Existence of bounded solutions of integral boundary value problems for singular differential equations on whole lines

Abstract

Boundary value problems of second-order singular differential equations with nonlinear operator Φ on whole lines are discussed. By applying the nonlinear alternative of Leray–Schauder-type fixed point theorem, some existence results of solutions for integral boundary value problems of differential equations on whole lines are established. The emphasis is put on the nonlinear operator [Φ(ρ(t)x′(t))]′ involved with the nonnegative function ρ that may satisfy ρ(0) = 0, the strictly increasing sup-multiplicative-like function Φ and the differential equations are defined on the whole line. Three examples and a remark are presented to illustrate the main theorems.