A relaxation theorem for a Banach space integral-inclusion with delays and shifts. (English) Zbl 0823.34023

Zhu, Qi Ji

J. Math. Anal. Appl. 188, No. 1, 1-24 (1994).

The paper deals with the integral inclusion system (1) \(x(t) = \int^ t_ a f(t,s,u(s)\), \(u(t - d_ 1), \dots, u(t - d_ k)ds + Q(t)\), \(u(t) \in F(t, \xi (x)(t))\) a.e. in \(T = [a,b]\), in Banach space. A relaxation theorem for this system is proved. The conditions are given for the set of relaxed solution pairs \((x,u)\) of (1) to be closed or compact as well as to be continuously dependent on various parameters. We note that relaxation theorems for finite and infinite dimensional systems with different type of inclusions were considered by many authors – see F. G. Clarke [Optimization and nonsmooth analysis (1983; Zbl 0582.49001)], J.-P. Aubin and A. Cellina [Differential inclusions (1984; Zbl 0538.34007)], A. A. Tolstonogov and P. I. Chugunov [Sib. Math. J. 24, 941-954 (1983); translation from Sib. Mat. Zh. 24, No. 6(142), 144-159 (1983; Zbl 0537.34011)], A. A. Tolstonogov [Sib. Math. J. 25, 131-143 (1984); translation from Sib. Mat. Zh. 25, No. 1(143), 159-173 (1984; Zbl 0537.34012)], N. S. Papageorgiou [Tohoku Math. J., II. Ser. 39, 505-517 (1987; Zbl 0647.34011)], H. Frankowska [C. R. Acad. Sci., Paris, Ser. I 308, 47-50 (1989; Zbl 0657.34059) and J. Differ. Equations 84, No. 1, 100-128 (1990; Zbl 0705.34016)], etc.

Reviewer: A.A.Kilbas (Minsk)

MSC:

34A60	Ordinary differential inclusions
34G20	Nonlinear differential equations in abstract spaces
34K30	Functional-differential equations in abstract spaces
45N05	Abstract integral equations, integral equations in abstract spaces
45G10	Other nonlinear integral equations

Keywords:

integral inclusion system; Banach space; relaxation theorem

Citations:

Zbl 0537.34011; Zbl 0554.34008; Zbl 0582.49001; Zbl 0538.34007; Zbl 0557.34008; Zbl 0537.34012; Zbl 0647.34011; Zbl 0657.34059; Zbl 0705.34016