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Functional differential inclusions with integral boundary conditions. (English) Zbl 1182.34006

Summary: We consider the second order functional differential inclusion
\[ x''(t)+\lambda x'(t)\in F(t,x_t)\text{ a.e. }t\in[0,1]\tag{1} \]
with initial function values,
\[ x(t)=\phi(t),\quad \in[-r,0],\tag{2} \]
and integral boundary conditions,
\[ x(1)=\int^1_0 g(x(s))\,ds\tag{3} \]
where \(F:[0,1]\times C([-r,0],\mathbb R)\to{\mathcal P}(\mathbb R)\) is a compact valued multivalued map, \({\mathcal P}(\mathbb R)\) is the family of all subsets of \(\mathbb R\), \(\lambda <0\), \(\phi\in C([-r,0],\mathbb R)\) and \(g:\mathbb R\to\mathbb R\) is continuous.
By using suitable fixed point theorems, we study the case when the right hand side has convex as well as nonconvex values.

MSC:

34A09 Implicit ordinary differential equations, differential-algebraic equations
34K10 Boundary value problems for functional-differential equations
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