Fixed point theorems for multivalued nonself Kannan-Berinde G-contraction mappings in complete metric spaces endowed with graphs

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1 The 22 nd Annual Meeting in Mathematics (AMM 2017) Department of Mathematics, Faculty of Science Chiang Mai University, Chiang Mai, Thailand Fixed point theorems for multivalued nonself Kannan-Berinde G-contraction mappings in complete metric spaces endowed with graphs Jenwit Puangpee and Suthep Suantai Department of Mathematics, Faculty of Science Chiang Mai University, Chiang Mai 50200, Thailand Abstract In this paper, we prove the existence theorems of fixed point for multivalued nonself Kannan-Berinde G-contraction mappings in complete metric spaces endowed with directed graphs and also give some examples to illustrate our main results. The main results obtained in this paper extend and generalize many important results in the literatures. Keywords: fixed point, nonself multivalued mappings, Kannan-Berinde G-contraction, edgepreserving, Rothe s boundary condition MSC: Primary 47H04; Secondary 47H10. 1 Introduction Let (X, d) be a metric space and CB(X) the set of all nonempty closed bounded subsets of X. Let A be a subset of X. The distance from x to A defined by D(x, A) := inf{d(x, y) : y A}. For A, B CB(X), we define { H(A, B) := max sup a A } D(a, B), sup D(b, A). b B It is called a Pompeiu-Hausdorff metric on CB(X) induced by the metric d on X. Let T : X 2 X (collection of all nonempty subsets of X) be a multivalued mapping. A point x in X is said to be a fixed point of T if x T x and the set of all fixed points of T denoted by F (T ) := {x X : x T x}. Theorem 1.1. (Banach s Contraction Principle) Let (X, d) be a complete metric space and let T : X X be a contraction mapping, i.e., there exists k [0, 1) such that Then T has a unique fixed point in X. d(t x, T y) kd(x, y) for all x, y X. Corresponding author. Speaker. address: jenwit.pp@hotmail.com (J. Puangpee), suthep.s@cmu.ac.th (S. Suantai) ANA-04-1

2 The Banach s contraction principle [1] or Banach s fixed point theorem is the fundamental well-known result which has been extended and generalized in many directions by many authors. In 1968, Kannan [2] introduced the new type of mappings (known as Kannan contraction mapping) as follows: Definition 1.2. Let (X, d) be a metric space. A mapping T : X X is called a Kannan mapping if there exists a [0, 1 2 ) such that d(t x, T y) a[d(x, T x) + d(y, T y)] for all x, y X. We can founded the results as the following in [2]. In 1969, Nadler [3] introduced the concept of Banach s contraction principle for a multivalued mapping and proved the following fixed point theorem for multivalued contraction mappings in complete metric spaces. Theorem 1.3. (Nadler s fixed point theorem) Let (X, d) be a complete metric space and let T be a map from X into CB(X). Suppose that T is a multivalued contraction mapping, i.e., there exists k [0, 1) such that Then there exists z X such that z T z. H(T x, T y) kd(x, y) for all x, y X. In 2007, M. Berinde and V. Berinde [4] extended Nadler s fixed point theorem to the new class of multivalued self mappings that is called a multivalued weak contraction or multivalued almost contraction and proved the existence fixed point theorems, see [4]. Definition 1.4. ([4]) Let (X, d) be a metric space and T : X CB(X) a multivalued mapping. Then T is said to be