Georgian Mathematical Journal 1(94), No. 4, 419-427

ON THE INITIAL VALUE PROBLEM FOR FUNCTIONAL DIFFERENTIAL SYSTEMS

V. ŠEDA AND J. ELIAŠ

Abstract. 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, x_{t}, …, x_{t}^{(p−1)}), b ≤ t < +∞,  (1)

we consider the initial value problem

x_{b}^{(k)} = 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 R_{δ}-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 R_{δ}-set in the Fréchet space of C^{p−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 R^{d}. Further, let H_{l} = C^{l}([−h, 0], R^{d}) be provided with the norm

∥x∥_{l} = max { ∑_{k=0}^{l} |x^{(k)}(s)| : −h ≤ s ≤ 0}

419 for each x ∈ H_{l} and l = 0, …, p − 1. For brevity ∥ · ∥_{p−1} will be denoted by ∥ · ∥.

1991 Mathematics Subject Classification. 34K05, 34K25.

1072-947X/94/0700-0419$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1994 Plenum Publishing Corporation

Let X = C^{p−1}([b, ∞), R^{d}) 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}^{∞} \frac{1}{2^{m}} \frac{p_{m}(x − y)}{1 + p_{m}(x − y)},

where

p_{m}(x) = sup { ∑_{k=0}^{p−1} |x^{(k)}(t)| : b ≤ t ≤ b + m}, x, y ∈ X, m ≥ 1.

Let X^{∗} = C^{p−1}([b − h, ∞), R^{d}) be a Fréchet space whose topology is determined by seminorms

p_{m}^{∗}(x) = sup { ∑_{k=0}^{p−1} |x^{(k)}(t)| : b − h ≤ t ≤ b + m}, x ∈ X^{∗}, m ≥ 1.

For x ∈ C([b − h, ∞), R^{d}) we shall denote by x_{t} ∈ H_{0} the function x_{t}(s) = x(t + s), s ∈ [−h, 0], t ≥ b. Clearly (x_{t})^{(k)}(s) = (x^{(k)})_{t}(s), s ∈ [−h, 0], k = 0, …, p − 1, and x ∈ X^{∗}, t ≥ b.

It is assumed throughout the paper that f ∈ X([b, ∞) × H_{p−1} × ⋯ × H_{0}, R^{d}), ψ ∈ H_{p−1}.

A solution x of (1), (2) is a function x ∈ X^{∗} such that x|_{[b,∞)} ∈ C^{p}([b, ∞), R^{d}) 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 R_{δ}-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 {U_{n}} of closed convex sets in E fulfilling the conditions (i) 0 ∈ U_{n} for each n ∈ N; (ii) lim_{n→∞} diam U_{n} = 0,

and a sequence {T_{n}} of maps T_{n} : M → E fulfilling the conditions (iii) T (x) − T_{n}(x) ∈ U_{n} for each x ∈ M and each n ∈ N; (iv) the map S_{n} = I − T_{n} is a homeomorphism of the set S_{n}^{−1}(U_{n}) onto

U_{n}.

Then the set F of all fixed points of the map T is a compact R_{δ}-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, φ_{n} ∈ C([b, ∞), [0, ∞)), and let the following conditions be fulfilled:

(v) For each t ∈ [b, ∞) the sequence {φ_{n}(t)} is nonincreasing and lim_{n→∞} φ_{n}(t) = 0. Let r_{k} ∈ R^{d}, k = 0, …, p − 1 and let

M ={x∈X : ∑_{k=0}^{p−1} |x^{(k)}(t)−r_{k}|≤φ(t), t≥b, x^{(k)}(b)=r_{k}, k =0, ..., p−1}.

