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Published in:
01-10-2016 | Original Research
Existence of two almost homoclinic solutions for p(t)-Laplacian Hamiltonian systems with a small perturbation
Published in: Journal of Applied Mathematics and Computing | Issue 1-2/2016
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Abstract
In this paper we study the existence of two almost homoclinic solutions for the following second order p(t)-Laplacian Hamiltonian systems with a small perturbation where \(t\in {\mathbb {R}}\), \(u\in {\mathbb {R}}^n\), \(p\in C({\mathbb {R}},{\mathbb {R}})\) with \(p(t)>1\), \(a\in C({\mathbb {R}},{\mathbb {R}})\), \(W\in C^1({\mathbb {R}}\times {\mathbb {R}}^n,{\mathbb {R}})\) and \(\nabla W(t,u)\) is the gradient of W(t, u) at u, \(f\in C({\mathbb {R}},{\mathbb {R}}^n)\) and belongs to \(L^{q(t)}({\mathbb {R}},{\mathbb {R}}^n)\). The point is that, assuming that a(t) is bounded in the sense that there are two constants \(0<\tau _1<\tau _2<\infty \) such that \(\tau _1\le a(t)\le \tau _2 \) for all \(t \in {\mathbb {R}}\), W(t, u) is of super-p(t) growth as \(|u|\rightarrow \infty \) and satisfies some other reasonable hypothesis, f is sufficiently small in \(L^{q(t)}({\mathbb {R}},{\mathbb {R}}^n)\), we provide one new criterion to ensure the existence of two almost homoclinic solutions. Recent results in the literature are extended and significantly improved.
$$\begin{aligned} \frac{d}{dt}\big (|\dot{u}(t)|^{p(t)-2} \dot{u}(t)\big )-a(t)|u(t)|^{p(t)-2}u(t)+\nabla W(t,u(t))=f(t), \end{aligned}$$