Perturbation of a convex-valued operator by a set-valued map of Hammerstein type with nonconvex values, and boundary value problems for functional-differential inclusions.

*(English. Russian original)*Zbl 0920.34017
Sb. Math. 189, No. 6, 821-848 (1998); translation from Mat. Sb. 189, No. 6, 3-32 (1998).

The first part of this article deals with the nonlinear operator inclusion
\[
x\in \Psi(x)+ V\Phi(x),
\]
where \(\Psi: C^n[a,b]\to \Omega(C^n[a, b])\) is a compact multivalued operator, \(\Phi: C^n[a,b]\to \Pi(C^n[a, b])\) a multivalued operator mapping bounded sets in sets with absolutely continuous norms, \(V: L^n[a,b]\to C^n[a,b]\) a linear integral operator with the kernel \(k(t,s)\) mapping weakly compact sets in compact ones; here \(C^n[a,b]\) and \(L^n[a,b]\) are the spaces of continuous and integrable \(n\)-dimensional vector-functions, \(\Omega(\cdot)\) is the metric space of nonempty convex closed bounded and convex with respect switching sets in a corresponding Banach space, \(\Pi(\cdot)\) the metric space of nonempty closed bounded and convex with respect switching sets.

The basic results are (1) an existence result based on the use of Michael’s selection theorem and Schauder fixed point principle, (2) a theorem on estimates of the distance between a solution to this inclusion and a fixed continuous vector function, (3) a theorem on coincidence the sets of solutions to the original inclusion and the ‘convexized’ inclusion \(x\in \Psi(x)+ V \overline{\text{co}} \Phi(x)\), and (4) some theorems described the structural properties of the set of solutions, including the ‘bang-bang’ principle.

The second part of the article is devoted to applications of all these results to the boundary value problem \[ {\mathcal L}x\in \Phi(x),\quad \ell(x)\in \varphi(x), \] with the linear continuous operator \({\mathcal L}: D^n[a,b]\to L^n[a,b]\), \(\ell: D^n[a,b]\to \mathbb{R}^n\), and multivalued operators \(\Phi: C^n[a,b]\to \Pi(L^n[a, b])\), \(\varphi: C^n[a,b]\to \Omega(\mathbb{R}^n)\) (\(D^n[a,b]\) is the space of absolutely continuous \(n\)-dimensional vector functions).

The basic results are (1) an existence result based on the use of Michael’s selection theorem and Schauder fixed point principle, (2) a theorem on estimates of the distance between a solution to this inclusion and a fixed continuous vector function, (3) a theorem on coincidence the sets of solutions to the original inclusion and the ‘convexized’ inclusion \(x\in \Psi(x)+ V \overline{\text{co}} \Phi(x)\), and (4) some theorems described the structural properties of the set of solutions, including the ‘bang-bang’ principle.

The second part of the article is devoted to applications of all these results to the boundary value problem \[ {\mathcal L}x\in \Phi(x),\quad \ell(x)\in \varphi(x), \] with the linear continuous operator \({\mathcal L}: D^n[a,b]\to L^n[a,b]\), \(\ell: D^n[a,b]\to \mathbb{R}^n\), and multivalued operators \(\Phi: C^n[a,b]\to \Pi(L^n[a, b])\), \(\varphi: C^n[a,b]\to \Omega(\mathbb{R}^n)\) (\(D^n[a,b]\) is the space of absolutely continuous \(n\)-dimensional vector functions).

Reviewer: P.Zabreiko (Minsk)

##### MSC:

34A60 | Ordinary differential inclusions |

47J05 | Equations involving nonlinear operators (general) |

34K10 | Boundary value problems for functional-differential equations |

47H04 | Set-valued operators |