GEORGIAN MATHEMATICAL JOURNAL: Vol. 5, No. 4, 1998, 341-360

ON SOME BOUNDARY VALUE PROBLEMS FOR AN ULTRAHYPERBOLIC EQUATION

S. KHARIBEGASHVILI

Abstract. The correct formulation of a characteristic problem and a Darboux type problem in the special weighted functional spaces for an ultrahyperbolic equation is investigated.

In the space of variables x_{1}, x_{2}, y_{1} and y_{2} we consider the ultrahyperbolic equation

u_{y_{1}y_{1}} + u_{y_{2}y_{2}} − u_{x_{1}x_{1}} − u_{x_{2}x_{2}} = F.  (1)

Denote by D : −y_{1} < x_{1} < y_{1}, 0 < y_{1} < +∞, a dihedral angle bounded by the characteristic surfaces S_{1} : x_{1} − y_{1} = 0, 0 ≤ y_{1} < +∞, and S_{2} : x_{1} + y_{1} = 0, 0 ≤ y_{1} < +∞, of equation (1).

We shall consider a characteristic problem formulated as follows: in the domain D find a solution u(x_{1}, x_{2}, y_{1}, y_{2}) of equation (1) by the boundary conditions

u |_{S_{i}}= f_{i}, i = 1, 2,  (2)

where f_{i}, i = 1, 2, are given real functions on S_{i} and (f_{1} − f_{2}) |_{S_{1}∩S_{2}}= 0. Characteristic problems formulated similarly were considered in [1–3]. Let G = {(x_{1}, ξ_{1}, y_{1}, ξ_{2}) ∈ R^{4} : −y_{1} < x_{1} < y_{1}, 0 < y_{1} < +∞;

−∞ < ξ_{i} < +∞, i = 1, 2}.

Denote by Φ^{k}(\overline{D}), k ≥ 2, the space of functions u(x_{1}, x_{2}, y_{1}, y_{2}) of the class C^{k}(\overline{D}) whose partial Fourier transforms û(x_{1}, ξ_{1}, y_{1}, ξ_{2}) with respect to the variables x_{2} and y_{2} are continuous functions in \overline{G} together with partial derivatives with respect to the variables x_{1} and y_{1} up to kth order inclusive and satisfy the following estimates: for any natural N there exist positive

341 integers C_{N} = C_{N}(x_{1}, y_{1}) and K_{N} = K_{N}(x_{1}, y_{1}) not depending on ξ_{1}, ξ_{2} such that the inequalities

1991 Mathematics Subject Classification. 35L20.

Key words and phrases. Ultrahyperbolic equation, characteristic problem, Darboux type problem.

1072-947X/98/0700-0341$15.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1998 Plenum Publishing Corporation

|∂^{i_{1}+i_{2}}/∂x_{1}^{i_{1}}∂y_{1}^{i_{2}} û(x_{1}, ξ_{1}, y_{1}, ξ_{2})| ≤ C_{N}e^{−N(|ξ_{1}|+|ξ_{2}|)}, 0 ≤ i_{1} + i_{2} ≤ k, (3)

are fulfilled for −y_{1} ≤ x_{1} ≤ y_{1}, 0 ≤ y_{1} < +∞ and |ξ_{1}| + |ξ_{2}| > K_{N}. Note that

C_{N}^{0}(x_{1}, y_{1}) = sup_{(x_{1}^{0},y_{1}^{0})∈I(x_{1},y_{1})} C_{N}(x_{1}^{0}, y_{1}^{0}) < +∞,

K_{N}^{0}(x_{1}, y_{1}) = sup_{(x_{1}^{0},y_{1}^{0})∈I(x_{1},y_{1})} K_{N}(x_{1}^{0}, y_{1}^{0}) < +∞,

where I(x_{1}, y_{1}) = {(x_{1}^{0}, y_{1}^{0}) ∈ R^{2} : 0 ≤ y_{1}^{0} − x_{1}^{0} ≤ y_{1} − x_{1}, 0 ≤ y_{1}^{0} + x_{1}^{0} ≤ y_{1} + x_{1}} is the closed rectangle.

