# zbMATH — the first resource for mathematics

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)
##### Keywords:
inclusion; linear integral operator