Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

A new result on Branciari metric space using (α, γ)-contractive mappings

A new result on Branciari metric space using (α, γ)-contractive mappings Topol. Algebra Appl. 2022; 10:103–112 Research Article Open Access Jayashree Patil, Basel Hardan, Ahmed A. Hamoud, Amol Bachhav, and Homan Emadifar* A new result on Branciari metric space using (α, )-contractive mappings https://doi.org/10.1515/taa-2022-0117 Received 27 May, 2022; accepted 16 July, 2022 Abstract: In this work, a new common xed point result by generalized contractive functions fullling the type of admissibility condition in a Hausdor Branciari metric space with the support of C-functions, was obtained. Keywords: xed point theorem; contractive mapping; admissible function MSC: 47H10; 47H09; 03D60 1 Introduction The Banach contraction principle, one of the important ndings in the development of the xed point theo- rems, was rst introduced by Banach in [1]. Banach contraction principle professes that, T has a unique xed point for all x, y 2 X where X is a complete metric space and τ 2 (0, 1) such that: τδ(x, y) ≥ δ(Tx, Ty), (1.1) The inequality (1.1) has been developed and expansive in sundry directions for example, the improvements made by Kannan, Chatterjee, Reich in {[2]–[4]} and many others. Newly we refer the improvements {[5]–[11]}. In 1976 Caristi used a lower semi-continuous mapping μ : X ! R to get noticeable expansive of Banach contraction principle, (see [12, 13]). In another context related to the development of metric spaces, Branciari in [14] established an equivalent for the Banach contraction principle on Branciari metric space. After that many researchers researched xed point theorems in this space. For more ideas about the xed point theo- rems in Branciari metric space, we cite {[15]–[18]}. One of the helpful tools for our result is the Hausdor space, it plays an important part in our theorem and its corollaries. Twinning the Branciari metric space and Hausdor space enhanced our theorem and its corollar- ies. Samet in [19] introduced an Interesting concept that contains applying discontinuous functions on the Banach contraction principle. This idea depends on α-admissible mappings. Due to this development, there are more studies in the literature that discuss these results (see. {[20]–[22]}). Two separate generalizations of α -admissible mappings were presented, Budhia in [23] used one of them on Branciari metric space, while Jayashree Patil: Department of Mathematics, Vasantrao Naik Mahavidyalaya, Cidco, Aurangabad, India, E-mail: jv.patil29@gmail.com Basel Hardan: Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad 431004, India, E-mail: bassil2003@gmail.com Ahmed A. Hamoud: Department of Mathematics, Computer Science, Taiz University, Taiz-380 015, Yemen, E-mail: drahmed985@yahoo.com Amol Bachhav: Navin Jindal School of Management, University of Texas at Dallas, Dallas, 75080, USA, E-mail: amol.bachhav@utdallas.edu *Corresponding Author: Homan Emadifar: Department of Mathematics, Hamedan Branch, Islamic Azad University, Hamedan, Iran, E-mail: homan_emadi@yahoo.com Open Access. © 2022 J. Patil et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 License. 104 Ë Jayashree Patil et al. Ansari in [24] used the other one on C -class mappings. We get a new result on Branciari metric space when we apply the researchers’ ideas. The interesting topics in xed point theorems are the discussion of the exis- tence and uniqueness of coincidence or common xed points of dierent functions in the specied spaces, (see {[25]–[30]}). In this paper, we get a new xed point result on Hausdor Branciarir metric space by the generalization of contractive mappings that fullled an admissibility condition by C-functions. 2 preliminaries In this section, we introduce the primary notions for making our major result Denition 2.1. [14] Let X be a non-empty set and let δ : X × X ! [0,∞) satises: (i) δ(x , x ) = 0 if and only if x = x ; 1 2 1 2 (ii) δ(x , x ) = δ(x , x ); 1 2 2 1 (iii) δ(x , y ) ≤ δ(x , x ) + δ(x , y ) + δ(y , y ). 1 2 1 2 2 1 1 2 Then (X, δ) is called Branciari metric space for all x , x 2 X and all distinct points y , y 2 X which distinct 1 2 1 2 from x , x . 1 2 Denition 2.2. [14] Let (X, δ) be a Branciari metric space. (i) A sequencefx g is called convergent to x 2 X if and only if δ(x , x) ! 0, as x ! ∞. i i (ii) A sequencefx g in X is called Cauchy if and only if for all ϵ > 0 there exists a natural number N(ϵ) such that δ(x , x ) < ϵ, for all i > j > N(ϵ). i j (iii) (X, δ) is called a complete Branciarir metric space if every Cauchy sequence is convergent in X. Recently published interesting papers on Branciari metric spaces, for example {[31]–[37]}. Denition 2.3. [19] Let T be a self mapping on a metric space (X, δ) and suppose α : X × X ! [0,∞) be a function. T is called a α-admissible function if α(Tx, Ty) ≥ 1 whenever α(x, y) ≥ 1,8x, y 2 X. Denition 2.4. [38] Let (X, δ) be a metric space, and let α, as in Denition 2.3. X is said to be α-orderly with respect to if for a sequencefx g in X with α(x , x ) ≥ (x , x ) for all i ≥ N and x ! x as i ! ∞, therefore, i i i+1 i i+1 i α(x , x) ≥ (x , x), for all i ≥ N. i i + + Denition 2.5. [39] A non-decreasing continuous function ψ : R ! R is called an altering distance map- ping if ψ(t) = 0 if and only if t = 0. Remark 2.1. We reference the class of altering distance functions by ϱ. Denition 2.6. [19] Let T be a self mapping on a metric space (X, δ). A map T is called a (α, ψ)-contractive mapping if there exist two functions α : X × X ! [0,∞) and ψ : [0, +∞) ! [0, +∞) such that for all x, y 2 X: α(x, y)δ(Tx, Ty) ≤ ψ(x, y). (2.1) For extra ideas of α-admissible and (α, ψ)-contractive mappings, see[19, 20]. Denition 2.7. [22] Let T be a self-mapping on a metric space (X, δ) and let α, : X × X ! [0,∞) are two mappings. A map T is called α-admissible with respect to if α(Tx, Ty) ≥ (Tx, Ty) where α(x, y) ≥ (x, y) for all x, y 2 X. Note that if (x, y) = 1 this denition leads us to Denition 2.3. Also, if we select α(x, y) = 1, so we say that T is a -sub admissible functions. + + Denition 2.8. [24] A function ψ : R × R ! R is called C-function, if A new result on Branciari metric space... Ë 105 i ψ(t , t ) ≤ t ; 1 2 1 ii ψ(t , t ) = t implicit, that either t = 0 or t = 0, 1 2 1 1 2 for all t , t 2 [0,∞). 