Abstract
Abstract This paper is concerned with the existence of solutions of a second-order impulsive differential equation with mixed boundary condition. We obtain sufficient conditions for the existence of a unique solution, at least one solution, at least two solutions and infinitely many solutions, respectively, by using critical point theorems. The main results are also demonstrated with examples. MSC:34B15, 34B18, 34B37, 58E30.
Highlights
1 Introduction Nowadays, with the rapid development of science and technology, many people have realized that the theory of impulsive differential equations is richer than the corresponding theory of differential equations but it represents a more natural framework for mathematical modeling of real world phenomena
It has become an effective tool to study some problems of biology, medicine, physics and so on [, ]
Significant progress has been made in the theory of systems of impulsive differential equations in recent twenty years
Summary
With the rapid development of science and technology, many people have realized that the theory of impulsive differential equations is richer than the corresponding theory of differential equations but it represents a more natural framework for mathematical modeling of real world phenomena. In the last few years, some researchers have studied the existence of solutions for impulsive differential equations with boundary conditions via variational methods [ – ]. We say that φ satisfies the (PS)c condition if the existence of a sequence {uk} in X, such that φ(uk) → c, φ (uk) → as k → ∞, implies that c is a critical value of φ.
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