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Non-local boundary value problems of arbitrary order. (English) Zbl 1165.34010

Many boundary value problems associated with nth-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

Tu(t)=Bu(t)+Fu(t)=: i=1 i=N β i [u]γ i (t)+ 0 1 k(t,s)g(s)f(s,u(s))ds,

under some conditions imposed on the kernel k, the functions g,γ 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[0,1], i.e. Stieltjes integrals

β i [u]= 0 1 u(s)dB j (s),

where B j are functions of bounded variation. These BCs contain as special cases multi-point BCs given by linear functionals β i [u]= i=1 m-2 β j i u(η i ) with some η i (0,1)· The β 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

u (4) (t)=g(t)f(t,u(t)),t(0,1)

with nonlocal BCs

u(0)=β 1 [u],u ' (0)=β 2 [u],u(1)=β 3 [u],u '' (1)=-β 4 [u]

and local BCs

u(0)=0,u ' (0)=0,u(1)=0,u '' (1)=0

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

MSC:
34B18Positive solutions of nonlinear boundary value problems for ODE
34B10Nonlocal and multipoint boundary value problems for ODE
47H11Degree theory (nonlinear operators)
47H30Particular nonlinear operators