Order Differential Inclusions Research Articles

Existence of solution of a system of non-linear differential inclusions with non-local, integral boundary conditions via fixed points of hybrid contractions

In the present article, we have introduced the notions of γ-admissibility for the pair of q-ROF set-valued maps and admissible hybrid q-ROF Z\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{Z}$\\end{document}-contraction. Notions introduced in the article generalizes the existing concepts in fuzzy literature. Common fixed point result for a pair of γ-admissible q-ROF mappings in b-metric spaces utilizing the introduced contraction is presented. A nontrivial example to support the obtained results is also included. As an application, we have discussed the existence of solution of system of non-linear n-th order differential inclusions with non-local and integral boundary conditions.

On second order differential inclusion driven by quasi-variational–hemivariational inequalities

In this paper we investigate an evolution problem (SDQHVI) which constitutes of the second order differential inclusion driven by a quasi-variational–hemivariational inequality (QHVI) with perturbation operator in Banach spaces. Unlike the existing literature on differential inclusions, on the one hand, the second order differential operator is not assumed to be compact; on the other hand, the solvability of the inequality is proved under non-coercive condition. More precisely, first, it is shown that the solution set of the QHVI with perturbation operator is nonempty, bounded, closed and convex under the KKM theorem and the Minty formulation. Then, an associated multivalued map with the solution set of the QHVI with perturbation operator is introduced, and we prove that it is upper semicontinuous and measurable. Finally, by utilizing Kakutani–Fan–Glicksberg fixed point theorem, a weak topology technique and properties of strong continuous cosine operators, we prove the existence of mild solutions for SDQHVI.

Sturm–Liouville Differential Inclusions with Set-Valued Reaction Term Depending on a Parameter

In this paper we study the controllability for a Cauchy problem governed by a nonlinear differential inclusion driven by a Sturm–Liouville type operator. In particular, the considered second order differential inclusion involves a set-valued reaction term depending on a parameter. The key tool in the proof of the controllability result we provide is a multivalued version of the theorem recently proved by Haddad–Yarou, here established for an initial conditions problem monitored by a nonlinear second order differential inclusion presenting the sum of two multimaps on the right-hand side. We thereby deduce the existence of a local admissible pair for the considered control problem, that is the existence of a couple of functions consisting of a control, which is a measurable function, and the correspondent trajectory, which is an absolutely continuous function with absolutely continuous derivative. Secondly, under appropriate assumptions on the involved multimaps, we obtain an increased regularity for the solutions produced by our existence result. This regularity is the same of that recently tested by Bonanno, Iannizzotto and Marras for a different type of problem, which however involves the Sturm–Liouville operator.

Further study on second order nonlocal problems monitored by an operator

In this note we prove the existence of mild solutions for nonlocal problems governed by semilinear second order differential inclusions which involves a nonlinear term driven by an operator. A first result is obtained in suitable Banach spaces in the lack of compactness both on the fundamental operator, generated by the linear part, and on the nonlinear multivalued term. This purpose is achieved by combining a fixed point theorem, a selection theorem and a containment theorem. Further we provide another existence result in reflexive spaces by using the classical Hahn–Banach theorem and a new selection proposition, proved here, for a multimap guided by an operator. This setting allows us to remove some assumptions required in the previous existence theorem. As a consequence of this last result we obtain the controllability of a problem driven by a wave equation on which an appropriate perturbation acts.

On convexity of reachable sets of second order differential inclusions

In control theory, there is growing interest in the evolution of sets, especially integral funnels and reachable sets at a certain time. In this paper, we establish sufficient conditions for the convexity of reachable sets for an object whose behavior is described by the second-order differential inclusions. This fact is proved using the concavity of the Hamilton function in the first argument. Further, in connection with the usefulness of the Hamilton function, some of its properties, such as continuity and Lipschitz property, are investigated. At the end of the article, the results obtained are demonstrated with some examples.

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.

