The Shrinking Projection Method for Solving Variational Inequality Problems and Fixed Point Problems in Banach Spaces

and Applied Analysis 3 4 if E is a reflexive, strictly convex, and smooth Banach space, then for all x, y ∈ E, φ ( x, y ) 0 iff x y. 1.7 For more details see 2, 3 . Let C be a closed convex subset of E, and let T be a mapping from C into itself. We denote by F T the set of fixed point of T . A point p in C is said to be an asymptotic fixed point of T 8 if C contains a sequence {xn}which converges weakly to p such that limn→∞‖xn − Txn‖ 0. The set of asymptotic fixed points of T will be denoted by F̂ T . A mapping T from C into itself is called nonexpansive if ‖Tx − Ty‖ ‖x − y‖ for all x, y ∈ C and relatively nonexpansive 9–11 if F̂ T F T and φ p, Tx φ p, x for all x ∈ C and p ∈ F T . The asymptotic behavior of relatively nonexpansive mappings which was studied in 9–11 is of special interest in the convergence analysis of feasibility, optimization, and equilibrium methods for solving the problems of image processing, rational resource allocation, and optimal control. The most typical examples in this regard are the Bregman projections and the Yosida type operators which are the cornerstones of the common fixed point and optimization algorithms discussed in 12 see also the references therein . The mapping T is said to be φ-nonexpansive if φ Tx, Ty ≤ φ x, y for all x, y ∈ C. T is said to be quasi-φ-nonexpansive if F T / ∅ and φ p, Tx ≤ φ p, x for all x ∈ C and p ∈ F T . Remark 1.1. The class of quasi-φ-nonexpansive is more general than the class of relatively nonexpansive mappings 9, 10, 13–15 which requires the strong restriction F̂ T F T . Next, we give some examples which are closed quasi-φ-nonexpansive 16 . Example 1.2. 1 Let E be a uniformly smooth and strictly convex Banach space and let A be a maximal monotone mapping from E to E such that its zero set A−10 is nonempty. The resolvent Jr J rA −1J is a closed quasi-φ-nonexpansive mapping from E onto D A and F Jr A−10. 2 Let ΠC be the generalized projection from a smooth, strictly convex, and reflexive Banach space E onto a nonempty closed convex subset C of E. ThenΠC is a closed and quasiφ-nonexpansive mapping from E onto C with F ΠC C. Iiduka and Takahashi 17 introduced the following algorithm for finding a solution of the variational inequality for an operator A that satisfies conditions C1 C2 in a 2 uniformly convex and uniformly smooth Banach space E. For an initial point x0 x ∈ C, define a sequence {xn} by xn 1 ΠCJ−1 Jxn − λnxn , ∀n ≥ 0. 1.8 where J is the duality mapping on E, and ΠC is the generalized projection of E onto C. Assume that λn ∈ a, b for some a, b with 0 < a < b < c2α/2 where 1/c is the 2 uniformly convexity constant of E. They proved that if J is weakly sequentially continuous, then the sequence {xn} converges weakly to some element z in V I A,C where z limn→∞ΠV I A,C xn . 4 Abstract and Applied Analysis The problem of finding a common element of the set of the variational inequalities for monotone mappings in the framework of Hilbert spaces and Banach spaces has been intensively studied by many authors; see, for instance, 18–20 and the references cited therein. On the other hand, one classical way to study nonexpansive mappings is to use contractions to approximate a nonexpansive mapping see 21 . More precisely, let t ∈ 0, 1 and define a contraction Gt : C → C by Gtx tx0 1 − t Tx for all x ∈ C, where x0 ∈ C is a fixed point in C. Applying Banach’s Contraction Principle, there exists a unique fixed point xt of Gt in C. It is unclear, in general, what is the behavior of xt as t → 0 even if T has a fixed point. However, in