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In developing iterative methods for approximating solutions of optimization problems, one of the major goals of researchers is to construct efficient iterative schemes. Over the years, different techniques have been devised by authors to achieve this goal. In this paper, we study a non-Lipschitz pseudomonotone variational inequality problem with common fixed points constraint of Bregman quasi-nonexpansive mappings. We introduce a new iterative method for approximating the solution of this problem in a more general framework of reflexive Banach spaces. Our method employs several techniques to achieve high level of efficiency. One of the techniques employed is the S-iteration process, a method known to be highly efficient in comparison with several of the well-known methods. Moreover, our proposed algorithm utilizes the inertial technique and a non-monotonic self-adaptive step size to guarantee high rate of convergence and easy implementation. Unlike several of the existing results on variational inequality problem with non-Lipschitz operator, the design of our method does not involve any linesearch technique. We obtain strong convergence result for the proposed algorithm without the sequentially weakly continuity condition often assumed by authors to guarantee convergence when solving pseudomonotone variational inequality problems. Furthermore, we apply our result to study utility-based bandwidth allocation problem and image restoration problem. Finally, we present several numerical experiments to demonstrate the efficiency of our proposed method in comparison with existing state-of-the-art methods. Our result extends and improves several of the recently announced results in this direction in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the solution of a class of monotone inclusion problems in reflexive Banach spaces. To reflect the geometry of the space and the operator, a more general proximal point iterative algorithm with Bregman divergence is proposed and some strong convergence results for the proposed scheme under standard assumptions are obtained. Meanwhile, the convex optimization problems and the critical point problems are studied in the applications, and the recovery of the sparse signal is simulated in the numerical experiments. [ABSTRACT FROM AUTHOR]

We mainly introduce a new self-adaptive extragradient method by using the inertial technique for solving variational inequality problems of pseudomonotone operators and fixed point problems of Bregman relatively nonexpansive mappings in real reflexive Banach spaces. Precisely, we show that the sequence generated by our iterative process converges strongly to a common element for the solution set of variational inequality problems and the set of fixed points of Bregman relatively nonexpansive mappings. Additionally, some numerical examples are given to show the effectiveness of our algorithm. The results obtained in this paper are the improvement and supplement of many recent ones in the field. [ABSTRACT FROM AUTHOR]

In this paper, we propose an inertial algorithm for solving split equality of monotone inclusion and f$$ f $$‐fixed point of Bregman relatively f$$ f $$‐nonexpansive mapping problems in reflexive real Banach spaces. Using the Bregman distance function, we prove a strong convergence theorem for the algorithm produced by the method in real reflexive Banach spaces. In addition, we provide some applications of our method and give numerical results to demonstrate the applicability and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

The purpose of this study is to introduce an inertial algorithm for approximating a solution of the split equality Hammerstein type equation problem in general reflexive real Banach spaces. Strong convergence results are established under the assumption that the associated mappings are monotone and uniformly continuous. The results in this paper generalize and improve many of the existing results in the literature in the sense that the underlying mappings are relaxed from Lipschitz continuous to uniformly continuous and the spaces under consideration are extended from Hilbert spaces to reflexive real Banach spaces with a more general problem which includes the Hammerstein type equation problems. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to construct an algorithm for approximating a common element of the set of solutions of a finite family of generalized mixed equilibrium problems, the set of f-fixed points of a finite family of f-pseudocontractive mappings and the set of solutions of a finite family of variational inequality problems for Lipschitz monotone mappings in real reflexive Banach spaces. [ABSTRACT FROM AUTHOR]

We investigated the existence and uniqueness of coupled best proximity points for some cyclic and semi-cyclic maps in a reflexive Banach space. We found sufficient conditions, ensuring the existence of coupled best proximity points in reflexive Banach spaces and some convexity types of conditions, ensuring uniqueness of the coupled best proximity points in strictly convex Banach spaces. We illustrate the results with examples and we present an application of one of the theorems in the modeling of duopoly markets, to have an existence of market equilibrium. We show that, in general, the iterative sequences can have chaotic behavior. [ABSTRACT FROM AUTHOR]

