Convergence Theorems for a Common Point of Solutions of Equilibrium and Fixed Point of Relatively Nonexpansive Multivalued Mapping Problems

and Applied Analysis 3 The study of fixed points for multi-valued nonexpansive mappings in relation to Hausdorff metric was introduced by Markin 4 see also 5 . Since then a lot of activity in this area and fixed point theory for multi-valued nonexpansive mappings has been developed which has some nontrivial applications in pure and applied sciences including control theory, convex optimization, differential inclusion, and economics see, e.g., 6 and references therein . Later, Lim 7 established the existence of fixed points for multi-valued nonexpansive mappings in uniformly convex Banach spaces. It is well known that the normal Mann’s iterative 8 algorithm has only weak convergence in an infinite-dimensional Hilbert space even for nonexpansive single-valued mappings. Consequently, in order to obtain strong convergence, one has to modify the normal Mann’s iteration algorithm, the so called hybrid projection iteration method is such a modification. The hybrid projection iteration algorithm HPIA was introduced initially by Haugazeau 9 in 1968. For 40 years, HPIA has received rapid developments. For details, the readers are referred to papers 10–12 and the references therein. In 2003, Nakajo and Takahashi 12 proposed the following modification of the Mann iteration method for a nonexpansive single-valued mapping T in a Hilbert space H: x0 ∈ C, chosen arbitrary, yn αnxn 1 − αn Txn, Cn { z ∈ C : ∥yn − z ∥∥ ≤ ‖xn − z‖ } , Qn {z ∈ C : 〈xn − z, x0 − xn〉 ≥ 0}, xn 1 PCn∩Qn x0 , n ≥ 0, 1.5 where C is a closed convex subset of H, PC denotes the metric projection from H onto C. They proved that if the sequence {αn} is bounded above from one then the sequence {xn} generated by 1.5 converges strongly to PF T x0 . In spaces more general than Hilbert spaces, Matsushita and Takahashi 11 proposed the following hybrid iteration method with generalized projection for relatively nonexpansive single-valued mapping T in a Banach space E: x0 ∈ C, chosen arbitrary, yn J−1 αnJxn 1 − αn JTxn , Cn { z ∈ C : φz, yn ) ≤ φ z, xn } , Qn {z ∈ C : 〈xn − z, Jx0 − Jxn〉 ≥ 0}, xn 1 ΠCn∩Qn x0 , n ≥ 0. 1.6 They proved the following convergence theorem. TheoremMT. Let E be a uniformly convex and uniformly smooth Banach space, letC be a nonempty closed convex subset of E, let T be a relatively nonexpansive single-valued mapping from C into itself, and let {αn} be a sequence of real numbers such that 0 ≤ αn < 1 and lim supn→∞αn < 1. 4 Abstract and Applied Analysis Suppose that {xn} is given by 1.6 , where J is the duality mapping on E. If F T is nonempty, then {xn} converges strongly toΠF T x0, whereΠF T · is the generalized projection from E onto F T . Let f : C × C → R be a bifunction, where R is the set of real numbers. The equilibrium problem for f is finding x∗ ∈ C such that fx∗, y ≥ 0, ∀y ∈ C. 1.7 The solution set of 1.7 is denoted by EP f . If f x, y 〈Ax, y − x〉, where A : C → C is a monotone mapping, then the problem 1.7 reduces to the system of variational inequality problem find an element x∗ ∈ C such that Ax∗, y − x∗ ≥ 0, ∀y ∈ C. 1.8 That is, the problem 1.8 is a special case of 1.7 . The set of solutions of inequality 1.8 is denoted by V I C,A . For solving the equilibrium problem for a bifunction f : C × C → R, we assume that f satisfies the following conditions: 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, limt→ 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 studied the problem of finding a common element of the set of fixed points of nonexpansive or relatively nonexpansive single-valued mapping and the set of solutions of an equilibrium problems in the frame work of Hilbert spaces and Banach spaces, respectively: see, for instance, 2, 13–21 and the references therein. In 22 , Kumam introduced the following iterative scheme in a Hilbert space: x0 ∈ H, un ∈ C such