Boundary value problems for first order multivalued differential systems.(English)Zbl 1117.34006

The authors investigate the solvability of the boundary value problem for the first order differential inclusion $x^{\prime }\in A(t)x+F\left ( t,x\right ),\quad t\in (0,1); \quad Mx(0)+Nx\left ( 1\right ) =0, \tag{*}$ where $$F\:[0,1]\times \mathbb R^n \to 2^{\mathbb R^n}$$ is a Carathéodory multifunction, $$A$$ is a continuous $$n\times n$$ matrix function, and $$M$$, $$N$$ are constant $$n\times n$$ matrices. Using the topological transversality theorem of A. Granas, fixed point theorems for multivalued mappings, and various differential inequalities, additional conditions on the nonlinearity $$F$$ are given which guarantee that (*) has at least one solution. A typical result is the following statement:
Suppose that $$F$$ is a $$L^1$$-Carathéodory multifunction whose values are bounded, closed, convex and nonempty subsets of $$\mathbb R^n$$ such that $$\| F(t,x)\| \leq \alpha (t) \psi (\| x\| )$$ for a.e.$$t\in I=[0,1]$$ and all $$x\in \mathbb R^n$$, where $$\alpha \in L^1(I)$$ and $$\psi \:[0, \infty )\to [0, \infty )$$ is a nondecreasing function such that $\limsup _{\varrho \to +\infty } \frac {\varrho } {\psi (\varrho )}=+\infty .$ Then (*) has at least one solution.

MSC:

 34A60 Ordinary differential inclusions 34B15 Nonlinear boundary value problems for ordinary differential equations 47H10 Fixed-point theorems
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