WEAK CONVERGENCE THEOREMS FOR EQUILIBRIUM PROBLEMS WITH NONLINEAR OPERATORS IN HILBERT SPACES

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1 Fixed Point Theory, 12(2011), No. 2, nodeacj/sfptcj.html WEAK CONVERGENCE THEOREMS FOR EQUILIBRIUM PROBLEMS WITH NONLINEAR OPERATORS IN HILBERT SPACES S. DHOMPONGSA, W. TAKAHASHI AND H. YINGTAWEESITTIKUL Department of Mathematics, Faculty of Science Chiang Mai University, Chiang Mai 50200, Thailand. Department of Mathematical and Computing Sciences Tokyo Institute of Technology, Ohokayama, Meguro-ku Tokyo , Japan. Department of Mathematics, Faculty of Science Chiang Mai University, Chiang Mai 50200, Thailand. Abstract. In this paper, we introduce an iterative sequence for finding a common element of the set of fixed points of a nonspreading mapping, the set of solutions of an equilibrium problem and the set of solutions of the variational inequality problem for a monotone and Lipschitz-continuous mapping. We show that the sequence converges weakly to a common element of the above three sets. Key Words and Phrases: Nonspreading mappings, monotone, Lipschitz-continuous mappings, variational inequalities, fixed points Mathematics Subject Classification: 47H05, 47H09, 47H Introduction Let C be a closed convex subset of a real Hilbert space H. Let f be a bifunction of C C into R, where R is the set of real numbers. The equilibrium problem for f : C C R is to find x C such that f(x, y) 0 for all y C. (1.1) The set of solutions (1.1) is denoted by EP (f). A mapping A of C into H is called monotone if Au Av, u v 0 for all u, v C. The variational inequality problem is to find u C such that Au, v u 0 for all v C. The set of solutions of the variational inequality problem is denoted by V I(C, A). A mapping A of C into H is called α-inverse-strongly monotone if there exists a positive real number α such that Au Av, u v α Au Av 2 for all u, v C. It is obvious that any α-inverse-strongly monotone mapping A is monotone and Lipschitz Corresponding author. 309

2 310 S. DHOMPONGSA, W. TAKAHASHI AND H. YINGTAWEESITTIKUL continuous; see, for example, [13]. A mapping S of C into itself is called nonexpansive if Su Sv u v for all u, v C. A mapping S of C into itself is called nonspreading (see [3, 4]) if 2 Su Sv 2 Su v 2 + Sv u 2, for all u, v C. We denote by F (S) the set of fixed points of S. Recently, in the case when S is a nonexpansive mapping, Nadezhkina and Takahashi [5] introduced an iterative process for finding a common element of the set F (S) and the set V I(C, A) for a monotone and Lipschitz-continuous mapping. On the other hand, Tada and Takahashi [9, 10] and Takahashi and Takahashi [11] obtained weak and strong convergence theorems for finding a common element of the set EP (f) and the set F (S) in a Hilbert space. In this paper, we prove weak convergence theorems for finding a common element of the set EP (f), the set V I(C, A) for a monotone and Lipschitz-continuous mapping and the set F (S) of a nonspreading mapping in a Hilbert space. 2. Preliminaries In this paper, we denote by N the set of positive integers and by R the set of real numbers. Let H be a real Hilbert space with inner product, and norm. x n x implies that {x n } converges strongly to x. x n x means that {x n } converges weakly to x. In a real Hilbert space H, we have λx + (1 λ)y 2 = λ x 2 + (1 λ) y 2 λ(1 λ) x y 2 for all x, y H and λ R; see [12]. Let C be a closed convex subset of H. Then, for every point x H, there exists a unique nearest point in C, denoted by P C x, such that x P C x x y for all y C. P C is called the metric projection of H onto C. We know that P C is a nonexpansive mapping of H onto C. It is also known that P C is characterized by the following properties: P C x C, x P C x, P C x y 0 (2.1) and x y 2 x P C x 2 + y P C x 2 (2.2) for all x H, y C. Let A be a monotone mapping of C into H. In the context of the variational inequality problem, this implies u V I(C, A) u = P C (u λau) for all λ > 0. It is also known that H satisfies the Opial condition [6]; i.e., for any sequence {x n } with x n x, the inequality lim inf x n x < lim inf x n y holds for every y H with y x. We also know that H has the Kadec-Klee property, that is, x n x and x n x imply x n x. In fact, from x n x 2 = x n 2 2 x n, x + x 2, we get that a Hilbert space has the Kadec.Klee property. An operator A : H 2 H is said to be monotone if x 1 x 2, y 1 y 2 0 whenever y 1 Ax 1 and y 2 Ax 2. Let A be a monotone, k-lipschitz-continuous mapping of C into H and let N C v be the normal cone to C at v C; i.e., N C v = {w H : v u, w 0, u C}. Define

