Mem. Differential Equations Math. Phys. 16(1999), 154–158

M. Grigolia

ON THE EXISTENCE AND UNIQUENESS OF SOLUTIONS OF THE GOURSAT PROBLEM FOR SYSTEMS OF FUNCTIONAL PARTIAL

DIFFERENTIAL EQUATIONS OF HYPERBOLIC TYPE

(Reported on October 12, 1998)

In the rectangle D_{ab} = [0, a] × [0, b] let us consider the system of functional partial differential equations of hyperbolic type

\frac{∂u_{1}(x, y)}{∂x} = f_{1}(u_{1}, u_{2})(x, y), \frac{∂u_{2}(x, y)}{∂y} = f_{2}(u_{1}, u_{2})(x, y)  (1)

with the initial conditions

u_{1}(0, y) = 0 for 0 ≤ y ≤ b, u_{2}(x, 0) = 0 for 0 ≤ x ≤ a,  (2)

where f_{i} : C(D_{ab}; R^{n}) × C(D_{ab}; R^{n}) → C(D_{ab}; R^{n}) (i = 1, 2) are continuous Volterra operators.

A particular case of the system (1) is, for example, the system of integro-differential equations

\begin{matrix}\frac{∂u_{1}(x, y)}{∂x} = g_{1}(x, y, u_{1}(α_{11}(x), β_{11}(y)), u_{2}(α_{12}(x), β_{12}(y)), \\ ∫_{0}^{β_{13}(y)}G_{1}(x, t)u_{1}(α_{13}(x), t)dt), \\ \frac{∂u_{2}(x, y)}{∂y} = g_{2}(x, y, u_{1}(α_{21}(x), β_{21}(y)), u_{2}(α_{22}(x), β_{22}(y)), \\ ∫_{0}^{α_{23}(x)}G_{2}(s, y)u_{2}(s, β_{23}(y))ds), \end{matrix}  (1′)

where g_{i} : D_{ab} × R^{n} × R^{n} × R^{n} → R^{n} and G_{i} : D_{ab} → R^{n×n} (i = 1, 2) are continuous vector and matrix functions, respectively, while α_{ik} : [0, a] → [0, a] and β_{ik} : [0, b] → [0, b] (i = 1, 2; k = 1, 2, 3) are continuous functions such that

α_{ik}(x) ≤ x for 0 ≤ x ≤ a, β_{ik}(y) ≤ y for 0 ≤ y ≤ b (i = 1, 2; k = 1, 2, 3). (3)

Along with (1′), (2) consider also the Goursat problem for the second order hyperbolic system with retarded arguments, i.e., the problem

\frac{∂^{2}u(x, y)}{∂x∂y} = g(x, y, u(α_{1}(x), β_{1}(y)), p(α_{2}(x), β_{2}(y)), q(α_{3}(x), β_{3}(y))),  (4)
u(0, y) = 0, for 0 ≤ y ≤ b, u(x, 0) = 0 for 0 ≤ x ≤ a,  (5)

1991 Mathematics Subject Classification. 35L45, 35L70.

Key words and phrases. Volterra operator, system of functional partial differential equations of hyperbolic type, second order hyperbolic system with retarded arguments, Goursat problem. where

p(x, y) = \frac{∂u(x, y)}{∂x} , q(x, y) = \frac{∂u(x, y)}{∂y} ,  (6)

g : D_{ab} × R^{n} × R^{n} × R^{n} → R^{n} is a continuous vector function, while α_{k} : [0, a] → [0, a] and β_{k} : [0, b] → [0, b] (k = 1, 2, 3) are continuous functions such that

α_{k}(x) ≤ x for 0 ≤ x ≤ a, β_{k}(y) ≤ y for 0 ≤ y ≤ b (k = 1, 2, 3).  (7)

Let the problem (4), (5) have a solution u. Suppose

u_{1}(x, y) = \frac{∂u(x, y)}{∂y} , u_{2}(x, y) = \frac{∂u(x, y)}{∂x} .