a multivalued almost contraction or multivalued (θ, L)-almost contraction if there exist two constants θ (0, 1) and L 0 such that H(T x, T y) θd(x, y) + L D(y, T x) for all x, y X. In 1972, Assad and Kirk [5] introduced and obtained the new fixed point theorem for nonself multivalued contraction mappings as the following theorem. Theorem 1.5. (Assad and Kirk s fixed point theorem) Let (X, d) be a complete and metrically convex metric space, K a nonempty closed subset of X, and let T : K CB(X) a multivalued contraction mapping. If T satisfies Rothe s type condition, that is, x K implies T x K, then T has a fixed point in K. In 2013, Alghamdi et al. [6] extended Theorem 1.5 to the multivalued nonself almost contractions on convex metric spaces. Theorem 1.6. ([6]) Let (X, d) be a complete convex metric space and K a nonempty closed subset of X. Suppose that T : K CB(X) is a multivalued almost contraction, that is, H(T x, T y) δd(x, y) + L D(y, T x) for all x, y K, with δ (0, 1) and some L 0 such that δ(1 + L) < 1. If T satisfies Rothe s type condition, then there exists z K such that z T z, that is, T has a fixed point in K. We now move on some basics and definitions in graph theory. Let G = (V (G), E(G)) be a directed graph such that V (G) a set of vertices of a graph and E(G) a set of its edges. Let := {(x, x) : x V (G)}, say that the diagonal of V (G) V (G), be a set of all loops in G. Assume that G has no parallel edges, thus one can identify G with the pair (V (G), E(G)). We denote by G 1 the directed graph obtained from G by reversing the direction of edges, that is, E(G 1 ) := {(x, y) V (G) V (G) : (y, x) E(G)}. In 2008, Jachymski [7] combined the concept of fixed point theory and graph theory to study fixed point theory in a metric space endowed with a directed graph. He also introduced a concept of G-contraction self mappings which generalized the regular contraction mapping. ANA-04-2

3 Definition 1.7. ([7]) Let (X, d) be a metric space, and G = (V (G), E(G)) a directed graph where V (G) = X and E(G) contains all loops, that is, E(G). We say that a mapping T : X X is a Banach G-contraction or simply G-contraction if T preserves edges of G, i.e., for any x, y X such that (x, y) E(G) implies (T x, T y) E(G), and there exists k (0, 1) such that d(t x, T y) kd(x, y) for all x, y X with (x, y) E(G). By using this concept and some conditions, he proved that F (T ) if and only if X T := {x X : (x, T x) E(G)} =. In 2011, Nicolae et al. [8] introduced the concept of G-contraction for a multivalued mapping T : X CB(X) and proved the fixed point results about this mapping in a complete metric space. In 2013, Dinevari and Frigon [9] introduced the new concept of G-contraction that is weaker than that of Nicolae et al. They proved that under some properties on a metric space, a multivalued G-contraction with closed values has a fixed point. In 2014, Tiammee and Suantai [10] introduced the concepts of edge-preserving multivalued mapping and a new type of multivalued almost G-contraction and then they proved the existence fixed point theorem on a metric space with a directed graph. Recently, Tiammee et al. [11] introduced and proved the fixed point theorems for multivalued nonself G-almost contractions in Banach spaces endowed with graphs, for more details on this see [11]. In this paper, we aim to study and prove some fixed point theorems for a new class of G-contraction for multivalued nonself mappings in a complete metric space endowed with a directed graph. Further, we also give some examples to illustrate our main results. 