3 WEAK CONVERGENCE THEOREMS FOR EQUILIBRIUM PROBLEMS 311 T v = { Av + N C v, if v C,, if v / C. Then, T is maximal monotone and 0 T v if and only if v V I(C, A); see [7]. For solving the equilibrium problem for a bifunction f : C C R, let us assume that f satisfies the following conditions: (A1) f(x, x) = 0 for all x C; (A2) f is monotone, i.e., f(x, y) + f(y, x) 0 for all x, y C; (A3) for each x, y, z C, lim t 0 f(tz + (1 t)x, y) f(x, y); (A4) for each x C, y f(x, y) is convex and lower semicontinuous. We know the following lemmas. Lemma 2.1. The following equality holds in a Hilbert space H: For u, v H, u v 2 = u 2 v 2 2 u v, v. Lemma 2.2. [1] Let C be a nonempty closed convex subset of H and let F be a bifunction of C C into R satisfying (A1)-(A4). Let r > 0 and x H. Then, there exists z C such that f(z, y) + 1 y z, z x 0, for all y C. r Lemma 2.3. [2] Assume that f : C C R satisfies (A1) (A4). For r > 0 and x 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 z H. Then, the following hold: 1. T r is single-valued and firmly nonexpansive, i.e., T r x T r y 2 T r x T r y, x y, for any x, y H; 2. F (T r ) = EP (f) ; 3. EP (f) is closed and convex. Lemma 2.4. [8] Let H be a real Hilbert space, let {α n } be a sequence of real numbers such that 0 < a α n b < 1 for all n {0, 1, 2,...} and let {v n } and {w n } be sequences in H. Suppose that there exists c 0 such that lim sup Then lim v n w n = 0. v n c, lim sup w n c and lim α nv n + (1 α n )w n = c. Lemma 2.5. [13] Let C be a nonempty closed convex subset of H. Let {x n } be a sequence in H. Suppose that x n+1 y x n y, for all y C and n {1, 2, 3,...}. Then {P C x n } converges strongly to some z 0 C.

4 312 S. DHOMPONGSA, W. TAKAHASHI AND H. YINGTAWEESITTIKUL 3. Main results In this section, we prove weak convergence theorems. Theorem 3.1. Let C be a closed convex subset of a Hilbert space H. Let f be a bifunction from C C to R satisfying (A1)-(A4) and let A be a monotone k-lipschitz continuous mapping of C into H and let S be a nonspreading mapping of C into itself such that F (S) V I(C, A) EP (f). Let {x n } be a sequence in C generated by x 1 = x C and f(u n, y) + 1 r n y u n, u n x n 0, y C, y n = P C (u n λ n Au n ), x n+1 = α n Sx n + (1 α n )P C (u n λ n Ay n ), where 0 < a λ n b < 1 k, 0 < c α n d < 1 and 0 < r r n. Then {x n } converges weakly to an element p F (S) V I(C, A) EP (f), where p = lim P F (S) V I(C,A) EP (f)x n. Proof. Put v n = P C (u n λ n Ay n ) for every n N. Let x F (S) V I(C, A) EP (f). Then x = Sx = P C (x λ n Ax ) = T rn x. From (2.2), we have v n x 2 u n λ n Ay n x 2 u n λ n Ay n v n = u n x 2 λ n Ay n 2 2 u n λ n Ay n x, λ n Ay n u n v n 2 + λ n Ay n u n λ n Ay n v n, λ n Ay n = u n x 2 u n v n x v n, λ n Ay n = u n x 2 u n v n 2 + 2λ n Ay n, x v n = u n x 2 u n v n 2 + 2λ n ( Ay n Ax, x y n + Ax, x y n + Ay n, y n v n ) u n x 2 u n v n 2 + 2λ n Ay n, y n v n = u n x 2 u n y n 2 2 u n y n, y n v n y n v n 2 + 2λ n Ay n, y n v n = u n x 2 u n y n 2 y n v n u n λ n Ay n y n, v n y n. From (2.1) and v n C, we have u n λ n Ay n y n, v n y n = u n λ n Au n y n, v n y n + λ n Au n λ n Ay n, v n y n λ n Au n λ n Ay n, v n y n λ n k u n y n v n y n.

