The Mild Modification of the (DL)-Condition and Fixed Point Theorems for Some Generalized Nonexpansive Mappings in Banach Spaces

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1 Int. Journal of Math. Analysis, Vol. 6, 2012, no. 19, The Mild Modification of the (DL)-Condition and Fixed Point Theorems for Some Generalized Nonexpansive Mappings in Banach Spaces Kanok Chuikamwong Department of Mathematics Faculty of Science and Technology Rambhai Barni Rajabhat University Chantaburi 22000, Thailand Naknimit Akkasriworn Department of Mathematics Faculty of Science and Technology Rambhai Barni Rajabhat University Chantaburi 22000, Thailand Kritsana Sokhuma Department of Mathematics Faculty of Science and Technology Muban Chom Bueng Rajabhat University Ratchaburi 70150, Thailand k sokhuma@yahoo.co.th Abstract A Banach space X is said to satisfy property (D) if there exists α [0, 1) such that for any nonempty weakly compact convex subset E of X, any sequence {x n } E which is regular relative to E, and any sequence {y n } A(E,{x n }) which is regular relative to E, we have r(e,{y n }) αr(e,{x n }). A this property is the mild modification of the (DL)-condition. Let X be a Banach space satisfying property (D) and let E be a weakly compact convex subset of X. If T : E E is a mapping satisfying condition (E) and (C λ ) for some λ (0, 1). We study the existence of a fixed point for this mapping.

2 934 K. Chuikamwong N. Akkasriworn and K. Sokhuma Keywords: (DL)-condition, Fixed point theorem, Generalized nonexpansive mapping, Banach space 1 Introduction A mapping T on a subset E of a Banach space X is called a nonexpansive mapping if Tx Ty x y for all x, y E. In 2008, Suzuki [1] introduced a condition (C) and proved fixed point theorems and convergence results for mappings satisfying the condition (C). Moreover, he presented that the condition (C) is weaker than nonexpansiveness and stronger than quasinonexpansiveness. Recently, Garcia-Falset et al. [2] introduced two new classes of generalized nonexpansive mapping and studied both the existence of fixed points and their asymptotic behavior. In 2006, Dhompongsa et al. [3] presented the (DL)-condition. This condition describes the ratio of the Chebyshev radius of the asymptotic center of a bounded sequence to its asymptotic radius. It guarantees the existence of fixed points for compact convex valued nonexpansive mappings. In 2009, Dhompongsa et al. [4] proved a fixed point theorem for mappings with condition (C) on a Banach space such that its asymptotic center in a bounded closed and convex subset of each bounded sequence is nonempty and compact. Moreover, they presented fixed point theorems for this class of mappings defined on weakly compact convex subsets of Banach spaces satisfying property (D). In this paper we study the existence of fixed points for mappings satisfying condition (E) and (C λ ) for some λ (0, 1) defined on weakly compact and convex subsets of spaces with property (D). 2 Preliminary Notes Let E be a nonempty closed convex subset of a Banach space X and {x n } a bounded sequence in X. For x X, define the asymptotic radius of {x n } at x as the number Let and r(x, {x n }) = lim sup x n x. r r(e,{x n }) := inf{r(x, {x n }):x E} A A(E,{x n }):={x E : r(x, {x n })=r}.

3 The mild modification of the (DL)-condition 935 The number r and the set A are, respectively, called the asymptotic radius and asymptotic center of {x n } relative to E. It is known that A(E,{x n })is nonempty, weakly compact and convex as E is [5]. The sequence {x n } is called regular relative to E if r(e,{x n })=r(e,{x n }) for each subsequence {x n } of {x n }. The following lemma was proved by Goebel [6] and Lim [7]. Lemma 2.1 Let {x n } and E be as above. Then there exists a subsequence of {x n } which is regular relative to E. The following definition is mild modification of the (DL)-condition. It was defined by Dhompongsa et al. [4]. Definition 2.2 A Banach space X is said to satisfy property (D) if there exists α [0, 1) such that for any nonempty weakly compact convex subset E of X, any sequence {x n } E which is regular relative to E, and any sequence {y n } A(E,{x n }) which is regular relative to E we have r(e,{y n }) αr(e,{x n }). Suzuki [1] defined a class of the generalized nonexpansive mapping as follows. Definition 2.3 Let T be a mapping on a subset E of a Banach space X. Then T is said to satisfy condition (C) if 1 x Tx x y implies Tx Ty x y 2 for all x, y E. The following definitions were defined by Garcia-Falset et al. [2]. Definition 2.4 Let E be a nonempty subset of a Banach space X. For μ 1 we say that a mapping T : E X satisfy condition (E μ ) on E if there exists μ 1 such that for all x, y E, x Ty μ x Tx + x y. We say that T satisfies condition (E) one whenever T satisfies (E μ ) for some μ 1. Form Lemma 7 in [1] we know that if T : E E satisfies condition (C) one, then is satisfies condition (E 3 ). Definition 2.5 For λ (0, 1) we say that a mapping T : E X satisfy condition (C λ ) on E if for all x, y E with λ x Tx x y one has that Tx Ty x y. The following lemma was proved by Goebel and Kirk [8]. Lemma 2.6 Let {z n } and {w n } be two bounded sequences in a Banach space X, and let 0 <λ<1. If for every natural number n we have z n+1 = λw n +(1 λ)z n and w n+1 w n z n+1 z n, then lim w n z n =0.

