Fixed Point Theorem of Uniformly Locally Geraghty Contractive Mappings on Connected Complete Metric Spaces

International Journal of Mathematical Analysis Vol. 11, 2017, no. 9, 445-456 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ijma.2017.7457 Fixed Point Theorem of Uniformly Locally Geraghty Contractive Mappings on Connected Complete Metric Spaces Ing-Jer Lin 1 and Chien-Lung Wu Department of Mathematics National Kaohsiung Normal University Kaohsiung 82444, Taiwan Copyright c 2017 Ing-Jer Lin and Chien-Lung Wu. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, we first introduce the concept of uniformly locally contractive mapping. Second, complete metric space is modified to be a connected complete metric space. Third, Our generalization of the Banach contraction principle replaces the global contraction hypothesis with the local contraction hypothesis. Then we investigate the behavior for mappings satisfying uniformly locally Geraghty contraction on connected complete metric space. Finally, we establish a new existence and uniqueness result of fixed point theorem. Mathematics Subject Classification: 41A50, 47H09, 47H10 Keywords: fixed point, uniformly locally contractive mappings, ɛ-chainable, connected complete metric space 1. Introduction Our generalization of the Banach contraction principle replaces the global contraction hypothesis with the local contraction hypothesis [8]. The aim of 1 Corresponding author

446 Ing-Jer Lin and Chien-Lung Wu this paper is to establish a new fixed point theorem for mappings satisfying uniformly locally Geraghty contraction in the setting of connected complete metric space. For a survey of Geraghty contraction, see articles of Rosa et al. [9], [10]. An example is provided to illustrate the main result. Our result generalizes Edelstein s and Geraghty s fixed point theorems. Fixed point theory plays a significant role in nonlinear functional analysis. It has been applied in physical sciences, Computing sciences and Engineering,etc. In 1922, Stefan Banach [1] proved a famous fixed point theorem for contractive mappings on complete metric spaces. It guarantees that the existence and uniqueness of fixed points of certain self mappings on complete metric spaces, and provides a constructive method to find fixed point. From then on, plenty of generalizations in various different aspects of the Banach contraction principle have been investigated by many scholars. Because of the importance of the Banach contraction principle, we begin with the theorem as following. Theorem 1.1. (B.C.P)[1] Let (X, d) be a complete metric space and T : X X be a selfmap. Assume that there exists a nonnegative number γ < 1 such that d(t x, T y) γd(x, y) for all x,y X. Then T has a unique fixed point in X. Moreover, for each x X the iterative sequence {T n x} converges to the fixed point. In 1973, Michael A. Geraghty[6] gave a generalization of Banach contraction principle for contraction condition of a functional form. Definition 1.2. Let S denotes the class of the functions β : [0, + ) [0, 1) which satisfies the condition β(t n ) 1 t n 0. Theorem 1.3.