5 WEAK CONVERGENCE THEOREMS FOR EQUILIBRIUM PROBLEMS 313 Hence, we have So, we have v n x 2 u n x 2 u n y n 2 y n v n 2 + 2λ n k u n y n v n y n u n x 2 u n y n 2 y n v n 2 + λ 2 nk 2 u n y n 2 + v n y n 2 = u n x 2 + (λ 2 nk 2 1) u n y n 2. x n+1 x 2 = α n (Sx n x ) + (1 α n )(v n x ) 2 α n Sx n x 2 + (1 α n ) v n x 2 α n x n x 2 + (1 α n ) u n x 2 + (1 α n )(λ 2 nk 2 1) u n y n 2 = α n x n x 2 + (1 α n ) T rn x n T rn x 2 + (1 α n )(λ 2 nk 2 1) u n y n 2 α n x n x 2 + (1 α n ) x n x 2 + (1 α n )(λ 2 nk 2 1) u n y n 2 = x n x 2 + (1 α n )(λ 2 nk 2 1) u n y n 2 (3.1) x n x 2. Hence { x n x } is bounded and nonincreasing. So, lim x n x exists. From (3.1), we obtain also u n y n 2 1 (1 α n )(1 λ 2 nk 2 ) ( x n x 2 x n+1 x 2 ), and y n v n 2 = P C (u n λ n Au n ) P C (u n λ n Ay n ) 2 u n λ n Au n (u n λ n Ay n ) 2 = λ n Ay n λ n Au n 2 λ 2 nk 2 y n u n 2. Then lim u n y n = 0 and lim y n v n = 0. Consider u n x 2 = T rn x n T rn x 2 T rn x n T rn x, x n x = u n x, x x n Then u n x 2 x n x 2 x n u n 2. = 1 2 ( u n x 2 + x n x 2 x n u n 2 ).

6 314 S. DHOMPONGSA, W. TAKAHASHI AND H. YINGTAWEESITTIKUL We have x n+1 x 2 α n x n x 2 + (1 α n ) v n x 2 α n x n x 2 + (1 α n ) u n x 2 α n x n x 2 + (1 α n ) x n x 2 (1 α n ) x n u n 2. So, we have x n u n 2 1 ( x n x 2 x n+1 x 2 ), which implies that 1 α n lim x n u n = 0. Since Sv n x v n x x n x, we have that lim sup Sv n x lim x n x. Further, we have lim α n(sx n x ) + (1 α n )(v n x ) = lim x n+1 x. By Lemma 2.4, we obtain lim Sx n v n = 0. From Sx n x n Sx n v n + v n y n + y n u n + u n x n, we get lim Sx n x n = 0. As {x n } is bounded, there exists a subsequence {x ni } of {x n } such that x ni p for some p C. Then v ni p Sv ni p. Next, to show p F (S), consider 2 Sv ni Sp 2 Sv ni p 2 + v ni Sp 2 = Sv ni p 2 + v ni Sx ni v ni Sv ni, Sv ni Sp + Sv ni Sp 2. Then Sx ni Sp 2 Sx ni p 2 + x ni Sx ni x ni Sx ni, Sx ni Sp. Suppose Sp p, From Opial s theorem [6] and lim Sx n x n = 0, we obtain lim inf i Sx n i p 2 < lim inf i Sx n i Sp 2 lim inf i ( Sx n i p 2 + x ni Sx ni x ni Sx ni, Sx ni Sp ) = lim inf i Sx n i p 2. This is a contradiction. Hence Sp = p. Next, to show p V I(C, A), let { Av + N C v, if v C, T v =, if v / C. Then, T is maximal monotone and 0 T v if and only if v V I(C, A). Let (v, w) G(T ). Then, we have w T v = Av + N C v and hence w Av N C v. So, we have v u, w Av 0 for all u C. On the other hand, from v n = P C (u n λ n Ay n ) and v C, we have u n λ n Ay n v n, v n v 0, and hence, v v n, v n u n λ n + Ay n 0.

