Periodic solution for second-order impulsive differential inclusions with relativistic operator

Suiming Shang,Zhanbing Bai,Yue Yue,Yu Tian

doi:10.1186/s13661-018-1088-6

Abstract

In this paper, the boundary value problem of a second-order impulsive differential inclusion involving a relativistic operator is studied. First, the singular problem is reduced to an equivalent non-singular problem in order to better apply the variational methods. Then the existence of a periodic solution is obtained by nonsmooth critical point theory. Moreover, the boundedness and nonnegativity of solutions are obtained by restricting the discontinuous nonlinear term.

Highlights

By physical experiments, a change of the mass of an object occurs when the velocity of the object is comparable with the speed of light
With the impulsive effects and differential inclusion taken into consideration, difficulties such as how to change the problem (1.7) into problem (1.6), how to deal with the non-differentiablity of the energy functional and how to prove the critical point of energy functional is classical solution of (1.7) have to be overcome
We obtained the existence of periodic solution by critical point theorem

Summary

Introduction

A change of the mass of an object occurs when the velocity of the object is comparable with the speed of light. We shall apply the critical point theorem [38] to obtain the existence and the property of weak solution for (1.7). With the impulsive effects and differential inclusion taken into consideration, difficulties such as how to change the problem (1.7) into problem (1.6), how to deal with the non-differentiablity of the energy functional and how to prove the critical point of energy functional is classical solution of (1.7) have to be overcome. We obtained the existence of periodic solution by critical point theorem. Lemma 2.3 ([38]) Let X be a real Banach space and let Φ, Ψ : X → R be two locally Lipschitz continuous functions. Since F is Lipschitz, there exists a positive constant B with –λΨ ◦(u; v) ≤ B v for any v ∈ HT This combining (3.2) implies | h, v | ≤ B v for any v ∈ HT.

Conclusion

Example