2 Preliminaries In this section, we recall some basic definitions and known results that are useful for the main results in our paper. Definition 2.1. A metric space (X, d) is said to be metrically convex or convex if for each x, y X with x y there exists z X, x z y, such that d(x, y) = d(x, z) + d(z, y). Proposition 2.2. ([12]) Let (X, d) be a complete and convex metric space, K a nonempty closed subset of X. If x K and y / K, then there exists a point z in the boundary of K, denote by K, such that d(x, y) = d(x, z) + d(z, y). We denote P [x, y] := {z K : d(x, y) = d(x, z) + d(z, y)}. Lemma 2.3. For A, B CB(X) and a A, then D(a, B) H(A, B). Lemma 2.4. Let A, B CB(X) and k > 1 be given. Then for a A, there exists b B such that d(a, b) kh(a, B). Property A. ([13]) For any sequence {x n } n N in X, if x n x X and (x n, x n+1 ) E(G) for all n N, then (x n, x) E(G) for any n N. In order to obtain our main results, we need to know the following definition of domination in a graph, see [14, 15]: Let G = (V (G), E(G)) be a directed graph. A set X V (G) is said to be a dominating set if for all v V (G) \ X, there exists x X such that (x, v) E(G) and we say that x dominates v or v is dominated by x. For each v V (G), a set X V (G) is dominated by v if (v, x) E(G) for all x X and we call that v dominates X if (v, x) E(G) for all x X. ANA-04-3

4 Definition 2.5. ([10]) Let X be a nonempty set, G = (V (G), E(G)) a graph such that V (G) = X and let T : X CB(X) be a multivalued mapping. Then T is said to be edge-preserving if for any x, y X such that 3 Main Results (x, y) E(G) implies (u, v) E(G) for all u T x and v T y. By using the concepts of almost contractions defined by M. Berinde and V. Berinde [4] and Kannan mappings by [2], we introduce a multivalued Kannan-Berinde G-contraction mapping in a metric space endowed with a directed graph and prove some fixed point theorems of this type of mappings. Definition 3.1. Let (X, d) be a metric space, K a nonempty subset of X and G := (V (G), E(G)) be a directed graph such that V (G) = K. A mapping T : K CB(X) is said to be a multivalued Kannan-Berinde G-contraction if there exist δ [0, 1), a [0, 1 3 ) and L 0 such that H(T x, T y) δd(x, y) + a[d(x, T x) + D(y, T y)] + L D(y, T x) for all x, y K with (x, y) E(G). We note that if a = 0, then a multivalued Kannan-Berinde G-contraction mapping is a multivalued G-almost contraction mapping introduced by Tiammee et al. [11] and moreover if E(G) = K K, then it becomes nonself almost contraction defined by Alghamdi et al. [6]. Theorem 3.2. Let (X, d) be a complete convex metric space and K a nonempty closed subset of X. Let G := (V (G), E(G)) be a directed graph such that V (G) = K. Suppose that K has Property A. If a map T : K CB(X) is a multivalued mapping satisfying the following properties: (i) there exists x 0 K such that (x 0, y) E(G) for some y T x 0 ; (ii) T is an edge-preserving mapping, that is, if (x, y) E(G), then (u, v) E(G) for all u T x and v T y; (iii) for each x K and y T x with y / K, (a) P [x, y] is dominated by x and (b) for each z P [x, y], z dominates T z; (iv) T has Rothe s boundary condition, that is, x K implies T x K; (v) T is a multivalued Kannan-Berinde G-contraction mapping with δ(1+a+l)+a(3+l) < 