The aim of this paper is to discuss Penot’s problem on a generalization of Caristi’s fixed point theorem. We settle this problem in the negative and we present some new theorems on the existence of fixed points of set-valued mappings in ordered metric spaces and reflexive Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new iterative method for solving the mixed equilibrium problems and the fixed point problems for a countable family of Bregman relatively nonexpansive mappings in reflexive Banach spaces. We prove that the sequence generated by the proposed iterative algorithm converges strongly to a common solution of the mentioned problems. Further, a numerical example of the iterative algorithm supporting our main result is presented. [ABSTRACT FROM AUTHOR]

We investigated the existence and uniqueness of coupled best proximity points for some cyclic and semi-cyclic maps in a reflexive Banach space. We found sufficient conditions, ensuring the existence of coupled best proximity points in reflexive Banach spaces and some convexity types of conditions, ensuring uniqueness of the coupled best proximity points in strictly convex Banach spaces. We illustrate the results with examples and we present an application of one of the theorems in the modeling of duopoly markets, to have an existence of market equilibrium. We show that, in general, the iterative sequences can have chaotic behavior.

The purpose of this paper is to study an inclusion problem which involves the sum of two monotone operators in a real reflexive Banach space. Using the technique of Bregman distance, we study the operator ResfT Af which is the composition of the resolvent of a maximal monotone operator T and the antiresolvent of a Bregman inverse strongly monotone operator A and prove that 0 ∊ Tx+Ax if and only if x is a fixed point of the composite operator ResTf Af. Consequently, weak and strong convergence results are given for the inclusion problem under study in a real reflexive Banach space. We apply our results to convex optimization and mixed variational inequalities in a real reflexive Banach space. Our results are new, interesting and extend many related results on inclusion problems from both Hilbert spaces and uniformly smooth and uniformly convex Banach spaces to more general reflexive Banach spaces. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to suggest a new algorithm for finding a common solution of a mixed equilibrium problem and a common fixed point of uniformly Bregman totally quasi-asymptotically nonexpansive mappings in reflexive Banach spaces. The strong convergence theorems under suitable control conditions are proven. [ABSTRACT FROM AUTHOR]

In this article, we introduce a new multi-step iteration for approximating a common fixed point of a finite class of multi-valued Bregman relatively nonexpansive mappings in the setting of reflexive Banach spaces. We prove a strong convergence theorem for the proposed iterative algorithm under certain hypotheses. Additionally, we also use our results for the solution of variational inequality problems and to find the zero points of maximal monotone operators. The theorems furnished in this work are new and well-established and generalize many well-known recent research works in this field. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new iterative process to approximate fixed points of nonexpansive maps in real Banach spaces having weakly continuous duality map and establish strong convergence theorems for the proposed iterative process. There is no compactness assumption on K or on T. Our results improve important recent results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce iterative algorithms for finding a zero of set-valued accretive operators by the viscosity approximation method based on Meir-Keeler-type contractions in a reflexive Banach space which admits a weakly continuous duality mapping. We obtain some strong convergence theorems under suitable conditions. As applications, we apply our results for finding common fixed point of nonexpansive semigroups and for solving equilibrium problem, optimization problem, and variational inequalities. [ABSTRACT FROM AUTHOR]

For a general variational equation with constraints we present both the stability of the corresponding Galerkin method and a collage theorem for a related inverse problem. In addition we demonstrate the applicability of the results by considering forward and inverse boundary value problems of some linear impulsive differential equations. [ABSTRACT FROM AUTHOR]

Under a mild regularity assumption, we derive an exact formula for the Fréchet coderivative and some estimates for the Mordukhovich coderivative of the normal cone mappings of perturbed generalized polyhedra in reflexive Banach spaces. Assume in addition that the generating elements are linearly independent and some qualification condition holds, the Lipschitzian stability of the parameterized variational inequalities over the right-hand side perturbed generalized polyhedra is characterized using the initial data. [ABSTRACT FROM AUTHOR]

In this article, we introduce a new multi-step iteration for approximating a common fixed point of a finite class of multi-valued Bregman relatively nonexpansive mappings in the setting of reflexive Banach spaces. We prove a strong convergence theorem for the proposed iterative algorithm under certain hypotheses. Additionally, we also use our results for the solution of variational inequality problems and to find the zero points of maximal monotone operators. The theorems furnished in this work are new and well-established and generalize many well-known recent research works in this field.

