Topological Properties Of Solution Sets Research Articles

Topological Properties of Solution Sets for τ-Fractional Non-Instantaneous Impulsive Semi-Linear Differential Inclusions with Infinite Delay

The knowledge of fractional calculus can be useful in developing models that allow us to better understand and manage some phenomena. In the present paper, we study the topological structure of the mild solution set for a semi-linear differential inclusion containing the τ-Caputo fractional derivative in the presence of non-instantaneous impulses and an infinite delay. We demonstrate that this set is non-empty and an Rδ-set. We use a recent result regarding the existence of solutions for τ-Caputo fractional semi-linear differential inclusions. Unlike many results, we do not suppose that the generating semigroup is compact. An illustrative example is given as an application of our results.

Some topological properties of solution sets in partially ordered set optimization

Some topological properties of solution sets in partially ordered set optimization

On topological properties of solution sets of semilinear fractional differential inclusions with non-convex right-hand side

On topological properties of solution sets of semilinear fractional differential inclusions with non-convex right-hand side

Topological properties of solution sets for nonlinear evolution hemivariational inequalities and applications

The purpose of this paper is to investigate the topological properties of solution sets for a class of nonlinear evolution hemivariational inequalities. We firstly obtain the nonemptiness and the compactness of the solution set for hemivariational inequalities by applying the Kakutani–KyFan fixed point theorem, Gronwall’s inequality and the multivalued analysis. Then by using Hyman’s theorem, we show that the solution set for presented problem is an Rδ set. Finally, some applications to infinite dimensional control systems are given.

Topological properties of solution sets for Riemann-Liouville fractional nonlocal delay control systems with noncompact semigroups and applications to approximate controllability

This paper investigates the topological properties of the mild solution set for a control system monitored by Riemann-Liouville fractional nonlocal delay evolution equations with noncompact semigroups. We first obtain the Rδ-property of the mild solution set by employing the method of the Hausdorff measure of noncompactness. Then, we apply the obtained result to show that the presented control problem admits a reachable invariant set under nonlinear perturbations. Furthermore, we apply the obtained result to characterize the approximate controllability of the presented control system. Finally, we provide an example to illustrate the application of the abstract results.

Topological Properties of Solution Sets for Hilfer Fractional Nonlocal Delay Control Systems and Applications

The aim of this paper is to investigate the topological properties of mild solution sets for a control system described by Hilfer fractional delay evolution equations with nonlocal initial conditions. We firstly obtain the nonemptiness, the compactness and -property of the mild solution set by applying Schauder’s fixed point theorem, a fixed point theorem of nonconvex valued maps and the fractional calculus. Then we apply this obtained result to show that the presented control problem has a reachable invariant set under nonlinear perturbations. Furthermore, we also apply the obtained results to characterize the approximate controllability of the presented control problem. Finally, we present an example to illustrate the application of abstract results.

Topological properties of solution sets for Sobolev-type fractional stochastic differential inclusions with Poisson jumps

ABSTRACTIn this paper, we investigate the topological properties of solution sets for Sobolev-type fractional stochastic differential inclusions with Poisson jumps in Caputo and Riemann–Liouville fractional derivatives of order , respectively. We show that the solution set is nonempty, compact and an -set under some suitable conditions, which implies that the solution set may not be a singleton, but in the point of view of algebraic topology, it is equivalent to a point in the sense that it has the same homology group as one-point space.

Topological properties of solution sets for stochastic evolution inclusions

ABSTRACTIn this paper, we investigate the topological structure of solution sets for stochastic evolution inclusions in Hilbert spaces when the semigroup is compact as well as noncompact. It is shown that the solution set is nonempty, compact, and an Rδ-set, which means that the solution set may not be a singleton but, from the point of view of algebraic topology, it is equivalent to a point, in the sense that it has the same homology group as one-point space. As applications of the obtained results, an example is given.

