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

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
Full Text: