In this paper, we combine the subgradient extragradient method with the Halpern method for finding a solution of a variational inequality involving a monotone Lipschitz mapping in Banach spaces. By using the generalized projection operator and the Lyapunov functional introduced by Alber, we prove a strong convergence theorem. We also consider the problem of finding a common element of the set o...

Assume that F is a nonlinear operator on a real Hilbert space H which is strongly monotone and Lipschitzian with constants η > 0 and κ > 0, respectively on a nonempty closed convex subset C of H . Assume also that C is the intersection of the fixed point sets of a finite number of nonexpansive mappings on H . We develop an implicit hybrid steepest-descent method which generates an iterative seq...

Let C be nonempty closed convex subset of real Hilbert space H. Consider C a nonexpansive semigroup I {T s : s ≥ 0}with a common fixed point, a contraction f with coefficient 0 < α < 1, and a strongly positive linear bounded operator A with coefficient γ > 0. Let 0 < γ < γ/α. It is proved that the sequence {xn} generated iteratively by xn I − αnA 1/tn ∫ tn 0 T s ynds αnγf xn , yn I − βnA xn βnγ...

Let K be a nonempty closed convex subset of a reflexive real Banach space E which has a uniformly Gâteaux differentiable norm. Assume that K is a sunny nonexpansive retract of E with Q as the sunny nonexpansive retraction. Let Ti : K → E, i = 1, . . . ,r, be a family of nonexpansive mappings which are weakly inward. Assume that every nonempty closed bounded convex subset of K has the fixed poin...

In this paper, we first introduce the class of generalized nonexpansive mappings in Banach spaces. This class contains both the classes of nonexpansive and ̨-nonexpansive mappings. In addition, we obtain some fixed point and coincidence point theorems for generalized nonexpansive mappings in uniformly convex Banach spaces. Our results extend some wellknown results in literature. 2010 Mathematic...

We introduce a general implicit algorithm for finding a common element of‎ ‎the set of solutions of systems of equilibrium problems and the set of common fixed points‎ ‎of a sequence of nonexpansive mappings and a continuous representation of nonexpansive mappings‎. ‎Then we prove the strong convergence of the proposed implicit scheme to the unique solution of the minimization problem on the so...

we introduce a general implicit algorithm for finding a common element of‎ ‎the set of solutions of systems of equilibrium problems and the set of common fixed points‎ ‎of a sequence of nonexpansive mappings and a continuous representation of nonexpansive mappings‎. ‎then we prove the strong convergence of the proposed implicit scheme to the unique solution of the minimization problem on the so...

We consider a hybrid projection algorithm basing on the shrinking projection method for two families of relatively weak quasi-nonexpansive mappings. We establish strong convergence theorems for approximating the common fixed point of the set of the common fixed points of such two families and the set of solutions of the variational inequality for an inverse-strongly monotone operator in the fra...

Let H be a real Hilbert space and let C be a nonempty closed convex subset of H . Let F :H →H be an operator such that for some constants k,η > 0, F is k-Lipschitzian and η-strongly monotone on C; that is, F satisfies the following inequalities: ‖Fx− Fy‖ ≤ k‖x− y‖ and 〈Fx− Fy,x− y〉 ≥ η‖x− y‖2 for all x, y ∈ C, respectively. Recall that T is nonexpansive if ‖Tx−Ty‖ ≤ ‖x− y‖ for all x, y ∈H . We ...