Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary value problems of local and nonlocal type.

*(English)*Zbl 1146.34020The 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.

\[ u(t)= Au(t)\equiv\int_0^1k(t,s)g(s)f(s,u(s))\,ds,\quad 0<t<1. \]

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.

Reviewer: Smail Djebali (Algiers)

##### MSC:

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 |