Convergence theorems for a finite family. of nonspreading and nonexpansive. multivalued mappings and equilibrium. problems with application

Transcription

1 Theoretical Mathematics & Applications, vol.3, no.3, 2013, ISSN: (print), (online) Scienpress Ltd, 2013 Convergence theorems for a finite family of nonspreading and nonexpansive multivalued mappings and equilibrium problems with application Hong Bo Liu 1 Abstract We introduce a finite family of nonspreading multivalued mappings and a finite family of nonexpansive multivalued mappings in Hilbert space. We establish some weak and strong convergence theorems of the sequences generated by our iterative process. Some new iterative sequences for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points it was introduced. The results improve and extend the corresponding results of Mohammad Eslamian (Optim. Lett., 7(3): , 2012). Mathematics Subject Classification: 47J05; 47H09; 49J25 Keywords: Equilibrium problem; onspreading mapping multivalued; nonexpansive multivalued mapping; common fixed point 1 School of Science, Southwest University of Science and Technology, Mianyang, Sichuan , P. R. China. Article Info: Received : May 17, Revised : July 10, 2013 Published online : September 1, 2013

2 50 Convergence theorems for a finite family of nonspreading 1 Introduction Throughout this paper, we denote by N and R the sets of positive integers and real numbers, respectively. Let E be a nonempty closed subset of a real Hilbert space H. Let N(E) and CB(E) denote the family of nonempty subsets and nonempty closed bounded subsets of E, respectively. The Hausdorff metric on CB(E) is defined by H(A 1, A 2 ) = max{ sup x A 1 d(x, A 2 ), sup y A 2 d(y, A 1 )} for A 1, A 2 CB(E), where d(x, A 1 ) = inf{ x y, y A 1 }. An element p E is called a fixed point of T : E N(E) if p T (p). The set of fixed points of T is represented by F (T ). A subset E H is called proximal if for each x H, there exists an element y H such that x y = d(x.e) = inf{ x z : z E}. We denote by K(E) and P (E) nonempty compact subsets and nonempty proximal bounded subsets of E, respectively. The multi-valued mapping T : E CB(E) is called nonexpansive if H(T x, T y) x y, for x, y E. The multi-valued mapping T : E CB(E) is called quasi-nonexpansive if F (T ) and H(T x, T p) x p, for x E p F (T ). Iterative process for approximating fixed points (and common fixed points) of nonexpansive multivalued mappings have been investigated by various authors (see[1], [2], [3]). Recently, Kohsaka and Takahashi (see[4], [5])introduced an important class of mappings which they called the class of nonspreading mappings. Let E be a subset of Hilbert space H, then they called a mapping T : E E nonspreading if 2 T x T y 2 T x y 2 + T y x 2, x, y E.

3 H. Liu 51 Lemoto and Takahashi [6] proved that T : E E is nonspreading if and only if T x T y 2 x y x T y, y T y, x, y E. Now, inspired by [4] and [5], we propose a definition as follows, Definition 1.1. The multivalued mapping T : E CB(E) is called nonspreading if 2 u x u y 2 u x y 2 + u y x 2, (1) for u x T x, u y T y, x, y E. By Takahashi [6], We get also the multivalued mapping T : E CB(E) is nonspreading if and only if u x u y 2 x y x u x, y u y, (2) for u x T x, u y T y, x, y E. In fact, 2 u x u y 2 u x y 2 + u y x 2 2 u x u y 2 u x x u x x, x y + x y 2 + u x x x u x, u x u y + u x u y 2 2 u x u y 2 2 u x x 2 + x y 2 + u x u y 2 +2 u x x, x u x (y u y ) u x u y 2 x y x u x, y u y. The equilibrium problem for φ : E E R is to find x E such that φ(x, y) 0, y E. The set of such solution is denoted by EP (φ). Given a mapping T : E CB(E), let φ(x, y) = x, y for all y E. The x EP (φ) if and only if x E is a solution of the variational inequality T x, y 0 for all y E. Numerous problems in physics, optimization, and economics can be reduced to find a solution of the equilibrium problem. Some methods have been proposed to solve the equilibrium problem. For instance, see Blum and Oettli [7], Combettes and Hirstoaga [8], Li and Li [9], Giannessi, Maugeri, and Pardalos [10], Moudafi and Thera [11] and Pardalos,Rassias and Khan [12]. In the

