Abstract
The existence and multiplicity of positive solutions are established for second-order periodic boundary value problem. Our results are based on the theory of a fixed point index for A-proper semilinear operators defined on cones due to Cremins. Our approach is different in essence from other papers and the main results of this paper are also new.
Highlights
We discuss the existence of positive solutions of the periodic boundary value problem PBVP for second-order differential equation x t f t, x, 0 < t < 1, x0 x1, x 0 x 1, 1.1 where f : 0, 1 × R → R is a continuous function
We are interested in positive solutions of 1.1, because we have been motivated by a problem from the Theory of Nonlinear Elasticity modelling radial oscillations of an elastic spherical membrane made up of a neo-Hookean material and subjected to an internal pressure
Torres 9 and Yao 10 obtained some results on the existence of positive solutions of a general periodic boundary value problem x t f t, x t, 0 < t < 2π, 1.4 x 0 x 2π, x 0 x 2π
Summary
We discuss the existence of positive solutions of the periodic boundary value problem PBVP for second-order differential equation x t f t, x , 0 < t < 1, x0 x1, x 0 x 1, 1.1 where f : 0, 1 × R → R is a continuous function. Our purpose here is to provide sufficient conditions for the existence of multiple positive solutions to the periodic boundary value problem 1.1 This will be done by applying the theory of a fixed point index for A-proper semilinear operators defined on cones obtained by Cremins 1. Torres 9 and Yao 10 obtained some results on the existence of positive solutions of a general periodic boundary value problem x t f t, x t , 0 < t < 2π, 1.4 x 0 x 2π , x 0 x 2π. In this case, the problem 1.4 has no Green function.
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