1 2 Denition 2.9. [40] Suppose T and F are self mappings on X, then i A point x 2 X is called a coincidence point of T and F if, Tx = Fx. In case ω = Tx = Fx, then ω is said to be a point of coincidence of T and F. ii A point ω 2 X is said to be a common xed point of T and F if ω = Tω = Fω. iii The self mappings T and F are said to be weakly compatible if they commute at their coincidence point that is, TFx = FTx whenever Fx = Tx. 3 Main results In this section, we will give a new xed point result using (α, ψ)-contractive mappings under the condition of α-admissibility on Hausdor Branciari metric space with the help of type C- functions. Our major result is: Theorem 3.1. Let (X, δ) be Hausdor Branciari metric space, let T, F : X ! X be weakly compatible fullling α-admissible mappings with respect to . Also let ψ 2 C-functions and ψ, ϑ 2 ϱ such that for all x, y 2 X: α(x, y) ≥ (x, y) ) ψ δ(Tx, Ty) ≤ ψ ψ ξ(x, y) , ϑ ξ(x, y) , (3.1) where, ξ(x, y) = supfjμTx − μFxj,jμTy − μFxj,jμTx − μFyj,jμTy − μFyjg, (3.2) such that for all x 2 X we have: μx = δ(Tx, Fy). (3.3) Suppose that: a there exists x 2 X such that α(Fx , Tx ) ≥ ((Fx , Tx ); 0 0 0 0 0 b α(x , x ) ≥ (x , x ) for all m ! ∞; i j i j m−1 m−1 m−1 m−1 c either T and F are continuous or α(x , ω) ≥ (x , ω) for all x 2 X, i 2 N, and for some ω 2 X. i i i Then there exists x 2 X such that T x = Fx = x for some n 2 N. That is, x is a periodic point, but if for each periodic point x satisfying α(Tx, Fx) ≥ (Tx, Fx), then T has a xed point. Moreover, the xed point is an unique if for all x, y 2 ψ(T) = fx 2 X : Tx = Fx = xg, such that α(x, y) ≥ (x, y). Proof. As, x 2 X, given α(Fx , Tx ) ≥ (Fx , Tx ). (3.4) 0 0 0 0 Consider the iteration T x = Tx = Fx = μx , (3.5) i−1 i i such that x ≠ x , for all i 2 N . So, by (3.5) and since T satised Denition 2.7 and use (3.4) we have i i+1 2 2 α(x, y) = α(Tx , T x ) ≥ (Tx , T x ) = (x, y) 0 0 0 0 By induction, we get α(x , x ) ≥ (x , x ),8i 2 N. i i+1 i i+1 106 Ë Jayashree Patil et al. We will follow an incremental approach to build and accomplish our proof of existence and uniqueness First Step. We shall show that δ(μx , μx ) is non-increasing. By inequality (3.1), we get i i+1 ψ(δ(μx , μx )) = ψ(δ(Tx , Tx )) ≤ ψ[ψ(ξ(x , x )), ϑ(ξ(x , x ))], (3.6) i i+1 i−1 i i−1 i i−1 i where, 8 9 > > jμTx − μFx j,jμTx − μFx j, < i i i−1 i−1 = ξ(x , x ) = sup i−1 i > > : ; jμTx − μFx j,jμTx − μFx j i i−1 i−1 i = supfjμx − μx j,jμx − μx j,jμx − μx j,jμx − μx jg i+1 i i i−1 i+1 i−1 i i = supfδ(μx , μx ), δ(μx , μx ), δ(μx , μx )g i+1 i i i−1 i+1 i−1 8 9 > > δ(μx , μx ) + δ(μx , μx ) < i+1 i i i−1 = ≤ sup > > : ; δ(μx + μx ) + δ(μx + μx ) i+1 i−1 i+2 i−1 = supfδ(μx , μx ) + δ(μx + μx )g i+1 i i+2 i−1 ≤ supfδ(μx , μx )g i+1 i Then, ξ(x , x ) = δ(μx , μx ), therefore i−1 i i+1 i ψ(δ(μx , μx )) = ψ(δ(Tx , Tx )) ≤ ψ[ψ(δ(μx , μx )), ϑ(δ(μx , μx ))] i i+1 i−1 i i+1 i i+1 i ≤ ψ(δ(μx , μx )). i+1 i From Denition 2.8, we nd: either ψ(δ(μx , μx )) = 0 or, i+1 i ϑ(δ(μx , μx )) = 0, i.e., δ(μx , μx ) = 0 but this is a contradiction, due to Tx ≠ Tx . i+1 i i+1 i i+1 i Second Step. We shall examine δ(μx , μx ) ! 0 as i ! ∞. By used the above proceedings. Then by i i+2 inequality (3.1), we have ψ δ(μx , μx ) = ψ δ(Tx , Tx ) i i+2 i−1 i+1 ≤ ψ ψ ξ(x , x ) , ϑ ξ(x , x ) i−1 i+1 i−1 i+1 ≤ ψ ξ(x , x ) . (3.7) i−1 i+1 Which could be δ(μx , μx ) ≤ ξ(x , x ). (3.8) i i+2 i−1 i+1 Where ψ is changing the functions’ space. We obtain A new result on Branciari metric space... Ë 107 8 9 > > μTx − μFx , μTx − μFx , < i+1 i+1 i−1 i−1 = ξ(x , x ) = sup i−1 i+1 > > : ; μTx − μFx , μTx − μFx i+1 i−1 i−1 i+1 8 9 > > jμx − μx j,jμx − μx j, < i+2 i+1 i i−1 = = sup > > : ; jμx − μx j,jμx − μx j i+2 i−1 i i+1 8 9 > > δ(μx , μx ), δ(μx , μx ), < i+2 i+1 i i−1 = = sup > > : ; δ(μx , μx ), δ(μx , μx ) i+2 i−1 i i+1 8 9 > > δ(μx , μx ) + δ(μx , μx ) < i+2 i+1 i i−1 = ≤ sup + > > : ; δ(μx , μx ) + δ(μx , μx ) i+2 i−1 i i+1 ≤ δ(μx , μx ) i+2 i−1 Then, ξ(x , x ) = δ(μx , μx ). Therefore by inequality (3.1), we obtain i−1 i i+2 i−1 ψ(δ(μx , μx ) = ψ(δ(Tx , Tx )) ≤ ψ[ψ(δ(μx , μx )), ϑ(δ(μx , μx ))]. i+2 i−1 i+1 i−2 i+2 i−1 i+2 i−1 From denition of C-functions, we nd: either ψ(δ(μx , μx )) = 0 or ϑ(δ(μx , μx )) = 0, i.e., δ(μx , μx ) = 0. Substitute that into an in- i+2 i−1 i+2 i−1 i+2 i−1 equality (3.8), we get δ(μx , μx ) = 0 when i ! ∞. Obviously, μx not necessity to be subsequent in arrange i+2 i i for all i 2 N of the convergence in X. Third Step. By contradiction, we shall prove thatfμx g is a Cauchy sequence. For that, the following lemma is helpful for the remainder of the theorem’s proof. Its proof is classic so, we skip it. Lemma 3.2. Let, (X, δ) be a Branciari meric space and letfμx g be a sequence in X such that lim (μx , μx ) = lim (μx , μx ) = 0. i i+1 i i+2 i!∞ i!∞ Where, x ≠ x , for all i ≠ j. If fμx g is not a Cauchy sequence, then there exist ϵ > 0 and two sub-sequences i j i μx , μx  fμx g, where m < j < i , m 2 N. Furthermore, i j i m m m m δ(x , x ) ≥ ϵ, δ(x , x ) < ϵ. i j i j m m m m−1 Thus, for the following sequences δ(μx , μx ), δ(μx , μx ), δ(μx , μx ), i i i i j j m m−1 m m−1 m m−1 it satises lim inf δ(μx , μx ) ≤ lim sup δ(μx , μx ) ≤ ϵ, i j i j m m m m i!∞ i!∞ lim inf δ(μx , μx ) ≤ lim sup δ(μx , μx ) ≤ ϵ, i i i i m m−1 m m−1 i!∞ i!∞ lim inf δ(μx , μx ) ≤ lim sup δ(μx , μx ) ≤ ϵ. j j j j m m−1 m m−1 i!∞ i!∞ 108 Ë Jayashree Patil et al. Now, lim δ(μx , μx ) = lim δ(Tx , Tx ), (3.9) i j i j m m m−1 m−1 m!∞ m!∞ for all i, j, m 2 N. by inequality (3.1) we have ψ(δ(μx , μx ) = ψ(δ(Tx , Tx )) i j i jm−1 m m m−1 ≤ ψ[ψ(ξ(x , x )), ϑ(ξ(x , x ))]. (3.10) i j i j m−1 m−1 m−1 m−1 Where, 8 9 > > jμTx − μFx j,jμTx − μFx j, < j j i i = m−1 m−1 m−1 m−1 ξ(x , x ) = sup i j m−1 m−1 > > : ; jμTx − μFx j,jμTx − μFx j j i i j m−1 m−1 m−1 m−1 8 9 > > jμx − μx j,jμx − μx j, < j j i i = m m−1 m m−1 = sup > > : ; jμx − μx j,jμx − μx j j i i j m m−1 m m−1 8 9 > > δ(μx , μx ), δ(μx , μx ), < j j i i = m m−1 m m−1 = sup > > : ; δ(μx , μx ), δ(μx , μx ) j i i j m m−1 m m−1 Then, by Lemma 3.2 and since, α(μx , μx ) ≥ (μx , μx ), j j j j m m−1 m m−1 also, α(μx , μx ) ≥ (μx , μx ). i i i i m m−1 m m−1 When i, m ! ∞, the inequality (3.10), will come like this ψ(ϵ) ≤ ψ[ψ(ϵ), ϑ(ϵ)] ≤ ψ(ϵ). (3.11) whereas the ψ, ϑ, ψ and μ are continuous functions. Therefore, we have ψ(ϵ) = 0 or ϑ(ϵ) = 0. Thus, ϵ = 0, which is a contradiction. We conclude thatfμx g is a Cauchy sequence. Since, X is a complete metric space, then μx ! x as i ! ∞, for some x 2 X. i * * Forth Step. If T and F are continuous mappings and by iteration relation (3.5), we nd μx = Fx = Tx = Tx , as i ! ∞. i+1 i+1 i * Then T and F have a periodical point. Since X is a Hausdor space, i.e., Tx = Fx = x . Otherwise, if X is an * * * α-regular with regard to , then by condition b, we obtain: ψ(δ(μTx , μTx )) ≤ ψ[ψ(ξ(x , x )), ϑ(ξ(x , x ))], (3.12) i * i * i * where, 8 9 > > jμTx − Fμx j,jμTx − Fμx j, < * * i i = ξ(x , x ) = sup > > : ; jμTx − Fμx j,jμTx − Fμx j * i i * n o = sup jμTx − μx j,jμx − μx j,jμTx − μx j,jμx − μx j * * i+1 i * i i+1 * n o = sup . δ(μTx , μx ), δ(μx , μx ), δ(μTx , μx ), δ(μx , μx ) * * i+1 i * i i+1 * A new result on Branciari metric space... Ë 109 Since, δ(μx , μx ) = δ(μx , μx ) = 0, when i ! ∞, thus i+1 i i+1 * lim ξ(x , x ) = δ(μTx , μx ) = δ(μTx , μx ). (3.13) i * * i * * i!