Differential Inclusion Problem Research Articles

Duality Problems with Second-Order Polyhedral Discrete and Differential Inclusions

The present paper deals with the theory of duality for the Mayer problem given by second-order polyhedral discrete and differential inclusions. First, we formulate the conditions of optimality in the form of Euler–Lagrange type inclusions and transversality conditions for the polyhedral problems with second-order discrete and differential inclusions. Second, we establish dual problems for discrete and differential inclusions based on the infimal convolution concept of convex functions and prove the results of duality. For both primary and dual problems, the Euler–Lagrange type inclusions are “duality relations” and that the dual problem for discrete-approximate problems bridges the gap between the dual problems of discrete and continuous problems. As a result, the passage to the limit in the dual problem with discrete approximations plays a substantial role in the following investigations, without any which can hardly ever be established by any duality to the continuous problem. Furthermore, the numerical results also are provided.

ON THE EXACT PENALTY OF HIGH ORDER FOR EXTREME PROBLEMS OF DIFFERENTIAL INCLUSIONS

In this paper, using theorems on the continuous dependence of the solution of differential inclusions on the perturbation, we obtain high-order exact penalty theorems for nonconvex extremal problems of differential inclusions in the space of Banach-valued absolutely continuous functions. Using the type of the distance function in the classes of 𝜙 − (𝜙, 𝜙, 𝜙, 𝜙, 𝜙) locally Lipschitz functions, the nonconvex extremal problem for differential inclusions is reduced to a variational problem and the necessary condition for the extremum of a high-order is obtained. The paper also shows that the used functions type of distance function satisfy the 𝜙 − (𝜙, 𝜙, 𝜙, 𝜙, 𝜙) locally Lipschitz conditions.

Boundary value problems for fractional-order differential inclusions in Banach spaces with nondensely defined operators

Boundary value problems for fractional-order differential inclusions in Banach spaces with nondensely defined operators

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.

Nonlinear boundary value problems for fractional differential inclusions with Caputo-Hadamard derivatives on the half line

The authors establish sufficient conditions for the existence of solutions to a boundary value problem for fractional differential inclusions involving the Caputo-Hadamard type derivative of order $ r \in (1, 2] $ on infinite intervals. Both cases of convex and nonconvex valued right hand sides are considered. The technique of proof involves fixed point theorems combined with a diagonalization method.

Boundary value problems for Hadamard-Caputo implicit fractional differential inclusions with nonlocal conditions

Boundary value problems for Hadamard-Caputo implicit fractional differential inclusions with nonlocal conditions

Periodic solutions for a differential inclusion problem involving the p(t)-Laplacian

Abstract In the present paper, we consider the nonlinear periodic systems involving variable exponent driven by p(t)-Laplacian with a locally Lipschitz nonlinearity. Our arguments combine the variational principle for locally Lipschitz functions with the properties of the generalized Lebesgue-Sobolev space. Applying the non-smooth critical point theory, we establish some new existence results.

Antiperiodic Boundary Value Problems for First-Order Impulsive Differential Inclusions

This paper discusses the antiperiodic boundary value problem for first-order impulsive differential inclusions. By using Martelli’s fixed point theorem with upper and lower solutions method, we establish the existence results.

Boundary value problems for fractional differential inclusions with nonlocal multipoint boundary conditions

We establish sufficient conditions for the existence of solutions of boundary value problems for a fractional differential inclusion with nonlocal multipoint boundary conditions involving the Caputo type derivative with order α∈(1,2]. Both convex- and nonconvex-valued right-hand sides are considered.

Boundary Value Problems for Hilfer Fractional Differential Inclusions with Nonlocal Integral Boundary Conditions

In this paper, we study boundary value problems for differential inclusions, involving Hilfer fractional derivatives and nonlocal integral boundary conditions. New existence results are obtained by using standard fixed point theorems for multivalued analysis. Examples illustrating our results are also presented.

On a fractional Caputo–Hadamard inclusion problem with sum boundary value conditions by using approximate endpoint property

In the present research manuscript, we introduce a novel general fractional boundary value inclusion problem and investigate the necessary and sufficient conditions for desired existence results. Indeed, more precisely, we formulate a class of fractional multiterm Caputo–Hadamard differential inclusion problem with two‐point sum boundary value conditions. In two separate theorems, we prove the existence results for mentioned inclusion problem by using α‐ψ‐contractive maps and approximate endpoint property. Finally, we provide an example to illustrate our main results.

