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Fixed point theorem for a sequence of multivalued nonself mappings in metrically convex metric spaces

Fixed point theorem for a sequence of multivalued nonself mappings in metrically convex metric... Topol. Algebra Appl. 2022; 10:1–12 Research Article Open Access David Aron and Santosh Kumar* Fixed point theorem for a sequence of multivalued nonself mappings in metrically convex metric spaces https://doi.org/10.1515/taa-2020-0108 Received 29 September, 2021; revised 6 December, 2021; accepted 2 January, 2022 Abstract: In this paper, a common xed point theorem is demonstrated for a sequence of multivalued map- pings which satisfy certain requirements in complete metric spaces. The results proved here will generalize and extend the results due to Ćirić [1]. Suitable examples are given at the end to support the results proved herein. Keywords: Multivalued mapping, non-self mapping, metrically convex metric spaces. PACS: 47H10, 54H25 1 Introduction In 1969, Nadler [2] studied xed points using the Hausdor metric for multivalued mappings. Assad and Kirk [3] extended the Banach contraction theorem from self mappings by giving its proof for non-self mappings. These results were employed by Rhoades [4] and a xed point theorem for a multivalued non-self mapping was proved. Ćirić [1] generalized and extended the theorem of Rhoades [4] by proving a xed point theorem for a continuous multivalued mapping. Imdad et al. [5] gave some common xed point theorems for nonself generalized hybrid contraction where they generalized several well known results in the literature. Later Imdad and Kumar [6] proved results for generalized T− contractive mappings. In 2013, Du et al. [7] established some xed/coincidence point theorems for multivalued non-selfmaps in the context of complete metric spaces. Kumam et al. [8] introduced a multi- valued cyclic generalized contraction by extending the Mizoguchi and Takahashi’s contraction for non-self mappings. In 2014, Ali et al. [9] introduced the notions of α− admissible and α−ψ− contractive type condition for nonself multivalued mappings. Dhage et al. [10] proved some common xed point theorems for sequences of nonself multivalued operators dened on a closed subset of a metrically convex metric space. Later on several authors worked in this direction. For more related results one can see [5, 6, 11–14] and the references therein. In this paper, throughout the discussion we will take ϱ as the metric distance between two points while ρ(x, A) denotes the distance from point x to a set A. David Aron: Department of Mathematics, College of Natural and Applied Sciences, University of Dar es Salaam, Tanzania; E-mail: d.aron.da6@gmail.com *Corresponding Author: Santosh Kumar: Department of Mathematics, College of Natural and Applied Sciences, University of Dar es Salaam, Tanzania; E-mail: drsengar2002@gmail.com Open Access. © 2022 Manuel David Aron and Santosh Kumar, published by De Gruyter. This work is licensed under the Creative Commons Attribution alone 4.0 License. 2 Ë David Aron and Santosh Kumar 2 Preliminaries In order to identify the major conclusions of this work, the following denitions and lemmas will be required. Denition 2.1. [2] Let (M, ϱ) be a metric space and CB(M) be the collection of all nonempty closed and bounded subsets of M. For A, B 2 CB(M), and u 2 M, dene ρ(u, A) = inffϱ(u, a) : a 2 Ag and H(A, B) = max sup ρ(a, B), sup ρ(A, b) . a2A b2B It can be easily veried that H is a metric on CB(M). H is called the Hausdor metric induced by ϱ. Denition 2.2. [15]. (i) A sequence fu g in a metric space (M, ϱ) is said to converge or to be convergent if there is an u 2 M such that lim ϱ(u , u) = 0. n!∞ (ii) A sequence fu g in a metric space (M, ϱ) is said to be Cauchy sequence if for every ϵ > 0 there is an N = N(ϵ) such that ϱ(u , u ) < ϵ, n m for every m, n > N. (iii) A metric space (M, ϱ) is said to be complete if every Cauchy sequence in M converges to an element of M. Convex property plays an important role in many areas such as functional analysis, optimization and control theory. This phenomena is well utilized by the researchers in metric xed point theory. The following is Assad and Kirk’s [3] denition of a metrically convex metric space. Denition 2.3. [3]. A metric space (M, ϱ) is said to be metrically convex if for any u, v 2 M with u ≠ v, there exists a point z 2 M,(u ≠ z ≠ v) such that ϱ(u, v) = ϱ(u, z) + ϱ(z, v). The following result is taken from Assad [3] where ∂K denotes the boundary of K. Lemma 2.1. [3]. If K is a nonempty closed subset of the complete and convex metric space M and if u 2 K, v 2 / K, then there exists a point z 2 ∂K, such that ϱ(u, v) = ϱ(u, z) + ϱ(z, v). The following result was proved by Assad and Kirk [3] as they gave sucient condition for a multi-valued non-self mappings from K into CB(M) to have a xed point. Theorem 2.1. [3]. Let M be a complete and convex metric space and K be a nonempty closed subset of M, and T : K ! CB(M) a contraction mapping such that ρ(T(u), T(v)) ≤ aϱ(u, v), (1) where a < 1. If Tu  K for each u 2 ∂K then there exists u 2 K such that u 2 T(u ) (i.e. T has a xed point 0 0 0 in K). The following denition is due to Khan [16] is useful in proving the main results: Denition 2.4. [16] Let K be a nonempty subset of a metric space (M, ϱ). A mapping T : K ! CB(M) is said to be continuous at u 2 K if for any ϵ > 0, there exists a d = d(ϵ) > 0 such that H(Tu, Tu ) < ϵ, whenever 0 0 ϱ(u, u ) < d. If T is continuous at every point of K, we say that T continuous at K. 0 Fixed point theorem for a sequence of multivalued ... Ë 3 Now, we present the following lemma which nds key applications in the proof our theorems in a convex metric space. Lemma 2.2. [17]. Let (M, ϱ) be a metric space. If A, B 2 CB(M) and α 2 A, then for any positive number σ < 1 there exists β = β(α) in B such that σϱ(α, β) ≤ H(A, B). (2) Rhoades [4] generalized the result of Itoh [18] by proving the following result for a multivalued non-self map- ping in a metrically convex complete metric space. Theorem 2.2. [4]. Let (M, ϱ) be a complete and metrically convex metric space and K be a nonempty closed subset of M. Let T : K ! CB(M) satisfy the following contractive condition: H(Tu, Tv) ≤ aϱ(u, v) + b maxfρ(u, Tu), ρ(v, Tv)g + c[ρ(u, Tv) + ρ(v, Tu)], (3) for all u, v 2 K, where a, b, c ≥ 0 and such that λ = [(1 + a + c)/(1 − b − c)] · [(a + b + c)/(1 − c)] < 1. If Tu  K for each u 2 ∂K, then there exists a z 2 K such that z 2 Tz. In the following theorem a wider class of multivalued non-self mappings than those in [4] and in [18] was considered. Theorem 2.3. [1]. Let (M, ϱ) be a complete and metrically convex metric space and K be a nonempty closed subset of M. Let T : K ! CB(M) be a mapping such that H(Tu, Tv) ≤ aϱ(u, v) + b maxfρ(u, Tu), ρ(v, Tv)g + c[ρ(u, Tv) + ρ(v, Tu)] + d[ρ(u, Tu) + ρ(v, Tv)] (4) where a, b, c, d ≥ 0 are such that λ = a + 2b + (3 + a)(c + d). (5) If Tu  K for each u 2 ∂K, then there exists w 2 K such that w 2 Tw. This paper aims to modify Theorem 2.3 so that it can be applied to a sequence of multivalued mappings in a metrically convex metric space. In doing so, our results will generalize several existing results in literature. 