∞ Substitute equation (3.13) into inequality (3.12) we get ψ(μTx , μx ) ≤ ψ[ψ δ(μTx , μx ) , ϑ δ(μTx , μx ) ], (3.14) * * * * * * where, i ! ∞. Thus, ψ δ(μTx , μx ) = 0 or ϑ δ(μTx , μx ) = 0, therefore δ(μTx , μx ) = 0. Hence, μTx = * * * * * * * μx , based on the denition of μ we conclude that T and F have a periodical point Tx = Fx = x . Thus, we * * * * have shown he validity of the c part of our theory. The showing that T and F have a common xed point of the periodical point, is what we will discuss in the following step. k k Fifth Step. Suppose μT ω = μF ω = ω, ω 2 X. Obviously, ω is a xed point of T where k = 1. We will prove k−1 k−1 μT ω = F ω = ω , where k > 1, 8k > 1, k 2 N. We have α(Tω, Fω) ≥ (Tω, Fω) for a periodical point k−1 k k−1 k ω. If potential, let μT ω ≠ μT ω and μF ω ≠ μF ω for all k > 1, k 2 N. Therefore, by inequalities (3.1): k−1 k k−2 k−1 k−2 k−1 ψ δ(T ω, T ω) ≤ ψ ψ ξ(T ω, T ω) , ϑ ξ(T ω, T ω) , (3.15) where, 8 9 k−1 k−1 k−2 k−2 > > jμTT ω − μFT ωj,jμTT ω − μFT ωj, < = k−2 k−1 ξ(T ω, T ω) = sup > > : k−1 k−2 k−2 k−1 ; jμTT ω − μFT ωj,jμTT ω − μFT ωj 8 9 k k−1 k−1 k−2 > > jμT ω − μT ωj,jμT ω − μT ωj, < = = sup > > : k k−2 k−1 k−1 ; jμT ω − μT ωj,jμT ω − μT ωj k k−1 k−1 k−2 k k−2 = supfδ(μT ω, μT ω), δ(μT ω, μT ω), δ(μT ω, μT ω)g. Now, we have three cases, k−2 k−1 k k−1 Case 1 If ξ(T ω, T ω) = δ(μT ω, μT ω) for some k. Then by (3.15) we have k−1 k k k−1 k k−1 ψ δ(μT ω, μT ω) ≤ ψ ψ δ(μT ω, μT ω) , ϑ δ(μT ω, μT ω) k k−1 ≤ ψ δ(μT ω, μT ω)). k k−1 k k−1 Hence, δ(μT ω, μT ω) = 0, by Denition 2.5 Thus, μT ω = μT ω, which is contradiction. k−2 k−1 k−1 k−2 Case 2 If ξ(T ω, T ω) = δ(μT ω, μT ω) for all k. Here, T is an α-admissible with regard to and α(Tω, Fω) ≥ (Tω, Fω), we obtain k−1 k k−1 k−2 k−1 k−2 ψ δ(μT ω, μT ω) ≤ ψ ψ δ(μT ω, μT ω) , ϑ δ(μT ω, μT ω) k−1 k−2 ≤ ψ δ(μT ω, μT ω) . k−1 k−2 + Consequently, δ(μT ω, μT ω) is a non-increasing sequence of R , we get ψ δ(ω, μTω) = ψ δ(μFω, μTω) k k+1 = ψ δ(μFT ω, μT ω) k−1 k ≤ ψ δ(μFT ω, μT ω) k−1 k−2 k−1 k−2 ≤ ψ ψ δ(μT ω, μT ω) , ϑ δ(μT ω, μT ω) k−1 k−2 ≤ ψ δ(μT ω, μT ω) ≤ ψ ψ δ(ω, μTω) , ϑ δ(μω, μTω) ≤ ψ δ(ω, μTω) 110 Ë Jayashree Patil et al. Thus, k−1 k k k−1 k k−1 ψ δ(μT ω, μT ω) ≤ ψ ψ δ(μT ω, μT ω) , ϑ δ(μT ω, μT ω) . k k−1 k k−1 Hence, μT ω = μT ω. And implicitly indicates that μF ω = μF ω which is contradiction. k−2 k−1 k k−2 Case 3 If ξ(T ω, T ω) = δ(μT ω, μT ω) for some k. Then by (3.15) we have k−1 k k k−2 k k−2 ψ δ(μT ω, μT ω) ≤ ψ ψ δ(μT ω, μT ω) , ϑ δ(μT ω, μT ω) k k−2 ≤ ψ δ(μT ω, μT ω)) k−2 k−1 k−1 k ≤ ψ δ(μT ω, μT ω) + δ(μT ω, μT ω) k−1 k From case 1 and case 2 we conclude that μT ω = μT ω, which is a contradiction. k−1 k−1 Accordingly, the claim that T ω = F ω = ω is not true. Hence, T and F have a common xed point ω. The uniqueness of the common xed point increases its strength and renders the solution not subject to other possibilities. Consequently, we nish our argument by explaining that the common xed point we have established is unique. This is what we will clarify in the nal step of the proof. Sixth Step. Suppose that ω , ω 2 X are two common xed points of T nd F such that ω ≠ ω . Use, (3.1) 1 2 1 2 and since, α(ω , ω ) ≥ (ω , ω ) we obtain 1 2 1 2 ψ(δ(ω , ω )) = ψ δ(μTω , μTω ) ≤ ψ ψ ξ(ω , ω ) , ϑ ξ(ω , ω ) , 1 2 1 2 1 2 1 2 where, ξ(ω , ω ) = supfjμTω − μFω j,jμTω − μFω j,jμTω − μFω j,jμTω − μFω jg. 1 2 2 2 1 1 2 1 1 2 Then, ξ(ω , ω ) = 0. Therefore, ψ(δ(ω , ω )) = 0. Consequently, ω = ω . 1 2 1 2 1 2 Thus we have proven the uniqueness of the common xed point that we have found, and by this, we have completed proving our result. Now, we will present the following corollaries which are derived from our main result. Corollary 3.3. Let (X, δ) be Hausdor Branciari metric space, let T, F : X ! X are weakly compatible fullling α-admissible mappings with respect to . Let ψ 2 C and ψ, ϑ 2 ϱ such that for all x, y 2 X: α(x, y) ≥ (x, y) ) ψ δ(Tx, Ty) ≤ ψ ξ(x, y) − ϑ ξ(x, y) , (3.16) where ξ(x, y) and the conditions a,b and c is the same as in Theorem 3.1. Then there exists x 2 X such that n n T x = F x = x, for some n 2 N that is, x is a periodic point, but if for each periodic point x satisfying α(Tx, Fx) ≥ (Tx, Fx), then T and F have a common xed point. Moreover, the xed point is an unique if for all x, y 2 ψ(T) = fx 2 X : Tx = Fx = xg, such that α(x, y) ≥ (x, y). Proof. Take, ψ(t − t ) = t − t and follow same method proving Theorem 3.1. 1 2 1 2 Corollary 3.4. Let (X, δ) be Hausdor Branciari metric space and let T, F : X ! X are weakly compatible fullling α-admissible mappings with respect to . Let ψ 2 C and ψ, ϑ 2 ϱ such that for all x, y 2 X α(x, y) ≥ (x, y) ) ψ δ(Tx, Ty) ≤ ξ(x, y) − ϑ ξ(x, y) , (3.17) where, ξ(x, y) and the conditions a,b and c is the same as in Theorem 3.1. Then there exists x 2 X such that n n T x = F x = x, for some n 2 N that is, x is a periodic point, but if for each periodic point x satisfying α(Tx, Fx) ≥ (Tx, Fx), then T and F have a common xed point. Moreover, the xed point is an unique if for all x, y 2 ψ(T) = fx 2 X : Tx = Fx = xg, such that α(x, y) ≥ (x, y). Proof. Consider, ψ(t) = t in Corollary 3.3. To support our results, we will provide the following example: A new result on Branciari metric space... Ë 111 Example 3.5. Let T and F be weakly compatible self-mappings on a Hausdor Branciari metric space X = [0, 1], such that 1 1 x + , x 2 [0, ), 3 3 Tx = 1 1 , x 2 [ , 1]. 