báo cáo hóa học: " Strong convergence theorems for system of equilibrium problems and asymptotically strict pseudocontractions in the intermediate sense" pot

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RESEA R C H Open Access Strong convergence theorems for system of equilibrium problems and asymptotically strict pseudocontractions in the intermediate sense Peichao Duan * and Jing Zhao * Correspondence: pcduancauc@126.com College of Science, Civil Aviation University of China, Tianjin 300300, PR China Abstract Let {S i } N i = 1 be N uniformly continuous asymptotically l i -strict pseudocontractions in the intermediate sense defined on a nonempty closed convex subset C of a real Hilbert space H. Consider the problem of finding a common element of the fixed point set of these mappings and the solution set of a system of equilibrium problems by using hybrid method. In this paper, we propose new iterative schemes for solving this problem and prove these schemes converge strongly. MSC: 47H05; 47H09; 47H10. Keywords: asymptotically strict pseudocontraction in the intermediate sense, system of equilibrium problem, hybrid method, fixed point 1. Introduction Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. A nonlinear mapping S : C ® C is a self mapping of C.Wedenotethesetoffixed points of S by F(S) (i.e., F(S)={x Î C : Sx = x}). Recall the following concepts. (1) S is uniformly Lipschitzian if there exists a constant L > 0 such that ||S n x −−S n y || ≤ L||x − y || for all inte g ers n ≥ 1andx, y ∈ C . (2) S is nonexpansive if ||Sx − S y || ≤ ||x − y || for all x, y ∈ C . (3) S is asymptotically nonexpansive if there exists a sequence k n of positive num- bers satisfying the property lim n®∞ k n = 1 and ||S n x − S n y || ≤ k n ||x − y || for all inte g ers n ≥ 1andx, y ∈ C . Duan and Zhao Fixed Point Theory and Applications 2011, 2011:13 http://www.fixedpointtheoryandapplications.com/content/2011/1/13 © 2011 Duan and Zhao; licensee Spring er. This is an Open Access article di stributed under the term s of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2 .0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (4) S is asymptotically nonexpansive in the intermediate sense [1] provided S is continuous and the following inequality holds: lim sup n→∞ sup x, y ∈C (||S n x − S n y|| − ||x − y||) ≤ 0 . (5) S is asymptotically l-strict pseudocontractive mapping [2] with sequence {g n }if there exists a constant l Î [0, 1) and a sequence {g n }in[0,∞) with lim n®∞ g n =0 such that ||S n x − S n y|| 2 ≤ ( 1+γ n ) ||x − y|| 2 + λ||x − S n x − ( y − S n y ) || 2 for all x, y Î C and n Î N. (6) S is asymptotically l-strict pseudocontractive mapping in the intermediate sense [3,4] with sequence {g n } if there exists a constant l Î [0, 1) and a sequence {g n }in [0, ∞) with lim n®∞ g n = 0 such that lim sup n→∞ sup x, y ∈C (||S n x − S n y|| 2 − (1 + γ n )||x − y|| 2 − λ||x − S n x − (y − S n y)|| 2 ) ≤ 0 (1:1) for all x, y Î C and n Î N. Throughout this paper, we assume that c n =max{0, sup x, y ∈C (||S n x − S n y|| 2 − (1 + γ n )||x − y|| 2 − λ||x − S n x − (y − S n y)|| 2 )} . Then, c n ≥ 0 for all n Î N, c n ® 0asn ® ∞ and (1.1) reduces to the relation | |S n x − S n y|| 2 ≤ ( 1+γ n ) ||x − y|| 2 + λ||x − S n x − ( y − S n y ) || 2 + c n (1:2) for all x, y Î C and n Î N. When c n =0foralln Î N in (1.2), then S is an asymptotically l-strict pseudocon- tractive mapping with sequence {g n }. We note that S is not necessarily uniformly L- Lipschitzian (see [4]), more examples can also be seen in [3]. Let {F k } be a countable famil y of bifunctions from C × C to ℝ,whereℝ is the set of rea l numbers. Combettes and Hirstoaga [5] considered the following system of equili- brium problems: Finding x ∈ C such that F k ( x, y ) ≥ 0, ∀k ∈  and ∀y ∈ C , (1:3) where Γ is an arbitrary index set. If Γ is a singleton, then problem (1.3) becomes the following equilibrium problem: Finding x ∈ C such that F ( x, y ) ≥ 0, ∀y ∈ C . (1:4) The solution set of (1.4) is denoted by EP(F). The problem (1.3) is very general in the sense that it includes, as special cases, opti- miz ation problems, variational inequalities, minimax pro blems, Nash equilibri um pro- blem in noncooperative games and others; see, for instance, [6,7] and the references therein. Some methods have been proposed to solve the equilibrium problem (1.3), related work can also be found in [8-11]. Duan and Zhao Fixed Point Theory and Applications 2011, 2011:13 http://www.fixedpointtheoryandapplications.com/content/2011/1/13 Page 2 of 13 For solving the equilibrium problem, let us assume that the bifunction F satisfies th e following conditions: (A1) F(x, x) = 0 for all x Î C; (A2) F is monotone, i.e.F(x, y)+F(y, x) ≤ 0 for any x, y Î C; (A3) for each x, y, z Î C, lim sup t®0 F(tz +(1-t)x, y) ≤ F(x, y); (A4) F(x,·) is convex and lower semicontionuous for each x Î C. Recall Mann’s iteration algorithm was introduced by Mann [12]. Since then, the con- struction of fixed points for nonexpansive mappings and asympt otically strict pseudo- contractions via Mann’ iteration a lgorithm has been extensively investigated by many authors (see, e.g., [2,6]). Mann’s iteration algorithm generates a sequence {x n } by the following manner: ∀x 0 ∈ C, x n+1 = α n x n + ( 1 − α n ) Sx n , n ≥ 0 , where a n is a real sequence in (0, 1) which satisfies certain control conditions. On the other hand, Qin et al. [13] introduced the following algorithm for a finite family of asymptotically l i -strict pseudocontractions. Let x 0 Î C and {α n } ∞ n = 0 be a sequence in (0, 1). The sequence {x n } by the following way: x 1 = α 0 x 0 +(1− α 0 )S 1 x 0 , x 2 = α 1 x 1 +(1− α 1 )S 2 x 1 , ··· x N = α N−1 x N−1 +(1− α N−1 )S N x N−1 , x N+1 = α N x N +(1− α N )S 2 1 x N , ··· x 2N = α 2N−1 x 2N−1 +(1− α 2N−1 )S 2 N x 2N−1 , x 2N+1 = α 2N x 2N +(1− α 2N )S 3 1 x 2N , ··· . It is called the explicit iterative sequence of a finite family of asymptotically l i -strict pseudocontractions {S 1 , S 2 , , S N }. Since, for each n ≥ 1, it can be written as n =(h -1) N + i,wherei = i(n) Î {1, 2, , N}, h = h(n) ≥ 1 is a positive integer and h(n) ® ∞,as n ® ∞. We can rewrite the above table in the following compact form: x n = α n−1 x n−1 +(1− α n−1 )S h( n ) i ( n ) x n−1 , ∀n ≥ 1 . Recently, S ahu et al. [4] introduced new iterative schemes for asymptotically strict pseudocontractive mappings in the intermediate sense. To be more precise, they proved the following theorem. Theorem 1.1. LetCbeanonemptyclosedconvexsubsetofarealHilbertspaceH and T: C ® C a uniformly continuous asymptotically -strict pseudocont ractive map- ping in the intermediate sense with sequence g n such that F(T) is nonempty and bou nded. Let a n be a sequence in [0, 1] such that 0<δ ≤ a n ≤ 1- for all n Î N. Let {x n } ⊂ C be sequences generated by the following (CQ) algorithm: ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ u = x 1 ∈ C chosen arbitrary, y n =(1− α n )x n + α n T n x n , C n = {z ∈ C : ||y n − z|| 2 ≤||x n − z|| 2 + θ n } , Q n = {z ∈ C : x n − z, u − x n ≥0}, x n+1 = P C n ∩Q n (u), for all n ∈ N, Duan and Zhao Fixed Point Theory and Applications 2011, 2011:13 http://www.fixedpointtheoryandapplications.com/content/2011/1/13 Page 3 of 13 where θ n = c n + g n Δ n and Δ n = sup {||x n - z||: z Î F(T)} < ∞. Then,{x n } converges strongly to P F(T) (u). Very recently, Hu and Cai [3] further considered the asymptotically strict pseudocon- tractive mappings in the intermediate sense concerning equilibrium problem. They obtained the following result in a real Hilbert space. Theorem 1.2. LetCbeanonemptyclosedconvexsubsetofarealHilbertspaceH and N ≥ 1 be an integer, j : C ® C be a bifunction satisfying (A1)-(A4) and A : C ® Hbeana-inverse-strongly monotone mapping. Let for each 1 ≤ i ≤ N, T i : C ® Cbea uniformly continuous k i -strictly asymptotically pseudo contractive mapping in the inter- media te sense for some 0 ≤ k i <1with sequences {g n,i } ⊂ [0, ∞) such that lim n®∞ g n,i = 0 and {c n,i } ⊂ [0, ∞) such that lim n®∞ c n,i =0.Let k = max{k i :1≤ i ≤ N}, g n = max{g n, i :1≤ i ≤ N} and c n = max{c n,i :1≤ i ≤ N}. Assume that F = ∩ N i =1 F(T i ) ∩ E P is nonempty and bounded. Let {a n } and {b n } b e sequences in [0, 1] such that 0<a ≤ a n ≤ 1, 0 <δ ≤ b n ≤ 1-kforallnÎ Nand0<b ≤ r n ≤ c <2a. Let {x n } and {u n } be sequences gener- ated by the following algorithm: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ x 0 ∈ C chosen arbitrary, u n ∈ C, such that φ(u n , y)+Ax n , y − u n  + 1 s y − u n , u n − x n ≥0, ∀y ∈ C , z n =(1− β n )u n + β n T h(n) i(n) u n , y n =(1− α n )u n + α n z n , C n = {v ∈ C : ||y n − v|| 2 ≤||x n − v|| 2 + θ n }, Q n = {v ∈ C : x n − v, x 0 − x n ≥0}, x n+1 = P C n ∩Q n x 0 , ∀n ∈ N ∪{0}, where θ n = c h ( n ) + γ h ( n ) ρ 2 n → 0 ,asn® ∞,wherer n = sup{||x n - v||: v Î F}<∞. Then,{x n } converges strongly to P F(T) x 0 . Motivated by Hu and Cai [3], Sahu et al. [4], and Duan [8], the main purpose of this paper is to introduce a new iterative process for finding a common element of the fixed point set of a finite family of asymptotically l i -strict pseudocontractions and the solution set of the problem (1.3). Using the hybrid method, we obtain strong conver- gence theorems that extend and improve the corresponding results [3,4,13,14]. We will adopt the following notations: 1. ⇀ for the weak convergence and ® for the strong convergence. 2. ω w (x n )={x : ∃x n j  x } denotes the weak ω-limit set of {x n }. 2. Preliminaries We need some facts and tools in a real Hilbert space H which are listed below. Lemma 2.1. Let H be a real Hilbert space. Then, the following identities hold. (i) ||x - y|| 2 =||x|| 2 -||y|| 2 -2〈x - y, y〉, ∀x, y Î H. (ii) ||tx +(1 - t)y|| 2 = t||x|| 2 +(1 - t)||y|| 2 - t(1 - t)||x - y|| 2 , ∀t Î [0, 1], ∀x, y Î H. Lemma 2.2. ([10]) LetHbearealHilbertspace.Givenanonemptyclosedconvex subset C ⊂ H and points x, y, z Î H and given also a real number a Î ℝ, the set {v ∈ C : || y − v|| 2 ≤||x − v|| 2 + z, v + a } is convex (and closed). Duan and Zhao Fixed Point Theory and Applications 2011, 2011:13 http://www.fixedpointtheoryandapplications.com/content/2011/1/13 Page 4 of 13 Lemma 2.3. ([15] ) Let C be a nonempty, closed and convex subset of H. Let {x n } be a sequence in H and u Î H. Let