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Existence of fast homoclinic orbits for a class of second-order non-autonomous problems

Qiongfen Zhang1*, Qi-Ming Zhang2 and Xianhua Tang3

Author Affiliations

1 College of Science, Guilin University of Technology, Guilin, Guangxi, 541004, P.R. China

2 College of Science, Hunan University of Technology, Zhuzhou, Hunan, 412000, P.R. China

3 School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan, 410083, P.R. China

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Boundary Value Problems 2014, 2014:89  doi:10.1186/1687-2770-2014-89

Published: 6 May 2014

Abstract

By applying the mountain pass theorem and the symmetric mountain pass theorem in critical point theory, the existence and multiplicity of fast homoclinic solutions are obtained for the following second-order non-autonomous problem: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M1">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M6">View MathML</a> are not periodic in t and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M7">View MathML</a> is a continuous function and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M8">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M9">View MathML</a>.

MSC: 34C37, 35A15, 37J45, 47J30.

Keywords:
fast homoclinic solutions; variational methods; critical point