4 52 Convergence theorems for a finite family of nonspreading recent years, the problem of finding a common element of the set of solutions of equilibrium problems and the set of fixed points of singlevalued nonexpansive mappings in the framework of Hilbert spaces has been intensively studied by many authors. In this paper, inspired by [13], we propose an iterative process for finding a common element of the set of solutions of equilibrium problem and the set of common fixed points of a countable family of nonexpansive multivalued mappings and a countable family of nonspreading multivalued mappings in the setting of real Hilbert spaces. We also prove the strong and weak convergence of the sequences generated by our iterative process. The results presented in the paper improve and extend the corresponding results in [15]. 2 Preliminaries In the sequel, we begin by recalling some preliminaries and lemmas which will be used in the proof. Let H be a real Hilbert spaces with the norm. It is also known that H satisfies Opial, s condition, i.e., for any sequence x n with x n x, the inequality lim inf holds for every y H with y x. x n x < lim inf x n y Lemma 2.1. (see[16]) Let H be a real Hilbert space, then the equality x y 2 = x 2 y 2 2 x y, y holds for all x, y H Lemma 2.2. (see[17]) Let H be a real Hilbert space. Then for all x, y, z H and α, β, γ [0, 1] with α + β + γ = 1, we have αx + βy + γz 2 = α x 2 + β y 2 + γ z 2 αβ x y 2 βγ y z 2 γα z x 2.

5 H. Liu 53 For solving the equilibrium problem, we assume that the bifunction φ : E E R satisfies the following conditions for all x, y, z E: (A 1 ) φ(x, x) = 0, (A 2 ) φ is monotone, i.e., φ(x, y) + φ(y, x) 0, (A 3 ) φ is upper-hemicontinuous, i.e.lim sup t 0 + φ(tz + (1 t)x, y) φ(x, y), (A 4 ) φ(x,.) = 0 is convex and lower semicontinuous. Lemma 2.3. (see[7]) Let E be a nonempty closed convex subset of H and let φ be a bifunction satisfying (A 1 ) (A 4 ). Let r > 0 and x H. Then, there existsz E such that φ(z, y) + 1 y z, z x 0, y E. r The following Lemma is established in [8]. Lemma 2.4. Let φ be a bifunction satisfying (A 1 ) (A 4 ). For r > 0 and x H, define a mapping T r : H E as follows T r x = {z E : φ(z, y) + 1 y z, z x 0, r y E}. Then, the following results hold (a) T r is single valued; (b) T r is firmly nonexpansive,i.e.,for any x, y H, T r x T r y 2 T r x T r y, x y ; (c) F (T r ) = EP (φ); (d) EP (φ) is closed and convex. The following Lemma is established in [3]. Lemma 2.5. Let E be a nonempty closed convex subset of a real Hilbert space H. Let T : E K(E) be a nonexpansive multivalued mapping. If x n p and lim d(x n, T x n ) = 0, then p T p.

6 54 Convergence theorems for a finite family of nonspreading Lemma 2.6. Let E be a nonempty closed convex subset of a real Hilbert space H. Let T : E E be a nonspreading multivalued mapping such that F (T ). Then S is demiclosed, i.e., x n p and lim d(x n, T x n ) = 0, we have p T p. Proof. Let {x n } E be a sequence such that x n p and lim d(x n, T x n ) = 0. Then the sequences {x n } and T x n are bounded. Suppose that p T p. From Definition 1.1 and Opial, s condition, we have lim inf x n p 2 < lim inf x n u p 2 = lim inf ( x n u xn 2 + u xn u p x n u xn, u xn u p ) lim inf ( x n u xn 2 + x n p x n u xn, p u p +2 x n u xn, u xn u p ) = lim inf x n p 2, where u p T p and u xn T x n. This is a contradiction. Hence we get the conclusion. 3 Main results Theorem 3.1. Let E be a nonempty closed convex subset of a real Hilbert space H, φ be a bifunction of E E into R satisfying (A 1 ) (A 2 ). Let for i = 1, 2, 3, N, f i : E E be a finite family of nonspreading multivalued mappings and T i : E K(E) be a finite family of nonexpansive multivalued mappings. Assume that F = N i=1 F (T i) F (f i ) EP (φ) and T i p = {p} for each p F. Let {x n } and {w n } be sequences generated initially by an arbitrary element x 1 E and φ(w n, y) + 1 r n y w n, w n x n 0, y E, x n+1 = α n w n + β n u n n + γ n z n, n 1, where u i n f i w n,z n T n w n, T n = T N(mod(N)) and