Then

u(x, y) = ∫_{0}^{y} u_{1}(x, t)dt, u(x, y) = ∫_{0}^{x} u_{2}(s, y)ds.

Therefore the vector function (u_{1}, u_{2}) is the solution of the problem (1′), (2), where

\begin{matrix}g_{i}(x, y, z_{1}, z_{2}, z_{3}) ≡ g(x, y, z_{3}, z_{2}, z_{1}), G_{i}(x, y) ≡ E (i = 1, 2), \\ α_{ik}(x) ≡ α_{4−k}(x), β_{ik}(x) ≡ β_{4−k}(x) (i = 1, 2; k = 1, 2, 3) \end{matrix}  (8)

and E is the unit n × n matrix.

The inverse assumption is obvious: if the identities (8) are fulfilled and the problem (1′), (2) has a solution (u_{1}, u_{2}), then the vector function u given by the equality

u(x, y) = ∫_{0}^{y} u_{1}(x, t)dt

is the solution of the problem (4), (5). Thus the problem (4), (5) is equivalent to the problem (1′), (2) for the case, where g_{i}, G_{i}, α_{ik}, and β_{ik} (i = 1, 2; k = 1, 2, 3) are given by the equalities (8).

In the cases, where α_{k}(x) ≡ x, β_{k}(y) ≡ y (k = 1, 2, 3), α_{ik}(x) ≡ x, β_{ik}(y) ≡ y (i = 1, 2; k = 1, 2, 3), the problems of the existence and uniqueness of a solution of the problems (4), (5), and (1′), (2) were investigated by many authors (see, e.g., [1–9] and references therein). However, the problem (1), (2) as well as the problems (1′), (2) and (4), (5) are studied insufficiently in the general case. The existence and uniqueness theorems formulated below concern with this case.

We shall use the following notation and definitions.

R^{n} is the space of n-dimensional real vectors in which under the norm ∥z∥ of an arbitrary vector z is meant a sum of absolute values of this vector.

R_{r}^{n} = {z ∈ R^{n} : ∥z∥ ≤ r}.

R^{n×n} is the space of real n × n matrices in which under the norm ∥Z∥ of an arbitrary matrix Z is meant maximal value among absolute values of components of this matrix.

C(D_{ab}; R^{n}) is the space of n-dimensional continuous vector functions z : D_{ab} → R^{n} with the norm

∥z∥_{a,b} = max {∥z(x, y)∥ : 0 ≤ x ≤ a, 0 ≤ y ≤ b}.

If z ∈ C(D_{ab}; R^{n}) and (x, y) ∈ D_{ab}, then

∥z∥_{x,y} = max {∥z(s, t)∥ : 0 ≤ s ≤ x, 0 ≤ t ≤ y}.

C^{(1,0)}(D_{ab}; R^{n}) is the space of continuous vector functions z : D_{ab} → R^{n} having the continuous partial derivative in the first argument.

C_{r,r_{0}}^{(1,0)}(D_{ab}; R^{n}) =

= \{z ∈ C_{r}(D_{ab}; R^{n}) : ∥z(a, y)∥ ≤ r, ∥\frac{∂z(x, y)}{∂x} ∥ ≤ r_{0} for (x, y) ∈ D_{ab}\}.

C^{(0,1)}(D_{ab}; R^{n}) is the space of continuous vector functions z : D_{ab} → R^{n} having the continuous partial derivative in the second argument.

C_{r,r_{0}}^{(0,1)}(D_{ab}; R^{n}) =

= \{z ∈ C^{(0,1)}(D_{ab}; R^{n}) : ∥z(a, y)∥ ≤ r, ∥\frac{∂z(x, y)}{∂y} ∥ ≤ r_{0} for (x, y) ∈ D_{ab}\}.

