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Published in:
02-06-2016 | Original Research
Solutions for a class of fractional Hamiltonian systems with a parameter
Published in: Journal of Applied Mathematics and Computing | Issue 1-2/2017
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Abstract
In this paper we are concerned with the existence of solutions for the following fractional Hamiltonian systems with a parameter
where \(\alpha \in (1/2,1)\), \(t\in {\mathbb {R}}\), \(u\in {\mathbb {R}}^n\), \(\lambda >0\) is a parameter, \(L\in C({\mathbb {R}},{\mathbb {R}}^{n^2})\) is a symmetric matrix for all \(t\in {\mathbb {R}}\), \(W\in C^1({\mathbb {R}}\times {\mathbb {R}}^n,{\mathbb {R}})\) and \(\nabla W(t,u)\) is the gradient of W(t, u) at u. The novelty of this paper is that, assuming L(t) is a symmetric and positive semi-definite matrix for all \(t\in {\mathbb {R}}\), that is, \(L(t)\equiv 0\) is allowed to occur in some finite interval T of \({\mathbb {R}}\), W(t, u) satisfies Ambrosetti–Rabinowitz condition and some other reasonable hypotheses, we show the existence of nontrivial solution of (FHS)\(_\lambda \), which vanishes on \({\mathbb {R}}\backslash T\) as \(\lambda \rightarrow \infty \), and converges to \(\tilde{u}\in H^\alpha ({\mathbb {R}})\); here \(\tilde{u}\in E_0^\alpha \) is a nontrivial solution of the Dirichlet BVP for fractional systems on the finite interval T. Recent results are generalized and significantly improved.