7 H. Liu 55 (i) {α n }, {β n }, {γ n } [a, 1] (0, 1), (ii) α n + β n + γ n = 1, n 1, (iii) r n > 0,and lim inf r n > 0. Then, the sequences {x n } and {w n } converge weakly to an element of F. Proof. (I) First, We claim that lim x n q exist for all q F. Indeed, let q F, then from the definition of T r in lemma 2.4,we have w n = T rn x n and therefore w n q = T rn x n T rn q x n q n 1. By Lemma 2.2 and Since for each i,f i is nonspearding multivalued mappings, we have x n+1 q 2 = α n w n + β n u n n + γ n z n q 2 α n w n q 2 + β n u n n q 2 + γ n z n q 2 α n β n w n u n n 2 α n γ n w n z n 2 = α n w n q 2 + β n u n n q 2 + γ n d 2 (z n, T n q) α n β n w n u n n 2 α n γ n w n z n 2 α n w n q 2 + β n u n n q 2 + γ n H 2 (T n w n, T n q) α n β n w n u n n 2 α n γ n w n z n 2 α n w n q 2 + β n w n q 2 + γ n w n q 2 α n β n w n u n n 2 α n γ n w n z n 2 = w n q 2 α n β n w n u n n 2 α n γ n w n z n 2 x n q 2 α n β n w n u n n 2 α n γ n w n z n 2. (3) This implies that x n+1 q x n q,hence we get lim x n q exist for all q F. (II) Next, we prove that lim w n w n+i = 0, for all i 1. From (3), we have α n β n w n u n n 2 x n q 2 x n+1 q 2.

8 56 Convergence theorems for a finite family of nonspreading It follows from our assumption that a 2 w n u n n 2 x n q 2 x n+1 q 2. Taking the limit as n, we have lim w n u n n = 0. Observing (3) again and our assumption, we have a 2 w n z n 2 x n q 2 x n+1 q 2. Taking the limit as n, we get lim w n z n = 0. Thus Also we have lim d(w n, T n w n ) lim w n z n = 0. (4) lim x n+1 w n lim β n w n u n n + lim γ n w n z n = 0. (5) Since w n = T rn x n, we have and hence w n q 2 = T rn x n T rn q 2 T rn x n T rn q, x n q = w n q, x n q = 1 2 ( w n q 2 + x n q 2 x n w n 2 ), w n q 2 x n q 2 x n w n 2. (6) Using (3) and the last inequality (6), we have and hence This imply that Using (5) and (7), we get x n+1 q 2 x n q 2 x n w n 2 x n w n 2 x n q 2 x n+1 q 2. lim x n w n = 0. (7) lim w n w n+1 lim ( x n+1 w n+1 + x n+1 w n ) = 0. (8)

9 H. Liu 57 It follows that lim w n w n+i = 0, i {1, 2, 3, N}. (9) Applying (7)-(9), we obtain that This also implies that lim x n+1 x n = 0. (10) lim x n+i x n = 0, i {1, 2, 3, N}. (11) (III) We claim that for i {1, 2, N}, Observing that for i N, lim d(w n, T i w n ) = lim d(w n, f i w n ) = 0 lim d(x n, T i x n ) = lim d(x n, f i x n ) = 0. lim d(w n, T n+i w n ) lim [ w n w n+i + d(w n+i, T n+i w n+i ) + H(T n+i w n+i, T n+i w n )] lim [2 w n w n+i + d(w n+i, T n+i w n+i )] = 0, which implies that the sequence we have N {d(w n, T n+i w n )} 0, i=1 Also we have lim d(w n, T i w n ) = lim d(w n, T n+(i n) w n ) = 0. lim d(w n, f n+i w n ) lim [ w n w n+i + d(w n+i, f n+i w n+i ) + H(f n+i w n+i, f n+i w n )] lim [ w n w n+i + w n+i u n+i n+i + un+i un+i n ] lim [ w n w n+i + w n+i u n+i n+i +( w n w n+i w n+i u n+i n+i, w n u n n+i ) 1 2 = 0