ν_{1}(z)(·, y) and ν_{2}(z)(·, x) are moduli of continuity of the vector function (x, y) → z(x, y) in the first and second arguments, i.e.,

ν_{1}(z)(δ, y) = max \{∥z(x, y) − z(\overline{x}, y)∥ : 0 ≤ x ≤ \overline{x} ≤ a, \overline{x} − x ≤ δ\},
ν_{2}(z)(δ, x) = max \{∥z(x, y) − z(x, \overline{y})∥ : 0 ≤ y ≤ \overline{y} ≤ b, \overline{y} − y ≤ δ\}.

Definition 1. We say that f : C(D_{ab}; R^{n}) × C(D_{ab}; R^{n}) → C(D_{ab}; R^{n}) is a Volterra operator if for any z_{i} and \overline{z}_{i} ∈ C(D_{ab}; R^{n}) (i = 1, 2) and (x, y) ∈ D_{ab} from the equalities

z_{i}(s, t) = \overline{z}_{i}(s, t) for 0 ≤ s ≤ x, 0 ≤ t ≤ y (i = 1, 2)

there follows the equality f(z_{1}, z_{2})(x, y) = f(\overline{z}_{1}, \overline{z}_{2})(x, y).

Definition 2. We say that the system (1) is evolutional if f_{i} : C(D_{ab}; R^{n})× C(D_{ab}; R^{n}) → C(D_{ab}; R^{n}) (i = 1, 2) are Volterra operators.

For example, the system (1′) is evolutional if the functions α_{ik} and β_{ik} (i = 1, 2; k = 1, 2, 3) satisfy the inequalities (3).

Throughout the remainder it will be assumed that f_{i} : C(D_{ab}; R^{n}) × C(D_{ab}; R^{n}) → C(D_{ab}; R^{n}) (i = 1, 2) are the continuous Volterra operators, i.e. the system (1) is evolutional.

By a solution of (1) we understand a vector function (u_{1}, u_{2}) ∈ C^{(1,0)}(D_{ab}; R^{n}) × C^{(0,1)}(D_{ab}; R^{n}) satisfying (1) on D_{ab}.

Theorem 1. Let there exist positive numbers r, r_{0}, and continuous nondecreasing functions ε : [0, +∞[ → [0, +∞[ , φ : [0, +∞[ →]0, +∞[, and ω_{i} : [0, +∞[ → [0, +∞[ (i = 1, 2) such that ε(0) = 0,

∫_{0}^{τ} \frac{ds}{φ(s)} > 2(a + b), r_{0} = φ(r),  (9)

ω_{i}(s) > 0 for s > 0, ∫_{0}^{1} \frac{ds}{ω_{i}(s)} = +∞ (i = 1, 2),  (10)

and for any z_{1} ∈ C_{r,r_{0}}^{(1,0)}(D_{ab}; R^{n}) and z_{2} ∈ C_{r,r_{0}}^{(0,1)}(D_{ab}; R^{n}) the conditions

∥f_{i}(z_{1}, z_{2})(x, y)∥ ≤ φ(∥z_{1}∥_{x,y} + ∥z_{2}∥_{x,y}) (i = 1, 2),  (11)
∥f_{1}(z_{1}, z_{2})(x, \overline{y}) − f_{1}(z_{1}, z_{2})(x, y)∥ ≤ ε(|y − \overline{y}|) + ω_{1}[ν_{2}(z_{1})(|y − \overline{y}|, x)],
∥f_{2}(z_{1}, z_{2})(\overline{x}, y) − f_{2}(z_{1}, z_{2})(x, y)∥ ≤ ε(|x − \overline{x}|) + ω_{2}[ν_{1}(z_{2})(|x − \overline{x}|, y)]

are fulfilled on D_{ab}. Then the problem (1), (2) has at least one solution.

