Multiple solutions of singular three-point boundary value problems on [0,∞)

In this paper, we study the existence of single and multiple solutions of three-point boundary value problems for the following nonlinear singular second-order differential equations View the MathML source{x″(t)−px′(t)−qx(t)+h(t)f(t,x(t))=0,t∈(0,+∞),ax(0)−bx′(0)−kx(ξ)=c≥0,limt→+∞x(t)ert=d≥0, Turn MathJax on where View the MathML sourcep,b≥0,a>k>0,q>0,0<ξ<+∞,r∈[0,p+p2+4q2], h:(0,+∞)→(0,+∞)h:(0,+∞)→(0,+∞) is continuous and may be singular at t=0t=0, f:[0,+∞)×[0,+∞)→(−∞,+∞)f:[0,+∞)×[0,+∞)→(−∞,+∞) is continuous and may take a negative value. By applying the technique of lower and upper solutions and the theory of topological degree, we obtain the conditions for the existence of at least one solution and at least three solutions respectively.