On the Initial Value Problem for Functional Differential Systems

For a system of functional differential equations of an arbitrary order the conditions are established for the initial value problem to be solvable on an infinite interval, and the structure of the set of solutions to this problem is studied. Introduction For the p-th order functional differential system x(p)(t) = f(t, xt, . . . , x (p−1) t ), b ≤ t < +∞, (1) we consider the initial value problem x b = ψ (k) (k = 0, . . . , p− 1). (2) The invesigation is based on Kubáček’s theorem [2] asserting that under certain conditions the set of all fixed points of the compact map in the Fréchet space is a compact Rδ-set. It is shown that some restrictions on the growth of the right-hand side of the functional differential system imply that the set of all solutions of the initial value problem for that system is a compact Rδ-set in the Fréchet space of Cp−1-functions. The result extends a similar theorem for first-order functional differential systems proved in [2] and the theorem for second-order functional differential systems proved in [3]. In the sequel we shall use the following notations and assumptions: Let h > 0, b ∈ R, d ∈ N , and let | · | be a norm in Rd. Further, let Hl = Cl([−h, 0], Rd) be provided with the norm ‖x‖l = max { l ∑ k=0 |x(k)(s)| : −h ≤ s ≤ 0 } 1991 Mathematics Subject Classification. 34K05, 34K25. 419 1072-947X/94/0700-0419$12.50/0 c © 1994 Plenum Publishing Corporation 420 V. ŠEDA AND J. ELIAŠ for each x ∈ Hl and l = 0, . . . , p − 1. For brevity ‖ · ‖p−1 will be denoted by ‖ · ‖. Let X = Cp−1([b,∞), Rd) be equipped with a topology of locally uniform convergence of the functions and of their p− 1 derivatives on [b,∞). In the Fréchet space X the topology is given by the metric d(x, y) = ∞ ∑ m=1 1 2m pm(x− y) 1 + pm(x− y) , where pm(x) = sup { p−1 ∑ k=0 |x(k)(t)| : b ≤ t ≤ b + m } , x, y ∈ X, m ≥ 1. Let X∗ = Cp−1([b − h,∞), Rd) be a Fréchet space whose topology is determined by seminorms pm(x) = sup { p−1 ∑ k=0 |x(k)(t)| : b− h ≤ t ≤ b + m } , x ∈ X∗, m ≥ 1. For x ∈ C([b − h,∞), Rd) we shall denote by xt ∈ H0 the function xt(s) = x(t + s), s ∈ [−h, 0], t ≥ b. Clearly (xt)(s) = (x)t(s), s ∈ [−h, 0], k = 0, . . . , p− 1, and x ∈ X∗, t ≥ b. It is assumed throughout the paper that f ∈ X([b,∞) × Hp−1 × · · · × H0, Rd), ψ ∈ Hp−1. A solution x of (1), (2) is a function x ∈ X∗ such that x ∣ ∣ [b,∞) ∈ Cp([b,∞), Rd) abd x satisfies (2) and the functional differential system (1) at each point t ≥ b. § 1. Auxiliary Propositions Now Kubáček’s theorem in [2] will be stated as Lemma 1. In that lemma the compact Rδ-set in the metric space (E, ρ) means a nonempty subset F of E which is homeomorphic to the intersection of a decreasing sequence of compact absolute retracts. By [1], p. 92, a metric space G is called an absolute retract when each continuous map f : K → G has a continuous extension g : H → G for each metric space H and each closed K ⊂ H. For example, a nonempty convex subset of the Fréchet space is an absolute retract. Lemma 1. Let M be a nonempty closed set in the Fréchet space (E, ρ), T : M → E a compact map (i.e., T is continuous and T (M) is a relatively compac set). Denote by S the map I − T where I is the identity map on E. Let there exist a sequence {Un} of closed convex sets in E fulfilling the conditions ON THE INITIAL VALUE PROBLEM 421 (i) 0 ∈ Un for each n ∈ N ; (ii) lim n→∞ diam Un = 0, and a sequence {Tn} of maps Tn : M → E fulfilling the conditions (iii) T (x)− Tn(x) ∈ Un for each x ∈ M and each n ∈ N ; (iv) the map Sn = I − Tn is a homeomorphism of the set S−1 n (Un) onto Un. Then the set F of all fixed points of the map T is a compact Rδ-set. In the special case E = X, ρ = d Lemma 1 implies Lemma 2. Let (X, d) be the Fréchet space given above; let φ,φn ∈ C([b,∞), [0,∞)), and let the following conditions be fulfilled: (v) For each t ∈ [b,∞) the sequence {φn(t)} is nonincreasing and lim n→∞ φn(t) = 0. Let rk ∈ Rd, k = 0, . . . , p− 1 and let M = { x∈X : p−1 ∑ k=0 |x(t)−rk|≤φ(t), t≥b, x(b)=rk, k=0, ..., p−1 } . It is assumed that T : M → X is a compact map with the property (T (x))(k)(b) = rk, k = 0, . . . , p − 1 for each x ∈ M and there exists a sequence {Tn} of compact maps Tn : M → X such that (Tn(x)(b) = rk, k = 0, . . . , p− 1 for each x ∈ M and (vi) p−1 ∑