Positive solutions of some higher order nonlocal boundary value problems. (English) Zbl 1201.34043

Webb, J. R. L.

Electron. J. Qual. Theory Differ. Equ. 2009, Spec. Iss. I, Paper No. 29, 15 p. (2009).

The paper studies the higher order differential equation
\[ u^{(n)}(t)+\lambda g(t)f(t,u(t))=0 \eqno (1) \]
subject to one of the following nonlocal boundary conditions
\[ u(0)=0,\quad u^{(k)}=0,\;2\leq k\leq n-1,\quad u'(1)=\alpha[u], \eqno (2) \]
\[ u(0)=0,\quad u^{(k)}=0,\;1\leq k\leq n-2,\quad u'(1)=\alpha[u], \eqno (3) \]
\[ u(0)=0,\quad u^{(k)}=0,\;1\leq k\leq n-2,\quad u''(1)=\alpha[u], \eqno (4) \]
where \(\alpha[u]\) is some functional given by the Riemann-Stieltjes integral \(\alpha[u]=\int_0^1 u(s)dA(s)\), with \(A\) a function of bounded variation. The author gets existence results by the standard methodology of seeking positive solutions as fixed points of the integral operator \(Su(t)=\int_0^1G(t,s)g(s)f(s,u(s))\,ds\), where \(G\) is the Green’s function corresponding to each BVP. Here, the Krasnoselskii type of cone, results of Webb and Lan involving comparison with the principal eigenvalue of a related linear problem and a unified method worked out by Webb and Infante are used. Some explicit examples with a calculation of the constants that are required by the theory are given.

Reviewer: Irena Rachůnková (Olomouc)

MSC:

34B18	Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15	Nonlinear boundary value problems for ordinary differential equations
34B10	Nonlocal and multipoint boundary value problems for ordinary differential equations
47N20	Applications of operator theory to differential and integral equations

Keywords:

positive solution; boundary value problem; nonlocal boundary condtions; fixed point index