Common fixed points of G-nonexpansive mappings on Banach spaces

Tripak Fixed Point Theory and Applications (2016) 2016:87 DOI 10.1186/s13663-016-0578-4

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Common fixed points of G-nonexpansive mappings on Banach spaces with a graph Orawan Tripak* *

Correspondence: [email protected] Department of Mathematics and Statistics, Faculty of Science, Prince of Songkla University, Songkhla, 90110, Thailand

Abstract In this paper, we prove the weak and strong convergence of a sequence {xn } generated by the Ishikawa iteration to some common fixed points of two G-nonexpansive mappings defined on a Banach space endowed with a graph. MSC: 47H09; 47E10; 47H10 Keywords: common fixed point; G-nonexpansive mappings; Ishikawa iteration; Banach space; directed graph

1 Introduction In , Banach proved a remarkable and powerful result called the Banach contraction principle. Because of its fruitful applications, the principle has been generalized in many directions. The recent version of the theorem was given in Banach spaces endowed with a graph. In , Jachymski [] gave a generalization of the Banach contraction principle to mappings on a metric space endowed with a graph. In , Aleomraninejad et al. [] presented some iterative scheme results for G-contractive and G-nonexpansive mappings on graphs. In , Alfuraidan and Khamsi [] defined the concept of G-monotone nonexpansive multivalued mappings defined on a metric space with a graph. In the same year, Alfuraidan [] gave a new definition of the G-contraction and obtained sufficient conditions for the existence of fixed points for multivalued mappings on a metric space with a graph, and also in [], he proved the existence of a fixed point of monotone nonexpansive mapping defined in a Banach space endowed with a graph. Recently, Tiammee et al. [] proved Browder’s convergence theorem for G-nonexpansive mapping in a Banach space with a directed graph. They also proved the strong convergence of the Halpern iteration for a G-nonexpansive mapping. Inspired by all aforementioned references, the author proves strong and weak convergence theorems for G-nonexpansive mappings using the Ishikawa iteration generated from arbitrary x in a closed convex subset C of a uniformly convex Banach space X endowed with a graph. 2 Preliminaries In this section, we recall some standard graph notations and terminology and also some needed results. © 2016 Tripak. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Tripak Fixed Point Theory and Applications (2016) 2016:87

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Let (X, d) be a metric space, and = {(x, x)|x ∈ X}. Consider a directed graph G for which the set V (G) of its vertices coincides with X and the set E(G) of its edges contains all loops. Assume that G has no parallel edges. Then G = (V (G), E(G)), and by assigning to each edge the distance between its vertices, G may be treated as a weighted graph. Definition . The conversion of a graph G is the graph obtained from G by reversing the direction of edges denoted by G– , and E G– = (x, y) ∈ X × X|(y, x) ∈ E(G) . Definition . Let x and y be vertices of a graph G. A path in G from x to y of length N (N ∈ N ∪ {}) is a sequence {xi }N i= of N +  vertices for which x = x,

xN = y,

and (xi , xi+ ) ∈ E(G) for i = , , . . . , N – .

Definition . A graph G is said to be connected if there is a path between any two vertices of the graph G. Definition . A directed graph G = (V (G), E(G)) is said to be transitive if, for any x, y, z ∈ V (G) such that (x, y) and (y, z) are in E(G), we have (x, z) ∈ E(G). The definition of a G-nonexpansive mapping is given as follows. Definition . Let C be a nonempty convex subset of a Banach space X, and G = (V (G), E(G)) a directed graph such that V (G) = C. Then a mapping T : C → C is G-nonexpansive (see [], Definition .(iii)) if it satisfies the following conditions. (i) T is edge-preserving. (ii) Tx – Ty ≤ x – y whenever (x, y) ∈ E(G) for any x, y ∈ C. Definition . ([]) Let C be a nonempty closed convex subset of a real uniformly convex Banach space X. The mappings Ti (i = , ) on C are said to satisfy Condition B if there exists a nondecreasing function f : [, ∞) → [, ∞) with f () =  and f (r) >  for all r >  such that, for all x ∈ C, max x – T x, x – T x ≥ f d(x, F) , where F = F(T ) ∩ F(T ) and F(Ti ) (i = , ) are the sets of fixed points of Ti . Definition . ([]) Let C be a subset of a metric space (X, d). A mapping T is semicompact if for a sequence {xn } in C with limn→∞ d(xn , Txn ) = , there exists a subsequence {xni } of {xn } such that xni → p ∈ C. Definition . A Banach space X is said to satisfy Opial’s property if the following inequality holds for any distinct elements x and y in X and for each sequence {xn } weakly convergent to x: lim inf xn – x < lim inf xn – y. n→∞

