Strong convergence theorems for a countable family of totally quasi-ϕ-asymptotically nonexpansive nonself mappings in Banach spaces with applications

Journal of Inequalities and Applications, Nov 2012

In this paper, we introduce a class of totally quasi-ϕ-asymptotically nonexpansive nonself mappings and study the strong convergence under a limit condition only in the framework of Banach spaces. Meanwhile, our results are applied to study the approximation problem of a solution to a system of equilibrium problems. The results presented in the paper improve and extend the corresponding results of Chang et al. (Appl. Math. Comput. 218:7864-7870, 2012). MSC: 47J05, 47H09, 49J25.

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Strong convergence theorems for a countable family of totally quasi-ϕ-asymptotically nonexpansive nonself mappings in Banach spaces with applications

Journal of Inequalities and Applications Strong convergence theorems for a countable family of totally quasi-φ -asymptotically nonexpansive nonself mappings in Banach spaces with applications Li Yi 0 0 School of Science, Southwest University of Science and Technology , Mianyang, Sichuan 621010 , P.R. China In this paper, we introduce a class of totally quasi-φ-asymptotically nonexpansive nonself mappings and study the strong convergence under a limit condition only in the framework of Banach spaces. Meanwhile, our results are applied to study the approximation problem of a solution to a system of equilibrium problems. The results presented in the paper improve and extend the corresponding results of Chang et al. (Appl. Math. Comput. 218:7864-7870, 2012). MSC: 47J05; 47H09; 49J25 generalized projection; quasi-φ-asymptotically nonexpansive nonself mapping; totally quasi-φ-asymptotically nonexpansive nonself mapping; iterative sequence; nonexpansive retraction 1 Introduction Assume that X is a real Banach space with the dual X*, D is a nonempty closed convex subset of X. We also denote by J the normalized duality mapping from X to X* which is J(x) = f * ∈ X* : x, f * = x  = f *  , x ∈ X, where ·, · denotes the generalized duality pairing. Let D be a nonempty closed subset of a real Banach space X. A mapping T : D → D is A Banach space X is said to be strictly convex if x+y ≤  for all x, y ∈ X with x = y =  and x = y. A Banach space is said to be uniformly convex if limn→∞ xn – yn =  for any two sequences {xn}, {yn} ⊂ X with xn = yn =  and limn→∞ xn+yn = . The norm of a Banach space X is said to be Gâteaux differentiable if for each x, y ∈ S(x), exists, where S(x) = {x : x = , x ∈ X}. In this case, X is said to be smooth. The norm of a Banach space X is said to be Fréchet differentiable if for each x ∈ S(x), the limit (.) is attained uniformly for y ∈ S(x) and the norm is uniformly Fréchet differentiable if the limit (.) is attained uniformly for x, y ∈ S(x). In this case, X is said to be uniformly smooth. A subset D of X is said to be a retract of X if there exists a continuous mapping P : X → D such that Px = x for all x ∈ X. It is well known that every nonempty closed convex subset of a uniformly convex Banach space X is a retract of X. A mapping P : X → D is said to be a retraction if P = P. It follows that if a mapping P is a retraction, then Py = y for all y in the range of P. A mapping P : X → D is said to be a nonexpansive retraction if it is nonexpansive