Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 149-162 Monotone Hybrid Methods for a Finite Family of Nonexpasive Multi-valued Maps and Equilibrium Problems ∗ Suthep Suantai and Watcharaporn Cholamjiak Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand Centre of Excellence in Mathematics, CHE, Si Ayutthaya Rd., Bangkok 10400, Thailand Email: scmti005@chiangmai.ac.th, c-wchp007@hotmail.com Abstract In this paper, we introduce a new monotone hybrid iterative scheme for finding a common element of the set of common fixed points of a finite family of nonexpansive multi- valued maps and the set of the solutions of the equilibrium problem in a Hilbert space. More- over, we also introduce a new iterative scheme for finding a common fixed point of a finite family of nonexpansive multi-valued maps in a Banach space. Strong convergence theorem of the proposed iteration is established. Keywords Nonexpansive multi-valued map Monotone hybrid method, 1. Introduction LetD be a nonempty convex subset of a Banach spaceE. Let f be a bifunction fromD×D to R, where R is the set of all real number. The equilibrium problem for f is to find x ∈ D such that f(x, y) ≥ 0 for all y ∈ D. The set of such solutions is denoted by EP (f). The set D is called proximinal if for each x ∈ E, there exists an element y ∈ D such that ‖x−y‖ = d(x,D), where d(x,D) = inf{‖x − z‖ : z ∈ D}. Let CB(D),K(D) and P (D) denote the families of nonempty closed bounded subsets, nonempty compact subsets, and nonempty proximinal bounded subsets of D, respectively. The Hausdorff metric on CB(D) is defined by H(A,B) = max { sup x∈A d(x,B), sup y∈B d(y,A) } for A,B ∈ CB(D). A single-valued map T : D → D is called nonexpansive if ‖Tx− Ty‖ ≤ ‖x − y‖ for all x, y ∈ D. A multi-valued map T : D → CB(D) is said to be nonexpansive if H(Tx, Ty) ≤ ‖x−y‖ for all x, y ∈ D. An element p ∈ D is called a fixed point of T : D → D (respectively, T : D → CB(D)) if p = Tp (respectively, p ∈ Tp). The set of fixed points of T is denoted by F (T ). The mapping T : D → CB(D) is called quasi-nonexpansive[18] if F (T ) 6= ∅ and H(Tx, Tp) ≤ ‖x − p‖ for all x ∈ D and all p ∈ F (T ). It is clear that every nonexpansive multi-valued map T with F (T ) 6= ∅ is quasi-nonexpansive. But there exist quasi-nonexpansive mappings that are not nonexpansive, see [17]. The mapping T : D → CB(D) is called hemicompact if, for any sequence {xn} in D such that d(xn, Txn) → 0 as n → ∞, there exists a subsequence {xnk } of {xn} such that xnk → ∗This research is supported by the Centre of Excellence in Mathematics, the commission on Higher Education, Thailand, the Thailand Research Fund and the Graduate School of Chiang Mai University for the financial support. ISSN 1078-6236 International Institute for General Systems Studies, Inc. 1 2 1 2 150 Suantai:Monotone Hybrid Methods for a Finite Family of Nonexpasive. . . . . . p ∈ D. We note that if D is compact, then every multi-valued mapping T : D → CB(D) is hemicompact. A mapping T : D → CB(D) is said to satisfy Condition (I) if there is a nondecreasing function f : [0,∞)→ [0,∞) with f(0) = 0, f(r) > 0 for r ∈ (0,∞) such that d(x, Tx) ≥ f(d(x, F (T ))) for all x ∈ D. A family {Ti : D → CB(D), i = 1, 2, ..., N} is said to satisfy Condition (II) if there is a nondecreasing function f : [0,∞)→ [0,∞) with f(0) = 0, f(r) > 0 for r ∈ (0,∞) such that d(x, Tix) ≥ f(d(x, N⋂ i=1 F (Ti))) for all i = 1, 2, ..., N and x ∈ D. In 1953, Mann [10] introduced the following iterative procedure to approximate a fixed point of a nonexpansive mapping T in a Hilbert space H: xn+1 = αnxn + (1− αn)Txn, ∀n ∈ N, (1) where the initial point x0 is taken in C arbitrarily and {αn} is a sequence in [0,1]. However, we note that Mann’s iteration process (1) has only weak convergence, in gen- eral; for instance, see [1, 7, 15]. In 2003, Nakajo and Takahashi [12] introduced the method which is the so-called CQ method to modify the process (1) so that strong convergence is guaranteed have recently been made. They also proved a strong convergence theorem for a nonexpansive mapping in a Hilbert space. Recently, Tada and Takahashi [20] proposed a new iteration for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping T in a Hilbert space H . In 2005, Sastry and Babu [16] proved that the Mann and Ishikawa iteration schemes for multi-valued map T with a fixed point p converge to a fixed point q of T under certain condi- tions. They also claimed that the fixed point q may be different from p. More precisely, they proved the following result for nonexpansive multi-valued map with compact domain. In 2007, Panyanak [13] extended the above result of Sastry and Babu [16] to uniformly convex Banach spaces but the domain of T remains compact. Later, Song and Wang [19] noted that there was a gap in the proofs of Theorem 3.1(see [13]) and Theorem 5 (see [16]). They further solved/revised the gap and also gave the affirmative answer to Panyanak [13] question using the following Ishikawa iteration scheme. In the main results, domain of T is still compact, which is a strong condition (see [19], Theorem 1) and T satisfies condition(I) (see [19], Theorem 1). In 2009, Shahzad and Zegeye [17] extended and improved the results of Panyanak [13], Sastry and Babu [16] and Song and Wang [19] to quasi-nonexpansive multi-valued maps. They also relaxed compactness of the domain of T and constructed an iteration scheme which removes Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 151 the restriction of T namely Tp = {p} for any p ∈ F (T ). The results provided an affirmative answer to Panyanak [13] question in a more general setting. In the main results, T satisfies condition(I)(see [17], Theorem 2.3) and T is hemicompact and continuous (see [17], Theorem 2.5). Question: How can we modify iteration process for a nonexpansive multi-valued map T which the domain of T is not necessary to be compact to obtain strong convergence theorems for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of T ? In the recent years, the problem of finding a common element of the set of solutions of equilibrium problems and the set of fixed points in the framework of Hilbert spaces and Banach spaces have been intensively studied by many authors, for instance, see [2, 3, 4, 5, 6, 8, 14, 20] and the references cited theorem. In this paper, we introduce a monotone hybrid iterative scheme for finding a common ele- ment of the set of a common fixed points of a finite family of nonexpasive multi-valued maps and the set of solutions of an equilibrium problem in a Hilbert space. Let D be nonempty, closed and convex subset of a Hilbert space H and αi n ∈ (0, 1) for all i = 0, 1, ...,m with∑m i=0 α i n = 1, ∀n ≥ 0 and rn ∈ (0,∞). For an initial point x0 ∈ D = C0, compute the sequence {xn} by the iterative process f(un, y) + 1 rn 〈y − un, un − xn〉 ≥ 0, ∀y ∈ D, yn = ∑m i=0 α i nz i n, z i n ∈ Tiun, ∀i = 1, 2, ...,m, z0n = un, Cn+1 = {z ∈ Cn : ‖yn − z‖ ≤ ‖xn − z‖}, xn+1 = PCn+1x0, n ≥ 0, (2) where Ti is a nonexpansive multi-valued map for all i = 1, 2, ...,m. 2. Preliminaries The following lemmas give some characterizations and a useful property of the metric projection PD in a Hilbert space. Let H be a real Hilbert space with inner product 〈·, ·〉 and norm ‖ · ‖. Let D be a closed and convex subset of H . For every point x ∈ H , there exists a unique nearest point in D, denoted by PDx, such that ‖x− PDx‖ ≤ ‖x− y‖, ∀y ∈ D. PD is called the metric projection of H onto D. We know that PD is a nonexpansive mapping of H onto D. Lemma 2.1. [11] Let D be a closed and convex subset of a real Hilbert space H and let PD be the metric projection from H onto D. Given x ∈ H and z ∈ D. Then z = PDx if and only if the following holds: 〈x− z, y − z〉 ≤ 0, ∀y ∈ D. Lemma 2.2. [12] Let D be a nonempty, closed and convex subset of a real Hilbert space H and PD : H → D be the metric projection from H onto D. Then the following inequality holds: ‖y − PDx‖2 + ‖x− PDx‖2 ≤ ‖x− y‖2, ∀x ∈ H, ∀y ∈ D. 152 Suantai:Monotone Hybrid Methods for a Finite Family of Nonexpasive. . . . . . Lemma 2.3. [11] Let H be a real Hilbert space. Then the following equations hold: (i) ‖x− y‖2 = ‖x‖2 − ‖y‖2 − 2〈x− y, y〉, ∀x, y ∈ H; (ii) ‖tx+ (1− t)y‖2 = t‖x‖2 + (1− t)‖y‖2 − t(1− t)‖x− y‖2, ∀t ∈ [0, 1] and x, y ∈ H . By using Lemma 2.3, we obtain the following lemma. Lemma 2.4. Let H be a real Hilbert space. Then for each m ∈ N ‖ m∑ i=1 tixi‖2 = m∑ i=1 ti‖xi‖2 − m∑ i=1,i 6=j titj‖xi − xj‖2, xi ∈ H and ti, tj ∈ [0, 1] for all i, j = 1, 2, ...,m with ∑m i=1 ti = 1. Lemma 2.5. [9]LetD be a nonempty, closed and convex subset of a real Hilbert spaceH . Given x, y, z ∈ H and also given a ∈ R, the set {v ∈ D : ‖y − v‖2 ≤ ‖x− v‖2 + 〈z, v〉+ a} is convex and closed. For solving the equilibrium problem, we assume the bifunction f : D × D → R satisfies the following conditions: (A1) f(x, x) = 0 for all x ∈ D; (A2) f is monotone, i.e., f(x, y) + f(y, x) ≤ 0 for all x, y ∈ D; (A3) for each x, y, z ∈ D, lim supt↓0 f(tz + (1− t)x, y) ≤ f(x, y); (A4) f(x, ·) is convex and lower semicontinuous for each x ∈ D. Lemma 2.6. [2] Let D be a nonempty, closed and convex subset of a real Hilbert space H . Let f be a bifunction from D ×D to R satisfying (A1)-(A4) and let r > 0 and x ∈ H . Then, there exists z ∈ D such that f(z, y) + 1 r 〈y − z, z − x〉 ≥ 0, for all y ∈ D. Lemma 2.7. [6] For r > 0, x ∈ H , defined a mapping Tr : H → 2D as follows: Tr(x) = { z ∈ D : f(z, y) + 1 r 〈y − z, z − x〉 ≥ 0, for all y ∈ D } . Then the followings hold: (1) Tr is single-value; (2) Tr is firmly nonexpansive, i.e., for any x, y ∈ H , ‖Trx− Try‖2 ≤ 〈Trx− Try, x− y〉; (3) F (Tr) = EP (f); (4) EP (f) is closed and convex. Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 153 Lemma 2.8. Let D be a closed and convex subset of a real Hilbert space H . Let T : D → CB(D) be a nonexpansive multi-valued map with F (T ) 6= ∅ and Tp = {p} for each p ∈ F (T ). Then F (T ) is a closed and convex subset of D. Proof. First, we will show that F (T ) is closed. Let {xn} be a sequence in F (T ) such that xn → x as n→∞. We have d(x, Tx) ≤ d(x, xn) + d(xn, Tx) ≤ d(x, xn) +H(Txn, Tx) ≤ 2d(x, xn). It follows that d(x, Tx) = 0, so x ∈ F (T ). Next, we show that F (T ) is convex. Let p = tp1 + (1− t)p2 where p1, p2 ∈ F (T ) and t ∈ (0, 1) . Let z ∈ Tp, by Lemma 2.3, we have ‖p− z‖2 = ‖t(z − p1) + (1− t)(z − p2)‖2 = t‖z − p1‖2 + (1− t)‖z − p2‖2 − t(1− t)‖p1 − p2‖2 = td(z, Tp1) 2 + (1− t)d(z, Tp2) 2 − t(1− t)‖p1 − p2‖2 ≤ tH(Tp, Tp1) 2 + (1− t)H(Tp, Tp2) 2 − t(1− t)‖p1 − p2‖2 ≤ t‖p− p1‖2 + (1− t)‖p− p2‖2 − t(1− t)‖p1 − p2‖2 = t(1− t)2‖p1 − p2‖2 + (1− t)t2‖p1 − p2‖2 − t(1− t)‖p1 − p2‖2 = 0, hence p = z. Therefore p ∈ F (T ). Lemma 2.9. [21] Let p > 1, r > 0 be two fixed numbers. Then a Banach space E is uniformly convex if and only if there exists a continuous, strictly increasing, and convex function g : [0,∞)→ [0,∞) with g(0) = 0 such that ‖λx+ (1− λ)y‖p ≤ λ‖x‖p + (1− λ)‖y‖p − ωp(λ)g(‖x− y‖), for all x, y ∈ Br(0) = {x ∈ E : ‖x‖ ≤ r} and λ ∈ [0, 1] where ωp(λ) = λ(1−λ)p+λp(1−λ). By using Lemma 2.9, we can prove the following Lemma by induction. Lemma 2.10. Let E be a uniformly convex Banach space and Br(0) = { x ∈ E : ‖x‖ ≤ r} be a closed ball of E. Then there exists a continuous strictly increasing convex function g : [0,∞)→ [0,∞) with g(0) = 0 such that ‖ m∑ i=1 αixi‖2 ≤ m∑ i=1 αi‖xi‖2 − α1α2g(‖x1 − x2‖), for all m ∈ N, xi ∈ Br(0) and αi ∈ [0, 1], i = 1, 2, ...,m with ∑m i=1 αi = 1. By interchanging the roles of vectors xi in Lemma 2.10 and summing the inequalities to- gether we obtain the following lemma. 154 Suantai:Monotone Hybrid Methods for a Finite Family of Nonexpasive. . . . . . Lemma 2.11. Let E be a uniformly convex Banach space and Br(0) = {x ∈ E : ‖x‖ ≤ r} be a closed ball of E. Then there exists a continuous strictly increasing convex function g : [0,∞)→ [0,∞) with g(0) = 0 such that for each j ∈ {1, 2, ...,m}, ‖ m∑ i=1 αixi‖2 ≤ m∑ i=1 αi‖xi‖2 − αj m− 1 ( m∑ i=1 αig(‖xj − xi‖) ) , for all m ∈ N, xi ∈ Br(0) and αi ∈ [0, 1] for all i = 1, 2, ...,m with ∑m i=1 αi = 1. Proof. Let j ∈ {1, 2, ...,m} be fixed. By Lemma 2.10, there is a continuous strictly increasing convex function g : [0,∞)→ [0,∞) with g(0) = 0 such that ‖α1x1 + α2x2 + α3x3 + ...+ αmxm‖2 ≤ m∑ i=1 αi‖xi‖2 − αjα1g(‖xj − x1‖) ‖α1x1 + α3x3 + α4x4 + ...+ α2x2‖2 ≤ m∑ i=1 αi‖xi‖2 − αjα2g(‖xj − x2‖) ... ‖α1x1 + α4x4 + α5x5 + ...+ α3x3‖2 ≤ m∑ i=1 αi‖xi‖2 − αjαj−1g(‖xj − xj−1‖) ‖α1x1 + α4x4 + α5x5 + ...+ α3x3‖2 ≤ m∑ i=1 αi‖xi‖2 − αjαj+1g(‖xj − xj+1‖) ... ‖α1x1 + αmxm + α2x2 + ...+ αm−1xm−1‖2 ≤ m∑ i=1 αi‖xi‖2 − αjαmg(‖xj − xm‖). By summing up above inequalities, we obtain ‖ m∑ i=1 αixi‖2 ≤ m∑ i=1 αi‖xi‖2 − αj m− 1 ( m∑ i=1 αig(‖xj − xi‖) ) . 3. Main Result First, we prove a strong convergence theorem for a finite family of nonexpansive multi- valued mappings which satisfies the condition (II) in a uniformly convex Banach space. Theorem 3.1. Let D be a nonempty, closed and convex subset of a uniformly convex Banach spaceE. Let Ti : D → CB(D) be a nonexpansive multi-valued map for all i = 1, 2, ...,m with⋂m i=1 F (Ti) 6= ∅ and Tip = {p} for each p ∈ ⋂m i=1 F (Ti). Assume that {Ti : i = 1, 2, ...,m} satisfies the condition (II) for all i = 1, 2, ...,m and αi n ∈ (0, 1) with 0 < lim infn→∞ α i n ≤ Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 155 lim supn→∞ α i n < 1 for all i = 0, 1, 2, ...,m. Let x0 ∈ D and let {xn} be the sequence in D generated by iteration process: xn+1 = m∑ i=0 αi nz i n, (3) where z0n = xn, zin ∈ Tixn