Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary value problems of local and nonlocal type. (English) Zbl 1146.34020

The aim of this paper is to prove the existence of multiple positive solutions to the Hammerstein integral equation
\[ u(t)= Au(t)\equiv\int_0^1k(t,s)g(s)f(s,u(s))\,ds,\quad 0<t<1. \]
The particular features of this problem lie in the fact that \(k\) is not necessarily symmetric and may further exhibit discontinuities in its second variable. The function \(g\) is \(L^1(0,1).\) The nonlinearity \(f\) is \(L^1_{\text{loc}}\) Carathéodory. The fixed point index on the positive cone of the Banach space of continuous functions on \([0,1]\) is computed for the mapping \(A.\) The discussion is given with respect to the relative position of the behavior of \(\lim\frac{f(u)}{u}\) as \(u\) tends to \(0^+\) or \(+\infty\) and the principal eigenvalue of the corresponding linear integral operator \[ Lu(t)\equiv\int_0^1k(t,s)g(s)u(s)\,ds,\quad 0<t<1. \] As a consequence, existence results for the integral equation are provided with application to the second-order differential equation \[ u''(t)+g(t)f(t,u(t))=0,\quad \text{a.e. on }\;[0,1]. \] A particular attention is then given to the case where \(k\) is symmetric which corresponds to the Green’s function when separated boundary value conditions are associated with this equation. The case of nonlocal multi-point boundary conditions is also discussed. The obtained existence theorems improve in some sense some results in the literature especially those with sub-linear and super-linear growth nonlinearities. Some example of applications illustrate this rich paper.


34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
47H10 Fixed-point theorems
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
45G10 Other nonlinear integral equations