GEORGIAN MATHEMATICAL JOURNAL: Vol. 2, No. 1, 1995, 1-8

CONTINUOUS TRANSFORMATIONS OF DIFFERENTIAL EQUATIONS WITH DELAYS

JAN ČERMÁK

Abstract. The aim of this paper is to find the class of continuous pointwise transformations (as general as possible) in the framework of which Kummer’s transformation z(t) = g(t)y(h(t)) represents the most general pointwise transformation converting every linear homogeneous differential equation of the nth order into an equation of the same type. Further, some forms of these equations having certain subspaces of solutions aer cobstructed.

Let I = [a, b), J = [c, d) be intervals, where b, d may be infinite. Further, let

t = f_{1}(x, y), z = f_{2}(x, y)

where (x, y) ∈ I × ℝ and denote by F = (f_{1}, f_{2}) a pointwise transformation of I × ℝ into U ⊂ ℝ^{2}.

In this article we shall study transformations F of a linear homogeneous differential equation of the nth order with m delays

y^{(n)}(x) + ∑_{i=0}^{n−1} p_{i}(x)y^{(i)}(x) + ∑_{i=0}^{n−1} ∑_{j=1}^{m} q_{ij}(x)y^{(i)}(τ_{j}(x)) = 0  (1)

on [x_{0}, b), where the initial set E_{x_{0}} = [a, x_{0}], p_{i}, q_{ij}, τ_{j} ∈ C^{0}([x_{0}, b)), τ_{j}(x) < x on [x_{0}, b), and q_{ij} ≢ 0 on [x_{0}, b) for a pair (i, j) (i = 0, …, n, j = 1, …, m). We wish to derive the form of such a transformation F which converts (in the sense of a pointwise transformation of solutions) every equation (1) into an equation of the same type, i.e.,

z^{(n)}(t) + ∑_{i=0}^{n−1} r_{i}(t)z^{(i)}(t) + ∑_{i=0}^{n−1} ∑_{j=1}^{m} s_{ij}(t)z^{(i)}(μ_{j}(t)) = 0  (2)

1 on [t_{0}, d), where E_{t_{0}} = [c, t_{0}], r_{i}, s_{ij}, μ_{j} ∈ C^{0}([t_{0}, d)), μ_{j}(t) < t on [t_{0}, d), and s_{ij} ≢ 0 on [t_{0}, d) for a pair (i, j) (i = 0, …, n, j = 1, …, m). Notice that solutions y(x) (resp. z(t)) of equation (1) (resp.(2)) are functions defined on the whole I (resp.J) and the space of solutions of both equations has an infinite dimension.

1991 Mathematics Subject Classification. 34K05, 34K15.

Key words and phrases. Differential equation, delay argument, transformation.

1072-947X/95/0100-0001$07.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1995 Plenum Publishing Corporation

Next, let W (y_{1}, …, y_{k})(x), where y_{1}, …, y_{k} ∈ C^{k−1}(I), be the Wronski determinant of functions y_{1}, …, y_{k} at x ∈ I.

We denote the following hypotheses concerning F: (H_{1}) F is a C^{n}-diffeomorphism of I × ℝ onto J × ℝ; (H′_{1}) F is a homeomorphism of I × ℝ onto J × ℝ; (H_{2}) for every equation (1) there exists an equation (2) such that F converts, pointwise,

(i) every nontrivial solution y(x) of (1) into a nontrivial solution z(t) of (2);

(i′) every nontrivial solution y(x) of (1) into a C^{n} function z(t) defined on J;

(ii) every k-tuple y_{1}, …, y_{k} of solutions of (1) satisfying W (y_{1}, …, y_{k})(x) ≠ 0 on I into a k-tuple z_{1}, …, z_{k} of solutions of (2) satisfying W (z_{1}, …, z_{k})(t) ≠ 0 on J, where k ∈ {2, …, n + 1} is a suitable number;

(iii) every function y ∘ τ_{j}, where y is a solution of (1), into a function z ∘ μ_{j}, where z is a solution of (2) (j = 1, …, m).

