Semilinear Differential Inclusions Research Articles

On the solution sets of differential inclusions and the periodic problem in Banach spaces

In this paper, the topological structure of the solution set of a constrained semilinear differential inclusion in a Banach space E is studied. It is shown that the set of all mild solutions, with values in a closed and, in general, thin subset D⊂ E, is an R δ -set provided natural boundary conditions and appropriate geometrical assumptions on D (which hold, e.g. when D is convex) are satisfied. Applications to the periodic problem and to the existence of equilibria are given.

An averaging method for singularly perturbed systems of semilinear differential inclusions with analytic semigroups

We consider a system of two semilinear parabolic inclusions depending on a small parameter ε>0 which is present both in front of the derivative in one of the two inclusions and in the nonlinear terms to model high-frequency inputs. The aim is to provide conditions in order to guarantee, for ε>0 sufficiently small, the existence of periodic solutions and in order to study their behaviour as ε tends to zero. Our assumptions permit the definition of upper semicontinuous, convex valued, compact vector operators whose fixed points represent the sought-after periodic solutions. The existence of fixed points is shown by using topological degree theory arguments.

Existence results for impulsive semilinear damped differential inclusions

In this paper we investigate the existence of mild solutions for rst and second order impulsive semilinear evolution inclusions in real separable Banach spaces. By using suitable xed point theorems, we study the case when the multivalued map has convex and nonconvex values.

Semilinear Impulsive Neutral Functional Differential Inclusions in Banach Spaces

The aim of this article is to study the existence of solutions to some classes of semilinear impulsive neutral functional differential inclusions in Banach spaces. We shall make use of a fixed point theorem for contraction multivalued maps due to Covitz and Nadler.

Viability for Semilinear Differential Inclusions via the Weak Sequential Tangency Condition

Let X be a reflexive and separable Banach space, A: D( A) ⊂ X → X the generator of a C 0-semigroup S( t) : X → X, t ≥ 0, D a locally weakly sequentially closed subset in X, and F: D → 2 X a nonempty, closed, convex, and bounded valued mapping which is weakly-weakly upper semi-continuous. The main result of the paper is: Theorem . Under the general assumptions above a necessary and sufficient condition in order that for each ξ ∈ D there exists at least one mild solution u of du dt t ∈ Au t + F u t satisfying u(0) = ξ is the so-called “weak sequential tangency condition” below. ( W S T C ) For each ξ ∈ D there exists y ∈ F(ξ) and two sequences ( t n ) n ∈ N in R + and ( p n ) n ∈ N in X such that t n → 0, p n → 0 and satisfying S( t n )ξ + t n ( y + p n ) ∈ D.

On semilinear differential inclusions with lower semicontinuous nonlinearities

In questo lavoro si dà un teorema di esistenza di soluzioni per un inclusione differenziale semilineare in uno spazio di Banach. Le ipotesi sono deboli in quanto si suppone che l'operatore lineare A sia il generatore di un semigruppo fortemente continuo, senza ipotesi di compattezza, e la non linerità multivoca sia semicontinua inferiormente, non necessariamente a valori convessi.

Topological structure of solution sets of differential inclusions: the constrained case

We survey and announce some current results on the existence, the viability, and the topological structure of the viable solutions of differential equations and inclusion in Banach spaces under set constraints. Some new results concerning semilinear differential inclusions with state variables constrained to the so‐called regular and strictly regular sets, together with their applications, are presented and discussed.

On a nonlocal Cauchy problem for differential inclusions

We establish sufficient conditions for the existence of solutions for semilinear differential inclusions, with nonlocal conditions. We rely on a fixed‐point theorem for contraction multivalued maps due to Covitz and Nadler andon the Schaefer′s fixed‐point theorem combined with lower semicontinuous multivalued operators with decomposable values.

Existence of monotone solutions for semilinear differential inclusions

Let X be a reflexive and separable Banach space, \(A:D(A)\subset X\to X\) the generator of a C 0-semigroup \(S(t):X\to X, t\ge 0, D\) a locally weakly closed set in \(X,F:D\to 2^X\) a nonempty, closed, convex and bounded valued mapping which is weakly-weakly upper semicontinuous. Let \(\preceq\) be a preorder on D, characterized by the set-valued map \(P:D\to 2^D\), defined by \(P(\xi )=\lbrace \eta\in D;\ \ \xi\preceq\eta \rbrace \) whose graph is weakly\(\times\)weakly sequentially closed. The main result of the paper is:¶¶Theorem. Under the general assumptions above a necessary and a sufficient condition in order that for each \( \xi \in D \) there exists at least one mild solution u of¶¶\( {du\over dt}(t) \in Au(t) + F(u(t))\ \ t\ge 0\)¶¶satisfying \(u(0)=\xi\) and \(u(s)\preceq u(t)\) for each s≤ t is the so called “bounded w-monotonicity condition” below.¶\((Bw\cal MC)\) There exists a locally bounded function \({\cal M} : D \to R_+^* \) such that for each \( \xi \in D \) there exists \(y \in F(\xi)\) such that for each \( \delta > 0\) and each weak neighborhood V of 0, there exist \(t\in (\,0,\delta \,]\) and \(p \in V\) with \(\Vert p\Vert\le {\cal M}(\xi )\) and satisfying \(S(t)\xi +t(y+p)\in P(\xi )\).