The spaces Φ^{k}(S_{i}), i = 1, 2, are introduced similarly. Note that the trace u |_{S_{i}} of the function u from the space Φ^{k}(\overline{D}) belongs to the space Φ^{k}(S_{i}).

When considering a solution of problem (1), (2) in the space Φ^{k}(\overline{D}), k ≥ 2, it will be assumed that F ∈ Φ^{k−1}(\overline{D}), f_{i} ∈ Φ^{k}(S_{i}), i = 1, 2.

If u is a solution of problem (1), (2) of the class Φ^{k}(\overline{D}), then after the Fourier transformation with respect to the variables x_{2} and y_{2}, equation (1) and conditions (2) can be rewritten as

v_{y_{1}y_{1}} − v_{x_{1}x_{1}} + (ξ_{1}^{2} − ξ_{2}^{2})v = F_{0},  (4)
v |_{l_{j}}= g_{j}, j = 1, 2,  (5)

where v, F_{0}, g_{1}, g_{2} are respectively the Fourier transforms of the functions u, F, f_{1}, f_{2} with respect to the variables x_{2} and y_{2}, i.e.,

v(x_{1}, y_{1}, ξ_{1}, ξ_{2}) = \frac{1}{2π} ∫_{−∞}^{∞} ∫_{−∞}^{∞} u(x_{1}, x_{2}, y_{1}, y_{2})e^{−ix_{2}ξ_{1}−iy_{2}ξ_{2}}dx_{2} dy_{2},

F_{0}(x_{1}, y_{1}, ξ_{1}, ξ_{2}) = \frac{1}{2π} ∫_{−∞}^{∞} ∫_{−∞}^{∞} F (x_{1}, x_{2}, y_{1}, y_{2})e^{−ix_{2}ξ_{1}−iy_{2}ξ_{2}}dx_{2} dy_{2},

g_{j}(x_{1}, y_{1}, ξ_{1}, ξ_{2}) = \frac{1}{2π} ∫_{−∞}^{∞} ∫_{−∞}^{∞} g_{j}(x_{1}, x_{2}, y_{1}, y_{2})e^{−ix_{2}ξ_{1}−iy_{2}ξ_{2}}dx_{2} dy_{2},

(x_{1}y_{1}) ∈ l_{j}, j = 1, 2,

and l_{1} : x_{1} − y_{1} = 0, 0 ≤ y_{1} < +∞, l_{2} : x_{1} + y_{1} = 0, 0 ≤ y_{1} < +∞, are the beams lying in the plane of variables x_{1}, y_{1} and outgoing from the origin O(0, 0).

Remark. Thus by the Fourier transformation with respect to the variables x_{2} and y_{2} the spatial problem (1), (2) is reduced to the Goursat plane problem (4), (5) with the parameters ξ_{1} and ξ_{2} lying in the domain D_{0} : −y_{1} < x_{1} < y_{1}, 0 < y_{1} < +∞ of the plane of variables x_{1}, y_{1}. It is easy to see that in the class of functions defined by inequalities (3), this reduction is equivalent, i.e., problem (1), (2) is equivalent to problem (4), (5).

Using for the functions v, F_{0}, g_{i} the previous notation, in terms of the new variables

x = \frac{1}{2}(y_{1} + x_{1}), y = \frac{1}{2}(y_{1} − x_{1}),  (6)

problem (4), (5) takes the form

\frac{∂^{2}v}{∂x∂y} + (ξ_{1}^{2} − ξ_{2}^{2})v = F_{0},  (7)
v |_{γ_{j}}= g_{j}, j = 1, 2.  (8)

Here the solution v = v(x, y, ξ_{1}, ξ_{2}) of equation (7) is considered in the domain Ω_{0} : 0 < x < +∞, 0 < y < +∞ of the plane of variables x, y, which is the image of the domain D_{0} for the linear transform (6), and γ_{1} : y = 0, 0 ≤ x < +∞, and γ_{2} : x = 0, 0 ≤ y < +∞, are the images l_{1} and l_{2} for the same transform (6).