q = P C u. Suppose that {x n } is such that ω w (x n ) ⊂ Cand satisfies the following condition | |x n − u|| ≤ ||u − q || f or all n . Then, x n ® q. Lemma 2.4. ([4]) Let C be a nonem pty closed convex subset of a real Hilbert space H and T : C ® C a continuous asymptotically -strict pseudocontractive mapping in the intermediate sense. Then I - T is demiclosed at zero in the sense that if {x n } is a sequence in C such that x n ⇀ x Î Candlim sup m®∞ lim sup n®∞ ||x n - T m x n || = 0, then (I - T)x =0. Lemma 2.5. ([4]) Let C be a nonempty subset of a Hilbert space H and T : C ® Can asymptotically  - strict pseudocontractive mapping in the intermediate sense with sequence {g n }. Then ||T n x − T n y|| ≤ 1 1 − κ (κ||x − y|| +  (1+(1− κ)γ n )||x − y|| 2 +(1− κ)c n ) for all x, y Î C and n Î N. Lemma 2.6. ([6]) Let C be a nonempty closed convex subset of H, let F be bifunction from C × Ctoℝ satisfying (A1)-(A4) and let r >0and x Î H. Then there exists z Î C such that F(z , y)+ 1 r y − z, z − x≥0, for all y ∈ C . Lemma 2.7. ([5]) For r >0,x Î H, define a mapping T r : H ® C as follows: T r (x)={z ∈ C | F(z, y)+ 1 r y − z, z − x≥0, ∀y ∈ C } for all x Î H. Then, the following statements hold: (i) T r is single-valued; (ii) T r is firmly nonexpansive, i.e., for any x, y Î H, ||T r x − T r y || 2 ≤T r x − T r y , x − y  ; (iii) F(T r )=EP(F); (iv) EP(F) is closed and convex. 3. Main result Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H and N ≥ 1 be an integer, let F k ,kÎ {1,2, M}, be a bifunction from C × Ctoℝ which satisfies conditions (A1)-(A4). Let, for each 1 ≤ i ≤ N, S i : C ® C be a uniformly contin- uous asymptotically l i -strict pseudocontractive mapping in the intermediate sense for some 0 ≤ l i <1with sequences {g n,i } ⊂ [0, ∞) such that lim n®∞ g n,i =0and {c n,i } ⊂ [0, ∞) such that lim n®∞ c n,i =0.Let l =max{l i :1≤ i ≤ N}, g n =max{g n,i :1≤ i ≤ N} and c n =max{c n,i :1≤ i ≤ N}. Assume that  = ∩ N i=1 F(S i ) ∩ (∩ M k =1 EP(F k ) ) is nonempt y Duan and Zhao Fixed Point Theory and Applications 2011, 2011:13 http://www.fixedpointtheoryandapplications.com/content/2011/1/13 Page 5 of 13 and bounded. Let {a n } and {b n } b e sequences in [0, 1] such that 0<a ≤ a n ≤ 1, 0 <δ ≤ b n ≤ 1-l for all n Î N and {r k,n } ⊂ (0, ∞) satisfies lim inf n®∞ r k,n >0for a l l k Î {1, 2, M}. Let {x n } and {u n } be sequences generated by the following algorithm: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ x 1 ∈ C chosen arbitrary, u n = T F M r M,n T F M−1 r M−1,n ···T F 2 r 2,n T F 1 r 1,n x n , z n =(1− β n )u n + β n S h(n) i(n) u n , y n =(1− α n )u n + α n z n , C n = {v ∈ C : ||y n − v|| 2 ≤||x n − v|| 2 + θ n } , Q n = {v ∈ C : x n − v, x 1 − x n ≥0}, x n+1 = P C n ∩ Q n x 1 , ∀n ∈ N, (3:1) where θ n = c h ( n ) + γ h ( n ) ρ 2 n → 0 ,asn® ∞,wherer n =sup{||x n - v|| : v Î Ω}<∞. Then {x n } converges strongly to P Ω x 1 . Proof. Denote  k n = T F k r k,n T F 2 r 2 , n T F 1 r 1 ,n for every k Î {1, 2, , M} and  0 n = I for all n Î N. Therefore u n =  M n x n . The proof is divided into six steps. Step 1. The sequence {x n } is well defined. It is obvious that C n is closed and Q n is closed and convex for every n Î N.From Lemma 2.2, we also get that C n is convex. Take p Î Ω, since for each k Î {1, 2, , M}, T F k r k ,n is nonexpansive, p = T F k r k , n p and u n =  M n x n , we have ||u n − p|| = || M n x n −  M n p|| ≤ ||x n − p|| for all n ∈ N . (3:2) It follows