In this paper we prove a strong convergence of both implicit and explicit shcemes to a common fixed point of finite family of nonexpansive maps which is also a unique solution of some variational inequality problem in Banach space. The result presented here improve and unify several important results recently announced. [ABSTRACT FROM AUTHOR]

Abstract: The main purpose of this paper is to extend to the setting of locally convex spaces the study of the mixed variational formulation of some elliptic boundary value problems, the so-called Babuška–Brezzi theory. This study consists of characterizing the existence of a solution and giving conditions that guarantee the stability of the corresponding Galerkin method. [Copyright &y& Elsevier]

Abstract: Let X be a reflexive Banach space which does not have the Kadec–Klee property. Then there exists a compact mapping f from the unit ball of X to the dual space such that and for every . [Copyright &y& Elsevier]

We prove that if X is a Banach space and $${f : X \rightarrow \mathbb{R} \cup \{+\infty\}}$$ is a proper function such that f − ℓ attains its minimum for every ℓ ε X, then the sublevels of f are all relatively weakly compact in X. As a consequence we show that a Banach space X where there exists a function $${f : X \rightarrow \mathbb{R}}$$ such that f − ℓ attains its minimum for every ℓ ε X is reflexive. We also prove that if $${f : X \rightarrow \mathbb{R} \cup \{+\infty\}}$$ is a weakly lower semicontinuous function on the Banach space X and if for every continuous linear functional ℓ on X the set where the function f − ℓ attains its minimum is convex and non-empty then f is convex. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we introduce the implicit and explicit viscosity iteration schemes for nonexpansive semigroups . Additionally, it proves that the proposed iterative schemes converge strongly to a unique common fixed point of in the framework of reflexive and strictly convex Banach space, which solves some variational inequality. The main results of this paper improve and generalize recent known results in the current literature. [Copyright &y& Elsevier]

Abstract: Usually, results on -strongly accretive operators involve various assumptions on the function , such as continuity, being positive, and/or being strictly increasing. However, by redefining this function, we are able to eliminate all these additional assumptions, and leave only the positiveness condition: for . This fact allows us to extend a theorem of Deimling and obtain a general result for the solvability of a variational inequality for -strongly pseudo-contractive mappings. [Copyright &y& Elsevier]

Abstract: In this paper, a new system of generalized mixed implicity equilibrium problems involving non-monotone set-valued mappings is introduced and studied in real reflexive Banach spaces. Following the idea of Moudafi, we consider a system of generalized equations problems and show its equivalence with the system of generalized mixed implicity equilibrium problems. By using a fixed point formulation of the system of generalized equation problems, a new iterative algorithm for solving the system of generalized mixed implicity equilibrium problems is suggested and analyzed. The strong convergence of the iterative sequences generated by the algorithm is proved under suitable conditions. These results are new and unify and generalize some recent results in this field. [Copyright &y& Elsevier]

Abstract: A new iterative method for approximating fixed points of bounded and continuous pseudocontractive mapping is proposed and a strong convergence theorem is obtained. As an application, we prove that a slight modification of our new scheme could be employed for approximating zeros of bounded and continuous accretive operators. Our theorems extend and unify most of the results that have been proved for this class of mappings. [Copyright &y& Elsevier]

Abstract: This paper concerns the study of solution maps to parameterized variational inequalities over generalized polyhedra in reflexive Banach spaces. It has been recognized that generalized polyhedral sets are significantly different from the usual convex polyhedra in infinite dimensions and play an important role in various applications to optimization, particularly to generalized linear programming. Our main goal is to fully characterize robust Lipschitzian stability of the aforementioned solution maps entirely via their initial data. This is done on the basis of the coderivative criterion in variational analysis via efficient calculations of the coderivative and related objects for the systems under consideration. The case of generalized polyhedra is essentially more involved in comparison with usual convex polyhedral sets and requires developing elaborated techniques and new proofs of variational analysis. [Copyright &y& Elsevier]

Abstract: We introduce a new class of mappings, called -cyclic -contractions, which contains the-cyclic contraction mappings as a subclass. Then, convergence and existence results of best proximity points for -cyclic -contraction mappings are obtained. Moreover, we prove results of the existence of best proximity points in a reflexive Banach space. These results are generalizations of the results of Al-Thagafi and Shahzad (2009) . [Copyright &y& Elsevier]

Abstract: In this paper, we provide a comprehensive study of coderivative formulas for normal cone mappings. This allows us to derive necessary and sufficient conditions for the Lipschitzian stability of parametric variational inequalities in reflexive Banach spaces. Our development not only gives an answer to the open questions raised in Yao and Yen (2009) , but also establishes generalizations and complements of the results given in Henrion et al. (2010)  and Yao and Yen (2009) . [Copyright &y& Elsevier]