With our assumptions for the functions F_{0}, g_{1}, g_{2}, problem (7), (8) has a unique solution v of the class C^{2}(\overline{Ω}_{0}) which has the form [4]

v(x, y, ξ_{1}, ξ_{2}) = R(x, 0; x, y)g_{1}(x, ξ_{1}, ξ_{2}) + R(0, y; x, y)g_{2}(y, ξ_{1}, ξ_{2}) −

−R(0, 0; x, y)g_{1}(0, ξ_{1}, ξ_{2}) − ∫_{0}^{x} \frac{∂R(ξ, 0; x, y)}{∂ξ}g_{1}(ξ, ξ_{1}, ξ_{2})dξ −

−∫_{0}^{y} \frac{∂R(0, η; x, y)}{∂η}g_{2}(η, ξ_{1}, ξ_{2})dη+∫_{0}^{x} dξ ∫_{0}^{y} R(ξ, η; x, y)F_{0}(ξ, η, ξ_{1}, ξ_{2})dη, (9)

where g_{1}(x, ξ_{1}, ξ_{2}) = v(x, 0, ξ_{1}, ξ_{2}), g_{2}(y, ξ_{1}, ξ_{2}) = v(0, y, ξ_{1}, ξ_{2}) are the Goursat data for v, and R(ξ, η; x, y) is the Riemann function for equation (7). For simplicity, we omit the dependence of R on the parameters ξ_{1} and ξ_{2}.

It is well known that the Riemann function R(ξ, η; x, y) for equation (7) can be expressed in terms of the Bessel function J_{0} of zero order as follows:

R(ξ, η; x, y) = J_{0}(\sqrt{4(ξ_{1}^{2} − ξ_{2}^{2})(x − ξ)(y − η)}) =
= ∑_{k=0}^{∞}(ξ_{2}^{2} − ξ_{1}^{2})^{k} \frac{(x − ξ)^{k}(y − η)^{k}}{(k!)^{2}} .  (10)

The representation [6] J_{0}(z) = \frac{1}{2π} ∫_{−π}^{π} exp(iz sin θ)dθ readily implies

|R(ξ, η; x, y)| ≤ \begin{cases}1 for |ξ_{1}| ≥ |ξ_{2}|, \\ exp \sqrt{4|ξ_{1}^{2} − ξ_{2}^{2}|(x − ξ)(y − η)} for |ξ_{1}| < |ξ_{2}|. \end{cases} (11)

Using the equalities

J′_{0}(z) = −\frac{z}{2π} ∫_{−π}^{π} cos^{2} θ exp(iz sin θ)dθ,

\frac{dJ_{0}(2λ\sqrt{νx})}{dx} = −\frac{λ^{2}ν}{π} ∫_{−π}^{π} cos^{2} θ exp(i2λ\sqrt{νx} sin θ)dθ, ν > 0,

we obtain

|\frac{dJ_{0}(2λ\sqrt{νx})}{dx}| ≤ \begin{cases}2λ^{2}ν for λ = 0, \\ 2|λ|^{2}ν exp 2|λ|\sqrt{νx} for Re λ = 0. \end{cases}

Hence for derivatives of R we easily derive the estimates

|\frac{∂R(ξ, 0; x, y)}{∂ξ}| ≤ \begin{cases}2(ξ_{1}^{2} − ξ_{2}^{2})\sqrt{y} for |ξ_{1}| ≥ |ξ_{2}|, \\ 2(ξ_{1}^{2} − ξ_{2}^{2})\sqrt{y} exp \sqrt{4(ξ_{1}^{2} − ξ_{2}^{2})(x − ξ)y} \\ for |ξ_{1}| < |ξ_{2}|, \end{cases}  (12)

|\frac{∂R(0, η; x, y)}{∂η}| ≤ \begin{cases}2(ξ_{1}^{2} − ξ_{2}^{2})\sqrt{x} for |ξ_{1}| ≥ |ξ_{2}|, \\ 2(ξ_{1}^{2} − ξ_{2}^{2})\sqrt{x} exp \sqrt{4(ξ_{1}^{2} − ξ_{2}^{2})x(y − η)} \\ for |ξ_{1}| < |ξ_{2}|. \end{cases}  (13)

