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Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011, Article ID 372975, 18 pages doi:10.1155/2011/372975 Research Article New Iterative Scheme for Finite Families of Equilibrium, Variational Inequality, and Fixed Point Problems in Banach Spaces Shenghua Wang1, 2 and Caili Zhou3 1 School of Applied Mathematics and Physics, North China Electric Power University, Baoding 071003, China 2 Department of Mathematics, Gyeongsang National University, Jinju 660-714, Republic of Korea 3 College of Mathematics and Computer, Hebei University, Baoding 071002, China Correspondence should be addressed to Shenghua Wang, sheng-huawang@hotmail.com Received 6 December 2010; Accepted 30 January 2011 Academic Editor: S. Al-Homidan Copyright q 2011 S. Wang and C. Zhou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We introduced a new iterative scheme for ﬁnding a common element in the set of common ﬁxed points of a ﬁnite family of quasi-φ-nonexpansive mappings, the set of common solutions of a ﬁnite family of equilibrium problems, and the set of common solutions of a ﬁnite family of variational inequality problems in Banach spaces. The proof method for the main result is simpliﬁed under some new assumptions on the bifunctions. 1. Introduction Throughout this paper, let R denote the set of all real numbers. Let E be a smooth Banach space and E∗ the dual space of E. The function φ : E × E → R is deﬁned by 2 2 φ x, y x − y, Jx y , ∀x, y ∈ E, 1.1 where J is the normalized dual mapping from E to E∗ deﬁned by x∗ ∈ E∗ : x, x∗ x∗ 2 2 Jx x , ∀x ∈ E. 1.2
2 Fixed Point Theory and Applications Let C be a nonempty closed and convex subset of E. The generalized projection Π : E → C is a mapping that assigns to an arbitrary point x ∈ E the minimum point of the function φ x, y , that is, ΠC x x, where x is the solution to the minimization problem φ x, x inf φ z, x . 1.3 z∈C x − y 2 and ΠC In Hilbert spaces, φ x, y PC , where PC is the metric projection. It is obvious from the deﬁnition of function φ that 2 2 y−x ≤ φ y, x ≤ y x , ∀x, y ∈ E. 1.4 We remark that if E is a reﬂexive, strictly convex and smooth Banach space, then for x, y ∈ E, φ x, y 0 if and only if x y. For more details on φ and Π, the readers are referred to 1–4 . Let T be a mapping from C into itself. We denote the set of ﬁxed points of T by F T . T is called to be nonexpansive if T x − T y ≤ x − y for all x, y ∈ C and quasi-nonexpansive if F T / ∅ and x − T y ≤ x − y for all x ∈ F T and y ∈ C. A point p ∈ C is called to be an asymptotic ﬁxed point of T 5 if C contains a sequence {xn } which converges weakly to p such that limn → ∞ xn − T xn 0. The set of asymptotic ﬁxed points of T is denoted by F T . The mapping T is said to be relatively nonexpansive 6–8 if F T F T and φ p, T x ≤ φ p, x for all x ∈ C and p ∈ F T . The mapping T is said to be φ-nonexpansive if φ T x, T y ≤ φ x, y for all x, y ∈ C. T is called to be quasi-φ-nonexpansive 9 if F T / ∅ and φ p, T x ≤ φ p, x for all x ∈ C and p ∈ F T . In 2005, Matsushita and Takahashi 10 introduced the following algorithm: x0 x ∈ C, J −1 αn Jxn yn 1 − αn J T xn , Cn z ∈ C : φ z, yn ≤ φ z, xn , 1.5 Qn {z ∈ C : xn − z, Jx − Jxn ≥ 0}, xn PCn ∩Qn x, ∀n ≥ 0, 1 where J is the duality mapping on E, T is a relatively nonexpansive mapping from C into itself, and {αn } is a sequence of real numbers such that 0 ≤ αn < 1 and lim supn → ∞ αn < 1 and proved that the sequence {xn } generated by 1.5 converges strongly to PF T x, where PF T is the generalized projection from C onto F T . Let f be a bifunction from C × C to R. The equilibrium problem for f is to ﬁnd p ∈ C such that f p, y ≥ 0, ∀y ∈ C. 1.6 We use EP f to denote the solution set of the equilibrium problem 1.6 . That is, EP f p ∈ C : f p, y ≥ 0, ∀y ∈ C . 1.7