3 Main Results In this section, we will prove common xed point theorem for a sequence of multi-valued non-self mappings. An extended and generalized version of the Theorem 2.3 is as follows: Theorem 3.1. Let (M, ϱ) be a complete and metrically convex metric space and K be a nonempty closed subset of M. LetfT g : K ! CB(M) be a sequence of multivalued mappings satisfying for i ≠ j, where i, j = 1, 2,··· , n=1 H(T (u), T (v)) ≤ aϱ(u, v) + b maxfρ(u, T (u)), ρ(v, T (v))g+ i j i j (6) c[ρ(u, T (v)) + ρ(v, T (u))] + d[ρ(u, T (u)) + ρ(v, T (v))], j i i j for all u, v 2 K, where a, b, c, d ≥ 0 are such that λ = a + 2b + (3 + a)(c + d) < 1. (7) ∞ ∞ IffT (u)g  K for each u 2 ∂K, then the sequencefT g has a common xed point in K (i.e. there exists n n n=1 n=1 w 2 K such that w 2 T (w)). n 4 Ë David Aron and Santosh Kumar Proof. The considered case is when T (∂K)  K, but not necessarily T (K)  K. The case T (K)  K is much n n n more simpler. In this case it follows that condition (7) can be simplied to λ = a + b + 2c + 2d < 1. Two sequences fu g and fu g in K and M, respectively, will be constructed in the following manner. Let 0 −1 α u 2 K and u = u 2 T (u ) be arbitrary. Let α be any xed number such that 0 < α < 2 . Put σ = λ . Then 0 1 1 0 from condition (7), σ < 1. By Lemma 2.2 we can choose u 2 T (u ) such that 2 1 0 0 σϱ(u , u ) ≤ H(T (u ), T (u )). 1 2 1 0 2 1 0 0 0 If u 2 K, put u = u . If u 2 / K, then, as M is metrically convex, we can choose u 2 ∂K such that 2 2 2 2 2 0 0 ϱ(u , u ) + ϱ(u , u ) = ϱ(u , u ). 1 2 2 2 1 2 Let u 2 T (u ) be such that 3 2 0 0 σϱ(u , u ) ≤ H(T (u ), T (u )). 2 1 3 2 2 3 By induction we may obtain sequencesfu g andfu g such that for n = 1, 2,··· , (i) u 2 T (u ) n n−1 0 0 (ii) σϱ(u , u ) ≤ H(T (u ), T (u )), where n n n n−1 n+1 n+1 0 0 (iii) u = u , if u 2 K, or n+1 n+1 n+1 0 0 0 (iv) u 2 ∂K, and ϱ(u , u ) + ϱ(u , u ) = ϱ(u , u ), if u 2 / K, for all n. Now let n n n+1 n+1 n+1 n+1 n+1 n+1 P = fu 2 fu g : u = u , ξ = 1, 2,···g; ξ ξ ξ Q = fu 2 fu g : u ≠ u , ξ = 1, 2,···g. ξ ξ Notice that if u 2 Q for some n, then u , u 2 P, since two consecutive terms offu g cannot be in Q. n n−1 n+1 n Here, the intention is to estimate the distance ϱ(u , u ). Thus, these cases must be considered. n n+1 Case 1. u 2 P and u 2 P. From assumption (ii) and (6) taking u = u and v = u we get n n+1 n−1 n 0 0 σϱ(u , u ) = σϱ(u , u ) ≤ H(T (u ), T (u )) ≤ aϱ(u , u ) n n n n n+1 n n+1 n−1 n+1 n−1 + b maxfρ(u , T (u )), ρ(u , T (u ))g + c[ρ(u , T (u )) n n n n n−1 n−1 n+1 n−1 n+1 + ρ(u , T (u ))] + d[ρ(u , T (u )) + ρ(u , T (u ))] n n n n n n−1 n−1 n−1 n+1 ≤ aϱ(u , u ) + b maxfϱ(u , u ), ϱ(u , u )g n n n n−1 n−1 n+1 + c[ϱ(u , u ) + ϱ(u , u )] + d[ϱ(u , u ) + ϱ(u , u )] n n n n n−1 n+1 n−1 n+1 ≤ aϱ(u , u ) + b maxfϱ(u , u ), ϱ(u , u )g n n n n−1 n−1 n+1 + c[ϱ(u , u ) + ϱ(u , u )] + d[ϱ(u , u ) + ϱ(u , u )]. n n n n n−1 n+1 n−1 n+1 It follows that σϱ(u , u ) ≤ (a + c + d)ϱ(u , u ) + b maxfϱ(u , u ), (8) n n n n+1 n−1 n−1 ϱ(u , u )g + (c + d)ϱ(u , u ). n n n+1 n+1 If the maximum of the coecient of b is ϱ(u , u ), then from (8) and by (7) we get n n+1 σϱ(u , u ) ≤ (a + c + d)ϱ(u , u ) + (b + c + d)ϱ(u , u ) n n n n+1 n−1 n+1 ≤ (a + b + 2c + 2d)ϱ(u , u ) ≤ λϱ(u , u ) ≤ λ ϱ(u , u ) n n+1 n n+1 n n+1 = σϱ(u , u ), n+1 which is a contradiction. Therefore, from (8) we obtain σϱ(u , u ) ≤ (a + b + c + d)ϱ(u , u ) + (c + d)ϱ(u , u ) n n n n+1 n−1 n+1 and hence a + b + c + d ϱ(u , u ) ≤ ϱ(u , u ). (9) n n n+1 n−1 σ − c − d Fixed point theorem for a sequence of multivalued ... Ë 5 Case 2. u 2 P and u 2 Q. Then by (iv) n+1 0 0 0 0 0 ϱ(u , u ) = ϱ(u , u ) − ϱ(u , u ) ≤ ϱ(u , u ) = ϱ(u , u ) n n n n+1 n+1 n+1 n+1 n+1 n n+1 Following the identical reasoning as in Case 1, we obtain a + b + c + d ϱ(u , u ) ≤ ϱ(u , u ). n n n+1 n−1 σ − c − d Case 3. u 2 Q and u 2 P. By applying the triangle inequality we have n n+1 0 0 0 0 0 ϱ(u , u ) ≤ ϱ(u , u ) + ϱ(u , u ) = ϱ(u , u ) + ϱ(u , u ). n n n n+1 n n n+1 n n n+1 Then from assumption (ii) and (6) we have 0 0 0 0 σϱ(u , u ) ≤ σϱ(u , u ) + σϱ(u , u ) ≤ σϱ(u , u ) +H(T (u ), T (u )) n n n n n n+1 n n n+1 n n−1 n+1 ≤ σϱ(u , u ) + aϱ(u , u ) + b maxfρ(u , T (u )), ρ(u , T (u ))g n n n n n n n−1 n−1 n−1 n+1 + c[ρ(u , T (u )) + ρ(u , T (u ))] + d[ρ(u , T (u )) + ρ(u , T (u ))] n n n n n n n−1 n+1 n−1 n−1 n−1 n+1 0 0 0 ≤ σϱ(u , u ) + aϱ(u , u ) + b maxfϱ(u , u ), ϱ(u , u )g n n n n n−1 n−1 n n+1 0 0 0 0 +c[ϱ(u , u ) + ϱ(u , u )] + d[ϱ(u , u ) + ϱ(u , u )]. (10) n n n−1 n+1 n n−1 n n+1 Since two consecutive terms offu g cannot be in Q, u 2 P. Then by condition (iv) n−1 0 0 ϱ(u , u ) + ϱ(u , u ) = ϱ(u , u ) n n n−1 n n−1 n and hence, α < λ = σ, we have 0 0 σϱ(u , u ) + aϱ(u , u ) ≤ σϱ(u , u ). n n n−1 n n−1 n Additionally, by the triangle inequality, 0 0 c[ϱ(u , u ) + ϱ(u , u )] ≤ c[ϱ(u , u ) + ϱ(u , u ) + ϱ(u , u )] n−1 n+1 n n n−1 n n n+1 n n = cϱ(u , u ) + cϱ(u , u ). n−1 n n n+1 0 0 Suppose that ϱ(u , u ) < ϱ(u , u ) = ϱ(u , u ). Then from (10) we get n n n−1 n n+1 n+1 0 0 σϱ(u , u ) ≤ σϱ(u , u ) + bϱ(u , u ) + (c + d)ϱ(u , u ) + (c + d)ϱ(u , u ) n n n n+1 n−1 n n+1 n−1 n n+1 and hence σ + c + d ϱ(u , u ) ≤ ϱ(u , u ). (11) n n+1 n−1 n σ − b − c − d If ϱ(u , u ) ≥ ϱ(u , u ), then, as 1 ≤ [(σ + c + d)/(σ − b − c − d)], we have again n−1 n n n+1 σ + c + d ϱ(u , u ) ≤ ϱ(u , u ). (12) n+1 n−1 n σ − b − c − d Therefore, (11) holds for all n. 0 0 0 0 Since u = u , it follows that ϱ(u , u ) = ϱ(u , u ). Then, as in Case 1, we have n−1 n−1 n n n−1 n−1 a + b + c + d ϱ(u , u ) ≤ ϱ(u , u ). (13) n−1 n n−2 n−1 σ − c − d By (11) and (13) we obtain a + b + c + d σ + c + d ϱ(u , u ) ≤ · ϱ(u , u ). (14) n n+1 n−2 n−1 σ − c − d σ − b − c − d 6 Ë David Aron and Santosh Kumar To verify this, recall from the inequality (7) which yields a + 2b + (3 + a)(c + d) < 1 ) a + 2b + 3c + 3d + ac + ad < 1 ) a + ac + ad + b + c + d < 1 − b − 2c − 2d ) a + ac + ad + b + c + d < 1 − b − c − d − c − d 2 2 ) a + ac + ad + b + bc + bd + c + c + cd + d + dc + d 2 2 < 1 − b − c − d − c + cb + c + cd − d + db + dc + d ) a(1 + c + d) + b(1 + c + d) + c(1 + c + d) + d(1 + c + d) < (1 − b − c − d) − c(1 − b − c − d) − d(1 − b − c − d) ) (a + b + c + d)(1 + c + d) < (1 − b − c − d)(1 − c − d) a + b + c + d 1 + c − d ) · < 1. 