Sequences of (ψ, φ)-weakly Contractive Mappings and Stability of Fixed Points

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1 Int. Journal of Math. Analysis, Vol. 7, 2013, no. 22, HIKARI Ltd, Sequences of (ψ, φ)-weakly Contractive Mappings and Stability of Fixed Points S. N. Mishra 1, Rajendra Pant 2 and R. Panicker 3 1,3 Department of Mathematics Walter Sisulu University, Mthatha 5117, South Africa 2 Department of Mathematics Visvesvaraya National Institute of Technology Nagpur , Maharashtra, India 1 swaminathmishra3@gmail.com, 2 pant.rajendra@gmail.com 3 rpanicker@wsu.ac.za Copyright c 2013 S. N. Mishra et al. This is an open access article 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 Stability results for a much wider class of (ψ,φ)-weakly contractive mappings are proved in a metric space. These results include a number of known results. Mathematics Subject Classification: 54H25; 47H10 Keywords: stability, (ψ, φ)-weakly contractive mappings; metric space 1 Introduction and Preliminaries Recently, Barbet and Nachi [5] (see also [4]) obtained some stability results using certain new notions of convergence over a variable domain in a metric space. The above results include the earlier results of Bonsall [6] and Nadler[19]. For some useful references on stability of fixed points, we refer to [1, 3, 21 25]. The results in [5] have been further generalized by Mishra et al. [14 18] in

2 1086 S. N. Mishra, Rajendra Pant and R. Panicker different settings. In this paper, we extend the results of Barbet and Nachi [5] to a much wider class of (ψ, φ)-weakly contractive mappings (see Remark 1.1 below) which include the well known contraction mappings and weakly contractive mappings (see [2] and [20] for details). Let (X, d) be a metric space and T : X X. Then T is called a contraction mapping if there exists a constant k (0, 1) such that for all x, y X. d(tx,ty) kd(x, y) (1.1) T : X X is called weakly contractive, if d(tx,ty) d(x, y) φ (d(x, y)) (1.2) for all x, y X, where φ :[0, ) [0, ) is a continuous and nondecreasing function such that φ(t) = 0 if and only if t =0. The above concept was introduced and studied first by Alber and Guerre- Delabriere [2] in a Hilbert space. As shown in [2], weakly contractive mappings possess a unique fixed point in a Hilbert space. Rhoades [20] extended the above result of [2] to complete metric spaces. T : X X is called (ψ, φ)-weakly contractive if ψ(d(tx,ty)) ψ(d(x, y)) φ(d(x, y)) (1.3) for all x, y X, where ψ, φ :[0, ) [0, ) are both continuous functions such that ψ(t),φ(t) > 0 for t (0, ) and ψ(0) = 0 = φ(0). In addition, φ is nondecreasing and ψ is increasing(strictly). The above class of mappings was initially introduced and studied by Dutta and Choudhury [10] where both φ and ψ were assumed to be nondecreasing. However, recently it was observed by Bose and Roychowdhury [7] (see also [9]) that, in the above definition the function ψ must be assumed to be strictly increasing. Remark 1.1. It is interesting to note that if one takes φ(t) =(1 k)t, where 0 <k<1, then (1.2) reduces to (1.1). When ψ(t) =t, then condition (1.3) recovers condition (1.2). Therefore (1.1) (1.2) (1.3). However, the above implication is not reversible (see [10, Example 2.2]. This shows the generality of (ψ, φ)-weakly contractive mappings. Now onwards, N will denote the set of real numbers and N = N { }.

3 Sequences of (ψ,φ)-weakly contractive mappings Barbet-Nachi type convergence The following notions of convergence are due to Barbet and Nachi [5] and present generalizations of the well known notions of pointwise and uniform convergence. Let {X n } be a family of nonempty subsets of X and {T n : X n X} a family of mappings. Then T is called a (G)-limit of the sequence {T n } or, equivalently {T n } satisfies the property (G), if the following condition holds: (G) Gr(T ) lim inf Gr(T n ): for every x X, there exist a sequence {x n } in X n such that: lim d(x n,x) = 0 and lim d(t n x n,t x)=0, n n where Gr(T ) stands for the graph of T. T is called a (G )-limit of the sequence {T n } or, equivalently {T n } satisfies the property (G ) if the following condition holds: (G ) Gr(T ) lim sup Gr(T n ): for every x X, there exist a sequence {x n } in X n such that: lim j d(x nj,x) = 0 and lim j d(t nj x nj,t x)=0, where {x nj } is a subsequence of {x n }. Further, T is called an (H)-limit of the sequence {T n } or, equivalently {T n } satisfies the property (H) if the following condition holds: (H) For all sequences {x n } in X n, there exists a sequence {y n } in X such that: lim n d(x n,y n ) = 0 and lim n d(t n x n,t y n )=0. Remark 2.1. The sequential form of limits as indicated in (G) and (G ) is obtained by using the definitions of sequences of sets and the graph of a function.