Fixed Point Theory and Applications 3 For studying the equilibrium problem, f is usually assumed to satisfy the following condi- tions: A1 f x, x 0 for all x ∈ C; A2 f is monotone, that is, f x, y f y, x ≤ 0 for all x, y ∈ C; A3 for each x, y, z ∈ C, lim supt → 0 f tz 1 − t x, y ≤ f x, y ; A4 for each x ∈ C, y → f x, y is convex and lower semicontinuous. Recently, many authors investigated the equilibrium problems in Hilbert spaces or Banach spaces; see, for example, 11–25 . In 20 , Qin et al. considered the following iterative scheme by a hybrid method in a Banach space: x0 ∈ E chosen arbitrarily, C1 C, x1 ΠC1 x0 , N J −1 αn,0 Jxn yn αn,i JTi xn , 1.8 i1 1 un ∈ C such that f un , y y − un , Jun − Jyn ≥ 0, ∀y ∈ C, rn Cn z ∈ Cn : φ z, un ≤ φ z, xn , 1 xn ΠCn 1 x0 , 1 where Ti : C → C is a closed quasi-φ-nonexpansive mapping for each i ∈ {1, 2, . . . , N }, αn,0 , {αn,1 }, . . . , {αn,N } are real sequences in 0, 1 satisfying N 0 αn,j 1 for each n ≥ 1 and j lim infn → ∞ αn,0 αn,i > 0 for each i ∈ {1, 2, . . . , N } and {rn } is a real sequence in a, ∞ with a > 0. N Then the authors proved that {xn } converges strongly to ΠF x0 , where F i 1 F Ti ∩ EP f . Very recently, Zegeye and Shahzad 25 introduced a new scheme for ﬁnding an element in the common ﬁxed point set of ﬁnite family of closed relatively quasi-nonexpansive mappings, common solutions set of ﬁnite family of equilibrium problems, and common solutions set of ﬁnite family of variational inequality problems for monotone mappings in a Banach space. More precisely, let fi : C×C → R, i 1, 2, . . . , L, be a ﬁnite family of bifunctions, Sj : C → C , j 1, . . . , D, a ﬁnite family of relatively quasi-nonexpansive mappings, and Ai : C → E∗ , i 1, 2, . . . , N , a ﬁnite family of continuous monotone mappings. For x ∈ E, deﬁne the mappings Frn , Trn : E → C by 1 Frn x z ∈ C : y − z, An z y − z, Jz − Jx ≥ 0, ∀y ∈ C , rn 1.9 1 Trn x z ∈ C : fn z, y y − z, Jz − Jx ≥ 0, ∀y ∈ C , rn
4 Fixed Point Theory and Applications where An An mod N , fn fn mod L and rn ⊂ c1 , ∞ for some c1 > 0. Zegeye and Shahzad 25 introduced the following scheme: x0 ∈ C0 C chosen arbitrarily, zn Frn xn , un Trn xn , 1.10 J −1 α0 Jxn yn α1 Jzn α2 JSn un , Cn z ∈ Cn : φ z, yn ≤ φ z, xn , 1 xn ΠCn 1 x0 , 1 where Sn Sn mod D , α0 , α1 , α2 ∈ 0, 1 such that α0 α1 α2 1. Further, they proved D F Sj ∩ N1 VI C, Ai ∩ that {xn } converges strongly to an element of F, where F j1 i L EP fl . l1 In this paper, motivated and inspired by the iterations 1.8 and 1.10 , we consider a new iterative process with a ﬁnite family of quasi-φ-nonexpansive mappings for a ﬁnite family of equilibrium problems and a ﬁnite family of variational inequality problems in a Banach space. More precisely, let {Si }N1 : C → C be a family of quasi-φ-nonexpansive 1 i mappings, {fi }N1 : C × C → R a ﬁnite family of bifunctions, and {Ai }N1 : C → E∗ a 3 2 i i N1 ∩ N1 EP fi ∩ ﬁnite family of continuous monotone mappings such that F i 1 F Si 2 i N3 N3 N2 i 1 VI C, Ai / ∅. Let {r1,i }i 1 ⊂ 0, ∞ and {r2,i }i 1 ⊂ 0, ∞ . Deﬁne the mappings Tr1,i , Fr2,i : E → C by 1 Tr1,i x z ∈ C : fi z, y y − z, Jz − Jx ≥ 0, ∀y ∈ C , i 1, . . . , N2 , 1.11 r1,i 1 Fr2,i x z ∈ C : y − z, Ai z y − z, Jz − Jx ≥ 0, ∀y ∈ C , i 1, . . . , N3 . 1.12 r2,i Consider the iteration x1 ∈ C chosen arbitrarily, N3 N1 N2 J −1 α0 Jxn yn α1 λ1,i JSi xn α2 λ2,i JTr1,i xn α3 λ3,i JFr2,i xn , i1 i1 i1 Cn v ∈ C : φ v, yn ≤ φ v, xn , 1.13 n Dn Ci , i1 xn ΠDn x1 , n ≥ 1, 1 where α0 , α1 , α2 , α3 are the real numbers in 0, 1 satisfying α0 α1 α2 α3 1 and for each Nj j 1, 2, 3, λj,1 , . . . , λj,Nj are the real numbers in 0, 1 satisfying λj,i 1. We will prove that i1
