Class Of Inclusions Research Articles

Existence results for Caputo type sequential fractional differential inclusions with nonlocal integral boundary conditions

In this paper, we study a new class of Caputo type sequential fractional differential inclusions with nonlocal Riemann–Liouville fractional integral boundary conditions. The existence of solutions for the given problem is established for the cases of convex and non-convex multivalued maps by using standard fixed point theorems. The obtained results are well illustrated with the aid of examples.

On a Class of Second-Order Differential Inclusions on the Positive Half-Line

Consider in a real Hilbert space H the differential equation (inclusion) (E): p(t)u''(t)+q(t)u'(t) \in Au(t) + f(t) a.e. in (0, \infty) , with the condition (B): u(0) = x \in \overline{D(A)} , where A :D(A)\subset H\rightarrow H is a (possibly set-valued) maximal monotone operator whose range contains 0 ; p, q\in L^{\infty}(0,\infty ) , such that \mathrm{ess} \inf \ p>0 , \frac{q}{p} is differentiable a.e., and \mathrm{ess} \inf \, \big[{(\frac{q}{p})}^2 + 2(\frac{q}{p})^{\prime}\big] >0 . We prove existence of a unique (weak or strong) solution u to (E), (B), satisfying a^{\frac{1}{2}}u \in L^{\infty}(0,\infty ;H) and t^{\frac{1}{2}}a^{\frac{1}{2}}u^{\prime} \in L^2(0,\infty ;H) , where a(t)=\exp{\big( \int_0^t \frac{q}{p}\, d\tau \big) } , showing in particular the behavior of u as t\rightarrow \infty .

Existence and regularity results for doubly nonlinear inclusions with nonmonotone perturbation

This paper deals with a class of nonlinear evolution inclusions arising from enthalpy formulation of heat conduction problems with phase change and nonmonotone source. Using time-discretization technique and monotone operators theory, we show the existence of weak solutions. Then we prove strong convergence of approximate solutions and lift regularity of the weak solutions under appropriate conditions. Moreover, application to physical model described by hemivariational inequality is given.

Boundary value problems for semilinear fuzzy impulsive differential inclusions based on semigroups in Banach spaces

A first-order semilinear fuzzy differential inclusion with fuzzy impulse characteristics and linear boundary conditions is considered in separable Banach spaces. By means of semigroup properties, stacking approach and the fixed point theorem for multivalued map due to Dhage, the existence results for fuzzy solution are established. Some general criteria are presented for a class of semilinear fuzzy differential inclusions including higher dimensional and infinite dimensional uncertain systems, and the conditions for existence of solutions in the results are concise and mild. Also, several examples are given to illustrate the utility and applicability.

New existence and stability results for partial fractional differential inclusions with multiple delay

We discuss the existence of solutions and Ulam's type stability concepts for a class of partial functional fractional differential inclusions with noninstantaneous impulses and a nonconvex valued right hand side in Banach spaces. An example is provided to

Unique pseudo-expectations for $C^{*}$-inclusions

Given an inclusion $\mathcal{D}\subseteq\mathcal{C}$ of unital $C^{*}$-algebras (with common unit), a unital completely positive linear map $\Phi$ of $\mathcal{C}$ into the injective envelope $I(\mathcal{D})$ of $\mathcal{D}$ which extends the inclusion of $\mathcal{D}$ into $I(\mathcal{D})$ is a pseudo-expectation. Pseudo-expectations are generalizations of conditional expectations, but with the advantage that they always exist. The set $\operatorname{PsExp}(\mathcal{C},\mathcal{D})$ of all pseudo-expectations is a convex set, and when $\mathcal{D}$ is Abelian, we prove a Krein–Milman type theorem showing that $\operatorname{PsExp}(\mathcal{C},\mathcal{D})$ can be recovered from its set of extreme points. In general, $\operatorname{PsExp}(\mathcal{C},\mathcal{D})$ is not a singleton. However, there are large and natural classes of inclusions (e.g., when $\mathcal{D}$ is a regular MASA in $\mathcal{C}$) such that there is a unique pseudo-expectation. Uniqueness of the pseudo-expectation typically implies interesting structural properties for the inclusion. For general inclusions of $C^{*}$-algebras with $\mathcal{D}$ Abelian, we give a characterization of the unique pseudo-expectation property in terms of order structure; and when $\mathcal{C}$ is Abelian, we are able to give a topological description of the unique pseudo-expectation property. As applications, we show that if an inclusion $\mathcal{D}\subseteq\mathcal{C}$ has a unique pseudo-expectation $\Phi$ which is also faithful, then the$C^{*}$-envelope of any operator space $\mathcal{X}$ with $\mathcal{D}\subseteq\mathcal{X}\subseteq\mathcal{C}$ is the $C^{*}$-subalgebra of $\mathcal{C}$ generated by $\mathcal{X}$; we also show that for many interesting classes of $C^{*}$-inclusions, having a faithful unique pseudo-expectation implies that $\mathcal{D}$ norms $\mathcal{C}$, although this is not true in general.

