-2017-47-2-571 24. Bachir, F.S., Said, A., Benbachir, M., Benchohra, M.: Hilfer–Hadamard fractional diﬀerential equations; existence and attractivity. Adv. Theory Nonlinear Anal. Appl 5(1), 49–57 (2021). https://doi.org/10.31197/atnaa.848928 25. Abdeljawad, T., Al-Mdallal, Q.M., Hammouchc, Z., Jarad, F.: Existence and uniqueness of positive solutions for boundary value problems of fractional diﬀerential equations. Adv. Theory Nonlinear Anal. Appl 4(4), 214–215 (2020). https://doi.org/10.31197/atnaa.810371 26. Tuan, N.H., Huynh, L.N., Baleanu, D., Can, N.H.: On a terminal value problem for a generalization of the fractional diﬀusion equation with hyper-Bessel operator. Math. Methods Appl. Sci. 43(6), 2858–2882 (2020). https://doi.org/10.1002/mma.6087 27. Can, N.H., Nikan, O., Rasoulizadeh, M.N., Gasimov, Y.S.: Numerical computation of the time nonlinear fractional generalized equal width model arising in shallow water channel. Therm. Sci. 24(1), 49–58 (2020). https://doi.org/10.2298/TSCI20S1049C 28. Tuan, N.H., Thuch, T.N., Can, N.H., O’Regan, D.: Regularization of a multidimensional diﬀusion equation with conformable time derivative and discrete data. Math. Methods Appl. Sci. 44(4), 2879–2891 (2021). https://doi.org/10.1002/mma.6133 29. Baleanu, D., Mohammadi, H., Rezapour, S.: Analysis of the model of HIV-1 infection of CD4 T-cell with a new approach of fractional derivative. Adv. Diﬀer. Equ. 2020, Article ID 71 (2020) 30. Rezapour, S., Mohammadi, H., Jajarmi, A.: A new mathematical model for Zika virus transmission. Adv. Diﬀer. Equ. 2020, 589 (2020). https://doi.org/10.1186/s13662-020-03044-7 31. Abbas, S., Benchohra, M.: Fractional order partial hyperbolic diﬀerential equations involving Caputo derivative. Stud. Univ. Babes–Bolyai ¸ Math. 57(4), 469–479 (2012) 32. Abbas, S., Benchohra, M.: Partial hyperbolic diﬀerential equations with ﬁnite delay involving the Caputo fractional derivative. Commun. Math. Anal. 7(2), 62–72 (2009) 33. Abbas, S., Benchohra, M., N’Guerekata, G.M.: Darboux problem for perturbed partial diﬀerential equations of fractional order with ﬁnite delay. Nonlinear Anal. Hybrid Syst. 3(4), 597–604 (2009). https://doi.org/10.1016/j.nahs.2009.05.001 34. Ahmad, B., Ntouyas, S.K., Tariboon, J.: A study of mixed Hadamard and Riemann–Liouville fractional integro-diﬀerential inclusions via endpoint theory. Appl. Math. Lett. 52, 9–14 (2016). https://doi.org/10.1016/j.aml.2015.08.002 35. Benchohra, M., Henderson, J., Mostefai, F.Z.: Weak solutions for hyperbolic partial fractional diﬀerential inclusions in Banach spaces. Comput. Math. Appl. 64(10), 3101–3107 (2012). https://doi.org/10.1016/j.camwa.2011.12.055 36. Etemad, S., Rezapour, S.: On the existence of solution for three variables partial fractional-diﬀerential equation and inclusion. J. Adv. Math. Stud. 8(2), 224–233 (2015) 37. Joannopoulos, J.D., Johnson, S.G., Winnn, J.N., Meade, R.D.: Photonic Crystals: Molding the Flow of Light. Princeton University Press, Princeton (2008) 38. Teschl, G.: Mathematical Methods in Quantum Mechanics: With Applications to Schrodinger Operators. Am. Math. Soc., New York (2014) 39. Ashrafyan, Y.: A new kind of uniqueness theorems for inverse Sturm–Liouville problems. Bound. Value Probl. 2017,79 (2017). https://doi.org/10.1186/s13661-017-0813-x 40. Adiguzel, R. S., Aksoy, U., Karapinar, E., Erhan, I. M.: On the solution of a boundary value problem associated with a fractional diﬀerential equation. Math. Method. Appl. Sci. (2020). https://doi.org/10.1002/mma.6652 41. Liu, Y., He, T., Shi, H.: Three positive solutions of Sturm–Liouville boundary value problems for fractional diﬀerential equations. Diﬀer. Equ. Appl. 5(1), 127–152 (2013) 42. Podlubny, I.: Fractional Diﬀerential Equations. Academic Press, San Diego (1999) 43. Deimling, K.: Multi-Valued Diﬀerential Equations. de Gruyter, Berlin (1992) 44. Aubin, J., Cellina, A.: Diﬀerential Inclusions: Set-Valued Maps and Viability Theory. Springer, Berlin (1984). https://doi.org/10.1007/978-3-642-69512-4 45. Samet, B., Vetro, C., Vetro, P.: Fixed point theorem for α-ψ contractive type mappings. Nonlinear Anal. 75(4), 2154–2165 (2012). https://doi.org/10.1016/j.na.2011.10.014 46. Mohammadi, B., Rezapour, S., Shahzad, N.: Some results on ﬁxed points of α-ψ-Ciric generalized multifunctions. Fixed Point Theory Appl. 2013, 24 (2013). https://doi.org/10.1186/1687-1812-2013-24

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Advances in Difference Equations – Springer Journals

Published: Jul 8, 2021

Keywords: α-ψ-contractive map; Inclusion problem; The Caputo derivative; The partial fractional Sturm–Liouville equation; Two variables partial differential equation; 34A08; 34A12

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On partial fractional Sturm–Liouville equation and inclusion

On partial fractional Sturm–Liouville equation and inclusion

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On partial fractional Sturm–Liouville equation and inclusion

On partial fractional Sturm–Liouville equation and inclusion

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