Fixed Point Theory and Applications 5 the sequence {xn } generated by 1.13 converges strongly to an element in F. In this paper, in order to simplify the proof, we will replace the condition A3 with A3’ : for each ﬁxed y ∈ C, f ·, y is continuous. Obviously, the condition A3’ implies A3 . Under the condition A3’ , we will show that each Tr1,i as well as Fr2,j , i 1, . . . , N2 , j 1, . . . , N3 is closed which is such that the proof for the main result of this paper is simpliﬁed. 2. Preliminaries The modulus of smoothness of a Banach space E is the function ρE : 0, ∞ → 0, ∞ deﬁned by x y x−y ρE τ −1: x 1; y τ. sup 2.1 2 The space E is said to be smooth if ρE τ > 0, for all τ > 0, and E is called uniformly smooth if and only if limτ → 0 ρE τ /τ 0. A Banach space E is said to be strictly convex if x y /2 < 1 for all x, y ∈ E with x y 1 and x / y. It is said to be uniformly convex if limn → ∞ xn − yn 0 for any two sequences {xn } and {yn } in E such that xn yn 1 and limn → ∞ xn yn /2 1. It is known that if a Banach space E is uniformly smooth, then its dual space E∗ is uniformly convex. A Banach space E is called to have the Kadec-Klee property if for any sequence {xn } ⊂ E and x ∈ E with xn x, where denotes the weak convergence, and xn → x , then xn − x → 0 as n → ∞, where → denotes the strong convergence. It is well known that every uniformly convex Banach space has the Kadec-Klee property. For more details on the Kadec-Klee property, the reader is referred to 3, 4 . Let C be a nonempty closed and convex subset of a Banach space E. A mapping S : C → C is said to be closed if for any sequence {xn } ⊂ C such that limn → ∞ xn x0 and limn → ∞ Sxn y0 , Sx0 y0 . Let A : D A ⊂ E → E∗ be a mapping. A is said to be monotone if for each x, y ∈ D A , the following inequality holds: x − y, Ax − Ay ≥ 0. 2.2 Let A be a monotone mapping from C into E∗ . The variational inequality problem on A is formulated as follows: ﬁnd a point u ∈ C such that v − u, Au ≥ 0, ∀v ∈ C. 2.3 The solution set of the above variational inequality problem is denoted by VI C, A .
6 Fixed Point Theory and Applications Next we state some lemmas which will be used later. Lemma 2.1 see 1 . Let C be a nonempty closed and convex subset of a smooth Banach space E and x ∈ E. Then, x0 ΠC x if and only if x0 − y, Jx − Jx0 ≥ 0 ∀y ∈ C. 2.4 Lemma 2.2 see 1 . Let E be a reﬂexive, strictly convex and smooth Banach space, C a nonempty closed and convex subset of E, and x ∈ E. Then φ y , ΠC x φ ΠC x, x ≤ φ y, x , ∀y ∈ C. 2.5 Lemma 2.3 see 20 . Let E be a strictly convex and smooth Banach space, C a nonempty closed and convex subset of E, and T : C → C a quasi-φ-nonexpansive mapping. Then F T is a closed and convex subset of C. Since the condition A3’ implies A3 , the following lemma is a natural result of 22, Lemmas 2.8 and 2.9 . Lemma 2.4. Let C be a closed and convex subset of a smooth, strictly convex and reﬂexive Banach space E. Let f be a bifunction from C × C → R satisfying (A1), (A2), (A3’), and (A4). Let r > 0 and x ∈ E. Then a there exists z ∈ C such that 1 f z, y y − z, Jz − Jx ≥ 0, ∀y ∈ C ; 2.6 r b deﬁne a mapping Tr : E → C by 1 Tr x z ∈ C : f z, y y − z, Jz − Jx ≥ 0, ∀y ∈ C . 2.7 r Then the following conclusions hold: 1 Tr is single-valued; 2 Tr is ﬁrmly nonexpansive, that is, for all x, y ∈ E, Tr x − Tr y, JTr x − JTr y ≤ Tr x − Tr y, Jx − Jy ; 2.8 3 F Tr EP f ; 4 Tr is quasi-φ-nonexpansive; 5 EP f is closed and convex; 6 φ p , Tr x φ Tr x, x ≤ φ p, x , for all p ∈ F Tr .