Nonlocal boundary value problems for Hilfer-type pantograph fractional differential equations and inclusions

In this paper, we study boundary value problems, involving the Hilfer fractional derivative, for pantograph fractional differential equations and inclusions supplemented by nonlocal integral boundary conditions. Existence and uniqueness results are obtained by using well-known fixed point theorems for single and multi-valued functions. Examples illustrating our results are also presented.

Second-order viability problems for differential inclusions with endpoint constraint and duality

ABSTRACT The paper deals with the optimal control of second-order viability problems for differential inclusions with endpoint constraint and duality. Based on the concept of infimal convolution and new approach to convex duality functions, we construct dual problems for discrete and differential inclusions and prove the duality results. It seems that the Euler–Lagrange type inclusions are ‘duality relations’ for both primary and dual problems. Finally, some special cases show the applicability of the general approach; duality in the control problem with second-order polyhedral DFIs and endpoint constraints defined by a polyhedral cone is considered.

Boundary value problems for Caputo-Hadamard fractional differential inclusions with Integral Conditions

Abstract For r ∈ (1, 2], the authors establish sufficient conditions for the existence of solutions for a class of boundary value problem for rth order Caputo-Hadamard fractional differential inclusions satisfying nonlinear integral conditions. Both cases of convex and nonconvex valued right hand sides are considered.

Infimal convolution and duality in convex optimal control problems with second order evolution differential inclusions

<p style='text-indent:20px;'>The paper deals with the optimal control problem described by second order evolution differential inclusions; to this end first we use an auxiliary problem with second order discrete and discrete-approximate inclusions. Then applying infimal convolution concept of convex functions, step by step we construct the dual problems for discrete, discrete-approximate and differential inclusions and prove duality results. It seems that the Euler-Lagrange type inclusions are "duality relations" for both primary and dual problems and that the dual problem for discrete-approximate problem make a bridge between them. At the end of the paper duality in problems with second order linear discrete and continuous models and model of control problem with polyhedral DFIs are considered.

Differential inclusions involving oscillatory terms

Motivated by mechanical problems where external forces are non-smooth, we consider the differential inclusion problem (Dλ)−Δu(x)∈∂F(u(x))+λ∂G(u(x))inΩ;u≥0,inΩ;u=0,on∂Ω,where Ω⊂Rn is a bounded open domain, and ∂F and ∂G stand for the generalized gradients of the locally Lipschitz functions F and G. In this paper we provide a quite complete picture on the number of solutions of (Dλ) whenever ∂F oscillates near the origin/infinity and ∂G is a generic perturbation of order p>0 at the origin/infinity, respectively. Our results extend in several aspects those of Kristály and Moroşanu (2010).

Infimal Convolution and Duality in Problems with Third-Order Discrete and Differential Inclusions

This paper is concerned with the Mayer problem for third-order evolution differential inclusions; to this end, first we use auxiliary problems with third-order discrete and discrete-approximate inclusions. In the form of Euler–Lagrange-type inclusions and transversality conditions, necessary and sufficient optimality conditions are derived. The principal idea of obtaining optimal conditions is locally adjoint mappings. Then, applying infimal convolution concept of convex functions, step by step, we construct the dual problems for third-order discrete, discrete-approximate and differential inclusions and prove duality results. It appears that the Euler–Lagrange-type inclusions are duality relations for both primary and dual problems and that the dual problem for discrete-approximate problem makes a bridge between the dual problems of discrete and continuous problems. As a result, the passage to the limit in the dual problem with discrete approximations plays a substantial role in the next investigations, without which it is hardly ever possible to establish any duality to continuous problem. In this way, relying to the described method for computation of the conjugate and support functions of discrete-approximate problems, a Pascal triangle with binomial coefficients can be successfully used for any “higher order” calculations. Thus, to demonstrate this approach, some semilinear problems with discrete and differential inclusions of third order are considered. These problems show that maximization in the dual problems is realized over the set of solutions of the Euler–Lagrange-type discrete/differential inclusions.

Differential inclusion problems with convolution and discontinuous nonlinearities

<p style='text-indent:20px;'>The paper investigates a new type of differential inclusion problem driven by a weighted (p, q)-Laplacian and subject to Dirichlet boundary condition. The problem fully depends on the solution and its gradient. The main novelty is that the problem exhibits simultaneously a nonlocal term involving convolution with the solution and a multivalued term describing discontinuous nonlinearities for the solution. Results stating existence, uniqueness and dependence on parameters are established.

On antiperiodic boundary value problem for semilinear fractional differential inclusion with deviating argument in Banach space

On antiperiodic boundary value problem for semilinear fractional differential inclusion with deviating argument in Banach space

On the impulsive Dirichlet problem for second-order differential inclusions

On the impulsive Dirichlet problem for second-order differential inclusions