Assuming q_{ij} ≡ 0 on [x_{0}, b) for each pair (i, j) in (1) we obtain a differential equation without any delay. The problem of the most general pointwise transformation converting any such equation (1) into an equation (2) (with s_{ij} ≡ 0 on [t_{0}, d)) was first solved by P.Stäckel in [1]. He proved that under hypotheses (H_{1}) and (H_{2})(i) F has the form

t = f(x), z = g(t)y for n ≥ 2 t = f(x), z = g(t)y^{λ}, λ > 0 for n = 1

where f is a C^{n}-diffeomorphism of I onto J, g ∈ C^{n}(J), g(t) ≠ 0 on J. Recently M.Čadek has shown (see [2]) that the assumption of differentiability of F is not necessary and the form of F remains preserved also under (H′_{1}), (H_{2})(i′), and (H_{2})(ii), where k = n (for more details see [3]).

Provided q_{ij} ≢ 0 on [x_{0}, b) for a pair (i, j) V.Tryhuk proved in [4] that assuming (H_{1}), (H_{2})(i), and (H_{2})(iii), F has the form t = f(x), z = g(t)y with g and f having the same properties as above and, moreover, f′(x) > 0 on I and f ∘ τ_{j} = μ_{j} ∘ f on I for j = 1, …, m. The aim of this paper is to weaken the assumption of differentiability of F as M.Čadek did for equations without delays.

Proposition 1. Let hypotheses (H′_{1}) and (H_{2})(i′) be fulfilled. Then f_{1} is a homeomorphism between I and J not depending on y.

CONTINUOUS TRANSFORMATIONS OF DIFFERENTIAL EQUATIONS 3

Proof. Suppose on the contrary that there exist t_{0}^{∗} ∈ J, z_{1}, z_{2} ∈ ℝ such that

(x_{1}, y_{1}) = F^{−1}(t_{0}^{∗}, z_{1}), (x_{2}, y_{2}) = F^{−1}(t_{0}^{∗}, z_{2}),

where x_{1} ≠ x_{2}. Choose x_{0} ∈ I, x_{0} > max(x_{1}, x_{2}) and let y ∈ C^{n}(I) be a function such that y(x_{i}) = y_{i} (i = 1, 2) and y(x) ≠ 0 on [x_{0}, b). Now we can define τ [x_{0}, b) ⟶^{onto} I as an arbitrary continuously increasing function satisfying τ(x) < x on [x_{0}, b) and denote V := τ^{−1}([a, x_{0}]). Then put

p(x) = −\frac{1}{y(x)}(q(x)y(τ(x)) + y^{(n)}(x)) on V

where q is an arbitrary continuous function on V and

q(x) = −\frac{1}{y(τ(x))}(p(x)y(x) + y^{(n)}(x)) on [x_{0}, b) − V

where p is an arbitrary continuous function on [x_{0}, b) − V satisfying

lim_{x→τ^{−1}(x_{0})} p(x) = p(τ^{−1}(x_{0})).

Thus p, q are well-defined continuous functions on [x_{0}, b) and y is a solution of the equation

y^{(n)}(x) + p(x)y(x) + q(x)y(τ(x)) = 0 on[x_{0}, b).

This contradicts the hypothesis (H_{2})(i′) because the image of the function y(x) is not a function.

Notation. Put h := f_{1}^{−1}, g(t, y(h(t)) := f_{2}(h(t), y(h(t)). Then a transformation F converting a solution y(x) into a solution z(t) can be rewritten as

z(t) = g(t, y(h(t)).  (3)

Proposition 2. Let transformation (3) satisfy (H_{2})(iii). Then

(a) μ_{j}(t) = h^{−1}(τ_{j}(h(t)) on [t_{0}, d) (j = 1, …, m);

(b) if lim_{x→b^{−}} τ_{j}(x) = b, then lim_{t→d^{−}} μ_{j}(t) = d (j = 1, …, m);

(c) h is an increasing homeomorphism between J and I.

Proof. The pointwise transformation (3) converts y(τ_{j}(x)) into z(μ_{j}(t)), i.e.,

g(μ_{j}(t), y(h(μ_{j}(t))) = z(μ_{j}(t)) = g(h^{−1}(τ_{j}(h(t)), τ_{j}(h(t))),

which implies (a).

Further, if lim_{x→b^{−}} τ_{j}(x) = b, then lim_{t→d^{−}} h^{−1}(τ_{j}(h(t)) = lim_{x→b^{−}} h^{−1}(x) = d.

Finally, μ_{j}(t) < t and h(μ_{j}(t)) = τ_{j}(h(t)) < h(t) holds for all t ∈ J, hence h, being a homeomorphism, is an increasing one.