and it is a retraction from X to D. Next, we assume that X is a smooth, strictly convex and reflexive Banach space and D is a nonempty closed convex subset of X. In the sequel, we always use φ : X × X → R+ to denote the Lyapunov functional defined by φ(x, y) = x  –  x, Jy + y , x, y ∈ X. φ(y, x) = φ(y, z) + φ(z, x) +  z – y, Jx – Jz , x, y, z ∈ X, for all λ ∈ [, ] and x, y, z ∈ X. Following Alber [], the generalized projection D : X → D is defined by ∀x ∈ X. Many problems in nonlinear analysis can be reformulated as a problem of finding a fixed point of a nonexpansive mapping. In the sequel, we denote the strong convergence and weak convergence of the sequence {xn} by xn → x and xn x, respectively. Lemma . (see []) Let X be a smooth, strictly convex and reflexive Banach space and D be a nonempty closed convex subset of X. Then the following conclusions hold: (a) φ(x, y) =  if and only if x = y; (b) φ(x, Dy) + φ( Dy, y) ≤ φ(x, y), ∀x, y ∈ D; (c) if x ∈ X and z ∈ D, then z = Dx if and only if z – y, Jx – Jz ≥ , ∀y ∈ D. Remark . (see []) Let D be the generalized projection from a smooth, reflexive and strictly convex Banach space X onto a nonempty closed convex subset D of X. Then D is a closed and quasi-φ-nonexpansive from X onto D. Remark . (see []) If H is a real Hilbert space, then φ(x, y) = x – y , and metric projection of H onto D. Definition . Let P : X → D be a nonexpansive retraction. () A nonself mapping T : D → X is said to be quasi-φ-nonexpansive if F(T) = , and φ p, T(PT)n–x ≤ φ(p, x), ∀x ∈ D, p ∈ F(T), ∀n ≥ ; () A nonself mapping T : D → X is said to be quasi-φ-asymptotically nonexpansive if F(T) = , and there exists a real sequence kn ⊂ [, +∞), kn →  (as n → ∞), such that φ p, T(PT)n–x ≤ knφ(p, x), ∀x ∈ D, p ∈ F(T), ∀n ≥ ; () A nonself mapping T : D → X is said to be totally quasi-φ-asymptotically nonexpansive if F(T) = , and there exist nonnegative real sequences {vn}, {μn} with vn, μn →  (as n → ∞) and a strictly increasing continuous function ζ : R+ → R+ with ζ () =  such that φ p, T(PT)n–x ≤ φ(p, x) + vnζ φ(p, x) + μn, ∀x ∈ D, ∀n ≥ , p ∈ F(T). Remark . From the definitions, it is obvious that a quasi-φ-nonexpansive nonself mapping is a quasi-φ-asymptotically nonexpansive nonself mapping, and a quasi-φasymptotically nonexpansive nonself mapping is a totally quasi-φ-asymptotically nonexpansive nonself mapping, but the converse is not true. Example . (see []) Let H be a real Hilbert space, D be a nonempty closed and convex subset of H and f : D × D → R be a bifunction satisfying the conditions: (A) f (x, x) = , ∀x ∈ D; (A) f (x, y)+f (y, x) ≤ , ∀x, y ∈ D; (A) for each x, y, z ∈ D, limt→ f (tz +(–t)x, y) ≤ f (x, y); (A) for each given x ∈ D, the function y –→ f (x, y) is convex and lower semicontinuous. The so-called equilibrium problem for f is to find an x* ∈ D such that f (x*, y) ≥ , ∀y ∈ D. The set of its solutions is denoted by EP(f ). Let r > , x ∈ H and define a mapping Tr : D → D ⊂ H as follows: then () Tr is single-valued, and so z = Tr(x); () Tr is a relatively nonexpansive nonself mapping, therefore it is a closed quasi-φ-nonexpansive nonself mapping; () F(Tr) = EP(f ) and F(Tr) is a nonempty and closed convex subset of D; () Tr : D → D is nonexpansive. Since F(Tr) is nonempty, and so it is a quasi-φ-nonexpansive nonself mapping from D to H, where φ(x, y) = x – y , x, y ∈ H. Now, we give an example of a