7 WEAK CONVERGENCE THEOREMS FOR EQUILIBRIUM PROBLEMS 315 Therefore, from w Av N C v and v n C, we have v v ni, w v v ni, Av v v ni, Av v v ni, v n i u ni λ ni + Ay ni = v v ni, Av Av ni + v v ni, Av ni Ay ni v v ni, v n i u ni λ ni v v ni, Av ni Ay ni v v ni, v n i u ni λ ni. Since x ni p, lim v n x n = 0, lim v n u n = 0, lim y n v n = 0 and A is Lipschitz continuous. So we obtain v p, w 0. Since T is maximal monotone, we have p T 1 0 and hence p V I(C, A). Let us show p EP (f). Since f(u ni, y) + 1 y u ni, u ni x ni 0 for all y C. r ni From (A2), we also have 1 y u ni, u ni x ni f(y, u ni ) r ni and hence y u ni, u n i x ni r ni f(y, u ni ). From lim u n x n = 0, we get u ni p. Since u n i x ni 0, it follows by (A4) r ni that 0 f(y, p) for all y C. For t with 0 < t 1 and y C, let y t = ty + (1 t)p. Since y, p C, we have y t C and hence f(y t, p) 0. So, from (A1) and (A4) we have 0 = f(y t, y t ) tf(y t, y) + (1 t)f(y t, p) tf(y t, y) and hence 0 f(y t, y). From (A3), we have 0 f(p, y) for all y C and hence p EP (f). Thus p F (S) V I(C, A) EP (f). Let {x nj } be another subsequence of {x n } such that x nj p. Then p F (S) V I(C, A) EP (f). Assume p p. Then we have lim x n p = lim inf x n i p i < lim inf x n i p i = lim x n p = lim x n j p j < lim x n j p j = lim x n p. This is a contradiction. Thus p = p and x n p F (S) V I(C, A) EP (f).

8 316 S. DHOMPONGSA, W. TAKAHASHI AND H. YINGTAWEESITTIKUL Put p n = P F (S) V I(C,A) EP (f) x n. We show p = lim p n. From p n = P F (S) V I(C,A) EP (f) x n and p F (S) V I(C, A) EP (f), we have p p n, p n x n 0. By Lemma 2.5, {p n } converges strongly to some p 0 F (S) V I(C, A) EP (f). Then we have p p 0, p 0 p 0 and hence p = p 0. This completes the proof. Next, we prove another weak convergence theorem which is different from Theorem 3.1. Theorem 3.2. Let C be a closed convex subset of a Hilbert space H. Let f be a bifunction from C C to R satisfying (A1)-(A4) and let A be a monotone and k- Lipschitz continuous mapping of C into H and let S be a nonspreading mapping of C into itself such that F (S) V I(C, A) EP (f). Let {x n } be a sequence in C generated by x 1 = x C and f(u n, y) + 1 r n y u n, u n x n 0, y C, y n = P C (u n λ n Au n ), x n+1 = α n x n + (1 α n )SP C (u n λ n Ay n ), where 0 < a λ n b < 1 k, 0 < c α n d < 1 and 0 < r r n. Then {x n } converges weakly to an element p F (S) V I(C, A) EP (f), where p = lim P F (S) V I(C,A) EP (f)x n. Proof. Put v n = P C (u n λ n Ay n ) for every n N. Let x F (S) V I(C, A) EP (f). Then x = Sx = P C (x λ n Ax ) = T rn x. As in the proof of Theorem 3.1, we have that Thus v n x 2 u n x 2 + (λ 2 nk 2 1) u n y n 2. x n+1 x 2 = α n (x n x ) + (1 α n )(Sv n x ) 2 α n x n x 2 + (1 α n ) Sv n x 2 α n x n x 2 + (1 α n ) v n x 2 α n x n x 2 + (1 α n ) u n x 2 + (1 α n )(λ 2 nk 2 1) u n y n 2 = α n x n x 2 + (1 α n ) T rn x n T rn x 2 + (1 α n )(λ 2 nk 2 1) u n y n 2 α n x n x 2 + (1 α n ) x n x 2 + (1 α n )(λ 2 nk 2 1) u n y n 2 = x n x 2 + (1 α n )(λ 2 nk 2 1) u n y n 2 (3.2) x n x 2. Hence { x n x } is bounded and nonincreasing. So, lim x n x exists.