4 936 K. Chuikamwong N. Akkasriworn and K. Sokhuma 3 Main Results In this section, we prove our main results. Lemma 3.1 Let T be a mapping defined on a bounded convex subset E of a Banach space X. Assume that T satisfies condition (C λ ). Define a sequence {x n } in E by x 1 E and x n+1 = λt x n +(1 λ)x n for n N, where λ (0, 1). Then {x n } is an approximate fixed point sequence for T, i.e., lim Tx n x n =0. Proof. From the assumption, we have λ x n Tx n = x n x n+1 for all n N. By the condition (C λ ), we have Tx n Tx n+1 x n x n+1. Now we can apply Lemma 2.6 to conclude that lim Tx n x n =0. Theorem 3.2 Let E be a nonempty bounded closed convex subset of a Banach space X. Let T : E E be a mapping satisfying condition (E) and (C λ ) for some λ (0, 1). Then T has a fixed point. Proof. Define a sequence {x n } by x 1 E and x n+1 = λt x n +(1 λ)x n for all n N, where λ (0, 1). Let A(E,{x n })={w}, we have w E. Since T satisfies the condition (E) we obtain x n Tw μ x n Tx n + x n w for some μ 1. From Lemma 3.1 and taking limit superior on both sides in above inequality, we obtain lim sup x n Tw lim sup x n w. By the uniqueness of the asymptotic center, we get Tw = w. Theorem 3.3 Let T be a mapping on a closed subset E of a Banach space X. Assume that T satisfies condition (C λ ) for some λ (0, 1). Then Fix(T ) is closed.

5 The mild modification of the (DL)-condition 937 Proof. Let {x n } be a sequence in Fix(T ) converging to some point x E. Since λ Tx n x n =0 x n x for n N and T satisfies condition (C λ ), then we have Tx n Tx x n x. It follows that lim sup x n Tx = lim sup Tx n Tx x n x =0. lim sup That is, {x n } converges to Tx. We have x = Tx. Hence Fix(T ) is closed. We give convergence theorem for mapping satisfying condition (C λ ) and (E). Theorem 3.4 Let E be a nonempty bounded closed convex subset of a Banach space X and let T : E E be a mapping satisfying condition (C λ ) and (E). Define a sequence {x n } in E by x 1 E and x n+1 = λt x n +(1 λ)x n for n N, where λ (0, 1). Then lim x n p exists for all p Fix(T ). Proof. From Theorem 3.2, we know that F ix(t ) is nonempty. Let p Fix(T ). Since λ Tp p =0 x n p for n N and T satisfies condition (C λ ), then we have Tx n Tp x n p. It follow that This is x n+1 p = λt x n +(1 λ)x n p = λt x n +(1 λ)x n p + λp λp = λt x n λp +(1 λ)(x n p) λ Tx n p +(1 λ) x n p = λ Tx n Tp +(1 λ) x n p λ x n p +(1 λ) x n p = x n p. x n+1 p x n p. Hence { x n p } is bounded and decreasing for all p Fix(T ). Thus we obtain the desired result. Lemma 3.5 Let E be a nonempty bounded closed convex subset of a Banach space X, and T : E E be a mapping satisfying condition (E) and (C λ ) for some λ (0, 1). Then A(E,{x n }) is invariant under T.