Theorem 2. Let there exist positive numbers r, r_{0}, and continuous nondecreasing functions φ : [0, +∞[ →]0, +∞[ and ω : [0, +∞[ → [0, +∞[ such that for any z_{1} and \overline{z}_{1} ∈ C_{r}^{(1,0)}(D_{ab}; R^{n}) and z_{2} and \overline{z}_{2} ∈ C_{r}^{(0,1)}(D_{ab}; R^{n}) along with (11) the condition

∥f_{i}(z_{1}, z_{2})(x, y)−f_{i}(\overline{z}_{1}, \overline{z}_{2})(x, y)∥≤ω(∥z_{1} − \overline{z}_{1}∥_{x,y} +∥z_{2} − \overline{z}_{2}∥_{x,y}) (i = 1, 2)

holds on D_{ab}. Moreover, let

ω(0) = 0, ω(s) > 0 for s > 0, ∫_{0}^{1} \frac{ds}{ω(s)} = +∞,  (12)

and let the condition (9) hold. Then the problem (1), (2) has at most one solution.

From Theorems 1 and 2 we obtain the following propositions on the solvability and unique solvability of the problems (1′), (2) and (4), (5).

Corollary 1. Let there exist positive numbers l, r, and continuous nondecreasing functions φ : [0, +∞[ →]0, +∞[ and ω_{i} : [0, +∞[ → [0, +∞[ (i = 1, 2) such that

∫_{0}^{β_{13}(y)}∥G_{1}(x, t)∥ dt ≤ l, ∫_{0}^{α_{23}(x)}∥G_{2}(s, y)∥ ds ≤ l for (x, y) ∈ D_{ab} (i = 1, 2) (13)

and the conditions

∥g_{i}(x, y, z_{1}, z_{2}, z_{3})∥ ≤ φ(∥z_{1}∥ + ∥z_{2}∥ + ∥z_{3}∥) (i = 1, 2),  (14)
∥g_{1}(x, y, z_{1}, z_{2}, z_{3}) − g_{1}(x, y, \overline{z}_{1}, z_{2}, z_{3})∥ ≤ ω_{1}(∥z_{1} − \overline{z}_{1}∥),
∥g_{1}(x, y, z_{1}, z_{2}, z_{3}) − g_{1}(x, y, z_{1}, \overline{z}_{2}, z_{3})∥∥ ≤ ω_{2}(∥z_{2} − \overline{z}_{2}∥)

hold on D_{ab} × R_{r}^{n} × R_{r}^{n} × R_{r}^{n}. Moreover, let

∫_{0}^{r} \frac{ds}{φ(s)} > 2(1 + l)(a + b)  (15)

and the functions α_{ik}, β_{ik}, ω_{i} (i = 1, 2; k = 1, 2, 3) satisfy the conditions (3) and (10). Then the problem (1′), (2) has at least one solution.

Corollary 2. Let there exist positive numbers l, r, and continuous nondecreasing functions φ : [0, +∞[ →]0, +∞[ and ω : [0, +∞[ → [0, +∞[ such that the matrix functions G_{i} (i = 1, 2) satisfy the inequalities (13), and along with (14) the condition

∥g_{i}(x, y, z_{1}, z_{2}, z_{3}) − g_{i}(x, y, \overline{z}_{1}, \overline{z}_{2}, \overline{z}_{3})∥ ≤
≤ ω(∥z_{1} − \overline{z}_{1}∥ + ∥z_{2} − \overline{z}_{2}∥ + ∥z_{3} − \overline{z}_{3}∥) (i = 1, 2)

holds on D_{ab} ×R_{r}^{n} ×R_{r}^{n} ×R_{r}^{n}. Moreover, let the functions α_{ik}, β_{ik} (i = 1, 2; k = 1, 2, 3), ω and φ satisfy the conditions (3), (12) and (15). Then the problem (1′), (2) has one and only one solution.

