Compact Convex Values Research Articles

Two Positive Solutions for Elliptic Differential Inclusions

The existence of two positive solutions for an elliptic differential inclusion is established, assuming that the nonlinear term is an upper semicontinuous set-valued mapping with compact convex values having subcritical growth. Our approach is based on variational methods for locally Lipschitz functionals. As a consequence, a multiplicity result for elliptic Dirichlet problems having discontinuous nonlinearities is pointed out.

Stieltjes Differential Inclusions with Periodic Boundary Conditions without Upper Semicontinuity

We are studying first order differential inclusions with periodic boundary conditions where the Stieltjes derivative with respect to a left-continuous non-decreasing function replaces the classical derivative. The involved set-valued mapping is not assumed to have compact and convex values, nor to be upper semicontinuous concerning the second argument everywhere, as in other related works. A condition involving the contingent derivative relative to the non-decreasing function (recently introduced and applied to initial value problems by R.L. Pouso, I.M. Marquez Albes, and J. Rodriguez-Lopez) is imposed on the set where the upper semicontinuity and the assumption to have compact convex values fail. Based on previously obtained results for periodic problems in the single-valued cases, the existence of solutions is proven. It is also pointed out that the solution set is compact in the uniform convergence topology. In particular, the existence results are obtained for periodic impulsive differential inclusions (with multivalued impulsive maps and finite or possibly countable impulsive moments) without upper semicontinuity assumptions on the right-hand side, and also the existence of solutions is derived for dynamic inclusions on time scales with periodic boundary conditions.

Measure Differential Inclusions: Existence Results and Minimum Problems

We focus on a very general problem in the theory of dynamic systems, namely that of studying measure differential inclusions with varying measures. The multifunction on the right hand side has compact non-necessarily convex values in a real Euclidean space and satisfies bounded variation hypotheses with respect to the Pompeiu excess (and not to the Hausdorff-Pompeiu distance, as usually in literature). This is possible due to the use of interesting selection principles for excess bounded variation set-valued mappings. Conditions for the minimization of a generic functional with respect to a family of measures generated by equiregulated left-continuous, nondecreasing functions and to associated solutions of the differential inclusion driven by these measures are deduced, under constraints only on the initial point of the trajectory.

Decompositions of Weakly Compact Valued Integrable Multifunctions

We give a short overview on the decomposition property for integrable multifunctions, i.e., when an “integrable in a certain sense” multifunction can be represented as a sum of one of its integrable selections and a multifunction integrable in a narrower sense. The decomposition theorems are important tools of the theory of multivalued integration since they allow us to see an integrable multifunction as a translation of a multifunction with better properties. Consequently, they provide better characterization of integrable multifunctions under consideration. There is a large literature on it starting from the seminal paper of the authors in 2006, where the property was proved for Henstock integrable multifunctions taking compact convex values in a separable Banach space X. In this paper, we summarize the earlier results, we prove further results and present tables which show the state of art in this topic.

A new algorithm for solving multi-valued variational inequality problems

In this paper, we present a new algorithm for solving multi-valued variational inequality problems, which combines the subgradient extragradient algorithm with inertial algorithm. We prove that the algorithm is globally convergent when the multi-valued mapping is continuous and pseudomonotone with nonempty compact convex values. And the convergence rate of this algorithm is Q-linear convergence.

Conditional nonlinear expectations

Let Ω be a Polish space with Borel σ-field F and countably generated sub σ-field G⊂F. Denote by L(F) the set of all bounded F-upper semianalytic functions from Ω to the reals and by L(G) the subset of G-upper semianalytic functions. Let E(⋅|G):L(F)→L(G) be a sublinear increasing functional which leaves L(G) invariant. It is shown that there exists a G-analytic set-valued mapping PG from Ω to the set of probabilities which are concentrated on atoms of G with compact convex values such that E(X|G)(ω)=supP∈PG(ω)EP[X] if and only if E(⋅|G) is pointwise continuous from below and continuous from above on the continuous functions. Further, given another sublinear increasing functional E(⋅):L(F)→R which leaves the constants invariant, the tower property E(⋅)=E(E(⋅|G)) is characterized via a pasting property of the representing sets of probabilities, and the importance of analytic functions is explained. Finally, it is characterized when a nonlinear version of Fubini’s theorem holds true and when the product of a set of probabilities and a set of kernels is compact.

On the Cauchy problem for a class of differential inclusions with applications

ABSTRACT Our main result is the following: let be a multifunction, and assume that there exists a neglegible subset , satisfying a certain geometrical condition, such that the restriction of F to is bounded, lower semicontinuous with non-empty closed values, and its range belongs to a certain family defined below. Then, there exists a bounded multifunction such that G is upper semicontinuous with non-empty compact convex values, and every generalized solution of is a solution of . Such a result improves a celebrated result by A. Bressan, valid for lower semicontinuous multifunctions. We point out that a multifunction F satisfying our assumptions can fail to be lower semicontinuous even at all points . We derive some existence and qualitative results for the Cauchy problem associated to such a class of multifunctions. As an application, we prove existence and qualitative results for the implicit Cauchy problem , , with f discontinuous in u.

A projection-type method for solving multi-valued variational inequalities and fixed point problems

In this paper, we introduce a projection algorithm for finding a common element of the set Fix(T) of fixed points of a -strict pseudocontraction mapping and the set S of solutions of a multi-valued variational inequality problem without monotonicity. We show that the sequence generated by our method is globally convergent to some common element of Fix(T) and S, provided that the mapping F is continuous with nonempty compact convex values. Preliminary numerical experiments are reported.