1 − b − c − d 1 − c − d Since σ < 1, we have a + b + c + d σ + c + d h = · < 1. (15) σ − b − c − d σ − c − d We conclude that h < 1. By (8), (9) and (10) we conclude that in all cases (n−1)/2 ϱ(u , u ) ≤ h maxfϱ(u , u ), ϱ(u , u )g (16) n n+1 n−2 n−1 n−1 n for all n ≥ 2, where h is given by (15). Now it is easy to show by induction that from the inequality (16), we have (n−1)/2 ϱ(u , u ) ≤ h maxfϱ(u , u ), ϱ(x , x )g. n n+1 0 1 1 2 It follows that for any m > n > N, X N/2 ϱ(u , u ) ≤ ϱ(u , u ) ≤ maxfϱ(u , u ), ϱ(u , u )g. n m ξ ξ+1 0 1 1 2 1/2 h − h ξ=N In light of this evidence it is clear that fu g is a Cauchy sequence. Since M is complete and K is closed, fu g converges to some point w 2 K. From the way in which thefu g were chosen, there exists a subsequence n n fu g offu g such that u 2 P (k = 1, 2,··· , ). Then for n = 1, 2,··· , we have n n n k k ρ(u , T (w)) ≤ H(T (u ), T (u)) n n n n −1 n k k ≤ aϱ(u , w) + b maxfρ(u , T (u )), ρ(w, T (w))g n −1 n −1 n n −1 n k k k + c[ρ(u , T (w)) + ρ(w, T (u ))] + d[ρ(u , T (u )) n −1 n n n −1 n −1 n n −1 k k k k k k + ρ(w, T (w))] ≤ aϱ(u , w) + b maxfϱ(u , u ), ρ(w, T (w))g n −1 n −1 n n k k + c[ρ(u , T (w)) + ϱ(w, u )] + d[ϱ(u , u ) + ρ(w, T (w))]. n −1 n n n −1 n n k k k k Taking the limit as k ! ∞ yields ρ(w, T (w)) ≤ (b + c + d)ρ(w, T (w)), n n which implies, as b + c + d < 1, that ρ(w, T (w)) = 0. Since T (w) is closed, w 2 T (w). n n n Remark 3.1. If a, b, c, d ≥ 0, then a + 2b + (3 + a)(c + d) < 1 if and only if [(a + b + c + d)/(1− b − c − d)]· [(1 + c + d)/(1 − c − d)] < 1. Hence putting T = T for n = 1, 2,··· , in our theorem, the result of Ćirić [1] is obtained. n Fixed point theorem for a sequence of multivalued ... Ë 7 Because every Banach space is metrically convex, the following corollary applies to singlevalued mappings. Corollary 3.1. Let C be a Banach space and K be a nonempty closed subset of C. Let f (n = 1, 2,··· ) be a sequence of single-valued mappings of K into C. Suppose that there are nonnegative real numbers a, b, c, d with a + 2b + (3 + c)(c + b) < 1 such that kf (u)− f (v)k ≤ aku− vk + bfku− f (u)k +kv− f (v)kg + cfku− f (v)k +kv− f (u)kg + dfku− f (u)k +kv− f (v)kg i j i j j i i j for all u, v 2 K and for all i, j = 1, 2,··· . If f (∂K)  K for each n = 1, 2,··· , then there exists a unique common xed point z 2 K. Example 3.1. Let M = [0,∞) with the usual metric and K = [0, 2]. Dene T : K ! CB(M) as T (u) = n n nu [0, ]. Since T (0) = 0 for all n 2 N, we get T(0) = 0. For u > 0, we have (2n+1) " # " #! nu nv H(T (u), T (v)) = H 0, , 0, i j 2n + 1 2n + 1 ( " # " #! nu nv = max sup ρ 0, , 0, , 2n + 1 2n + 1 " # " #!) nv nu sup ρ 0, , 0, . (17) 2n + 1 2n + 1 " # " #! ( " #! nu nv nv sup ρ 0, , 0, = max ρ 0, 0, , 2n + 1 2n + 1 2n + 1 " #!) nu nv ρ , 0, 2n + 1 2n + 1 ( ) nu = max 0, 2n + 1 nu = . (18) 2n + 1 " # " #! ( " #! nv nu nu sup ρ 0, , 0, = max ρ 0, 0, , 2n + 1 2n + 1 2n + 1 " #!) nv nu ρ , 0, 2n + 1 2n + 1 ( ) nv n(v − u) = max , 2n + 1 2n + 1 n(v − u) = . (19) 2n + 1 Using (18) and (19) in (17) we get ( ) nu n(v − u) H(T (u), T (v)) = max , i j 2n + 1 2n + 1 n(v − u) = . 2n + 1 8 Ë David Aron and Santosh Kumar The same way, we calculate the following metrics ϱ(u, v) = ju − vj, ! ( ! h i nu ρ(u, T (u)) = ρ u, 0, = min ρ u, 0 , 2n + 1 !) nu u(n + 1) ρ u, = . 2n + 1 2n + 1 ! ( ! !) h i nv nv ρ(v, T (v)) = ρ y, 0, = min ρ v, 0 , ρ v, , 2n + 1 2n + 1 ( ) (n + 1)v (n + 1)v = min v, = . 2n + 1 2n + 1 ! ( ! !) h i nv nv ρ(u, T (v)) = ρ u, 0, = min ρ u, 0 , ρ u, , 2n + 1 2n + 1 ( ) (2u − v)n + u (2u − v)n + u = min u, = . 2n + 1 2n + 1 ! ( ! !) h i nu nu ρ(v, T (u)) = ρ v, 0, = min ρ v, 0 , ρ v, 2n + 1 2n + 1 ( ) (2v − u)n + uv (2v − u)n + v = min v, = . 2n + 1 2n + 1 Using (6) with the above inequalities, It follows that, ( ) n(v − u) u(n + 1) v(n + 1) ≤ aju − vj + b max , + 2n + 1 2n + 1 2n + 1 " # n(2u − v) + u n(2v−) + u c + + 2n + 1 2n + 1 " # u(n + 1) v(n + 1) d + , 2n + 1 2n + 1 v(n + 1) (u + v)(n + 1) ≤ aju − vj + b + c + 2n + 1 2n + 1 (u + v)(n + 1) d , 2n + 1 v(n + 1) (u + v)(n + 1) ≤ aju − vj + b + (c + d) . 2n + 1 2n + 1 By choosing the appropriate values of a, b, c, d and n, we conclude that the condition in (6) is satised. Further, since fT (u)g = [0, 1]  K for ∂K = f0, 2g, then T posses a xed point in K, which is a contradiction for a n=1 multivalued map. Next, we shall give a xed point theorem for a continuous multi-valued mapping, by weakening the condition (7), not requiring that the constant λ be less than 1. Denition 2.4 is required to prove our theorem. We will also need some literature available in the work of Assad [19]. Theorem 3.2. Let (M, ϱ) be a complete and metrically convex metric space and K a nonempty compact subset of M. LetfT g : K ! CB(M) be a sequence of continuous mappings satisfying for i ≠ j, where i, j = 1, 2,··· n=1 and for all u, v 2 K with u ≠ v, H(T (u), T (v)) ≤ aϱ(u, v) + b maxfρ(u, T (u)), ρ(v, T (v))g+ (20) i j i j Fixed point theorem for a sequence of multivalued ... Ë 9 c[ρ(u, T (v)) + ρ(v, T (u))] + d[ρ(u, T (u)) + ρ(v, T (v))], j i i j where a, b, c, d ≥ 0 and such that λ = a + 2b + (3 + a)(c + d) < 1. (21) IffT (u)g  K for each u 2 ∂K, then there exists w 2 K such that w 2 T (w). n n n=1 Proof. Consider f : K ! R dened by f (u) = ρ(u, T (u)), u 2 K. Since for each u, v 2 K, ρ(u, T (u)) ≤ i i i i ϱ(u, v) + ρ(v, T (u)); ρ(v, T (u)) ≤ ρ(v, T (v)) + H(T (u), T (v)) we have i i j i j jf (u) − f (v)j ≤ jρ(u, T (u)) − ρ(v, T (v))j i j i j = jρ(u, T (u)) − ρ(v, T (u)) + ρ(v, T (u)) − ρ(v, T (v))j i i i j ≤ jρ(u, T (u)) − ρ(v, T (u))j +jρ(v, T (u)) − ρ(v, T (v))j i i i j ≤ ϱ(u, v) +H(T (u), T (v)) i j Hence, the continuity offT g implies that f (u) is continuous on the compact set K. Let z 2 K such that f (z) = minff (u) : u 2 Kg, (22) i i such that ρ(z, T (z)) ≤ ρ(u, T (u)) (23) i i for each u 2 K. We are required to show that ϱ(z, T (z)) = 0. Assume to the contrary that f (u) > 0 for all i i u 2 K. Letfu g be a sequence in T (z) such that lim ϱ(z, u ) = ρ(z, T (z)). (24) n!∞ Suppose at rst that there exists an innite subsequence of fu g which is contained in a compact subset of K. Then there exists a subsequencefu g which converges to some u 2 K. Since T (z) is closed, u 2 T (z). i 0 i 0 i Thus ϱ(z, u ) = ρ(z, T (z)). From condition (20) we obtain, as ρ(z, T (z)) > 0 implies that z ≠ 0. 0 i i ρ(u , T (u )) ≤ H(T (z), T (u )) 0 j 0 i j 0 < aϱ(z, u ) + b maxfρ(z, T (z)), ρ(u , T (u ))g 0 i 0 j 0 + c[ρ(z, T (u )) + ρ(u , T (z))] + d[ρ(z, T (z)) + ρ(u , T (u )] j 0 0 i i 0 j 0 and hence, as ρ(z, T (u )) < ϱ(z, u ) + ρ(u , T (u )) = ρ(z, T (z)) + ρ(u , T (u )), j 0 0 0 j 0 i 0 j 0 we have ρ(u , T (u )) < aρ(z, T (z)) + bρ(u , T (u )) + (c + d)[ρ(z, T (z)) + ρ(u , T (u ))]. 0 j 0 i 0 j 0 i 0 j 0 Hence, using that ρ(z, T (z)) ≤ ρ(u , T (u )) and a + b + 2c + 2d ≤ 1, we have i 0 j 0 ρ(u , T (u )) < (a + b + 2c + 2d)ρ(u , T (u )) ≤ ρ(u , T (u )), 0 j 0 0 j 0 0 j 0 a contradiction. Suppose now that u 2 / K for all suciently large n. Since M is metrically convex and z 2 K, for each such u n n there exists v 2 ∂K such that ϱ(z, v ) + ϱ(v , u ) = ϱ(z, u ). (25) n n n n Since K is compact, we may suppose, for the sake of convenience, thatfv g converges to some v 2 ∂K. Since f is continuous, lim ρ(v , T (v )) = ρ(v , T (v )). (26) n n j 0 j 0 n!