that f ( un, y ) 1 rn 〈 y − un, un − yn 〉 ≥ 0, ∀y ∈ C, wn αnxn 1 − αn Tun, Cn {z ∈ H : ‖wn − z‖ ≤ ‖xn − z‖}, Qn {z ∈ C : 〈xn − z, x0 − xn〉 ≥ 0}, xn 1 PCn 1∩Qn x0 , n ≥ 0, 1.9 for finding a common element of the set of fixed point of nonexpansive single-valued mapping T and set of solution of equilibrium problems. Abstract and Applied Analysis 5 In the case that E is a Banach space, Takahashi and Zembayashi 16 introduced the following iterative scheme which is called the shrinking projection method:and Applied Analysis 5 In the case that E is a Banach space, Takahashi and Zembayashi 16 introduced the following iterative scheme which is called the shrinking projection method: x0 ∈ C, chosen arbitrary, yn J−1 αnJxn 1 − αn JTxn , un ∈ C such that f ( un, y ) 1 rn 〈 y − un, Jun − Jyn 〉 ≥ 0, ∀y ∈ C, Cn 1 { z ∈ Cn : φ z, un ≤ φ z, xn } , xn 1 ΠCn 1 x0 , n ≥ 0, 1.10 where J is the duality mapping on E, ΠC is the generalized projection from E onto C and T is relatively nonexpansive single-valued mapping. They proved that the sequence {xn} converges strongly to a common element of the set of fixed point of relatively nonexpansive single-valued mapping and set of solution of equilibrium problem under appropriate conditions. We remark that the computation of xn 1 in 1.9 and 1.10 is not simple because of the involvement of computation of Cn 1 from Cn for each n ≥ 0. More recently, Homaeipour and Razani 3 studied the following iterative scheme for a fixed point of relatively nonexpansive multi-valued mapping in uniformly convex and uniformly smooth Banach space E: x0 ∈ C, chosen arbitrary, xn 1 ΠCJ−1 αnJxn 1 − αn Jzn , zn ∈ Txn, n ≥ 0, 1.11 where {αn} ⊂ 0, 1 for all n ≥ 0 and lim infn→∞αn 1 − αn > 0. They proved that if J is weakly sequentially continuous then the sequence {xn} converges weakly to a fixed point of T . Furthermore, it is shown that the scheme converges strongly to a fixed point of T if interior of F T is nonempty. But it is worth mentioning that the convergence of the scheme is either weak or it requires that the interior of F T is nonempty. In this paper, motivated by Kumam 22 , Takahashi and Zembayashi 16 , and Homaeipour and Razani 3 , we construct an iterative scheme which converges strongly to a common point of set of solutions of equilibrium problem and set of fixed points of finite family of relatively nonexpansivemulti-valuedmappings in Banach spaces. Our scheme does not involve computation of Cn and Qn, for each n ≥ 0, and the requirement that the interior of F is nonempty is dispensed with. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators. 6 Abstract and Applied Analysis 2. Preliminaries Let E be a normed linear space with dimE ≥ 2. The modulus of smoothness of E is the function ρE : 0,∞ → 0,∞ defined by ρE τ : sup {∥ ∥x y ∥ ∥ ∥ ∥x − y∥ 2 − 1 : ‖x‖ 1;∥y∥ τ } . 2.1 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 limt→ 0 ρE t /t 0. The modulus of convexity of E is the function δE : 0, 2 → 0, 1 defined by δE : inf { 1 − ∥ ∥∥ x y 2 ∥ ∥∥ : ‖x‖ ∥y ∥∥ 1; ∥x − y∥ } . 2.2 E is called uniformly convex if and only if δE > 0, for every ∈ 0, 2 . In the sequel, we will need the following results. Lemma 2.1 see 1 . Let K be a nonempty closed and convex subset of a real reflexive, strictly convex, and smooth Banach space E and let x ∈ E. Then for all y ∈ K, φ ( y,ΠKx ) φ ΠKx, x ≤ φ ( y, x ) . 2.3 We make use of the function V : E × E∗ → R, defined by V x, x∗ ‖x‖ − 2〈x, x∗〉 ‖x∗‖2, ∀x ∈ E, x∗ ∈ E∗, 2.4 studied by Alber 1 . That is, V x, x∗ φ x, J−1x∗ for all x ∈ E and x∗ ∈ E∗. We know the following lemma. Lemma 2.2 see 1 . Let E be reflexive strictly convex and smooth Banach space with E∗ as its dual. Then V x, x∗ 2 〈 J−1x∗ − x, y∗ 〉 ≤ V x, x∗ y∗, 2.5 for all x ∈ E and x∗, y∗ ∈ E∗. Lemma 2.3 see 1 . Let C be a convex subset of a real smooth Banach space E. Let x ∈ E. Then x0 ΠCx if and only if 〈z − x0, Jx − Jx0〉 ≤ 0, ∀z ∈ C. 2.6 Abstract and Applied Analysis 7 Lemma 2.4 see 23 . Let