Corollary 3. Let there exist positive numbers l, r, and continuous nondecreasing functions φ : [0, +∞[ →]0, +∞[ and ω_{i} : [0, +∞[ → [0, +∞ (i = 1, 2) such that

α_{1}(x) ≤ l for 0 ≤ x ≤ a, β_{1}(y) ≤ l for 0 ≤ y ≤ b,  (16)

and the conditions

∥g(x, y, z_{1}, z_{2}, z_{3})∥ ≤ φ(∥z_{1}∥ + ∥z_{2}∥ + ∥z_{3}∥),  (17)
∥g(x, y, z_{1}, z_{2}, z_{3}) − g(x, y, z_{1}, \overline{z}_{2}, z_{3})∥ ≤ ω_{1}(∥z_{2} − \overline{z}_{2}∥),
∥g(x, y, z_{1}, z_{2}, z_{3}) − g(x, y, z_{1}, z_{2}, \overline{z}_{3})∥ ≤ ω_{2}(∥z_{3} − \overline{z}_{3}∥)

hold on D_{ab} × R_{r}^{n} × R_{r}^{n} × R_{r}^{n}. Moreover, let the functions α_{k}, β_{k} (k = 1, 2, 3), ω_{i} (i = 1, 2) and φ satisfy the conditions (7), (10), and (15). Then the problem (4), (5) has at least one solution.

In the case, where α_{k}(x) ≡ x and β_{k}(y) ≡ y (k = 1, 2, 3), the results of Hartman– Wintner [1] and Alexiewicz–Orlicz [3] concerning the solvability of the Goursat problem follow from Corollary 3.

Corollary 4. Let there exist positive numbers l, r, and continuous nondecreasing functions φ : [0, +∞[ →]0, +∞[ and ω : [0, +∞[ → [0, +∞[ such that the inequalities (7) and (16) are fulfilled and along with (17) the condition

∥g(x, y, z_{1}, z_{2}, z_{3})−g(x, y, \overline{z}_{1}, \overline{z}_{2}, \overline{z}_{3})∥≤ω(∥z_{1}−\overline{z}_{1}∥+∥z_{2}−\overline{z}_{2}∥+∥z_{3}−\overline{z}_{3}∥)

holds on D_{ab} ×R^{n} ×R^{n} ×R^{n}. Moreover, let the functions ω and φ satisfy the conditions (12) and (15). Then the problem (4), (5) has one and only one solution.

References

1. Ph. Hartman and A. Wintner, On hyperbolic partial differential equation. Amer. J. Math. 74(1952), No. 4, 834–864.

2. C. Ciliberto, Il problema di Darboux per una equazione di tipo iperbolico in due variabili. Ricerche Mat. 4(1955), No. 1, 15–29.

3. A. Alexiewicz and W. Orlicz, Some remarks on the existence and uniqueness of solutions of the hyperbolic equation \frac{∂^{2}z}{∂x∂y} = f(x, y, z, \frac{∂z}{∂x} , \frac{∂z}{∂y}). Studia Math. 15(1956), No. 2, 201–215.

4. A. Zitarosa, Su alcuni sistemi iperbolici di equazioni a derivate parziali del primo ordine. Recerche Mat. 8(1959), No. 2, 240–270.

5. J. Persson, Theorems for continuous solutions of generalized Goursat problems. Matematiche 24(1969), No. 1, 108–136.

6. J. Persson, Uniqueness and Goursat problems. Ann. Mat. Pura Appl. 28(1969), 173–182.

7. W. Walter, Differential and integral inequalities. Springer-Verlag, Berlin, Heidelberg, New York, 1970.

8. K. Deimling, A Carathéodory theory for systems of integral equations. Ann. Mat. Pura Appl. 85(1970), 217–260.

9. M. P. Grigolia, On solvability of Goursat problem. (Russian) Trudy Tbiliss. Gos. Univ. 204(1978), 91–105.

Author’s address:

Chair of Differential and Integral Equations

Faculty of Mechanics & Mathematics

I. Javakhishvili Tbilisi State University

2, University St., Tbilisi 380043

Georgia