×

Boundary value problems for \(n\)-th order differential inclusions with four-point integral boundary conditions. (English) Zbl 1252.34024

The paper studies an \(n\)-th order differential inclusion with four-point integral boundary conditions of the form \[ x^{(n)}\in F(t,x),\quad t\in (0,1), \]
\[ x(0)=\alpha \int_0^ax(s)\,ds,\;x'(0)=0,\;x''(0)=0,\dotsc,x^{(n-2)}(0)=0, \]
\[ x(1)=\beta \int_b^1x(s)\,ds,\quad 0<a<b<1, \] where \(F:[0,1]\times {\mathbb R}\to {\mathcal P}({\mathbb R})\) is a set-valued map and \(\alpha ,\beta \in {\mathbb R}\).
Three existence results are obtained for the problem considered. The first result relies on the nonlinear alternative of Leray-Schauder type, the second result essentially uses the Bressan-Colombo selection theorem for lower semicontinuous set-valued maps with decomposable values and the third result is based on the Covitz-Nadler contraction principle for set-valued maps.

MSC:

34A60 Ordinary differential inclusions
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
PDFBibTeX XMLCite
Full Text: DOI