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An existence theorem on unbounded intervals for a class of second order functional differential inclusions in Banach spaces. (English) Zbl 1025.34085
The author proves an existence result for solutions on infinite intervals for an initial value problem for second-order functional-differential inclusions of the form $(\rho(t)y'(t))'\in F(t,y_{t}), \quad \text{a.e.}\quad t\in [0,\infty),\quad y_{0}=\phi, \;y'(0)=\eta,$ where $$F:[0,\infty)\times C([-r,0],E)\to 2^{E}$$ is a bounded, closed, convex-valued multivalued map, $$\rho\in C([0,\infty),\mathbb{R}_{+}),$$ $$\phi\in C([-r,0],E)$$ and $$E$$ a real Banach space. The strategy is to convert the problem to an appropriate fixed-point problem and use the fixed-point theorem due to Ma, which is an extension to multivalued maps on locally convex topological spaces of Schaefer’s theorem.

##### MSC:
 34K30 Functional-differential equations in abstract spaces 34A60 Ordinary differential inclusions 34K05 General theory of functional-differential equations
##### Keywords:
differential inclusion; initial value problems; fixed-point