9 WEAK CONVERGENCE THEOREMS FOR EQUILIBRIUM PROBLEMS 317 From (3.2), we obtain also and u n y n 2 1 (1 α n )(1 λ 2 nk 2 ) ( x n x 2 x n+1 x 2 ), y n v n 2 = P C (u n λ n Au n ) P C (u n λ n Ay n ) 2 u n λ n Au n (u n λ n Ay n ) 2 = λ n Ay n λ n Au n 2 λ 2 nk 2 y n u n 2. Then lim u n y n = 0 and lim y n v n = 0. Consider u n x 2 = T rn x n T rn x 2 T rn x n T rn x, x n x = u n x, x x n Then u n x 2 x n x 2 x n u n 2. We have = 1 2 ( u n x 2 + x n x 2 x n u n 2 ). x n+1 x 2 α n x n x 2 + (1 α n ) v n x 2 α n x n x 2 + (1 α n ) u n x 2 α n x n x 2 + (1 α n ) x n x 2 (1 α n ) x n u n 2 = x n x 2 (1 α n ) x n u n 2. So, we have x n u n 2 1 ( x n x 2 x n+1 x 2 ), 1 α n which implies that lim x n u n = 0. Since Sv n x v n x x n x, we have lim sup Sv n x lim x n x. Further, we have lim α n(x n x ) + (1 α n )(Sv n x ) = lim x n+1 x. By Lemma 2.4, we obtain lim Sv n x n = 0. From Sv n v n Sv n x n + x n u n + u n y n + y n v n, we get lim Sv n v n = 0. As {x n } is bounded, there exists a subsequence {x ni } of {x n } such that x ni p for some p C. Then v ni p and Sv ni p.

10 318 S. DHOMPONGSA, W. TAKAHASHI AND H. YINGTAWEESITTIKUL Next, to show p F (S), consider 2 Sv ni Sp 2 Sv ni p 2 + v ni Sp 2 = Sv ni p 2 + v ni Sv ni 2 2 v ni Sv ni, Sv ni Sp + Sv ni Sp 2. Then Sv ni Sp 2 Sv ni p 2 + v ni Sv ni 2 2 v ni Sv ni, Sv ni Sp. Suppose Sp p, From Opial condition and lim Sv n v n = 0, we obtain lim inf i Sv n i p 2 < lim inf i Sv n i Sp 2 lim inf i ( Sv n i p 2 + v ni Sv ni 2 2 v ni Sv ni, Sv ni Sp ) = lim inf i Sv n i p 2. This is a contradiction. Hence Sp = p. We can now follow the proof of Theorem Applications Using Theorems 3.1 and 3.2, we prove four theorems in a real Hilbert space. Corollary 4.1. Let C be a closed convex subset of a Hilbert space H. Let A be a monotone and k-lipschitz continuous mapping of C into H and let S be a nonspreading mapping of C into itself such that F (S) V I(C, A). Let {x n } be a sequence in C generated by x 1 = x C, y n = P C (x n λ n Ax n ), x n+1 = α n Sx n + (1 α n )P C (x n λ n Ay n ), where 0 < a λ n b < 1 k, 0 < c α n d < 1. Then {x n } converges weakly to p F (S) V I(C, A), where p = lim P F (S) V I(C,A)x n. Proof. Putting f(x, y) = 0 for all x, y C and r n = 1 in Theorem 3.1, we obtain the desired result. Corollary 4.2. Let C be a closed convex subset of a Hilbert space H. Let A be a monotone k-lipschitz continuous mapping of C into H and let S be a nonspreading mapping of C into itself such that F (S) V I(C, A) EP (f). Let {x n } be a sequence in C generated by x 1 = x C, y n = P C (x n λ n Ax n ), x n+1 = α n x n + (1 α n )SP C (x n λ n Ay n ),