the case of T having a fixed point, Browder 21 proved that the net {xt} defined by xt tx0 1− t Txt for all t ∈ 0, 1 converges strongly to an element of F T which is nearest to x0 in a real Hilbert space. Motivated by Browder 21 , Halpern 22 proposed the following iteration process: x0 ∈ C, xn 1 αnx0 1 − αn Txn, n 0 1.9 and proved the following theorem. Theorem H. Let C be a bounded closed convex subset of a Hilbert space H and let T be a nonexpansive mapping on C. Define a real sequence {αn} in 0, 1 by αn n−θ, 0 < θ < 1. Define a sequence {xn} by 1.9 . Then {xn} converges strongly to the element of F T which is the nearest to u. Recently, Martinez-Yanes and Xu 23 have adapted Nakajo and Takahashi’s 24 idea to modify the process 1.9 for a single nonexpansive mapping T in a Hilbert space H: x0 x ∈ C chosen arbitrary, yn αnx0 1 − αn Txn, Cn { v ∈ C : ∥yn − v ∥2 ‖xn − v‖ αn ( ‖x0‖ 2〈xn − x0, v〉 )} , Qn {v ∈ C : 〈xn − v, x0 − xn〉 0}, xn 1 PCn∩Qnx0, 1.10 where PC denotes the metric projection ofH onto a closed convex subsetC ofH. They proved that if {αn} ⊂ 0, 1 and limn→∞αn 0, then the sequence {xn} generated by 1.10 converges strongly to PF T x. In 15 see also 13 , Qin and Su improved the result of Martinez-Yanes and Xu 23 fromHilbert spaces to Banach spaces. To be more precise, they proved the following theorem. TheoremQS. Let E be a uniformly convex and uniformly smooth Banach space, let C be a nonempty closed convex subset of E, and let T : C → C be a relatively nonexpansive mapping. Assume that Abstract and Applied Analysis 5 {αn} is a sequence in 0, 1 such that limn→∞αn 0. Define a sequence {xn} in C by the following algorithm:and Applied Analysis 5 {αn} is a sequence in 0, 1 such that limn→∞αn 0. Define a sequence {xn} in C by the following algorithm: x0 x ∈ C chosen arbitrary, yn J−1 αnJx0 1 − αn JTxn , Cn { v ∈ C : φv, yn ) ≤ αnφ ( v, yn ) 1 − αn φ v, xn } , Qn {v ∈ C : 〈xn − v, Jx0 − Jxn〉 0}, xn 1 ΠCn∩Qnx0, 1.11 where J is the single-valued duality mapping on E. If F T is nonempty, then {xn} converges to ΠF T x0. In 14 , Plubtieng and Ungchittrakool introduced the following hybrid projection algorithm for a pair of relatively nonexpansive mappings: x0 x ∈ C chosen arbitrary, zn J−1 ( β 1 n Jxn β 2 n JTxn β 3 n JSxn ) , yn J−1 αnJx0 1 − αn Jzn , Hn { z ∈ C : φz, yn ) φ z, xn αn ( ‖x0‖ 2〈z, Jxn − Jx〉 )} , Wn {z ∈ C : 〈xn − z, Jx − Jxn〉 0}, xn 1 PHn∩Wnx, n 0, 1, 2, . . . , 1.12 where {αn}, {β 1 n }, {β 2 n }, and {β 3 n } are sequences in 0, 1 satisfying β 1 n β 2 n β 3 n 1 for all n ∈ N ∪ {0} and T, S are relatively nonexpansive mappings and J is the single-valued duality mapping on E. They proved, under appropriate conditions on the parameters, that the sequence {xn} generated by 1.12 converges strongly to a common fixed point of T and S. Very recently, Qin et al. 25 introduced a new hybrid projection algorithm for two families of quasi-φ-nonexpansive mappings which are more general than relatively nonexpansive mappings to have strong convergence theorems in the framework of Banach spaces. To be more precise, they proved the following theorem. Theorem QCKZ. Let E be a uniformly convex and uniformly smooth Banach space, and let C be a nonempty closed convex subset of E. Let {Si}i∈I and {Ti}i∈I be two families of closed quasi-φnonexpansive mappings of C into itself with F : ⋂ i∈IF Ti ∩ ⋂ i∈IF Si being nonempty, where 6 Abstract and Applied Analysis I is an index set. Let the sequence {xn} be generated by the following manner: x0 x ∈ C chosen arbitrary, zn,i J−1 ( β 1 n,i Jxn β 2 n,i JTixn β 3 n,i JSixn ) , yn,i J−1 αn,iJx0 1 − αn,i Jzn,i , Cn,i { u ∈ C : φu, yn,i ) φ u, xn αn,i ( ‖x0‖ 2〈u, Jxn − Jxn〉 )} ,