Abstract
In this paper, we study solvability of a class of second-order differential equations in a conservative Liénard form subject to periodic boundary conditions. Results on existence of non-trivial T-periodic solutions or positive T-periodic solutions are obtained respectively. Applications of the theorems are shown by examples. The results are proved by applying the coincidence degree theory for semilinear operator equations.
Highlights
We study existence of non-trivial solutions, or positive solutions of the following second-order periodic Boundary Value Problem (BVP): x00 þ f ðtÞx 1⁄4 gðxÞ; ð1:1Þ
(1.1) represents a class of conservative systems, where there is no damping force. It considers the special case where the restoring force can be expressed by a t-dependent linear part, and an x-dependent non-linear part
Only positive solutions are relevant, and so effort has gone into investigating existence of positive periodic solutions to equations like (1.3); for example, see [9,10,11]
Summary
We study existence of non-trivial solutions, or positive solutions of the following second-order periodic Boundary Value Problem (BVP): x00 þ f ðtÞx 1⁄4 gðxÞ; ð1:1Þ xð0Þ 1⁄4 xðT Þ; x0ð0Þ 1⁄4 x0ðT Þ; ð1:2Þ where g : R ! R and f : 1⁄20; T ! R are continuous functions. Since the early 1920’s, when Liénard first studied equations of the form x00 + f(x)x0 + x = 0 with periodic boundary conditions [3], many generalizations fitting the form (1.3) have been investigated (see, e.g., [1, 4,5,6,7,8] and the references therein) This is due to the wide application of these models to oscillatory systems arising in Physics and Engineering. The study of positive solutions for BVPs requires different techniques and stronger conditions than solution existence We prove both existence of a positive solution and existence of at least one solution using the same theorem from the coincidence degree theory for semilinear operator equations.
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