It is assumed that T : M → X is a compact map with the property (T (x))^{(k)}(b) = r_{k}, k = 0, …, p − 1 for each x ∈ M and there exists a sequence {T_{n}} of compact maps T_{n} : M → X such that (T_{n}(x)^{(k)}(b) = r_{k}, k = 0, …, p − 1 for each x ∈ M and

(vi) ∑_{k=0}^{p−1} |(T_{n}(x))^{(k)}(t) − (T (x))^{(k)}(t)| ≤ φ_{n}(t), x ∈ M, t ≥ b;

(vii) for each n ∈ N there exists a function φ_{∗n} ∈ C([b, ∞), [0, ∞)) such that φ_{∗n} + φ_{n} ≤ φ on [b, ∞) and

∑_{k=0}^{p−1} |(T_{n}(x))^{(k)}(t) − r_{k}| ≤ φ_{∗n}(t), x ∈ M, t ≥ b;

(viii) the map S_{n} = I − T_{n} is injective on M, where I is the identity on X.

Then the set F of all fixed points of the map T is a compact R_{δ}.

Proof. The set

U_{n} = {x ∈ X : ∑_{k=0}^{p−1} |x^{(k)}(t)|≤φ_{n}(t), t ≥ b, x^{(k)}(b) = 0, k = 0, …, p − 1}

is convex and closed on X for each n ∈ N. We shall show that the sequence {U_{n}} satisfies all conditions of Kubáček’s theorem when E = X, ρ = d and thus Lemma 2 will follow from Lemma 1.

Clearly, (i) is fulfilled. As to the condition (ii), we choose an arbitrary ε > 0. Then there is an m_{0} ∈ N such that ∑_{m=m_{[}+1}^{∞}(1/2^{m}) < ε/2. The condition (v) and the Dini theorem imply that the sequence {φ_{n}} converges on [b, ∞) locally uniformly to 0. Therefore for ε > 0 and m_{0} ∈ N there is an n_{0} ∈ N such that g_{m}(φ_{n}) = sup{|φ_{n}(t)| : b ≤ t ≤ b + m} ≤ ε/4 m_{0} for n ≥ n_{0} and m = 1, 2, …, m_{0}. Hence for n ≥ n_{0} and x, y ∈ U_{n} we have

d(x, y) = ∑_{m=1}^{∞} \frac{1}{2^{m}} \frac{p_{m}(x − y)}{1 + p_{m}(x − y)} ≤ ∑_{m=1}^{m_{0}} p_{m}(x − y) + ∑_{m=m_{0}+1}^{∞} \frac{1}{2^{m}} ≤

≤ ∑_{m=1}^{m_{0}} 2g_{m}(φ_{n}) + ∑_{m=m_{0}+1}^{∞} \frac{1}{2^{m}} ≤ 2m_{0} \frac{ε}{4m_{0}} + \frac{ε}{2} = ε.

This implies that (ii) is satisfied. The assumption (iii) follows from (vi) and from the definition of T, T_{n}, n ∈ N.

Now we show the inclusion U_{n} ⊂ S_{n}(M). The condition (viii) then implies that S_{n} is the bijection of S_{n}^{−1}(U_{n}) onto U_{n} and is fulfilled, since the continuity of S_{n}^{−1}|_{U_{n}} is the consequence of the compactness of T_{n}.

Thus we have to prove that for each y ∈ U_{n} there is an x_{y} ∈ M such that x_{y} − T_{n}(x_{y}) = y. This means that the map P_{n}(x) = y + T_{n}(x) has a fixed point for each y ∈ U_{n}. The condition (vii) implies

∑_{k=0}^{p−1} |y^{(k)}(t) + (T_{n}(x))^{(k)}(t) − r_{k}| ≤ ∑_{k=0}^{p−1} |y^{(k)}(t)| +

+ ∑_{k=0}^{p−1} |(T_{n}(x))^{(k)}(t) − r_{k}| ≤ φ_{n}(t) + φ_{∗n}(t) ≤ φ(t), t ≥ b,

and (P_{n}(t))^{(k)}(b) = r_{k}, k = 0, …, p−1, for each x ∈ M. Therefore P_{n}(M) ⊂ M. As M is a closed convex bounded set and P_{n}(M) ⊂ M is a compact map, by Tikhonov’s fixed point theorem, P_{n} has a fixed point.

