Multiplicity Of Positive Solutions To Singular Superlinear Second-order Periodic Boundary Value Problems

Many authors have investigated the existence of positive solutions for differential equations with seperated boundary value problems, for examples, see, [1-10, 15,20] , the proof of the existence of the positive solutions is based on an application of a fixed point theorem for completely continuous operators in cones and the positive properties of the Green's function. However, there are few results on second-order nonlinear periodic differential equations except. Recently, the authors of [14, 16, 17] considered the above problem and gave the sufficient conditions about the existence of single and multiple solutions by employing the norm-type expansion and compression theorem in cones due to Krasnoselskii. In this paper, we are devoted to establish the multiplicity of positive solutions to superlinear attractive singular equations with periodic boundary conditions. It is proved that such a problem has at least two positive solutions under our reasonable conditions. Our nonlinearity may be singular in its dependent variable and superlinear at infinity. The proof relies on a nonlinear alternative of Leray-Schaudcr type and on Krasnoselskii fixed point theorem on compression and expansion of cones. The Green Function is also important in the proof. The existence of the first solution is obtained using a nonlinear alternative of Leray-Schauder, and the second one is found using a fixed point theorem in cones.Besides fixed point theorems in a cone used in the existence problems, another tool--the method of upper and lower solutions-is also used in the literature. In fact, the method of upper and lower solutions is much more frequently used.