n→∞


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Definition . Let X be a Banach space. A mapping T with domain D and range R in X is demiclosed at  if, for any sequence {xn } in D such that {xn } converges weakly to x ∈ D and {Txn } converges strongly to , we have Tx = . Lemma . ([]) Let X be a uniformly convex Banach space, and {αn } a sequence in [δ,  – δ] for some δ ∈ (, ). Suppose that sequences {xn } and {yn } in X are such that lim supn→∞ xn ≤ c, lim supn→∞ yn ≤ c and lim supn→∞ αxn + ( – αn )yn = c for some c ≥ . Then limn→∞ xn – yn = . Lemma . ([]) Let X be a Banach space, and R >  be a fixed number. Then X is uniformly convex if and only if there exists a continuous, strictly increasing, and convex function g : [, ∞) → [, ∞) with g() =  such that λx + ( – λ)y ≤ λx + ( – λ)y – λ( – λ)g x – y for all x, y ∈ BR () = {x ∈ X|x ≤ R} and λ ∈ [, ]. Lemma . ([]) Let X be a Banach space that satisfies Opial’s property, and let {xn } be a sequence in X. Let x, y in X be such that limn→∞ xn – x and limn→∞ xn – y exist. If {xnj } and {xnk } are subsequences of {xn } that converge weakly to x and y, respectively, then x = y.

3 Main results Throughout the section, we let C be a nonempty closed convex subset of a Banach space X endowed with a directed graph G such that V (G) = C and E(G) is convex. We also suppose that the graph G is transitive. The mappings Ti (i = , ) are G-nonexpansive from C to C with F = F(T ) ∩ F(T ) nonempty. Let {xn } be a sequence generated from arbitrary x ∈ C, xn+ = ( – αn )xn + αn T yn , yn = ( – βn )xn + βn T xn , where {αn } and {βn } are real sequences in [, ]. We first begin by proving the following useful results. Proposition . Let z ∈ F be such that (x , z ), (y , z ), (z , x ), and (z , y ) are in E(G). Then (xn , z ), (yn , z ), (z , xn ), (z , yn ), and (xn , yn ) are in E(G). Proof We divide the proof into three parts. In the first part, with the assumption (x , z ), (y , z ) ∈ E(G), we will show by induction that (xn , z ), (yn , z ) ∈ E(G). Then, with the assumption (z , x ), (z , y ) ∈ E(G), we will again prove by induction that (z , xn ), (z , yn ) ∈ E(G). In the third part, we combine these two results using transitivity of G to get the statement in the proposition. Let (x , z ) and (y , z ) ∈ E(G). Then (T y , z ), (T x , z ) ∈ E(G) since Ti (i = , ) are edge-preserving. By the convexity of E(G) and (T y , z ), (x , z ) ∈ E(G), we have (x , z ) ∈ E(G). Then, by edge-preserving of T , (T x , z ) ∈ E(G). Again, by the convexity of E(G) and (T x , z ), (x , z ) ∈ E(G), we get (y , z ) ∈ E(G) and then (T y , z ) ∈ E(G). Next, we assume that (xk , z ), (yk , z ) ∈ E(G). Then (T xk , z ), (T yk , z ) ∈


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E(G) since Ti (i = , ) are edge-preserving. Since E(G) is convex, (xk+ , z ) ∈ E(G). Indeed, α(T yk , z ) + ( – α)(xk , z ) = αT yk + ( – α)xk , z = (xk+ , z ) ∈ E(G). Since T is edge-preserving, (T xk+ , z ) ∈ E(G). Using the convexity of E(G), we get (yk+ , z ) ∈ E(G). To be explicit, β(T xk+ , z ) + ( – β)(xk+ , z ) = βT xk+ + ( – β)xk+ , z = (yk+ , z ) ∈ E(G). Hence, by induction, (xn , z ), (yn , z ) ∈ E(G) for all n ≥ . Using a similar argument, we can show that (z , xn ), (z , yn ) ∈ E(G) under the assumption that (z , x ), (z , y ) ∈ E(G). Therefore, (xn , yn ) ∈ E(G) by the transitivity of G. Lemma . Let z ∈ F. Suppose that (x , z ), (y , z ), (z , x ), (z , y ) ∈ E(G) for arbitrary x in C. Then limn→∞ xn – z exists. Proof Notice that xn+ – z = ( – αn )xn + αn T yn – z ≤ ( – αn )xn – z + αn T yn – z ≤ ( – αn )xn – z + αn yn – z = ( – αn )xn – z + αn ( – βn )xn – ( – βn )z + βn (T xn – z ) ≤ ( – αn )xn – z + αn ( – βn )xn – z + αn βn xn – z = ( – αn )xn – z + αn xn – z = xn – z . Thus, limn→∞ xn – z exists. In particular, the sequence {xn } is bounded.

