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

Under certain conditions a comparison is made between solutions of a pair of initial value problems in a Banach space. This comparison includes and unifies several recent results on differential inequalities in Banach spaces.

This paper studies fractional differential equations (FDEs) with mixed fractional derivatives. Existence, uniqueness, stability, and asymptotic results are derived.

We consider an extension of the notion of well‐posedness by perturbations, introduced by Zolezzi (1995, 1996) for a minimization problem, to a class of generalized mixed variational inequalities in Banach spaces, which includes as a special case the class of mixed variational inequalities. We establish some metric characterizations of the well‐posedness by perturbations. On the other hand, it is also proven that, under suitable conditions, the well‐posedness by perturbations of a generalized mixed variational inequality is equivalent to the well‐posedness by perturbations of the corresponding inclusion problem and corresponding fixed point problem. Furthermore, we derive some conditions under which the well‐posedness by perturbations of a generalized mixed variational inequality is equivalent to the existence and uniqueness of its solution.

The BCQ and the Abadie CQ for infinite systems of convex inequalities in Banach spaces are characterized in terms of the upper semi–continuity of the convex cones generated by the subdifferentials of active convex functions. Some relationships with other constraint qualifications such as the CPLV and the Slate condition are also studied. Applications in best approximation theory are provided.

Motivated and inspired by Korpelevich′s and Noor′s extragradient methods, we suggest an extragradient method by using the sunny nonexpansive retraction which has strong convergence for solving the generalized variational inequalities in Banach spaces.

This paper aims to establish the Tikhonov regularization method for generalized mixed variational inequalities in Banach spaces. For this purpose, we firstly prove a very general existence result for generalized mixed variational inequalities, provided that the mapping involved has the so-called mixed variational inequality property and satisfies a rather weak coercivity condition. Finally, we establish the Tikhonov regularization method for generalized mixed variational inequalities. Our findings extended the results for the generalized variational inequality problem (for short, GVIP(F, K)) in \(R^n\) spaces (He in Abstr Appl Anal, 2012) to the generalized mixed variational inequality problem (for short, GMVIP\((F,\phi , K)\)) in reflexive Banach spaces. On the other hand, we generalized the corresponding results for the generalized mixed variational inequality problem (for short, GMVIP\((F,\phi ,K)\)) in \(R^n\) spaces (Fu and He in J Sichuan Norm Univ (Nat Sci) 37:12–17, 2014) to reflexive Banach spaces.

Analysis of an adhesive contact problem for viscoelastic materials with long memory

The Green’s function is widely used in solving boundary value problems for differential equations, to which many mathematical and physical problems are reduced. In particular, solutions of partial differential equations by the Fourier method are reduced to boundary value problems for ordinary differential equations. Using the Green's function for a homogeneous problem, one can calculate the solution of an inhomogeneous differential equation. Knowing the Green's function makes it possible to solve a whole class of problems of finding eigenvalues in quantum field theory. The developed method for constructing the Green’s function of boundary value problems for ordinary linear differential equations is described. An algorithm and program for calculating the Green's function of boundary value problems for differential equations of the second and third orders in an explicit analytical form are presented. Examples of computing the Green's function for specific boundary value problems are given. The fundamental system of solutions of ordinary differential equations with singular points needed to construct the Green's function is calculated in the form of generalized power series with the help of the developed programs in the Maple environment. An algorithm is developed for constructing the Green's function in the form of power series for second-order and third-order differential equations with given boundary conditions. Compiled work programs in the Maple environment for calculating the Green functions of arbitrary boundary value problems for differential equations of the second and third orders. Calculations of the Green's function for specific third-order boundary value problems using the developed program are presented. The obtained approximate Green’s function is compared with the known expressions of the exact Green’s function and very good agreement is found

In this paper, we study a class of partial differential variational inequalities. A general stability result for the partial differential variational inequality is provided in the case the perturbed parameters are involved in both the nonlinear mapping and the set of constraints. The main tools are theory of semigroups, theory of monotone operators, and variational inequality techniques.

We study the following generalized additive functional inequality , associated with linear mappings in Banach spaces. Moreover, we prove the Hyers-Ulam-Rassias stability of the above generalized additive functional inequality, associated with linear mappings in Banach spaces.

The main goal of this paper is to analyze and study a new class of dual abstract systems that consists of differential hemivariational inequalities systematized by an evolutionary hemivariational inequality accumulated with a fractional differential equation of order 1<q<2 in a Banach space. The existence of a solution to the mentioned problem is proved by employing Rothe’s method and a surjectivity result. An example is provided to validate the results.

In this paper we give some new existence theorems for nonlinear random equations and inequalities involving operators of monotone type in Banach spaces. A random Hammerstein integral equation is also studied. In order to obtain random solutions we use some results from the existing deterministic theory as well as from the theory of measurable multifunctions and, in particular, the measurable selection theorems of Kuratowski/Ryll‐Nardzewski and of Saint‐Beuve.

This paper studies local and global error bounds for a convex inequality defined by a proper convex function in a Banach space. The concept of weak basic constraint qualification (weak BCQ) is introduced to control the normal directions at a boundary point of the solution set. Local and global error bounds are characterized by a direction‐length decomposition condition, which provides a way to independently verify the weak BCQ and the length control of the subdifferential. To further characterize global error bounds, the segment extension property is proposed and studied. It is shown that the verification of the direction‐length condition for global error bounds can be sufficiently carried out on any subset having the segment extension property instead of the entire boundary. This leads to a simple formula for the smallest global error bound. In the Euclidean space, the verification of the condition and the computation of the smallest global error bound can be carried out on the set of extreme points.

Existence and uniqueness results for boundary value problems of higher order fractional integro-differential equations involving gronwall's inequality in banach spaces

The aim of this paper is to establish existence and boundedness theorems for perturbed variational inequalities defined by a set-valued mapping without any kind of monotonicity in Banach spaces. The first result is shown that if a coercivity condition holds, then the solution set of a variational inequality perturbed along a direction is nonempty and uniformly bounded. Second, by employing the Minty variational inequalities perturbed by a nonlinear mapping without monotonicity, we prove the boundedness result for the corresponding perturbed variational inequalities under a kind of coercivity condition.