1. Then T has a fixed point in K. Proof. Since δ(1 + a + L) + a(3 + L) < 1, there exists k > 1 such that (1 + ka + kl)(kδ + ka) () 2 < 1. We construct two sequences {x n } and {y n } as follows: Let x 0 K be such that (x 0, y 1 ) E(G) for some y 1 T x 0. If y 1 K, we denote x 1 = y 1. Consider in case y 1 / K. By Proposition 2.2, there exists x 1 P [x 0, y 1 ] such that d(x 0, y 1 ) = d(x 0, x 1 ) + d(x 1, y 1 ). By (iii) (a), that is, P [x 0, y 1 ] is dominated by x 0, we obtain (x 0, x 1 ) E(G). Moreover, we have x 1 K, and, by Lemma 2.4, there exists y 2 T x 1 such that d(y 1, y 2 ) kh(t x 0, T x 1 ). ANA-04-4

5 If y 2 K, we denote x 2 = y 2. When y 1 K, that is, x 1 = y 1 T x 0 and x 2 = y 2 T x 1, using T is edge-preserving, we get that (x 1, x 2 ) E(G). When y 1 / K, that is, x 1 P [x 0, y 1 ], by (iii) (b), we also get (x 1, x 2 ) E(G). Otherwise, if y 2 / K, then there exists x 2 P [x 1, y 2 ] such that d(x 1, y 2 ) = d(x 1, x 2 ) + d(x 2, y 2 ). Thus (x 1, x 2 ) E(G) and x 2 K, by Lemma 2.4, there exists y 3 T x 2 such that d(y 2, y 3 ) kh(t x 1, T x 2 ). Continuing the arguments, we can construct two sequences {x n } and {y n } such that (a) y n+1 T x n ; (b) d(y n, y n+1 ) kh(t x n 1, T x n ), where x n = y n if y n K; d(x n 1, y n ) = d(x n 1, x n ) + d(x n, y n ) if y n / K. Now we show that the sequence {x n } is a Cauchy sequence. Suppose that P := {x i {x n } : x i = y i, i = 1, 2,...} Q := {x i {x n } : x i y i, i = 1, 2,...}. Note that {x n } K and (x n, x n+1 ) E(G) for all n N. Moreover, if x i Q, then x i 1 and x i+1 belong to the set P. By the assumption (iv), we cannot have two consecutive terms of {x n } in the set Q. Now, for n 2, we consider the distance d(x n, x n+1 ). The three possibilities must be considered as follows: Case 1. x n P and x n+1 P : Then x n = y n, x n+1 = y n+1. Since (x n 1, x n ) E(G), by (v), we have d(x n, x n+1 ) = d(y n, y n+1 ) This implies that kh(t x n 1, T x n ) kδd(x n 1, x n ) + ka[d(x n 1, T x n 1 ) + D(x n, T x n )] + kl D(x n, T x n 1 ). kδd(x n 1, x n ) + kad(x n 1, y n ) + kad(x n, y n+1 ) = kδd(x n 1, x n ) + kad(x n 1, x n ) + kad(x n, x n+1 ). d(x n, x n+1 ) ( ) kδ + ka d(x n 1, x n ). Case 2. x n P and x n+1 Q: Then x n = y n and d(x n, y n+1 ) = d(x n, x n+1 ) + d(x n+1, y n+1 ). We note that d(y n, y n+1 ) kh(t x n 1, T x n ) So we get that Then kδd(x n 1, x n ) + ka[d(x n 1, T x n 1 ) + D(x n, T x n )] + kl D(x n, T x n 1 ). kδd(x n 1, x n ) + kad(x n 1, x n ) + kad(y n, y n+1 ). d(y n, y n+1 ) ( ) kδ + ka d(x n 1, x n ). d(x n, x n+1 ) = d(x n, y n+1 ) d(x n+1, y n+1 ) d(x n, y n+1 ) = d(y n, y n+1 ). Then we obtain that d(x n, x n+1 ) ( ) kδ + ka d(x n 1, x n ). ANA-04-5

6 Case 3. x n Q and x n+1 P : Then x n 1 P. So we have x n 1 = y n 1, x n+1 = y n+1, and d(x n 1, y n ) = d(x n 1, x n ) + d(x n, y n ). Since (x n, x n+1 ) E(G), y n T x n 1 for all n N and kδ < 1, we obtain Hence d(y n, y n+1 ) kh(t x n 1, T x n ) kδd(x n 1, x n ) + ka[d(x n 1, T x n 1 ) + D(x n, T x n )] + kl D(x n, T x n 1 ). kδd(x n 1, x n ) + kad(x n 1, y n ) + kad(x n, y n+1 ) + kld(x n, y n ) = kδd(x n 1, x n ) + kad(x n 1, y n ) + kad(x n, x n+1 ) + kld(x n, y n ). d(x n, x n+1 ) d(x n, y n ) + d(y n, x n+1 ) = d(x n, y n ) + d(y n, y n+1 ) d(x n, y n ) + kδd(x n 1, x n ) + kad(x n 1, y n ) + kad(x n, x n+1 ) + kld(x n, y n ) (1 + kl)[d(x n, y n ) + d(x n 1, x n )] + kad(x n 1, y n ) + kad(x n, x n+1 ) = (1 + kl)d(x