Differential Inclusion Problem Research Articles

A Survey on Existence Results for Boundary Value Problems of Nonlinear Fractional Differential Equations and Inclusions

In this survey paper, we shall establish sufficient conditions for the existence and uniqueness of solutions for various classes of initial and boundary value problem for fractional differential equations and inclusions involving the Caputo fractional derivative. The both cases of convex and nonconvex valued right hand side are considered. The topological structure of the set of solutions is also considered.

Bound sets approach to boundary value problems for vector second-order differential inclusions

A continuation principle is established for the solvability of vector second-order boundary value problems associated with upper-Carathéodory differential inclusions. For Floquet second-order problems, this principle is combined with a bound sets approach. The viability result is also obtained in this way.

Systems of variational inclusion problems and differential inclusion problems with applications

In this paper, we study the existence theorems of systems of variational inclusion problems. From these existence results, we study the existence theorems of systems of variational differential inclusion problems, mathematical program with systems of variational inclusion constraints, and mathematical program with systems of equilibrium constraints.

TOPOLOGICAL DEGREE METHODS IN BOUNDARY VALUE PROBLEMS FOR DEGENERATE FUNCTIONAL DIFFERENTIAL INCLUSIONS WITH INFINITE DELAY

We consider the general boundary value problem for a degenerate semilinear functional differential inclusion in a Banach space with infinite delay. We construct the multivalued integral operator whose fixed points are mild solutions of the above problem and study its properties. We apply the topological degree method to obtain the general existence principle and consider some particular cases, including Cauchy and periodic problems.

Nonlinear boundary value problems for differential inclusions with Caputo fractional derivative

In this paper, we shall establish sufficient conditions for the existence of solutions for a class of boundary value problem for fractional differential inclusions involving the Caputo fractional derivative of order $\alpha\in (1,2]$. The both cases of convex valued and nonconvex valued right hand side are considered.

Three solutions for a Neumann-type differential inclusion problem involving the p(x)-Laplacian

In this paper we consider Neumann-type differential inclusion problems involving the p ( x ) -Laplacian of the type { − div ( | ∇ u | p ( x ) − 2 ∇ u ) + | u | p ( x ) − 2 u ∈ λ ∂ F ( x , u ) in Ω , ∂ u ∂ γ = 0 on ∂ Ω . Applying a version of the non-smooth three-critical-points theorem we obtain the existence of three solutions of the problem in W 0 1 , p ( x ) ( Ω ) .

Existence of periodic and non-periodic solutions to systems of boundary value problems for first-order differential inclusions with super-linear growth

We establish some new sufficient conditions under which periodic and non-periodic solutions exist for systems of nonlinear, first-order differential inclusions. Our new results are significant for two main reasons: they apply to differential inclusions that may have a right-hand side that grows super-linearly in the second variable and thus our ideas extend known linear-growth results; and our results also apply to systems of first-order differential inclusions, thus our work extends previous research on scalar-valued inclusions and may lead to a deeper understanding of higher-order differential inclusions. Our approach is based on novel differential inequalities and the well-known nonlinear alternative of Leray–Schauder. Some new results for ordinary differential equations with Carathéodory single-valued right-hand sides are also obtained.

Nonlinear boundary value problems for second order differential inclusions

In this paper, we study a class of nonlinear value boundary problems for second order differential inclusions with nonlinear perturbations, which satisfy the generalized Hartman condition weaker than that considered in some papers. Using techniques from multivalued analysis, theory of monotone operators and fixed points, we prove the existence of solutions in both “convex” and “nonconvex” cases. Our framework can be incorporated with Dirichlet, Neumann, and mixed boundary problems.

Infinitely many solutions for a Neumann-type differential inclusion problem involving the p ( x )-Laplacian

In this paper we consider a differential inclusion problem involving the p ( x ) -Laplacian of the type { − div ( | ∇ u | p ( x ) − 2 ∇ u ) + λ ( x ) | u | p ( x ) − 2 u ∈ ∂ F ( x , u ) + ∂ G ( x , u ) in Ω , ∂ u ∂ γ = 0 on ∂ Ω . We prove the existence of infinitely many solutions of this problem under suitable hypotheses by applying a non-smooth Ricceri-type variational principle and the theory of the variable exponent Sobolev spaces.

