Impulsive semilinear differential inclusions: Topological structure of the solution set and solutions on non-compact domains

Existence results for impulsive differential equations with nonlocal conditions via measures of noncompactness

“…We use some hypothesis in [18,19], and using the method of Hausdorff's measure of noncompactness, we give the existence of integral solution of impulsive differential equation with nonlocal conditions (1.1)-(1.3). The results obtained in this paper are generalizations of the results given by [12,18,19,23,24].…”

Nonlocal Impulsive Cauchy Problems for Evolution Equations

“…According to the proof of Step 1, we know that {Q n1 u n ; n ≥ 1}| h,t 2 , {Q F1 u n ; n ≥ 1}| h,t 2 , {Q n4 u n ; n ≥ 1}| h,t 2 3.39 are all precompact in PC h, t 2 , X and Q F2 : PC h, t 2 , X → PC h, t 2 , X is Lipschitz continuous with constant M 0 L LC 1−β K β /β. Next, we will show that {Q n2 u n ; n ≥ 1}| h,t 2 is precompact in PC h, t 2 , X .…”

Mild solutions to nonlocal impulsive differential inclusions governed by a noncompact semigroup

“…For more details on this theory and on its applications we refer to the monographs of Lakshmikantham et al [18], Benchohra et al [6], the papers of [1,10,14,23] and the references therein. In [10], Cardinali et al considered the impulsive semilinear differential inclusions under the assumptions of the measure of noncompactness with multivalued perturbations F. Moreover, Byszewski and Lakshmikantham [9] introduced the nonlocal Cauchy problems as the corresponding models that can describe the phenomena more accurately than the classical initial condition u(0) = u 0 alone.…”

“…In [10], Cardinali et al considered the impulsive semilinear differential inclusions under the assumptions of the measure of noncompactness with multivalued perturbations F. Moreover, Byszewski and Lakshmikantham [9] introduced the nonlocal Cauchy problems as the corresponding models that can describe the phenomena more accurately than the classical initial condition u(0) = u 0 alone. Therefore, it has been studied extensively under various conditions on A and F by several authors (see [3,8,20]).…”