Boundary Value Problems for ψ-Hilfer Type Sequential Fractional Differential Equations and Inclusions with Integral Multi-Point Boundary Conditions

In the present article, we study a new class of sequential boundary value problems of fractional order differential equations and inclusions involving ψ-Hilfer fractional derivatives, supplemented with integral multi-point boundary conditions. The main results are obtained by employing tools from fixed point theory. Thus, in the single-valued case, the existence of a unique solution is proved by using the classical Banach fixed point theorem while an existence result is established via Krasnosel’skiĭ’s fixed point theorem. The Leray–Schauder nonlinear alternative for multi-valued maps is the basic tool to prove an existence result in the multi-valued case. Finally, our results are well illustrated by numerical examples.

Optimization of boundary value problems for higher order differential inclusions and duality

The paper is mainly devoted to the theory of duality of boundary value problems (BVPs) for differential inclusions of higher orders. For this, on the basis of the apparatus of locally conjugate mappings in the form of Euler–Lagrange-type inclusions and transversality conditions, sufficient optimality conditions are obtained. Wherein remarkable is the fact that inclusions of Euler–Lagrange type for prime and dual problems are “duality relations”. To demonstrate this approach, the optimization of some third-order semilinear BVPs and polyhedral fourth-order BVPs is considered. These problems show that sufficient conditions and dual problems can be easily established for problems of any order.

On nonlocal problems for semilinear second order differential inclusions without compactness

Existence of mild solutions for a nonlocal abstract problem driven by a semilinear second order differential inclusion is studied in Banach spaces in the lack of compactness both on the fundamental system generated by the linear part and on the nonlinear multivalued term. The method used for proving our existence theorems is based on the combination of a fixed point theorem and a selection theorem developed by ourselves with an approach that uses De Blasi measure of noncompactness and the weak topology. As application of our existence result we present the study of the controllability of a problem guided by a wave equation.

On bounded solutions of semilinear fractional order differential inclusions in Hilbert spaces

On bounded solutions of semilinear fractional order differential inclusions in Hilbert spaces

Non-autonomous Second Order Differential Inclusions with a Stabilizing Effect

In this paper we prove the existence of mild solutions for a problem governed by a semilinear non-autonomous second order differential inclusion where a stabilization of the solution is expected due to the control of the reaction term. In order to obtain our existence theorem, first we study a more general problem with a differential inclusion which involves a perturbation guided by an operator N :I rightarrow C(C(I;X);X), where X is a Banach space. Finally we show an illustrative example of application of our results to a problem involving a wave equation.

Fractional order differential inclusions on an unbounded domain with infinite delay

We provide sufficient conditions for the existence of solutions to initial value problems, for partial hyperbolic differential inclusions of fractional order involving Caputo fractional derivative with infinite delay by applying the nonlinear alternative of Frigon type for multivalued admissible contraction in Frechet spaces.

Optimal control of higher order viable differential inclusions and duality

The paper is devoted to the duality of the Mayer problem for -th order viable differential inclusions with endpoint constraints, where is an arbitrary natural number. Thus, this paper for constructing the dual problems to viable differential inclusions of any order with endpoint constraints can make a great contribution to the modern development of optimal control theory. For this, using locally conjugate mappings in the form of Euler–Lagrange type inclusions and transversality conditions, sufficient optimality conditions are obtained. It is noteworthy that the Euler–Lagrange type inclusions for both primary and dual problems are ‘duality relations’. To demonstrate this approach, some semilinear problems and polyhedral optimization with fourth order differential inclusions are considered. These problems show that sufficient conditions and dual problems can be easily established for problems of a reasonable order.

Optimal control of higher order differential inclusions with functional constraints

The present paper studies the Mayer problem with higher order evolution differential inclusions and functional constraints of optimal control theory (PFC); to this end first we use an interesting auxiliary problem with second order discrete-time and discrete approximate inclusions (PFD). Are proved necessary and sufficient conditions incorporating the Euler–Lagrange inclusion, the Hamiltonian inclusion, the transversality and complementary slackness conditions. The basic concept of obtaining optimal conditions is locally adjoint mappings and equivalence results. Then combining these results and passing to the limit in the discrete approximations we establish new sufficient optimality conditions for second order continuous-time evolution inclusions. This approach and results make a bridge between optimal control problem with higher order differential inclusion (PFC) and constrained mathematical programming problems in finite-dimensional spaces. Formulation of the transversality and complementary slackness conditions for second order differential inclusions play a substantial role in the next investigations without which it is hardly ever possible to get any optimality conditions; consequently, these results are generalized to the problem with an arbitrary higher order differential inclusion. Furthermore, application of these results is demonstrated by solving some semilinear problem with second and third order differential inclusions.