6 938 K. Chuikamwong N. Akkasriworn and K. Sokhuma Proof. Let z A(E,{x n }). By the condition (E), we have x n Tz μ Tx n x n + x n z. By Lemma 3.1, we get lim Tx n x n = 0. It follows that, r(tz,{x n }) = lim sup x n Tz μ lim sup = lim sup = r(z, {x n }). Therefore, we obtain Tz A(E,{x n }). Tx n x n + lim sup x n z x n z Lemma 3.6 Let T be a mapping on a subset E of a Banach space X. Let E be a nonempty bounded closed convex. Assume that T satisfies condition (E). Then holds for all x, y E. x Tx 2 x y + μ Ty y Proof. Let x, y E. By Lemma 2.4, we have x Tx x y + y Tx x y + μ Ty y + y x = 2 x y + μ Ty y. Theorem 3.7 Let X be a Banach space satisfying property (D) and let E be a weakly compact convex subset of X. If T : E E is a mapping satisfying condition (E) and (C λ ) for some λ (0, 1), then T has a fixed point. Proof. Let {x 0 n } be an approximate fixed point sequence for T in E. By the boundedness of {x 0 n} we can assume, by Lemma 2.1 that {x 0 n} is regular relative to E. Let A 0 = A(E,{x 0 n }). By Lemma 3.5, A0 is invariant under T. Thus, Lemma 3.1 provides us an approximate fixed point sequence {x 1 n } for T in A 0. We assume again that {x 1 n} is regular relative to A 0. Since X satisfies property (D), we obtain, for some α [0, 1), r(e,{x 1 n }) αr(e,{x0 n }). Continue the procedure to obtain a regular sequence relative to E {x m n } in A m 1 such that, for each m 1, lim Txm n xm n =0,

7 The mild modification of the (DL)-condition 939 and where A m := A(E,{x m n }). Therefore r(e,{x m n }) αr(e,{xm 1 n }), r(e,{x m n }) αr(e,{x m 1 n }) α m r(e,{x 0 n}). Let {x m = x m+1 m+1} be the diagonal sequence of the sequences {x m n }. We obtain {x m } is a Cauchy sequence. Indeed, for each m 1 we have for all positive integer n, Hence x m m xm n x m m xm n xm m xm 1 k + x m 1 k x m n for all k. lim sup k Thus, for each n, x m m xm 1 k + lim sup x m 1 k x m n 2r(E,{xm 1 k }). k x m 1 x m x m 1 x m n + xm n x m = x m m xm n + xm n xm+1 2r(E,{x m 1 k })+ x m n x m+1 m+1. Taking upper limit as n, we have x m 1 x m 2r(E,{x m 1 n })+r(e,{x m n }) 3α m 1 r(e,{x 0 n}). Since α<1, we have {x m } is Cauchy. Thus there exist x E such that x m converges to x. Next, we show that x is a fixed point of T. For each m 1, Lemma 3.6 implies that x m Tx m 2 x m n xm+1 m+1 + μ Txm n xm n. Taking upper limit as n, x m Tx m 2lim sup x m n xm+1 =2r(E,{xm n }) 2αm r(e,{x 0 n }). Hence lim m x m Tx m =0. Finally, we see, by Lemma 3.6 once again, that x Tx 2 x x m + μ x m Tx m. we get x Tx =0. Thusx Fix(T ).

8 940 K. Chuikamwong N. Akkasriworn and K. Sokhuma Corollary 3.8 Let X be a Banach space satisfying property (D) and let E be a weakly compact convex subset of X. If T : E E is a mapping satisfying condition (C), then T has a fixed point. Proof. Since T satisfying condition (C). We have T satisfying condition (C λ ) for some λ (0, 1) and E 3. So the result follows Theorem 3.7. ACKNOWLEDGEMENTS. We would like to thank the Rambhai Barni Rajabhat University and Muban Chom Bueng Rajabhat University, Thailand for supporting by grant fund. References [1] T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 340(2008) [2] J. Garcia-Falset, Enrique Llorens-Fuster and T. Suzuki, Fixed point theory for a class of generalized nonexpansive mappings, J. Math. Anal. Appl. 375(2011) [3] S. Dhompongsa, A. Kaewcharoen and A. Kaewkhao, The Domingues- Lorenzo condition and multivalued nonexpansive mappings, Nonlinear Analysis. 64(2006) [4] S. Dhompongsa, W. Inthakon and A. Kaewkhao, Edelstein s method and fixed point theorem for some generalized nonexpansive mappings, J. Math. Anal. Appl. 350(2009) [5] K. Goebel and W.A. Kirk, Topic in Metric Fixed point Theory, Cambridge University Press, Cambridge, [6] K. Goebel, On a fixed point theorem for multivalued nonexpansive mappings, Ann. Univ. Mariae Curie-Sklodowska 29(1975) [7] T.C. Lim, A fixed point theorem for multivalued nonexpansive mappings in a uniformly convex Banach space, Bull. Amer. Math. Soc. 80(1974) [8] K. Goebel and W.A. Kirk, Iteration processes for nonexpansive mappings, Topological Methods in Nonlinear Functionl Analysis, Amer. Math. Soc. 21(1983) Received: November, 2011