Inequalities In Banach Spaces Research Articles

Three novel inertial subgradient extragradient methods for quasi-monotone variational inequalities in Banach spaces

Three novel inertial subgradient extragradient methods for quasi-monotone variational inequalities in Banach spaces

New algorithms for solving pseudo-monotone variational inequalities in Banach spaces

In this paper, we introduce new algorithms for finding a solution of a variational inequality problem involving pseudo-monotone operator which is also a fixed point of a Bregman relatively inexpensive mapping in $p$-uniformly convex and uniformly smooth Banach spaces that are more general than Hilbert spaces. We prove weak and strong convergence theorems for proposed algorithms. Finally, we give some numerical experiments for supporting our main results.

On fuzzy fractional differential inclusion driven by variational–hemivariational inequality in Banach spaces

The aim of this paper is to examine an evolution problem (FFDIVHVI) involving a fuzzy fractional differential inclusion and a variational–hemivariational inequality (VHVI) in Banach spaces. First, we show a uniqueness and existence theorem for VHVI under the theory of monotone operators and the surjectivity theorem. Then, by utilizing fixed point theorem for multivalued contraction mapping and fuzzy set theory, we establish the existence result for FFDIVHVI. In addition, it is proven that the collection of all mild trajectories of FFDIVHVI exhibits compactness. Finally, we illustrate the applicability of the abstract theory by a nonlinear quasistatic thermoelastic frictional contact problem for which we provide existence results.

Global existence and attractivity for Riemann-Liouville fractional semilinear evolution equations involving weakly singular integral inequalities

In this paper, we obtain several results on the global existence, uniqueness and attractivity for fractional evolution equations involving the Riemann-Liouville type by exploiting some results on weakly singular integral inequalities in Banach spaces. Some boundedness conditions of the nonlinear term are considered to obtain the main results that generalize and improve some well-known works.

The Stability of Multi-Coefficients Pexider Additive Functional Inequalities in Banach Spaces

The Hyers–Ulam stability of multi-coefficients Pexider additive functional inequalities in Banach spaces is investigated. In order to do this, the fixed point method and the direct method are used.

Fuzzy fractional delay differential inclusions driven by hemivariational inequalities in Banach spaces

This paper investigates a novel class of nonlinear dynamical fuzzy systems referred to as fuzzy fractional delay differential hemivariational inequalities (FFDDHVIs) in Banach spaces. These systems integrate fractional fuzzy differential inclusions with delay and hemivariational inequalities. The existence theorem for the HVIs is established based on the KKM theorem. Moreover, specific propositions are proved, covering the superpositional measurability and the upper semicontinuity for the HVIs. Next, by using the fixed points theorem, we establish the existence and compactness of mild solution sets for the FFDDHVIs under certain mild conditions. Finally, As an illustrative application, we investigate a frictional quasistatic contact problem for viscoelastic materials, in which the friction and contact conditions are described by the Clarke generalized gradient of nonconvex and nonsmooth functionals.

Discrete logarithmic Sobolev inequalities in Banach spaces

AbstractLet be the discrete hypercube equipped with the uniform probability measure . We prove that if is a Banach space of finite cotype and , then every function satisfies the dimension‐free vector‐valued logarithmic Sobolev inequality The finite cotype assumption is necessary for the conclusion to hold. This estimate is the hypercube counterpart of a result of Ledoux (1988) in Gauss space and the optimal vector‐valued version of a deep inequality of Talagrand (1993). As an application, we use such vector‐valued logarithmic Sobolev inequalities to derive new lower bounds for the bi‐Lipschitz distortion of nonlinear quotients of the Hamming cube into Banach spaces with prescribed Rademacher type.

An upper bound for a condition number theorem of variational inequalities

Nonlinear variational inequalities in Banach spaces are considered. A notion of (absolute) condition number with respect to the right-hand side is introduced. A distance among variational inequalities is defined. We prove that the distance to suitably restricted ill-conditioned variational inequalities is bounded from above by a multiple of the reciprocal of the condition number. By using an analogous lower bound of the companion paper [14], we obtain a full condition number theorem for variational inequalities. The particular case of convex optimization problems is also considered. Known results dealing with optimization problems are thereby generalized.

Projected solutions for generalized quasivariational inequalities without compact constraint multimap

In this paper we study the existence of projected solutions for a class of generalized quasivariational inequalities in Banach spaces. Generalized quasivariational inequalities are defined by a principal operator and a constrained set determined by a multivalued map. We obtain existence results without assuming that either the constraint multimap or the principal operator are compact. We present two types of existence results: one is obtained considering the principal operator of class S+ and in the other we assume that the constrained multivalued map is condensing. Furthermore, for both these types of results, first the upper semicontinuity associated to pseudomonotonicity on the principal operator are required, then much weaker hypotheses of regularity and monotonicity, such us the upper sign continuity and the properly quasimonotonicity, are taken into account.