n 1, y n ) + kad(x n 1, y n ) + kad(x n, x n+1 ), which implies that ( ) 1 + ka + kl d(x n, x n+1 ) d(x n 1, y n ) = ( 1 + ka + kl Since x n 1 P and x n Q, it follows from Case 2 that ( ) kδ + ka d(y n 1, y n ) d(x n 2, x n 1 ). Thus Since h := (1+ka+kL)(kδ+ka) (1 ka) 2 d(x n, x n+1 ) (1 + ka + kl)(kδ + ka) () 2 d(x n 2, x n 1 ). < 1, we obtain that d(x n, x n+1 ) hd(x n 2, x n 1 ). ) d(y n 1, y n ). Combining Case 1, 2 and 3, we obtain that for n 2, { hd(x n 1, x n ) when x n, x n+1 P or x n P, x n+1 Q; d(x n, x n+1 ) = hd(x n 2, x n 1 ) when x n Q, x n+1 P. By induction, it follows from Assad and Kirk see [5], for n 2, Now, for m > n, we obtain d(x n, x n+1 ) h (n 1)/2 max{d(x 0, x 1 ), d(x 1, x 2 )}. d(x n, x m ) d(x n, x n+1 ) + d(x n+1, x n+2 ) d(x m 1, x m ) = (h (n 1)/2 + h n/ h (m 2)/2 ) max{d(x 0, x 1 ), d(x 1, x 2 )}. Since h < 1, we obtain that {x n } is a Cauchy sequence in K. Since X is complete and K is closed, there exists x K such that lim n x n = x. Moreover, from (x n, x n+1 ) E(G) for all n N and by Property A, we have (x n, x) E(G) for all n N. From the construction of {x n }, there is an infinite subsequence {x nj } of {x n } such ANA-04-6

7 that {x nj } P. Then x nj = y nj T x nj 1. Now, we show that x T x. Since lim j x nj = x, we have 0 D(x, T x nj 1) d(x, x nj ). So, D(x, T x nj 1) 0 as j. Then for each j N, D(x, T x) d(x, x nj ) + D(x nj, T x) Hence d(x, x nj ) + H(T x nj 1, T x) d(x, x nj ) + δd(x nj 1, x) + a[d(x nj 1, T x nj 1) + D(x, T x)] + L D(x, T x nj 1) d(x, x nj ) + δd(x nj 1, x) + ad(x nj 1, x nj ) + ad(x, T x) + L D(x, T x nj 1). (1 a)d(x, T x) d(x, x nj ) + δd(x nj 1, x) + ad(x nj 1, x nj ) + L D(x, T x nj 1) d(x, x nj ) + δd(x nj 1, x) + ad(x nj 1, x) + ad(x, x nj ) + L D(x, T x nj 1). Letting j, we obtain (1 a)d(x, T x) = 0. Since a [ 0, 1 3), we get that D(x, T x) = 0. Hence x T x, that is, T has a fixed point in K. This completes the proof. Remark 3.3. As a consequence of Theorem 3.2, we can conclude the following results as our special cases: (i) If we put a = 0 in Theorem 3.2, we obtain Theorem 5 of [11]. (ii) If we put a = 0 and E(G) = K K in Theorem 3.2, we obtain Theorem 9 of [6]. (iii) If we put a = L = 0 and E(G) = K K in Theorem 3.2, we obtain Theorem 1 of [5]. Next, we give an example to illustrate Theorem 3.2. Example 3.4. Let X = R and K = [0, 1] {2} endowed with usual metric d(x, y) = x y. Let G = (V (G), E(G)) be a graph consisting of V (G) := K and E(G) := { ( )} {( 1 (0, 0), 9, 0 0, 1 ) ( ) ( n, 3 n, 0, 3 m, 1 ) } 3 n : m, n N\{2}. Notice that K has Property A. Define a mapping T : K CB(X) by {1, 2} if x = 2; T x = { 1 if x = 1 { } 9 ; 0, x 9 otherwise. We see that K = {0, 1, 2} T (0), T (1) and T (2) are subset of K, which implies T satisfies Roth s boundary condition. We choose x 0 = 0 K and y = 0 {0} = T x 0, and so (x 0, y ) = (0, 0) E(G). Since the only x K and y T x with y / K are x = 1 9 and y = 1, we get that P [ 1 9, ] 1 ( = {0}. Further, we have 1 9, 0), (0, 0) E(G). Hence P [x, y] is dominated by x and 0 dominates T (0). Next, we prove that T is edge-preserving. Let (x, y) E(G). Then we get that x, y {0} { 1 : k N\{2} }. We obtain that T x, T y 3 k {0} { 1 : k N\{1, 2, 4} }. Then for all u T x and v T y, we get that (u, v) E(G), 3 k which show that T is edge-preserving. Now we claim that T is a multivalued Kannan-Berinde G-contraction mapping. Let (x, y) E(G). We will discuss the following three possible cases. ANA-04-7