Fixed Point Theory and Applications 7 Remark 2.5. Let A : C → E∗ be a continuous monotone mapping and deﬁne f x, y y− x, Ax for all x, y ∈ C. It is easy to see that f satisﬁes the conditions A1 , A2 , A3’ , and A4 and EP f VI C, A . Hence, for every real number r > 0, if deﬁning a mapping Fr : E → C by 1 Fr x z ∈ C : y − z, Az y − z, Jz − Jx ≥ 0, ∀y ∈ C , 2.9 r then Fr satisﬁes all the conclusions in Lemma 2.4. See 25, Lemma 2.4 . Lemma 2.6 see 26 . Let p > 1 and s > 0 be two ﬁxed real numbers. Then a Banach space E is uniformly convex if and only if there exists a continuous strictly increasing convex function g : 0, ∞ with g 0 0 such that p p 2 λx 1−λ y ≤λ x 1−λ y − wp λ g x−y 2.10 λ 1 − λ p. λp 1 − λ for all x, y ∈ Bs 0 {x ∈ E : x ≤ s} and λ ∈ 0, 1 , where wp λ The following lemma can be obtained from Lemma 2.6 immediately; also see 20, Lemma 1.9 . Lemma 2.7 see 20 . Let E be a uniformly convex Banach space, s > 0 a positive number, and Bs 0 a closed ball of E. There exists a continuous, strictly increasing and convex function g : 0, ∞ with g0 0 such that 2 N N 2 αi xi ≤ αi xi − αj αk g xj − xk , j, k ∈ {1, 2, . . . , N } with j / k 2.11 i1 i1 N for all x1 , x2 , . . . , xN ∈ Bs 0 {x ∈ E : x ≤ s} and α1 , α2 , . . . , αN ∈ 0, 1 such that αi 1. i1 Lemma 2.8. Let C be a closed and convex subset of a uniformly smooth and strictly convex Banach space E. Let f : C × C → R be a bifunction satisfying (A1), (A2), (A3’), and (A4). Let r > 0 and Tr : E → C be a mapping deﬁned by 2.7 . Then Tr is closed. Proof. Let {xn } ⊂ E converge to x and {Tr xn } converge to x. To end the conclusion, we need to prove that Tr x x. Indeed, for each xn , Lemma 2.4 shows that there exists a unique zn ∈ C such that zn Tr xn , that is, 1 f zn , y y − zn , Jzn − Jxn ≥ 0, ∀y ∈ C. 2.12 r
8 Fixed Point Theory and Applications Since E is uniformly smooth, J is continuous on bounded set note that {xn } and {zn } are both bounded . Taking the limit as n → ∞ in 2.12 , by using A3’ , we get 1 f x, y y − x, J x − Jx ≥ 0, ∀y ∈ C, 2.13 r which implies that Tr x x. This completes the proof. 3. Main Results Theorem 3.1. Let C be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space E which has the Kadec-Klee property. Let {Si }N11 : C → C be a family of closed i quasi-φ-nonexpansive mappings, {fi }N21 : C × C → R a ﬁnite family of bifunctions satisfying the i conditions (A1), (A2), (A3’), and (A4), and {Ai }N31 : C → E∗ a ﬁnite family of continuous monotone i ∩ N21 EP fi ∩ N31 VI C, Ai / ∅. Let {r1,i }N21 , {r2,i }N31 ⊂ N1 mappings such that F i 1 F Si i i i i 0, ∞ . Let {xn } be a sequence generated by the following manner: x1 ∈ C chosen arbitrarily, N1 zn λ1,i JSi xn , i1 N2 un λ2,i JTr1,i xn , i1 N3 wn λ3,i JFr2,i xn , 3.1 i1 J −1 α0 Jxn yn α1 zn α2 un α3 wn , Cn z ∈ C : φ v, yn ≤ φ v, xn , n Dn Ci , i1 xn ΠDn x1 , n ≥ 1, 1 where Tr1,i i 1, 2, . . . , N2 and Fr2,j j 1, 2, . . . , N3 are deﬁned by 1.11 and 1.12 , α0 , α1 , α2 , α3 are the real numbers in 0, 1 satisfying α0 α1 α2 α3 1 and for each j 1, 2, 3, λj,1 , . . . , λj,Nj Nj are the real numbers in 0, 1 satisfying i 1 λj,i 1. Then the sequence {xn } converges strongly to ΠF x1 , where ΠF is the generalized projection from E onto F. Proof. First we prove that Dn is closed and convex for each n ≥ 1. From the deﬁnition of Cn , it is obvious that Cn is closed. Moreover, since φ v, yn ≤ φ v, xn is equivalent to 2 v, Jxn − Jyn − xn 2 yn 2 ≥ 0, it follows that Cn is convex for each n ≥ 1. By the deﬁnition of Dn , we can conclude that Dn is closed and convex for each n ≥ 1.