3 3 Fx = x, 8x 2 X. Let, α, : X × X ! R , such that, α(x, y) = 3, (x, y) = 2, 8x, y 2 X. Suppose δ : X × X ! [0, 1). Consider, ψ : [0,∞) × [0,∞) ! R and ψ, ϑ : [0,∞) ! [0,∞) dened as ψ(t , t ) = t − t , ψ(t) = t/3 and 1 2 2 1 ϑ(t) = 4t/5. Then, all the terms of Theorem 3.1 are satised. Hence, x = be a unique common xed point of T and F on X. Conict of interest: Authors declare no conict of interest. References [1] S. Banach, Sur les operations dans les ensembles abstraits et leur applications aux equations integrales, Fundam. Math. 3(1922), no. 1, 133–181. [2] R. Kannan, Some results on xed points, Bulletin of the Calcutta Mathematical Society, 60(1968), 71–76. [3] S. Chatterjee, Fixed point theorem, C. R. Acad, Bulgar Sci. 25(1972), 727-730. [4] S. Reich, Fixed points of contractive functions, Bollettino della Unione Matematica Italiana, (1972), 26–42. [5] W. Alfaqih, M. Imdad, R. Gubran and I. Khan, Relation-theoretic coincidence and common xed point results under (F, R) - contractions with an application, Fixed Point Theory and Applications, 2019 (2019), no. 1, 1–18. [6] A. Biahdillah and S. Surjanto, Application of Banach Contraction Principle in Complex Valued Rectangular b-Metric Space, Journal of Physics: Conference Series, 1490(2020), no. 1, 1–11. [7] H. Isik, B. Mohammadi, M. Reza Haddadi and V. Parvaneh, On a New Generalization of Banach Contraction Principle with Application, Mathematics, 9 (2019), no. 7, P. 862. [8] J. Patil, B. Hardan, M. Abdo, A. Chaudhari and A. Bachhav, Generalized Fractional Dierential Equations by Using a Fixed Point Theorem For Generalized Contractive Type, Dynamics of Continuous, Discrete and Impulsive Systems Series B: Appli- cations and Algorithms, 28(2021), no. 2, 77-88. [9] J. Patil, B. Hardan, A. Chaudhari and A. Bachhav, Approximate xed points for n-Linear functional (μ, σ)-nonexpansive Map- pings on n-Banach spaces, J. Math. Anal and Model. 1(2020), no. 1, 20–32. [10] J. Patil, B. Hardan, M. Abdo, A. Chaudhari, and A. Bachhav, A xed point theorem for Hardy-Rogers type on generalized fractional dierential equations, Advances in the Theory of Nonlinear Analysis and its Applications, 4(2020), no. 4, 407– [11] J. Patil and B. Hardan, On Fixed Point Theorems in Complete Metric Space, Journal of Computer and Mathematical Sciences, 10(2019), no. 7, 1419–1425. [12] J. Caristi, Fixed point theorems for mappings satisfying inwardness conditions, Trans. Amer. Math. Soc. 215(1976), 241-251. [13] J. Caristi, Fixed point theory and inwardness conditions, Applied Nonlinear Analysis, (1979) 479–483. [14] A. Branciari, A xed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces, Publ. Math. Debre- cen, 57(2000), 31–37. [15] H. Aydi, C. Chen and E. Karapinar, Interpolative Ćirić-Reich-Rus Type Contractions via the Branciari Distance, Mathematics, 7(2019), no. 1, p 84. [16] H. Aydi, N. Tas, N. Özgür, and N. Mlaiki, Fixed-discs in rectangular metric spaces, Symmetry, 11(2019), no. 2, p 294. [17] I. R. Sarma, J. M. Rao and S. S. Rao, Contractions over generalized metric spaces, Journal of Nonlinear Sciences and Appli- cations, 3(2009), no. 2, 180–182. [18] N. Souayah, H. Aydi, T. Abdeljawad, and N. Mlaiki, Best proximity point theorems on rectangular metric spaces endowed with a graph, Axioms, 8(2019), no. 1, p 17. [19] B. Samet, C. Vetro,and P. Vetro, Fixed point theorems for α, -contractive type mappings, Nonlinear Anal. Theory Methods Appl. 76(2012), no. 4, 2154–2165. [20] H. Alsulami, S. Chandok, M. Taoudi and I. Erhan, Some xed point theorems for (ϕ, ψ)-rational type contractive mappings, Fixed Point Theory Appl. 2015(2015), p. 97. [21] D. Gopal, P. Kumam and M. Abbas, Background and Recent Developments of Metric Fixed Point Theory, CRC Press, (2017). [22] P. Salimi, A. Latif and N. Hussain, Modied α − ψ-contractive mappings with applications, Fixed Point Theory Appl. 2013(2013), no. 1, 1–19. 112 Ë Jayashree Patil et al. [23] L. Budhia, M. Kir, D. Gopal and H. Kiziltunç, New xed point results in rectangular metric space and application to fractional calculus, Tbil. Math. J. 10(2017), no. 1, 91–104. [24] A. Ansari, Note on α, ψ-contractive type mappings and related xed point, in The 2nd regional conference on mathematics and applications, Tonekabon, Iran, September, Noor Univ. 2014(2014), 377–380. [25] I. Altun, H. Sahin and D. Turkoglu, Caristi-type xed point theorems and some generalizations on M-metric space, Bulletin of the Malaysian Mathematical Sciences Society, 43(2020), no. 3, 2647–2657. [26] Y. Cho, J. Martínez-Moreno, A. Roldán and C. Roldán, Multidimensional xed point theorems under (ψ, ϕ)-contractive con- ditions in partially ordered complete metric spaces, Journal of Computational and Applied Mathematics, 273(2013), 76–87. [27] M. Nazam, M. Arshad and M. Postolache, Coincidence and common xed point theorems for four mappings satisfying (α , F)-contraction, Nonlinear Analysis: Modelling and Control, 23(2018), no. 5, 664–690. [28] N. Tiwari, P. Tripahti and M. Singh, Coincidence and Fixed Point in b-Metric Space, International Journal of Advanced Science and Technology, 29(2020), no. 2, 1121 – 1124. [29] A. Tomar and S. Upadhyay, Coincidence and common xed point theorems for faintly compatible maps, journal of applied and engineering mathematics, Journal of Applied and Engineering Mathematics, 7(2017), no. 1, 25-32. [30] H. Wu, Common Coincidence Points and Common Fixed Points in Fuzzy Semi-Metric Spaces, Mathematics, 6(2018), no. 2, p 29. [31] T. Abdeljawad, E. Karapinar, S. K. Panda and N. Maliki, Solutions of boundary value problems on extended- Branciary b- distance, Journal of Inequalities and Applications, 2020(2020), no. 1, 1–16. [32] I. M. Erhan, E. Karapinar and T. Sekulić, Fixed points of (ψ−ϕ)-contractions on rectangular metric spaces, Fixed Point Theory and Applications, 2012(2020), no. 1, 1–12. [33] S. Gülyaz, E. Karapinar and I. Erhan, Generalized alpha-Meir-Keeler contraction mappings on Branciari b-metric spaces, Fiomat, 31(2017), no. 17, 5445–5456. [34] E. Karapinar, Discussion on α, ψ-contractions on generalized metric spaces, Abstract and Applied Analysis, 2014(2014), 7 page. [35] E. Karapinar, Some xed results on Branciari metric spaces via implicit functions, Caprathian Journal of Mathematics, 31(2015), no. 3339-348. [36] E. Karapinar and D. Zhang, Properties and principles in Branciari distance, Journal of xed Point Theorey and Applications, 21(2019), no. (3), 1–18. [37] J. Patil, B. Hardan, A. Hamoud, A. Bachhav, H. Emadifar and H. Günerhan, Generalizing contractive mappings on rectangular metric space, Advances in Mahtematical Physics, 2022(2022), 10 pages. [38] L. Budhia, H. Aydi, A. Ansari and D. Gopala, Some new xed point results in rectangular metric spaces with an application to fractional-order functional dierential equations, Nonlinear Anal. Model. Control, 25(2020), no. 4, 580–597. [39] M. S. Khan, M. Swaleh and S. Sessa, Fixed point theorems by altering distances between the points, Bull. Aust. Math. Soc. 30(1984), no. 1, 1–9. [40] M. Arshad, J. Ahmad and E. Karapinar, Some Common Fixed Point Results in Rectangular Metric Spaces, International Journal of Analysis, 2013(2013), 7 pages. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Topological Algebra and its Applications de Gruyter

A new result on Branciari metric space using (α, γ)-contractive mappings

Download PDF
10 pages

Loading next page...
