Semilinear Differential Inclusions Research Articles

Fractional semilinear differential inclusions

In this paper, we first present a version of Filippov’s Theorem for fractional semilinear differential inclusions of the form, {(cDαy−Ay)(t)∈F(t,y(t)), a.e. t∈[0,b],y(0)=y0∈X, where cDα is the Caputo fractional derivative and F is a set-valued map. After several existence results, the compactness of solutions sets is also investigated. Some results from topological fixed point theory together with notions of measure of noncompactness are used.

The solution sets for second order semilinear impulsive multivalued boundary value problems

A second order semilinear differential inclusion with some boundary continuous and impulse characteristics in a separable Banach space is considered. Some existence theorems for mild solutions are given, when the multivalued nonlinearity of the inclusion is only a locally integrably bounded upper-Carathéodory map with convex and weakly compact values. Then the compactness of the set of all mild solutions for the problem is proved. The results are obtained by using the theory of continuous cosine families of bounded linear operators and a fixed point theorem for multivalued maps due to Agarwal, Meehan and O’Regan. A corresponding result for closed graph of composition is extended.

Existence and controllability results for some impulsive partial functional differential inclusion

In this work, we prove the existence of mild and extremal mild solutions for first-order semilinear non densely defined impulsive functional differential inclusions in separable Banach spaces with local and nonlocal conditions. Firstly, we show the existence of mild and extremal solutions. Secondly, we study the controllability of a semilinear functional differential inclusions.

On the Dimension of the Solution Set for Semilinear Fractional Differential Inclusions

We investigate the existence and dimension of the solution set for a nonlocal problem of semilinear fractional differential inclusions. The main tools of our study include some well‐known results on multivalued maps.

On the existence of mild solutions for nonconvex fractional semilinear differential inclusions

We establish some Filippov type existence theorems for solutions of fractional semilinear differential inclusions involving Caputo’s fractional derivative in Banach spaces.

Approximate controllability of differential inclusions in Hilbert spaces

In this paper, controllability for the system originating from semilinear functional differential equations in Hilbert spaces is studied. We consider the problem of approximate controllability of semilinear differential inclusion assuming that semigroup, generated by the linear part of the inclusion, is compact and under the assumption that the corresponding linear system is approximately controllable. By using resolvent of controllability Gramian operator and fixed point theorem, sufficient conditions have been formulated and proved. An example is presented to illustrate the utility and applicability of the proposed method.

Existence Results for Stochastic Semilinear Differential Inclusions with Nonlocal Conditions

We discuss existence results of mild solutions for stochastic differential inclusions subject to nonlocal conditions. We provide sufficient conditions in order to obtain a priori bounds on possible solutions of a one-parameter family of problems related to the original one. We, then, rely on fixed point theorems for multivalued operators to prove our main results.

Impulsive mild solutions for semilinear differential inclusions with nonlocal conditions in Banach spaces

In this paper we deal with the existence of impulsive mild solutions for semilinear differential inclusions with nonlocal conditions, where the linear part generates an evolution system and the nonlinearity satisfies the lower Scorza–Dragoni property. Our theorems extend the existence propositions proved by Fan in 2010. An example is presented.

Existence and controllability results for fractional semilinear differential inclusions

In this paper, we prove the existence and controllability results for fractional semilinear differential inclusions involving the Caputo derivative in Banach spaces. The results are obtained by using fractional calculation, operator semigroups and Bohnenblust–Karlin’s fixed point theorem. At last, an example is given to illustrate the theory.

Approximate weak invariance for semilinear differential inclusions in Banach spaces

Abstract In this paper we give a criterion for a given set K in Banach space to be approximately weakly invariant with respect to the differential inclusion x′(t) ∈ Ax(t) + F(x(t)), where A generates a C 0-semigroup and F is a given multi-function, using the concept of a tangent set to another set. As an application, we establish the relation between approximate solutions to the considered differential inclusion and solutions to the relaxed one, i.e., x′(t) ∈ Ax(t) + $\overline {co} $ F(x(t)), without any Lipschitz conditions on the multi-function F.

Lower semi-continuity of the solution set for semilinear differential inclusions

In this paper we provide sufficient conditions for the solution set of a semilinear differential inclusion to be lower semi-continuous. We give an application to the propagation of continuity of the minimum time function associated with the given differential inclusion and the target zero.

Nonlocal cauchy problems and their controllability for semilinear differential inclusions with lower Scorza-Dragoni nonlinearities

In this paper we prove the existence of mild solutions and the controllability for semilinear differential inclusions with nonlocal conditions. Our results extend some recent theorems.

Necessary and Sufficient Conditions for Local Invariance for Semilinear Differential Inclusions

We establish new necessary and sufficient conditions for local invariance referring to semilinear differential inclusions, conditions expressed in terms of a new suitable defined tangency concept. An application to the study of the Hamilton-Jacobi-Bellman equation is given.

