Solution Sets for Differential Equations and Inclusions

Solution set for fractional differential equations with Riemann-Liouville derivative

“…From the choice of U , there is no y ∈ ∂U such that y = γF (y) for some γ ∈ (0, 1). As a consequence of the nonlinear alternative of Leray-Schauder type [7,10], we deduce that F has a fixed point y in U which is a solution of the problem (1.1)). Now, prove that S(f, c) is compact.…”

“…Topological structure of the solution set for ordinary differential equations and inclusions is developed the recent monographs [9,10] and in [11,17].…”

Existence, Uniqueness, Compactness of the Solution Set, and Dependence on a Parameter for an Impulsive Periodic Boundary Value Problem

“…(see (6)) which implies that y is a solution of (1)- (4). If y is a fixed point of N , then (H 8 )-(H 9 ) imply that y(t) ≥ 0 for each t ∈ J.…”

“…Such phenomena are often modeled by impulsive differential equations and inclusions. Some good discussions and results on impulsive equations can be found in the monographs by Bainov, Lakshmikantham, and Simeonov [3], Djebali, Gorniewicz, and Ouahab [6], Graef, Henderson, and Ouahab [7], and Perestyuk, Plotnikov, Samoilenko, and Skripnik [13] and the references therein. Using fixed point arguments, existence as well as uniqueness results for some classes of impulsive boundary differential equations and dynamic equations were given by Benchohra and Eloe [4], Graef and Ouahab [9], and Nieto [12], among others.…”

Advanced Functional Evolution Equations and Inclusions

“…For more details on multi-valued maps we refer to the books of Deimling [101], Djebali et al [104], Górniewicz [123], Hu and Papageorgiou [143], and Tolstonogov [181]. Definition 1.10.…”