Boundary value problems for \(n\)-th order differential inclusions with four-point integral boundary conditions (Q2901944)

The paper studies an \(n\)-th order differential inclusion with four-point integral boundary conditions of the form NEWLINE\[NEWLINE x^{(n)}\in F(t,x),\quad t\in (0,1), NEWLINE\]NEWLINE NEWLINE\[NEWLINE x(0)=\alpha \int_0^ax(s)\,ds,\;x'(0)=0,\;x''(0)=0,\dotsc,x^{(n-2)}(0)=0, NEWLINE\]NEWLINE NEWLINE\[NEWLINE x(1)=\beta \int_b^1x(s)\,ds,\quad 0<a<b<1, NEWLINE\]NEWLINE where \(F:[0,1]\times {\mathbb R}\to {\mathcal P}({\mathbb R})\) is a set-valued map and \(\alpha ,\beta \in {\mathbb R}\).NEWLINENEWLINEThree 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.