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The Mann iterates behave well for nonexpansive mappings for any initial guess in the domain. Our aim in this article is to extend this method to a broad class of inexact fixed point algorithms generated by nearly nonexpansive sequences in Banach spaces and to locate the weak limit of the iterates by its initial guesses. Due to the inexactness, our algorithms become efficiently applicable for a wider class of problems. As applications, we give convergence theorems for finding solutions of variational inclusion problems and constrained multiple-sets split feasibility problems. Our results are significant refinements and improvements of the corresponding results in the literature. [ABSTRACT FROM AUTHOR]

We propose implicit and explicit iterative algorithms for finding a common element of the set of solutions of the minimization problem for a convex and continuously Fréchet differentiable functional, the set of solutions of a finite family of generalized mixed equilibrium problems, and the set of solutions of a finite family of variational inequalities for inverse strong monotone mappings in a real Hilbert space. We prove that the sequences generated by the proposed algorithms converge strongly to a common element of three sets, which is the unique solution of a variational inequality defined over the intersection of three sets under very mild conditions. [ABSTRACT FROM AUTHOR]

We propose two iterative algorithms for finding a common element of the set of solutions of finite generalized mixed equilibrium problems, the set of solutions of finite variational inclusions for maximal monotone and inverse strong monotone mappings, and the set of common fixed points of infinite nonexpansive mappings and an asymptotically K-strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove some strong and weak convergence theorems for the proposed iterative algorithms under suitable conditions. [ABSTRACT FROM AUTHOR]

We suggest and analyze relaxed extragradient iterative algorithms with regularization for finding a common element of the solution set of a general system of variational inequalities, the solution set of a split feasibility problem, and the fixed point set of a strictly pseudocontractive mapping defined on a real Hilbert space. Here the relaxed extragradient methods with regularization are based on the well-known successive approximation method, extragradient method, viscosity approximation method, regularization method, and so on. Strong convergence of the proposed algorithms under some mild conditions is established. Our results represent the supplementation, improvement, extension, and development of the corresponding results in the very recent literature. [ABSTRACT FROM AUTHOR]

We study in this paper the existence and the approximation of solutions of variational inequalities involving generalized pseudocontractive mappings in Banach spaces. The convergence analysis of a proposed hybrid iterative method for approximating common zeros or fixed points of a possibly infinitely countable or uncountable family of such operators will be conducted within the conceptual framework of the "viscosity approximation technique" in reflexive Banach spaces with uniform Gâteaux differentiable norms. This technique should make existing or new results in solving variational inequalities more applicable. [ABSTRACT FROM AUTHOR]

We introduce a new implicit iterative scheme with perturbation for finding the approximate solutions of a hierarchical variational inequality, that is, a variational inequality over the common fixed point set of a finite family of nonexpansive mappings. We establish some convergence theorems for the sequence generated by the proposed implicit iterative scheme. In particular, necessary and sufficient conditions for the strong convergence of the sequence are obtained. [ABSTRACT FROM AUTHOR]

We deal with a common fixed point problem for a family of quasinonexpansive mappings defined on a Hilbert space with a certain closedness assumption and obtain strongly convergent iterative sequences to a solution to this problem. We propose a new type of iterative scheme for this problem. A feature of this scheme is that we do not use any projections, which in general creates some difficulties in practical calculation of the iterative sequence. We also prove a strong convergence theorem by the shrinking projection method for a family of such mappings. These results can be applied to common zero point problems for families of monotone operators. [ABSTRACT FROM AUTHOR]

Let E be a uniformly convex and uniformly smooth Banach space with the dual E* and let T : E → 2E* be a maximal monotone operator. By using the technique of resolvent operators and by using modified Ishikawa iteration and modified Halpern iteration for relatively non-expansive mappings, we suggest and analyse two iterative algorithms for finding an element x ∈ E such that 0 ∈ T(x). Strong convergence theorems for such iterative algorithms are proved. The ideas of these algorithms are applied to solve the problem of finding a minimizer of a convex function on E. [ABSTRACT FROM AUTHOR]

We consider an iterative scheme for finding a common element of the set of solutions of a pseudomonotone, Lipschitz-continuous variational inequality problem and the set of common fixed points of N nonexpansive mappings. The proposed iterative method combines two well-known schemes: extragradient and approximate proximal methods. We derive a necessary and sufficient condition for weak convergence of the sequences generated by the proposed scheme. [ABSTRACT FROM AUTHOR]

This paper introduces an Ishikawa type iterative algorithm for finding approximating solutions of a class of multi-valued variational inclusion problems. Characterization of strong convergence of this iterative method is established. [ABSTRACT FROM AUTHOR]

In this paper, the hybrid steepest descent methods are extended to develop new iterative schemes for finding the zeros of bounded, demicontinuous and φ-strongly accretive mappings in uniformly smooth Banach spaces. Two iterative schemes are proposed. Strong convergence results are established and applications to variational inequalities are given. [ABSTRACT FROM AUTHOR]

We propose a relaxed composite implicit iteration process for finding approximate common fixed points of a finite family of strictly pseudocontractive mappings in Banach spaces. Several convergence results for this process are established. [ABSTRACT FROM AUTHOR]