4 1088 S. N. Mishra, Rajendra Pant and R. Panicker 3 Stability results for (G)- convergence We now give a sufficient condition for uniqueness of the (G)- limit mapping of a sequence of mappings satisfying condition (1.3) Proposition 3.1. Let X be a metric space, {X n } a family of nonempty subsets of X and {T n : X n X} a sequence of (ψ, φ)-weakly contractive mappings. If T : X X is a (G)-limit of {T n }, then T is unique. Proof. Let T,T : X X be two (G)- limits of the sequence {T n }. Then for any point x X, there exist two sequences {x n } and {y n } in converging to x such that {T n x n } and {T n y n } converge to T x and T x respectively. Thus lim d(t nx n,t x) = 0 and lim d(t n y n,t x)=0. (3.1) Since {x n } and {y n } converge to x, we have Further, d(x n,y n ) d(x n,x)+d(y n,x) 0asn. (3.2) d(t x, T x) d(t x, T n x n )+d(t n x n,t n y n )+d(t n y n,t x). (3.3) Since T n is (ψ, φ)-weakly contractive for each n N, which implies that ψ (d(t n x n,t n y n )) ψ (d(x n,y n )) φ (d(x n,y n )), ψ (d(t n x n,t n y n )) ψ (d(x n,y n )). As ψ is increasing, from the above inequality we have From (3.3) and (3.4) we get, X n d(t n x n,t n y n ) d(x n,y n ). (3.4) d(t x, T x)) d(t x, T n x n )+d(x n,y n )+d(t n y n,t x). Letting n and using (3.1) and (3.2), the above expression tends to zero and we deduce that T x = T x.

5 Sequences of (ψ,φ)-weakly contractive mappings 1089 The following result in [5, Proposition 1] follows directly from Proposition 3.1. Corollary 3.2. Let X be a metric space, {X n } a family of nonempty subsets of X and {T n : X n X} a sequence of k-contraction mappings. If T : X X is a (G)-limit of {T n }, then T is unique. Proof. It comes from Proposition 3.1 by taking ψ(t) =t and φ(t) =(1 k)t, 0 <k<1. We now prove the following theorem which is our first stability result. Theorem 3.3. Let X be a metric space, {X n } a family of nonempty subsets of X and {T n : X n X} a family of mappings satisfying the property (G) such that for all n N, T n is a (ψ, φ)-weakly contractive mapping. If for all n N, x n is a fixed point of T n, then the sequence {x n } converges to x. Proof. Let x n be a fixed point of T n for each n N. Then, since the property (G) holds and x X, there exists a sequence {y n } in X n such that y n x and T n y n T x. Therefore ψ (d(x n,x )) = ψ (d(t n x n,t x )) ψ (d(t n x n,t n y n )+d(t n y n,t x )). Taking the limit as n and using the continuity of ψ and φ, we obtain lim n,x )) lim ψ (d(t n x n,t n y n )) lim [ψ (d(x n,y n )) φ (d(x n,y n ))] (by condition (1.3)) lim [ψ (d(x n,x )+d(y n,x ))] lim [φ (d(x n,x )+d(y n,x ))] = lim ψ (d(x n,x )) lim φ (d(x n,x )). Thus lim n,x )) 0. By the property of φ, we get lim d(x n,x ) = 0. Hence the conclusion follows. The following result in [5, Theorem 2] follows from the above theorem in view of Remark 1.1. Corollary 3.4. Let X be a metric space, {X n } a family of nonempty subsets of X and {T n : X n X} a family of mappings satisfying the property (G) such that for all n N, T n is a k-contraction from X n into X. If for all n N, x n is a fixed point of T n, then the sequence {x n } converges to x.