from the definition of S i and Lemma 2.1(ii), we get ||z n − p|| 2 = ||(1 − β n )(u n − p)+β n (S h(n) i(n) u n − p)|| 2 =(1− β n )||u n − p|| 2 + β n ||S h(n) i(n) u n − p|| 2 − β n (1 − β n )||S h(n) i(n) u n − u n || 2 ≤ (1 − β n )||u n − p|| 2 + β n  ||(1 + γ h(n) )||u n − p|| 2 + λ||S h(n) i(n) u n − u n || 2 + c h(n)  − β n (1 − β n )||S h(n) i(n) u n − u n || 2 ≤ (1 + γ h(n) )||u n − p|| 2 − β n (1 − β n − λ)||S h(n) i(n) u n − u n || 2 + β n c h(n) ≤ (1 + γ h ( n ) )||u n − p|| 2 + β n c h ( n ) . (3:3) By virtue of the convexity of ||·|| 2 , one has | |y n − p|| 2 = || ( 1 − α n )( u n − p ) + α n ( z n − p ) || 2 ≤ ( 1 − α n ) ||u n − p|| 2 + α n ||z n − p|| 2 . (3:4) Substituting (3.2) and (3.3) into (3.4), we obtain | |y n − p|| 2 ≤ (1 − α n )||u n − p|| 2 + α n  (1 + γ h(n) )||u n − p|| 2 + β n c h(n)  ≤||u n − p|| 2 + γ h(n) ||u n − p|| 2 + β n c h(n) ≤||u n − p|| 2 + γ h(n) ||x n − p|| 2 + c h(n) ≤||u n − p|| 2 + θ n ≤||x n − p || 2 + θ n . (3:5) It follows that p Î C n for all n Î N. Thus, Ω ⊂ C n . Next, we prove that Ω ⊂ Q n for all n Î N by induction. For n = 1, we have Ω ⊂ C = Q 1 . Assume that Ω ⊂ Q n for some n ≥ 1. Since x n+1 = P C n ∩ Q n x 1 , we obtain Duan and Zhao Fixed Point Theory and Applications 2011, 2011:13 http://www.fixedpointtheoryandapplications.com/content/2011/1/13 Page 6 of 13  x n+1 − z, x 1 − x n+1  ≥ 0, ∀z ∈ C n ∩ Q n . As Ω ⊂ C n ⋂ Q n by induction assumption, the inequality holds, in particular, for all z Î Ω. This together with the definition of Q n+1 implies that Ω ⊂ Q n +1 . Hence Ω ⊂ Q n holds for all n ≥ 1. Thus Ω ⊂ C n ⋂ Q n and therefore the sequence {x n } is well defined. Step 2. Set q = P Ω x 1 , then | |x n+1 − x 1 || ≤ || q − x 1 || for all n ∈ N . (3:6) Since Ω is a nonempt y closed convex subset of H, there exists a unique q Î Ω such that q = P Ω x 1 . From x n+1 = P C n ∩ Q n x 1 , we have | |x n+1 − x 1 || ≤ ||v − x 1 || for all v ∈ C n ∩ Q n ,foralln ∈ N . Since q Î Ω ⊂ C n ⋂ Q n , we get (3.6). Therefore, {x n } is bounded. So are {u n } and {y n }. Step 3. The following limits hold: l im n →∞ ||u n − u n+i || =0, l im n →∞ ||x n − x n+i || =0;∀i = 1, 2, , N. From the definition o f Q n ,wehave x n = P Q n x 1 , which together with the fact that x n+1 Î C n ⋂ Q n ⊂ Q n implies that || x n − x 1 || ≤ || x n+1 − x 1 || ,  x n − x n+1 , x 1 − x n  ≥ 0 . (3:7) This shows that the sequence {||x n - x 1 ||} is nondecreasing. Since { x n }isbounded, the limit of {||x n - x 1 ||} exists. It follows from Lemma 2.1(i) and (3.7) that | |x n+1 − x n || 2 = ||x n+1 − x 1 − (x n − x 1 )|| 2 = ||x n+1 − x 1 || 2 −||x n − x 1 || 2 − 2x n − x n+1 , x 1 − x n  ≤ || x n+1 − x 1 || 2 − || x n − x 1 || 2 . Noting that lim n®∞ ||x n - x 1 || exists, this implies lim n → ∞ ||x n − x n+1 || =0 . (3:8) It is easy to get || x n+i − x n || → 0, ∀ i = 1, 2, , N ,asn →∞ . (3:9) Since x n+1 Î C n , we have || y n − x n+1 || 2 ≤||x n − x n+1 || 2 + θ n . So, we get lim n®∞ ||y n - x n+1 || = 0. It follows that | | y n − x n || ≤ || y n − x n+1 || + ||x n − x n+1 || → 0, as n →∞ . (3:10) Next we will show that lim n →∞ || k n x n −  k−1 n x n || =0, k =1,2, , M . (3:11) Duan and Zhao Fixed Point Theory and Applications 2011, 2011:13 http://www.fixedpointtheoryandapplications.com/content/2011/1/13 Page 7 of 13 Indeed, for p Î Ω, it follows from the firmly nonexpansivity of T F k r k ,n that for each k Î {1, 2, , M}, we have || k n x n − p|| 2 = ||T