In this paper we are concerned with the existence of solutions for the following fractional Hamiltonian systems with a parameter [graphic not available: see fulltext]where α∈(1/2,1)\documentclass[12pt]{minimal}
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\begin{document}$$\alpha \in (1/2,1)$$\end{document}, t∈R\documentclass[12pt]{minimal}
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\begin{document}$$t\in {\mathbb {R}}$$\end{document}, u∈Rn\documentclass[12pt]{minimal}
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\begin{document}$$u\in {\mathbb {R}}^n$$\end{document}, λ>0\documentclass[12pt]{minimal}
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\begin{document}$$\lambda >0$$\end{document} is a parameter, L∈C(R,Rn2)\documentclass[12pt]{minimal}
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\begin{document}$$L\in C({\mathbb {R}},{\mathbb {R}}^{n^2})$$\end{document} is a symmetric matrix for all t∈R\documentclass[12pt]{minimal}
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\begin{document}$$t\in {\mathbb {R}}$$\end{document}, W∈C1(R×Rn,R)\documentclass[12pt]{minimal}
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\begin{document}$$W\in C^1({\mathbb {R}}\times {\mathbb {R}}^n,{\mathbb {R}})$$\end{document} and ∇W(t,u)\documentclass[12pt]{minimal}
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\begin{document}$$\nabla W(t,u)$$\end{document} 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∈R\documentclass[12pt]{minimal}
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\begin{document}$$t\in {\mathbb {R}}$$\end{document}, that is, L(t)≡0\documentclass[12pt]{minimal}
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\begin{document}$$L(t)\equiv 0$$\end{document} is allowed to occur in some finite interval T of R\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {R}}$$\end{document}, W(t, u) satisfies Ambrosetti–Rabinowitz condition and some other reasonable hypotheses, we show the existence of nontrivial solution of (FHS)λ\documentclass[12pt]{minimal}
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\begin{document}$$_\lambda $$\end{document}, which vanishes on R\T\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {R}}\backslash T$$\end{document} as λ→∞\documentclass[12pt]{minimal}
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\begin{document}$$\lambda \rightarrow \infty $$\end{document}, and converges to u~∈Hα(R)\documentclass[12pt]{minimal}
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\begin{document}$$\tilde{u}\in H^\alpha ({\mathbb {R}})$$\end{document}; here u~∈E0α\documentclass[12pt]{minimal}
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\begin{document}$$\tilde{u}\in E_0^\alpha $$\end{document} is a nontrivial solution of the Dirichlet BVP for fractional systems on the finite interval T. Recent results are generalized and significantly improved.