[6] Let (X, d) be a complete metric space and T : X X be a selfmap. Assume that there exists β S such that d(t x, T y) β(d(x, y))d(x, y) for all x,y X. Then T has a unique fixed point in X. Moreover, for each x X the iterative sequence {T n x} converges to the fixed point. 2. Preliminaries Michael Edelstein [4] asked whether the Banach contraction principle could be altered so as to be valid when contraction condition is assumed to hold for

Fixed point theorem of uniformly locally Geraghty contractive mappings 447 sufficiently closed points only. In 1961, he generalized the Banach contraction principle for mapping of a uniformly locally contraction. This brings a view point of locally contraction which is different from globally point of view. In addition, see Dumitru [3] and Hu [7] for related researches. Definition 2.1. A map T : X X is said to be locally contractive, denoted (LC), if for every x X there exist numbers ɛ x > 0 and λ x [0, 1) such that for all y, z X with y, z B(x; ɛ x ) implies d(t y, T z) λ x d(y, z). Moreover, T is uniformly locally contractive, denoted (ULC), if the same ɛ and λ work for all x X, which we also indicate by saying that T is (ɛ, λ)-(ulc). Definition 2.2. For ɛ > 0, we say that (X, d) is ɛ-chainable, provided for every p, q X there exists a finite sequence s = x 0, x 1, x 2,..., x n, referred to as an ɛ-chain from p to q, such that x 0 = p, x n = q, and d(x i, x i+1 ) < ɛ for all i < n. The length of the ɛ-chain s is defined as length(s) = i=n 1 i=0 d(x i, x i+1 ). Theorem 2.3.[4] Let (X, d) be a complete ɛ-chainable metric space. If T : X X is an (ɛ, λ)-uniformly locally contractive mapping, then T has a unique fixed point. The following is an important theorem we will use in Main results. Krzysztof Chris Ciesielski et al.[2] provided a proof for it. Theorem 2.4.[5] Connected spaces are ɛ-chainable for any ɛ > 0. In order to develop our fixed point theorem, we define a map called uniformly locally Geraghty contractive mapping. Definition 2.5. A map T : X X is said to be uniformly locally Geraghty contractive, denoted (ULGC), if there exist ɛ > 0 and β S such that for all x, y X with d(x, y) < ɛ implies d(t x, T y) β(d(x, y))d(x, y). In this case, we say that T is (ɛ, β)-(ulgc). In this paper, we generalize Edelstein s and Geraghty s fixed point theorems to develop our theory. First, metric space is altered to be a connected complete metric space. Second, we modify contraction mapping to be a uniformly locally contraction of a functional form. 3. Main results In this section, we investigate the behavior of uniformly locally Geraghty contractive mappings in the setting of connected complete metric space. Then