We introduce an implicit iteration scheme with a perturbed mapping for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points of finitely many nonexpansive mappings in a Hilbert space. Then, we establish some convergence theorems for this implicit iteration scheme which are connected with results by Xu and Ori (Numer. Funct. Analysis Optim. 22:767–772, ), Zeng and Yao (Nonlinear Analysis, Theory, Methods Appl. 64:2507–2515, ) and Takahashi and Takahashi (J. Math. Analysis Appl. 331:506–515, ). In particular, necessary and sufficient conditions for strong convergence of this implicit iteration scheme are obtained. [ABSTRACT FROM AUTHOR]

Let C be a nonempty closed convex subset of a real Banach space E. Let S: C → C be a quasi-nonexpansive mapping, let T: C → C be an asymptotically demicontractive and uniformly Lipschitzian mapping, and let F:= { x ε C: Sx = x and Tx = x} ≠ 0. Let { x n } n≥0 be the sequence generated from an arbitrary x 0 ε C by We prove the necessary and sufficient conditions for the strong convergence of the iterative sequence { x n } to an element of F. These extend and improve the recent results of Moore and Nnoli. [ABSTRACT FROM AUTHOR]

Assume that F is a nonlinear operator on a real Hilbert space H which is η-strongly monotone and κ-Lipschitzian 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 construct an iterative algorithm with variable parameters which generates a sequence {Xn} from an arbitrary initial point X0 ϵ H. The sequence {Xn} is shown to converge in norm to the unique solution u∗ of the variational inequality ⟨F(u∗), v - u∗⟩ ⩾ 0, ∀v ϵ C. [ABSTRACT FROM AUTHOR]

This paper introduces an iterative method for finding approximate solutions of a set-valued mixed quasivariational inequality in the setting of a Banach space. Existence of a solution of this rather general problem and the convergence of the proposed iterative method to a solution are established. [ABSTRACT FROM AUTHOR]

The auxiliary principle technique is extended to study the generalized strongly nonlinear mixed variational-like inequality problem for set-valued mappings without compact values. We establish first the existence of a solution of the related auxiliary problem. Then, the iterative algorithm for solving that problem is given by using this existence result. Moreover, the existence of a solution of the original problem and the convergence of iterative sequences generated by the algorithm are both derived. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider the split feasibility problem (SFP) in infinite-dimensional Hilbert spaces, and study the relaxed extragradient methods for finding a common element of the solution set of SFP and the set of fixed points of a nonexpansive mapping . Combining Mann’s iterative method and Korpelevich’s extragradient method, we propose two iterative algorithms for finding an element of . On one hand, for , the identity mapping, we derive the strong convergence of one iterative algorithm to the minimum-norm solution of the SFP under appropriate conditions. On the other hand, we also derive the weak convergence of another iterative algorithm to an element of under mild assumptions. [Copyright &y& Elsevier]

Abstract: The present paper is divided into two parts. In the first part, we introduce implicit and explicit iterative schemes for finding the fixed point of a nonexpansive mapping defined on the closed convex subset of a real Hilbert space. We establish results on the strong convergence of the sequences generated by the proposed schemes to a fixed point of a nonexpansive mapping. Such a fixed point is also a solution of a variational inequality defined on the set of fixed points. In the second part, we propose implicit and explicit iterative schemes for finding the approximate minimizer of a constrained convex minimization problem and prove that the sequences generated by our schemes converge strongly to a solution of the constrained convex minimization problem. Such a solution is also a solution of a variational inequality defined over the set of fixed points of a nonexpansive mapping. The results of this paper extend and improve several results presented in the literature in the recent past. [Copyright &y& Elsevier]

Abstract: We introduce a new composite iterative scheme to approximate a zero of an -accretive operator defined on uniform smooth Banach spaces and a reflexive Banach space having a weakly continuous duality map. It is shown that the iterative process in each case converges strongly to a zero of . The results presented in this paper substantially improve and extend the results due to Ceng et al. [L.C. Ceng, H.K. Xu, J.C. Yao, Strong convergence of a hybrid viscosity approximation method with perturbed mappings for nonexpansive and accretive operators, Taiwanese J. Math. (in press)], Kim and Xu [T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal. 61 (2005) 51–60] and Xu [H.K. Xu, Strong convergence of an iterative method for nonexpansive and accretive operators, J. Math. Anal. Appl. 314 (2006) 631–643]. Our work provides a new approach for the construction of a zero of -accretive operators. [Copyright &y& Elsevier]

Abstract: In this paper, we propose an iterative scheme for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a strict pseudo-contraction mapping in the setting of real Hilbert spaces. We establish some weak and strong convergence theorems of the sequences generated by our proposed scheme. Our results combine the ideas of Marino and Xu’s result [G. Marino, H.K. Xu, Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl. 329 (2007) 336–346], and Takahashi and Takahashi’s result [S. Takahashi, W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2007) 506–515]. In particular, necessary and sufficient conditions for strong convergence of our iterative scheme are obtained. [Copyright &y& Elsevier]