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About sign-constancy of Green's functions for impulsive second order delay equations

Domoshnitsky A., Landsman G., Yanetz S.

Opuscula Mathematica

2014

Vol. 34, no. 2

339--362

We consider the following second order differential equation with delay [formula] In this paper we find necessary and sufficient conditions of positivity of Green's functions for this impulsive equation coupled with one or two-point boundary conditions in the form of theorems about differential inequalities. By choosing the test function in these theorems, we obtain simple sufficient conditions. For example, the inequality [formula] is a basic one, implying negativity of Green's function of two-point problem for this impulsive equation in the case 0<γi≤1, 0<δi≤1 for i=1,…,p.

Nonlocal impulsive problems for fractional differential equations with time-varying generating operators in Banach spaces

Wang J., Yang Y., Wei W.

Opuscula Mathematica

2010

Vol. 30, no. 3

361-381

In this paper, we study the existence and uniqueness of the PC-mild solution for a class of impulsive fractional differential equations with time-varying generating operators and nonlocal conditions. By means of the generalized Ascoli-Arzela Theorem given by us and the fixed point theorem, some existence and uniqueness results are obtained. Finally, an example is given to illustrate the theory.