448 Ing-Jer Lin and Chien-Lung Wu we establish a new existence result of fixed point theorem and give a constructive process to find the unique fixed point. Theorem 3.1 If (X, d) is a connected complete metric space and T : X X is (η, β)-(ulgc), then T has a unique fixed point in X. Moreover, for each x 0 X the iterative sequence {x n } n N {0} in X converges to the fixed point. Proof. Since T is (η, β)-(ulgc), there exist η > 0 and β S such that for all y, z X with d(y, z) < η implies d(t y, T z) β(d(y, z))d(y, z). Let x 0 X and set x 1 = T (x 0 ). If x 1 = x 0, then x 0 is the fixed point of T. If x 1 x 0. By Theorem 2.4., there exist a positive number ɛ and a sequence z (1,0), z (1,1), z (1,2),..., z (1,k) such that x 0 = z (1,0), x 1 = z (1,k), and d(z (1,i 1), z (1,i) ) < ɛ < η i = 1, 2, 3,..., k. (3.1) Applying that T is (η, β)-(ulgc) on (3.1). We have d(t z (1,i 1), T z (1,i) ) β(d(z (1,i 1), z (1,i) ))d(z (1,i 1), z (1,i) ) < d(z (1,i 1), z (1,i) ) < ɛ < η i = 1, 2, 3,..., k In addition, we set z (2,i 1) = T (z (1,i 1) ) i = 1, 2, 3,..., k + 1. Particularly, x 1 = z (1,k) = T (x 0 ) = T (z (1,0) ) = z (2,0) and x 2 = T (x 1 ) = T (z (1,k) ) = z (2,k). Thus, we construct a sequence z (2,0), z (2,1), z (2,2),..., z (2,k) such that x 1 = z (2,0), x 2 = z (2,k), and d(z (2,i 1), z (2,i) ) < ɛ < η i = 1, 2, 3,..., k. (3.2) If x 2 = x 1, then x 1 is the fixed point of T. If x 2 x 1. Applying that T is (η, β)-(ulgc) on (3.2). We have d(t z (2,i 1), T z (2,i) ) β(d(z (2,i 1), z (2,i) ))d(z (2,i 1), z (2,i) ) < d(z (2,i 1), z (2,i) ) < ɛ < η i = 1, 2, 3,..., k In addition, we set z (3,i 1) = T (z (2,i 1) ) i = 1, 2, 3,..., k + 1. Particularly, x 2 = z (2,k) = T (x 1 ) = T (z (2,0) ) = z (3,0) and x 3 = T (x 2 ) = T (z (2,k) ) = z (3,k). Thus, we construct a sequence z (3,0), z (3,1), z (3,2),..., z (3,k) such that x 2 = z (3,0), x 3 = z (3,k), and d(z (3,i 1), z (3,i) ) < ɛ < η i = 1, 2, 3,..., k. (3.3)

Fixed point theorem of uniformly locally Geraghty contractive mappings 449 Continuing in this way. If x n0 +1 = x n0 for some n 0 N, then T (x n0 ) = x n0. That is, x n0 is the fixed point of T and we are finished. So we may assume that x n+1 x n for each n N. Thus we obtain a diagram as follows: where x 0 = z (1,0), z (1,1), z (1,2),..., z (1,k) = x 1 x 1 = z (2,0), z (2,1), z (2,2),..., z (2,k) = x 2 x 2 = z (3,0), z (3,1), z (3,2),..., z (3,k) = x 3 x n 1 = z (n,0), z (n,1), z (n,2),..., z (n,k) = x n x n = z (n+1,0), z (n+1,1), z (n+1,2),..., z (n+1,k) = x n+1 d(z (n+1,i 1), z (n+1,i) ) β(d(z (n,i 1), z (n,i) ))d(z (n,i 1), z (n,i) ) (3.4) < d(z (n,i 1), z (n,i) ) < ɛ < η (3.5) i = 1, 2, 3,..., k n N Claim1: i = 1, 2, 3,..., k, lim d(z (n,i 1), z (n,i) ) = 0 When i = 1. According to (3.5), we have d(z (n+1,0), z (n+1,1) ) < d(z (n,0), z (n,1) ) n N. So sequence { d(z (n,0), z (n,1) ) } is strictly decreasing. Thus lim d(z (n,0), z (n,1) ) = α exists. Suppose that α 0. That is, α > 0. Then d(z (n,0), z (n,1) ) α > 0 n N. According to (3.4), we have d(z (n+1,0), z (n+1,1) ) β(d(z d(z (n,0), z (n,1) ) (n,0), z (n,1) )) < 1 