Next, we assume without loss of generality that for the functions F_{0}, g_{1}, g_{2} estimates (3) hold with respect to ξ_{1}, ξ_{2}, with the same integers C_{N}, C_{N}^{0}, K_{N}, and K_{N}^{0}. Then, using (11)–(13), for the solution v(x, y, ξ_{1}, ξ_{2}) of problem (7), (8) written in form (9) we obtain for |ξ_{1}| + |ξ_{2}| > K_{N}^{0} the following estimates:

|v(x, y, ξ_{1}, ξ_{2})| ≤ |g_{1}(x, ξ_{1}, ξ_{2})| + |g_{2}(y, ξ_{1}, ξ_{2})| + |g_{1}(0, ξ_{1}, ξ_{2}) ×

× exp \sqrt{4|ξ_{1}^{2} − ξ_{2}^{2}|xy} + 2|ξ_{2}^{2} − ξ_{1}^{2}|\sqrt{y} exp \sqrt{4|ξ_{2}^{2} − ξ_{1}^{2}|xy} ×

× ∫_{0}^{x} |g_{1}(ξ, ξ_{1}, ξ_{2})|dξ + 2|ξ_{2}^{2} − ξ_{1}^{2}|\sqrt{x} exp \sqrt{4|ξ_{2}^{2} − ξ_{1}^{2}|xy} ×

× ∫_{0}^{y} |g_{2}(η, ξ_{1}, ξ_{2})|dη + exp \sqrt{4|ξ_{2}^{2} − ξ_{1}^{2}|xy} ∫_{0}^{x} dξ ∫_{0}^{y} |F_{0}(ξ, η, ξ_{1}, ξ_{2})|dη ≤

≤ 2C_{N}^{0} exp(−N(|ξ_{1}| + |ξ_{2}|) + C_{N}^{0} exp \sqrt{4|ξ_{1}^{2} − ξ_{2}^{2}|xy} ×

× exp[−N(|ξ_{1}| + |ξ_{2}|)] + 2|ξ_{2}^{2} − ξ_{1}^{2}|\sqrt{y}x exp \sqrt{4|ξ_{2}^{2} − ξ_{1}^{2}|xy}C_{N}^{0} ×

× exp[−N(|ξ_{1}| + |ξ_{2}|)] + 2|ξ_{2}^{2} − ξ_{1}^{2}|\sqrt{x}y exp \sqrt{4|ξ_{2}^{2} − ξ_{1}^{2}|xy}C_{N}^{0} ×

× exp[−N(|ξ_{1}| + |ξ_{2}|)] + xy exp \sqrt{4|ξ_{2}^{2} − ξ_{1}^{2}|xy}C_{N}^{0} ×

× exp[−N(|ξ_{1}| + |ξ_{2}|)] ≤ [3 + 2(\sqrt{x}y + \sqrt{y}x)|ξ_{2}^{2} − ξ_{1}^{2}| + xy] ×
×C_{N}^{0} exp 2\sqrt{xy}(|ξ_{1}| + |ξ_{2}|) exp[−N(|ξ_{1}| + |ξ_{2}|)].  (14)

Now choose a least natural number N_{0} = N_{0}(x, y) such that

N_{0} > N + 2\sqrt{xy} + 1.  (15)

Taking into account that

A_{N}(x, y) = sup_{(ξ_{1},ξ_{2})∈R^{2}} C_{N_{0}}^{0}[3 + 2(\sqrt{x}y + \sqrt{y}x)|ξ_{2}^{2} − ξ_{1}^{2}| + xy] ×