for all i = 1, 2, ...,m and ∑m i=0 α i n = 1. Then {xn} converges strongly to a common fixed point of Ti, i = 1, 2, ...,m. Proof. Let p ∈ ⋂m i=1 F (Ti). By the nonexpansiveness of Ti, we have ‖xn+1 − p‖ ≤ m∑ i=0 αi n‖zin − p‖ = α0 n‖xn − p‖+ m∑ i=1 αi nd(zin, Tip) ≤ α0 n‖xn − p‖+ m∑ i=1 αi nH(Tixn, Tip) ≤ ‖xn − p‖, (4) which implies that limn→∞ ‖xn − p‖ exists. For each i = 1, 2, ...,m, we have ‖zin − p‖ = d(zin, Tip) ≤ H(Tixn, Tip) ≤ ‖xn − p‖. It follows that {‖zin − p‖} is bounded for all i = 1, 2, ...,m. Put r = max1≤i≤m{supn ‖zin− p‖}. By Lemma 2.11, there is a continuous strictly increasing convex function g : [0,∞)→ [0,∞) with g(0) = 0 such that ‖xn+1 − p‖2 ≤ m∑ i=0 αi n‖zin − p‖2 − α0 n m m∑ i=1 αi ng ( ‖zin − xn‖ ) = α0 n‖xn − p‖2 + m∑ i=1 αi nd(zin, Tip) 2 − α0 n m m∑ i=1 αi ng ( ‖zin − xn‖ ) ≤ α0 n‖xn − p‖2 + m∑ i=1 αi nH(Tixn, Tip) 2 − α0 n m m∑ i=1 αi ng ( ‖zin − xn‖ ) ≤ ‖xn − p‖2 − α0 n m m∑ i=1 αi ng ( ‖zin − xn‖ ) . It follows that α0 n m m∑ i=1 αi ng ( ‖zin − xn‖ ) ≤ ‖xn − p‖2 − ‖xn+1 − p‖2. This implies that g ( ‖zin − xn‖ ) → 0 as n → ∞ for all i = 1, 2, ...,m. Since g is continuous strictly increasing with g(0) = 0, we can conclude that ‖zin − xn‖ → 0 as n → ∞ for all i = 1, 2, ...,m. Also d(xn, Tixn) ≤ ‖zin − xn‖ → 0 as n → ∞ for all i = 1, 2, ...,m. Since that {Ti}mi=1 satisfies the condition (II), we have d(xn, ⋂m i=1 F (Ti))→ 0. Thus there is a 156 Suantai:Monotone Hybrid Methods for a Finite Family of Nonexpasive. . . . . . subsequence {xnk } of {xn} such that ‖xnk − pk‖ < 1 2k for some {pk} ⊂ ⋂m i=1 F (Ti) and all k. From (4), we obtain ‖xnk+1 − pk‖ ≤ ‖xnk − pk‖ < 1 2k . Next, we shall show that {pk} is Cauchy sequence in D. Notice that ‖pk+1 − pk‖ ≤ ‖pk+1 − xnk+1 ‖+ ‖xnk+1 − pk‖ < 1 2k+1 + 1 2k < 1 2k−1 . This implies that {pk} is Cauchy sequence in D and thus converges to q ∈ D. Since d(pk, Tiq) ≤ H(Tiq, Tipk) ≤ ‖q − pk‖ for all i = 1, 2, ...,m and pk → q as n→∞, it follows that d(q, Tiq) = 0 for all i = 1, 2, ...,m and thus q ∈ ⋂m i=1 F (Ti) and {xnk } converges strongly to q. Since limn→∞ ‖xn − q‖ exists, it follows that {xn} converges strongly to q. This completes the proof. Note that in Theorem 3.1 in order to have strong convergence of the iterative sequence {xn} defined by (3), we need to assume that {Ti}mi=1 satisfy the condition (II). In the following theorem, we introduce a new monotone hybrid iterative scheme (2) for finding a common element of the set of a common fixed points of a family of nonexpasive multi-valued maps and the set of solutions of an equilibrium problem in a Hilbert space, and we prove strong convergence of the sequence {xn} defined by (2) without the condition (II). Theorem 3.2. Let D be a nonempty, closed and convex subset of a real Hilbert space H . Let f be a bifunction from D × D to R satisfying (A1)-(A4) and let Ti : D → CB(D) be nonexpansive multi-valued maps for all i = 1, 2, ...,m with ⋂m i=1 F (Ti) ∩ EP (f) 6= ∅ and Tip = {p} for each p ∈ ⋂m i=1 F (Ti). Assume that αi n ∈ (0, 1) with 0 < lim infn→∞ α i n ≤ lim supn→∞ α i n < 1 for all i = 0, 1, 2, ...,m and rn ∈ (0,∞) with lim infn→∞ rn > 0. Then the sequence {xn} generated by (2) converges strongly to P⋂m i=1 F (Ti)∩EP (f)x0. Proof. We split the proof into six steps. Step 1. Show that PCn+1x0 is well defined for every x0 ∈ D. By Lemma 2.8, we obtain that ⋂m i=1 F (Ti) is a closed and convex subset of