n N. Since and d(z (n+1,0), z (n+1,1) ) lim d(z (n,0), z (n,1) ) lim d(z (n+1,0), z (n+1,1) ) lim d(z (n,0), z (n,1) ) lim β(d(z (n,0), z (n,1) )) 1 = α α = 1. It follows from Squeeze Theorem that lim β(d(z (n,0), z (n,1) )) = 1. This implies that lim d(z (n,0), z (n,1) ) = 0, but contradicts that α 0. Therefore, lim d(z (n,0), z (n,1) ) = 0. By similar arguments above, when i = 2, 3, 4,..., k, we have lim d(z (n,i 1), z (n,i) ) = 0. Claim2: lim d(x n 1, x n ) = 0

450 Ing-Jer Lin and Chien-Lung Wu Since d(x n 1, x n ) = d(z (n,0), z (n,k) ) d(z (n,0), z (n,1) ) + d(z (n,1), z (n,2) ) + d(z (n,2), z (n,3) ) +... + d(z (n,k 1), z (n,k) ). It follows from Claim1 that lim d(x n 1, x n ) lim d(z (n,0), z (n,1) ) + lim d(z (n,1), z (n,2) ) + lim d(z (n,2), z (n,3) ) +... + lim d(z (n,k 1), z (n,k) ) = 0 + 0 + 0 +... + 0 = 0. By Squeeze Theorem, lim d(x n 1, x n ) = 0. Claim3: {x n } is a Cauchy sequence. Suppose not. Then there exists a positive number ɛ 0 such that i N, q i > p i i d(x pi, x qi ) ɛ 0. So we obtain two sequences {p i } and {q i } satisfying q i > p i i and d(x pi, x qi ) ɛ 0 i N. For t = 1, we have q 1 > p 1 1 such that d(x p1, x q1 ) ɛ 0. Set n 1 = p 1 and m 1 = min{ j N : d(x j, x n1 ) ɛ 0, j > n 1 }. Then d(x n1, x m1 ) ɛ 0 and d(x n1, x m1 1) < ɛ 0. For t = 2, from {p i } choose p i1 > m 1. Set n 2 = p i1 and m 2 = min{ j N : d(x j, x n2 ) ɛ 0, j > n 2 }. Then d(x n2, x m2 ) ɛ 0 and d(x n2, x m2 1) < ɛ 0. For t = 3, from {p i } choose p i2 > m 2. Set n 3 = p i2 and m 3 = min{ j N : d(x j, x n3 ) ɛ 0, j > n 3 }. Then d(x n3, x m3 ) ɛ 0 and d(x n3, x m3 1) < ɛ 0. Continuing in this way, we obtain two subsequences {x ni } and {x mi } of {x n } satisfying m i > n i, d(x ni, x mi ) ɛ 0 and d(x ni, x mi 1) < ɛ 0 i N. Case 1: ɛ 0 < ɛ. Since ɛ 0 d(x ni, x mi ) d(x ni, x mi 1)+d(x mi 1, x mi ) < ɛ 0 +d(x mi 1, x mi ) i N. So lim ɛ 0 lim d(x ni, x mi ) lim ɛ 0 + lim d(x mi 1, x mi ). It follows from Claim2 that ɛ 0 lim d(x ni, x mi ) ɛ 0 + 0. Thus, lim d(x ni, x mi ) = ɛ 0. Since ɛ 0 < ɛ. So we can set ˆɛ = ɛ ɛ 0 2. For this ˆɛ > 0. It follows from lim d(x ni, x mi ) = ɛ 0 that there exists a positive integer N such that d(x ni, x mi ) ɛ 0 < ˆɛ whenever i N. This implies that d(x ni, x mi ) < ɛ 0 + ˆɛ = ɛ + ɛ 0 2 < ɛ whenever i N. (3.6) By applying that T is (η, β)-(ulgc) on (3.6), we have d(x ni +1, x mi +1) = d(t x ni, T x mi ) β(d(x ni, x mi ))d(x ni, x mi ) whenever i N.

Fixed point theorem of uniformly locally Geraghty contractive mappings 451 Thus, d(x ni +1, x mi +1) d(x ni, x mi ) β(d(x ni, x mi )) whenever i N. Notice that lim d(x ni +1, x mi +1) = ɛ 0. Indeed, since d(x ni +1, x mi +1) d(x ni +1, x ni )+ d(x ni, x mi 1)+d(x mi 1, x mi )+d(x mi, x mi +1) i N. We have lim sup d(x ni +1, x ni )+lim supd(x ni, x mi 1)+lim sup lim sup = 0 + lim sup d(x ni +1, x mi +1) d(x mi 1, x mi )+lim supd(x mi, x mi +1) d(x ni, x mi 1) + 0 + 0 = lim sup d(x ni, x mi 1) ɛ 0. That is, d(x ni +1, x mi +1) ɛ 0. On the