Topological properties of solution sets of fractional stochastic evolution inclusions

In this paper, we investigate the topological structure for the solution set of Caputo type neutral fractional stochastic evolution inclusions in Hilbert spaces. We introduce the concept of mild solutions for fractional neutral stochastic inclusions and show that the solution set is nonempty compact and R_{delta}-set, which means that the solution set may not be a singleton but, from the point of view of algebraic topology, it is equivalent to a point in the sense that it has the same homology group as one-point space. Finally, we illustrate the obtained theory with the aid of an example.

Topological properties of solution sets for partial functional evolution inclusions

This paper deals with functional evolution inclusions of neutral type in Banach space when the semigroup is compact as well as noncompact. The topological properties of the solution set is investigated. It is shown that the solution set is nonempty, compact and an Rδ-set which means that the solution set may not be a singleton but, from the point of view of algebraic topology, it is equivalent to a point, in the sense that it has the same homology group as one-point space. As a sample of application, we consider a partial differential inclusion at end of the paper.

On topological properties of solution sets of non Lipschitzian quantum stochastic differential inclusions

In this paper, we establish results on continuous mappings of the space of the matrix elements of an arbitrary nonempty set of pseudo solutions of non Lipschitz quantum Stochastic differential inclusion (QSDI) into the space of the matrix elements of its solutions. we show that under the non Lipschitz condition, the space of the matrix elements of solutions is still an absolute retract, contractible, locally and integrally connected in an arbitrary dimension. The results here generalize existing results in the literature.

THE EXISTENCE OF SOLUTIONS AND WELL-POSEDNESS FOR BILEVEL MIXED EQUILIBRIUM PROBLEMS IN BANACH SPACES

In this paper, a new class of bilevel mixed equilibrium problems (for short, (BMEP)) is introduced and investigated in reflexive Banach space and some topological properties of solution sets for the lower level mixed equilibrium problem and the problem (BMEP) are established without coercivity. Subsequently, we construct a new iterative algorithm which can directly compute some solutions of the problem (BMEP). Some strong convergence theorems of the sequence generated by the proposed algorithm are also presented. Finally, the well-posedness and generalized well-posedness for the problem (BMEP) are introduced by an $\epsilon$-bilevel mixed equilibrium problem. Also, we explore the sufficient and necessary conditions for (generalized) well-posedness of the problem (BMEP) and show that, under some suitable conditions, the well-posedness and generalized well-posedness of (BMEP) are equivalent to the uniqueness and existence of its solutions, respectively. These results are new and improve some recent results in this field.

Some Topological Properties of Solution Sets in a Second Order Differential Inclusion with m-point Boundary Conditions

We consider a class of m-point (m > 3) second order boundary value problem (\(\cal{P}_{F}\)) in a separable Banach space E of the form Open image in new window where F is a closed valued mapping. Under suitable compactness conditions on F we prove that the \(W ^{2, 1}_E([0, 1])\)-solutions of (\(\cal{P}_{F}\)) is compact and is a retract in \(C^1_E([0, 1])\). A new existence result of \(W ^{2, 1}_E([0, 1])\)-solutions and a related relaxation problem are also provided, here F is no longer bounded.

On the aronszajn property for a differential equation of fractional order in Banach spaces

Abstract In this paper we investigate some topological properties of solution sets of a differential equation of fractional order in Banach spaces. Our assumptions and proofs are expressed in terms of measures of noncompactness.

Fractional functional differential equations with causal operators in Banach spaces

In this paper, we study the fractional functional differential equations with causal operators in an arbitrary separable Banach space. By means of the techniques of the measure of noncompactness, the existence and continuation of solutions are given. Further, some topological properties of solution sets are discussed and the existence of optimal solutions of the associated control problem are presented. An example is given to illustrate the result.

On the Aronszajn Property for an Implicit Differential Equation of Fractional Order

In this paper we investigate some topological properties of solution sets of an implicit differential equation of fractional order in Banach spaces.

On topological properties of solution sets for differential equations in locally convex spaces

On topological properties of solution sets for differential equations in locally convex spaces