Nonsequential approximate solutions in abstract problems of attainability

The problem of constructing attraction sets in a topological space is considered in the case when the choice of the asymptotic version of the solution is subject to constraints in the form of a nonempty family of sets. Each of these sets must contain an “almost entire” solution (for example, all elements of the sequence, starting from some number, when solution-sequences are used). In the paper, problems of the structure of the attraction set are investigated. The dependence of attraction sets on the topology and the family determining “asymptotic” constraints is considered. Some issues concerned with the application of Stone-Čech compactification and the Wallman extension are investigated.

     

Extensions of abstract problems of attainability: Nonsequential version

Extension constructions of the problem of attainability in a topological space are studied. The constructions are based on compactification of the whole space of solutions or some of its fragments.     

Representation of attraction elements in abstract attainability problems with asymptotic constraints

We consider an abstract attainability problem with asymptotic constraints in a topological space. We construct an extension in the class of ultrafilters of widely interpreted measurable spaces. We study an example of a static problem on the asymptotic attainability in the class of layer functions.     

Attraction sets in abstract attainability problems: Equivalent representations and basic properties

We consider attainability problems in topological spaces under asymptotic constraints.     

Existence of solutions for generalized variational inequality problems in Banach spaces

In this paper, we prove the existence of solutions of generalized variational inequality for upper semicontinuous multivalued mappings with compact contractible values over compact convex subsets in a reflexive Banach space with a Fréchet differentiable norm. Moreover, we give some conditions that guarantee the existence of solutions of generalized variational inequality for upper semicontinuous multivalued mappings with compact contractible values over unbounded closed convex subsets. The result obtained in this paper improves and extends the recent ones announced by Yu and Yang [J. Yu, H. Yang, Existence of solutions for generalized variational inequality problems, Nonlinear Anal., 71 (2009) e2327-e2330] and many others.     

Christensen measurable solutions of generalized Cauchy functional equations

Aequationes mathematicae -     

Multifunctions on abstract measurable spaces and application to stochastic decision theory

Summary The main results are some very general theorems about measurable multifunctions on abstract measurable spaces with compact values in a separable metric space. It is shown that measurability is equivalent to the existence of a pointwise dense countable family of measurable selectors, and that the intersection of two compact-valued measurable multifunctions is measurable. These results are used to obtain a Filippov type implicit function theorem, and a general theorem concerning the measurability of y(t)=min f({t} × Γ(t)) when f is a real valued function and Γ a compact valued multifunction. An application to stochastic decision theory is given generalizing a result of Benes. The research in this paper was partially supported by University of Kansas General Research Fund Grants 3918-5038 and 3199-5038. Entrata in Redazione il 20 dicembre 1972.     

Existence of solutions of generalized vector equilibrium problems in reflexive Banach spaces

In this paper, a generalized vector equilibrium problem with set-valued maps defined on a reflexive Banach space is considered. By using the recession method, we first give the conditions under which the solution set is non-empty, convex and weakly compact, and then extend it to the strong generalized vector equilibrium problem. This facilitates generalizing and modifying various existence theorems. Furthermore, the topological properties of the solution set are studied and it is shown that the solution set includes some boundary points.     

Strong solutions for differential equations in abstract spaces

Let (E,F) be a locally convex space. We denote the bounded elements of E by . In this paper, we prove that if B_Eb is relatively compact with respect to the F topology and f:I×E_b→E_b is a measurable family of F-continuous maps then for each x₀∈E_b there exists a norm-differentiable, (i.e. differentiable with respect to the ∥·∥_F norm) local solution to the initial valued problem u_t(t)=f(t,u(t)), u(t₀)=x₀. All of this machinery is developed to study the Lipschitz stability of a nonlinear differential equation involving the Hardy-Littlewood maximal operator.     

Elliptic problems in generalized Orlicz-Musielak spaces

We consider a strongly nonlinear monotone elliptic problem in generalized Orlicz-Musielak spaces. We assume neither a ??₂ nor ?₂-condition for an inhomogeneous and anisotropic N-function but assume it to be log-H?lder continuous with respect to x. We show the existence of weak solutions to the zero Dirichlet boundary value problem. Within the proof the L ^??-truncation method is coupled with a special version of the Minty-Browder trick for non-reflexive and non-separable Banach spaces.     

Second order three-point boundary value problems in abstract spaces

In this paper we investigate the existence of solutions of the nonhomogeneous three-point boundaryvalue problem We search for solutions of the above problem in the Banach space of continuous functions C（[O, 1], E） with the Pettis integrability assumptions imposed on $. Some classes of Pettis-integrable functions are described in the paper and exploited in the proofs of main results. We stress on a class of pseudo-solutions of considered problem. Our results extend previous results of the same type for both Bochner and Pettis integrability settings. Similar results are also proved for differential inclusions i.e. when f is a multivalued function.     

Interpolative construction and the generalized cotype of abstract Lorentz spaces

The Banach operator ideals generated by an interpolative construction depending on concave functions are studied. These ideals are described in terms of factorization through abstract interpolation Lorentz spaces. The abstract notion of Rademacher type and cotype for operators between Banach spaces is introduced. It is shown that abstract interpolation Lorentz spaces that appeared in the factorization theorem are of the generalized Rademacher cotype determined by Orlicz sequence spaces.     

Fixed point results for generalized quasicontraction mappings in abstract metric spaces

In this paper, we introduce the concept of generalized quasicontraction mappings in an abstract metric space. By using this concept, we construct an iterative process which converges to a unique fixed point of these mappings. The result presented in this paper generalizes the Banach contraction principle in the setting of metric space and a recent result of Huang–Zhang for contractions. We also validate our main result by an example.     

Fixed point results for generalized quasicontraction mappings in abstract metric spaces

In this paper, we introduce the concept of generalized quasicontraction mappings in an abstract metric space. By using this concept, we construct an iterative process which converges to a unique fixed point of these mappings. The result presented in this paper generalizes the Banach contraction principle in the setting of metric space and a recent result of Huang-Zhang for contractions. We also validate our main result by an example.     

Iterative algorithm of solutions for a system of generalized mixed implicity equilibrium problems in reflexive Banach spaces

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.