Periodic solutions of a class of non‐autonomous second‐order differential inclusions systems

Daniel Paşca

doi:10.1155/s1085337501000525

Abstract

Using an abstract framework due to Clarke (1999), we prove the existence of periodic solutions for second‐order differential inclusions systems.

Highlights

Using an abstract framework due to Clarke (1999), we prove the existence of periodic solutions for second-order differential inclusions systems
For f2 we can apply Theorem 3.5 under Hypothesis 3.4 with the same cast of characters, but ft (x, y) = L2(t, x, y) = (1/2) y 2 + F2(t, x). It results that the integrand L2(t, x, y) is measurable in t for a given element (x, y) of Y and locally Lipschitz in (x, y) for each t ∈ [0, T ]
It follows like in the proof of Theorem 2.1 that φ is coercive by (ix), which completes the proof

Summary

Introduction

U(0) − u(T ) = u(0) − u(T ) = 0, where T > 0 and F : [0, T ] × Rn → R satisfies the following assumption: (A) F (t, x) is measurable in t for each x ∈ Rn and continuously differentiable in x for a.e. t ∈ [0, T ], and there exist a ∈ C(R+, R+), b ∈ L1(0, T ; R+) such that. Wu and Tang in [4] proved the existence of solutions for problem (1.1) when F = F1 + F2 and F1, F2 satisfy some assumptions. We will consider problem (1.1) in a more general sense. Our results represent the extensions to systems with discontinuity (we consider the generalized gradients unlike continuously gradient in classical results)

Main results

Preliminary results

Proofs of the theorems