Lemma. Consider y_{1}, …, y_{k} ∈ C^{n}(I) for some k ∈ {1, …, n + 1} such that W (y_{1}, …, y_{k})(x) ≠ 0 on I. Then there exists an equation (1) having y_{1}, …, y_{k} as a k-tuple of linearly independent solutions.

Proof. First let 1 ≤ k ≤ n. Suppose that there exists i ∈ {1, …, k} and x_{0} ∈ I such that y_{i}^{(n)}(x_{0}) ≠ 0. Define τ : [x_{0}, b) ⟶^{onto} I as an arbitrary continuous function satisfying τ(x) < x on [x_{0}, b) and consider the following linear system for unknown functions q_{j}(x) in the form

∑_{j=0}^{k−1} y_{l}^{(j)}(τ(x))q_{j}(x) = −y_{l}^{(n)}(x),

where 1 ≤ l ≤ k, x ∈ [x_{0}, b). The matrix of this system is regular for every x ∈ [x_{0}, b) because W (y_{1}, …, y_{k})(x) ≠ 0 on I. Hence, the system has a unique solution q_{0}(x), …, q_{k−1}(x) continuous on [x_{0}, b) and, moreover, the relation y_{i}^{(n)} ≢ 0 implies q_{j} ≢ 0 on I for some j ∈ {0, 1, …, k − 1}. Thus we get that y_{1}, …, y_{k} are solutions of an equation with a delay τ

y^{(n)}(x) + ∑_{j=0}^{k−1} q_{j}(x)y^{(j)}(τ(x)) = 0 on [x_{0}, b).

If y_{i}^{(n)} ≡ 0 on I for every 1 ≤ i ≤ k, then the k-tuple y_{1}, …, y_{k} obviously satisfies the equation

y^{(n)}(x) + y^{(n−1)}(x) − y^{(n−1)}(τ(x)) = 0 on [x_{0}, b)

for every x_{0} ∈ I and any suitable delay τ.

Now consider the case k = n + 1. Introduce the system of functions {f_{x}, x ∈ I} by the relation

f_{x}(u) := c_{1}(x)y_{1}(u) + ... + c_{n+1}(x)y_{n+1}(u), u ∈ I,

where

c_{i}(x) = (−1)^{n+i−1} det (\begin{matrix}y_{1}(x) & y′_{1}(x) & … & y_{1}^{(n−1)}(x) \\ ⋮ & ⋮ & ⋮ & ⋮ \\ y_{i−1}(x) & y′_{i−1}(x) & … & y_{i−1}^{(n−1)}(x) \\ y_{i+1}(x) & y′_{i+1}(x) & … & y_{i+1}^{(n−1)}(x) \\ ⋮ & ⋮ & ⋮ & ⋮ \\ y_{n+1}(x) & y′_{n+1}(x) & … & y_{n+1}^{(n−1)}(x) \end{matrix})

(i = 1, …, n + 1). Note that for every x ∈ I there exists i ∈ {1, …, k} such that c_{i}(x) ≠ 0 and denote V_{x} := {u ∈ I, f_{x}(u) = 0} for x ∈ I. We show that V_{x} has no accumulation point for any x ∈ I.

Let u_{m}, u^{∗} ∈ V_{x} for some x ∈ I such that u_{m} → u^{∗}. Then f_{x}(u_{m}) = f_{x}(u^{∗}) = 0 and with respect to Rolle’s theorem we get

f_{x}(u^{∗}) = \frac{df_{x}}{du} (u^{∗}) = ... = \frac{d^{n}f_{x}}{du^{n}} (u^{∗}) = 0

what contradicts the assumption W (y_{1}, …, y_{n+1})(u^{∗}) ≠ 0.

Further, choose x_{0} ∈ I, x_{0} > a, such that a ∉ V_{x_{0}} and introduce a continuous function τ : [x_{0}, b) → I fulfilling τ(x_{0}) = a, τ(x) < x and τ(x) ∉ V_{x} for every x ∈ [x_{0}, b). This is possible because V_{x} consists of only a finite number of points in [a, x]. Now consider the system with unknown functions p_{i}(x), q(x)

∑_{i=0}^{n−1} y_{l}^{(i)}(x)p_{i}(x) + y_{l}(τ(x))q(x) = −y_{l}^{(n)}(x),

where 1 ≤ l ≤ n + 1 and x ∈ [x_{0}, b). Since

W_{τ} (y_{1}, …, y_{n+1})(x) :=

= det (\begin{matrix}y_{1}(x) & … & y_{1}^{(n−1)}(x) & y_{1}(τ(x)) \\ ⋮ & ⋮ & ⋮ & ⋮ \\ y_{n+1}(x) & … & y_{n+1}^{(n−1)}(x) & y_{n+1}(τ(x)) \end{matrix}) = f_{x}(τ(x)) ≠ 0

on [x_{0}, b), the considered system has uniquely determined solutions p_{i}(x), q(x) continuous on [x_{0}, b) and, moreover,

q(x) = −\frac{W (y_{1}, …, y_{n+1})(x)}{W_{τ} (y_{1}, …, y_{n+1})(x)} ≠ 0 on [x_{0}, b).

