Weakly relaxed α-pseudomonotonicity and equilibrium problem in Banach spaces

In this paper, we introduce the weakly relaxed ??-pseudomonotone mapping which generalizes the class of pseudomonotone mappings. Some examples are given which show that the weakly relaxed ??-pseudomonotonicity is a proper generalization of pseudomonotonicity for both scalar and vector valued bi-functions. Using the KKM technique, we establish the existence of solutions of equilibrium problems and vector equilibrium problems with weakly relaxed ??-pseudomonotonicity in the reflexive Banach spaces. The present work extends some corresponding results of the variational-like inequalities (Fang and Huang in J. Optim. Theory Appl. 118:327?C338, 2003; Lee and Lee in J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math. 11: 231?C242, 2004) to equilibrium problems.     

Variational-type inequalities for (η, θ, δ)-pseudomonotone-type set-valued mappings in nonreflexive Banach spaces

In this paper, we introduce (η, θ, δ)-pseudomonotone-type set-valued mappings and consider the existence of solutions to variational-type inequality problems for (η, θ, δ)-pseudomonotone-type set-valued mappings in nonreflexive Banach spaces.     

On the ℂ-differentiability of mappings of Banach spaces

We study the monogeneity conditions for -differentiable mappings of domains of a Banach space and establish criteria of -differentiability of a mapping at a point.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 10, pp. 1336–1342, October, 1994.     

General proximal point algorithm involving η-maximal accretiveness framework in Banach spaces

First, based on η-maximal accretiveness, a generalization to Rockafellar’s theorem (1976) in the context of approximating a solution to a general inclusion problem involving a multivalued η-maximal accretive mapping using the proximal point algorithm in a q-uniformly smooth Banach space setting is considered. Then an application to a minimization problem of a functional is examined. The general framework for η-maximal accretiveness generalizes the general theory of multivalued maximal monotone mappings.      

A Denjoy–Wolff theorem for compact holomorphic mappings in reflexive Banach spaces

Using the Kobayashi distance, we establish a Denjoy–Wolff theorem for compact holomorphic self-mappings of a bounded and strictly convex domain in a complex reflexive Banach space.     

A Banach–Zarecki Theorem for functions with values in Banach spaces

A Banach–Zarecki Theorem for a Banach space-valued function  \(F : [0,1] \rightarrow X\) with compact range is presented. We define the strong absolute continuity ( \(sAC_{||.||_{F}}\) ) and the bounded variation ( \(BV_{||.||_{F}}\) ) of \(F\) with respect to the Minkowski functional \(||.||_{F}\) associated to the closed absolutely convex hull \(C_{F}\) of \(F([0,1])\) . It is proved that \(F\) is \(sAC_{||.||_{F}}\) if and only if \(F\) is \(BV_{||.||_{F}}\) , weak continuous on \([0,1]\) and satisfies the weak property \((N)\) .     

Weighted variational inequalities in non-pivot Hilbert spaces with applications

We introduce variational inequalities defined in non-pivot Hilbert spaces and we show some existence results. Then, we prove regularity results for weighted variational inequalities in non-pivot Hilbert space. These results have been applied to the weighted traffic equilibrium problem. The continuity of the traffic equilibrium solution allows us to present a numerical method to solve the weighted variational inequality that expresses the problem. In particular, we extend the Solodov-Svaiter algorithm to the variational inequalities defined in finite-dimensional non-pivot Hilbert spaces. Then, by means of a interpolation, we construct the solution of the weighted variational inequality defined in a infinite-dimensional space. Moreover, we present a convergence analysis of the method.     

Existence and approximation of fixed points of multivalued generalized α-nonexpansive mappings in Banach spaces

Numerical Algorithms - We consider the theoretical and numerical aspects of the quadrature rules associated with a sequence of polynomials generated by a special RII recurrence relation. We also...     

Geometry of Banach spaces with property β

We prove that every Banach space can be isometrically and 1-complementably embedded into a Banach space which satisfies property β and has the same character of density. Then we show that, nevertheless, property β satisfies a hereditary property. We study strong subdifferentiability of norms with property β to characterize separable polyhedral Banach spaces as those admitting a strongly subdifferentiable β norm. In general, a Banach space with such a norm is polyhedral. Finally, we provide examples of non-reflexive spaces whose usual norm fails property β and yet it can be approximated by norms with this property, namely (L ₁[0,1], ‖·‖1) and (C(K), ‖·‖∗) whereK is a separable Hausdorff compact space To the memory of A. Plans Supported in part by DGICYT grant PB 94-0243 and DGICYT PB 96-0607.     

Existence and algorithm of solutions for general set-valued Noor variational inequalities with relaxed (μ,ν)-cocoercive operators in Hilbert spaces

The purpose of this paper is to suggest and analyze a number of iterative algorithms for solving the generalized set-valued variational inequalities in the sense of Noor in Hilbert spaces. Moreover, we show some relationships between the generalized set-valued variational inequality problem in the sense of Noor and the generalized set-valued Wiener-Hopf equations involving continuous operator. Consequently, by using the equivalence, we also establish some methods for finding the solutions of generalized set-valued Wiener-Hopf equations involving continuous operator. Our results can be viewed as a refinement and improvement of the previously known results for variational inequality theory.     

Approximating fixed points of generalized α-nonexpansive mappings in Banach spaces by new faster iteration process

Numerical Algorithms - In this paper, we introduce a new iterative scheme to approximate fixed point of generalized α-nonexpansive mappings and then, we prove that the proposed iteration...     

On the split common fixed point problem for strict quasi-ϕ-pseudocontractive mappings in Banach spaces

Numerical Algorithms - The purpose of this paper is to propose an algorithm for solving the split common fixed point problem for strict quasi-?-pseudocontractive mappings in Banach spaces. It...     

Hölder type inequalities in Lorentz spaces

We study optimal Hölder type inequalities for the Lorentz spaces L ^p,s (R, μ), in the range 1 < p < ∞, 1 ≤ s ≤ ∞, for both the maximal and the dual norms. These estimates also give sharp results for the corresponding associate norms.     

Grüss type inequalities in normed spaces

Generalizations of the classic Grüss inequality in normed spaces are given. The results are illustrated with some specific examples.     

Moreau’s decomposition in Banach spaces

Moreau’s decomposition is a powerful nonlinear hilbertian analysis tool that has been used in various areas of optimization and applied mathematics. In this paper, it is extended to reflexive Banach spaces and in the context of generalized proximity measures. This extension unifies and significantly improves upon existing results.     

A steepest-descent Krasnosel’skii–Mann algorithm for a class of variational inequalities in Banach spaces

In this paper, in order to solve a variational inequality problem over the fixed point set of a nonexpansive mapping on uniformly smooth or reflexive and strictly convex Banach spaces with a uniformly Gâteaux differentiable norm, we investigate an explicit iteration method, based on the steepest-descent and Krasnosel’skii–Mann algorithms. We also show that some modifications of the last and Halpern-type algorithms are special cases of our result.