Abstract
In this paper, we study the existence of solutions for systems of random semilinear impulsive differential equations. The existence results are established by means of a new version of Perov’s, a nonlinear alternative of Leray-Schauder’s fixed point principles combined with a technique based on vector-valued metrics and convergent to zero matrices. Also, we give a random abstract formulation to Sadovskii’s fixed point theorem in a vector-valued Banach space. Examples illustrating the results are included.
Highlights
1 Introduction Many evolution processes are characterized by the fact that at certain moments of time, they experience change of state abruptly in a form of shocks, harvesting, natural disasters, etc. These phenomena involve short term perturbations from continuous and smooth dynamics, whose duration is negligible in comparison with the duration of entire evolution. This has been the main reason for the development of a new branch of the theory of ordinary differential equations called impulsive differential equations
2.2 Random variable and some selection theorems we introduce notations, definitions, and preliminary facts from multivalued analysis and random variable which are used throughout this paper
As a consequence of Theorem . , we present versions of Schaefer’s fixed point theorem and the nonlinear alternative Leray-Schauder-type theorem for β-condensing operators in a generalized Banach space
Summary
Many evolution processes are characterized by the fact that at certain moments of time, they experience change of state abruptly in a form of shocks, harvesting, natural disasters, etc. ([ ]) Let ( , ), Y be a separable generalized metric space and F : → Pcl(Y ) be measurable multivalued. ([ ]) Let ( , F ) be a measurable space, X be a real separable generalized Banach space and F : × X → X be a continuous random operator, and let M(ω) ∈ Mn×n(R+) be a random variable matrix such that, for every ω ∈ , the matrix M(ω) converges to and d F(ω, x ), F(ω, x ) ≤ M(ω)d(x , x ) for each x , x ∈ X, ω ∈. ([ ]) Let X be a separable generalized metric space and G : × X → X be a mapping such that G(·, x) is measurable for all x ∈ X and G(ω, ·) is continuous for all ω ∈. ⎪⎪⎪⎪⎪⎩.(x. .m+ (t, ω), ym+ (t, ω)), if t ∈ [ , t ], if t ∈
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