10 58 Convergence theorems for a finite family of nonspreading which imply that lim d(w n, f i w n ) = 0, i {1, 2, N}. Hence for i {1, 2, N}, we have and lim n, T i x n ) lim [ x n w n + d(w n, T i w n ) + H(T i w n, T i x n )] lim [2 x n w n + d(w n, T i x n )] = 0, lim n, f i x n ) lim [ x n w n + d(w n, f i w n ) + H(f i w n, f i x n )] lim [ x n w n + d(w n, f i w n ) +( w n x n w n u i n, x n u i n ) 1 2 = 0. (IV) Now we prove that ω w (x n ) F where ω w (x n ) = {x H : x ni x, {x ni } {x n }}. Since {x n } is bounded and H is reflexive,ω w (x n ) is nonempty. Let µ ω w (x n ) is be an arbitrary element. Then there exists a subsequence {x ni {x n } converging weakly to µ. From (3.5) we get that w ni µ as i. We show that µ EP (φ). Since w n = T rn x n, we have φ(w n, y) + 1 y w n, w n x n 0, y E. r n By (A 4 ), we have and hence 1 r n y w n, w n x n φ(y, w n ), y w ni, w n i x ni r ni φ(y, w ni ). Therefore, we have φ(y, µ) 0 for all y E. For t (0, 1], let y t = ty + (1 t)µ.since y, µ E and E is convex, we have y t E and hence φ(y t, µ) 0. Form (A 1 ), (A 4 ) we have φ(y t, y t ) = φ(y t, ty + (1 t)µ) tφ(y t, y) + (1 t)φ(y t, µ) tφ(y t, y),

11 H. Liu 59 which gives φ(y t, y) 0. From (A 3 ) we have φ(µ, y) 0, and hence µ EP (φ). By Lemma 2.5 and 2.6, we have µ N i=1 F (T i) F (f i ). Therefore ω w (x n ) F. (V) We show that {x n } and {w n } converge weakly to an element of F. To verify that the aberration is valid, it is sufficient to show that ω w (x n ) is a single point set. Let µ 1, µ 2 ω w (x n ), and {x nk }, {x nm } {x n }, such that x nk µ 1, x nm µ 2. Since lim x n q exists for all q F, and µ 1, µ 2 F, we have lim x n µ 1 and lim x n µ 2. Now let µ 1 µ 2, then by Opial, s property, lim x n µ 1 = lim x nk µ 1 k < lim x nk µ 2 k = lim x n µ 2 = lim x n m µ 2 m < lim x n m µ 1 m = lim x n µ 1, which is a contradiction. Therefore µ 1 = µ 2, This show that ω w (x n ) is a single point set. This complete the proof of Theorem 3.1. References [1] Y. Song and H. Wang, Convergence of iterative algorithms for multivalued mappings in Banach spaces, Nonlinear Anal., 70, (2009), [2] N. Shahzad and H. Zegeye, On Mann and Ishikawa iteration schemes for multivalued maps in Banach space, Nonlinear Anal., 71, (2009), [3] M. Eslamian and A. Abkar, One-step iterative process for a finite family of multivalued mappings, Math. Comput. Modell., 54, (2011),

12 60 Convergence theorems for a finite family of nonspreading [4] F. Kohsaka and W. Takahashi, Fixed point theorems for a class of nonlinear mappings relate to maximal monotone operators in Banach spaces, Arch. Math. (Basel)., 91, (2008), [5] F. Kohsaka and W. Takahashi, Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces, SIAM J. Optim., 19, (2008), [6] S. Iemoto and W. Takahashi, Approximating commom fixed points of nonexpansive mappings and nonspreading mappings in a Hilbert space, Nonlinear Anal., 71, (2009), [7] E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student., 63, (1994), [8] P.L. Combettes and S.A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6, (2005), [9] X.B. Li and S.J. Li, Existence of solutions for generalized vector quasiequilibrium problems, Optim. Lett., 4, (2010), [10] F. Giannessi, G. Maugeri and P.M. Pardalos, Equilibrium problems: nonsmooth optimization and variational inequality models, Kluwer Academics Publishers, Dordrecht, Holland, [11] A. Moudafi and M. Thera, Proximal and dynamical approaches to equilibrium problems, In: Lecture Note in Economics and Mathematical Systems, Springer-Verlag, New York, [12] P.M. Pardalos, T.M. Rassias and A.A. Khan, Nonlinear Analysis and Variational Problems, Springer, Berlin, [13] L.C. Ceng, S. Al-Homidan, Q.H. Ansari and J.C. Yao, An iterative scheme for equilibrium problems and fixed point problems of strict pseudocontraction mappings, J. Comput. Appl. Math., 223, (2009), [14] J.F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer Verlag, New York, 2000.

13 H. Liu 61 [15] Mohammad Eslamian, Convergence theorems for nonspreading mappings and nonexpansive multivalued mappings and equilibrium problems, Optim. Lett., 7(3), (2012), [16] G. Marino and H.K. Xu, Weak and strong convergence theorems for strict pseudo-contractions in Hilbert space, J. Math. Anal. Appl., 329, (2007), [17] M.O. Osilike and D.I. Igbokwe, Weak and strong convergence theorems for fixed points of pseudocontractions and solutions of monotone type operator equations, Comput. Math. Appl., 40, (2000),