Now the function φ in Lemma 2 will be given as a solution of an integral equation. The existence and some properties of the solution to that equation will be discussed in the following two lemmas.

Lemma 3. Let ψ ∈H_{p−1}, ω ∈C([b, ∞),[0, ∞)), g ∈C([0, ∞),[0, ∞)) be a nondecreasing function. Further, let

\begin{matrix}σ(t) ≡ 0, b ≤ t < ∞, if p = 1, and \\ σ(t) = ∑_{l=0}^{p−2} ( ∑_{k=l+1}^{p−1} \frac{|ψ^{(k)}(0)|}{(k − l)!}(t − b)^{k−l}, b ≤ t < ∞, if p ≥ 2, \end{matrix} (3)

K(t, s) = ∑_{l=0}^{p−1} \frac{(t − s)^{p−1−l}}{(p − 1 − l)!} , b ≤ s ≤ t.  (4)

Then the following statements hold:

1. A solution φ ∈ C([b, ∞), [0, ∞)) of the integral equation

φ(t) = σ(t) + ∫_{b}^{t} K(t, s)ω(s)g(∥ψ∥ + φ(s)) ds, b ≤ t < ∞,  (5)

exists and satisfies

0 ≤ φ(t) ≤ λ(t), b ≤ t < ∞,  (6)

if and only if there exists a function λ ∈ C([b, ∞), [0, ∞)) such that

λ(t) ≥ σ(t) + ∫_{b}^{t} K(t, s)ω(s)g(∥ψ∥ + λ(s)) ds, b ≤ t < ∞,  (7)

i.e., if and only if there exists an upper solution λ of (5).

2. A solution φ of (5) (whenever it exists) is a nondecreasing function in [b, ∞).

3. If (5) has a solution φ and ω_{1} ∈ C([b, ∞), (0, ∞)) satisfies 0 ≤ ω_{1}(t) ≤ ω(t) for b ≤ t < ∞, then the equation

φ(t) = σ(t) + ∫_{b}^{t} K(t, s)ω_{1}(s)g(∥ψ∥ + φ(s)) ds  (8)

has a solution φ_{1} such that 0 ≤ φ_{1}(t) ≤ φ(t), b ≤ t < ∞.

Proof. 1. The necessity is clear. To prove the sufficiency we shall proceed by the method of steps. Hence we prove by mathematical induction that for each m = 1, 2, …:

(a) There exists a solution y_{m} ∈ C([b, b + m], [0, ∞)) of (5) satisfying the inequalities 0 ≤ y_{m}(t) ≤ λ(t), b ≤ t ≤ b + m, and

(b) y_{m+1}(t) = y_{m}(t), b ≤ t ≤ b + m.

Consider the partially ordered Banach space X_{1} = C([b, b+1], R) with the sup-norm where z_{1} ≤ z_{2} if and only if z_{1}(t) ≤ z_{2}(t) for each t ∈ [b, b+1] and each pair z_{1}, z_{2} from that space. Then, by definition, the interval ⟨z_{1}, z_{2}⟩ = {y ∈ X_{1} : z_{1}(t) ≤ y(t) ≤ z_{2}(t), b ≤ t ≤ b + 1}. The operator U_{1} : X_{1} → X_{1} defined by

U_{1}(t) = σ(t) + ∫_{b}^{t} K(t, s)ω(s)g(∥ψ∥ + y(s)) ds, b ≤ t ≤ b + 1,

is completely continuous, nondecreasing, and, in view of (7), maps the interval ⟨0, λ|_{[b,b+1]}⟩ into itself. Hence by Schauder’s fixed point theorem U_{1} has a fixed point Y_{1} satisfying (6) on [b, b + 1].