Lemma . If X is uniformly convex, {αn }, {βn } ⊂ [δ,  – δ] for some δ ∈ (,  ), and (x , z ), (y , z ), (z , x ), (z , y ) ∈ E(G) for arbitrary x in C and z ∈ F, then lim xn – T xn =  = lim xn – T xn .

n→∞

n→∞

Proof Let z ∈ F. Then, by the boundedness of {xn } and {T xn } there exists r >  such that xn – z , yn – z ∈ Br () for all n ≥ . Put c = limn→∞ xn – z . If c = , then by the G-nonexpansiveness of Ti (i = , ) we have xn – Ti xn ≤ xn – z + z – Ti xn ≤ xn – z + z – xn . Therefore, the result follows. Suppose that c > . Hence, by Lemma . together with the G-nonexpansiveness of T , we have  yn – z  = ( – βn )xn + βn T xn – z  = βn (T xn – z ) + ( – βn )(xn – z )


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≤ βn T xn – z  + ( – βn )xn – z  – βn ( – βn )g T xn – xn ≤ βn xn – z  + ( – βn )xn – z  = xn – z  . Thus, lim sup yn – z ≤ lim sup xn – z ≤ c. n→∞

n→∞

Notice also that  xn+ – z  = ( – αn )xn + αn T yn – z

≤ αn yn – z  + ( – αn )xn – z  – αn ( – αn )g T yn – xn ≤ xn – z  – αn ( – αn )g T yn – xn ≤ xn – z  – δ  g T yn – xn .

Thus, δ  g T yn – xn ≤ xn – z  – xn+ – z  . This implies that limn→∞ g(T yn – xn ) = , and since g is strictly increasing and continuous at , lim T yn – xn = .

()

n→∞

Since T is G-nonexpansive, we have xn – z ≤ xn – T yn + T yn – T z ≤ xn – T yn + yn – z . Taking lim inf yields c ≤ lim inf yn – z . n→∞

Hence, we have lim yn – z = c.

n→∞

Since lim βn (T xn – z ) + ( – βn )(xn – z ) = lim yn – z = c

n→∞

and lim sup T xn – z ≤ c, n→∞

n→∞


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by Lemma . we have lim T xn – xn = .

n→∞

()

By the G-nonexpansiveness of T together with xn – yn ≤ T xn – xn we have T xn – xn ≤ T xn – T yn + T yn – xn ≤ xn – yn + T yn – xn ≤ T xn – xn + T yn – xn . Using () and (), limn→∞ T xn – xn = . Hence, the lemma is proved.

Lemma . Suppose that X satisfies the Opial’s property and that (x , z ), (y , z ) are in E(G) for z ∈ F and arbitrary x ∈ C. Then I – Ti (i = , ) are demiclosed. Proof Suppose that {xn } is a sequence in C that converges weakly to q. From Lemma . we have limn→∞ xn – Ti xn = . Suppose for contradiction that q = Ti q. Then, by Opial’s property we have lim sup xn – q < lim sup xn – Ti q n→∞

n→∞

≤ lim sup xn – Ti xn + Ti xn – Ti q n→∞

≤ lim sup xn – q, n→∞

a contradiction. Hence, Ti q = q, so the conclusion holds.

Theorem . Suppose X is uniformly convex, {αn }, {βn } ⊂ [δ,  – δ] for some δ ∈ (,  ), Ti (i = , ) satisfy Condition B, F is dominated by x , F dominates x , and (x , z), (y , z), (z, x ), (z, y ) ∈ E(G) for each z ∈ F and arbitrary x ∈ C. Then {xn } converges strongly to a common fixed point of Ti . Proof Let z ∈ F. Recall the following facts from Lemma .: (i) {xn } is bounded; (ii) limn→∞ xn – z exists; (iii) xn+ – z ≤ xn – z for all n ≥ . They imply that d(xn+ , F) ≤ d(xn , F). Thus limn→∞ d(xn , F) exists. Since each Ti (i = , ) satisfies Condition B and limn→∞ xn – Ti xn = , we have lim f d(xn , F) = 

n→∞


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and then lim d(xn , F) = .

n→∞

Hence, there are a subsequence {xnj } of {xn } and a sequence {zj } ⊂ F satisfying xnj – zj ≤

