Abstract
In this paper, we study a class of second-order neutral impulsive functional differential equations. Under certain conditions, we establish the existence of multiple periodic solutions by means of critical point theory and variational methods. We propose an example to illustrate the applicability of our result.
Highlights
Introduction and Main ResultsIn this paper we consider a class of second-order neutral impulsive functional differential equations u (t − τ) − u (t − τ)+ λf (t, u (t), u (t − τ), u (t − 2τ)) = 0, t ≠ tj, t ∈ J = [0, 2kτ], (1)Δu = Ij (u), j = 1, 2, . . . , l, u (0) − u (2kτ) = u (0) − u (2kτ) = 0, where f ∈ C(R4, R), Ij ∈ C(R, R), and 0 = t0 < t1 < t2 < ⋅ ⋅ ⋅ < tl < tl+1 = 2kτ
The necessity to study delay differential equations is due to the fact that these equations are useful mathematical tools in modeling many real processes and phenomena studied in biology, medicine, chemistry, physics, engineering, economics, and so forth [1, 2]
Impulsive differential equation is richer than the corresponding theory of differential equations and represents a more natural framework for mathematical modeling of real world phenomena
Summary
In this paper we consider a class of second-order neutral impulsive functional differential equations u (t − τ) − u (t − τ). People generally consider impulses in positions u and u for the second-order differential equation u = f(t, u, u). The existence of periodic solutions of delay differential equations has been focused on by many researchers [3–6]. Some researchers have studied the existence of solutions for delay differential equations via variational methods [11–13]. Some researchers, by using critical point theory, have studied the existence of solutions for boundary value problems, periodic solutions, and homoclinic orbits of impulsive differential systems [14–19]. We aim to establish existence of multiple periodic solutions for the second-order neutral impulsive. Mathematical Problems in Engineering functional differential equation (1) by using critical point theory and variational methods.
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