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Non-local boundary value problems of arbitrary order. (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 $\left[0,1\right]$ may be linear, nonlinear, integral, local, nonlocal,...

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

$Tu\left(t\right)=Bu\left(t\right)+Fu\left(t\right)=:\sum _{i=1}^{i=N}{\beta }_{i}\left[u\right]{\gamma }_{i}\left(t\right)+{\int }_{0}^{1}k\left(t,s\right)g\left(s\right)f\left(s,u\left(s\right)\right)ds,$

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

${\beta }_{i}\left[u\right]={\int }_{0}^{1}u\left(s\right)d{B}_{j}\left(s\right),$

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 ${\eta }_{i}\in \left(0,1\right)·$ 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 $C\left[0,1\right]$ 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

${u}^{\left(4\right)}\left(t\right)=g\left(t\right)f\left(t,u\left(t\right)\right),\phantom{\rule{1.em}{0ex}}t\in \left(0,1\right)$

with nonlocal BCs

$u\left(0\right)={\beta }_{1}\left[u\right],\phantom{\rule{0.277778em}{0ex}}{u}^{\text{'}}\left(0\right)={\beta }_{2}\left[u\right],\phantom{\rule{0.277778em}{0ex}}u\left(1\right)={\beta }_{3}\left[u\right],\phantom{\rule{0.277778em}{0ex}}{u}^{\text{'}\text{'}}\left(1\right)=-{\beta }_{4}\left[u\right]$

and local BCs

$u\left(0\right)=0,\phantom{\rule{0.277778em}{0ex}}{u}^{\text{'}}\left(0\right)=0,\phantom{\rule{0.277778em}{0ex}}u\left(1\right)=0,\phantom{\rule{0.277778em}{0ex}}{u}^{\text{'}\text{'}}\left(1\right)=0$

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

MSC:
 34B18 Positive solutions of nonlinear boundary value problems for ODE 34B10 Nonlocal and multipoint boundary value problems for ODE 47H11 Degree theory (nonlinear operators) 47H30 Particular nonlinear operators