Multiplicity of solutions for impulsive differential equation on the half-line via variational methods

Abstract

In this paper, the existence of solutions for a second-order impulsive differential equation with two parameters on the half-line is investigated. Applying variational methods, we give some new criteria to guarantee that the impulsive problem has at least one classical solution, three classical solutions and infinitely many classical solutions, respectively. Some recent results are extended and significantly improved. Two examples are presented to demonstrate the application of our main results.

Multiple positive solutions for a second-order boundary value problem on the half-line

“…This requires first to give the variational structure of the problem, then to obtain a Harnack type inequality, and finally to guarantee the invariance, compactness and compression conditions from the general critical point theorems which are used. This way, our approach completely differs from those in [8] and [9].…”

Positive Periodic Solutions for a First-Order Nonlinear Neutral Differential Equation with Impulses on Time Scales

“…For important research on impulsive differential equations or systems, Akgl and Zafer [19] obtained prescribed asymptotic behavior of second-order impulsive differential equations via principal and nonprincipal solutions; principal and nonprincipal solutions of impulsive differential equations were studied in [20]; Akgöl and Zafer [21] obtained asymptotic integration of second-order impulsive differential equations; the authors of [22] investigated multiplicity results for second order impulsive differential equations by variational methods. For more details about impulsive differential equations, see [23][24][25][26][27].…”

Oscillation Criteria for Solutions of Neutral Differential Equations of Impulses Effect with Positive and Negative Coefficients

“…Mohamad et al (8) obtained sufficient conditions for the oscillation of all solutions of neutral differential equations with variable delays, while H. Chen et al (9) studied the existence of solutions for impulsive differential equations. Isaac (10) classifies non-oscillatory solutions of impulsive differential equations of the second order.…”