Fixed Point Theory and Applications 9 Next, we prove that F ⊂ Dn for each n ≥ 1. From Lemma 2.4 and Remark 2.5, we see that each Tr1,i i 1, 2, . . . , N2 and Fr2,j j 1, 2, . . . , N3 are quasi-φ-nonexpansive. Hence, for any p ∈ F, we have φ p, J −1 α0 Jxn φ p, yn α1 zn α2 un α3 wn 2 2 p − 2 p, α0 Jxn α1 zn α2 un α3 wn α0 Jxn α1 zn α2 un α3 wn 2 ≤p − 2α0 p, Jxn − 2α1 p, zn − 2α2 p, un 2 2 2 2 − 2α3 p, wn α0 xn α1 zn α2 un α3 wn N1 N2 2 ≤p − 2α0 p, Jxn − 2α1 λ1,i p, JSi xn − 2α2 λ2,i p, JTr1,i xn i1 i1 N3 N1 2 2 − 2α3 λ3,i p, JFr2,i xn α0 xn α1 λ1,i J Si xn i1 i1 N3 N2 2 2 α2 λ2,i J Tr1,i xn α3 λ3,i J Fr2,i xn 3.2 i1 i1 N1 N2 α0 φ p, xn α1 λ1,i φ p, Si xn α2 λ2,i φ p, Tr1,i xn i1 i1 N3 α3 λ3,i φ p, Fr2,i xn i1 N1 N2 ≤ α0 φ p, xn α1 λ1,i φ p, xn α2 λ2,i φ p, xn i1 i1 N3 α3 λ3,i φ p, xn i1 φ p, xn , which implies that F ⊂ Cn for each n ≥ 1. So, it follows from the deﬁnition of Dn that F ⊂ Dn for each n ≥ 1. Therefore, the sequence {xn } is well deﬁned. Also, from Lemma 2.2 we see that φ xn 1 , x1 φ ΠDn x1 , x1 ≤ φ p, x1 − φ p, xn ≤ φ p, x1 , 3.3 1 for each p ∈ F. This shows that the sequence {φ xn , x1 } is bounded. It follows from 1.4 that the sequence {xn } is also bounded.
10 Fixed Point Theory and Applications x∗ . Since Dn Since E is reﬂexive, we may, without loss of generality, assume that xn ∗ is closed and convex for each n ≥ 1, we can conclude that x ∈ Dn for each n ≥ 1. By the deﬁnition of {xn }, we see that φ xn , x1 ≤ φ x∗ , x1 . 3.4 It follows that φ x∗ , x1 ≤ lim inf φ xn , x1 ≤ lim sup φ xn , x1 ≤ φ x∗ , x1 . 3.5 n→∞ n→∞ This implies that φ x∗ , x1 . lim φ xn , x1 3.6 n→∞ x∗ as n → ∞. In view of the Kadec-Klee property of E, we get Hence, we have xn → that x∗ . lim xn 3.7 n→∞ By the construction of Dn , we have that Dn ⊂ Dn and xn ΠDn 1 x1 ⊂ Dn . It follows 1 2 from Lemma 2.2 that φ xn 2 , xn φ xn 2 , ΠDn x1 1 ≤ φ xn 2 , x1 − φ ΠDn x1 , x1 3.8 φ xn 2 , x1 − φ xn 1 , x1 . n Letting n → ∞, we obtain that φ xn 2 , xn → 0. In view of xn ∈ Dn Cn , we have i1 1 1 xn 1 ∈ Cn and hence φ xn 1 , yn ≤ φ xn 1 , xn . 3.9 It follows that lim φ xn 1 , yn 0. 3.10 n→∞ From 1.4 , we see that yn −→ x∗ as n → ∞. 3.11
Fixed Point Theory and Applications 11 Hence, J yn −→ J x∗ as n → ∞. 3.12 This implies that the sequence {Jyn } is bounded. Note that reﬂexivity of E implies reﬂexivity of E∗ . Thus, we may assume that Jyn y ∈ E∗ . Furthermore, reﬂexivity of E implies that there exists x ∈ E such that y J x. Then, it follows that 2 2 φ xn 1 , yn xn − 2 xn 1 , Jyn yn 1 3.13 2 2 xn − 2 xn 1 , Jyn J yn . 1 Take lim inf on both sides of 3.13 over n and use weak lower semicontinuity of norm to get that 0 ≥ x∗ − 2 x∗ , y 2 2 y x∗ − 2 x∗ , Jx 2 2 Jx 3.14 x∗ − 2 x∗ , Jx 2 2 x φ x∗ , x , which implies that x∗ x. Hence, y J x∗ . It follows that Jyn Jx∗ . Now, from 3.12 and ∗ ∗ Kadec-Klee property of E , we obtain that Jyn → Jx as n → ∞. Then the demicontinuity of J −1 implies that yn x∗ . Now, from 3.11 and the fact that E has the Kadec-Klee property, we obtain that limn → ∞ yn x∗ . Note that xn − yn ≤ xn − x∗ x∗ − yn . 3.15 It follows that lim xn − yn 0. 3.16 n→∞ Since J is uniformly norm-to-norm continuous on any bounded sets, we have lim J xn − Jyn 0. 3.17 n→∞