8 Case 1. If (x, y) = (0, 0), then H(T (0), T (0)) = 0. Case 2. If (x, y) = ( 1 9, 0), we have ({ H(T x, T y) = H 1 } ), {0} = 1 δ a 4 + L 1 = δ [D + a ( 1 9, { 1 }) ] ( { + D(0, {0}) + L D 0, 1 }) = δd(x, y) + a[d(x, T x) + D(y, T y)] + L D(y, T x), where L 1, 0 δ < 1 and 0 a < 1 3 such that δ(1 + a + L) + a(3 + L) < 1. Case 3. If (x, y) = ( 0, 1 ) ( 3 or 1 n 3, 0 ) or ( 1 n 3, 1 ) m 3 for all m, n N\{2}, we have n ({ H(T x, T y) = H 0, x } {, 0, y }) 9 9 = x 9 y 9 x 9 + y 9 = 1 ( x + 8 ) 9 y = 1 [( x x ) ( + y y )] = 1 [ ( { D x, 0, x }) ( { + D y, 0, y })] δd(x, y) + a[d(x, T x) + D(y, T y)] + L D(y, T x), where a = 1 8, 0 δ < 1 and L 0 such that δ(1 + a + L) + a(3 + L) < 1. Now, by summarizing all cases, we can conclude that T is a multivalued Kannan-Berinde G- contraction mapping with a = 1 8, L = 1 and 0 δ < (3 + 1) = 4 17, which the condition δ(1 + a + L) + a(3 + L) < 1 is also satisfied. Therefore, T is a multivalued Kannan-Berinde G-contraction that satisfies all assumptions in Theorem 3.2. Then there exist z K such that z T z. Notice that F (T ) = {0, 2}. Acknowledgment. The authors would like to thank the Development and Promotion for Science and Technology talents project (DPST), Thailand research fund under the project RTA and Chiang Mai University, Chiang Mai, Thailand for the financial support. References [1] S. Banach, Sur les operations dans les ensembles abstraits et leur application aux equations itegrales, Fundam. Math. 3 (1922), [2] R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc. 10 (1968), [3] S. B. Nadler, Multi-valued contraction mappings, Pac. J. Math. 30 (1969), [4] M. Berinde and V. Berinde, On a general class multi-valued weakly Picard mappings, J. Math. Anal. Appl. 326 (2007), ANA-04-8

9 [5] N. A. Assad and W. A. Krik, Fixed point theorems for set-valued mappings of contractive type, Pac. J. Math. 43 (1972), [6] M. A. Alghamdi, V. Berinde and N. Shahzad, Fixed points of multivalued nonself almost contractions, J. Appl. Math (2013), Article ID [7] J. Jachymski, The contraction principle for mappings on a metric space with a graph, Proceedings of the American Mathematical Society 136 (2008), [8] A. Nicolae, D. O Regan and A. Petrusel, Fixed point theorems for singlevalued and multivalued generalized contractions in metric spaces endowed with a graph, Georgian Math. J. 18 (2011), [9] T. Dinevari and M. Frigon, Fixed point results for multivalued contractions on a metric space with a graph, J. Math. Anal. Appl. 405 (2013), [10] J. Tiammee and S. Suantai, Coincidence point theorems for graph-preserving multi-valued mappings, Fixed Point Theory Appl (2014), article 70. [11] J. Tiammee, P. Charoensawan and S. Suantai, Fixed point theorems for multivalued nonself G-almost contractions in Banach spaces endowed with graphs, J. Appl. Math (2017), Article ID [12] W. A. Kirk, P. S. Srinivasan and P. Veeramani, Fixed points for mappings satisfying cyclical contractive conditions, Fixed Point Theory 4 (2003), no. 1, [13] I. Beg and A. R. Butt, Fixed point for set-valued graph contractive mappings, J. Inequal. Appl (2013), article 252. [14] J. Bang-Jensen and G. Gutin, Digraph Theory, Algorithms and Applications, Springer Monographs in Mathematics, Springer-Verlag London Ltd., London, [15] C. Pang, R. Zhang, Q. Zhang and J. Wang, Dominating sets in directed graphs, Information Sciences 180 (2010), ANA-04-9