Positive solutions of a second-order integral boundary value problem

AbstractIn this paper we study the existence of positive solutions of a second-order integral boundary value problems for ordinary differential equations. Our results presented here unify, generalize and substantially improve the existing results in the literature. Moreover, it is worthwhile to point out that our method will dispense with constructing a new Green function

Positive solutions for a system of nnth-order nonlinear boundary value problems

In this paper, we investigate the existence, multiplicity and uniqueness of positive solutions for the following system of $n$th-order nonlinear boundary value problems $$\begin{cases} u^{(n)}(t)+f(t,u(t),v(t))=0,0<t<1,\\v^{(n)}(t)+g(t,u(t),v(t))=0, 0<t<1,\\ u(0)=u'(0)=\ldots=u^{(n-2)}(0)=u(1)=0,\\ v(0)= v'(0)=\ldots=v^{(n-2)}(0)=v(1)=0. \end{cases}$$ Based on a priori estimates achieved by using Jensen's integral inequality, we use fixed point index theory to establish our main results. Our assumptions on the nonlinearities are mostly formulated in terms of spectral radii of associated linear integral operators. In addition, concave and convex functions are utilized to characterize coupling behaviors of $f$ and $g$, so that we can treat the three cases: the first with both superlinear, the second with both sublinear, and the last with one superlinear and the other sublinear