 
/lp/de-gruyter/a-new-result-on-branciari-metric-space-using-contractive-mappings-prKAR1eXzM
Publisher
de Gruyter
Copyright
© 2022 Jayashree Patil et al., published by De Gruyter
ISSN
2299-3231
eISSN
2299-3231
DOI
10.1515/taa-2022-0117
Publisher site
See Article on Publisher Site

Abstract

Topol. Algebra Appl. 2022; 10:103–112 Research Article Open Access Jayashree Patil, Basel Hardan, Ahmed A. Hamoud, Amol Bachhav, and Homan Emadifar* A new result on Branciari metric space using (α, )-contractive mappings https://doi.org/10.1515/taa-2022-0117 Received 27 May, 2022; accepted 16 July, 2022 Abstract: In this work, a new common xed point result by generalized contractive functions fullling the type of admissibility condition in a Hausdor Branciari metric space with the support of C-functions, was obtained. Keywords: xed point theorem; contractive mapping; admissible function MSC: 47H10; 47H09; 03D60 1 Introduction The Banach contraction principle, one of the important ndings in the development of the xed point theo- rems, was rst introduced by Banach in [1]. Banach contraction principle professes that, T has a unique xed point for all x, y 2 X where X is a complete metric space and τ 2 (0, 1) such that: τδ(x, y) ≥ δ(Tx, Ty), (1.1) The inequality (1.1) has been developed and expansive in sundry directions for example, the improvements made by Kannan, Chatterjee, Reich in {[2]–[4]} and many others. Newly we refer the improvements {[5]–[11]}. In 1976 Caristi used a lower semi-continuous mapping μ : X ! R to get noticeable expansive of Banach contraction principle, (see [12, 13]). In another context related to the development of metric spaces, Branciari in [14] established an equivalent for the Banach contraction principle on Branciari metric space. After that many researchers researched xed point theorems in this space. For more ideas about the xed point theo- rems in Branciari metric space, we cite {[15]–[18]}. One of the helpful tools for our result is the Hausdor space, it plays an important part in our theorem and its corollaries. Twinning the Branciari metric space and Hausdor space enhanced our theorem and its corollar- ies. Samet in [19] introduced an Interesting concept that contains applying discontinuous functions on the Banach contraction principle. This idea depends on α-admissible mappings. Due to this development, there are more studies in the literature that discuss these results (see. {[20]–[22]}). Two separate generalizations of α -admissible mappings were presented, Budhia in [23] used one of them on Branciari metric space, while Jayashree Patil: Department of Mathematics, Vasantrao Naik Mahavidyalaya, Cidco, Aurangabad, India, E-mail: jv.patil29@gmail.com Basel Hardan: Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad 431004, India, E-mail: bassil2003@gmail.com Ahmed A. Hamoud: Department of Mathematics, Computer Science, Taiz University, Taiz-380 015, Yemen, E-mail: drahmed985@yahoo.com Amol Bachhav: Navin Jindal School of Management, University of Texas at Dallas, Dallas, 75080, USA, E-mail: amol.bachhav@utdallas.edu *Corresponding Author: Homan Emadifar: Department of Mathematics, Hamedan Branch, Islamic Azad University, Hamedan, Iran, E-mail: homan_emadi@yahoo.com Open Access. © 2022 J. Patil et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 License. 104 Ë Jayashree Patil et al. Ansari in [24] used the other one on C -class mappings. We get a new result on Branciari metric space when we apply the researchers’ ideas. The interesting topics in xed point theorems are the discussion of the exis- tence and uniqueness of coincidence or common xed points of dierent functions in the specied spaces, (see {[25]–[30]}). In this paper, we get a new xed point result on Hausdor Branciarir metric space by the generalization of contractive mappings that fullled an admissibility condition by C-functions. 2 preliminaries In this section, we introduce the primary notions for making our major result Denition 2.1. [14] Let X be a non-empty set and let δ : X × X ! [0,∞) satises: (i) δ(x , x ) = 0 if and only if x = x ; 1 2 1 2 (ii) δ(x , x ) = δ(x , x ); 1 2 2 1 (iii) δ(x , y ) ≤ δ(x , x ) + δ(x , y ) + δ(y , y ). 1 2 1 2 2 1 1 2 Then (X, δ) is called Branciari metric space for all x , x 2 X and all distinct points y , y 2 X which distinct 1 2 1 2 from x , x . 1 2 Denition 2.2. [14] Let (X, δ) be a Branciari metric space. (i) A sequencefx g is called convergent to x 2 X if and only if δ(x , x) ! 0, as x ! ∞. i i (ii) A sequencefx g in X is called Cauchy if and only if for all ϵ > 0 there exists a natural number N(ϵ) such that δ(x , x ) < ϵ, for all i > j > N(ϵ). i j (iii) (X, δ) is called a complete Branciarir metric space if every Cauchy sequence is convergent in X. Recently published interesting papers on Branciari metric spaces, for example {[31]–[37]}. Denition 2.3. [19] Let T be a self mapping on a metric space (X, δ) and suppose α : X × X ! [0,∞) be a function. T is called a α-admissible function if α(Tx, Ty) ≥ 1 whenever α(x, y) ≥ 1,8x, y 2 X. Denition 2.4. [38] Let (X, δ) be a metric space, and let α, as in Denition 2.3. X is said to be α-orderly with respect to if for a sequencefx g in X with α(x , x ) ≥ (x , x ) for all i ≥ N and x ! x as i ! ∞, therefore, i i i+1 i i+1 i α(x , x) ≥ (x , x), for all i ≥ N. i i + + Denition 2.5. [39] A non-decreasing continuous function ψ : R ! R is called an altering distance map- ping if ψ(t) = 0 if and only if t = 0. Remark 2.1. We reference the class of altering distance functions by ϱ. Denition 2.6. [19] Let T be a self mapping on a metric space (X, δ). A map T is called a (α, ψ)-contractive mapping if there exist two functions α : X × X ! [0,∞) and ψ : [0, +∞) ! [0, +∞) such that for all x, y 2 X: α(x, y)δ(Tx, Ty) ≤ ψ(x, y). (2.1) For extra ideas of α-admissible and (α, ψ)-contractive mappings, see[19, 20]. Denition 2.7. [22] Let T be a self-mapping on a metric space (X, δ) and let α, : X × X ! [0,∞) are two mappings. A map T is called α-admissible with respect to if α(Tx, Ty) ≥ (Tx, Ty) where α(x, y) ≥ (x, y) for all x, y 2 X. Note that if (x, y) = 1 this denition leads us to Denition 2.3. Also, if we select α(x, y) = 1, so we say that T is a -sub admissible functions. + + Denition 2.8. [24] A function ψ : R × R ! R is called C-function, if A new result on Branciari metric space... Ë 105 i ψ(t , t ) ≤ t ; 1 2 1 ii ψ(t , t ) = t implicit, that either t = 0 or t = 0, 1 2 1 1 2 for all t , t 2 [0,∞). 