totally quasi-φ-asymptotically nonexpansive nonself mapping. Example . (see []) Let D be a unit ball in a real Hilbert space l, and let T : D → l be a nonself mapping defined by T : (x, x, . . .) → , x, ax, ax, . . . ∈ l, ∀(x, x, . . .) ∈ D, where {ai} is a sequence in (, ) such that i∞= ai =  . It is proved in Goebal and Kirk [] that (i) Tx – Ty ≤  x – y , ∀x, y ∈ D; (ii) T nx – T ny ≤  jn= aj, ∀x, y ∈ D, n ≥ . Let √k = , √kn =  jn= aj, n ≥ , then limn→∞ kn = . Letting νn = kn –  (n ≥ ), ζ (t) = t (t ≥ ) and {μn} be a nonnegative real sequence with μn → , then from (i) and (ii) we have T nx – T ny  ≤ x – y  + νnζ x – y  + μn, ∀x, y ∈ D. Since D is a unit ball in a real Hilbert space l, it follows from Remark . that φ(x, y) = x – y , ∀x, y ∈ D. The above inequality can be written as φ T nx, T ny ≤ φ(x, y) + νnζ φ(x, y) + μn, ∀x, y ∈ D. φ p, T (PT )n–x ≤ φ(p, x) + νnζ φ(p, x) + μn, ∀p ∈ F(T ), x ∈ D, where P is the nonexpansive retraction. This shows that the mapping T defined as above is a totally quasi-φ-asymptotically nonexpansive nonself mapping. Lemma . (see []) Let X be a uniformly convex and smooth Banach space, and let {xn} and {yn} be two sequences of X such that {xn} and {yn} are bounded; if φ(xn, yn) → , then xn – yn → . Lemma . Let X be a smooth, strictly convex and reflexive Banach space and D be a nonempty closed convex subset of X. Let T : D → X be a totally quasi-φ-asymptotically nonexpansive nonself mapping with μ = , then F(T ) is a closed and convex subset of D. Proof Let {xn} be a sequence in F(T ) such that xn → p. Since T is a totally quasi-φasymptotically nonexpansive nonself mapping, we have for all n ∈ N . Therefore, φ(p, Tp) = lim φ(xn, Tp) ≤ lim φ(xn, p) + vζ φ(xn, p) = φ(p, p) = . n→∞ n→∞ By Lemma ., we obtain Tp = p. So, we have p ∈ F(T ). This implies F(T ) is closed. Let p, q ∈ F(T ) and t ∈ (, ), and put w = tp + ( – t)q. We prove that w ∈ F(T ). Indeed, in view of the definition of φ, let {un} be a sequence generated by u = Tw, u = T (PT )w, u = T (PT )w, . . . , un = T (PT )n–w = TPun–, we have = w  –  tp + ( – t)q, Jun + un  = w  + tφ(p, un) + ( – t)φ(q, un) – t p  – ( – t) q . ≤ t φ(p, w) + vnζ φ(p, w) + μn + ( – t) φ(q, w) + vnζ φ(q, w) + μn = t p  –  p, Jw + w  + vnζ φ(p, w) + μn = t p  + ( – t) q  – w  + tvnζ φ(p, w) + ( – t)vnζ φ(q, w) + μn. Substituting (.) into (.) and simplifying it, we have φ(w, un) ≤ tvnζ φ(p, w) + ( – t)vnζ φ(q, w) + μn →  (as n → ∞). Hence, we have un → w. This implies that un+ → w. Since TP is closed and un+ = T (PT )nw = TPun, we have TPw = w. Since w ∈ C, and so Tw = w, i.e., w ∈ F(T ). This implies F(T ) is convex. This completes the proof of Lemma .. Definition . () (see []) A countable family of nonself mappings {Ti} : D → X is said to be uniformly quasi-φ-asymptotically nonexpansive if i∞= F(Ti) = , and there exist nonnegative real sequences kn ⊂ [, +∞), kn → , such that for each i ≥ , ∀x ∈ D, ∀n ≥ , p ∈ F(T ). () A countable family of nonself mappings {Ti} : D → X is said to be uniformly totally quasi-φ-asymptotically nonexpansive if i∞= F(Ti) = , and there exist nonnegative real sequences {vn}, {μn} with vn, μn →  (as n → ∞) and a strictly increasing continuous function ζ : R+ → R+ with ζ () =  such that for each i ≥ , φ p, Ti(PTi)n–x ≤ φ(p, x) + vnζ φ(p, x) + μn, ∀x ∈ D, ∀n ≥ , p ∈ F(T ). () (see []) A nonself mapping T : D → X is said to be uniformly L-Lipschitz