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The method of upper and lower solutions for second order differential inclusions with integral boundary conditions. (English) Zbl 1205.34013

The authors study a class of nonlinear boundary value problems for second order differential inclusions. In particular, they consider the problem
\[ x''(t)+\lambda x'(t)\in F(t,x(t)) \text{ a.e. on } T=[0,1], \]
\[ x(0)=a, \quad x(1)=\displaystyle{\int_0^1 g(x(s))\,ds}, \]
where \(F:[0,1]\times \mathbb R \to 2^{\mathbb R}\) is a Carathéodory multivalued map with non empty, compact and convex values, \(\lambda>0\), \(a\in\mathbb R\) and \(g:\mathbb R \to \mathbb R\) is a continuous and nondecreasing function. By using fixed point techniques combined with the method of lower and upper solutions, the authors prove an existence result.

MSC:

34A60 Ordinary differential inclusions
34B15 Nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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