D. Since EP (f) is also closed and convex, then ⋂m i=1 F (Ti) ∩ EP (f) is a closed and convex subset of D. From the definition of Cn+1, it follows from Lemma 2.5 that Cn+1 is closed and convex for each n ≥ 0. Let v ∈ ⋂m i=1 F (Ti) ∩ EP (f). From un = Trnxn, we have ‖un − v‖ = ‖Trnxn − Trnv‖ ≤ ‖xn − v‖, (5) for every n ≥ 0. From this, we have ‖yn − v‖ = ‖ m∑ i=0 αi nz i n − v‖ ≤ m∑ i=0 αi n‖zin − v‖ = α0 n‖un − v‖+ m∑ i=1 αi nd(zin, Tiv) ≤ α0 n‖un − v‖+ m∑ i=1 αi nH(Tiun, Tiv) ≤ ‖un − v‖ ≤ ‖xn − v‖. (6) Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 157 So, we have v ∈ Cn+1, thus ⋂m i=1 F (Ti)∩EP (f) ⊂ Cn+1. Therefore PCn+1x0 is well defined. Step 2. Show that limn→∞ ‖xn − x0‖ exists. Since ⋂m i=1 F (Ti) ∩ EP (f) is a nonempty, closed and convex subset of H , there exists a unique v ∈ ⋂m i=1 F (Ti) ∩ EP (f) such that v = P⋂m i=1 F (Ti)∩EP (f)x0. From xn = PCnx0, Cn+1 ⊂ Cn and xn+1 ∈ Cn, ∀n ≥ 0, we get ‖xn − x0‖ ≤ ‖xn+1 − x0‖, ∀n ≥ 0. On the other hand, as ⋂m i=1 F (Ti) ∩ EP (f) ⊂ Cn, we obtain ‖xn − x0‖ ≤ ‖v − x0‖, ∀n ≥ 0. It follows that the sequence {xn} is bounded and nondecreasing. Therefore limn→∞ ‖xn−x0‖ exists. Step 3. Show that xn → w ∈ D as n→∞. For m > n, by the definition of Cn, we see that xm = PCmx0 ∈ Cm ⊂ Cn. By Lemma 2.2, we get ‖xm − xn‖2 ≤ ‖xm − x0‖2 − ‖xn − x0‖2. From Step 2, we obtain that {xn} is Cauchy. Hence, there exists w ∈ D such that xn → w as n→∞. Step 4. Show that ‖zin − xn‖ → 0 as n→∞ for every i = 1, 2, ..,m. From xn+1 ∈ Cn+1, we have ‖xn − yn‖ ≤ ‖xn − xn+1‖+ ‖xn+1 − yn‖ ≤ 2‖xn − xn+1‖ → 0 (7) 158 Suantai:Monotone Hybrid Methods for a Finite Family of Nonexpasive. . . . . . as n→∞. For v ∈ ⋂m i=1 F (Ti) ∩ EP (f), by Lemma 2.4 we have ‖yn − v‖2 = ‖ m∑ i=0 αi nz i n − v‖2 ≤ m∑ i=0 αi n‖zin − v‖2 − m∑ i=1 α0 nα i n‖zin − un‖2 ≤ α0 n‖un − v‖2 + m∑ i=1 αi nd(zin, Tiv)2 − m∑ i=1 α0 nα i n‖zin − un‖2 ≤ α0 n‖un − v‖2 + m∑ i=1 αi nH(Tiun, Tiv)2 − m∑ i=1 α0 nα i n‖zin − un‖2 ≤ ‖un − v‖2 − m∑ i=1 α0 nα i n‖zin − un‖2 ≤ ‖xn − v‖2 − m∑ i=1 α0 nα i n‖zin − un‖2. This implies that m∑ i=1 α0 nα i n‖zin − un‖2 ≤ ‖xn − v‖2 − ‖yn − v‖2 ≤ M‖xn − yn‖, where M = supn≥0{‖xn − v‖+ ‖yn − v‖}. By our assumptions and (7), we obtain ‖zin − un‖ → 0 as n→∞, ∀i = 1, 2, ...,m. (8) From Lemma 2.7, we obtain ‖un − v‖2 = ‖Trnxn − Trnv‖2 ≤ 〈Trnxn − Trnv, xn − v〉 = 〈un − v, xn − v〉 = 1 2 { ‖un − v‖2 + ‖xn − v‖2 − ‖xn − un‖2 } , hence ‖un − v‖2 ≤ ‖xn − v‖2 − ‖xn − un‖2. Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 159 Therefore, by Lemma 2.4, we get ‖yn − v‖2 = ‖ m∑ i=0 αi nz i n − v‖2 ≤ m∑ i=0 αi n‖zin − v‖2 ≤ α0 n‖un − v‖2 + m∑ i=1 αi nd(zin, Tiv)2 ≤ α0 n‖un − v‖2 + m∑ i=1 αi nH(Tiun, Tiv)2 ≤ ‖un − v‖2 ≤ ‖xn − v‖2 − ‖xn − un‖2. It follows that ‖xn − un‖2 ≤ ‖xn − v‖2 − ‖yn − v‖2 ≤ M‖xn − yn‖, where M = supn≥0{‖xn − v‖+ ‖yn − v‖}. From (7), we obtain ‖xn − un‖ → 0 as n→∞. (9) From (8) and (9), we have ‖xn − zin‖ ≤ ‖xn − un‖+ ‖un − zin‖ → 0 as n→∞. (10) Step 5. Show that w ∈ ⋂m i=1 F (Ti) ∩ EP (f). From (9) and lim infn→∞ rn > 0, we get ‖xn − un rn ‖ = 1 rn ‖xn − un‖ → 0 , n→∞. (11) From xn → w as n → ∞ and (9), we obtain also that un → w. We shall show that w ∈ EP (f). By un = Trnxn, we get f(un, y) + 1 rn 〈y − un, un − xn〉 ≥ 0, ∀y ∈ D. From the monotonicity of f , we have 1 rn 〈y − un, un − xn〉 ≥ f(y, un), ∀y ∈ D, hence 〈y − un, un − xn rn 〉 ≥ f(y, un), ∀y ∈ D. From (11) and condition (A4), we have 0 ≥ f(y, w), ∀y ∈ D. 160 Suantai:Monotone Hybrid Methods for a Finite Family of Nonexpasive. . . . . . For t with 0 < t ≤ 1 and y ∈ D, let yt = ty+ (1− t)w. Since y, w ∈ D and D is convex, then yt ∈ D and hence f(yt, w) ≤ 0. So, we have 0 = f(yt, yt) ≤ tf(yt, y) + (1− t)f(yt, w) ≤ tf(yt, y). Dividing by t, we obtain f(yt, y) ≥ 0, ∀y ∈ D. Letting t ↓ 0 and from (A3), we get f(w, y) ≥ 0, ∀y ∈ D. Therefore, we obtain w ∈ EP (f). Next, we will show that w ∈ ⋂m i=1 F (Ti). For each i = 1, 2, ...,m, we have d(w, Tiw) ≤ ‖w − xn‖+ ‖xn − zin‖+ d(zin, Tiw) ≤ ‖w − xn‖+ ‖xn − zin‖+H(Tiun, Tiw) ≤ ‖w − xn‖+ ‖xn − zin‖+ ‖un − w‖. It follows from Step 4 that d(w, Tiw) = 0 and thus w ∈ F (Ti) for all i = 1, 2, ...,m. Step 6. Show that w = P⋂m i=1 F (Ti)∩EP (f)x0. Since xn = PCnx0, by Lemma 2.1, we have 〈z − xn, x0 − xn〉 ≤ 0 for all z ∈ Cn. Since w ∈ ⋂m i=1 F (Ti) ∩ EP (f) ⊂ Cn, we get 〈z − w, x0 − w〉 ≤ 0 for all z ∈ ⋂m i=1 F (Ti)∩EP (f). Again by Lemma 2.1, we obtain thatw = P⋂m i=1 F (Ti)∩EP (f)x0. This completes the proof. Corollary 3.3. Let D be a nonempty, closed and convex subset of a real Hilbert space H . Let f be a bifunction from D × D to R satisfying (A1)-(A4) and let T : D → CB(D) be a nonexpansive multi-valued map for all i = 1, 2, ...,m with F (T ) ∩ EP (f) 6= ∅ and Tp = {p} for each p ∈ F (T ). Assume that βn ∈ (0, 1) with 0 < lim infn→∞ βn ≤ lim supn→∞ βn < 1 and rn ∈ (0,∞) with lim infn→∞ rn > 0. For an initial point x0 ∈ D = C0, compute the sequence {xn} by the iterative process f(un, y) + 1 rn 〈y − un, un − xn〉 ≥ 0, ∀y ∈ D, yn = βnun + (1− βn)zn, zn ∈ Tun, Cn+1 = {z ∈ Cn : ‖yn − z‖ ≤ ‖xn − z‖}, xn+1 = PCn+1x0, n ≥ 0, Then the sequence {xn} converges strongly to PF (T )∩EP (f)x0. Proof. Putting T1 = T and Ti = I for i = 2, 3, ...,m, where I : D → CB(D) such that Id = {d} for all d ∈ D in Theorem 3.2, we obtain the desired result directly from Theorem 3.2. Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 161 Corollary 3.4. Let D be a nonempty, closed and convex subset of a real Hilbert space H . Let T : D → CB(D) be a nonexpansive multi-valued map for all i = 1, 2, ...,m with F (T ) ∩ EP (f) 6= ∅ and Tp = {p} for each p ∈ F (T ). Assume that βn ∈ (0, 1) with 0 < lim infn→∞ βn ≤ lim supn→∞ βn < 1. For an initial point x0 ∈ D = C0, compute the sequence {xn} by the iterative process yn = βnxn + (1− βn)zn, zn ∈ Txn, Cn+1 = {z ∈ Cn : ‖yn − z‖ ≤ ‖xn − z‖}, xn+1 = PCn+1x0, n ≥ 0, Then the sequence {xn} converges strongly to PF (T )x0. Proof. Putting f(x, y) = 0 for all x, y ∈ D in Corollary 3.3, we obtain the desired result directly from Corollary 3.3. The main result of this paper holds true under the assumption that Tp = {p} for all p ∈ F (T ). This condition was introduced by Shahzad and Zegeye [17]. The following example gives an example of a nonexpansive multi-valued map T which satisfies the property that Tp = {p} for all p ∈ F (T ) and Tx is not a singleton for all x /∈ F (T ). Example. Consider D = [0, 1]× [0, 1] with the usual norm. Define T : D → CB(D) by T (x, y) =  {(x, 0)}, x 6= 0, y = 0 {(0, y)}, x = 0, y 6= 0 {(x, 0), (0, y)}, x, y 6= 0 {(0, 0)}, x, y = 0. Open problem: Can we drop the condition that Tp = {p} for all p ∈ F (T ) in the main result of this paper ? References [1] Bauschke,H. H., Matoušková,E., Reich,S. (2004). Projection and proximal point methods: convergence results and counterexamples. Nonlinear Anal. 56 (2004) 715-738. [2] Blum,E., & Oettli,W. (1994). From optimization and variational inequalities to equilib- rium problems. Math. Student 63 (1994) 123-145. [3] Ceng,L.