F k r k,n  k−1 n x n − T F k r k,n p|| 2 ≤ k n x n − p,  k−1 n x n − p = 1 2 (|| k n x n − p|| 2 + || k−1 n x n − p|| 2 −|| k n x n −  k−1 n x n || 2 ) . Thus we get || k n x n − p|| 2 ≤|| k−1 n x n − p|| 2 −|| k n x n −  k−1 n x n || 2 , k =1,2, , M , which implies that for each k Î {1, 2, , M}, | | k n x n − p|| 2 ≤|| 0 n x n − p|| 2 −|| k n x n −  k−1 n x n || 2 −|| k−1 n x n −  k−2 n x n || 2 −···−|| 2 n x n −  1 n x n || 2 −|| 1 n x n −  0 n x n || 2 ≤||x n − p|| 2 −|| k n x n −  k−1 n x n || 2 . (3:12) Therefore, by the convexity of ||·|| 2 , (3.5) and the nonexpansivity of T F k r k ,n , we get | |y n − p|| 2 ≤||u n − p|| 2 + θ n = || M n x n −  M n p|| 2 + θ n ≤|| k n x n − p|| 2 + θ n ≤||x n − p|| 2 −|| k n x n −  k−1 n x n || 2 + θ n . It follows that || k n x n − k−1 n x n || 2 ≤||x n −p|| 2 −||y n −p|| 2 +θ n ≤||x n −y n ||(||x n −p||+ ||y n −p||)+θ n . (3:13) From (3.10) and (3.13), we obtain (3.11). Then, we have ||u n − x n || ≤ ||u n −  M−1 n x n || + || M−1 n x n −  M−2 n x n || + ···+ || 1 n x n − x n || → 0 . (3:14) Combining (3.8) and (3.14), we have || u n+1 − u n || ≤ || u n+1 − x n+1 || + || x n+1 − x n || + || x n − u n || → 0, as n →∞ . (3:15) It follows that || u n+i − u n || → 0, ∀ i = 1, 2, , N ,asn →∞ . (3:16) Step 4. Show that ||u n - S i u n || ® 0, ||x n - S i x n || ® 0, as n ® ∞; ∀i Î {1, 2, , N}. Since, for any positive integer n ≥ N,itcanbewrittenasn =(h(n)-1)N + i(n), where i(n) Î {1, 2, , N}. Observe that | |u n − S n u n || ≤ ||u n − S h(n) i(n) u n || + ||S h(n) i(n) u n − S n u n || = ||u n − S h(n) i ( n ) u n || + ||S h(n) i ( n ) u n − S i(n) u n || . (3:17) From (3.10), (3.14), the conditions 0 <a ≤ a n ≤ 1 and 0 <δ ≤ b n ≤ 1-l, we obtain ||S h(n) i(n) u n − u n || = 1 β n ||z n − u n || = 1 α n β n ||y n − u n || ≤ 1 a δ (||y n − x n || + ||u n − x n ||) → 0, as n →∞ . (3:18) Duan and Zhao Fixed Point Theory and Applications 2011, 2011:13 http://www.fixedpointtheoryandapplications.com/content/2011/1/13 Page 8 of 13 Next, we prove that lim n →∞ ||S h(n)−1 i(n) u n − u n || =0 . (3:19) It is obvious that the relations hold: h(n)=h(n - N)+1,i(n)=i(n - N). Therefore, ||S h( n ) −1 i(n) u n − u n || ≤ ||S h( n ) −1 i(n) u n − S h( n ) −1 i(n−N) u n−N+1 || + ||S h( n ) −1 i(n−N) u n−N+1 − S h( n−N ) i(n−N) u n−N | | + ||S h(n−N) i(n−N) u n−N − u n−N || + ||u n−N − u n−N+1 || + ||u n−N+1 − u n || = ||S h(n)−1 i(n) u n − S h(n)−1 i(n) u n−N+1 || + ||S h(n−N) i(n−N) u n−N+1 − S h(n−N) i(n−N) u n−N || + ||S h(n−N) i ( n−N ) u n−N − u n−N || + ||u n−N − u n−N+1 || + ||u n−N+1 − u n ||. (3:20) Applying Lemma 2.5 and (3.16), we get (3.19). Using the uniformly continuity of S i , we obtain lim n →∞ ||S h(n) i(n) u n − S i(n) u n || =0 , (3:21) this together with (3.17) yields lim n → ∞ ||u n − S n u n || =0 . We also have || u n − S n+i u n || ≤ || u n − u n+i || + || u n+i − S n+i u n+i || + || S n+i u n+i − S n+i u n || → 0, as n →∞ , for any i =1,2, N, which gives that lim n → ∞ ||u n − S i u n || =0;∀i =1,2, N . (3:22) Moreover, for each i Î {1, 2, N}, we obtain that || x n − S i x n || ≤ || x n − u n || + || u n − S i u n || + || S i u n − S i x n || → 0, as n →∞ . (3:23) Step 5. The following implication holds: ω w ( x n ) ⊂  . (3:24) We first show that ω w (x n ) ⊂∩ N i =1 F(S i ) .Tothisend,wetakeω Î ω w (x n ) and assume that x n j ω as j ® ∞ for some subsequence {x n j } of x n . Note that S i is uniformly continuous and (3.23), we see that ||x n − S m i x n || → 0 , for all m Î N. So by Lemma 2.4, it follows that ω ∈∩ N i =1 F(S i ) and hence ω w (x n ) ⊂∩ N i =1 F(S i ) . Next we will show that ω ∈∩ M k =1 EP(F k ) . Indeed, by Lemma 2.6, we have that for each k = 1, 2, , M, F k ( k n x n , y)+ 1 r n y −  k n x n ,  k n x n −  k−1 n x n ≥0, ∀y ∈ C . From (A2), we get 1 r n y −  k n x n ,  k n x n −  k−1 n x n ≥F k (y,  k n x n ), ∀y ∈ C . Hence, y −  k n j x n j ,  k n j x n j −  k−1 n j x n j r n j ≥F k (y,  k n j x n j ), ∀y ∈ C . Duan and Zhao Fixed Point Theory and Applications 2011, 2011:13 http://www.fixedpointtheoryandapplications.com/content/2011/1/13 Page 9 of 13 From (3.11), we obtain that  k n j x n j  ω as j ® ∞ for each k = 1, 2, , M (especially, u n j =  M n j x n j ). Together with (3.11) and (A4) we have, for each k = 1, 2, , M, that 0 ≥ F k ( y, ω ) , ∀y ∈ C . For any, 0 <t ≤ 1 and y Î C, let y t = ty +(1-t)ω. Since y Î C and ω Î C, we obtain that y t Î C and hence F k (y t , ω) ≤ 0. So, we have 0=F k ( y t , y t ) ≤ tF k ( y t , y ) + ( 1 − t ) F k ( y t , ω ) ≤ tF k ( y t , y ). Dividing by t, we get, for each k = 1, 2, , M, that F k ( y t , y ) ≥ 0, ∀y ∈ C . Letting t ® 0 and from (A3), we get F k ( ω, y ) ≥ 0 for all y Î C and ω Î EP(F k ) for each k = 1, 2, , M, i.e., ω ∈∩ M k =1 EP(F k ) . Hence (3.24) holds. Step 6. Show that x n ® q = P Ω x 1 . From (3.6), (3.24) and Lemma 2.3, we conclude that x n ® q, where q = P Ω x 1 . □ Corollary 3.2. LetCbeanonemptyclosedconvexsubsetofarealHilbertspaceH and N ≥ 1 be an integer, let F be a bifunction from C × Ctoℝ which satisfies condi- tions (A1)-(A4).Let,foreach1 ≤ i ≤ N, S i : C ® C be a uniformly continuous l i -strict asymptotically pseudocontractive mapping in the in termediate sense for some 0 ≤ l i <1 with sequences {g n,i } ⊂ [0, ∞) such that lim n®∞ g n,i =0and {c n,i } ⊂ [0, ∞) such that lim n®∞ c n , i =0.Letl =max{l i :1≤ i ≤ N}, g n =max{g n, i :1≤ i ≤ N} and c n =max {c n,i :1≤ i ≤ N}. Assume that  = ∩ N i =1 F(S i ) ∩ EP(F ) is non empty and bounded. Let {a n } and {b n } be sequences in [0, 1] such that 0 < a ≤ a n ≤ 1,0 <δ ≤ b n ≤ 1-l for all n Î N and {r n } ⊂ (0,∞) satisfies lim inf n®∞ r n > 0 for all k Î {1, 2, M}. Let {x n } and {u n } be sequences generated by the following algorithm: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ x 1 ∈ C chosen arbitrary, u n = T F r n x n , z n =(1− β n )u n + β n S h(n) i(n) u n , y n =(1− α n )u n + α n z n , C n = {v ∈ C : ||y n − v|| 2 ≤||x n − v|| 2 + θ n } , Q n = {v ∈ C : x n − v, x 1 − x n ≥0}, x n+1 = P C n ∩Q n x 1 , ∀n ∈ N, (3:25) where θ n = c h ( n ) + γ h ( n ) ρ 2 n → 0 ,asn ® ∞, where r n = sup{||x n - v|| : v Î Ω}<∞ . Then {x n } converges strongly to P Ω x 1 . Proof. Putting M = 1, we can draw the desired conclusion from Theorem 3.1. □ Remark 3.3. Corollary 3.2 extends the theorem of Tada and Takahashi [14] from a nonexpansive mapping to a finite family of asymptotically l i -strict pseudocontractive mappings in the intermediate sense. Corollary 3.4. Let C be a nonempty closed convex subset of a real Hilbert space H and N ≥ 1 be an integer, let, for each 1 ≤ i ≤ N, S i :C® C be a uniformly continuous l i -strict asymptotically pseudocontractive mapping in the intermediate sense for some 0 ≤ l i <1 with sequences {g n,i } ⊂ [0, ∞) such that lim n®∞ g n,i =0and {c n,i } ⊂ [0, ∞) such that lim n®∞ c n,i =0. Let l= max{l i :1≤ i ≤ N}, g n = max{g n,i :1≤ i ≤ N} and c n = max{c n,i :1 Duan and Zhao Fixed Point Theory and Applications 2011, 2011:13 http://www.fixedpointtheoryandapplications.com/content/2011/1/13 Page 10 of 13 [...]