× exp [ − (|ξ_{1}| + |ξ_{2}|)] < +∞,

from (14) and (15), for |ξ_{1}| + |ξ_{2}| > K_{N_{0}}^{0}, we obtain

|v(x, y, ξ_{1}, ξ_{2})| ≤ [3 + 2(\sqrt{x}y + \sqrt{y}x)|ξ_{2}^{2} − ξ_{1}^{2}| + xy]C_{N_{0}}^{0} exp 2\sqrt{xy} ×

×(|ξ_{1}| + |ξ_{2}|) exp[−N(|ξ_{1}| + |ξ_{2}|)] ≤ C_{N_{0}}^{0}[3 + 2(\sqrt{x}y + \sqrt{y}x)|ξ_{2}^{2} − ξ_{1}^{2}| +

+xy] exp 2\sqrt{xy}(|ξ_{1}| + |ξ_{2}|) exp[−(|ξ_{1}| + |ξ_{2}|)] exp [ − 2\sqrt{xy}(|ξ_{1}| + |ξ_{2}|)] ×

× exp [ − (N_{0} − N − 2\sqrt{xy} − 1)(|ξ_{1}| + |ξ_{2}|)] ×
× exp[−N(|ξ_{1}| + |ξ_{2}|)] ≤ A_{N}(x, y) exp[−N(|ξ_{1}| + |ξ_{2}|)].  (16)

The latter inequality implies estimate (3) for i_{1} = i_{2} = 0. The proof of this estimate for i_{1} + i_{2} > 0 is similar. Thus we have

Theorem 1. For any F ∈ Φ^{k−1}(\overline{D}), f_{i} ∈ Φ^{k}(S_{i}), i = 1, 2, problem (1),

(2) is uniquely solvable in the class Φ^{k}(\overline{D}), k ≥ 2.

We denote by D_{1} : −k_{2}y_{1} < x_{1} < k_{1}y, 0 < y_{1} < +∞, 0 < k_{i} < 1, i = 1, 2, a dihedral angle bounded by the surfaces S_{1}^{0} : x_{1} − k_{1}y_{1} = 0, 0 ≤ y_{1} < +∞ and S_{2}^{0} : x_{1} + k_{2}y_{1} = 0, 0 ≤ y_{1} < +∞.

Let us consider a multidimensional variant of the second Darboux problem fomulated as follows: in the domain D_{1} find a solution u(x_{1}, y_{1}, x_{2}, y_{2}) of equation (1) by the boundary conditions

(M_{1}u_{x_{1}} + N_{1}u_{y_{1}} + M̃_{1}u_{x_{2}} + Ñ_{1}u_{y_{2}} + S̃_{1}u)|_{S_{1}^{0}} = f_{1},  (17)
(M_{2}u_{x_{1}} + N_{2}u_{y_{1}} + M̃_{2}u_{x_{2}} + Ñ_{2}u_{y_{2}} + S̃_{2}u)|_{S_{2}^{0}} = f_{2},  (18)

where M_{i}, N_{i}, M̃_{i}, Ñ_{i}, S̃_{i}, i = 1, 2, are given real functions on S_{i}^{0}; the coefficients M_{i}, N_{i}, M̃_{i}, Ñ_{i}, S̃_{i}, i = 1, 2, depend only on the variables x_{1} and y_{1}.

Note that some multidimensional variants of the second Darboux problem for a wave equation are investigated in [7, 8].

Denote by Φ^{∘}_{α}^{k}(\overline{D}_{1}), k ≥ 2, α ≥ 0, the space of functions u(x_{1}, x_{2}, y_{1}, y_{2}) of the class C^{k}(\overline{D}_{1}) for which

∂^{i_{1}+i_{2}}/∂x_{1}^{i_{1}}∂y_{1}^{i_{2}}u(0, x_{2}, 0, y_{2}) = 0,
−∞ < x_{2} < +∞, −∞ < y_{2} < +∞ 0 ≤ i_{1} + i_{2} ≤ k,

and whose partial Fourier transforms û(x_{1}, ξ_{1}, y_{1}, ξ_{2}) with respect to the variables x_{2} and y_{2} are continuous functions in