Suppose now that there exists a solution y_{m} of (5) on [b, b+m]. Consider the space X_{m+1} = C([b, b + m + 1], R) with the sup-norm and with the natural ordering. Let U_{m+1} be the operator given by the right-hand side of (5) on [b, b + m + 1]. U_{m+1} is completely continuous, nondecreasing, and maps the interval ⟨0, λ|_{[b,b+m+1]}⟩ = {y ∈ X_{m+1} : 0 ≤ y(t) ≤ λ(t), b ≤ t ≤ b + m + 1} into itself. Similarly, U_{m+1} maps the closed and convex set Y_{m+1} = {x ∈ X_{m+1} : x(t) = y_{m}(t), b ≤ t ≤ b + m} into itself. Hence U_{m+1}(⟨0, λ|_{[b,b+m+1]}⟩ ∩ Y_{m+1}) ⊂ ⟨ 0, λ|_{[b,b+m+1]}⟩ ∩ Y_{m+1} and there exists a fixed point y_{m+1} of U_{m+1} in ⟨0, λ|_{[b,b+m+1]}⟩ ∩ Y_{m+1}. This is the sought-for function y_{m+1} with the properties (a) and (b). Then the function φ(t) = y_{m}(t) for b ≤ t ≤ b + m, m = 1, 2, …, is a solution of (5) in [b, ∞), satisfying (6).

2. The statement follows from (3), (4) and (5).

3. Since each solution φ of (5) is an upper solution of (8), Statement 1 implies Statement 3.

The existence of an upper solution λ of (5) is provided by

Lemma 4. Let ψ, σ, and K have the same meaning as in Lemma 3 and let g ∈ C([0, ∞), (0, ∞)) be a nondecreasing function. Then λ ∈ C([b, ∞, [0, ∞)) is an upper solution of (5) if there is a function ρ ∈ C([b, ∞), [0, ∞)) such that

λ(t) = σ(t) + ∫_{b}^{t} K(t, s)ρ(s) ds, b ≤ t < ∞,  (9)

and

0 ≤ ω(t) ≤ \frac{ρ(t)}{g(∥ψ∥ + λ(t))}, b ≤ t < ∞.  (10)

Proof. By combining (9), (10) we get that λ determined by (9) is an upper solution of (5) in [b, ∞).

Remark 1. Similarly, the necessary and sufficient condition for λ ∈ C([b, ∞, [0, ∞)) to be a solution of (5) is that there exists a function ρ ∈ C([b, ∞, [0, ∞)) such that (9) is fulfilled and

ω(t) = \frac{ρ(t)}{g(∥ψ∥ + λ(t))}, b ≤ t < ∞.

§ 2. The Main Theorem

The main theorem reads as follows.

Theorem 1. Let ψ ∈ H_{p−1}, f ∈ C([b, ∞) × H_{p−1} × ⋯ × h_{0}, R^{d}). Let, further, ω ∈ C([b, ∞), [0, , ∞)), g ∈ C([0, ∞), [0, ∞)) be a nondecreasing function, and let

(ix) |f(t, χ_{t}, …, χ_{t}^{(p−1)}| ≤ ω(t)g(∥χ_{t}∥) for each (t, χ) ∈ [b, ∞) × M^{∗}, where

M^{∗} = {x ∈ X^{∗} : ∑_{k=0}^{p−1} |x^{(k)}(t) − ψ^{(k)}(0)| ≤ φ(t) for t ≥ b and

x_{b}^{(k)} = ψ^{(k)}, k = 0, …, p − 1},

and φ is a solution of equation (5) where the functions σ and K are determined by (3) and (4), respectively.

Then the problem (1), (2) has a solution x lying on M^{∗} and the set F^{∗} is a compact R_{δ}-set in the space X^{∗}.

Proof. Consider the set

M = {x ∈ X : ∑_{k=0}^{p−1} |x^{(k)}(t) − ψ^{(k)}(0)| ≤ φ(t) for t ≥ b and

x^{(k)}(b) = ψ^{(k)}(0), k = 0, …, p − 1}.