∞ 10 Ë David Aron and Santosh Kumar By using the triangle inequalities, (20) and (25), we have, as z ≠ v , ρ(v , T (v )) ≤ ϱ(v , u ) + ρ(u , T (v )) ≤ ϱ(z, u ) − ϱ(z, v ) +H(T (z), T (v )) n n n n n n n n n j j i j < ϱ(z, u ) − ϱ(z, v ) + aϱ(z, v ) + b maxfρ(z, T (z)), ρ(v , T (v ))g n n n n n i j + c[ρ(z, T (v )) + ρ(v , T (z))] + d[ρ(z, T (z)) + ρ(v , T (v ))] n n n n j i i j ≤ ϱ(z, u ) + b maxfρ(z, T (z)), ρ(v , T (v ))g n n n i j + c[ϱ(z, v ) + ρ(v , T (v )) + ϱ(v , u )] + d[ρ(z, T (z)) + ρ(v , T (v ))] n n n n n n n j i j = ϱ(z, u ) + bρ(v , T (v )) n n n + c[ϱ(z, u ) + ρ(v , T (v ))] + d[ρ(z, T (z)) + ρ(v , T (v ))]. n n n n n j i j Taking the limit as n increases towards innity and considering the limits (24) and (26) we obtain ρ(v , T (v )) ≤ ρ(z, T (z)) + bρ(v , T (v )) 0 j 0 i 0 j 0 + c[ρ(z, T (z)) + ρ(v , T (v ))] + d[ρ(z, T (z)) + ρ(v , T (v ))]. i 0 j 0 i 0 j 0 Hence ρ(v , T (v )) ≤ [(1 + c + d)/(1 − b − c − d)]ρ(z, T (z)). (27) 0 0 j i Since v 2 K, T (v )  K. Thus T (v ) is compact and so there exists w 2 T (v ) such that ϱ(v , w) = 0 j 0 j 0 j 0 0 ρ(v , T (v )). 0 j 0 From condition (20), as f (v ) > 0 implies that w ≠ y , we have j 0 0 ρ(w, T (w)) ≤ H(T (v ), T (w)) i j 0 i < aϱ(w, v ) + b maxfρ(w, T (w)), ρ(v , T (v ))g + cρ(v , T (w)) 0 i 0 j 0 0 i + d[ρ(v , T (v )) + ρ(w, T (w))]. 0 j 0 i Since ρ(v , T (w)) ≤ ϱ(v , w) + ρ(w, T (w)) = ρ(v , T (v )) + ρ(w, T (w)), we have 0 i 0 i 0 j 0 i ρ(w, T (w)) ≤ aρ(v , T (v )) + b maxfρ(w, T (w)), ρ(v , T (v ))g i 0 j 0 i 0 j 0 + (c + d)fρ(w, T (w)), ρ(v , T (v ))g i 0 j 0 and hence a + b + c + d 1 + c + d ρ(w, T (w)) ≤ · ρ(v , T (v )) i 0 j 0 a − c − d 1 − b − c − d = [(a + b + c + d)/(1 − c − d)]ρ(y , T (y )) 0 0 So by condition (27) we have a + b + c + d 1 + c + d ρ(w, T (w)) ≤ · ρ(z, T (z)) (28) i i 1 − c − d 1 − b − c − d Since (a + b + c + d)(1 + c + d) a + 2b + (3 + a)(d + c) − b − (c + d)(2 − b − c − d) (1 − c − d)(1 − b − c − d) 1 − b − (c + d)(2 − b − c − d) Taking into consideration condition (21) we get a + b + c + d 1 + c + d · < 1. 1 − c − d 1 − b − c − d Thus by condition (28) we have ρ(w, T (w)) ≤ ρ(z, T (z)), i i a contradiction with condition (23). Therefore, ρ(z, T (z)) = 0. Hence, as T (z) is closed, z 2 T (z). i i i The following corollary is obtained for Banach spaces: Fixed point theorem for a sequence of multivalued ... Ë 11 Corollary 3.2. Let K be a nonempty compact subset of a Banach space C and f be a sequence of single-valued mappings of K into C for all n 2 N. Suppose that there are nonnegative reals a, b, c, d with a+2b+(3+c)(c+b) < 1 such that for any x, y 2 K and for all i, j = 1, 2,··· , kf (u)− f (v)k ≤ aku− vk + bfku− f (u)k +kv− f (v)kg + cfku− f (v)k +kv− f (u)kg + dfku− f (u)k +kv− f (v)kg i j i j j i i j If f (∂K)  K for each n = 1, 2,··· , then there exists a unique common xed point of f in K. n n Example 3.2. Let M = [0,∞) with the usual metric and K = [0, 1]. Dene fT g : K ! CB(M) as T (u) = n n nu [0, ] for all u 2 K. Since T (0) = 0 for all n 2 N, we get T(0) = 0. For u > 0, we have (2n+1) nu u lim = lim = , n!∞ (2n + 1) n!∞ 2 + 1/n which yields T(u) = u/2 for u > 0. HencefT g converges pointwise to g where 0 if u = 0 g(u) = u/2 if u > 0 and u is continuous for all u 2 [0, 1]. Thus, K is compact. Moreover, we see that T (∂K) = T (f0, 1g) = f0, 1g n n =) T (0) and T (1) are subset of K, n n which implies fT g satises T (∂K)  K. By direct computation as in Example 3.1 we can show that all condi- n n tions of Theorem 3.2 are satised. Therefore, F(T ) = 0. Referring from Theorem 2.3, the following result can also be obtained: Theorem 3.3. Let (M, ϱ) be a complete and metrically convex metric space, K a nonempty closed subset of M and let S = fT g be a family of multivalued mappings of K into CB(M). Suppose that there exists some T 2 S j j2J i such that for each T 2 S H(T (u), T (v)) ≤ a ϱ(u, v) + b maxfρ(u, T (u)), ρ(v, T (v))g+ i j j j i j (29) c [ρ(u, T (v)) + ρ(v, T (u))] + d [ρ(u, T (u)) + ρ(v, T (v))], j j i j i j where a , b , c , d ≥ 0 are positive real numbers satisfying j j j j λ = a + 2b + (3 + a )(c + d ) < 1. j j j j j j If T u  K for each u 2 ∂K and for each T 2 S, then a family S has a common xed point in K such that j j w 2 T (w) for all T 2 S. j j Proof. Suppose that T is an arbitrary but xed member of S. Then, from Theorem ?? with T = T there exists j j 0 0 a point in K, say w, which is a xed point of T . Let T 2 S, T ≠ T , be arbitrary. Then from equation (29) we have j j j ρ(w, T (w)) ≤ H(T (w), T (w)) j i j ≤ b maxf0, ρ(w, T (w))g + (c + d )[ρ(w, T (w)) + 0], j j j j j and hence (1 − b − c − d )ρ(w, T (w)) ≤ 0. j j j j Since b + c + d ≤ λ < 1, we have ρ(w, T (w)) = 0. Hence w 2 T (w). Clearly, w is a xed point of a family j j j j j j S. 12 Ë David Aron and Santosh Kumar 4 Conclusion The main contribution of this study to xed point theory is the xed point result given in Theorem 3.1 , 3.2 and Theorem 3.3. These theorems proved for sequence of multivalued mappings which satisfy certain re- quirements in complete metric spaces. These results generalizes many recent known results in convex metric spaces in the literature. The results proved here gives generalization of the results due to Ćirić [1]. Suitable examples are given to support the results proved herein. Conict of interest: Both authors declare that they have no conicts of interest. Research involving human participants and/or animals: The authors declare that there is no human participants and / or animals involved in this research. Funding: Authors declare that there is no funding available for this research. Acknowledgments: The authors would like to thank and acknowledge the learned reviewer for their valuable comments. References [1] Ćirić, L. J., Fixed point theorems for set-valued non-self mappings on convex metric spaces, Math. Balk. 20, 207–217, (2006). [2] Nadler, S. B., Multivalued contraction mappings, Pac. J. Math. 30, 475–488, (1969). [3] Assad, N. A. and Kirk, W. A., Fixed point theorems for multivalued mappings of contractive type, Pac. J. Math. 43, 553–562, (1972). [4] Rhoades, B. E., A xed point theorem for a multi-valued non-self mapping, Comment. Math. Univ. Caroline, 37, 401–404, (1996). [5] Imdad, M., Ahmad, A, and Kumar, S., On nonlinear non-self hybrid contraction, Radovi Matematicki., 10(2), 233-244, (2001) (MR1981096), (New Name: Sarajevo Journal of Mathematics). [6] Imdad, M. and Kumar, S., Rhoades-type xed-point theorems for a pair of non-self mappings, Jour. of computers Math. Appl. 46, 919–927, (2003) (MR2020449 (2004h:47082)). [7] Du, W. S., Karapınar, E., and Shahzad, N., The study of xed point theory for various multivalued non-self-maps, Abstract and Applied Analysis (Vol. 2013), 2013. [8] Kumam, P., Aydi, H., Karapınar, E., and Sintunavarat, W., Best proximity points and extension of Mizoguchi-Takahashi’s xed point theorems, Fixed Point Theory and Applications, 2013(1), 1-15, (2013). [9] Ali, M. U., Kamran, T., and Karapinar, E., A new approach to (α, ψ)− contractive nonself multivalued mappings, Journal of Inequalities and Applications, 2014(1), 1-9, (2014). [10] Dhage, B. C., Dolhare, U. P., and ELSome, Adrian Petrus, Common Fixed Point Theorems for Sequences of Nonself Multivalued operators in