Topological Structure Of Solution Sets Research Articles

Topological Structure of Solution Sets of Fractional Control Delay Problem

This paper is concerned with the existence of a mild solution for the fractional delay control system. Firstly, we will study the control problem. Then, we will deal with the topological structure of the solution set consisting of the compactness and Rσ property. We will derive a mild solution to the above delay control problem by using the Laplace transform method.

Volterra nonautonomous evolution inclusions: topological structure of solution sets and applications

This paper concerns with a class of Volterra nonautonomous evolution inclusions with time delay effect defined on a noncompact interval. It involves a family of (possibly unbounded) operators, which generates a noncompact evolution family. In the framework of Fréchet spaces, the topological structure of the solution set for evolution inclusion is considered. Moreover, we obtain geometric features of the corresponding solution map. Our results extend essentially the existing ones which do not involve Volterra integral in nonlinearity. We finally present a concrete example to illustrate the abstract results.

Topological structure of solution sets for hybrid impulsive fractional differential inclusion in banach algebra

The aim of this paper is to study the topological structure of solution sets of hybrid impulsive fractional differential inclusions on the weighted space of piecewise continuous functions, we apply dhage fixed point theorem for multivalued operator with analysis tools to prove existence result. The paper concludes with an example to illustrate the feasibility of our main result.

A Filippov's Theorem and Topological Structure of Solution Sets for Fractional q-Difference Inclusions

A Filippov's Theorem and Topological Structure of Solution Sets for Fractional q-Difference Inclusions

Existence results on impulsive stochastic semilinear differential inclusions

In this paper, we present some existence results of mild solutions and study the topological structure of solution sets for the following first-order impulsive stochastic semilinear differential inclusions driven by Poisson jumps with periodic boundary conditions.We consider the cases in which the right hand side can be either convex . The results are obtained by using fixed point theorems for multivalued mappings, more precisely, the technique is based on fixed point theorem a nonlinear alternative of Leray-Schauder's fixed point theorem in generalised metric and Banach spaces.

Some New Results for the Sobolev-Type Fractional Order Delay Systems with Noncompact Semigroup

The topological structure of solution sets for the Sobolev-type fractional order delay systems with noncompact semigroup is studied. Based on a fixed point principle for multivalued maps, the existence result is obtained under certain mild conditions. With the help of multivalued analysis tools, the compactness of the solution set is also obtained. Finally, we apply the obtained abstract results to the partial differential inclusions.

Topological structure of solution sets for control problems governed by semilinear fractional impulsive evolution equations with nonlocal conditions

Abstract This article investigates the topological structural of the mild solution set for a control problem monitored by semilinear fractional impulsive evolution equations with nonlocal conditions. The $R_{\delta }$-property of the mild solution set is obtained by applying the measure of noncompactness and a fixed point theorem of condensing maps and a fixed point theorem of nonconvex valued maps. Then this result is applied to prove that the presented control problem has a reachable invariant set under nonlinear perturbations. The obtained results are also applied to characterize the approximate controllability of the presented control problem.

Disconnectedness and unboundedness of the solution sets of monotone vector variational inequalities

In this paper, we investigate the topological structure of solution sets of monotone vector variational inequalities (VVIs). We show that if the weak Pareto solution set of a monotone VVI is disconnected, then each connected component of the set is unbounded. Similarly, this property holds for the proper Pareto solution set. Two open questions on the topological structure of the solution sets of (symmetric) monotone VVIs are raised at the end of the paper.

"Existence and topological structure of solution sets for φ-Laplacian impulsive stochastic differential systems "

In this article, we present results on the existence and the topological struc- ture of the solution set for initial-value problems for the rst-order impulsive dierential equation with innite Brownian motions are proved.The approach is based nonlinear alternative Leary-Schauder type theorem in generalized Banach spaces

Topological structure of solution sets for fractional evolution inclusions of Sobolev type

The paper is devoted to establishing the solvability and topological property of solution sets for the fractional evolution inclusions of Sobolev type. We obtain the existence of mild solutions under the weaker conditions that the semigroup generated by -AE^{-1} is noncompact as well as F is weakly upper semicontinuous with respect to the second variable. On the same conditions, the topological structure of the set of all mild solutions is characterized. More specifically, we prove that the set of all mild solutions is compact and the solution operator is u.s.c. Finally, an example is given to illustrate our abstract results.