other hand, since d(x ni, x mi ) d(x ni, x ni +1)+ lim sup d(x ni +1, x mi +1) + d(x mi +1, x mi ) i N. We have d(x ni, x mi ) d(x ni, x ni +1) d(x mi +1, x mi ) d(x ni +1, x mi +1) i N. So lim inf lim inf d(x n i +1, x mi +1) lim inf [ d(x n i, x mi ) d(x ni, x ni +1) d(x mi +1, x mi ) ] d(x n i, x mi ) + lim sup That is, ɛ 0 lim inf d(x ni, x ni +1) + lim sup lim inf d(x mi +1, x mi ) = lim inf d(x n i, x mi ) + 0 + 0 = ɛ 0. d(x n i +1, x mi +1). Thus, lim sup d(x n i +1, x mi +1). It follows from lim inf that lim d(x ni +1, x mi +1) = ɛ 0. Since and d(x ni +1, x mi +1) lim d(x ni, x mi ) lim d(x n i +1, x mi +1) lim d(x n i, x mi ) d(x ni +1, x mi +1) ɛ 0 d(x n i +1, x mi +1) lim sup lim β(d(x ni, x mi )) 1 = ɛ 0 ɛ 0 = 1. d(x ni +1, x mi +1) So lim β(d(x ni, x mi )) = 1. This implies that lim d(x ni, x mi ) = 0, but contradicts that ɛ 0 > 0. Case 2: ɛ 0 ɛ. By Claim2 that lim d(x n 1, x n ) = 0. There exists a positive integer L such that d(x n 1, x n ) < ɛ m L n whenever n L where m L > n L L with L d(x nl, x ml ) ɛ 0. So ɛ 0 d(x nl, x ml ) d(x nl, x nl +1) + d(x nl +1, x nl +2) + d(x nl +2, x nl +3) +... + d(x ml 1, x ml ) < ɛ m L n + ɛ L m L n + ɛ L m L n + L... + ɛ m L n = ɛ. This contradicts to ɛ 0 ɛ. L According to arguments case 1 and case 2, {x n } is a Cauchy sequence. Because of (X, d) is complete. Sequence {x n } must converge to a point in X, say a. Claim4: a = T (a). Fact1: d(a, T a) < ξ ξ > 0 with ξ < ɛ. Let ξ > 0 be given with ξ < ɛ. Since lim x n = a. So there exists a positive

452 Ing-Jer Lin and Chien-Lung Wu integer K such that d(x n, a) < ξ 2 < ɛ < η whenever n K. In particular, d(x K, a) < ξ 2 < ɛ < η (3.7) and d(x K+1, a) < ξ 2 < ɛ < η. By applying that T is (η, β)-(ulgc) on (3.7), we have So Thus, d(t x K, T a) β(d(x K, a))(d(x K, a)) < d(x K, a). d(x K+1, T a) < d(x K, a) < ξ 2. d(a, T a) d(a, x K+1 ) + d(x K+1, T a) < ξ 2 + ξ 2 = ξ. Since ξ > 0 is arbitrary and ξ < ɛ. Therefore, d(a, T a) < ξ ξ > 0 with ξ < ɛ. Fact2: If d(a, T a) < ξ ξ > 0 with ξ < ɛ, then d(a, T a) = 0. Suppose not. That is, d(a, T a) > 0. By Archimedean Principle, there exists a positive integer N 1 such that 1 N 1 < d(a, T a). By Archimedean Principle again, there exists a positive integer N 2 such that 1 N2 < ɛ. Set M = max{n 1, N 2 }. So 1 M 1 N 1 < d(a, T a), and 1 M 1 N 2 < ɛ. This means that there exists a number 1 M with 0 < 1 M < ɛ such that 1 M < d(a, T a), but contradicts that d(a, T a) < ξ ξ > 0 with ξ < ɛ. Therefore, d(a, T a) = 0. i.e. a = T a. Claim5: a is unique. Suppose that there exist a and b in X such that T (a) = a and T (b) = b. By Theorem 2.4., there exist a positive number ɛ and a sequence c (1,0), c (1,1), c (1,2),..., c (1,l) such that a = c (1,0), b = c (1,l), and d(c (1,i 1), z (1,i) ) < ɛ < η i = 1, 2, 3,..., l. (3.8) By applying that T is (η, β)-(ulgc) on (3.8), we have d(t c (1,i 1), T c (1,i) ) β(d(c (1,i 1), c (1,i) ))d(c (1,i 1), c (1,i) ) < d(c (1,i 1), c (1,i) ) i = 1, 2, 3,..., l.

