First order impulsive differential inclusions with periodic conditions

In this paper, we present an impulsive version of Filippov's Theorem for the first-order nonresonance impulsive differential inclusion $$\begin{array}{rlll} y'(t)-\lambda y(t) &\in& F(t,y(t)), &\hbox{ a.e. } \, t\in J\backslash \{t_{1},\ldots,t_{m}\},\\ y(t^+_{k})-y(t^-_k)&=&I_{k}(y(t_{k}^{-})), &k=1,\ldots,m,\\ y(0)&=&y(b), \end{array}$$ where $J=[0,b]$ and $F: J \times \mathbb{R}^n\to{\cal P}(\mathbb{R}^n)$ is a set-valued map. The functions $I_k$ characterize the jump of the solutions at impulse points $t_k$ ($k=1,\ldots,m.$). Then the relaxed problem is considered and a Filippov-Wasewski result is obtained. We also consider periodic solutions of the first order impulsive differential inclusion $$\begin{array}{rlll} y'(t) &\in& \varphi(t,y(t)), &\hbox{ a.e. } \, t\in J\backslash \{t_{1},\ldots,t_{m}\},\\ y(t^+_{k})-y(t^-_k)&=&I_{k}(y(t_{k}^{-})), &k=1,\ldots,m,\\ y(0)&=&y(b), \end{array}$$ where $\varphi: J\times \mathbb{R}^n\to{\cal P}(\mathbb{R}^n)$ is a multi-valued map. The study of the above problems use an approach based on the topological degree combined with a Poincar\'e operator

Existence results for nondensely defined semilinear functional differential inclusions in Fréchet spaces

In this paper, a recent Frigon nonlinear alternative for contractive multivalued maps in Fréchet spaces, combined with semigroup theory, is used to investigate the existence of integral solutions for first order semilinear functional differential inclusions. An application to a control problem is studied. We assume that the linear part of the differential inclusion is a nondensely defined operator and satisfies the Hille-Yosida condition

Initial boundary value problems for second order impulsive functional differential inclusions

In this paper we investigate the existence of solutions for initial and boundary value problems for second order impulsive functional differential inclusions. We shall rely on a fixed point theorem for contraction multivalued maps due to Covitz and Nadler

Oscillatory and nonoscillatory solutions for first order impulsive differential inclusions

summary:In this paper we discuss the existence of oscillatory and nonoscillatory solutions of first order impulsive differential inclusions. We shall rely on a fixed point theorem of Bohnenblust-Karlin combined with lower and upper solutions method

Structure of solutions sets and a continuous version of Filippov's theorem for first order impulsive differential inclusions with periodic conditions

In this paper, the authors consider the first-order nonresonance impulsive differential inclusion with periodic conditions $$\begin{array}{rlll} y'(t)-\lambda y(t) &\in& F(t,y(t)), &\hbox{ a.e. } \, t\in J\backslash \{t_{1},\ldots,t_{m}\},\\ y(t^+_{k})-y(t^-_k)&=&I_{k}(y(t_{k}^{-})), &k=1, 2, \ldots,m,\\ y(0)&=&y(b), \end{array}$$ where $J=[0,b]$ and $F: J\times \mathbb{R}^n\to{\cal P}(\mathbb{R}^n)$ is a set-valued map. The functions $I_k$ characterize the jump of the solutions at impulse points $t_k$ ($k=1, 2, \ldots,m$). The topological structure of solution sets as well as some of their geometric properties (contractibility and $R_\delta$-sets) are studied. A continuous version of Filippov's theorem is also proved

Filippov's theorem for impulsive differential inclusions with fractional order

In this paper, we present an impulsive version of Filippov's Theorem for fractional differential inclusions of the form: $$\begin{array}{rlll} D^{\alpha}_*y(t) &\in& F(t,y(t)), &\hbox{ a.e. } \, t\in J\backslash \{t_{1},\ldots,t_{m}\},\ \alpha\in(1,2],\\ y(t^+_{k})-y(t^-_k)&=&I_{k}(y(t_{k}^{-})), &k=1,\ldots,m,\\ y'(t^+_{k})-y'(t^-_k)&=&\overline{I}_{k}(y'(t_{k}^{-})), &k=1,\ldots,m,\\ y(0)&=&a,\ y'(0)=c,\ \end{array}$$ where $J=[0,b],$ $D^{\alpha}_*$ denotes the Caputo fractional derivative and $F$ is a set-valued map. The functions $I_k,\overline{I}_k$ characterize the jump of the solutions at impulse points $t_k$ ($k=1,\ldots,m$)

Existence results for impulsive dynamic inclusions on time scales

In this paper, we investigate the existence of solutions and extremal solutions for a first order impulsive dynamic inclusion on time scales. By using suitable fixed point theorems, we study the case when the right hand side has convex as well as nonconvex values

Impulsive stochastic functional differential inclusions driven by a fractional Brownian motion with infinite delay

In this paper, we prove the existence of mild solutions for the following first-order impulsive semilinear stochastic functional differential inclusions driven by a fractional Brownian motion with infinite delay in the case where the right hand side is convex or nonconvex-valued. The results are obtained by using two fixed point theorems for multivalued mappings.Ministerio de Economía y Competitividad (España) MTM2011-22411Junta de Andalucía. Consejería de Innovación, Ciencia y Empresa 2010/FQM314Junta de Andalucía P12-FQM-149

Existence and stability results for semilinear systems of impulsive stochastic differential equations with fractional Brownian motion

Some results on the existence and uniqueness of mild solution for a system of semilinear impulsive differential equations with infinite fractional Brownian motions are proved. The approach is based on Perov’s fixed point theorem and a new version of Schaefer’s fixed point theorem in generalized Banach spaces. The relationship between mild and weak solutions and the exponential stability of mild solutions are investigated as well. The abstract theory is illustrated with an example.Fondo Europeo de Desarrollo RegionalMinisterio de Economía y CompetitividadConsejería de Innovación, Ciencia y Empresa (Junta de Andalucía

Impulsive neutral functional differential equations driven by a fractional Brownian motion with unbounded delay

In this paper, we prove the local and global existence and attractivity of mild solutions for stochastic impulsive neutral functional differential equations with infinite delay, driven by fractional Brownian motion.Fondo Europeo de Desarrollo RegionalMinisterio de Economía y CompetitividadJunta de Andalucí