Metrically Convex Metric Spaces, Fixed Point Theory, Volume 4, No. 2, 143-158, (2003). [11] Kumar, S., Approximating xed point of generalized non-expansive mappings, International J. of Math. Sci. & Engg. Appls., Vol. 6, 361–367, (2012). [12] Chauhan, S., Imdad, M., Karapınar, E., and Fisher, B., An integral type xed point theorem for multi-valued mappings em- ploying strongly tangential property, Journal of the Egyptian Mathematical Society, 22(2), 258-264, (2014). [13] Ali, M. U., Kamran, T., and Karapinar, E., Discussion on α− Strictly Contractive Nonself Multivalued Maps, Vietnam Journal of Mathematics, 44(2), 441-447, (2016). [14] Ali, M. U., Kamran, T., and Karapınar, E., Further discussion on modied multivalued alpha–contractive type mapping, Filo- mat, 29(8), 1893-1900, (2015). [15] Kreyszig, E., Introductory functional analysis with applications, John Wily & Sons.Inc., 1978 [16] Khan, M. S., Common xed point theorems for multivalued mappings, Pacic J. Math. 95, 337–347, (1981). [17] Ćirić, L. J., Fixed points for generalized multi-valued mappings, Mat. Vesnik. 24, 265–272, (1972). [18] Itoh, S., Multivalued generalized contractions and xed point theorems, Comment, Math. Univ. Carolinae 18, 247-–258, (1977). [19] Assad, N. A., Fixed point theorems for set valued transformations on compact set, Boll. Un. Math. Ital. 4, 1–7, (1973). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Topological Algebra and its Applications de Gruyter

Fixed point theorem for a sequence of multivalued nonself mappings in metrically convex metric spaces

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Topol. Algebra Appl. 2022; 10:1–12 Research Article Open Access David Aron and Santosh Kumar* Fixed point theorem for a sequence of multivalued nonself mappings in metrically convex metric spaces https://doi.org/10.1515/taa-2020-0108 Received 29 September, 2021; revised 6 December, 2021; accepted 2 January, 2022 Abstract: In this paper, a common xed point theorem is demonstrated for a sequence of multivalued map- pings which satisfy certain requirements in complete metric spaces. The results proved here will generalize and extend the results due to Ćirić [1]. Suitable examples are given at the end to support the results proved herein. Keywords: Multivalued mapping, non-self mapping, metrically convex metric spaces. PACS: 47H10, 54H25 1 Introduction In 1969, Nadler [2] studied xed points using the Hausdor metric for multivalued mappings. Assad and Kirk [3] extended the Banach contraction theorem from self mappings by giving its proof for non-self mappings. These results were employed by Rhoades [4] and a xed point theorem for a multivalued non-self mapping was proved. Ćirić [1] generalized and extended the theorem of Rhoades [4] by proving a xed point theorem for a continuous multivalued mapping. Imdad et al. [5] gave some common xed point theorems for nonself generalized hybrid contraction where they generalized several well known results in the literature. Later Imdad and Kumar [6] proved results for generalized T− contractive mappings. In 2013, Du et al. [7] established some xed/coincidence point theorems for multivalued non-selfmaps in the context of complete metric spaces. Kumam et al. [8] introduced a multi- valued cyclic generalized contraction by extending the Mizoguchi and Takahashi’s contraction for non-self mappings. In 2014, Ali et al. [9] introduced the notions of α− admissible and α−ψ− contractive type condition for nonself multivalued mappings. Dhage et al. [10] proved some common xed point theorems for sequences of nonself multivalued operators dened on a closed subset of a metrically convex metric space. Later on several authors worked in this direction. For more related results one can see [5, 6, 11–14] and the references therein. In this paper, throughout the discussion we will take ϱ as the metric distance between two points while ρ(x, A) denotes the distance from point x to a set A. David Aron: Department of Mathematics, College of Natural and Applied Sciences, University of Dar es Salaam, Tanzania; E-mail: d.aron.da6@gmail.com *Corresponding Author: Santosh Kumar: Department of Mathematics, College of Natural and Applied Sciences, University of Dar es Salaam, Tanzania; E-mail: drsengar2002@gmail.com Open Access. © 2022 Manuel David Aron and Santosh Kumar, published by De Gruyter. This work is licensed under the Creative Commons Attribution alone 4.0 License. 2 Ë David Aron and Santosh Kumar 2 Preliminaries In order to identify the major conclusions of this work, the following denitions and lemmas will be required. Denition 2.1. [2] Let (M, ϱ) be a metric space and CB(M) be the collection of all nonempty closed and bounded subsets of M. For A, B 2 CB(M), and u 2 M, dene ρ(u, A) = inffϱ(u, a) : a 2 Ag and H(A, B) = max sup ρ(a, B), sup ρ(A, b) . a2A b2B It can be easily veried that H is a metric on CB(M). H is called the Hausdor metric induced by ϱ. Denition 2.2. [15]. (i) A sequence fu g in a metric space (M, ϱ) is said to converge or to be convergent if there is an u 2 M such that lim ϱ(u , u) = 0. n!∞ (ii) A sequence fu g in a metric space (M, ϱ) is said to be Cauchy sequence if for every ϵ > 0 there is an N = N(ϵ) such that ϱ(u , u ) < ϵ, n m for every m, n > N. (iii) A metric space (M, ϱ) is said to be complete if every Cauchy sequence in M converges to an element of M. Convex property plays an important role in many areas such as functional analysis, optimization and control theory. This phenomena is well utilized by the researchers in metric xed point theory. The following is Assad and Kirk’s [3] denition of a metrically convex metric space. Denition 2.3. [3]. A metric space (M, ϱ) is said to be metrically convex if for any u, v 2 M with u ≠ v, there exists a point z 2 M,(u ≠ z ≠ v) such that ϱ(u, v) = ϱ(u, z) + ϱ(z, v). The following result is taken from Assad [3] where ∂K denotes the boundary of K. Lemma 2.1. [3]. If K is a nonempty closed subset of the complete and convex metric space M and if u 2 K, v 2 / K, then there exists a point z 2 ∂K, such that ϱ(u, v) = ϱ(u, z) + ϱ(z, v). The following result was proved by Assad and Kirk [3] as they gave sucient condition for a multi-valued non-self mappings from K into CB(M) to have a xed point. Theorem 2.1. [3]. Let M be a complete and convex metric space and K be a nonempty closed subset of M, and T : K ! CB(M) a contraction mapping such that ρ(T(u), T(v)) ≤ aϱ(u, v), (1) where a < 1. If Tu  K for each u 2 ∂K then there exists u 2 K such that u 2 T(u ) (i.e. T has a xed point 0 0 0 in K). The following denition is due to Khan [16] is useful in proving the main results: Denition 2.4. [16] Let K be a nonempty subset of a metric space (M, ϱ). A mapping T : K ! CB(M) is said to be continuous at u 2 K if for any ϵ > 0, there exists a d = d(ϵ) > 0 such that H(Tu, Tu ) < ϵ, whenever 0 0 ϱ(u, u ) < d. If T is continuous at every point of K, we say that T continuous at K. 0 Fixed point theorem for a sequence of multivalued ... Ë 3 Now, we present the following lemma which nds key applications in the proof our theorems in a convex metric space. Lemma 2.2. [17]. Let (M, ϱ) be a metric space. If A, B 2 CB(M) and α 2 A, then for any positive number σ < 1 there exists β = β(α) in B such that σϱ(α, β) ≤ H(A, B). (2) Rhoades [4] generalized the result of Itoh [18] by proving the following result for a multivalued non-self map- ping in a metrically convex complete metric space. Theorem 2.2. [4]. Let (M, ϱ) be a complete and metrically convex metric space and K be a nonempty closed subset of M. Let T : K ! CB(M) satisfy the following contractive condition: H(Tu, Tv) ≤ aϱ(u, v) + b maxfρ(u, Tu), ρ(v, Tv)g + c[ρ(u, Tv) + ρ(v, Tu)], (3) for all u, v 2 K, where a, b, c ≥ 0 and such that λ = [(1 + a + c)/(1 − b − c)] · [(a + b + c)/(1 − c)] < 1. If Tu  K for each u 2 ∂K, then there exists a z 2 K such that z 2 Tz. In the following theorem a wider class of multivalued non-self mappings than those in [4] and in [18] was considered. Theorem 2.3. [1]. Let (M, ϱ) be a complete and metrically convex metric space and K be a nonempty closed subset of M. Let T : K ! CB(M) be a mapping such that H(Tu, Tv) ≤ aϱ(u, v) + b maxfρ(u, Tu), ρ(v, Tv)g + c[ρ(u, Tv) + ρ(v, Tu)] + d[ρ(u, Tu) + ρ(v, Tv)] (4) where a, b, c, d ≥ 0 are such that λ = a + 2b + (3 + a)(c + d). (5) If Tu  K for each u 2 ∂K, then there exists w 2 K such that w 2 Tw. This paper aims to modify Theorem 2.3 so that it can be applied to a sequence of multivalued mappings in a metrically convex metric space. In doing so, our results will generalize several existing results in literature. 3 Main Results In this section, we will prove common xed point theorem for a sequence of multi-valued non-self mappings. An extended and generalized version of the Theorem 2.3 is as follows: Theorem 3.1. Let (M, ϱ) be a complete and metrically convex metric space and K be a nonempty closed subset of M. LetfT g : K ! CB(M) be a sequence of multivalued mappings satisfying for i ≠ j, where i, j = 1, 2,··· , n=1 H(T (u), T (v)) ≤ aϱ(u, v) + b maxfρ(u, T (u)), ρ(v, T (v))g+ i j i j (6) c[ρ(u, T (v)) + ρ(v, T (u))] + d[ρ(u, T (u)) + ρ(v, T (v))], j i i j for all u, v 2 K, where a, b, c, d ≥ 0 are such that λ = a + 2b + (3 + a)(c + d) < 1. (7) ∞ ∞ IffT (u)g  K for each u 2 ∂K, then the sequencefT g has a common xed point in K (i.e. there exists n n n=1 n=1 w 2 K such that w 2 T (w)). n 4 Ë David Aron and Santosh Kumar Proof. The considered case is when T (∂K)  K, but not necessarily T (K)  K. The case T (K)  K is much n n n more simpler. In this case it follows that condition (7) can be simplied to λ = a + b + 2c + 2d < 1. Two sequences fu g and fu g in K and M, respectively, will be constructed in the following manner. Let 0 −1 α u 2 K and u = u 2 T (u ) be arbitrary. Let α be any xed number such that 0 < α < 2 . Put σ = λ . Then 0 1 1 0 from condition (7), σ < 1. By Lemma 2.2 we can choose u 2 T (u ) such that 2 1 0 0 σϱ(u , u ) ≤ H(T (u ), T (u )). 1 2 1 0 2 1 0 0 0 If u 2 K, put u = u . If u 2 / K, then, as M is metrically convex, we can choose u 2 ∂K such that 2 2 2 2 2 0 0 ϱ(u , u ) + ϱ(u , u ) = ϱ(u , u ). 1 2 2 2 1 2 Let u 2 T (u ) be such that 3 2 0 0 σϱ(u , u ) ≤ H(T (u ), T (u )). 2 1 3 2 2 3 By induction we may obtain sequencesfu g andfu g such that for n = 1, 2,··· , (i) u 2 T (u ) n n−1 0 0 (ii) σϱ(u , u ) ≤ H(T (u ), T (u )), where n n n n−1 n+1 n+1 0 0 (iii) u = u , if u 2 K, or n+1 n+1 n+1 0 0 0 (iv) u 2 ∂K, and ϱ(u , u ) + ϱ(u , u ) = ϱ(u , u ), if u 2 / K, for all n. Now let n n n+1 n+1 n+1 n+1 n+1 n+1 P = fu 2 fu g : u = u , ξ = 1, 2,···g; ξ ξ ξ Q = fu 2 fu g : u ≠ u , ξ = 1, 2,···g. ξ ξ Notice that if u 2 Q for some n, then u , u 2 P, since two consecutive terms offu g cannot be in Q. n n−1 n+1 n Here, the intention is to estimate the distance ϱ(u , u ). Thus, these cases must be considered. n n+1 Case 1. u 2 P and u 2 P. From assumption (ii) and (6) taking u = u and v = u we get n n+1 n−1 n 0 0 σϱ(u , u ) = σϱ(u , u ) ≤ H(T (u ), T (u )) ≤ aϱ(u , u ) n n n n n+1 n n+1 n−1 n+1 n−1 + b maxfρ(u , T (u )), ρ(u , T (u ))g + c[ρ(u , T (u )) n n n n n−1 n−1 n+1 n−1 n+1 + ρ(u , T (u ))] + d[ρ(u , T (u )) + ρ(u , T (u ))] n n n n n n−1 n−1 n−1 n+1 ≤ aϱ(u , u ) + b maxfϱ(u , u ), ϱ(u , u )g n n n n−1 n−1 n+1 + c[ϱ(u , u ) + ϱ(u , u )] + d[ϱ(u , u ) + ϱ(u , u )] n n n n n−1 n+1 n−1 n+1 ≤ aϱ(u , u ) + b maxfϱ(u , u ), ϱ(u , u )g n n n n−1 n−1 n+1 + c[ϱ(u , u ) + ϱ(u , u )] + d[ϱ(u , u ) + ϱ(u , u )]. n n n n n−1 n+1 n−1 n+1 It follows that σϱ(u , u ) ≤ (a + c + d)ϱ(u , u ) + b maxfϱ(u , u ), (8) n n n n+1 n−1 n−1 ϱ(u , u )g + (c + d)ϱ(u , u ). n n n+1 n+1 If the maximum of the coecient of b is ϱ(u , u ), then from (8) and by (7) we get n n+1 σϱ(u , u ) ≤ (a + c + d)ϱ(u , u ) + (b + c + d)ϱ(u , u ) n n n n+1 n−1 n+1 ≤ (a + b + 2c + 2d)ϱ(u , u ) ≤ λϱ(u , u ) ≤ λ ϱ(u , u ) n n+1 n n+1 n n+1 = σϱ(u , u ), n+1 which is a contradiction. Therefore, from (8) we obtain σϱ(u , u ) ≤ (a + b + c + d)ϱ(u , u ) + (c + d)ϱ(u , u ) n n n n+1 n−1 n+1 and hence a + b + c + d ϱ(u , u ) ≤ ϱ(u , u ). (9) n n n+1 n−1 σ − c − d Fixed point theorem for a sequence of multivalued ... Ë 5 Case 2. u 2 P and u 2 Q. Then by (iv) n+1 0 0 0 0 0 ϱ(u , u ) = ϱ(u , u ) − ϱ(u , u ) ≤ ϱ(u , u ) = ϱ(u , u ) n n n n+1 n+1 n+1 n+1 n+1 n n+1 Following the identical reasoning as in Case 1, we obtain a + b + c + d ϱ(u , u ) ≤ ϱ(u , u ). n n n+1 n−1 σ − c − d Case 3. u 2 Q and u 2 P. By applying the triangle inequality we have n n+1 0 0 0 0 0 ϱ(u , u ) ≤ ϱ(u , u ) + ϱ(u , u ) = ϱ(u , u ) + ϱ(u , u ). n n n n+1 n n n+1 n n n+1 Then from assumption (ii) and (6) we have 0 0 0 0 σϱ(u , u ) ≤ σϱ(u , u ) + σϱ(u , u ) ≤ σϱ(u , u ) +H(T (u ), T (u )) n n n n n n+1 n n n+1 n n−1 n+1 ≤ σϱ(u , u ) + aϱ(u , u ) + b maxfρ(u , T (u )), ρ(u , T (u ))g n n n n n n n−1 n−1 n−1 n+1 + c[ρ(u , T (u )) + ρ(u , T (u ))] + d[ρ(u , T (u )) + ρ(u , T (u ))] n n n n n n n−1 n+1 n−1 n−1 n−1 n+1 0 0 0 ≤ σϱ(u , u ) + aϱ(u , u ) + b maxfϱ(u , u ), ϱ(u , u )g n n n n n−1 n−1 n n+1 0 0 0 0 +c[ϱ(u , u ) + ϱ(u , u )] + d[ϱ(u , u ) + ϱ(u , u )]. (10) n n n−1 n+1 n n−1 n n+1 Since two consecutive terms offu g cannot be in Q, u 2 P. Then by condition (iv) n−1 0 0 ϱ(u , u ) + ϱ(u , u ) = ϱ(u , u ) n n n−1 n n−1 n and hence, α < λ = σ, we have 0 0 σϱ(u , u ) + aϱ(u , u ) ≤ σϱ(u , u ). n n n−1 n n−1 n Additionally, by the triangle inequality, 0 0 c[ϱ(u , u ) + ϱ(u , u )] ≤ c[ϱ(u , u ) + ϱ(u , u ) + ϱ(u , u )] n−1 n+1 n n n−1 n n n+1 n n = cϱ(u , u ) + cϱ(u , u ). n−1 n n n+1 0 0 Suppose that ϱ(u , u ) < ϱ(u , u ) = ϱ(u , u ). Then from (10) we get n n n−1 n n+1 n+1 0 0 σϱ(u , u ) ≤ σϱ(u , u ) + bϱ(u , u ) + (c + d)ϱ(u , u ) + (c + d)ϱ(u , u ) n n n n+1 n−1 n n+1 n−1 n n+1 and hence σ + c + d ϱ(u , u ) ≤ ϱ(u , u ). (11) n n+1 n−1 n σ − b − c − d If ϱ(u , u ) ≥ ϱ(u , u ), then, as 1 ≤ [(σ + c + d)/(σ − b − c − d)], we have again n−1 n n n+1 σ + c + d ϱ(u , u ) ≤ ϱ(u , u ). (12) n+1 n−1 n σ − b − c − d Therefore, (11) holds for all n. 0 0 0 0 Since u = u , it follows that ϱ(u , u ) = ϱ(u , u ). Then, as in Case 1, we have n−1 n−1 n n n−1 n−1 a + b + c + d ϱ(u , u ) ≤ ϱ(u , u ). (13) n−1 n n−2 n−1 σ − c − d By (11) and (13) we obtain a + b + c + d σ + c + d ϱ(u , u ) ≤ · ϱ(u , u ). (14) n n+1 n−2 n−1 σ − c − d σ − b − c − d 6 Ë David Aron and Santosh Kumar To verify this, recall from the inequality (7) which yields a + 2b + (3 + a)(c + d) < 1 ) a + 2b + 3c + 3d + ac + ad < 1 ) a + ac + ad + b + c + d < 1 − b − 2c − 2d ) a + ac + ad + b + c + d < 1 − b − c − d − c − d 2 2 ) a + ac + ad + b + bc + bd + c + c + cd + d + dc + d 2 2 < 1 − b − c − d − c + cb + c + cd − d + db + dc + d ) a(1 + c + d) + b(1 + c + d) + c(1 + c + d) + d(1 + c + d) < (1 − b − c − d) − c(1 − b − c − d) − d(1 − b − c − d) ) (a + b + c + d)(1 + c + d) < (1 − b − c − d)(1 − c − d) a + b + c + d 1 + c − d ) · < 1. 