\overline{G}_{1} = \{(x_{1}, ξ_{1}, y_{1}, ξ_{2}) ∈ R^{4} : −k_{2}y_{1} ≤ x_{1} ≤ k_{1}y_{1},
0 ≤ y_{1} < +∞; −∞ < ξ_{i} < +∞, i = 1, 2}

together with partial derivatives with respect to the variables x_{1} and y_{1} up to kth order inclusive and satisfy the following estimates: for any natural N there exist positive integers C_{N} = C_{N}(x_{1}, y_{1}) and K_{N} = K_{N}(x_{1}, y_{1}) not depending on ξ_{1}, ξ_{2} such that for −k_{2}y_{1} ≤ x_{1} ≤ k_{1}y_{1}, 0 ≤ y_{1} < +∞, and |ξ_{1}| + |ξ_{2}| > K_{N} we have the inequalities

|∂^{i_{1}+i_{2}}/∂x_{1}^{i_{1}}∂y_{1}^{i_{2}} û(x_{1}, ξ_{1}, y_{1}, ξ_{2})| ≤ C_{N}y_{1}^{k+α−i_{1}−i_{2}}e^{−N(|ξ_{1}|+|ξ_{2}|)}, (19)
0 ≤ i_{1} + i_{2} ≤ k,

with

C_{N}^{0}(x_{1}, y_{1}) = sup_{(x_{1}^{0},y_{1}^{0})∈I_{1}(x_{1},y_{1})} C_{N}(x_{1}^{0}, y_{1}^{0}) < +∞,

K_{N}^{0}(x_{1}, y_{1}) = sup_{(x_{1}^{0},y_{1}^{0})∈I_{1}(x_{1},y_{1})} K_{N}(x_{1}^{0}, y_{1}^{0}) < +∞,

where I_{1}(x_{1}, y_{1}) = {(x_{1}^{0}, y_{1}^{0}) ∈ R^{2} : −k_{2}y_{1}^{0} ≤ x_{1}^{0} ≤ k_{1}y_{1}^{0}, y_{1}^{0} − x_{1}^{0} ≤ y_{1} − x_{1}, y_{1}^{0} + x_{1}^{0} ≤ y_{1} + x_{1}, −k_{2}y_{1} ≤ x_{1} ≤ k_{1}y_{1}} is the closed rectangle.

The spaces Φ^{∘}_{α}^{k}(S_{i}^{0}), i = 1, 2, are derived in a similar manner. Note that the trace u|_{S_{i}^{0}} of the function u from the space Φ^{∘}_{α}^{k}(\overline{D}_{1}) belongs to the space

Φ^{∘}_{α}^{k}(S_{i}^{0}).

Remark. When considering problem (1), (17), (18) in the class Φ^{∘}_{α}^{k}(\overline{D}_{1}), we require of the functions F, f_{i}, i = 1, 2, and the coefficients M_{i}, N_{i}, M̃_{i}, Ñ_{i}, S̃_{i}, i = 1, 2, in the boundary conditions (17), (18) that

F ∈ Φ^{∘}_{α}^{k−1}(\overline{D}_{1}), f_{i} ∈ Φ^{∘}_{α}^{k−1}(S_{i}^{0}), i = 1, 2;

M_{i}, N_{i}, M̃_{i}, Ñ_{i}, S̃_{i} ∈ C^{k−1}(S_{i}^{0}), i = 1, 2.

Using the notation u_{y_{1}} = v_{1}, u_{x_{1}} = v_{2}, u_{y_{2}} = v_{3}, u_{x_{2}} = v_{4}, we reduce equation (1) to the first-order system

u_{y_{1}} = v_{1},  (20)
v_{1y_{1}} + v_{3y_{2}} − v_{2x_{1}} − v_{4x_{2}} = F,  (21)
v_{2y_{1}} − v_{1x_{1}} = 0,  (22)
v_{3y_{1}} − v_{1y_{2}} = 0,  (23)
v_{4y_{1}} − v_{1x_{2}} = 0,  (24)

and write the boundary conditions (17) and (18) in the form

(M_{1}v_{2} + N_{1}v_{1} + M̃_{1}v_{4} + Ñ_{1}v_{3} + S̃_{1}u)|_{S_{1}} = f_{1},  (25)
(M_{2}v_{2} + N_{2}v_{1} + M̃_{2}v_{4} + Ñ_{2}v_{3} + S̃_{2}u)|_{S_{2}} = f_{2}.  (26)

