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Existence of positive homoclinic solutions for damped differential equations. (English) Zbl 1379.34038

Summary: This paper is concerned with the existence of positive homoclinic solutions for the second-order differential equation \[ u''+cu'-a(t)u+f(t,u)=0, \] where \(c\geq 0\) is a constant and the functions \(a\) and \(f\) are continuous and not necessarily periodic in \(t\). Under other suitable assumptions on \(a\) and \(f\), we obtain the existence of positive homoclinic solutions in both cases sub-quadratic and super-quadratic by using critical point theorems.

MSC:

34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
37C60 Nonautonomous smooth dynamical systems
58E50 Applications of variational problems in infinite-dimensional spaces to the sciences
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