Generalized Solutions of Functional Differential Inclusions

We consider the initial value problem for a functional differential inclusion with a Volterra multivalued mapping that is not necessarily decomposable inL1n[a,b]. The concept of the decomposable hull of a set is introduced. Using this concept, we define a generalized solution of such a problem and study its properties. We have proven that standard results on local existence and continuation of a generalized solution remain true. The question on the estimation of a generalized solution with respect to a given absolutely continuous function is studied. The density principle is proven for the generalized solutions. Asymptotic properties of the set of generalized approximate solutions are studied.

Boundary value problems for differential inclusions with fractional order

Boundary value problems for differential inclusions with fractional order

Numerical fixed grid methods for differential inclusions

Numerical methods for initial value problems for differential inclusions usually require a discretization of time as well as of the set valued right hand side. In this paper, two numerical fixed grid methods for the approximation of the full solution set are proposed and analyzed. Convergence results are proved which show the combined influence of time and (phase) space discretization.

Initial value problems for singular and nonsmooth second order differential inclusions

We study the existence of W2,1 solutions for singular and nonsmooth initial value problems of the type whereT > 0 is a priori fixed, x0, x1 ∈ ℝ, and F: [0, T ] × ℝ → 𝒫(ℝ) \ {∅︁} is a multivalued mapping. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Impulsive Boundary Value Problems for Sturm-Liouville Type Differential Inclusions

In this paper, the authors investigate the existence of solutions of impulsive boundary value problems for Sturm-Liouville type differential inclusions which admit non-convex-valued multifunctions on right hand side. Two results under weaker conditions are presented. The methods rely on a fixed point theorem for contraction multi-valued maps due to Covitz and Nadler and Schaefer’s fixed point theorem combined with lower semi-continuous multi-valued operators with decomposable values.

Impulsive Boundary Value Problems for First-order Ordinary Differential Inclusions

In this paper, we investigate the existence of solutions for impulsive first order ordinary differential inclusions which admitting nonconvex valued right hand side. We present two classes of results. In the first one, we rely on a fixed point theorem for contraction multivalued maps due to Covitz and Nadler, and for the second one, we use Schaefer's fixed point theorem combined with lower semi-continuous multivalued operators with decomposable values under weaker conditions.

Baire category and boundary value problems for ordinary and partial differential inclusions under Carathéodory assumptions

By using the Baire category method we prove an existence result for boundary value problem of Dirichlet type, for non-convex ordinary differential inclusions under Caratheodory assumptions. By counterexamples we show that an analogous existence result is non-valid for non-convex partial differential inclusions.

Differential inclusions and the control of forest fires

This paper introduces a new class of variational problems for differential inclusions, motivated by the control of forest fires. The area burned by the fire at time t > 0 is modelled as the reachable set for a differential inclusion x ˙ ∈ F ( x ) , starting from an initial set R 0 . To block the fire, a wall can be constructed progressively in time, at a given speed. In this paper, we study the possibility of constructing a wall which completely encircles the fire. Moreover, we derive necessary conditions for an optimal strategy, which minimizes the total area burned by the fire.

P-Laplacian Inclusions via Fixed Points for Multifunctions in Posets

We consider the Dirichlet problem for elliptic differential inclusions involving the p-Laplacian and governed by multivalued nonlinearities bounded above and below by single-valued monotone functions not necessarily continuous. The aim of this note is to provide existence and comparison results for the problem under consideration without imposing further regularity assumptions. The main tools used in the proof of our results are existence and comparison results for extremal solutions of (single-valued) nonlinear elliptic equations, and an abstract comparison principle for fixed points of multivalued operators in partially ordered sets (posets).

A new approach for solving of optimal nonlinear control problems

In this paper, we are going to consider a nonlinear optimal control problem (NOC). First we change the (NOC) problem to an optimal differential inclusion problem (ODI), then by defining new control variables, (ODI) problem is converted to an optimal control problem where it is linear in term of control variable and we determine the approximation of this control problem, then by using measure theory, we may convert the new optimal control problem to an infinite dimensional linear programming problem. We approximate the latest one to a finite linear programming problem, then by using of the optimal solution of the linear programming we obtain an approximation of the state and feedback control of the original (ONC) problem.

On Systems of Boundary Value Problems for Differential Inclusions

Herein we consider the existence of solutions to second-order, two-point boundary value problems (BVPs) for systems of ordinary differential inclusions. Some new Bernstein–Nagumo conditions are presented that ensure a priori bounds on the derivative of solutions to the differential inclusion. These a priori bound results are then applied, in conjunction with appropriate topological methods, to prove some new existence theorems for solutions to systems of BVPs for differential inclusions. The new conditions allow of the treatment of systems of BVPs without growth restrictions.