Existence of positive solutions for a system of nonlinear second-order integral boundary value problems (Q260853)

The paper is concerned with the solvability of the following system of nonlinear second-order differential equations: NEWLINENEWLINE\[NEWLINE \begin{gathered} (a(t)u'(t))'-b(t)u(t)+f(t,v(t))=0,\;0<t<1,\\ NEWLINE(c(t)v'(t))'-d(t)v(t)+g(t,u(t))=0,\;0<t<1 \end{gathered} NEWLINE\]NEWLINE NEWLINEwith the Riemann-Stieljes integral boundary conditionsNEWLINENEWLINE\[NEWLINE \begin{gathered} \alpha u(0)-\beta a(0)u'(0)=\int_0^1u(s)dH_1(s),\;\gamma u(1)+\delta a(1)u'(1)=\int_0^1u(s)dH_2(s),\\ NEWLINE\tilde\alpha v(0)-\tilde\beta c(0)v'(0)=\int_0^1v(s)dK_1(s),\;\tilde\gamma v(1)+\tilde\delta c(1)v'(1)=\int_0^1v(s)dK_2(s). \end{gathered} NEWLINE\]NEWLINE NEWLINEUsing appropriate Green's function, the problem is reformulated into a fixed point problem for a compact operator taking into account the generalized Sturm-Liouville boundary conditions at the end-points. The fixed point index theory is then employed to show existence of at least one solution (Theorems 3.1, 3.2) or two solutions (Theorem 3.3) in the positive cone of continuous functions on the interval \([0,1]\). The functions \(f\) and \(g\) satisfy nonlinear growth conditions including sub-linearities and super-linearities. An example is included to illustrate Theorem 3.