From here we get that functions y_{1}, …, y_{n+1} form linearly independent solutions of the equation

y^{(n)}(x) + ∑_{i=0}^{n−1} p_{i}(x)y^{(i)}(x) + q(x)y(τ(x)) = 0 on [x_{0}, b).  (4)

Consequence. Consider y_{1}, y_{2} ∈ C^{1}(I), W (y_{1}, y_{2})(x) ≠ 0 on I and let V_{x} = {u ∈ I, y_{1}(x)y_{2}(u) − y_{2}(x)y_{1}(u) = 0} for every x ∈ I. Choose x_{0} ∈ I, x_{0} > a, and suppose that τ : [x_{0}, b) ⟶^{onto} I is a continuous function satisfying τ(x) < x on [x_{0}, b). Then y_{1}, y_{2} are solutions of the equation

y′(x) + p(x)y(x) + q(x)y(τ(x)) = 0  on [x_{0}, b)  (5)

if and only if τ(x) ∉ V_{x} for every x ∈ [x_{0}, b). Moreover, τ with such a property always exists and the functions p, q are then uniquely determined by the relations

p(x) = \frac{y_{1}(τ(x))y′_{2}(x) − y_{2}(τ(x))y′_{1}(x)}{y_{1}(x)y_{2}(τ(x)) − y_{2}(x)y_{1}(τ(x))} on I,

q(x) = \frac{y′_{1}(x)y_{2}(x) − y_{1}(x)y′_{2}(x)}{y_{1}(x)y_{2}(τ(x)) − y_{2}(x)y_{1}(τ(x))} on I.

Proof. The system

y_{l}(x)p(x) + y_{l}(τ(x))q(x) = −y′_{l}(x), l = 1, 2, on [x_{0}, b),

has (necessarily unique) continuous solution p(x), q(x) if and only if y_{1}(x)y_{2}(τ(x)) − y_{2}(x)y_{1}(τ(x)) ≠ 0 for every x ∈ [x_{0}, b), i.e., τ(x) ∉ V_{x} for every x ∈ [x_{0}, b). Since the form of equation (4) agrees with the form of (5) for n = 1, the existence of a suitable τ follows from the previous Lemma.

Example. Consider the functions y_{1}(x) = 1, y_{2}(x) = x with the nonzero Wronski determinant on an interval I. Then V_{x} = {x} and according to the previous statement all the equations (5) having prescribed functions 1, x as solutions are of the form

y′(x) + \frac{1}{x − τ(x)}y(x) − \frac{1}{x − τ(x)}y(τ(x)) = 0 on [x_{0}, b),

where τ is any continuous delay mapping [x_{0}, b) onto I.

On the other hand, it is easy to see that some k-tuples of functions with a nonzero Wronski determinant on I are not solutions of any equation (5) for k ≥ 3. For example, functions 1, x, x^{2} have this property because the function y(x) = x^{2} does not satisfy the previous equation. Of course, they can be solutions of an equation of the first order with more delays, e.g.,

y′(x) − \frac{τ_{1} + τ_{2}}{τ_{1}τ_{2}} y(x) − \frac{τ_{2}}{τ_{1}(τ_{1} − τ_{2})}y(x − τ_{1}) + \frac{τ_{1}}{τ_{2}(τ_{1} − τ_{2})}y(x − τ_{2}) = 0,

where τ_{1}, τ_{2} > 0, τ_{1} ≠ τ_{2}.

Theorem. Let the hypotheses (H′_{1}), (H_{2})(i′), (ii), (iii) be fulfilled. Then transformation (3) has the form

z(t) = g(t)y(h(t)),  (3′)

where g ∈ C^{n}(J), g(t) ≠ 0 on J, h is a C^{n}-diffeomorphism of J onto I, h′(t) > 0 on J and τ_{j}(h(t)) = h(μ_{j}(t)) on J (j = 1, …, m).

