Multiple positive solutions to some second-order integral boundary value problems with singularity on space variable

Abstract

This article deals with integral boundary value problems of the second-order differential equations {u″(t)+a(t)u′(t)+b(t)u(t)+f(t,u(t))=0,t∈J+,u(0)=∫01g(s)u(s)ds,u(1)=∫01h(s)u(s)ds,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\left \\{ \\textstyle\\begin{array}{lcl} u''(t)+a(t)u'(t)+b(t)u(t)+f(t,u(t))=0,\\quad t\\in J_{+},\\\\ u(0)= \\int_{0}^{1}g(s)u(s)\\,\\text{d}s,\\qquad u(1)=\\int _{0}^{1}h(s)u(s)\\,\\text{d}s, \\end{array}\\displaystyle \\right .$$\\end{document} where ain C(J), bin C(J, R_{-}), fin C(J_{+}times R_{+}, R^{+}) and g, hin L^{1}(J) are nonnegative. The result of the existence of two positive solutions is established by virtue of fixed point index theory on cones. Especially, the nonlinearity f permits the singularity on the space variable.