Existence and multiplicity of solutions for second-order impulsive differential equations on the half-line
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RESEARCH
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Existence and multiplicity of solutions for second-order impulsive differential equations on the half-line Lizhao Yan1,2 , Jian Liu3* and Zhiguo Luo2 *
Correspondence: [email protected] School of Economics and Management, Changsha University of Science and Technology, Changsha, Hunan 410076, P.R. China Full list of author information is available at the end of the article 3
Abstract In this paper, we use variational methods to investigate the solutions of impulsive differential equations on the half-line. The conditions for the existence and multiplicity of solutions are established. The main results are also demonstrated with examples. Keywords: variational methods; impulsive differential equations; boundary value problem; half-line
1 Introduction Impulsive differential equations arising from the real world describe the dynamics of processes in which sudden, discontinuous jumps occur. Such processes are naturally seen in biology, medicine, mechanics, engineering, chaos theory and so on. Due to their significance, a great deal of work has been done in the theory of impulsive differential equations [–]. In this paper, we consider the following second-order impulsive differential equations on the half-line: ⎧ ⎪ ⎪ ⎨–u (t) + u(t) = μf (t, u(t)), t = tj , a.e. t ∈ [, +∞), –u (tj ) = Ij (u(tj )), j = , , . . . , n, ⎪ ⎪ ⎩ u () = , u (+∞) = ,
(.)
where = t < t < t < · · · < tn < ∞, u (tj ) = u (tj+ ) – u (tj– ) for u (tj± ) = limt→t± u (t), j = j , , . . . , n, u (+∞) = limt→+∞ u (t). In recent years, boundary value problems (BVPs) for impulsive differential equations in an infinite interval have been studied extensively and many results for the existence of solutions, positive solutions, multiple solutions have been obtained [–]. The main methods used for the infinite interval problems are upper and lower solutions techniques, fixed point theorems and the coincidence degree theory of Mawhin in a special Banach space. On the other hand, many researchers used variational methods to study the existence of solutions for impulsive boundary value problems on the finite intervals [–]. However, to the best of our knowledge, the study of solutions (in particular the multiplicity of solutions) for impulsive boundary value problems on the half-line using a varia©2013 Yan et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Yan et al. Advances in Difference Equations 2013, 2013:293 http://www.advancesindifferenceequations.com/content/2013/1/293
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tional method has received considerably less attention. In [], Chen and Sun studied the following equations: ⎧ ⎪ ⎪ ⎨–u (t) + u(t) = λf (t, u(t)), t = tj , a.e. t ∈ [, +∞), –u (tj ) = Ij (u(tj )), j = , , . . . , l, ⎪ ⎪ ⎩ + u ( ) = g(u()), u (+∞) = ,
(.)
where λ is
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