On semilinear differential inclusions in Banach spaces with nondensely defined operators

We consider a semilinear differential inclusion in a Banach space assuming that its linear part is a nondensely defined Hille–Yosida operator whereas Caratheodory-type multivalued nonlinearity satisfies a regularity condition expressed in terms of the Hausdorff measure of noncompactness. We apply the theory of integrated semigroups and the fixed point theory of condensing multivalued maps to obtain local and global existence results and to prove the continuous dependence of the solutions set on initial data. An application to an optimization problem for a feedback control system is given.

Lyapunov pairs for multi-valued semi-linear evolutions

We give here a characterization of a Lyapunov pair for a multi-valued semi-linear evolution equation on a Banach space by means of an appropriate lower contingent derivative. The contingent derivative introduced in this paper is related to a new concept of tangency introduced recently in [O. Cârjă, M. Necula, I.I. Vrabie, Necessary and sufficient conditions for viability for semilinear differential inclusions, Trans. Amer. Math. Soc. 361 (2009) 343–390]. As an application, we give a controllability result and a Lipschitz estimate of the corresponding minimum time function under a Petrov-like condition.

First-order periodic impulsive semilinear differential inclusions: Existence and structure of solution sets

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 semilinear differential inclusions with periodic boundary conditions: { ( y ′ − A y ) ( t ) ∈ F ( t , y ( t ) ) , a.e. t ∈ J ∖ { t 1 , … , t m } , y ( t k + ) − y ( t k − ) = I k ( y ( t k − ) ) , k = 1 , … , m , y ( 0 ) = y ( b ) where J = [ 0 , b ] and 0 = t 0 < t 1 < ⋯ < t m < t m + 1 = b ( m ∈ N ∗ ) A is the infinitesimal generator of a C 0 -semigroup T on a separable Banach space E and F is a multi-valued map. The functions I k characterize the jump of the solutions at impulse points t k ( k = 1 , … , m ). We will have to distinguish between the cases when either or neither 1 lies in the resolvent of T ( b ) . Accordingly, the problem is either formulated as a fixed point problem for an integral operator or for a Poincaré translation operator. Our existence results rely on fixed point theory and on a new nonlinear alternative for compact u.s.c. maps respectively. Then, we present some existence results and investigate the topological structure of the solution set. A continuous version of Filippov’s theorem is provided and the continuous dependence of solutions on parameters in both convex and nonconvex cases are examined.

Semilinear differential inclusions via weak topologies

The paper deals with the multivalued initial value problem x ′ ∈ A ( t , x ) x + F ( t , x ) for a.a. t ∈ [ a , b ] , x ( a ) = x 0 in a separable, reflexive Banach space E. The nonlinearity F is weakly upper semicontinuous in x and the investigation includes the case when both A and F have a superlinear growth in x. We prove the existence of local and global classical solutions in the Sobolev space W 1 , p ( [ a , b ] , E ) with 1 < p < ∞ . Introducing a suitably defined Lyapunov-like function, we are able to investigate the topological structure of the solution set. Our main tool is a continuation principle in Frechét spaces and we prove the required pushing condition in two different ways. Some examples complete the discussion.

BVP for Carathéodory inclusions in Hilbert spaces: sharp existence conditions and applications

This article concerns an existence result for Floquet boundary value problems associated to semilinear differential inclusions with Carathéodory right hand side in a Hilbert space. We apply a continuation principle and we require a sharp (i.e., localized on the boundary) transversality condition. We give an application to a nonlinear partial differential inclusion with periodic conditions.

Evolution equations on locally closed graphs and applications

Let X be a Banach space, let A : D ( A ) ⊆ X → X be the infinitesimal generator of a C 0 -semigroup, let I be a nonempty, bounded interval and let K : I ↝ X be a given multi-valued function. By using the concept of A - quasi-tangent set introduced by Cârjă, Necula, Vrabie [O. Cârjă, M. Necula, I.I. Vrabie, Necessary and sufficient conditions for viability for a semilinear differential inclusions, Trans. Amer. Math. Soc., 361 (2009) 343–390; O. Cârjă, M. Necula, I.I. Vrabie, Viability, Invariance and Applications, North-Holland a Mathematics Studies, vol. 207, Elsevier, 2007] and using a tangency condition expressed in the terms of this concept, we establish a necessary and sufficient condition for mild viability referring to evolution inclusions of the form u ′ ( t ) ∈ A u ( t ) + F ( t , u ( t ) ) , where F is a multi-function defined on the graph of K . As applications, we deduce a sufficient condition for null-controllability and a comparison result for semilinear evolution inclusions.

A Survey on Semilinear Differential Equations and Inclusions Involving Riemann-Liouville Fractional Derivative

We establish sufficient conditions for the existence of mild solutions for some densely defined semilinear functional differential equations and inclusions involving the Riemann-Liouville fractional derivative. Our approach is based on the -semigroups theory combined with some suitable fixed point theorems.