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.


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