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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 $$ x'(t)\in b(t)x(t)+F(t,x(t)),\quad t\in (0,1),\quad x(0)=x(1) \tag1$$ and also with the first-order differential inclusion subject to the non-periodic conditions $$x'(t)\in F(t,x(t)),\quad t\in (0,1), \quad Ax(0)+Bx(1)=0.\tag2$$ Here, $I=[0,1]$, $F:I\times \bbfR^n \to 2^{\bbfR^n}$ is a Carathéodory multifunction, $b: I\to \bbfR$ is continuous and does not vanish on the whole interval $I$. Further, $A$, $B$ are $n\times n$ matrices with real elements such that $\det(A+B)\not=0$, and either $\det(A)\not=0$, $\Vert A^{-1}B\Vert <1$, or $\det(B)\not=0$, $\Vert B^{-1}A\Vert <1$. 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.

34A60Differential inclusions
34B15Nonlinear boundary value problems for ODE
47N20Applications of operator theory to differential and integral equations
Full Text: DOI
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