Optimality conditions for higher order polyhedral discrete and differential inclusions

The problems considered in this paper are described in polyhedral multi-valued mappings for higher order(s-th) discrete (PDSIs) and differential inclusions (PDFIs). The present paper focuses on the necessary and sufficient conditions of optimality for optimization of these problems. By converting the PDSIs problem into a geometric constraint problem, we formulate the necessary and sufficient conditions of optimality for a convex minimization problem with linear inequality constraints. Then, in terms of the Euler-Lagrange type PDSIs and the specially formulated transversality conditions, we are able to obtain conditions of optimality for the PDSIs. In order to obtain the necessary and sufficient conditions of optimality for the discrete-approximation problem PDSIs, we reduce this problem to the form of a problem with higher order discrete inclusions. Finally, by formally passing to the limit, we establish the sufficient conditions of optimality for the problem with higher order PDFIs. Numerical approach is developed to solve a polyhedral problem with second order polyhedral discrete inclusions.

Corrigendum to: Positive solutions for a fourth order differential inclusion with boundary values, published in Differential Equations and Applications Vol. 8 No. 1 (2016), 21-31, by John S. Spraker

Corrigendum to: Positive solutions for a fourth order differential inclusion with boundary values, published in Differential Equations and Applications Vol. 8 No. 1 (2016), 21-31, by John S. Spraker

Finite-Time Synchronization of Memristor-Based Fractional Order Cohen-Grossberg Neural Networks

This paper is concerned with the finite-time synchronization (FTS) of memristor-based fractional order Cohen-Grossberg neural networks (MFCGNNs) with time-varying delays. Under the frame of fractional order differential inclusion and set-valued map, some new sufficient conditions to guarantee the FTS of MFCGNNs are established by means of constructing two different Lyapunov functions based on L1-norm in Theorem 1 and Lp-norm in Theorem 2. Via applying the asymptotic expansion property of Mittag-Leffler function, we propose a new estimation method of the settling time for synchronization which is less conservative than previous researches. Meanwhile, we deeply discuss the influence factor of settling time for synchronization. Finally, two numerical examples are given to demonstrate the effectiveness of obtained results.

On the convergence of the continuous gradient projection method

We investigate the long time behaviour of the solutions to the first order differential inclusion where is the subgradient of a given convex and continuous function defined on a real Hilbert space , the operator is the orthogonal projection onto a closed, nonempty and convex subset Q of , and is an absolutely continuous function. We establish that if the objective function Φ has at least one minimizer over Q and behaviours, for t large enough, like for some constant then any solution to (the above equation) converges weakly to a minimizer of Φ over Q and satisfies the following fast decay property: where . Moreover, we prove the strong convergence of the solutions under some simple geometrical assumptions on the function Φ.

Stability analysis of fractional order memristive discontinuous neural networks with partial state control

This paper investigates exponential stability of fractional order memristive discontinuous neural networks (FMDNNs). Under the framework of the fractional order Filippov solution and differential inclusion theory, the global existence of the solution for the FMDNNs is studied by a given growth conditions. Based on fractional stability theoryand the properties of the Mittag Leffler function, some new criteria for the stability of FMDNNs are obtained by using effective partial state impulsive control, which only needs to control a small fraction of the states. At every impulsive moment, these states of the trajectory far away from the desired trajectory will be firstly controlled. The relations between stable region and fractional order α, control parameters and control rate are discussed. Finally, a numerical simulation is given to verify the effectiveness of the theoretical analysis.

Positive solutions for a fourth order differential inclusion based on the Euler-Bernoulli equation for a Cantilever beam

Positive solutions for a fourth order differential inclusion based on the Euler-Bernoulli equation for a Cantilever beam