Fixed point theorem of uniformly locally Geraghty contractive mappings 453 In addition, we set c (2,i 1) = T (c (1,i 1) ) i = 1, 2, 3,..., l + 1. Thus we construct a sequence c (2,0), c (2,1), c (2,2),..., c (2,l) such that a = c (2,0), b = c (2,l), and d(c (2,i 1), z (2,i) ) < ɛ < η i = 1, 2, 3,..., l. (3.9) By applying that T is (η, β)-(ulgc) on (3.9), we have d(t c (2,i 1), T c (2,i) ) β(d(c (2,i 1), c (2,i) ))d(c (2,i 1), c (2,i) ) < d(c (2,i 1), c (2,i) ) i = 1, 2, 3,..., l. In addition, we set c (3,i 1) = T (c (2,i 1) ) i = 1, 2, 3,..., l + 1. Thus we construct a sequence c (3,0), c (3,1), c (3,2),..., c (3,l) such that a = c (3,0), b = c (3,l), and d(c (3,i 1), z (3,i) ) < ɛ < η i = 1, 2, 3,..., l. where a = c (3,0), b = c (3,l), and d(c (3,i 1), c (3,i) ) < ɛ < η i = 1, 2, 3,..., l. Continuing in this way, we obtain a diagram as follows: where In addition, a = c (1,0), c (1,1), c (1,2),..., c (1,l) = b a = c (2,0), c (2,1), c (2,2),..., c (2,l) = b a = c (3,0), c (3,1), c (3,2),..., c (3,l) = b a = c (n,0), c (n,1), c (n,2),..., c (n,l) = b a = c (n+1,0), c (n+1,1), c (n+1,2),..., c (n+1,l) = b d(c (n+1,i 1), c (n+1,i) ) β(d(c (n,i 1), c (n,i) ))d(c (n,i 1), c (n,i) ) < d(c (n,i 1), c (n,i) ) < ɛ < η i = 1, 2, 3,..., l n N d(a, b) d(c (n,0), c (n,1) )+d(c (n,1), c (n,2) )+d(c (n,2), c (n,3) )+...+d(c (n,l 1), c (n,l) ) n N.

454 Ing-Jer Lin and Chien-Lung Wu Then lim d(a, b) lim d(c (n,0), c (n,1) )+ lim d(c (n,1), c (n,2) )+ lim d(c (n,2), c (n,3) )+...+ lim d(c (n,l 1), c (n,l) ) Applying the conclusion of Claim1: i = 1, 2, 3,..., l, lim d(c (n,i 1), c (n,i) ) = 0. We have d(a, b) 0. This implies that d(a, b) = 0. Therefore, a = b. i.e. a is the unique fixed point of T. Example 3.2. Let C[0, 1] denote the metric space of all continuous functions f : [0, 1] R, with the metric d(f, g) = f(x) g(x). Then (C[0, 1], d) is a connected complete metric space. We define T : C[0, 1] C[0, 1] by T (f(x)) = 1 0 sup x [0,1] (x t)f(t)dt f C[0, 1]. Then T is (η, β)-(ulgc) { where η = 1 2 and β : [0, ) [0, 1) with β(u) = 0, if u = 0 1. 1 + u, if u > 0 Indeed, whenever f and g are in C[0, 1] with d(f, g) < η, we have d(f, g) + 1 < η + 1 = 2 3. So β(d(f, g)) = 1 1 + d(f, g) > 3 2 > 2 1. Thus, d(t f(x), T g(x)) = T f(x) T g(x) sup x [0,1] = sup 1 x [0,1] 0 1 (x t)[f(t) g(t)] dt sup x [0,1] 1 0 x t f(t) g(t) dt sup x t sup f(t) g(t) dt = d(f, g)( sup x t dt) = 1 d(f, g) x [0,1] 0 t [0,1] x [0,1] 0 2 < β(d(f, g))d(f, g) whenever f and g are in C[0, 1] with d(f, g) < η. By applying our Theorem 3.1., T has a unique fixed point h in C[0, 1] such that T (h) = h. Corollary 3.3. Assume that (X, d) is a complete metric space and T : X X is (η, β)-(ulgc). If X has a finite number of components, then T n 0 = T T has a fixed point for some n 0 N. Proof. Since X has a finite number of components. We can assume that m C 1, C 2, C 3,..., C m are distinct components of X. Then X = C i and each C i is connected and closed. It follows from T is (η, β)-(ulgc) that T : X X is a continuous map. Now, observe component C 1. Becouse of connectedness is preserved by continuity, we have T n [C 1 ] is connected for all n N. This implies that there exist two positive integers i 0 and j 0 with i 0 < j 0 such that T i 0 [C 1 ] and T j 0 [C 1 ] are contained in the same component of X, call it C. Then 1 i=1

