Positive solutions to second-order singular nonlocal problems: existence and sharp conditions

Abstract

Abstract In this paper we consider sharp conditions on ω and f for the existence of C1[0,1] $C^{1}[0,1]$ positive solutions to a second-order singular nonlocal problem u″(t)+ω(t)f(t,u(t))=0 $u''(t)+\omega (t)f(t,u(t))=0$, u(0)=u(1)=∫01g(t)u(t)dt $u(0)=u(1)=\int _{0} ^{1}g(t)u(t)\,dt$; it turns out that this case is more difficult to handle than two point boundary value problems and needs some new ingredients in the arguments. On the technical level, we adopt the topological degree method.

Keywords

• Sharp conditions
• Singular boundary value problems with integral boundary conditions
• Hölder’s inequality
• Fixed point theorems
• Positive solutions