Controllability results for a class of fractional semilinear integro-differential inclusions via resolvent operators

In this paper, we consider a class of fractional integro-differential inclusions in Banach spaces. This paper deals with the controllability for fractional integro-differential control systems. First, we establishes a set of sufficient conditions for the controllability of fractional semilinear integro-differential inclusions in Banach spaces via resolvent operators. We use Bohnenblust–Karlin’s fixed point theorem to prove our main results. Further, we extend the result to study the controllability concept with nonlocal conditions. An example is also given to illustrate our main results.

Existence and Ulam Stability for Partial Impulsive Discontinuous Fractional Differential Inclusions in Banach Algebras

In this paper, we investigate some existence and Ulam’s type stability concepts of fixed point inclusions for a class of partial discontinuous fractional-order differential inclusions with impulses in Banach Algebras. Our results are obtained using weakly Picard operators theory.

A note on Clifford-Klein forms

We consider the problem of finding Clifford-Klein forms in a class of homogeneous spaces determined by inclusions of real Lie algebras of a special type which we call strongly regular. This class of inclusions is described in terms of their Satake diagrams. For example, the complexifications of such inclusions contain the class of subalgebras generated by automorphisms of finite order. We show that the condition of strong regularity implies the restriction on the real rank of subalgebras. This in part explains why the known examples of Clifford-Klein forms are rare. We make detailed calculations of some known examples from the point of view of the Satake diagrams.

Existence of local solutions for a class of delayed neural networks with discontinuous activations

In this paper, we study the existence of local solutions for a class of differential inclusions, which right-hand set-valued functions may not be upper semicontinuous. Moreover, we employ our results to obtain the existence of local solutions for a class of neural networks.

Sensitivity of Optimal Solutions to Control Problems for Second Order Evolution Subdifferential Inclusions.

In this paper the sensitivity of optimal solutions to control problems described by second order evolution subdifferential inclusions under perturbations of state relations and of cost functionals is investigated. First we establish a new existence result for a class of such inclusions. Then, based on the theory of sequential Gamma -convergence we recall the abstract scheme concerning convergence of minimal values and minimizers. The abstract scheme works provided we can establish two properties: the Kuratowski convergence of solution sets for the state relations and some complementary Gamma -convergence of the cost functionals. Then these two properties are implemented in the considered case.

Asymptotically periodic solutions to some second-order evolution and difference equations

In this paper, we examine the asymptotic periodicity of the solutions of some second-order differential inclusions on associated with maximal monotone operators in a Hilbert space , whose forcing terms are periodic functions perturbed by functions from . It is worth pointing out that strong solutions do not exist in general, so we need to consider weak solutions for this class of evolution inclusions. Similar second-order difference inclusions are also addressed. Our main results on asymptotic periodicity represent significant extensions of the previous theorems proved by R.E. Bruck (1980) and B. Djafari Rouhani (2012).

On approximate controllability of fractional stochastic neutral integro-differential inclusions with infinite delay

In this paper, the approximate controllability of a class of fractional stochastic neutral integro-differential inclusions with infinite delay in Hilbert spaces is considered. By using the Hölder inequality, the -resolvent operator and fixed point strategy, a new set of necessary and sufficient conditions are formulated which guarantees the approximate controllability of the nonlinear fractional stochastic system. The results are obtained under the assumption that the associated linear system is approximately controllable. Finally, an example is provided to illustrate the proposed theory.

