*(English)*Zbl 1165.34010

Many boundary value problems associated with $n$th-order differential equations may be formulated as fixed point problems for nonlinear operators involving kernels, say Green’s functions. In general, the boundary conditions at the end-points of some bounded interval $[0,1]$ may be linear, nonlinear, integral, local, nonlocal,...

In this paper, the authors consider the existence of positive fixed points of the integral operator

under some conditions imposed on the kernel $k,$ the functions $g,{\gamma}_{i}$ and the nonlinearity $f\xb7$ The integer $N$ lies between 0 and the order of the underlying differential equation. The boundary operator involve linear functionals on $C[0,1]$, i.e. Stieltjes integrals

where ${B}_{j}$ are functions of bounded variation. These BCs contain as special cases multi-point BCs given by linear functionals ${\beta}_{i}\left[u\right]={\sum}_{i=1}^{m-2}{\beta}_{j}^{i}u\left({\eta}_{i}\right)$ with some ${\eta}_{i}\in (0,1)\xb7$ The ${\beta}_{j}^{i}$ may change sign and non-homogeneous conditions are also considered. The authors first prove cone invariance of some mappings in the space of continuous functions $C[0,1]$ and fixed point index results are obtained in theses cones. Some hypotheses on the nonlinearity $f$ including sublinear and upperlinear growths are assumed and an existence result is then derived. Also, a nonexistence result is proved. Finally, the authors illustrate their existence results by giving details for the weakly singular fourth-order equation

with nonlocal BCs

and local BCs

for which the Green’s function is easily determined. The paper ends with the case of non-homogeneous BCs.