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Perturbation of a single-valued operator by a multi-valued mapping of Hammerstein type with nonconvex images. (English. Russian original) Zbl 0999.47035

Russ. Math. 43, No. 3, 1-13 (1999); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1999, No. 3, 3-16 (1999).
The authors study the inclusion \(x\in f(x)+ V\Phi(x)\), where \(f: C([a,b],\mathbb{R}^n)\to C([a,b],\mathbb{R}^n)\) is compact, \(\Phi: C([a,b],\mathbb{R}^n)\to L^1([a,b],\mathbb{R}^n)\) is multivalued with certain additional properties, and \(V: L^1([a,b],\mathbb{R}^n)\to C([a,b],\mathbb{R}^n)\) is a linear integral operator. The main tool is a fixed point result due to the first author [Mat. Sb. 183, No. 10, 63-86 (1992; Zbl 0772.34017)].

MSC:

47H04 Set-valued operators
28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections
34A60 Ordinary differential inclusions
45N05 Abstract integral equations, integral equations in abstract spaces
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
47J05 Equations involving nonlinear operators (general)

Citations:

Zbl 0772.34017
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