Some new viability results for semilinear differential inclusions

Let X be a reflexive and separable Banach space, $ A : D(A) \subset X \rightarrow X $ the infinitesimal generator of a C 0-semigroup $ S(t) : X \rightarrow X, t \geq 0 $ , D a locally weakly closed subset in X and $ F : D \rightarrow 2^X $ a nonempty, closed, convex and bounded valued mapping which is weakly-weakly upper semi-continuous. The main result of the paper is:¶Theorem. Under the general assumptions above a necessary and suffcient condition in order that for each $ \xi \in D $ there exists at least one mild solution u of¶¶ $ {du \over dt} (t) \in Au(t) + F(u(t)) $ ¶satisfying $ u(0) = \xi $ is the so called "bounded w-tangency condition" below. $ (Bw {\cal TC}) $ . There exists a locally bounded function $ {\cal M} : D \rightarrow {\bf R}^*_+ $ enjoying the property that for each $ \xi \in D $ there exists $ y \in F(\xi) $ such that for each $ \delta \gt 0 $ and each weak neighborhood V of 0 there exist $ t \in ({\rm 0},\delta ] $ and $ p \in V $ with $ \parallel p \parallel \leq {\cal M}(\xi) $ and satisfying¶¶ $ S(t)\xi + t(y + p) \in D. $

On the Solvability of Systems of Inclusions Involving Noncompact Operators

We consider the solvability of a system \[ \left \{ {\begin {array}{*{20}{c}} {y \in \bar F(x,y),} \\ {x \in \bar G(x,y)} \\ \end {array} } \right .\] of set-valued maps in two different cases. In the first one, the map $(x,y) - \circ \bar F(x,y)$ is supposed to be closed graph with convex values and condensing in the second variable and $(x,y) - \circ \bar G(x,y)$ is supposed to be a permissible map (i.e. composition of an upper semicontinuous map with acyclic values and a continuous, single-valued map), satisfying a condensivity condition in the first variable. In the second case $\bar F$ is as before with compact, not necessarily convex, values and $\bar G$ is an admissible map (i.e. it is composition of upper semicontinuous acyclic maps). In the latter case, in order to apply a fixed point theorem for admissible maps, we have to assume that the solution set $x - \circ S(x)$ of the first equation is acyclic. Two examples of applications of the abstract results are given. The first is a control problem for a neutral functional differential equation on a finite time interval; the second one deals with a semilinear differential inclusion in a Banach space and sufficient conditions are given to show that it has periodic solutions of a prescribed period.

On the solvability of systems of inclusions involving noncompact operators

We consider the solvability of a system \[ { y ∈ F ¯ ( x , y ) , x ∈ G ¯ ( x , y ) \left \{ {\begin {array}{*{20}{c}} {y \in \bar F(x,y),} \\ {x \in \bar G(x,y)} \\ \end {array} } \right . \] of set-valued maps in two different cases. In the first one, the map ( x , y ) − ∘ F ¯ ( x , y ) (x,y) - \circ \bar F(x,y) is supposed to be closed graph with convex values and condensing in the second variable and ( x , y ) − ∘ G ¯ ( x , y ) (x,y) - \circ \bar G(x,y) is supposed to be a permissible map (i.e. composition of an upper semicontinuous map with acyclic values and a continuous, single-valued map), satisfying a condensivity condition in the first variable. In the second case F ¯ \bar F is as before with compact, not necessarily convex, values and G ¯ \bar G is an admissible map (i.e. it is composition of upper semicontinuous acyclic maps). In the latter case, in order to apply a fixed point theorem for admissible maps, we have to assume that the solution set x − ∘ S ( x ) x - \circ S(x) of the first equation is acyclic. Two examples of applications of the abstract results are given. The first is a control problem for a neutral functional differential equation on a finite time interval; the second one deals with a semilinear differential inclusion in a Banach space and sufficient conditions are given to show that it has periodic solutions of a prescribed period.

A priori estimates for operational differential inclusions

We prove a set-valued Gronwall lemma and a relaxation theorem for the semilinear differential inclusion x′ ϵ Ax + F( t, x), x(0) = x 0, where A is the infinitesimal generator of a C 0-semigroup on a separable Banach space X and F: [0, T] × X ↦ X is a set-valued map. This allows us to investigate infinitesimal generators of reachable sets and variational inclusions. The results are applied to a semilinear optimal control problem with end point constraints.