Integro-differential equations with multivalued solutions (Q2722238)

This paper deals with the following system of integro-differential equations:NEWLINE\[NEWLINED_hX(t)=F\left(t,X(t),\int_{0}^{t}\Phi(t,s,X(s)) ds\right),\quad X(0)=X^0, \tag{1}NEWLINE\]NEWLINE where \(X(\cdot):\mathbb R \to \text{Conv}(\mathbb R^n), F(\cdot,\cdot,\cdot):\mathbb R\times \text{Conv}(\mathbb R^n)\times \text{Conv}(\mathbb R^n)\to\text{Conv}(\mathbb R^n), \Phi(\cdot,\cdot,\cdot):\mathbb R\times \mathbb R\times \text{Conv}(\mathbb R^n) \to \text{Conv}(\mathbb R^n)\) are convex-set-valued mappings, \(D_h\) stands for the Hukuhara derivative [cf. \textit{M. Hukuhara}, Funkt. Ekvacioj, Ser. Int. 10, 205-223 (1967; Zbl 0161.24701)], and the integral is considered in the sense of \textit{R. J. Aumann} [J. Math. Anal. Appl. 12, 1-12 (1965; Zbl 0163.06301)]. Under the assumption that the mappings \(F, \Phi \) are continuous and Lipschitzian with respect to the Hausdorff metrics, the authors prove the existence of a unique local solution for the equation (1).NEWLINENEWLINENEWLINENext the equation NEWLINE\[NEWLINED_hX=\varepsilon F\left(t,X,\int_{0}^{t}\Phi(t,s,X(s)) ds\right),\quad X(0)=X^0, \tag{2}NEWLINE\]NEWLINE containing a small parameter \(\varepsilon \) is considered. Let \(\Phi_1(t,X):=\int_{0}^{t}\Phi(t,s,X(s)) ds\). Suppose that there exists the limit \(\lim_{T\to \infty }\int_{0}^{T}F(t,X(t),\Phi_1(t,X)) dt=:\overline F(X)\). The authors prove the analogue of the first Bogolyubov theorem [cf. \textit{N. Bogolyubov}, On certain statistical methods in mathematical physics, Akad. Nauk Ukrainskij SSR (1945; Zbl 0063.00496)] about the closeness between solutions of the equation (2) and the corresponding averaged equation \(D_hY=\varepsilon \overline F(Y)\), \(Y(0)=X^0\).