-C., & Yao,J.-C. (2008). A hybrid iterative scheme for mixed equilibrium prob- lems and fixed point problems. J. Comput. Appl. Math. 214 (2008) 186-201 [4] Cholamjiak,P. (2009). A hybrid iterative scheme for equilibrium problems, variational inequality problems and fixed point problems in Banach spaces. Fixed Point Theory Appl. Volume 2009 (2009). Article ID 719360. 18 pages. [5] Cholamjiak,P. & Suantai,S. (2009). A new hybrid algorithm for variational inclusions, generalized equilibrium problems and a finite family of quasi-nonexpansive mappings. Fixed Point Theory Appl. Volume 2009 (2009). Article ID 350979. 20 pages. 162 Suantai:Monotone Hybrid Methods for a Finite Family of Nonexpasive. . . . . . [6] Combettes,P.L. & Hirstoaga,S.A. (2005). Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6 (2005) 117-136. [7] Genal,A. & Lindenstrass,J. (1975). An example concerning fixed points. Israel J. Math. 22 (1975) 81-86. [8] Kangtunyakarn,A. & Suantai,S. (2009). Hybrid iterative scheme for generalized equilib- rium problems and fixed point problems of finite family of nonexpansive mappings, Non- linear Analysis: Hybrid Systems. 3 (2009) 296-309. [9] Kim,T.H. & Xu,H.K. (2006). Strongly convergence of modified Mann iterations for with asymptotically nonexpansive mappings and semigroups, Nonlinear Anal. 64 (2006) 1140- 1152. [10] Mann,W.R. (1953). Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953) 506-510. [11] Marino,G. & Xu,H.K. (2007). Weak and strong convergence theorems for strict pseudo- contractions in Hilbert spaces. J. Math. Anal. Appl. 329 (2007) 336-346. [12] Nakajo,K. & Takahashi,W. (2003). Strongly convergence theorems for nonexpansive map- pings and nonexpansive semigroups. J. Math. Anal. Appl. 279 (2003) 372-379. [13] Panyanak,B. (2007). Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces. Comput. Math. Appl. 54 (2007) 872-877. [14] Peng,J.-W. , Liou,Y.-C. , Yao,J.-C. (2009). An iterative algorithm combining viscosity method with parallel method for a generalized equilibrium problem and strict pseudocon- tractions. Fixed Point Theory Appl. Volume 2009 (2009). Article ID 794178. 21 pages. [15] Reich,S. (1979). Weak convergence theorems for nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 67 (1979) 274-276. [16] Sastry,K.P.R. & Babu,G.V.R. (2005). Convergence of Ishikawa iterates for a multivalued mappings with a fixed point. Czechoslovak Math. J. 55 (2005) 817-826. [17] Shahzad,N. & Zegeye,H. (2009). On Mann and Ishikawa iteration schemes for multi- valued maps in Banach spaces. Nonlinear Analysis 71 (2009) 838-844. [18] Shiau,C. , Tan,K.K. , Wong,C.S. (1975). Quasi-nonexpansive multi-valued maps and se- lection. Fund. Math. 87 (1975) 109-119. [19] Song,Y. & Wang,H. (2008). Erratum to ”Mann and Ishikawa iterative processes for mul- tivalued mappings in Banach spaces”[Comput. Math. Appl. 54 (2007) 872-877]. Comput. Math. Appl. 55 (2008) 2999-3002. [20] Tada,A. & Takahashi,W. (2007). Weak and strong convergence theorems for a nonexpan- sive mapping and an equilibrium problem. J. Optim Theory Appl. 133 (2007) 359-370. [21] Xu,H.K. (1991). Inequality in Banach spacees with applications. Nonlinear. Anal. 16 (1991) 1127-1138.