Nonlocal Impulsive Cauchy Problems for Evolution Equations

Abstract

Of concern is the existence of solutions to nonlocal impulsive Cauchy problems for evolution equations. Combining the techniques of operator semigroups, approximate solutions, noncompact measures and the fixed point theory, new existence theorems are obtained, which generalize and improve some previous results since neither the Lipschitz continuity nor compactness assumption on the impulsive functions is required. An application to partial differential equations is also presented.

On Existence of Solutions of Impulsive Nonlinear Functional Neutral Integro-Differential Equations With Nonlocal Condition

“…We are generalizing the results reported in [2], [11], [14]. The main tool used in our analysis is based on an application of the Banach contraction theorem and the theory of fractional power of operators.…”

“…On the other hand, the theory of functional differential equations with nonlocal conditions has been extensively studied in the literature, see [1], [4]- [10], [12]- [14], [19], as they have applications in physics and many other areas of applied mathematics. The nonlocal condition is more precise for describing natural phenomena than the classical condition because more information is taken into account, thereby decreasing the negative effects incurred by a possibly single measurement taken at initial time.…”

Nonlocal Problems for Fractional Differential Equations via Resolvent Operators

“…Many authors developed different techniques and methods to solve this problem. For more details on this topic, we refer to [10,11,[22][23][24][25][26][27][28][29][30][31][32][33] and references therein.…”