12 Fixed Point Theory and Applications Since E is uniformly smooth, we know that E∗ is uniformly convex. In view of Lemma 2.7, we see that, for any p ∈ F, φ p, J −1 α0 Jxn φ p, yn α1 zn α2 un α3 wn 2 2 p − 2 p, α0 Jxn α1 zn α2 un α3 wn α0 Jxn α1 zn α2 un α3 wn N1 2 2 2 ≤p − 2 p, α0 Jxn α1 zn α2 un α3 wn α0 xn α1 λ1,i Si xn i1 N3 N2 2 2 α2 λ2,i Tr1,i xn α3 λ3,i Fr2,i xn − α0 α1 λ1,1 g J xn − JS1 xn i1 i1 N1 N2 3.18 α0 φ p, xn α1 λ1,i φ p, Si xn α2 λ2,i φ p, Tr1,i xn i1 i1 N3 α3 λ3,i φ p, Fr2,i xn − α0 α1 λ1,1 g J xn − JS1 xn i1 N3 N1 N2 ≤ α0 φ p, xn α1 λ1,i φ p, xn α2 λ2,i φ p, xn α3 λ3,i φ p, xn i1 i1 i1 − α0 α1 λ1,1 g J xn − JS1 xn φ p, xn − α0 α1 λ1,1 g J xn − JS1 xn . It follows that α0 α1 λ1,1 g J xn − JS1 xn ≤ φ p, xn − φ p, yn . 3.19 Note that 2 2 φ p, xn − φ p, yn xn − yn − 2 p, Jxn − Jyn 3.20 ≤ xn − yn xn yn 2p J xn − Jyn . It follows from 3.16 and 3.17 that φ p, xn − φ p, yn −→ 0 as n −→ ∞. 3.21 By 3.19 , 3.21 , and α0 α1 λ1,1 > 0, we have g J xn − JS1 xn −→ 0 as n −→ ∞. 3.22
Fixed Point Theory and Applications 13 It follows from the property of g that J xn − JS1 xn −→ 0 as n −→ ∞. 3.23 Since xn → x∗ as n → ∞ and J : E → E∗ is demicontinuous, we obtain that Jxn Jx∗ ∈ E∗ . Note that | Jxn − J x∗ | | xn − x∗ | ≤ xn − x∗ . 3.24 This implies that J x∗ . lim J xn 3.25 n→∞ Since E∗ enjoys the Kadec-Klee property, we see that lim J xn − Jx∗ 0. 3.26 n→∞ Note that J S1 xn − Jx∗ ≤ J S1 xn − Jxn J xn − Jx∗ . 3.27 From 3.23 and 3.26 , we arrive at lim J S1 xn − Jx∗ 0. 3.28 n→∞ Note that J −1 : E∗ → E is demicontinuous. It follows that S1 xn x∗ . On the other hand, since | S1 xn − x∗ | | J S1 xn − J x∗ | ≤ J S1 xn − Jx∗ , 3.29 → x∗ as n → ∞. Since E enjoys the Kadec-Klee prop- by 3.28 we conclude that S1 xn erty, we obtain that lim S1 xn − x∗ 0. 3.30 n→∞ By repeating 3.18 – 3.30 , we also can get lim Si xn − x∗ 0, i 2, . . . , N1 , 3.31 n→∞ lim Tr1,i xn − x∗ 0, i 1, . . . , N2 , 3.32 n→∞ lim Fr2,i xn − x∗ 0, i 1, . . . , N3 . 3.33 n→∞
14 Fixed Point Theory and Applications Since each Si is closed, by 3.30 and 3.31 we conclude that Si x∗ x∗ , that is, ∗ x ∈ F Si , i 1, 2, . . . , N1 . On the other hand, Lemma 2.4, Remark 2.5, and Lemma 2.8 show that Tr1,i i 1, 2, . . . , N2 and Fr2,i i 1, 2, . . . , N3 are closed. So, by 3.32 and 3.33 we have Tr1,i x∗ x∗ i 1, 2, . . . , N2 and Fr2,i x∗ x∗ i 1, 2, . . . , N3 . Now, it follows from Lemma 2.4 and Remark 2.5 that F Tr1,i EP fi i 1, 2, . . . , N2 and F Fr2,i VI C, Ai ∗ ∗ i 1, 2, . . . , N3 . Hence, x ∈ EP fi i 1, 2, . . . , N2 and x ∈ VI C, Ai i 1, 2, . . . , N3 . Therefore, x∗ ∈ F. Finally, we prove that x∗ ΠF x1 . From xn 1 ΠDn x1 , by Lemma 2.1, we see that xn − p, Jx1 − Jxn ≥ 0, ∀p ∈ Dn . 3.34 1 1 Since F ⊂ Dn for each n ≥ 1, we have xn − p, Jx1 − Jxn ≥ 0, ∀p ∈ F . 3.35 1 1 Letting n → ∞ in 3.35 , we see that x∗ − p, Jx1 − Jp ≥ 0, ∀p ∈ F . 