In this paper, we introduce and study a new system of generalized set-valued strongly nonlinear mixed variational-like inequalities problems and its related auxiliary problems in reflexive Banach spaces. The auxiliary principle technique is applied to study the existence and iterative algorithm of solutions for the system of generalized set-valued strongly nonlinear mixed variational-like inequalities problems. Firstly, we prove the existence and uniqueness of solutions of the auxiliary problems for the system of generalized set-valued strongly nonlinear mixed variational-like inequalities problems. Secondly, an iterative algorithm for solving the system of generalized set-valued strongly nonlinear mixed variational-like inequalities problems is constructed by using this existence and uniqueness result. Finally, we show the existence of solutions of the system of generalized set-valued strongly nonlinear mixed variational-like inequalities problems and discuss the convergence analysis of this algorithm. These results improve, unify and generalize many corresponding known results given in literatures. [ABSTRACT FROM AUTHOR]

Abstract: In this work, we study the following Generalized Variational Inequality Problem (for short, GVIP): Given a closed convex set in a reflexive Banach space with the dual , a multifunction , and a vector , find such that there exists satisfying By using generalized projection and the well-known Fan–KKM Theorem, we prove existence results for solutions of GVIP. Our results extend some recent results from the literature. [Copyright &y& Elsevier]

Abstract: In the theory of ordered spaces and in microeconomic theory two important notions, the notion of the base for a cone which is defined by a continuous linear functional and the notion of the budget set are equivalent. In economic theory the maximization of the preference relation of a consumer on any budget set defines the demand correspondence which at any price vector indicates the preferred vectors of goods and this is one of the fundamental notions of this theory. Contrary to the finite-dimensional economies, in the infinite-dimensional ones, the existence of the demand correspondence is not ensured. In this article we show that in reflexive spaces (and in some other classes of Banach spaces), there are only two classes of closed cones, i.e. cones whose any budget set is bounded and cones whose any budget set is unbounded. Based on this dichotomy result, we prove that in the first category of these cones the demand correspondence exists and that it is upper hemicontinuous. We prove also a characterization of reflexive spaces based on the existence of the demand correspondences. [Copyright &y& Elsevier]

Abstract: Let be a reflexive Banach space with a uniformly Gâteaux differentiable norm, let be a nonempty closed convex subset of , and let be a continuous pseudocontraction which satisfies the weakly inward condition. For any contraction map on , and every nonempty closed convex and bounded subset of having the fixed point property for nonexpansive self-mappings, it is shown that the path , in , defined by is continuous and strongly converges to the fixed point of , which is the unique solution of some co-variational inequality. If, in particular, is a Lipschitz pseudocontractive self-mapping of , it is also shown, under appropriate conditions on the sequences of real numbers , that the iteration process: , , strongly converges to the fixed point of , which is the unique solution of the same co-variational inequality. Our results propose viscosity approximation methods for Lipschitz pseudocontractions. [Copyright &y& Elsevier]

Abstract: In this paper, we consider completely generalized nonlinear quasi-variational-like inclusions in Banach spaces and propose an Ishikawa type iterative algorithm for computing their approximate solutions by applying the new notion of -proximal mapping given in [R. Ahmad, A.H. Siddiqi, Z. Khan, Proximal point algorithm for generalized multi-valued nonlinear quasi-variational-like inclusions in Banach spaces, Appl. Math. Comput. 163 (2005) 295–308]. We prove that the approximate solutions obtained by the proposed algorithm converge to the exact solution of our quasi-variational-like inclusions. The results presented in this paper extend and improve the corresponding results of [R. Ahmad, A.H. Siddiqi, Z. Khan, Proximal point algorithm for generalized multi-valued nonlinear quasi-variational-like inclusions in Banach spaces, Appl. Math. Comput. 163 (2005) 295–308; X.P. Ding, F.Q. Xia, A new class of completely generalized quasi-variational inclusions in Banach spaces, J. Comput. Appl. Math. 147 (2002) 369–383; N.J. Huang, Generalized nonlinear variational inclusions with non-compact valued mappings, Appl. Math. Lett. 9(3) (1996) 25–29; A. Hassouni, A. Moudafi, A perturbed algorithm for variational inclusions, J. Math. Anal. Appl. 185(3) (1994) 706–712]. Some special cases are also discussed. [Copyright &y& Elsevier]

