The General Hybrid Approximation Methods for Nonexpansive Mappings in Banach Spaces

and Applied Analysis 3 Recently, Marino and Xu 11 mixed the iterative method 1.6 and the viscosity approximation method 1.7 and considered the following general iterative method: xn 1 I − λnA Txn λnγf xn , n ≥ 0, 1.9 where A is a strongly positive bounded linear operator on H. They proved that if the sequence {λn} of parameters satisfies the following appropriate conditions: limn→∞λn 0, ∑∞ n 1 λn ∞, and either ∑∞ n 1 |λn 1 − λn| < ∞ or limn→∞ λn/λn 1 1, then the sequence {xn} generated by 1.9 converges strongly to the unique solution x∗ in H of the variational inequality 〈( A − γf)x∗, x − x∗〉 ≥ 0, x ∈ H, 1.10 which is the optimality condition for theminimization problem:minx∈F T 1/2 〈Ax, x〉−h x , where h is a potential function for γf i.e., h′ x γf x for x ∈ H . Very recently, Wangkeeree et al. 12 extended Marino and Xu’s result to the setting of Banach spaces and obtained the strong convergence theorems in a reflexive Banach space which admits a weakly continuous duality mapping. Let E be a reflexive Banach space which admits a weakly continuous duality mapping Jφ with gauge φ such that φ is invariant on 0, 1 . Let T : E → E be a nonexpansive mapping with F T / ∅, f a contraction with coefficient 0 < α < 1 and A a strongly positive bounded linear operator with coefficient γ > 0 and 0 < γ < γφ 1 /α. Define the net {xt} by xt tγf xt I − tA Txt. 1.11 It is proved in 12 that {xt} converges strongly as t → 0 to a fixed point x̃ of T which solves the variational inequality 〈( A − γf)x̃, Jφ x̃ − z 〉 ≤ 0, z ∈ F T . 1.12 On the other hand, Ceng et al. 13 introduced the iterative approximation method for solving the variational inequality generated by two strongly positive bounded linear operators on a real Hilbert space H. Let f : H → H be a contraction with coefficient 0 < α < 1, and let A,B : H → H be two strongly positive bounded linear operators with coefficient γ ∈ 0, 1 and β > 0, respectively. Assume that 0 < γα < β, {λn} is a sequence in 0, 1 , {μn} is a sequence in 0,min{1, ‖B‖−1} . Starting with an arbitrary initial x0 ∈ H, define a sequence {xn} recursively by xn 1 1 − λnA Txn λn 1 [ Txn − μn 1 ( BTxn − γf xn )] , n ≥ 0. 1.13 It is proved in 13, Theorem 3.1 that if the sequences {λn} and {μn} satisfy the following conditions: C1 limn→∞λn 0, C2 ∑∞ n 1 λn ∞, C3 ∑∞ n 1 |λn 1 − λn| < ∞ or limn→∞ λn/λn 1 1, C4 1 − γ / β − γα < limn→∞μn μ < 2 − γ / β − γα , 4 Abstract and Applied Analysis then the sequence {xn} generated by 1.13 converges strongly to the unique solution x̃ in H of the variational inequality 〈( A − I μ(B − γf))x̃, x̃ − z〉 ≤ 0, z ∈ F T . 1.14 Observe that if B I and μn 1 for all n ≥ 1, then algorithm 1.13 reduces to 1.9 . Moreover, the variational inequality 1.14 reduces to 1.10 . Furthermore, the applications of these results to constrained generalized pseudoinverse are studied. In this paper, motivated by Marino and Xu 11 , Wangkeeree et al. 12 , and Ceng et al. 13 , we introduce two general iterative approximation methods one implicit and one explicit for finding a fixed point of a nonexpansive mapping which solving the variational inequality generated by two strongly positive bounded linear operators. Strong convergence theorems of the proposed iterative methods are obtained in a reflexive Banach space which admits a weakly continuous duality mapping. The results presented in this paper improve and extend the corresponding results announced by Marino and Xu 11 , Wangkeeree et al. 12 , and Ceng et al. 13 , and many others. 2. Preliminaries Throughout this paper, let E be a real Banach space and E∗ its dual space. We write xn ⇀ x resp. xn⇀∗x to indicate that the sequence {xn} weakly resp. weak∗ converges to x; as usual, xn → x will symbolize strong convergence. Let UE {x ∈ E : ‖x‖ 1}. A Banach space E is said to uniformly convex if, for any ε ∈ 0, 2 , there exists δ > 0 such that, for any x, y ∈ UE, ‖x−y‖ ≥ ε implies ‖ x y /2‖ ≤ 1−δ. It is known that a uniformly convex Banach space is reflexive and strictly convex see also 14 . A Banach space E is said to be smooth if the limit limt→ 0 ‖x ty‖ − ‖x‖ /t exists for all x, y ∈ UE. It is also said to be uniformly smooth if the limit is attained uniformly for x, y ∈ UE. By a gauge function φ, we mean a continuous strictly increasing function φ : 0,∞ → 0,∞ such that φ 0 0 and φ t → ∞ as t → ∞. Let E∗ be the dual space of E. The duality mapping Jφ : E → 2E associated to a gauge function φ is defined by Jφ x { f∗ ∈ E∗ : 〈x, f∗〉 ‖x‖φ ‖x‖ ,∥∥f∗∥∥ φ ‖x‖ }, ∀x ∈ E. 2.1 In particular, the duality mapping with the gauge function φ t t, denoted by J , is referred to as the normalized duality mapping. Clearly, there holds the relation Jφ x φ ‖x‖ /‖x‖ J x for all x / 0 see 15 . Browder 15 initiated the study of certain classes of nonlinear operators by means of the duality mapping Jφ. Following Browder 15 , we say that a Banach space E has a weakly continuous duality mapping if there exists a gauge φ for which the duality mapping Jφ x is single valued and continuous from the weak topology to the weak∗ topology; that is, for any {xn} with xn ⇀ x, the sequence {Jφ xn } converges weakly∗ to Jφ x . It is known that l has a weakly continuous duality mapping with a gauge function φ t tp−1 for all 1 < p < ∞. Set