Clearly, the restriction P : X^{∗} → X determined by P (x) = x|_{[b,∞)} is a homeomorphism of M^{∗} onto M. Let the map T : m → X be determined by

T (x)(t) = ∑_{k=0}^{p−1} \frac{ψ^{(k)}(0)}{k!}(t − b)^{k} +

+∫_{b}^{t} \frac{(t − s)^{p−1}}{(p − 1)!} f[s, (P^{−1}x)_{s}, ..., (P^{−1}x)_{s}^{(p−1)}] ds, x∈M, t≥b, (11)

where P^{−1} is the inverse of P |_{M^{∗}}. Then F^{∗} = P^{−1}(F), where F is the set of all fixed points of the map T. Since the homeomorphic image of the compact R_{δ}-set is again a compact R_{δ}-set, it is sufficient to prove that F is a compact R_{δ}-set in the space X. This will be done by using Lemma 2 where we put r_{k} = ψ^{(k)}(0), k = 0, …, p − 1. Due to (ix) the maps T_{n} : M → X determined by

T_{n}(x)(t) = \{ \begin{matrix}∑_{k=0}^{p−1} \frac{ψ^{(k)}(0)}{k!}(t − b)^{k} for b ≤ t ≤ b + \frac{1}{n}, \\ ∑_{k=0}^{p−1} \frac{ψ^{(k)}(0)}{k!}(t − b)^{k} + ∫_{b}^{t− \frac{1}{n}} \frac{(t− \frac{1}{n} −s)^{p−1}}{(p−1)!} × \\ ×f[s, (P^{−1}x)_{s}, …, (P^{−1}x)_{s}^{(p−1)}] ds \\ for b + \frac{1}{n} ≤ t < ∞ and x ∈ M \end{matrix}  (12)

are, together with T, compact and, again by (ix),

∑_{l=0}^{p−1} |(T_{n}(x))^{(l)}(t) − (T (x))^{(l)}(t)| ≤

≤ \{ \begin{matrix}∑_{l=0}^{p−1} ∫_{b}^{t} \frac{(t−s)^{p−1−l}}{(p−1−l)!} ω(s)g(∥(p^{−1}x)_{s}∥)ds, b ≤ t ≤ b + \frac{1}{n}, \\ ∑_{l=0}^{p−1} \{ ∫_{b}^{t− \frac{1}{n}} \frac{(t−s)^{p−1−l}−(t− \frac{1}{n} −s)^{p−1−l}}{(p−1−l)!} ω(s)g(∥(p^{−1}x)_{s}∥)ds+ \\ + ∫_{t− \frac{1}{n}}^{t} \frac{(t−s)^{p−1−l}}{(p−1−l)!} ω(s)g(∥(p^{−1}x)_{s}∥)\}ds, b + \frac{1}{n} ≤ t < ∞. \end{matrix}

By Lemma 3 φ is nondecreasing in [b, ∞) and therefore ∥(P^{−1}x)_{s}∥ ≤ ψ + φ(s) for each b ≤ s < ∞, x ∈ M. Hence, using also (4), we get

∑_{l=0}^{p−1} |(T_{n}(x))^{(l)}(t) − (T (x))^{(l)}(t)| ≤ φ_{n}(t), t ≥ b, x ∈ M,

where

φ_{n}(t) = \{ \begin{matrix}∫_{b}^{t} K(t, s)ω(s)g(∥ψ∥ + φ(s))ds, b ≤ t ≤ b + \frac{1}{n}, \\ ∫_{b}^{t} K(t, s)ω(s)g(∥ψ∥ + φ(s))ds − ∫_{b}^{t− \frac{1}{n}} K(t − \frac{1}{n}, s)ω(s)× \\ ×(∥ψ∥ + φ(s))ds, b + \frac{1}{n} ≤ t < ∞, n ∈ N. \end{matrix}

Clearly, φ_{n} ∈ C([b, ∞), [0, ∞)) and the relations φ_{n+1}(t) ≤ φ_{n}(t), lim_{n→∞} φ_{n}(t) = 0 can be proved for each t ≥ b. Therefore these functions satisfy the assumptipons (v), (vi) of Lemma 2.