 . j

Put nj+ = nj + k for some k ≥ . Then xnj+ – zj ≤ xnj +k– – zj ≤ xnj – zj ≤

 . j

Hence, zj+ – zj ≤

 , j+

so that {zj } is a Cauchy sequence. We assume that zj → q ∈ C as n → ∞. Since F is closed, q ∈ F. Hence, we have xnj → q as j → ∞, and since limn→∞ xn – q exists, the conclusion follows. Theorem . Suppose that X is uniformly convex, {αn }, {βn } ⊂ [δ,  – δ] for some δ ∈ (,  ), one of Ti (i = , ) is semicompact, F is dominated by x , F dominates x , and (x , z ), (y , z ), (z , x ), (z , y ) ∈ E(G) for z ∈ F and arbitrary x ∈ C. Then {xn } converges strongly to a common fixed point of Ti . Proof Suppose that T is semicompact; by Lemma . and Lemma . we have a bounded sequence {xn }, and limn→∞ xn – Ti xn = . Hence, by the semicompactness of T there exist q ∈ C and a subsequence {xnj } of {xn } such that xnj → q as j → ∞ and limn→∞ xnj – Ti xnj = . Notice that q – Ti q ≤ q – xnj + xnj – Ti xnj + Ti xnj – Ti q ≤ q – xnj + xnj – Ti xnj + xnj – q →  as n → ∞. Hence, q ∈ F. Since limn→∞ d(xn , F) = , it follows, by repeating the same argument as in the proof of Theorem ., that {xn } converges strongly to a common fixed point of Ti (i = , ), and the proof is complete. Theorem . Suppose that X is uniformly convex, {αn }, {βn } ⊂ [δ,  – δ] for some δ ∈ (,  ). If X satisfies Opial’s property, I – Ti is demiclosed at zero for each i, F is dominated by x , F dominates x , and (x , z ), (y , z ), (z , x ), (z , y ) ∈ E(G) for z ∈ F and arbitrary x ∈ C, then {xn } converges weakly to a common fixed point of Ti . Proof Note that by Lemma ., for each q ∈ F, lim xn – q

n→∞

exists.

()


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Let {xnk } and {xnj } be subsequences of the sequence {xn } with two weak limits q and q , respectively. Notice that, by Lemma ., xnj – Ti xnj →  as n → ∞ and xnk – Ti xnk →  as n → ∞. Hence, Ti q = q and Ti q = q . By Lemma . we have q , q ∈ F. In particular, q = q by Lemma .. Therefore, {xn } converges weakly to a common fixed point in F. Competing interests The author declares that she has no competing interests. Acknowledgements The author is grateful to Professor Suthep Suantai for valuable suggestion and comments. The author would also like to thank the anonymous reviewers for their helpful comments. Received: 12 February 2016 Accepted: 12 August 2016 References 1. Jachymski, J: The contraction principle for mappings on a metric space with a graph. Proc. Am. Math. Soc. 136, 1359-1373 (2008) 2. Aleomraninejad, SMA, Rezapour, S, Shahzad, N: Some fixed point result on a metric space with a graph. Topol. Appl. 159, 659-663 (2012) 3. Alfuraidan, MR, Khamsi, MA: Fixed points of monotone nonexpansive mappings on a hyperbolic metric space with a graph. Fixed Point Theory Appl. (2015). doi:10.1186/s13663-015-0294-5 4. Alfuraidan, MR: Remarks on monotone multivalued mappings on a metric space with a graph. J. Inequal. Appl. (2015). doi:10.1186/s13660-015-0712-6 5. Alfuraidan, MR: Fixed points of monotone nonexpansive mappings with a graph. Fixed Point Theory Appl. (2015). doi:10.1186/s13663-015-0299-0 6. Tiammee, J, Kaekhao, A, Suantai, S: On Browder’s convergence theorem and Halpern iteration process for G-nonexpansive mappings in Hilbert spaces endowed with graph. Fixed Point Theory Appl. (2015). doi:10.1186/s13663-015-0436-9 7. Shahzad, N, Al-Dubiban, R: Approximating common fixed points of nonexpansive mappings in Banach spaces. Georgian Math. J. 13(3), 529-537 (2006) 8. Schu, J: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bull. Aust. Math. Soc. 43(1), 153-159 (1991) 9. Xu, HK: Inequalities in Banach spaces with applications. Nonlinear Anal. 16(12), 1127-1138 (1991) 10. Suantai, S: Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings. J. Math. Anal. Appl. 331, 506-517 (2005)