... out the proof of convergence of the theorems and realization of numerical examples JZ carried out the check of the manuscript All authors read and approved the final manuscript Competing interests The authors declare that they have no competing interests Received: 22 January 2011 Accepted: 5 July 2011 Published: 5 July 2011 References 1 Bruck, RE, Kuczumow, T, Reich, S: Convergence of iterates of asymptotically. .. mappings in Banach spaces with the uniform opial property Colloq Math 65, 169–179 (1993) 2 Kim, TH, Xu, HK: Convergence of the modified Mann’s iteration method for asymptotically strict pseudocontractions Nonlinear Anal 68, 2828–2836 (2008) doi:10.1016/j.na.2007.02.029 3 Hu, CS, Cai, G: Convergence theorems for equilibrium problems and fixed point problems of a finite family of asymptotically k -strict. .. variational inequalities to equilibrium problems Math Stud 63, 123–145 (1994) 7 Colao, V, Marino, G, Xu, HK: An iterative method for finding common solutions of equilibrium and fixed point problems J Math Anal Appl 344, 340–352 (2008) doi:10.1016/j.jmaa.2008.02.041 8 Duan, PC: Convergence theorems concerning hybrid methods for strict pseudocontractions and systems of equilibrium problems J Inequal Appl (2010)... theorems for system of equilibrium problems and asymptotically strict pseudocontractions in the intermediate sense Fixed Point Theory and Applications 2011 2011:13 Submit your manuscript to a journal and benefit from: 7 Convenient online submission 7 Rigorous peer review 7 Immediate publication on acceptance 7 Open access: articles freely available online 7 High visibility within the field 7 Retaining the. .. v|| : v Î Ω} 0 Therefore, T is an asymptotically k -strict pseudocontractive mapping in the intermediate sense n+1 We apply it to In the algorithm (3.1), set Fk (x, y) = 0, N = 1, βn = 1 − k, αn = 2n find the fixed point of T of Example 4.1 Under the above assumptions,... mappings in the intermediate sense Comput Math Appl (2010) 4 Sahu, DR, Xu, HK, Yao, JC: Asymptotically strict pseudocontractive mappings in the intermediate sense Nonlinear Anal 70, 3502–3511 (2009) doi:10.1016/j.na.2008.07.007 5 Combettes, PL, Hirstoaga, SA: Equilibrium programming in Hilbert spaces J Nonlinear Convex Anal 6, 117–136 (2005) 6 Blum, E, Oettli, W: From optimization and variational inequalities . R C H Open Access Strong convergence theorems for system of equilibrium problems and asymptotically strict pseudocontractions in the intermediate sense Peichao Duan * and Jing Zhao * Correspondence: pcduancauc@126.com College. of the fixed point set of a finite family of asymptotically l i -strict pseudocontractions and the solution set of the problem (1.3). Using the hybrid method, we obtain strong conver- gence theorems. Duan and Zhao: Strong convergence theorems for system of equilibrium problems and asymptotically strict pseudocontractions in the intermediate sense. Fixed Point Theory and Applications 2011 2011:13. Submit

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  • Abstract

  • 1. Introduction

  • 2. Preliminaries

  • 3. Main result

  • 4. Numerical result

  • Acknowledgements

  • Authors' contributions

  • Competing interests

  • References

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