1 2 Denition 2.9. [40] Suppose T and F are self mappings on X, then i A point x 2 X is called a coincidence point of T and F if, Tx = Fx. In case ω = Tx = Fx, then ω is said to be a point of coincidence of T and F. ii A point ω 2 X is said to be a common xed point of T and F if ω = Tω = Fω. iii The self mappings T and F are said to be weakly compatible if they commute at their coincidence point that is, TFx = FTx whenever Fx = Tx. 3 Main results In this section, we will give a new xed point result using (α, ψ)-contractive mappings under the condition of α-admissibility on Hausdor Branciari metric space with the help of type C- functions. Our major result is: Theorem 3.1. Let (X, δ) be Hausdor Branciari metric space, let T, F : X ! X be weakly compatible fullling α-admissible mappings with respect to . Also let ψ 2 C-functions and ψ, ϑ 2 ϱ such that for all x, y 2 X: α(x, y) ≥ (x, y) ) ψ δ(Tx, Ty) ≤ ψ ψ ξ(x, y) , ϑ ξ(x, y) , (3.1) where, ξ(x, y) = supfjμTx − μFxj,jμTy − μFxj,jμTx − μFyj,jμTy − μFyjg, (3.2) such that for all x 2 X we have: μx = δ(Tx, Fy). (3.3) Suppose that: a there exists x 2 X such that α(Fx , Tx ) ≥ ((Fx , Tx ); 0 0 0 0 0 b α(x , x ) ≥ (x , x ) for all m ! ∞; i j i j m−1 m−1 m−1 m−1 c either T and F are continuous or α(x , ω) ≥ (x , ω) for all x 2 X, i 2 N, and for some ω 2 X. i i i Then there exists x 2 X such that T x = Fx = x for some n 2 N. That is, x is a periodic point, but if for each periodic point x satisfying α(Tx, Fx) ≥ (Tx, Fx), then T has a xed point. Moreover, the xed point is an unique if for all x, y 2 ψ(T) = fx 2 X : Tx = Fx = xg, such that α(x, y) ≥ (x, y). Proof. As, x 2 X, given α(Fx , Tx ) ≥ (Fx , Tx ). (3.4) 0 0 0 0 Consider the iteration T x = Tx = Fx = μx , (3.5) i−1 i i such that x ≠ x , for all i 2 N . So, by (3.5) and since T satised Denition 2.7 and use (3.4) we have i i+1 2 2 α(x, y) = α(Tx , T x ) ≥ (Tx , T x ) = (x, y) 0 0 0 0 By induction, we get α(x , x ) ≥ (x , x ),8i 2 N. i i+1 i i+1 106 Ë Jayashree Patil et al. We will follow an incremental approach to build and accomplish our proof of existence and uniqueness First Step. We shall show that δ(μx , μx ) is non-increasing. By inequality (3.1), we get i i+1 ψ(δ(μx , μx )) = ψ(δ(Tx , Tx )) ≤ ψ[ψ(ξ(x , x )), ϑ(ξ(x , x ))], (3.6) i i+1 i−1 i i−1 i i−1 i where, 8 9 > > jμTx − μFx j,jμTx − μFx j, < i i i−1 i−1 = ξ(x , x ) = sup i−1 i > > : ; jμTx − μFx j,jμTx − μFx j i i−1 i−1 i = supfjμx − μx j,jμx − μx j,jμx − μx j,jμx − μx jg i+1 i i i−1 i+1 i−1 i i = supfδ(μx , μx ), δ(μx , μx ), δ(μx , μx )g i+1 i i i−1 i+1 i−1 8 9 > > δ(μx , μx ) + δ(μx , μx ) < i+1 i i i−1 = ≤ sup > > : ; δ(μx + μx ) + δ(μx + μx ) i+1 i−1 i+2 i−1 = supfδ(μx , μx ) + δ(μx + μx )g i+1 i i+2 i−1 ≤ supfδ(μx , μx )g i+1 i Then, ξ(x , x ) = δ(μx , μx ), therefore i−1 i i+1 i ψ(δ(μx , μx )) = ψ(δ(Tx , Tx )) ≤ ψ[ψ(δ(μx , μx )), ϑ(δ(μx , μx ))] i i+1 i−1 i i+1 i i+1 i ≤ ψ(δ(μx , μx )). i+1 i From Denition 2.8, we nd: either ψ(δ(μx , μx )) = 0 or, i+1 i ϑ(δ(μx , μx )) = 0, i.e., δ(μx , μx ) = 0 but this is a contradiction, due to Tx ≠ Tx . i+1 i i+1 i i+1 i Second Step. We shall examine δ(μx , μx ) ! 0 as i ! ∞. By used the above proceedings. Then by i i+2 inequality (3.1), we have ψ δ(μx , μx ) = ψ δ(Tx , Tx ) i i+2 i−1 i+1 ≤ ψ ψ ξ(x , x ) , ϑ ξ(x , x ) i−1 i+1 i−1 i+1 ≤ ψ ξ(x , x ) . (3.7) i−1 i+1 Which could be δ(μx , μx ) ≤ ξ(x , x ). (3.8) i i+2 i−1 i+1 Where ψ is changing the functions’ space. We obtain A new result on Branciari metric space... Ë 107 8 9 > > μTx − μFx , μTx − μFx , < i+1 i+1 i−1 i−1 = ξ(x , x ) = sup i−1 i+1 > > : ; μTx − μFx , μTx − μFx i+1 i−1 i−1 i+1 8 9 > > jμx − μx j,jμx − μx j, < i+2 i+1 i i−1 = = sup > > : ; jμx − μx j,jμx − μx j i+2 i−1 i i+1 8 9 > > δ(μx , μx ), δ(μx , μx ), < i+2 i+1 i i−1 = = sup > > : ; δ(μx , μx ), δ(μx , μx ) i+2 i−1 i i+1 8 9 > > δ(μx , μx ) + δ(μx , μx ) < i+2 i+1 i i−1 = ≤ sup + > > : ; δ(μx , μx ) + δ(μx , μx ) i+2 i−1 i i+1 ≤ δ(μx , μx ) i+2 i−1 Then, ξ(x , x ) = δ(μx , μx ). Therefore by inequality (3.1), we obtain i−1 i i+2 i−1 ψ(δ(μx , μx ) = ψ(δ(Tx , Tx )) ≤ ψ[ψ(δ(μx , μx )), ϑ(δ(μx , μx ))]. i+2 i−1 i+1 i−2 i+2 i−1 i+2 i−1 From denition of C-functions, we nd: either ψ(δ(μx , μx )) = 0 or ϑ(δ(μx , μx )) = 0, i.e., δ(μx , μx ) = 0. Substitute that into an in- i+2 i−1 i+2 i−1 i+2 i−1 equality (3.8), we get δ(μx , μx ) = 0 when i ! ∞. Obviously, μx not necessity to be subsequent in arrange i+2 i i for all i 2 N of the convergence in X. Third Step. By contradiction, we shall prove thatfμx g is a Cauchy sequence. For that, the following lemma is helpful for the remainder of the theorem’s proof. Its proof is classic so, we skip it. Lemma 3.2. Let, (X, δ) be a Branciari meric space and letfμx g be a sequence in X such that lim (μx , μx ) = lim (μx , μx ) = 0. i i+1 i i+2 i!∞ i!∞ Where, x ≠ x , for all i ≠ j. If fμx g is not a Cauchy sequence, then there exist ϵ > 0 and two sub-sequences i j i μx , μx  fμx g, where m < j < i , m 2 N. Furthermore, i j i m m m m δ(x , x ) ≥ ϵ, δ(x , x ) < ϵ. i j i j m m m m−1 Thus, for the following sequences δ(μx , μx ), δ(μx , μx ), δ(μx , μx ), i i i i j j m m−1 m m−1 m m−1 it satises lim inf δ(μx , μx ) ≤ lim sup δ(μx , μx ) ≤ ϵ, i j i j m m m m i!∞ i!∞ lim inf δ(μx , μx ) ≤ lim sup δ(μx , μx ) ≤ ϵ, i i i i m m−1 m m−1 i!∞ i!∞ lim inf δ(μx , μx ) ≤ lim sup δ(μx , μx ) ≤ ϵ. j j j j m m−1 m m−1 i!∞ i!∞ 108 Ë Jayashree Patil et al. Now, lim δ(μx , μx ) = lim δ(Tx , Tx ), (3.9) i j i j m m m−1 m−1 m!∞ m!∞ for all i, j, m 2 N. by inequality (3.1) we have ψ(δ(μx , μx ) = ψ(δ(Tx , Tx )) i j i jm−1 m m m−1 ≤ ψ[ψ(ξ(x , x )), ϑ(ξ(x , x ))]. (3.10) i j i j m−1 m−1 m−1 m−1 Where, 8 9 > > jμTx − μFx j,jμTx − μFx j, < j j i i = m−1 m−1 m−1 m−1 ξ(x , x ) = sup i j m−1 m−1 > > : ; jμTx − μFx j,jμTx − μFx j j i i j m−1 m−1 m−1 m−1 8 9 > > jμx − μx j,jμx − μx j, < j j i i = m m−1 m m−1 = sup > > : ; jμx − μx j,jμx − μx j j i i j m m−1 m m−1 8 9 > > δ(μx , μx ), δ(μx , μx ), < j j i i = m m−1 m m−1 = sup > > : ; δ(μx , μx ), δ(μx , μx ) j i i j m m−1 m m−1 Then, by Lemma 3.2 and since, α(μx , μx ) ≥ (μx , μx ), j j j j m m−1 m m−1 also, α(μx , μx ) ≥ (μx , μx ). i i i i m m−1 m m−1 When i, m ! ∞, the inequality (3.10), will come like this ψ(ϵ) ≤ ψ[ψ(ϵ), ϑ(ϵ)] ≤ ψ(ϵ). (3.11) whereas the ψ, ϑ, ψ and μ are continuous functions. Therefore, we have ψ(ϵ) = 0 or ϑ(ϵ) = 0. Thus, ϵ = 0, which is a contradiction. We conclude thatfμx g is a Cauchy sequence. Since, X is a complete metric space, then μx ! x as i ! ∞, for some x 2 X. i * * Forth Step. If T and F are continuous mappings and by iteration relation (3.5), we nd μx = Fx = Tx = Tx , as i ! ∞. i+1 i+1 i * Then T and F have a periodical point. Since X is a Hausdor space, i.e., Tx = Fx = x . Otherwise, if X is an * * * α-regular with regard to , then by condition b, we obtain: ψ(δ(μTx , μTx )) ≤ ψ[ψ(ξ(x , x )), ϑ(ξ(x , x ))], (3.12) i * i * i * where, 8 9 > > jμTx − Fμx j,jμTx − Fμx j, < * * i i = ξ(x , x ) = sup > > : ; jμTx − Fμx j,jμTx − Fμx j * i i * n o = sup jμTx − μx j,jμx − μx j,jμTx − μx j,jμx − μx j * * i+1 i * i i+1 * n o = sup . δ(μTx , μx ), δ(μx , μx ), δ(μTx , μx ), δ(μx , μx ) * * i+1 i * i i+1 * A new result on Branciari metric space... Ë 109 Since, δ(μx , μx ) = δ(μx , μx ) = 0, when i ! ∞, thus i+1 i i+1 * lim ξ(x , x ) = δ(μTx , μx ) = δ(μTx , μx ). (3.13) i * * i * * i!