1 − b − c − d 1 − c − d Since σ < 1, we have a + b + c + d σ + c + d h = · < 1. (15) σ − b − c − d σ − c − d We conclude that h < 1. By (8), (9) and (10) we conclude that in all cases (n−1)/2 ϱ(u , u ) ≤ h maxfϱ(u , u ), ϱ(u , u )g (16) n n+1 n−2 n−1 n−1 n for all n ≥ 2, where h is given by (15). Now it is easy to show by induction that from the inequality (16), we have (n−1)/2 ϱ(u , u ) ≤ h maxfϱ(u , u ), ϱ(x , x )g. n n+1 0 1 1 2 It follows that for any m > n > N, X N/2 ϱ(u , u ) ≤ ϱ(u , u ) ≤ maxfϱ(u , u ), ϱ(u , u )g. n m ξ ξ+1 0 1 1 2 1/2 h − h ξ=N In light of this evidence it is clear that fu g is a Cauchy sequence. Since M is complete and K is closed, fu g converges to some point w 2 K. From the way in which thefu g were chosen, there exists a subsequence n n fu g offu g such that u 2 P (k = 1, 2,··· , ). Then for n = 1, 2,··· , we have n n n k k ρ(u , T (w)) ≤ H(T (u ), T (u)) n n n n −1 n k k ≤ aϱ(u , w) + b maxfρ(u , T (u )), ρ(w, T (w))g n −1 n −1 n n −1 n k k k + c[ρ(u , T (w)) + ρ(w, T (u ))] + d[ρ(u , T (u )) n −1 n n n −1 n −1 n n −1 k k k k k k + ρ(w, T (w))] ≤ aϱ(u , w) + b maxfϱ(u , u ), ρ(w, T (w))g n −1 n −1 n n k k + c[ρ(u , T (w)) + ϱ(w, u )] + d[ϱ(u , u ) + ρ(w, T (w))]. n −1 n n n −1 n n k k k k Taking the limit as k ! ∞ yields ρ(w, T (w)) ≤ (b + c + d)ρ(w, T (w)), n n which implies, as b + c + d < 1, that ρ(w, T (w)) = 0. Since T (w) is closed, w 2 T (w). n n n Remark 3.1. If a, b, c, d ≥ 0, then a + 2b + (3 + a)(c + d) < 1 if and only if [(a + b + c + d)/(1− b − c − d)]· [(1 + c + d)/(1 − c − d)] < 1. Hence putting T = T for n = 1, 2,··· , in our theorem, the result of Ćirić [1] is obtained. n Fixed point theorem for a sequence of multivalued ... Ë 7 Because every Banach space is metrically convex, the following corollary applies to singlevalued mappings. Corollary 3.1. Let C be a Banach space and K be a nonempty closed subset of C. Let f (n = 1, 2,··· ) be a sequence of single-valued mappings of K into C. Suppose that there are nonnegative real numbers a, b, c, d with a + 2b + (3 + c)(c + b) < 1 such that kf (u)− f (v)k ≤ aku− vk + bfku− f (u)k +kv− f (v)kg + cfku− f (v)k +kv− f (u)kg + dfku− f (u)k +kv− f (v)kg i j i j j i i j for all u, v 2 K and for all i, j = 1, 2,··· . If f (∂K)  K for each n = 1, 2,··· , then there exists a unique common xed point z 2 K. Example 3.1. Let M = [0,∞) with the usual metric and K = [0, 2]. Dene T : K ! CB(M) as T (u) = n n nu [0, ]. Since T (0) = 0 for all n 2 N, we get T(0) = 0. For u > 0, we have (2n+1) " # " #! nu nv H(T (u), T (v)) = H 0, , 0, i j 2n + 1 2n + 1 ( " # " #! nu nv = max sup ρ 0, , 0, , 2n + 1 2n + 1 " # " #!) nv nu sup ρ 0, , 0, . (17) 2n + 1 2n + 1 " # " #! ( " #! nu nv nv sup ρ 0, , 0, = max ρ 0, 0, , 2n + 1 2n + 1 2n + 1 " #!) nu nv ρ , 0, 2n + 1 2n + 1 ( ) nu = max 0, 2n + 1 nu = . (18) 2n + 1 " # " #! ( " #! nv nu nu sup ρ 0, , 0, = max ρ 0, 0, , 2n + 1 2n + 1 2n + 1 " #!) nv nu ρ , 0, 2n + 1 2n + 1 ( ) nv n(v − u) = max , 2n + 1 2n + 1 n(v − u) = . (19) 2n + 1 Using (18) and (19) in (17) we get ( ) nu n(v − u) H(T (u), T (v)) = max , i j 2n + 1 2n + 1 n(v − u) = . 2n + 1 8 Ë David Aron and Santosh Kumar The same way, we calculate the following metrics ϱ(u, v) = ju − vj, ! ( ! h i nu ρ(u, T (u)) = ρ u, 0, = min ρ u, 0 , 2n + 1 !) nu u(n + 1) ρ u, = . 2n + 1 2n + 1 ! ( ! !) h i nv nv ρ(v, T (v)) = ρ y, 0, = min ρ v, 0 , ρ v, , 2n + 1 2n + 1 ( ) (n + 1)v (n + 1)v = min v, = . 2n + 1 2n + 1 ! ( ! !) h i nv nv ρ(u, T (v)) = ρ u, 0, = min ρ u, 0 , ρ u, , 2n + 1 2n + 1 ( ) (2u − v)n + u (2u − v)n + u = min u, = . 2n + 1 2n + 1 ! ( ! !) h i nu nu ρ(v, T (u)) = ρ v, 0, = min ρ v, 0 , ρ v, 2n + 1 2n + 1 ( ) (2v − u)n + uv (2v − u)n + v = min v, = . 2n + 1 2n + 1 Using (6) with the above inequalities, It follows that, ( ) n(v − u) u(n + 1) v(n + 1) ≤ aju − vj + b max , + 2n + 1 2n + 1 2n + 1 " # n(2u − v) + u n(2v−) + u c + + 2n + 1 2n + 1 " # u(n + 1) v(n + 1) d + , 2n + 1 2n + 1 v(n + 1) (u + v)(n + 1) ≤ aju − vj + b + c + 2n + 1 2n + 1 (u + v)(n + 1) d , 2n + 1 v(n + 1) (u + v)(n + 1) ≤ aju − vj + b + (c + d) . 2n + 1 2n + 1 By choosing the appropriate values of a, b, c, d and n, we conclude that the condition in (6) is satised. Further, since fT (u)g = [0, 1]  K for ∂K = f0, 2g, then T posses a xed point in K, which is a contradiction for a n=1 multivalued map. Next, we shall give a xed point theorem for a continuous multi-valued mapping, by weakening the condition (7), not requiring that the constant λ be less than 1. Denition 2.4 is required to prove our theorem. We will also need some literature available in the work of Assad [19]. Theorem 3.2. Let (M, ϱ) be a complete and metrically convex metric space and K a nonempty compact subset of M. LetfT g : K ! CB(M) be a sequence of continuous mappings satisfying for i ≠ j, where i, j = 1, 2,··· n=1 and for all u, v 2 K with u ≠ v, H(T (u), T (v)) ≤ aϱ(u, v) + b maxfρ(u, T (u)), ρ(v, T (v))g+ (20) i j i j Fixed point theorem for a sequence of multivalued ... Ë 9 c[ρ(u, T (v)) + ρ(v, T (u))] + d[ρ(u, T (u)) + ρ(v, T (v))], j i i j where a, b, c, d ≥ 0 and such that λ = a + 2b + (3 + a)(c + d) < 1. (21) IffT (u)g  K for each u 2 ∂K, then there exists w 2 K such that w 2 T (w). n n n=1 Proof. Consider f : K ! R dened by f (u) = ρ(u, T (u)), u 2 K. Since for each u, v 2 K, ρ(u, T (u)) ≤ i i i i ϱ(u, v) + ρ(v, T (u)); ρ(v, T (u)) ≤ ρ(v, T (v)) + H(T (u), T (v)) we have i i j i j jf (u) − f (v)j ≤ jρ(u, T (u)) − ρ(v, T (v))j i j i j = jρ(u, T (u)) − ρ(v, T (u)) + ρ(v, T (u)) − ρ(v, T (v))j i i i j ≤ jρ(u, T (u)) − ρ(v, T (u))j +jρ(v, T (u)) − ρ(v, T (v))j i i i j ≤ ϱ(u, v) +H(T (u), T (v)) i j Hence, the continuity offT g implies that f (u) is continuous on the compact set K. Let z 2 K such that f (z) = minff (u) : u 2 Kg, (22) i i such that ρ(z, T (z)) ≤ ρ(u, T (u)) (23) i i for each u 2 K. We are required to show that ϱ(z, T (z)) = 0. Assume to the contrary that f (u) > 0 for all i i u 2 K. Letfu g be a sequence in T (z) such that lim ϱ(z, u ) = ρ(z, T (z)). (24) n!