Operator extrapolation method for variational inequalities in Banach spaces

Operator extrapolation method for variational inequalities in Banach spaces

A general differential quasi variational–hemivariational inequality: Well-posedness and application

In the paper we analyze a general differential quasi variational-hemivariational inequality in Banach spaces which couples the Cauchy problem for an ordinary differential equation with a quasi variational-hemivariational inequality considered with a unilateral constraint and history-dependent operators. The latter appear in all of the data: the governing nonlinear operator, the generalized directional derivative of a locally Lipschitz potential, a convex potential, and the ordinary differential equation. The results on the well-posedness and regularity of solutions are proved by exploiting a fixed point approach. The theory is illustrated by an application to a quasistatic generalized Signorini unilateral frictional contact problem for viscoplastic materials with a nonsmooth multivalued contact condition.

On variational–hemivariational inequalities in Banach spaces

This paper is devoted to a well-posedness analysis of elliptic variational–hemivariational inequalities in Banach spaces. The differential operator associated with the variational–hemivariational inequality is assumed to be strongly monotone of a general order, in contrast to that in the majority of existing references on this subject where the differential operator is assumed to be strongly monotone of order 2. Moreover, the solution existence is proved with an approach more accessible to applied mathematicians and engineers, instead of through an abstract surjectivity result for pseudomonotone operators in existing references. Equivalent minimization principles are established for certain variational–hemivariational inequalities, which are valuable for developing efficient numerical algorithms. The theoretical results are applied to the analysis of a mixed hemivariational inequality in the study of a generalized Newtonian fluid flow problem involving a nonsmooth slip boundary condition of friction type. Existence and uniqueness of both the velocity and pressure unknowns are shown for the mixed hemivariational inequalities.

On the Approximation of Some Evolutionary Inclusions in Banach Spaces

This paper deals with an original application of method of finite difference approximations for evolutionary variational inequalities in Banach spaces. The main result of the study is the suggestion of a procedure and the demonstration of a theorem that, with a different approach, respect to the well known methods present in literature, proves some properties of the solutions of evolutionary variational inequalities.

Second order evolutionary problems driven by mixed quasi-variational–hemivariational inequalities

In this paper we introduce a differential quasi-variational inequality which consists of a second order partial differential equation involving history-dependent operators and a mixed quasi-variational–hemivariational inequality in Banach spaces. At first, by using the KKM theorem and monotonicity arguments, we show that the solution set of the mixed quasi-variational–hemivariational inequality is nonempty, bounded, closed and convex. Then, we establish the measurability and upper semicontinuity of the solution set with respect to the time variable and state variable. Finally, based on the theory of strongly continuous cosine operators and a fixed point theorem for condensing set-valued operators, we build the existence of mild solutions for the differential quasi-variational–hemivariational inequality.

Strong convergent inertial Tseng’s extragradient method for solving non-Lipschitz quasimonotone variational inequalities in Banach spaces

Strong convergent inertial Tseng’s extragradient method for solving non-Lipschitz quasimonotone variational inequalities in Banach spaces

Boundary value problems of higher order fractional integro-differential equations involving Gronwall's inequality in Banach spaces

Boundary value problems of higher order fractional integro-differential equations involving Gronwall's inequality in Banach spaces

Stability of a new additive functional inequality in Banach spaces

Stability of a new additive functional inequality in Banach spaces

A study of nonlocal fractional delay differential equations with hemivariational inequality

<abstract><p>In this paper, we study an abstract system of fractional delay differential equations of order $ 1 < q < 2 $ with a hemivariational inequality in Banach spaces. To establish the existence of a solution to the abstract inequality, we employ the Rothe technique in conjunction with the surjectivity of multivalued pseudomonotone operators and features of the Clarke generalized gradient. Further, to show the existence of the fractional differential equation, we use the fractional cosine family and fixed point theorem. Finally, we include an example to elaborate the effectiveness of the findings.</p></abstract>

Aspects of the Kahane–Salem–Zygmund inequalities in Banach spaces

The main aim of this work is to discuss several different approaches to the celebrated Kahane–Salem–Zygmund inequalities. In particular, we prove estimates for exponential Orlicz norms of averages sup _{1le jle N}Big |sum _{i=1}^K a_i(j) gamma _iBig |,, where (a_i(j)) in ell _infty ^N, , 1 le i le K and the (gamma _i) form a sequence of real or complex subgaussian random variables. Lifting these inequalities to finite dimensional Banach spaces, we get some new Kahane–Salem–Zygmund type inequalities—in particular, for spaces of subgaussian random polynomials and multilinear forms on finite dimensional Banach spaces, and also for subgaussian random Dirichlet polynomials.

A new class of differential quasivariational inequalities with an application to a quasistatic viscoelastic frictional contact problem

The overarching goal of this paper is to introduce and investigate a new nonlinear system driven by a nonlinear differential equation, a history-dependent quasivariational inequality and a parabolic variational inequality in Banach spaces. Such a system can be used to model quasistatic frictional contact problems for viscoelastic materials with long memory, damage and wear. By using the Banach fixed point theorem, we prove an existence and uniqueness theorem of solution for such a system under some mild conditions. As a novel application, we obtain a unique solvability of a quasistatic viscoelastic frictional contact problem with long memory, damage and wear.