continuous if there exists a constant L >  such that T (PT )n–x – T (PT )n–y ≤ L x – y , ∀x, y ∈ D, ∀n ≥ . Considering the strong and weak convergence of asymptotically nonexpansive self or nonself mappings, relatively nonexpansive, quasi-φ-nonexpansive and quasi-φ-asymptotically nonexpansive self or nonself mappings have been considered extensively by several authors in the setting of Hilbert or Banach spaces (see [–]). The purpose of this paper is to modify the Halpern and Mann-type iteration algorithm for a family of totally quasi-φ-asymptotically nonexpansive nonself mappings to have the strong convergence under a limit condition only in the framework of Banach spaces. As an application, we utilize our results to study the approximation problem of solution to a system of equilibrium problems. The results presented in the paper improve and extend the corresponding results of Chang et al. [, , , ], Su et al. [], Kiziltunc et al. [], Yildirim et al. [], Yang et al. [], Wang [, ], Pathak et al. [], Thianwan [], Qin et al. [], Hao et al. [], Guo et al. [], Nilsrakoo et al. [] and others. 2 Main results Theorem . Let X be a real uniformly smooth and uniformly convex Banach space, D be a nonempty closed convex subset of X. Let {Ti} : D → X be a family of uniformly totally quasiφ-asymptotically nonexpansive nonself mappings with sequences {vn}, {μn}, with vn, μn →  (as n → ∞), and a strictly increasing continuous function ζ : R+ → R+ with ζ () =  such that for each i ≥ , {Ti} : D → X is uniformly Li-Lipschitz continuous. Let {αn} be a sequence in [, ] and {βn} be a sequence in (, ) satisfying the following conditions: (i) limn→∞ αn = ; (ii)  < limn→∞ inf βn ≤ limn→∞ sup βn < . Let xn be a sequence generated by ⎧⎪ x ∈ X is arbitrary; D = D, ⎪⎪⎪⎨⎪ yn,i = J–[αnJx + ( – αn)(βnJxn + ( – βn)JTi(PTi)n–xn)] (i ≥ ), where ξn = vn supp∈F ζ (φ(p, xn)) + μn, F = i∞= F(Ti), Dn+ is the generalized projection of X onto Dn+. If F is nonempty, then {xn} converges strongly to F x. Proof (I) First, we prove that F and Dn are closed and convex subsets in D. In fact, by Lemma . for each i ≥ , F(Ti) is closed and convex in D. Therefore, F is a closed and convex subset in D. By the assumption that D = D is closed and convex, suppose that Dn is closed and convex for some n ≥ . In view of the definition of φ, we have i≥ i≥ z ∈ D : sup φ(z, yn,i) ≤ αnφ(z, x) + ( – αn)φ(z, xn) + ξn ∩ Dn i≥ z ∈ D : αn z, Jx + ( – αn) z, Jxn –  z, Jyn,i ≤ αn x  + ( – αn) xn  – yn,i  ∩ Dn. This shows that Dn+ is closed and convex. The conclusions are proved. (II) Next, we prove that F ⊂ Dn for all n ≥ . In fact, it is obvious that F ⊂ D. Suppose that F ⊂ Dn. Let wn,i = J–(βnJxn + ( – βn)JTi(PTi)n–xn). Hence for any u ∈ F ⊂ Dn, by (.), we have ≤ βnφ(u, xn) + ( – βn) φ(u, xn) + vnζ φ(u, xn) + μn where ξn = vn supp∈F ζ (φ(p, xn)) + μn. This shows that u ∈ F ⊂ Dn+ and so F ⊂ Dn. The conclusion is proved. (III) Now, we prove that {xn} converges strongly to some point p*. Since xn = Dn x, from Lemma .(c), we have xn – y, Jx – Jxn ≥ , ∀y ∈ Dn. Again since F ⊂ Dn, we have xn – u, Jx – Jxn ≥ , ∀u ∈ F . It follows from Lemma .