3.36 In view of Lemma 2.1, we can obtain that x∗ ΠF x1 . This completes the proof. Remark 3.2. Obviously, the proof process of x∗ ∈ N1 EP fi ∩ N1 VI C, Ai is simple since 3 2 i i we replace the condition A3 with A3’ which is such that Tr1,i and Fr2,j i 1, 2, . . . , N2 , j 1, 2, . . . , N3 are closed. In fact, although the condition A3’ is stronger than A3 , it is not easier to verify the condition A3 than verify the condition A3’ . Hence, from this point, the condition A3’ is acceptable. On the other hand, the deﬁnition of Dn is of some interest. If Si S for each i 1, 2, . . . , N1 , fi f for each i 1, 2, . . . , N2 and Ai A for each i 1, 2, . . . , N3 , then Theorem 3.1 reduces to the following result. Corollary 3.3. Let C be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space E which has the Kadec-Klee property. Let S : C → C be a closed quasi-φ- nonexpansive mapping, f : C × C → R a bifunction satisfying the conditions (A1), (A2), (A3’), and (A4) and A : C → E∗ a continuous monotone mapping such that F F S ∩ EP f ∩ VI C, A / ∅. Let r1 , r2 ⊂ 0, ∞ . Let {xn } be a sequence deﬁned by the following manner: x1 ∈ C chosen arbitrarily, J −1 α0 Jxn yn α1 JSxn α2 JTr1 xn α3 JFr2 xn , Cn z ∈ C : φ v, yn ≤ φ v, xn , 3.37 n Dn Ci , i1 xn ΠDn x1 , n ≥ 1, 1
Fixed Point Theory and Applications 15 where Tr1 and Fr2 are deﬁned by 1.11 and 1.12 with r1,i r1 i 1, 2, . . . , N2 and r2,j r2 j 1, 2, . . . , N3 , α0 , α1 , α2 , α3 are the real numbers in 0, 1 satisfying α0 α1 α2 α3 1. Then the sequence {xn } converges strongly to PF x1 , where ΠF is the generalized projection from E onto F. Corollary 3.4. Let C be a nonempty closed and convex subset of a Hilbert space H. Let {Si }N11 : C → i C be a family of closed quasi-nonexpansive mappings, {fi }N21 : C × C → R a ﬁnite family of bifunctions i satisfying the conditions (A1)–(A4), and {Ai }N31 : C → H a ﬁnite family of continuous monotone i ∩ N21 EP fi ∩ N31 VI C, Ai / ∅. Let {r1,i }N21 , {r2,i }N31 ⊂ N1 mappings such that F i 1 F Si i i i i 0, ∞ . Deﬁne a sequence {xn } by the following manner: x1 ∈ C chosen arbitrarily, N1 zn λ1,i Si xn , i1 N2 un λ2,i Tr1,i xn , i1 N3 wn λ3,i Fr2,i xn , 3.38 i1 yn α0 xn α1 zn α2 un α3 wn , Cn z ∈ C : v − yn ≤ v − xn , n Dn Ci , i1 xn PDn x1 , n ≥ 1, 1 where {Tr1,i }N1 and {Fr1,i }N1 are deﬁned by 1.11 and 1.12 with J I (I is the identity mapping), 2 3 i i α0 , α1 , α2 , α3 are the real numbers in 0, 1 satisfying α0 α1 α2 α3 1 and for each j 1, 2, 3, Nj λj,1 , . . . , λj,Nj are the real numbers in 0, 1 satisfying i 1 λj,i 1. Then the sequence {xn } converges strongly to PF x1 , where PF is the projection from H onto F. Proof. By the proof of Theorem 3.1, we have xn → x∗ as n → ∞, lim Si xn − xn 0, i 1, 2, . . . , N1 , n→∞ lim Tr1,i xn − xn 0, i 1, 2, . . . , N2 , 3.39 n→∞ lim Fr2,i xn − xn 0, i 1, 2, . . . , N3 . n→∞ Since each Si is closed, we can conclude that x∗ ∈ F Si , i 1, 2, . . . , N1 . Note that in a Hilbert space, a ﬁrmly-nonexpansive mapping is also nonexpansive. Hence, Tr1 ,i and Fr2,j are nonexpansive for each i 1, 2, . . . , N2 and j 1, 2, . . . , N3 . By demiclosed principle, we can conclude that x∗ ∈ F Tr 1,i EP fi and x∗ ∈ F Fr 2,i VI C, Aj for each i 1, 2, . . . , N2 ∗ and j 1, 2, . . . , N3 . That is, x ∈ F. Then by the ﬁnal part of proof of Theorem 3.1, we have xn → x∗ PF x1 . This completes the proof.