11 WEAK CONVERGENCE THEOREMS FOR EQUILIBRIUM PROBLEMS 319 where 0 < a λ n b < 1 k, 0 < c α n d < 1. Then {x n } converges weakly to p F (S) V I(C, A), where p = lim P F (S) V I(C,A)x n. Proof. Putting f(x, y) = 0 for all x, y C and r n = 1 in Theorem 3.2, we obtain the desired result. Corollary 4.3. Let C be a closed convex subset of a Hilbert space H. Let f be a bifunction from C C to R satisfying (A1)-(A4) and let S be a nonspreading mapping of C into itself such that F (S) EP (f). Let {x n } be a sequence in C generated by { x 1 = x C, x n+1 = α n Sx n + (1 α n )T rn x n, where 0 < c α n d < 1 and 0 < r r n. Then {x n } converges weakly to p F (S) EP (f), where p = lim P F (S) EP (f)x n. Proof. Putting A = 0 in Theorem 3.1, we obtain the desired result. Corollary 4.4. Let C be a closed convex subset of a Hilbert space H. Let f be a bifunction from C C to R satisfying (A1)-(A4) and let S be a nonspreading mapping of C into itself such that F (S) EP (f). Let {x n } be a sequence in C generated by { x 1 = x C, x n+1 = α n x n + (1 α n )ST rn x n, where 0 < c α n d < 1 and 0 < r r n. Then {x n } converges weakly to p F (S) EP (f), where p = lim P F (S) EP (f)x n. Proof. Putting A = 0 in Theorem 3.2, we obtain the desired result. Acknowledgement The authors would like to thank the Thailand Research Fund (grant BRG ) and the Development and Promotion of Science and Technology Talent Project (DPST) for their support. References [1] E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student, 63(1994), [2] P.L. Combettes and A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6(2005), [3] F. Kosaka and W. Takahashi, Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces, SIAM. J. Optim., 19(2008), [4] F. Kosaka and W. Takahashi, Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces, Arch. Math. (Basel), 91(2008), [5] N. Nadezhkina and W. Takahashi, Weak convergence theorem by an extragradient Method for Nonexpansive mappings and Monotone Mappings, J. Optim. Theory Appl., 128(2006), [6] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73(1967), [7] R.T. Rockafellar, On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc., 149(1970),

12 320 S. DHOMPONGSA, W. TAKAHASHI AND H. YINGTAWEESITTIKUL [8] J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc., 43(1991), [9] A. Tada and W. Takahashi, Strong convergence theorem for an equilibrium problem and a nonexpansive mapping, in: Nonlinear Analysis and Convex Analysis, (W. Takahashi and T. Tanaka-Eds.), Yokohama Publishers, Yokohama, 2007, [10] A. Tada and W. Takahashi, Weak and strong convergence theorems for a nonexpansive mapping and equilibrium problem, J. Optim. Theory Appl., 133(2007), [11] S. Takahashi and W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl., 331(2007), [12] W. Takahashi, Introduction to Nonlinear and Convex Analysis, Yokohama Publishers, Yokohama, [13] W. Takahashi and M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 118(2003), Received: March 4, 2010; Accepted: June 11, 2010.