∞ Suppose at rst that there exists an innite subsequence of fu g which is contained in a compact subset of K. Then there exists a subsequencefu g which converges to some u 2 K. Since T (z) is closed, u 2 T (z). i 0 i 0 i Thus ϱ(z, u ) = ρ(z, T (z)). From condition (20) we obtain, as ρ(z, T (z)) > 0 implies that z ≠ 0. 0 i i ρ(u , T (u )) ≤ H(T (z), T (u )) 0 j 0 i j 0 < aϱ(z, u ) + b maxfρ(z, T (z)), ρ(u , T (u ))g 0 i 0 j 0 + c[ρ(z, T (u )) + ρ(u , T (z))] + d[ρ(z, T (z)) + ρ(u , T (u )] j 0 0 i i 0 j 0 and hence, as ρ(z, T (u )) < ϱ(z, u ) + ρ(u , T (u )) = ρ(z, T (z)) + ρ(u , T (u )), j 0 0 0 j 0 i 0 j 0 we have ρ(u , T (u )) < aρ(z, T (z)) + bρ(u , T (u )) + (c + d)[ρ(z, T (z)) + ρ(u , T (u ))]. 0 j 0 i 0 j 0 i 0 j 0 Hence, using that ρ(z, T (z)) ≤ ρ(u , T (u )) and a + b + 2c + 2d ≤ 1, we have i 0 j 0 ρ(u , T (u )) < (a + b + 2c + 2d)ρ(u , T (u )) ≤ ρ(u , T (u )), 0 j 0 0 j 0 0 j 0 a contradiction. Suppose now that u 2 / K for all suciently large n. Since M is metrically convex and z 2 K, for each such u n n there exists v 2 ∂K such that ϱ(z, v ) + ϱ(v , u ) = ϱ(z, u ). (25) n n n n Since K is compact, we may suppose, for the sake of convenience, thatfv g converges to some v 2 ∂K. Since f is continuous, lim ρ(v , T (v )) = ρ(v , T (v )). (26) n n j 0 j 0 n!∞ 10 Ë David Aron and Santosh Kumar By using the triangle inequalities, (20) and (25), we have, as z ≠ v , ρ(v , T (v )) ≤ ϱ(v , u ) + ρ(u , T (v )) ≤ ϱ(z, u ) − ϱ(z, v ) +H(T (z), T (v )) n n n n n n n n n j j i j < ϱ(z, u ) − ϱ(z, v ) + aϱ(z, v ) + b maxfρ(z, T (z)), ρ(v , T (v ))g n n n n n i j + c[ρ(z, T (v )) + ρ(v , T (z))] + d[ρ(z, T (z)) + ρ(v , T (v ))] n n n n j i i j ≤ ϱ(z, u ) + b maxfρ(z, T (z)), ρ(v , T (v ))g n n n i j + c[ϱ(z, v ) + ρ(v , T (v )) + ϱ(v , u )] + d[ρ(z, T (z)) + ρ(v , T (v ))] n n n n n n n j i j = ϱ(z, u ) + bρ(v , T (v )) n n n + c[ϱ(z, u ) + ρ(v , T (v ))] + d[ρ(z, T (z)) + ρ(v , T (v ))]. n n n n n j i j Taking the limit as n increases towards innity and considering the limits (24) and (26) we obtain ρ(v , T (v )) ≤ ρ(z, T (z)) + bρ(v , T (v )) 0 j 0 i 0 j 0 + c[ρ(z, T (z)) + ρ(v , T (v ))] + d[ρ(z, T (z)) + ρ(v , T (v ))]. i 0 j 0 i 0 j 0 Hence ρ(v , T (v )) ≤ [(1 + c + d)/(1 − b − c − d)]ρ(z, T (z)). (27) 0 0 j i Since v 2 K, T (v )  K. Thus T (v ) is compact and so there exists w 2 T (v ) such that ϱ(v , w) = 0 j 0 j 0 j 0 0 ρ(v , T (v )). 0 j 0 From condition (20), as f (v ) > 0 implies that w ≠ y , we have j 0 0 ρ(w, T (w)) ≤ H(T (v ), T (w)) i j 0 i < aϱ(w, v ) + b maxfρ(w, T (w)), ρ(v , T (v ))g + cρ(v , T (w)) 0 i 0 j 0 0 i + d[ρ(v , T (v )) + ρ(w, T (w))]. 0 j 0 i Since ρ(v , T (w)) ≤ ϱ(v , w) + ρ(w, T (w)) = ρ(v , T (v )) + ρ(w, T (w)), we have 0 i 0 i 0 j 0 i ρ(w, T (w)) ≤ aρ(v , T (v )) + b maxfρ(w, T (w)), ρ(v , T (v ))g i 0 j 0 i 0 j 0 + (c + d)fρ(w, T (w)), ρ(v , T (v ))g i 0 j 0 and hence a + b + c + d 1 + c + d ρ(w, T (w)) ≤ · ρ(v , T (v )) i 0 j 0 a − c − d 1 − b − c − d = [(a + b + c + d)/(1 − c − d)]ρ(y , T (y )) 0 0 So by condition (27) we have a + b + c + d 1 + c + d ρ(w, T (w)) ≤ · ρ(z, T (z)) (28) i i 1 − c − d 1 − b − c − d Since (a + b + c + d)(1 + c + d) a + 2b + (3 + a)(d + c) − b − (c + d)(2 − b − c − d) (1 − c − d)(1 − b − c − d) 1 − b − (c + d)(2 − b − c − d) Taking into consideration condition (21) we get a + b + c + d 1 + c + d · < 1. 1 − c − d 1 − b − c − d Thus by condition (28) we have ρ(w, T (w)) ≤ ρ(z, T (z)), i i a contradiction with condition (23). Therefore, ρ(z, T (z)) = 0. Hence, as T (z) is closed, z 2 T (z). i i i The following corollary is obtained for Banach spaces: Fixed point theorem for a sequence of multivalued ... Ë 11 Corollary 3.2. Let K be a nonempty compact subset of a Banach space C and f be a sequence of single-valued mappings of K into C for all n 2 N. Suppose that there are nonnegative reals a, b, c, d with a+2b+(3+c)(c+b) < 1 such that for any x, y 2 K and for all i, j = 1, 2,··· , kf (u)− f (v)k ≤ aku− vk + bfku− f (u)k +kv− f (v)kg + cfku− f (v)k +kv− f (u)kg + dfku− f (u)k +kv− f (v)kg i j i j j i i j If f (∂K)  K for each n = 1, 2,··· , then there exists a unique common xed point of f in K. n n Example 3.2. Let M = [0,∞) with the usual metric and K = [0, 1]. Dene fT g : K ! CB(M) as T (u) = n n nu [0, ] for all u 2 K. Since T (0) = 0 for all n 2 N, we get T(0) = 0. For u > 0, we have (2n+1) nu u lim = lim = , n!∞ (2n + 1) n!∞ 2 + 1/n which yields T(u) = u/2 for u > 0. HencefT g converges pointwise to g where 0 if u = 0 g(u) = u/2 if u > 0 and u is continuous for all u 2 [0, 1]. Thus, K is compact. Moreover, we see that T (∂K) = T (f0, 1g) = f0, 1g n n =) T (0) and T (1) are subset of K, n n which implies fT g satises T (∂K)  K. By direct computation as in Example 3.1 we can show that all condi- n n tions of Theorem 3.2 are satised. Therefore, F(T ) = 0. Referring from Theorem 2.3, the following result can also be obtained: Theorem 3.3. Let (M, ϱ) be a complete and metrically convex metric space, K a nonempty closed subset of M and let S = fT g be a family of multivalued mappings of K into CB(M). Suppose that there exists some T 2 S j j2J i such that for each T 2 S H(T (u), T (v)) ≤ a ϱ(u, v) + b maxfρ(u, T (u)), ρ(v, T (v))g+ i j j j i j (29) c [ρ(u, T (v)) + ρ(v, T (u))] + d [ρ(u, T (u)) + ρ(v, T (v))], j j i j i j where a , b , c , d ≥ 0 are positive real numbers satisfying j j j j λ = a + 2b + (3 + a )(c + d ) < 1. j j j j j j If T u  K for each u 2 ∂K and for each T 2 S, then a family S has a common xed point in K such that j j w 2 T (w) for all T 2 S. j j Proof. Suppose that T is an arbitrary but xed member of S. Then, from Theorem ?? with T = T there exists j j 0 0 a point in K, say w, which is a xed point of T . Let T 2 S, T ≠ T , be arbitrary. Then from equation (29) we have j j j ρ(w, T (w)) ≤ H(T (w), T (w)) j i j ≤ b maxf0, ρ(w, T (w))g + (c + d )[ρ(w, T (w)) + 0], j j j j j and hence (1 − b − c − d )ρ(w, T (w)) ≤ 0. j j j j Since b + c + d ≤ λ < 1, we have ρ(w, T (w)) = 0. Hence w 2 T (w). Clearly, w is a xed point of a family j j j j j j S. 12 Ë David Aron and Santosh Kumar 4 Conclusion The main contribution of this study to xed point theory is the xed point result given in Theorem 3.1 , 3.2 and Theorem 3.3. These theorems proved for sequence of multivalued mappings which satisfy certain re- quirements in complete metric spaces. These results generalizes many recent known results in convex metric spaces in the literature. The results proved here gives generalization of the results due to Ćirić [1]. Suitable examples are given to support the results proved herein. Conict of interest: Both authors declare that they have no conicts of interest. Research involving human participants and/or animals: The authors declare that there is no human participants and / or animals involved in this research. Funding: Authors declare that there is no funding available for this research. 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Journal

Topological Algebra and its Applications – de Gruyter

Published: Jan 1, 2022

Keywords: Multivalued mapping; non-self mapping; metrically convex metric spaces; 47H10; 54H25

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