Topological structure of solution sets to asymptotic $n$-th order vector boundary value problems

Topological structure of solution sets to asymptotic $n$-th order vector boundary value problems

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 structure of solution set for a class of fractional neutral evolution equations on the half-line

A topological structure of the set of all mild solutions of fractional neutral evolution equations with finite delay on the half-line is investigated. We show that the solution set is an R$_\delta$-set. It is proved on compact intervals by establishing a result on topological structure of fixed point set of Krasnosel'skiĭ type operators. Next, using the inverse limit method, we obtain the same result on the half-line.

Fractional delay control problems: topological structure of solution sets and its applications

This paper deals with the control problems of semilinear fractional delay evolution equations. Under rather mild conditions, we study the topological structure of solution sets (compactness and Rδ-property). Then, the information about the structure is employed to show the invariance of a reachability set of the control problem under nonlinear perturbations. Finally, we present an example to illustrate the feasibility of the abstract results.

Topological structure of solution sets for impulsive differential inclusions in Fréchet spaces

In this paper, we consider the existence of solutions as well as the topological and geometric structure of solution sets for first-order impulsive differential inclusions in some Fréchet spaces. Both the initial and terminal problems are considered. Using ingredients from topology and homology, the topological structures of solution sets (closedness and compactness) as well as some geometric properties (contractibility, acyclicity, A R and R δ ) are investigated. Some of our existence results are obtained via the method of taking the inverse system limit on noncompact intervals.

Topological structure of solution sets to asymptotic boundary value problems

Topological structure is investigated for second-order vector asymptotic boundary value problems. Because of indicated obstructions, the R δ -structure is firstly studied for problems on compact intervals and then, by means of the inverse limit method, on non-compact intervals. The information about the structure is furthermore employed, by virtue of a fixed-point index technique in Fréchet spaces developed by ourselves earlier, for obtaining an existence result for nonlinear asymptotic problems. Some illustrating examples are supplied.

On the existence, uniqueness and topological structure of solution sets to a certain fractional differential equation

The main goal of this paper is to prove two Aronszajn type theorems for some initial value problems formulated in terms of fractional derivatives. Moreover, we are going to establish a theorem on the existence and uniqueness of positive solutions.

Topological structure of solution sets to differential problems in Fréchet spaces

Using projective limit realizations of Fréchet spaces, we study the topological structure of solution sets for set differential equations and differential inclusions in Fréchet spaces. We apply suitable fixed point results for limit maps induced by maps o

Structure of solutions sets and a continuous version of Filippov's theorem for first order impulsive differential inclusions with periodic conditions

In this paper, the authors consider the first-order nonresonance impulsive differential inclusion with periodic conditions y 0 (t) �y(t) 2 F(t,y(t)), a.e. t 2 J\{t1,...,tm}, y(t + ) y(t k ) = Ik(y(t k )), k = 1,2,... ,m, y(0) = y(b), where J = [0,b] and F : J × R n ! P(R n ) is a set-valued map. The functions Ik characterize the jump of the solutions at impulse points tk (k = 1,2,... ,m). The topological structure of solution sets as well as some of their geometric properties (contractibility and R�-sets) are studied. A continuous version of Filippov’s theorem is also proved.

Filippov-Ważewski theorems and structure of solution sets for first order impulsive semilinear functional differential inclusions

In this paper, we first present an impulsive version of Filippov's Theorem for first-order semilinear functional differential inclusions of the form: $$ \cases (y'-Ay) \in F(t,y_t) &\text{a.e. } t\in J\setminus \{t_{1},\ldots,t_{m}\}, \\ y(t^+_{k})-y(t^-_k)=I_{k}(y(t_{k}^{-})) &\text{for } k=1,\ldots,m, \\ y(t)=\phi(t) &\text{for } t\in[-r,0], \endcases $$ where $J=[0,b]$, $A$ is the infinitesimal generator of a $C_0$-semigroup on a separable Banach space $E$ and $F$ is a set-valued map. The functions $I_k$ characterize the jump of the solutions at impulse points $t_k$ ($k=1,\ldots,m$). Then the convexified problem is considered and a Filippov-Wa{\plr z}ewski result is proved. Further to several existence results, the topological structure of solution sets -- closeness and compactness -- is also investigated. Some results from topological fixed point theory together with notions of measure on noncompactness are used. Finally, some geometric properties of solution sets, AR, $R_\delta$-contractibility and acyclicity, corresponding to Aronszajn-Browder-Gupta type results, are obtained.