Fixed point theorem of uniformly locally Geraghty contractive mappings 455 T j 0 i 0 [C] C. Set n 0 = j 0 i 0. We have T n 0 [C] C. Since C is closed and (X, d) is complete. So (C, d) is also complete. Notice that C is connected. Then, consider the restriction of T n 0 : X X to subset C X, denoted by T n 0 C. By applying our Theorem 3.1. to T n 0 C : C C, T n 0 has a fixed point in (C, d). Therefore, we find a positive integer n 0 such that T n 0 has a fixed point in (X, d). Example 3.4. Let X = [2, 4] [6, 8] [10, 12] [14, 16] with d be usual 9 + x 2, if x [2, 4] 11 + x metric and T : X X be defined by T (x) = 2, if x [6, 8] 1 + x 2, if x [10, 12] x 2 5, if x [14, 16]. Then, (X, d) is a complete metric space with finite number of components and T : X { X is (η, β)-(ulgc) where η = 1 and β : [0, ) 0, if u = 0 [0, 1) with β(u) = ln(1 + u). Indeed, for d(x, y) = 0, we u, if u > 0 have d(t x, T y) = β(d(x, y))d(x, y). For all d(x, y) (0, 1), notice that function ln(1 + u) is continuous on [0, d(x, y)] and differentiable on (0, d(x, y)). It follows from Mean Value Theorem that there exists a number c (0, d(x, y)) such ln(1 + d(x, y)) that β(d(x, y)) = = d(x, y) 1 + 1 c > 2 1. Then, d(t x, T y) = 2 1 d(x, y) < β(d(x, y))d(x, y). Thus, d(t x, T y) β(d(x, y))d(x, y) whenever x and y are in X with d(x, y) < η. By applying Corollary 3.3., there exists a positive integer n 0 such that T n 0 has fixed point in X. Acknowledgements. The authors wish to express their hearty thanks to Professor Wei-Shih Du for his valuable suggestions and comments. References [1] Stefan Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fundamenta Mathematicae, 3 (1922), 133-181. [2] Krzysztof Chris Ciesielski and Jakub Jasinski, On fixed points of locally and pointwise contracting maps, Topology and its Applications, 204 (2016), 70-78. https://doi.org/10.1016/j.topol.2016.02.011 [3] Dan Dumitru, Attractors of infinite iterated function systems containing contraction type functions, Annals of the Alexandru Ioan Cuza University- Mathematics, (2013). https://doi.org/10.2478/v10157-012-0044-5

456 Ing-Jer Lin and Chien-Lung Wu [4] Michael Edelstein, An extension of Banach s contraction principle, Proceedings of the American Mathematical Society, 12 (1961), 7-10. https://doi.org/10.2307/2034113 [5] Ryszard Engelking, General Topology, Heldermann, Berlin, 1989. [6] Michael A. Geraghty, On contractive mappings, Proceedings of the American Mathematical Society, 40 (1973), 604-608. https://doi.org/10.2307/2039421 [7] Thakyin Hu, Fixed point theorems for multi-valued mappings, Canadian Mathematical Bulletin, 23 (1980), no. 2, 193-197. https://doi.org/10.4153/cmb-1980-026-2 [8] Peter K.F. Kuhfittig, Fixed points of locally contractive and nonexpansive set-valued mappings, Pacific Journal of Mathematics, 65 (1976), 399-403. https://doi.org/10.2140/pjm.1976.65.399 [9] Vincenzo La Rosa and Pasquale Vetro, Fixed points for Geraghty-contractions in partial metric spaces, Journal of Nonlinear Sciences and Applications, 7 (2014), 1-10. [10] K.P.R. Sastry, G.V.R. Babu, K.K.M. Sarma and P. H. Krishna, Fixed points of Geraghty contractions in partially ordered metric spaces under the influence of altering distances, International Journal of Mathematical Archive, 5 (2014), no. 10, 185-195. Received: April 15, 2017; Published: May 3, 2017