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Existence of periodic and non-periodic solutions to systems of boundary value problems for first-order differential inclusions with super-linear growth. (English) Zbl 1165.34004

The paper deals with the first-order differential inclusion subject to the periodic conditions

${x}^{\text{'}}\left(t\right)\in b\left(t\right)x\left(t\right)+F\left(t,x\left(t\right)\right),\phantom{\rule{1.em}{0ex}}t\in \left(0,1\right),\phantom{\rule{1.em}{0ex}}x\left(0\right)=x\left(1\right)\phantom{\rule{2.em}{0ex}}\left(1\right)$

and also with the first-order differential inclusion subject to the non-periodic conditions

${x}^{\text{'}}\left(t\right)\in F\left(t,x\left(t\right)\right),\phantom{\rule{1.em}{0ex}}t\in \left(0,1\right),\phantom{\rule{1.em}{0ex}}Ax\left(0\right)+Bx\left(1\right)=0·\phantom{\rule{2.em}{0ex}}\left(2\right)$

Here, $I=\left[0,1\right]$, $F:I×{ℝ}^{n}\to {2}^{{ℝ}^{n}}$ is a Carathéodory multifunction, $b:I\to ℝ$ is continuous and does not vanish on the whole interval $I$. Further, $A$, $B$ are $n×n$ matrices with real elements such that $det\left(A+B\right)\ne 0$, and either $det\left(A\right)\ne 0$, $\parallel {A}^{-1}B\parallel <1$, or $det\left(B\right)\ne 0$, $\parallel {B}^{-1}A\parallel <1$. The authors provide new sufficient conditions under which solutions of problem (1) or problem (2) exist. The results apply to differential inclusions that may have a right-hand side with a super-linear growth in its second variable and also apply to systems of first-order differential inclusions. The proofs are based on novel differential inequalities and the Leray-Schauder nonlinear alternative. Some new results for ordinary differential equations with Carathéodory single-valued right-hand sides are also obtained.

##### MSC:
 34A60 Differential inclusions 34B15 Nonlinear boundary value problems for ODE 47N20 Applications of operator theory to differential and integral equations