Description of the Attainable Sets of One-Dimensional Differential Inclusions

The role played by the attainable set of a differential inclusion, in the study of dynamic control systems and fuzzy differential equations, is widely acknowledged. A procedure for estimating the attainable set is rather complicated compared to the numerical methods for differential equations. This article addresses an alternative approach, based on an optimal control tool, to obtain a description of the attainable sets of differential inclusions. In particular, we obtain an exact delineation of the attainable set for a large class of nonlinear differential inclusions.

Global attractor for a class of functional differential inclusions with Hille–Yosida operators

We study the dynamics for a class of functional differential inclusions whose linear part generates an integrated semigroup. Some techniques of measure of noncompactness are deployed to prove the global solvability and the existence of a compact global attractor for the m-semiflow generated by our system. The obtained results generalize recent ones in the same direction.

On the ‘bang-bang’ principle for a class of fractional semilinear evolution inclusions

In this paper, we study the problem of the existence of solutions for a class of fractional semilinear evolution inclusions with non-convex right-hand side. We mainly focus on the existence of the extreme solution and the relationship of the solution sets between the original problem and the convexified problem. An example is provided to illustrate the application of the obtained theory.

Controllability for systems governed by semilinear evolution inclusions without compactness

In this paper we study the controllability for a class of semilinear differential inclusions in Banach spaces. Since we assume the regularity of the nonlinear part with respect to the weak topology, we do not require the compactness of the evolution operator generated by the linear part. As well we are not posing any conditions on the multivalued nonlinearity expressed in terms of measures of noncompactness. We are considering the usual assumption on the controllability of the associated linear problem. Notice that, in infinite dimensional spaces, the above mentioned compactness of the evolution operator and linear controllability condition are in contradiction with each other. We suppose that the nonlinear term has convex, closed, bounded values and a weakly sequentially closed graph when restricted to its second argument. This regularity setting allows us to solve controllability problem under various growth conditions. As application, a controllability result for hyperbolic integro-differential equations and inclusions is obtained. In particular, we consider controllability of a system arising in a model of nonlocal spatial population dispersal and a system governed by the second order one-dimensional telegraph equation.

On integro‐differential inclusions with operator‐valued kernels

AbstractWe study integro‐differential inclusions in Hilbert spaces with operator‐valued kernels and give sufficient conditions for the well‐posedness. We show that several types of integro‐differential equations and inclusions are covered by the class of evolutionary inclusions, and we therefore give criteria for the well‐posedness within this framework. As an example, we apply our results to the equations of visco‐elasticity and to a class of nonlinear integro‐differential inclusions describing phase transition phenomena in materials with memory. Copyright © 2014 John Wiley & Sons, Ltd.

Anti-periodic extremal problems for a class of nonlinear evolution inclusions in R N

The aim of this paper is to establish the existence of anti-periodic solutions to the following nonlinear anti-periodic problem: a.e. , , in where denotes the extremal point set of the multifunction , and is a nonlinear map from to . Sufficient conditions for the existence of extremal solutions are presented. Also, we prove that the extremal point set of this problem is compact in and dense in the solution set of nonlinear evolution problems with a convex valued perturbation which is multivalued. We apply our results on the control system with a priori feedback.

Necessary and sufficient condition for multistability of neural networks evolving on a closed hypercube

The paper considers nonsmooth neural networks described by a class of differential inclusions termed differential variational inequalities (DVIs). The DVIs include the relevant class of neural networks, introduced by Li, Michel and Porod, described by linear systems evolving in a closed hypercube of Rn. The main result in the paper is a necessary and sufficient condition for multistability of DVIs with nonsymmetric and cooperative (nonnegative) interconnections between neurons. The condition is easily checkable and provides a sharp bound between DVIs that can store multiple patterns, as asymptotically stable equilibria, and those for which this is not possible. Numerical examples and simulations are presented to confirm and illustrate the theoretic findings.