16 Fixed Point Theory and Applications Let H be a Hilbert space and C a nonempty closed and convex subset of H . A mapping T : C → H is called a pseudocontraction if for all x, y ∈ C, 2 2 2 Tx − Ty ≤ x−y I−T x− I −T y , 3.40 or equivalently, I − T x − I − T y, x − y ≥ 0. 3.41 Let A I − T , where T : C → H is a pseudocontraction. Then A is a monotone mapping and A−1 0 F T . Moreover, F T VI C, A . Indeed, it is easy to see that F T ⊂ VI C, A . Let u ∈ VI C, A . We have v − u, Au ≥ 0, i.e., v − u, I − T u ≥ 0, 3.42 for all v ∈ C. Take v T u. Then we have T u − u, I − T u ≥ 0. That is, − u − T u 2 ≥ 0. This shows that u T u, which implies that VI C, A ⊂ F T . So, F T VI C, A . Based this, we have following result. Corollary 3.5. Let C be a nonempty closed and convex subset of a Hilbert space H. Let {Si }N11 : i C → C be a family of closed quasi-nonexpansive mappings, {fi }N21 : C × C → R a ﬁnite i family of bifunctions satisfying the conditions (A1)–(A4), and {Ti }N31 : C → H a ﬁnite family of i N3 N1 N2 continuous pseudocontractions such that F ∩ ∩ / ∅. Let F Si EP fi F Ti i1 i1 i1 {r1,i }N21 , {r2,i }N31 ⊂ 0, ∞ . Deﬁne a sequence {xn } by the following manner: i i x1 ∈ C chosen arbitrarily, N1 zn λ1,i Si xn , i1 N2 un λ2,i Tr1,i xn , i1 N3 wn λ3,i Fr2,i xn , 3.43 i1 yn α0 xn α1 zn α2 un α3 wn , Cn z ∈ C : v − yn ≤ v − xn , n Dn Ci , i1 xn PDn x1 , n ≥ 1, 1
Fixed Point Theory and Applications 17 where {Tr1,i }N1 are deﬁned by 1.11 with J I and Fr2,i is deﬁned by 2 i 1 Fr2,i x z ∈ C : y − x, I − Ti x y − z, z − x ≥ 0 ∀y ∈ C , i 1, 2, . . . , N3 , r2,i 3.44 α0 , α1 , α2 , α3 are the real numbers in 0, 1 satisfying α0 α1 α2 α3 1 and for each j 1, 2, 3, Nj λj,1 , . . . , λj,Nj are the real numbers in 0, 1 satisfying i 1 λj,i 1. Then the sequence {xn } converges strongly to PF x1 , where PF is the projection from H onto F. If Si S, fj f , and Tk T for each i 1, 2, . . . , N1 , j 1, 2, . . . , N2 , and k 1, 2, . . . , N3 , then Corollary 3.5 reduced the following result. Corollary 3.6. Let C be a nonempty closed and convex subset of a Hilbert space H. Let S : C → C be a closed quasi-nonexpansive mapping, f : C × C → R a bifunction satisfying the conditions (A1)– (A4), and T : C → H a continuous pseudocontraction such that F F S ∩ EP f ∩ F T / ∅. Let r1 , r2 ⊂ 0, ∞ . Deﬁne a sequence {xn } by the following manner: x1 ∈ C chosen arbitrarily, J −1 α0 xn yn α1 Sxn α2 Tr1 xn α3 Fr2 xn , Cn z ∈ C : v − yn ≤ v − xn , 3.45 n Dn Ci , i1 xn PDn x1 , n ≥ 1, 1 where Tr1 is deﬁned by 1.11 with J I and r1,i r1 i 1, 2, . . . , N2 , Fr2 is deﬁned by 3.44 r2,j r2 j 1, 2, . . . , N3 , and α0 , α1 , α2 , α3 are the real numbers in 0, 1 satisfying α0 α1 α2 α3 1. Then the sequence {xn } converges strongly to PF x1 , where PF is the projection from H onto F. Acknowledgment This work was supported by the Natural Science Foundation of Hebei Province A2010001482 . References 1 Y. I. Alber, “Metric and generalized projection operators in Banach spaces: properties and applications,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, vol. 178 of Lecture Notes in Pure and Appl. Math., pp. 15–50, Dekker, New York, NY, USA, 1996. 2 Ya. I. Alber and S. Reich, “An iterative method for solving a class of nonlinear operator equations in Banach spaces,” Panamerican Mathematical Journal, vol. 4, no. 2, pp. 39–54, 1994. 3 I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, vol. 62 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1990. 4 W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, Japan, 2000, Fixed point theory and Its application. 5 S. Reich, “A weak convergence theorem for the alternating method with Bregman distances,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, A. G. Kartsatos, Ed., vol. 178 of Lecture Notes in Pure and Appl. Math., pp. 313–318, Dekker, New York, NY, USA, 1996.
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