∞ Substitute equation (3.13) into inequality (3.12) we get ψ(μTx , μx ) ≤ ψ[ψ δ(μTx , μx ) , ϑ δ(μTx , μx ) ], (3.14) * * * * * * where, i ! ∞. Thus, ψ δ(μTx , μx ) = 0 or ϑ δ(μTx , μx ) = 0, therefore δ(μTx , μx ) = 0. Hence, μTx = * * * * * * * μx , based on the denition of μ we conclude that T and F have a periodical point Tx = Fx = x . Thus, we * * * * have shown he validity of the c part of our theory. The showing that T and F have a common xed point of the periodical point, is what we will discuss in the following step. k k Fifth Step. Suppose μT ω = μF ω = ω, ω 2 X. Obviously, ω is a xed point of T where k = 1. We will prove k−1 k−1 μT ω = F ω = ω , where k > 1, 8k > 1, k 2 N. We have α(Tω, Fω) ≥ (Tω, Fω) for a periodical point k−1 k k−1 k ω. If potential, let μT ω ≠ μT ω and μF ω ≠ μF ω for all k > 1, k 2 N. Therefore, by inequalities (3.1): k−1 k k−2 k−1 k−2 k−1 ψ δ(T ω, T ω) ≤ ψ ψ ξ(T ω, T ω) , ϑ ξ(T ω, T ω) , (3.15) where, 8 9 k−1 k−1 k−2 k−2 > > jμTT ω − μFT ωj,jμTT ω − μFT ωj, < = k−2 k−1 ξ(T ω, T ω) = sup > > : k−1 k−2 k−2 k−1 ; jμTT ω − μFT ωj,jμTT ω − μFT ωj 8 9 k k−1 k−1 k−2 > > jμT ω − μT ωj,jμT ω − μT ωj, < = = sup > > : k k−2 k−1 k−1 ; jμT ω − μT ωj,jμT ω − μT ωj k k−1 k−1 k−2 k k−2 = supfδ(μT ω, μT ω), δ(μT ω, μT ω), δ(μT ω, μT ω)g. Now, we have three cases, k−2 k−1 k k−1 Case 1 If ξ(T ω, T ω) = δ(μT ω, μT ω) for some k. Then by (3.15) we have k−1 k k k−1 k k−1 ψ δ(μT ω, μT ω) ≤ ψ ψ δ(μT ω, μT ω) , ϑ δ(μT ω, μT ω) k k−1 ≤ ψ δ(μT ω, μT ω)). k k−1 k k−1 Hence, δ(μT ω, μT ω) = 0, by Denition 2.5 Thus, μT ω = μT ω, which is contradiction. k−2 k−1 k−1 k−2 Case 2 If ξ(T ω, T ω) = δ(μT ω, μT ω) for all k. Here, T is an α-admissible with regard to and α(Tω, Fω) ≥ (Tω, Fω), we obtain k−1 k k−1 k−2 k−1 k−2 ψ δ(μT ω, μT ω) ≤ ψ ψ δ(μT ω, μT ω) , ϑ δ(μT ω, μT ω) k−1 k−2 ≤ ψ δ(μT ω, μT ω) . k−1 k−2 + Consequently, δ(μT ω, μT ω) is a non-increasing sequence of R , we get ψ δ(ω, μTω) = ψ δ(μFω, μTω) k k+1 = ψ δ(μFT ω, μT ω) k−1 k ≤ ψ δ(μFT ω, μT ω) k−1 k−2 k−1 k−2 ≤ ψ ψ δ(μT ω, μT ω) , ϑ δ(μT ω, μT ω) k−1 k−2 ≤ ψ δ(μT ω, μT ω) ≤ ψ ψ δ(ω, μTω) , ϑ δ(μω, μTω) ≤ ψ δ(ω, μTω) 110 Ë Jayashree Patil et al. Thus, k−1 k k k−1 k k−1 ψ δ(μT ω, μT ω) ≤ ψ ψ δ(μT ω, μT ω) , ϑ δ(μT ω, μT ω) . k k−1 k k−1 Hence, μT ω = μT ω. And implicitly indicates that μF ω = μF ω which is contradiction. k−2 k−1 k k−2 Case 3 If ξ(T ω, T ω) = δ(μT ω, μT ω) for some k. Then by (3.15) we have k−1 k k k−2 k k−2 ψ δ(μT ω, μT ω) ≤ ψ ψ δ(μT ω, μT ω) , ϑ δ(μT ω, μT ω) k k−2 ≤ ψ δ(μT ω, μT ω)) k−2 k−1 k−1 k ≤ ψ δ(μT ω, μT ω) + δ(μT ω, μT ω) k−1 k From case 1 and case 2 we conclude that μT ω = μT ω, which is a contradiction. k−1 k−1 Accordingly, the claim that T ω = F ω = ω is not true. Hence, T and F have a common xed point ω. The uniqueness of the common xed point increases its strength and renders the solution not subject to other possibilities. Consequently, we nish our argument by explaining that the common xed point we have established is unique. This is what we will clarify in the nal step of the proof. Sixth Step. Suppose that ω , ω 2 X are two common xed points of T nd F such that ω ≠ ω . Use, (3.1) 1 2 1 2 and since, α(ω , ω ) ≥ (ω , ω ) we obtain 1 2 1 2 ψ(δ(ω , ω )) = ψ δ(μTω , μTω ) ≤ ψ ψ ξ(ω , ω ) , ϑ ξ(ω , ω ) , 1 2 1 2 1 2 1 2 where, ξ(ω , ω ) = supfjμTω − μFω j,jμTω − μFω j,jμTω − μFω j,jμTω − μFω jg. 1 2 2 2 1 1 2 1 1 2 Then, ξ(ω , ω ) = 0. Therefore, ψ(δ(ω , ω )) = 0. Consequently, ω = ω . 1 2 1 2 1 2 Thus we have proven the uniqueness of the common xed point that we have found, and by this, we have completed proving our result. Now, we will present the following corollaries which are derived from our main result. Corollary 3.3. Let (X, δ) be Hausdor Branciari metric space, let T, F : X ! X are weakly compatible fullling α-admissible mappings with respect to . Let ψ 2 C and ψ, ϑ 2 ϱ such that for all x, y 2 X: α(x, y) ≥ (x, y) ) ψ δ(Tx, Ty) ≤ ψ ξ(x, y) − ϑ ξ(x, y) , (3.16) where ξ(x, y) and the conditions a,b and c is the same as in Theorem 3.1. Then there exists x 2 X such that n n T x = F x = x, for some n 2 N that is, x is a periodic point, but if for each periodic point x satisfying α(Tx, Fx) ≥ (Tx, Fx), then T and F have a common xed point. Moreover, the xed point is an unique if for all x, y 2 ψ(T) = fx 2 X : Tx = Fx = xg, such that α(x, y) ≥ (x, y). Proof. Take, ψ(t − t ) = t − t and follow same method proving Theorem 3.1. 1 2 1 2 Corollary 3.4. Let (X, δ) be Hausdor Branciari metric space and let T, F : X ! X are weakly compatible fullling α-admissible mappings with respect to . Let ψ 2 C and ψ, ϑ 2 ϱ such that for all x, y 2 X α(x, y) ≥ (x, y) ) ψ δ(Tx, Ty) ≤ ξ(x, y) − ϑ ξ(x, y) , (3.17) where, ξ(x, y) and the conditions a,b and c is the same as in Theorem 3.1. Then there exists x 2 X such that n n T x = F x = x, for some n 2 N that is, x is a periodic point, but if for each periodic point x satisfying α(Tx, Fx) ≥ (Tx, Fx), then T and F have a common xed point. Moreover, the xed point is an unique if for all x, y 2 ψ(T) = fx 2 X : Tx = Fx = xg, such that α(x, y) ≥ (x, y). Proof. Consider, ψ(t) = t in Corollary 3.3. To support our results, we will provide the following example: A new result on Branciari metric space... Ë 111 Example 3.5. Let T and F be weakly compatible self-mappings on a Hausdor Branciari metric space X = [0, 1], such that 1 1 x + , x 2 [0, ), 3 3 Tx = 1 1 , x 2 [ , 1]. 