A projection-type algorithm for solving generalized mixed variational inequalities

We propose a projection-type algorithm for generalized mixed variational inequality problem in Euclidean space ℝn. We establish the convergence theorem for the proposed algorithm, provided the multi-valued mapping is continuous and f-pseudomonotone with nonempty compact convex values on dom(f), where f: ℝn → ℝU{+∞} is a proper function. The algorithm presented in this paper generalize and improve some known algorithms in literatures. Preliminary computational experience is also reported.

A subgradient extragradient algorithm for solving multi-valued variational inequality

In this paper, we propose a subgradient extragradient method for solving multi-valued variational inequality. It is showed that the method converges globally to a solution of multi-valued variational inequality, provided the multi-valued mapping is pseudomonotone with nonempty compact convex values. Convergence rate is analyzed. Preliminary computational experience is also reported.

Relations among Henstock, McShane and Pettis integrals for multifunctions with compact convex values

Fremlin (Ill J Math 38:471–479, 1994) proved that a Banach space valued function is McShane integrable if and only if it is Henstock and Pettis integrable. In this paper we prove that the result remains valid also in case of multifunctions with compact convex values being subsets of an arbitrary Banach space (see Theorem 3.4). Di Piazza and Musial (Monatsh Math 148:119–126, 2006) proved that if \(X\) is a separable Banach space, then each Henstock integrable multifunction which takes as its values convex compact subsets of \(X\) is a sum of a McShane integrable multifunction and a Henstock integrable function. Here we show that such a decomposition is true also in case of an arbitrary Banach space (see Theorem 3.3). We prove also that Henstock and McShane integrable multifunctions possess Henstock and McShane (respectively) integrable selections (see Theorem 3.1).

Extremal solutions of differential inclusions via Baire category: A dual approach

Let F be a continuous multifunction on Rn with compact convex values. For any vector w, let Fw(x)⊆F(x) be the subset of points which maximize the inner product with w. Call W the set of all continuous functions w:[0,T]↦Rn with the following property: all solutions to the Cauchy problem x˙(t)∈Fw(t)(x(t)), x(0)=0, are also solutions to x˙(t)∈extF(x(t)). We prove that W is residual in C([0,T];Rn).

An algorithm for solving a multi-valued variational inequality

We propose a new extragradient method for solving a multi-valued variational inequality. It is showed that the method converges globally to a solution of the multi-valued variational inequality, provided the multi-valued mapping is continuous with nonempty compact convex values. Preliminary computational experience is also reported. MSC: 47H04; 47H10; 47J20; 47J25

A Projection-Type Method for Multivalued Variational Inequality

We propose a projection-type method for multivalued variational inequality. The iteration sequence generated by the algorithm is proven to be globally convergent to a solution, provided that the multivalued mapping is continuous with nonempty compact convex values. Moreover, we present a necessary and sufficient condition on the nonemptiness of the solution set. Preliminary computational experience is also reported.

Equivariant degree of convex-valued maps applied to set-valued BVP

Abstract An equivariant degree is defined for equivariant completely continuous multivalued vector fields with compact convex values. Then it is applied to obtain a result on existence of solutions to a second order BVP for differential inclusions carrying some symmetries.

Smooth Morse–Lyapunov Functions of Strong Attractors for Differential Inclusions

This paper is concerned with a smooth converse Lyapunov theorem for Morse decompositions of strong attractors of differential inclusion $x'(t)\in F(x(t))$, where $F$ is an upper semicontinuous multivalued mapping on $\mathbb{R}^m$ with compact convex values. Roughly speaking, let there be given a strong attractor $\mathscr{A}$ of the system with attraction basin $\Omega$ and Morse decomposition $\mathcal{M}=\{M_1,\ldots,M_l\}$. We will construct a radially unbounded function $V\in C^\infty(\Omega)$ such that (1) $V$ is constant on each Morse set $M_k$ and (2) $V$ is strictly decreasing along any solution of the system in $\Omega$ outside the Morse sets.

Strict feasibility of pseudomonotone set-valued variational inequalities

In this article, we use degree theory developed in Kien et al. [B.T. Kien, M.-M. Wong, N.C. Wong, and J.C. Yao, Degree theory for generalized variational inequalities and applications, Eur. J. Oper. Res. 193 (2009), pp. 12–22.] to prove a result on the existence of solutions to set-valued variational inequality under a weak coercivity condition, provided that the set-valued mapping is upper semicontinuous with nonempty compact convex values. If the set-valued mapping is pseudomonotone in the sense of Karamardian and upper semicontinuous with nonempty compact convex values, it is shown that the set-valued variational inequality is strictly feasible if and only if its solution set is nonempty and bounded.

A New Projection Algorithm for Generalized Variational Inequality

We propose a new projection algorithm for generalized variational inequality with multivalued mapping. Our method is proven to be globally convergent to a solution of the variational inequality problem, provided that the multivalued mapping is continuous and pseudomonotone with nonempty compact convex values. Preliminary computational experience is also reported.

Exceptional family of elements for generalized variational inequalities

This paper introduces a new concept of exceptional family of elements for a finite-dimensional generalized variational inequality problem. Based on the topological degree theory of set-valued mappings, an alternative theorem is obtained which says that the generalized variational inequality has either a solution or an exceptional family of elements. As an application, we present a sufficient condition to ensure the existence of a solution to the variational inequality. The set-valued mapping is assumed to be upper semicontinuous with nonempty compact convex values.

An algorithm for generalized variational inequality with pseudomonotone mapping

We suggest an algorithm for a variational inequality with multi-valued mapping. The iteration sequence generated by the algorithm is proven to converge to a solution, provided the multi-valued mapping is continuous and pseudomonotone with nonempty compact convex values.