Infinitely many solutions for an instantaneous and non-instantaneous fourth-order differential system with local assumptions

TL;DR

This paper addresses the multiplicity of solutions for a fourth-order impulsive differential system that combines instantaneous and non-instantaneous impulses. It employs a variational framework built around a Kajikiya variant of Clark's theorem, augmented by a local (near-zero) subquadratic growth condition and a cut-off technique to handle locality and the perturbation-induced loss of symmetry, allowing a small parameter $\varepsilon$ to be nonzero. Under the constraint $β<\frac{2e^{H_0}}{T^2}$ and with the nonlinear terms $f_i$ and impulsive terms $I_i$ odd and locally subquadratic, the paper proves that for any $k$ there exists $\varepsilon(k)>0$ so that at least $k$ distinct $L^{\infty}$-small solutions exist; when $\varepsilon=0$, infinitely many solutions exist. The results extend prior work by accommodating a nonzero perturbation and a broader local regime, using a well-defined modified functional $\tilde{J}_\varepsilon$ whose critical points correspond to weak solutions, and providing an explicit example confirming the theory.

Abstract

We investigate a class of fourth-order differential systems with instantaneous and non-instantaneous impulses. Our technical approach is mainly based on a variant of Clark's theorem without the global assumptions. Under locally subquadratic growth conditions imposed on the nonlinear terms $f_i(t,u)$ and impulsive terms $I_i$, combined with perturbations governed by arbitrary continuous functions of small coefficient $\varepsilon$, we establish the existence of multiple small solutions. Specifically, the system exhibits infinitely many solutions in the case where $\varepsilon=0$.