6 1090 S. N. Mishra, Rajendra Pant and R. Panicker When X n = X for all n N, X is complete and ψ(t) =t, φ(t) =(1 k)t, for all t>0 and k (0, 1), then we get the following result of Bonsall [6, Theorem 2] as a consequence of Theorem 3.3. Corollary 3.5. Let X be a complete metric space, and {T n : X X} a family of k-contraction mappings with Lipschitz constant k<1 and such that the sequence {T n } converges pointwise to T. Then for all n N, T n has a unique fixed point x n and the sequence {x n } converges to x. The following theorem proves the existence of a fixed point for a (G)-limit of a sequence of (ψ, φ)-weakly contractive mappings. Theorem 3.6. Let X be a metric space, {X n } a family of nonempty subsets of X and {T n : X n X} a family of mappings satisfying the property (G) such that for any n N, T n is a (ψ, φ)-weakly contractive mapping. Assume that for any n N, x n is a fixed point of T n. Then T admits a fixed point {x n } converges and lim x n X {x n } admits a subsequence converging to a point of X. Proof. The necessary part follows from Theorem 3.3. To prove the sufficiency, let {x nj } be a subsequence of {x n } such that lim x nj = x X. By the j property (G), there exists a sequence {y n } in X n such that y n x and T n y n T x as n. For any j N, we have d(x,t x ) d(x,x nj )+d(t nj x nj,t nj y nj )+d(t nj y nj,t x ). (3.5) By condition (1.3), which implies that Then, from (3.5), ψ ( d(t nj x nj,t nj y nj ) ) ψ ( d(x nj,y nj ) ) φ ( d(x nj,y nj ) ) ψ ( d(x nj,y nj ) ), d(t nj x nj,t nj y nj ) d(x nj,y nj ). d(x,t x ) d(x,x nj )+d(x nj,y nj )+d(t nj y nj,t x ) d(x,x nj )+d(x nj,x )+d(y nj,x )+d(t nj y nj,t x ). Now passing over to the limit as j, we deduce that T x = x.

7 Sequences of (ψ,φ)-weakly contractive mappings 1091 Remark 3.7. Under the assumptions of Theorem3.6, and if, 1. lim inf X n X (ie; the limit of any convergent sequence {z n } is in X ), then T admits a fixed point {x n } converges. 2. lim sup X n X (ie; the cluster point of any sequence {z n } is in X ), then T admits a fixed point {x n } admits a convergent subsequence. Proposition 3.8. Let X be a metric space, {X n } a family of nonempty subsets of X and {T n : X n X} a family of mappings satisfying the property (G) and such that, for any n N, T n is a (ψ, φ)- weakly contractive mapping. Then T is (ψ, φ)-weakly contractive. Proof. Given two points x and y in X. By the property (G), there exist two sequences {x n } and {y n } in X n converging respectively to x and y such that the sequences {T n x n } and {T n y n } converge respectively to T x and T y. For any n N, by triangle inequality, we have d(t x, T y) d(t x, T n x n )+d(t n x n,t n y n )+d(t n y n,t y). Since ψ is increasing, ψ (d(t x, T y)) ψ (d(t x, T n x n )+d(t n x n,t n y n )+d(t n y n,t y)). Letting n, and using the continuity of both ψ and φ we have, ψ (d(t x, T y)) lim ψ (d(t n x n,t n y n )) lim [ψ (d(x n,y n )) φ (d(x n,y n ))] = ψ (d(x, y)) φ (d(x, y)), and the conclusion holds. The following result in [5, Proposition 4] follows from Proposition 3.8. Corollary 3.9. Let X be a metric space, {X n } a family of nonempty subsets of X and {T n : X n X} a family of mappings satisfying property (G) such that, for any n N, T n is a k n -contraction. Then T is a k-contraction where {k n } is a bounded (resp.convergent) sequence with k =: sup k n (resp. lim k n ). n n X n X n

8 1092 S. N. Mishra, Rajendra Pant and R. Panicker Under a compactness assumption, the existence of a fixed point of the (G)- limit mapping can be obtained from the existence of fixed points of the (ψ, φ)-weakly contractive mappings T n. Theorem Let {X n } be a family of nonempty subsets of a metric space X and {T n : X n X} a family of mappings satisfying the property (G) and such that, for any n N, T n is a (ψ, φ)-weakly contractive mapping. Assume that lim sup X n X and X n is relatively compact. If for any n N, T n admits a fixed point x n, then the (G)-limit mapping T admits a fixed point x and the sequence {x n } converges to x. Proof. Let x n be the fixed point of T n for each n N. From the compactness condition, there exists a convergent subsequence {x nj } of {x n }. Now, by Remark 3.7, T admits a fixed point x and by Theorem 3.6, the sequence {x n } converges to x. As a consequence of Theorem 3.10 and Remark 2.1, we have the following result in [5, Theorem7]. Corollary Let {X n } be a family of nonempty subsets of a metric space X and {T n : X n X} a family of mappings satisfying the property (G) such that for any n N, T n is a k-contraction. Assume that lim sup X n X and X n is relatively compact. If for any n N, T n admits a fixed point x n, then the (G)-limit mapping T admits a fixed point. Now we present a stability result for a sequence of mappings {T n } satisfying the property (G ). Theorem Let {X n } be a family of nonempty subsets of a metric space X and {T n : X n X} a family of (ψ, φ)-weakly contractive mappings satisfying the property (G ). If for any n N, x n is a fixed point of T n, then x is a cluster point of the sequence {x n }. Proof. By the property (G ), there exists a sequence {y n } in X n which has a subsequence {y nj } such that y nj x and T nj y nj T x as j.we have which implies that d(x nj,x ) d(t nj x nj,t nj y nj )+d(t nj y nj,t x ), ψ ( d(x nj,x ) ) ψ ( d(t nj x nj,t nj y nj )+d(t nj y nj,t x ) ).