Further,

∑_{l=0}^{p−1} |(T_{n}(x))^{(l)}(t) − φ^{(l)}(0)| ≤ φ_{∗n}(t), t ≥ b, x ∈ M,

where

φ_{∗n} = \{ \begin{matrix}σ(t), b ≤ t ≤ b + \frac{1}{n}, \\ σ(t) + ∫_{b}^{t− \frac{1}{n}} K(t − \frac{1}{n}, s)ω(s)g(∥ψ∥ + φ(s))ds, \\ b + \frac{1}{n} ≤ t < ∞, \end{matrix}

n ∈ N, σ and K are the functions introduced by (3) and (4). The functions φ_{∗n} are nonnegative continuous functions on [b, ∞) and, by virtue of (5) we obtain φ_{n}(t) + φ_{∗n}(t) = φ(t), t ≥ b, n ∈ N. Hence the assumption (vii) of Lemma 2 is satisfied, too. Thus it remains for us to show that the assumption (viii) holds, and then Lemma 2 will imply the statement of Theorem 1.

Let n ∈ N be arbitrary but fixed. If x, y ∈ M, x ≠ y, then there exists a t_{0} ∈ [b, ∞) such that x(t_{0}) ≠ y(t_{0}). If b ≤ t_{0} ≤ b + \frac{1}{n}, then, taking into account (12), we obtain x(t_{0}) − T_{n}(x)(t_{0}) ≠ y(t_{0}) − T_{n}(y)(t_{0}). In the other case there is a t_{1} ≥ b + \frac{1}{n} such that t_{1} = sup\{τ > b : x(t) = y(t) for b ≤ t < τ\}. Then there exists t_{0} ∈ (t_{1}, t_{1} + \frac{1}{n}) such that x(t_{0}) ≠ y(t_{0}). By (12) we now have

T_{n}(x)(t_{0}) = ∑_{k=0}^{p=1} \frac{ψ^{(k)}(0)}{k!}(t_{0} − b)^{k} + ∫_{b}^{t_{0}− \frac{1}{n}} \frac{(t_{0} − \frac{1}{n} − s)^{p−1}}{(p − 1)!} ×

×f[s, (P^{−1}x)_{s}, …, (P^{−1}x)_{s}^{(p−1)}]ds = ∑_{k=0}^{p−1} \frac{ψ^{(k)}(0)}{k!}(t_{0} − b)^{k} +

+ ∫_{b}^{t_{0}− \frac{1}{n}} \frac{(t_{0} − \frac{1}{n} − s)^{p−1}}{(p − 1)!} f[s, (P^{−1}y)_{s}, …, (P^{−1}y)_{s}^{(p−1)}]ds = T_{n}(y)(t_{0})

and thus x(t_{0}) − T_{n}(x)(t_{0}) ≠ y(t_{0}) − T_{n}(y)(t_{0}), which we were to prove.

References

1. J. Dugundji and A. Granas, Fixed point theory. PWN, Warszawa, 1982.

2. Z. Kubáček, Remarks on the paper of K. Czarnowski and T. Pruszko “On the structure of fixed point sets. . ..” Preprint.

3. V. Šeda and Š. Belohorec, A remark on second-order functional differential system (submitted to Arch. Math.).

(Received 15.04.1993) Authors’ addresses:

Valter Šeda Jozef Eliaš Comenius University Slovak Technical University Department of Mathematical Analysis Department of Mathematics Mlynská dolina Mám. Slobody 17 842 15 Bratislava, Slovakia 812 31 Bratislava, Slovakia