3 3 Fx = x, 8x 2 X. Let, α, : X × X ! R , such that, α(x, y) = 3, (x, y) = 2, 8x, y 2 X. Suppose δ : X × X ! [0, 1). Consider, ψ : [0,∞) × [0,∞) ! R and ψ, ϑ : [0,∞) ! [0,∞) dened as ψ(t , t ) = t − t , ψ(t) = t/3 and 1 2 2 1 ϑ(t) = 4t/5. Then, all the terms of Theorem 3.1 are satised. Hence, x = be a unique common xed point of T and F on X. Conict of interest: Authors declare no conict of interest. References [1] S. Banach, Sur les operations dans les ensembles abstraits et leur applications aux equations integrales, Fundam. Math. 3(1922), no. 1, 133–181. [2] R. Kannan, Some results on xed points, Bulletin of the Calcutta Mathematical Society, 60(1968), 71–76. [3] S. Chatterjee, Fixed point theorem, C. R. Acad, Bulgar Sci. 25(1972), 727-730. [4] S. Reich, Fixed points of contractive functions, Bollettino della Unione Matematica Italiana, (1972), 26–42. [5] W. Alfaqih, M. Imdad, R. Gubran and I. Khan, Relation-theoretic coincidence and common xed point results under (F, R) - contractions with an application, Fixed Point Theory and Applications, 2019 (2019), no. 1, 1–18. [6] A. Biahdillah and S. Surjanto, Application of Banach Contraction Principle in Complex Valued Rectangular b-Metric Space, Journal of Physics: Conference Series, 1490(2020), no. 1, 1–11. [7] H. Isik, B. Mohammadi, M. Reza Haddadi and V. Parvaneh, On a New Generalization of Banach Contraction Principle with Application, Mathematics, 9 (2019), no. 7, P. 862. [8] J. Patil, B. Hardan, M. Abdo, A. Chaudhari and A. Bachhav, Generalized Fractional Dierential Equations by Using a Fixed Point Theorem For Generalized Contractive Type, Dynamics of Continuous, Discrete and Impulsive Systems Series B: Appli- cations and Algorithms, 28(2021), no. 2, 77-88. [9] J. Patil, B. Hardan, A. Chaudhari and A. Bachhav, Approximate xed points for n-Linear functional (μ, σ)-nonexpansive Map- pings on n-Banach spaces, J. Math. Anal and Model. 1(2020), no. 1, 20–32. [10] J. Patil, B. Hardan, M. Abdo, A. Chaudhari, and A. Bachhav, A xed point theorem for Hardy-Rogers type on generalized fractional dierential equations, Advances in the Theory of Nonlinear Analysis and its Applications, 4(2020), no. 4, 407– [11] J. Patil and B. Hardan, On Fixed Point Theorems in Complete Metric Space, Journal of Computer and Mathematical Sciences, 10(2019), no. 7, 1419–1425. [12] J. Caristi, Fixed point theorems for mappings satisfying inwardness conditions, Trans. Amer. Math. Soc. 215(1976), 241-251. [13] J. Caristi, Fixed point theory and inwardness conditions, Applied Nonlinear Analysis, (1979) 479–483. [14] A. Branciari, A xed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces, Publ. Math. Debre- cen, 57(2000), 31–37. [15] H. Aydi, C. Chen and E. Karapinar, Interpolative Ćirić-Reich-Rus Type Contractions via the Branciari Distance, Mathematics, 7(2019), no. 1, p 84. [16] H. Aydi, N. Tas, N. Özgür, and N. Mlaiki, Fixed-discs in rectangular metric spaces, Symmetry, 11(2019), no. 2, p 294. [17] I. R. Sarma, J. M. Rao and S. S. Rao, Contractions over generalized metric spaces, Journal of Nonlinear Sciences and Appli- cations, 3(2009), no. 2, 180–182. [18] N. Souayah, H. Aydi, T. Abdeljawad, and N. Mlaiki, Best proximity point theorems on rectangular metric spaces endowed with a graph, Axioms, 8(2019), no. 1, p 17. [19] B. Samet, C. Vetro,and P. Vetro, Fixed point theorems for α, -contractive type mappings, Nonlinear Anal. Theory Methods Appl. 76(2012), no. 4, 2154–2165. [20] H. Alsulami, S. Chandok, M. Taoudi and I. Erhan, Some xed point theorems for (ϕ, ψ)-rational type contractive mappings, Fixed Point Theory Appl. 2015(2015), p. 97. [21] D. Gopal, P. Kumam and M. Abbas, Background and Recent Developments of Metric Fixed Point Theory, CRC Press, (2017). [22] P. Salimi, A. Latif and N. Hussain, Modied α − ψ-contractive mappings with applications, Fixed Point Theory Appl. 2013(2013), no. 1, 1–19. 112 Ë Jayashree Patil et al. [23] L. Budhia, M. Kir, D. Gopal and H. Kiziltunç, New xed point results in rectangular metric space and application to fractional calculus, Tbil. Math. J. 10(2017), no. 1, 91–104. [24] A. Ansari, Note on α, ψ-contractive type mappings and related xed point, in The 2nd regional conference on mathematics and applications, Tonekabon, Iran, September, Noor Univ. 2014(2014), 377–380. [25] I. Altun, H. Sahin and D. Turkoglu, Caristi-type xed point theorems and some generalizations on M-metric space, Bulletin of the Malaysian Mathematical Sciences Society, 43(2020), no. 3, 2647–2657. [26] Y. Cho, J. Martínez-Moreno, A. Roldán and C. Roldán, Multidimensional xed point theorems under (ψ, ϕ)-contractive con- ditions in partially ordered complete metric spaces, Journal of Computational and Applied Mathematics, 273(2013), 76–87. [27] M. Nazam, M. Arshad and M. Postolache, Coincidence and common xed point theorems for four mappings satisfying (α , F)-contraction, Nonlinear Analysis: Modelling and Control, 23(2018), no. 5, 664–690. [28] N. Tiwari, P. Tripahti and M. Singh, Coincidence and Fixed Point in b-Metric Space, International Journal of Advanced Science and Technology, 29(2020), no. 2, 1121 – 1124. [29] A. Tomar and S. Upadhyay, Coincidence and common xed point theorems for faintly compatible maps, journal of applied and engineering mathematics, Journal of Applied and Engineering Mathematics, 7(2017), no. 1, 25-32. [30] H. Wu, Common Coincidence Points and Common Fixed Points in Fuzzy Semi-Metric Spaces, Mathematics, 6(2018), no. 2, p 29. [31] T. Abdeljawad, E. Karapinar, S. K. Panda and N. Maliki, Solutions of boundary value problems on extended- Branciary b- distance, Journal of Inequalities and Applications, 2020(2020), no. 1, 1–16. [32] I. M. Erhan, E. Karapinar and T. Sekulić, Fixed points of (ψ−ϕ)-contractions on rectangular metric spaces, Fixed Point Theory and Applications, 2012(2020), no. 1, 1–12. [33] S. Gülyaz, E. Karapinar and I. Erhan, Generalized alpha-Meir-Keeler contraction mappings on Branciari b-metric spaces, Fiomat, 31(2017), no. 17, 5445–5456. [34] E. Karapinar, Discussion on α, ψ-contractions on generalized metric spaces, Abstract and Applied Analysis, 2014(2014), 7 page. [35] E. Karapinar, Some xed results on Branciari metric spaces via implicit functions, Caprathian Journal of Mathematics, 31(2015), no. 3339-348. [36] E. Karapinar and D. Zhang, Properties and principles in Branciari distance, Journal of xed Point Theorey and Applications, 21(2019), no. (3), 1–18. [37] J. Patil, B. Hardan, A. Hamoud, A. Bachhav, H. Emadifar and H. Günerhan, Generalizing contractive mappings on rectangular metric space, Advances in Mahtematical Physics, 2022(2022), 10 pages. [38] L. Budhia, H. Aydi, A. Ansari and D. Gopala, Some new xed point results in rectangular metric spaces with an application to fractional-order functional dierential equations, Nonlinear Anal. Model. Control, 25(2020), no. 4, 580–597. [39] M. S. Khan, M. Swaleh and S. Sessa, Fixed point theorems by altering distances between the points, Bull. Aust. Math. Soc. 30(1984), no. 1, 1–9. [40] M. Arshad, J. Ahmad and E. Karapinar, Some Common Fixed Point Results in Rectangular Metric Spaces, International Journal of Analysis, 2013(2013), 7 pages.

Journal

Topological Algebra and its Applications – de Gruyter

Published: Jan 1, 2022

Keywords: fixed point theorem; contractive mapping; admissible function; 47H10; 47H09; 03D60

There are no references for this article.