9 Sequences of (ψ,φ)-weakly contractive mappings 1093 Taking the limit as j, lim j ψ ( d(x nj,x ) ) lim j ψ ( d(t nj x nj,t nj y nj ) ). Since each mapping T nj have is (ψ, φ)-weakly contractive and ψ is increasing, we lim j ψ ( d(x nj,x ) ) lim j [ ψ ( d(xnj,y nj ) ) φ ( d(x nj,y nj ) )] [ ( lim ψ d(xnj,x )+d(y nj,x ) ) φ ( d(x nj,x )+d(y nj,x ) )] j = lim ψ ( d(x nj,x ) ) lim φ ( d(x nj,x ) ). j j Hence, lim φ ( d(x nj,x ) ) =0. j Thus {x nj } converges to x, the fixed point of T. The following result in [5, Theorem 8] follows from Theorem Corollary Let {X n } a family of nonempty subsets of a metric space X and let {T n : X n X} a family of k-contraction mappings satisfying the property (G ). If for any n N, x n is a fixed point of T n, then x is a cluster point of the sequence {x n }. 4 Stability results for (H)- convergence Now, we present another stability result using the (H)-convergence. Theorem 4.1. Let X be a metric space, {X n } a family of nonempty subsets of a metric space X and let {T n : X n X} be a family of mappings satisfying the property (H) such that T is a (ψ, φ)-weakly contractive mapping. If for any n N, x n is a fixed point of T n, then the sequence {x n } converges to x. Proof. By property (H), there exists a sequence {y n } in X such that d(x n,y n ) 0 and d(t n x n,t y n ) 0. We have Since ψ is increasing, d(x n,x ) d(t n x n,t y n )+d(t y n,t x ). ψ (d(x n,x )) ψ (d(t n x n,t y n )+d(t y n,t x )).

10 1094 S. N. Mishra, Rajendra Pant and R. Panicker Taking the limit as n, lim n,x )) lim ψ (d(t y n,t x )) lim [ψ (d(y n,x )) φ (d(y n,x ))] lim [ψ(d(x n,y n )+d(x n,x ) φ(d(x n,y n )+d(x n,x ))] = lim ψ (d(x n,x )) lim φ (d(x n,x )). Thus lim φ (d(x n,x )) = 0, and hence the conclusion follows. The following result in [5, Theorem 11] follows directly from the above theorem. Corollary 4.2. Let X be a metric space, {X n } a family of nonempty subsets of a metric space X and let {T n : X n X} be a family of mappings satisfying the property (H) such that T is a k- contraction. If for any n N, x n is a fixed point of T n, then the sequence {x n } converges to x. References [1] S. P. Acharya, Convergence of a sequence of fixed points in a uniform space, Mat. Vesnik. 13(1976) (28), [2] Ya. I.Alber, and S. Guerre-Delabriere, Principle of weakly contractive mappings in Hilbert spaces. New results in operator theory and its applications, 7 22, Oper. Theory Adv. Appl., 98, Birkhäuser, Basel, [3] V. G. Angelov, A continuous dependence of fixed points of φ-contractive mappings in uniform spaces, Archivum Mathematicum (Brno) 28(1992), (3 4), [4] L. Barbet and K. Nachi, Convergence des points fixes de k-contractions (Convergence of fixed points of k-contractions), (2006) Preprint, University of Pau. [5] L. Barbet and K. Nachi, Sequences of contractions and convergence of fixed points, Monografias del Seminario Matemático García de Galdeano 33(2006), [6] F. F. Bonsall, Lectures on Some Fixed Point Theorems of Functional Analysis, Tata Institute of Fundamental Research, Bombay 1962.

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