(b) that for each u ∈ F and for each n ≥ , φ(xn, x) = φ( Dn x, x) ≤ φ(u, x) – φ(u, xn) ≤ φ(u, x). Therefore, {φ(xn, x)} is bounded, and so is {xn}. Since xn = Dn x and xn+ = Dn+ x ∈ Dn+ ⊂ Dn, we have φ(xn, x) ≤ φ(xn+, x). This implies that {φ(xn, x)} is nondecreasing. Hence, limn→∞ φ(xn, x) exists. By the construction of {Dn}, for any m ≥ n, we have Dm ⊂ Dn and xm = Dm x ∈ Dn. This shows that φ(xm, xn) = φ(xm, Dn x) ≤ φ(xm, x) – φ(xn, x) →  (as n → ∞). It follows from Lemma . that limn→∞ xm – xn = . Hence, {xn} is a Cauchy sequence in D. Since D is complete, without loss of generality, we can assume that limn→∞ xn = p* (some point in D). By the assumption, it is easy to see that nl→im∞ ξn = lim vn sup ζ φ(p, xn) + μn = . n→∞ p∈F Since xn → p*, it follows from (.) and Lemma . that expansive nonself mappings, we have φ p, Ti(PTi)n–xn ≤ φ(p, xn) + vnζ φ(p, xn) + μn, ∀x ∈ D, ∀n, i ≥ , p ∈ F(Ti). Since J– is uniformly continuous on each bounded subset of X*, it follows from (.) This implies that {Ti(PTi)n–xn} is uniformly bounded. ≤ xn + Ti(PTi)n–xn , this implies that {wn,i} is also uniformly bounded. for each i ≥ . By condition (ii), we have that Since J is uniformly continuous, this shows that for each i ≥ . Again, by the assumption that {Ti} : D → X is uniformly Li-Lipschitz continuous for each i ≥ , thus we have Ti(PTi)nxn – Ti(PTi)n–xn ≤ Ti(PTi)nxn – Ti(PTi)nxn+ + Ti(PTi)nxn+ – xn+ ≤ (Li + ) xn+ – xn + Ti(PTi)nxn+ – xn+ + xn – Ti(PTi)n–xn for each i ≥ . We get limn→∞ Ti(PTi)nxn – Ti(PTi)n–xn = . Since limn→∞ Ti(PTi)n–xn = P* and limn→∞ xn = p*, we have limn→∞ TiPTi(PTi)n–xn = p*. In view of the continuity of TiP, it yields that TiPp* = p*. Since p* ∈ C, it implies that Tip* = p*. By the arbitrariness of i ≥ , we have p* ∈ F . (V) Finally, we prove that p* = F x and so xn → F x = p*. Let w = F x. Since w ∈ F ⊂ Dn and xn = Dn x, we have φ(xn, x) ≤ φ(w, x). This implies that ⎪⎪⎪⎪⎨⎧⎪ yxn,i∈=XJ–is[aαrnbJxitr+ar(y;– αn)D(βn=JxDn+, ( – βn)JTi(PTi)n–xn)] (i ≥ ), where ξn = (kn – ) supp∈F φ(p, xn), F = i∞= F(Ti), Dn+ is the generalized projection of X onto Dn+. If F is nonempty, then {xn} converges strongly to F x. 3 Application In this section we utilize Corollary . to study a modified Halpern iterative algorithm for a system of equilibrium problems. We have the following result. Theorem . Let H be a real Hilbert space, D be a nonempty closed and convex subset of H. {αn}, (βn) be the same as in Theorem .. Let {fi} : D × D → R be a countable family of By Remark ., the following corollary is obtained. Corollary . Let X, D, {αn}, {βn} be the same as in Theorem .. Let {Ti} : D → X be a family of uniformly quasi-φ-asymptotically nonexpansive nonself mappings with the sequence kn ⊂ [, +∞), kn → , such that for each i ≥ , {Ti} : D → X is uniformly Li-Lipschitz continuous. Let xn be a sequence generated by F x. Therefore, xn → F x. The proof of Theorem . is be the family of mappings defined by (.), i.e., Let {xn} be the sequence generated by Tr,i(x) = z ∈ D, fi(z, y) + y – z, z – x ≥ , ∀y ∈ D , ∀x ∈ D ⊂ H. ∀y ∈ D, r > , i ≥ , ⎪⎪⎪⎪⎪⎪⎪ Dn+ = {z ∈ Dn : supi≥ z – yn,i  ≤ αn z – x  + ( – αn) z – xn }, ⎪⎩ xn+ = Dn+ x (n = , , . . .). If F = i∞= F(Tr,i) = , then {xn} converges strongly to F x, which is a common solution of the system of equilibrium problems for f . rewritten as follows: Therefore, the conclusion of Theorem . can be obtained from Corollary .. 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Li Yi. Strong convergence theorems for a countable family of totally quasi-ϕ-asymptotically nonexpansive nonself mappings in Banach spaces with applications, Journal of Inequalities and Applications, 2012, 268,