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Eigenvalues and non-negative solutions of a system with nonlocal BCs. (English) Zbl 1184.34027
The authors are concerned with the existence of nonnegative solutions for the following eigenvalue system of second-order differential equations:
\[ \begin{cases}\lambda u''(t)+g_1(t)f_1(v(t))=0 &\text{for }0<t<1\\ \lambda u''(t)+g_2(t)f_2(v(t))=0 &\text{for }0<t<1 \end{cases} \]
with the nonlocal boundary conditions
\[ \begin{cases} u(0)=0, &u(1)=\alpha[u]\\ v(0)=0, &v(1)=\beta[v] \end{cases}, \]
where \(\alpha[.]\) and \(\beta[.]\) are positive affine functionals, for instance:
\[ \alpha[u]=A_0+\int_0^1u(s)dA(s) \]
that involves the Lebesgue-Stieljes integral and includes as special cases the integral and multi-point boundary conditions. \(\lambda\) is a real parameter playing the role of an eigenvalue of the problem, the nonlinearities \(f_1\), \(f_2\) are positive, continuous, while \(g_1,\) \(g_2\) are non-negative \(L^1-\)functions. The authors are interested in the existence of non-negative solutions in a positive cone of the Banach space \(C[0,1]\). They first consider the case \(\lambda=1\) and prove existence of solutions for \(\lambda\) in a certain interval. The proofs use the classical fixed point index on cones of Banach spaces.

34B09 Boundary eigenvalue problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations