Compactness Of Solution Set Research Articles

Fractional elliptic obstacle systems with multivalued terms and nonlocal operators

In this paper, we introduce and study a fractional elliptic obstacle system, which is composed of two elliptic inclusions with fractional (pi, qi)-Laplace operators, nonlocal functions, and multivalued terms. The weak solution of fractional elliptic obstacle system is formulated by a fully nonlinear coupled system driven by two nonlinear and nonmonotone variational inequalities with constraints. The nonemptiness and compactness of solution set in the weak sense are proved via employing a surjectivity theorem to the multivalued operators formulated by the sum of a multivalued pseudomonotone operator and a maximal monotone operator.

Existence and Compactness Results for a System of Fractional Differential Equations

The existence and uniqueness, boundedness, and continuous dependence of solutions for fractional differential equations with Caputo fractional derivative is proven by Perov’s fixed point theorem in vector Banach spaces. We study the existence and compactness of solution sets and the u.s.c. of operator solutions.

Existence, compactness, and connectedness of solution sets to nonlocal and impulsive Cauchy problems

In this paper, by using some well‐known inequalities and Schaefer fixed point theorem, we show existence results for impulsive Cauchy problems with nonlocal conditions. The compactness of solution sets can be shown in some certain conditions. Moreover, connectedness of solution sets to impulsive Cauchy problems is shown.

Positive solutions for nonlinear singular elliptic equations of p-Laplacian type with dependence on the gradient

In this paper, we study a nonlinear Dirichlet problem of p-Laplacian type with combined effects of nonlinear singular and convection terms. An existence theorem for positive solutions is established as well as the compactness of solution set. Our approach is based on Leray–Schauder alternative principle, method of sub-supersolution, nonlinear regularity, truncation techniques, and set-valued analysis.

Existence and compactness of solutions to impulsive differential equations with nonlocal conditions

The new type of nonlinear integral inequalities of Volterra–Fredholm type for discontinuous functions is investigated. Then, by using these inequalities and Schaefer fixed‐point theorem, we present new existence results for impulsive semilinear differential equations with nonlocal conditions. Moreover, the compactness of solution sets can be shown in some certain conditions. Copyright © 2015 John Wiley & Sons, Ltd.

Generalized strong vector quasi-equilibrium problem with domination structure

In this paper, a generalized strong vector quasi-equilibrium problem with set-valued mapping and domination structure is presented, and under the condition of cone-continuity, existence of solutions and the compactness of solution set are established. As applications, existence of solutions for strong vector F-implicit variational inequality problem and strong vector F-implicit complementarity problem is derived, and Khan's main results are corrected.

Characterization of the Nonemptiness and Weak Compactness of Solution Sets of Nonconvex Vector Optimization Problems

Characterization of the Nonemptiness and Weak Compactness of Solution Sets of Nonconvex Vector Optimization Problems

A Filippov’s Theorem, Some Existence Results and the Compactness of Solution Sets of Impulsive Fractional Order Differential Inclusions

In this paper, we first present an impulsive version of Filippov's Theorem for fractional differential inclusions of the form, $$\begin{array}{lll} \quad \qquad D^{\alpha}_{*}y(t) & \in & F(t, y(t)), \quad\; {\rm a.e.}\ t\, \in \, J{\backslash} \{t_{1}, \ldots, t_{m}\}, \ \alpha\, \in \, (0,1], \\ y(t^{+}_{k}) - y(t^{-}_{k}) & = & I_{k}(y(t^{-}_{k})), \quad k = 1, \ldots, m, \\ \qquad \qquad y(0) & = & a,\end{array}$$ where J = [0, b], \({D^{\alpha}_{*}}\) denotes the Caputo fractional derivative and F is a set-valued map. The functions Ikcharacterize the jump of the solutions at impulse points tk (\({k = 1, \ldots , m}\)). In addition, several existence results are established, under both convexity and nonconvexity conditions on the multivalued right-hand side. The proofs rely on a nonlinear alternative of Leray-Schauder type and on Covitz and Nadler’s fixed point theorem for multivalued contractions. The compactness of solution sets is also investigated.

Some existence results and properties of solutions in quasilinear variational inequalities with general growths

This paper is about the existence and some properties of solutions of the boundary value problem: $$ \left\{ \begin{gathered} - div(a(|\nabla u|)\nabla u) = f(x,u)in\Omega \hfill \\ u = 0on\partial \Omega , \hfill \\ \end{gathered} \right. $$ with the principal term a(|∇u|)∇u having general growth. In the non-coercive case, a sub-supersolution approach is applied to get existence and enclosure results. Other properties such as compactness of solution sets and existence of extremal solutions are also derived.

Characterization of the Nonemptiness and Compactness of Solution Sets in Convex and Nonconvex Vector Optimization

As a consequence of an abstract theorem proved elsewhere, a vector Weierstrass theorem for the existence of a weakly efficient solution without any convexity assumption is established. By using the notion (recently introduced in an earlier paper) of semistrict quasiconvexity for vector functions and assuming additional structure on the space, new existence results encompassing many results appearing in the literature are derived. Also, when the cone defining the preference relation satisfies some mild assumptions (but including the polyhedral and icecream cones), various characterizations for the nonemptiness and compactness of the weakly efficient solution set to convex vector optimization problems are given. Similar results for a class of nonconvex problems on the real line are established as well.

THE CONNECTIVITY AND COMPACTNESS OF SOLUTION SET OF AN INTEGRAL EQUATION AND WEAK SOLUTION SET OF AN INITIAL-BOUNDARY VALUE PROBLEM

THE CONNECTIVITY AND COMPACTNESS OF SOLUTION SET OF AN INTEGRAL EQUATION AND WEAK SOLUTION SET OF AN INITIAL-BOUNDARY VALUE PROBLEM

Weak Compactness of Solution Sets to Stochastic Differential Inclusions with Non-Convex Right-Hand Sides

Necessary and sufficient conditions for the existence of weak solutions to stochastic differential inclusions with non-convex right-hand sides are given. The main results of the paper deal with the weak compactness of solution sets with respect to the convergence in distribution. The paper is related to author's paper [6], where the convex valued case is considered.

Characterizations of the Nonemptiness and Compactness of Solution Sets in Convex Vector Optimization

In this paper, various necessary and sufficient conditions are given for the nonemptiness and compactness of the weakly efficient solution set of a convex vector optimization problem.