Proof. According to the main result in [4] this statement holds under (H_{1}) and (H_{2})(i), (iii). Transformation (3′) obviously satisfies (H_{2})(ii) as well because of the formula

W (z_{1}, …, z_{k})(t) = (g(t))^{k}(h′(t))^{\frac{k(k−1)}{2}} W (y_{1}, …, y_{k})(h(t))

(see [5]). We show that by weakening the assumptions (H_{1}) and (H_{2})(i) to (H′_{1}) and (H_{2})(i′) the class of transformations does not become larger.

Let the hypothesis (H_{2})(ii) hold for a fixed k ∈ {2, …, n + 1}. Consider a linear differential equation of the kth order without a delay in the form

y^{(k)}(x) + a_{k−1}(x)y^{(k−1)}(x) + ... + a_{0}(x) = 0  on I,  (1′)

where a_{i}(x) ∈ C^{n−k}(I) (if k = n+1 then a_{i}(x) ∈ C^{0}(I)), and let y_{1}, …, y_{k} ∈ C^{n}(I) be a k-tuple of linearly independent solutions of (1′). According to the previous lemma there exists an equation (1) such that y_{1}, …, y_{k} are its solutions. Let transformation (3) convert equation (1) into an equation (2) and denote z_{i}(t) = g(t, y_{i}(h(t)) (i = 1, …, k). Then with respect to (H_{2})(ii) we get that z_{1}, …, z_{k} ∈ C^{n}(J) are solutions of (2) and W (z_{1}, …, z_{k})(t) ≠ 0 on J; hence there exists an equation

z^{(k)}(t) + b_{k−1}(t)z^{(k−1)}(t) + ... + b_{0}(t) = 0  on J  (2′)

having z_{1}, …, z_{k} as a fundamental system of solutions. Since equation (1′) can be arbitrarily chosen, Čadek’s result [2] shows that such a transformation has the form (3′), where g ∈ C^{n}(J), g(t) ≠ 0 on J, and h is a C^{n}-diffeomorphism of J on I. The remaining properties of h follow from Proposition 2.

Remark 1. Assuming that the hypothesis (H_{2})(ii) is fulfilled for k = 1 we obtain transformation (3) in the form z(t) = g(t)y^{λ}(h(t)), λ > 0, which does not agree with the form we wish to have.

Remark 2. We cannot represent each equation (1) by the space of its solutions as is possible for linear homogeneous equations without delays. Consider, e.g., all functions defined by the relation

y(x) = \begin{cases}φ(x) & for x ∈ [0, 1], \\ φ(1) & for x ∈ [1, 2], \\ φ(1)(3 − x − \frac{sin 2πx}{2π}) & for x ∈ [2, 3], \\ 0 & for x ≥ 3, \end{cases}

where φ is any continuous function defined on [0, 1] (cf. [6]). Then this set forms the space of solutions of the equations

y′(x) + q(x)y(τ(x)) = 0 on [1, ∞),

where

q(x) = \begin{cases}0 & for x ∈ (2n − 1, 2n) \\ 2(sin πx)^{2} & for x ∈ [2n, 2n + 1] \end{cases}

and τ may be any continuous delay mapping every interval [n; n+1] onto [n− 1; n], n = 1, 2, . . .. That is the reason why we cannot omit the assumption (H_{2})(iii) if we wish to preserve the validity of the relation τ_{j} ∘ h = h ∘ μ_{j} on J in the case of transformations of such an equation.

References

1. P. Stäckel, Über Transformationen von Differentialgleichungen. J. Reine Angew. Math. 111(1893), 290–302.

2. M. Čadek, Form of general pointwise transformations of linear differential equations. Czechoslovak Math. J. 35(110)(1985),617–624.

3. F. Neuman, Global Properties of Linear Differential Equations. Academia, Prague, 1991.

4. V. Tryhuk, The most general transformation of homogeneous retarded linear differential equations of the nth order. Math. Slovaka 33(1983), 15– 21.

6. J. K. Hale, Functional Differential Equations. Springer-Verlag, New York, 1971.

5. F. Neuman, Geometrical approach to linear differential equations of the nth order. Rend. Mat. 5(1972), 572–602.

(Received 14.09.1993)

Author’s address: Department of Mathematics Technical University of Brno Technická 2, 616 69 Brno Czech Republic