Multiplicity Of Positive Solutions To Second Order Differential Equation With Period Boundary Conditions

In this paper, we consider the following periodic boundary value problem:where x~[1](t) = p(t)x'(t). Throughout this paper we always assume that p(t) > 0, q(t) > 0, and the function f(t,x) satisfies f : [0,1] × (0, +∞)→ [0, +∞) is continuous, f may be singular at x = 0.Many authors have investigated the existence of positive solutions for differential equations with seperated boundary value problems, for examples, see, [4-13, 15,20]. Some 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 uses the positive properties of the Green's function. However, there are few results on second-order nonlinear periodic differential equations except [14, 16, 17]. 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 due to Krasnoselskii.In this paper we shall present a new existence theory for single and multiple positive periodic solutions for periodic boundary value problems by applying a well-known fixed point theorem in cones (see Theorem 2.3). The conditions in our main theorems can easily be checked in practice.