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The purpose of this paper is to find a common element in the intersection of the set of zeros of the inclusion problem of sum of two monotone mappings and the set of fixed points of a Bregman quasi nonexpansive mapping in a reflexive Banach space by using Bregman distance and shrinking projection method. Under suitable conditions, some strong convergence theorems are proved. As applications, we utilize our results to study the convex minimization problem, variational inequality problem. [ABSTRACT FROM AUTHOR]

In the present paper, we study the finite termination of sequences generated by inexact proximal point algorithms for finding zeroes of a maximal monotone (set-valued) operator $$T$$ on a Hilbert space. Under some mild conditions, we get that a sequence generated by inexact proximal point algorithm stops after a finite number of iterations. Our results extend the corresponding results in Rockafellar (SIAM J Control Optim 14:877-898, ). In particular, for optimization problems, our results improve corresponding results in Ferris (Math Progr 50:359-366, ). As applications, we obtain finite termination of projected gradient method. [ABSTRACT FROM AUTHOR]

The aim of this paper is to define a generalized monotone mapping, which is the sum of symmetric cocoercive mapping and symmetric monotone mapping. The resolvent operator associated with generalized monotone mapping is defined and some of its properties are shown. We solve a variational inclusion problem using these new concepts. For illustration, some examples are given. [ABSTRACT FROM AUTHOR]

In this article, we consider a monotone variational inequality with variational inequality constraint over the set of fixed points of a nonexpansive mapping, which is called a triple-hierarchical variational inequality (THVI). A relaxed hybrid steepest-descent method with variable parameters is introduced for solving the THVI. Strong convergence of the method to a unique solution of the THVI is studied under certain assumptions. We also investigate another monotone variational inequality with the variational inequality constraint over the set of common fixed points of a finite family of nonexpansive mappings, and present an iterative algorithm for solving such kind of problems. It is proven that under mild conditions the sequence generated by the proposed algorithm converges strongly to a unique solution of later problem. [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]

By using a stability theorem of S.M. Robinson and a scalarization method, we establish sufficient conditions for the upper semicontinuity of the solution maps of parametric monotone affine vector variational inequalities. As a by-product, some new topological properties of the solution sets of these problems are obtained. Our results imply several facts for the solution stability and connectedness of the solution sets of convex quadratic vector optimization problems and of linear fractional vector optimization problems. [ABSTRACT FROM AUTHOR]

Let C be a nonempty closed convex subset of a real Hilbert space H. Let F : C → H be a κ-Lipschitzian and η-strongly monotone operator with constants κ, η > 0, V, T : C → C be nonexpansive mappings with Fix(T) ≠ ∅ where Fix(T) denotes the fixed-point set of T, and f : C → H be a ρ-contraction with coefficient ρ ∈ [0, 1). Let 0 < µ < 2η/κ² and 0 < γ ≤ τ, where τ = √1 - µ(2η - µκ²). For each s, t ∈ (0, 1), let xs,t be a unique solution of the fixed-point equation xs,t PC[sγf(xs,t) + (I - sµF)(tV + (1 - t)T)xs,t].We derive the following conclusions on the behavior of the net {xs,t} along the curve t = t(s): (i) if t(s) = O(s), as s → 0, then xs,t(s) → z∞ strongly, which is the unique solution of the variational inequality of finding z∞ ∈ Fix(T) such that <[(µF - γf) + l(I - V)]z∞, x - z∞> ≥ 0, for all x ∈ Fix(T) and ?(ii) if t(s)/s → ∞, as s → 0, then xs,t(s) → x∞ strongly, which is the unique solution of some hierarchical variational inequality problem. [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]

The purpose of this paper is to introduce and study two hybrid proximal-point algorithms for finding a common element of the set of solutions of an equilibrium problem and the set of solutions to the equation 0∈ Tx for a maximal monotone operator T in a uniformly smooth and uniformly convex Banach space X. Strong and weak convergence results of these two hybrid proximal-point algorithms are established, respectively. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce and study two new hybrid proximal-point algorithms for finding a common element of the set of solutions to a generalized equilibrium problem and the sets of zeros of two maximal monotone operators in a uniformly smooth and uniformly convex Banach space. We established strong and weak convergence theorems for these two modified hybrid proximal-point algorithms, respectively. [ABSTRACT FROM AUTHOR]

We extend some results due to Thanh-Hao (Acta Math. Vietnam. 31: 283-289, 2006) and Noor (J. Optim. Theory Appl. 115:447-452, 2002). The first paper established a convergence theorem for the Tikhonov regularization method (TRM) applied to finite-dimensional pseudomonotone variational inequalities (VIs), answering in the affirmative an open question stated by Facchinei and Pang (Finite- Dimensional Variational Inequalities and Complementarity Problems, Springer, New York, 2003). The second paper discussed the application of the proximal point algorithm (PPA) to pseudomonotone VIs. In this paper, new facts on the convergence of TRM and PPA (both the exact and inexact versions of PPA) for pseudomonotone VIs in Hilbert spaces are obtained and a partial answer to a question stated in (Acta Math. Vietnam. 31:283-289, 2006) is given. As a byproduct, we show that the convergence theorem for inexact PPA applied to infinite-dimensional monotone variational inequalities can be proved without using the theory of maximal monotone operators. [ABSTRACT FROM AUTHOR]

We discuss the lower semicontinuity of the set of efficient solutions for parametric generalized systems with monotone bifunctions in real locally convex Hausdorff topological vector spaces. [ABSTRACT FROM AUTHOR]

We introduce the concept of positive proper efficient solutions to the generalized system in this paper. We show that, under some suitable conditions, the set of positive proper efficient solutions is dense in the set of efficient solutions to the generalized system. We discuss also the connectedness of the set of efficient solutions for the generalized system with monotone bifunctions in real locally convex Hausdorff topological vector spaces. [ABSTRACT FROM AUTHOR]

In this paper, we consider a generalized vector variational-like inequality problem (for short, GVVLIP), which includes generalized vector variational inequalities, vector variational inequalities and classical variational inequalities as special cases. The concepts of generalized C-pseudomonotone-like and generalized H-hemicontinuous-like operators are introduced. Some existence results for GVVLIP are obtained under the assumptions of generalized C-pseudomonotone-like property and generalized H-hemicontinuous-like property. These results appear to be new and interesting. New existence results of the classical variational inequality are also obtained. [ABSTRACT FROM AUTHOR]

The article presents a discussion that looks into the wide class of inexact proximal-type methods for handling the generalized variational inequality problem with maximal monotone operator in a reflexive Banach space. An extension of Kamimura and Takahashi's algorithm is also provided with the aim of developing a generic inexact proximal point algorithm. A convergence analysis which unifies a broad class of proximal-type methods associated with finding Hilbert spaces or Banach spaces is also included.

In this paper, we are interested in the existence of solutions for a class of noncoercive variational inequalities involving a p-Laplacian type operator. Our approach is based essentially on equilibrium problems and arguments from recession analysis. Our results are of two types: the first is obtained in a monotone framework; the second is obtained without a monotonicity assumption. [ABSTRACT FROM AUTHOR]