E be a uniformly convex Banach space and BR 0 be a closed ball of E. Then, there exists a continuous strictly increasing convex function g : 0,∞ → 0,∞ with g 0 0 such thatand Applied Analysis 7 Lemma 2.4 see 23 . Let E be a uniformly convex Banach space and BR 0 be a closed ball of E. Then, there exists a continuous strictly increasing convex function g : 0,∞ → 0,∞ with g 0 0 such that ‖α1x1 α2x2 · · · αNxN‖ ≤ N ∑ i 1 αi‖xi‖ − αiαjg ∥xi − xj ∥ ∥, 2.7 for i, j ∈ {1, . . . ,N}, αi ∈ 0, 1 such that ∑N i 1 αi 1, and xi ∈ BR 0 : {x ∈ E : ||x|| ≤ R}, for i 1, 2, . . . ,N. Lemma 2.5 see 24 . Let E be a real smooth and uniformly convex Banach space and let {xn} and {yn} be two sequences of E. If either {xn} or {yn} is bounded and φ xn, yn → 0 as n → ∞, then xn − yn → 0, as n → ∞. Proposition 2.6 see 3 . Let E be a strictly convex and smooth Banach space and C be a nonempty closed convex subset of E. Let T : C → N C be a relatively nonexpansive multi-valued mapping. Then F T is closed and convex. Lemma 2.7 see 16 . LetC be a nonempty, closed and convex subset of a uniformly smooth, strictly convex and reflexive real Banach space E. Let f be a bifunction from C × C to R which satisfies conditions (A1)–(A4). For r > 0 and x ∈ E, define the mapping Fr : E → C as follows: Frx : { z ∈ C : fz, y 1 r 〈 y − z, Jz − Jx ≥ 0, ∀y ∈ C } . 2.8 Then the following statements hold: 1 Fr is single-valued, 2 F Fr EP f , 3 φ q, Frx φ Frx, x ≤ φ q, x , for q ∈ F Fr , 4 EP f is closed and convex. Lemma 2.8 see 25 . Let {an} be sequences of real numbers such that there exists a subsequence {ni} of {n} such that ani < ani 1 for all i ∈ N. Then there exists a nondecreasing sequence {mk} ⊂ N such thatmk → ∞ and the following properties are satisfied by all (sufficiently large) numbers k ∈ N: amk ≤ amk 1, ak ≤ amk 1. 2.9 In fact,mk max{j ≤ k : aj < aj 1}. Lemma 2.9 see 26 . Let {an} be a sequence of nonnegative real numbers satisfying the following relation: an 1 ≤ ( 1 − βn ) an βnδn, n ≥ n0, for some n0 ∈ N, 2.10 where {βn} ⊂ 0, 1 and {δn} ⊂ R satisfying the following conditions: limn→∞βn 0, ∑∞ n 1 βn ∞, and lim supn→∞δn ≤ 0. Then, limn→∞an 0. 8 Abstract and Applied Analysis 3. Main Result Let C be a nonempty, closed and convex subset of a smooth, strictly convex and reflexive real Banach space E with dual E∗. Let f : C × C → R be a bifunction. For the rest of this paper, Frnx is a mapping defined as follows. For x ∈ E, let Frn :E → C be given by Frnx : { z ∈ C : fz, y 1 rn 〈 y − z, Jz − Jx ≥ 0, ∀y ∈ C } , 3.1 where {rn}n∈N ⊂ c1,∞ , for some c1 > 0. Theorem 3.1. Let C be a nonempty, closed and convex subset of a uniformly smooth and uniformly convex real Banach space E. Let f : C×C → R, be a bifunction which satisfies conditions (A1)–(A4). Let Ti : C → CB C , for i 1, 2, . . . ,N, be a finite family of relatively nonexpansive multi-valued mappings. Assume that F : ∩i 1F Ti ∩ EP f is nonempty. Let {xn} be a sequence generated by x0 w ∈ C, chosen arbitrarily, wn Frnxn, yn ΠCJ−1 αnJw 1 − αn Jwn , xn 1 J−1 ( βn,0Jwn N ∑ i 1 βn,iJun,i ) , un,i ∈ Tiyn, n ≥ 0, 3.2 where αn ∈ 0, 1 such that limn→∞αn 0, ∑∞ n 1 αn ∞, {βn,i} ⊂ a, b ⊂ 0, 1 , for i 1, 2, . . . ,N, satisfying βn,0 βn,1 · · · βn,N 1, for each n ≥ 0. Then {xn} converges strongly to an element of F. Proof. Since F is nonempty closed and convex, put x∗ : ΠFw. Now from 3.2 , Lemma 2.7 3 and property of φ, we get that φ ( x∗, yn ) φ ( x∗,ΠCJ−1 αnJw 1 − αn Jwn ) ≤ φ ( x∗, J−1 αnJw 1 − αn Jwn ) ‖x∗‖2 − 2〈x∗, αnJw 1 − αn Jwn〉 ‖αnJw 1 − αn Jwn‖ ≤ ‖x∗‖2 − 2αn〈x∗, Jw〉 − 2 1 − αn 〈x∗, Jwn〉 αn‖w‖ 1 − αn ‖wn‖ ≤ αnφ x∗, w 1 − αn φ x∗, wn αnφ x∗, w 1 − αn φ x∗, Frnxn ≤ αnφ x∗, w 1 − αn φ x∗, xn . 3.3 Abstract and Applied Analysis 9 Now, from 3.2 , Lemma 2.7 3 , relatively nonexpansiveness of Ti, property of φ and 3.3 , we have thatand Applied Analysis 9 Now, from 3.2 , Lemma 2.7 3 , relatively nonexpansiveness of Ti, property of φ and 3.3 , we have that φ x∗, xn 1 φ ( x∗, J−1 ( βn,0Jwn N ∑