In this paper, we study the class of mixed variational- like inequalities in reflexive Banach spaces. By applying a minimax inequality due to the author, some existence and uniqueness theorems for solutions of mixed variational-like inequalities are proved. Next, by applying the auxiliary problem technique, we suggest an innovative iterative algorithm to compute approximate solutions of the mixed variational-like inequality. Finally, convergence criteria are also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a new class of generalized nonlinear variational-like inequalities in reflexive Banach spaces. By applying the auxiliary principle technique due to Glowinski-Lions-Tremolières (Ref 1) and the minimax inequality due to Ding-Tan (Ref 2), we establish existence and uniqueness theorems for solutions of generalized nonlinear variational-like inequalities; also, we suggest two general algorithms and prove the convergence of the iterative sequences generated by the algorithms. Our results extend, improve, and unify several known results due to Cohen, Ding, Bose, Parida-Sahoo-Kumar, Dien, Ding-Tarafdar, Noor, and others. [ABSTRACT FROM AUTHOR]

Abstract: We construct a family () of reflexive Banach spaces with long (countable as well as uncountable) transfinite bases but with no unconditional basic sequences. The method we introduce to achieve this allows us to considerably control the structure of subspaces of the resulting spaces as well as to precisely describe the corresponding spaces on non-strictly singular operators. For example, for every pair of countable ordinals , we are able to decompose every bounded linear operator from to as the sum of a diagonal operator and an strictly singular operator. We also show that every finite-dimensional subspace of any member of our class can be moved by and ()-isomorphism to essentially any region of any other member or our class. Finally, we find subspaces X of such that the operator space is quite rich but any bounded operator T from X into X is a strictly singular pertubation of a scalar multiple of the identity. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce the group decision problems in a bilevel programming structure, and its corresponding optimization with equilibrium constraints (MPEC for short) problems. For these models, we establish new existence results, which extend some known results to infinite-dimensional space setting. We also establish the links between group decision problems in a bilevel programming structure and its corresponding MPEC problems in Banach spaces. An interesting example about the mean-variance portfolio optimization is given and the existence and uniqueness results for its corresponding MPEC problems are obtained. [Copyright &y& Elsevier]

In this paper, we define a new notion of Jη-proximal mapping for a nonconvex, lower semicontinuous, η-subdifferentiable proper functional in Banach spaces. The existence and Lipschitz continuity of Jη-proximal mapping of a lower semicontinuous, η-subdifferentiable proper functional are proved. By applying this notion, we introduce and study generalized multivalued nonlinear quasi-variational-like inclusions in reflexive Banach spaces and propose a proximal point algorithm for finding the approximate solutions of this class of variational inclusions. The convergence criteria of the iterative sequences generated by our algorithm is discussed. Several special cases are also given. [Copyright &y& Elsevier]

It is shown that every l.s.c. closed-and-convex valued mapping Φ :X→2Y, where X is a hereditarily collectionwise normal and countably paracompact space and Y is a reflexive Banach space has a single-valued continuous selection. [Copyright &y& Elsevier]

Sequences of cubature formulas with a joint countable set of nodes are studied. Each cubature formula under consideration has only a finite number of nonzero weights. We call a sequence of such kind a multicubature formula. For a given reflexive Banach space it is shown that there is a unique optimal multicubature formula and the sequence of the norm of optimal error functionals is monotonically decreasing to 0 as the number of the formula nodes tends to infinity. [ABSTRACT FROM AUTHOR]

Variational inequalities associated with monotone operators (possibly nonlinear and multivalued) and convex sets (possibly unbounded) are studied in reflexive Banach spaces. A variety of results are given which relate to a stability concept involving a natural parameter. These include characterizations useful as criteria for stable existence of solutions and also several characterizations of surjectivity. The monotone complementarity problem is covered as a special case, and the results are sharpened for linear monotone complementarity and for generalized linear programming. [ABSTRACT FROM AUTHOR]

We show that any Banach space X that has a sequence of unit vectors weakly converging to 0 has a closed and bounded convex set that is not remotal. This extends the main result of Sababheh and Khalil, (Numer. Funct. Anal. Optim. 2008; 29:1166-1170). [ABSTRACT FROM AUTHOR]