Along with conditions (25), (26) we shall also consider the boundary conditions

(u_{x_{1}} − v_{2})|_{S_{1}^{0}∪S_{2}^{0}} = 0,  (27)
(u_{x_{2}} − v_{4})|_{S_{1}^{0}∪S_{2}^{0}} = 0,  (28)
(u_{y_{2}} − v_{3})|_{S_{1}^{0}∪S_{2}^{0}} = 0.  (29)

Clearly, if u is a regular solution of problem (1), (17), (18) of the class Φ^{∘}_{α}^{k}(\overline{D}_{1}), then the system of functions u, v_{1}, v_{2}, v_{3}, v_{4} will be a regular solution of the boundary value problem (20)–(29), where v_{i} ∈ Φ^{∘}_{α}^{k−1}(\overline{D}_{1}), i = 1, …, 4. Conversely, let the system of functions u, v_{1}, v_{2}, v_{3}, v_{4} of the class Φ^{∘}_{α}^{k−1}(\overline{D}_{1}) be a solution of problem (20)–(29). We shall prove that then v_{1} = u_{y}, v_{2} = u_{x_{1}}, v_{3} = u_{y_{2}}, v_{4} = u_{x_{2}} and therefore the function u is a solution of problem (1), (17), (18) in the class Φ^{∘}_{α}^{k}(\overline{D}_{1}). Indeed, by equality (22) we have

(u_{x_{1}} − v_{2})_{y_{1}} = (u_{y_{1}})_{x_{1}} − v_{2y_{1}} = v_{1x_{1}} − v_{2y_{1}} = 0,

which by virtue of the boundary condition (27) implies that v_{2} ≡ u_{x_{1}} in \overline{D}_{1}. Further, the use of equality (23) gives

(u_{y_{2}} − v_{3})_{y_{1}} = (u_{y_{1}})_{y_{2}} − v_{3y_{1}} = v_{1y_{2}} − v_{3y_{2}} = 0,

which by virtue of the boundary condition (29) implies that v_{3} ≡ u_{y_{2}} in \overline{D}_{1}.

Finally, on account of (24) and (28) we find by a similar reasoning that v_{4} ≡ u_{x_{2}} in \overline{D}_{1}.

Thus problem (1), (17), (18) in the class Φ^{∘}_{α}^{k}(\overline{D}_{1}) is equivalent to the problem of finding a system of functions u, v_{1}, v_{2}, v_{3}, v_{4} of the class Φ^{∘}_{α}^{k−1}(\overline{D}_{1}) satisfying (20)–(29).

If u, v_{1}, v_{2}, v_{3}, v_{4} are a solution of problem (20)–(29), then, after the Fourier transformation with respect to the variables x_{2} and y_{2}, equations (20)–(24) and boundary conditions (25)–(29) take the form

û_{y_{1}} = v̂_{1},  (30)
v̂_{1y_{1}} − v̂_{2x_{1}} + iξ_{2}v̂_{3} − iξ_{1}v̂_{4} = F̂,  (31)
v̂_{2y_{1}} − v̂_{1x_{1}} = 0,  (32)
v̂_{3y_{1}} − iξ_{2}v̂_{1} = 0,  (33)
v̂_{4y_{1}} − iξ_{1}v̂_{1} = 0,  (34)
(M_{1}v̂_{2} + N_{1}v̂_{1} + M̃_{1}v̂_{4} + Ñ_{1}v̂_{3} + S̃_{1}û)|_{l_{1}} = f̂_{1},  (35)
(M_{2}v̂_{2} + N_{2}v̂_{1} + M̃_{2}v̂_{4} + Ñ_{2}v̂_{3} + S̃_{2}û)|_{l_{2}} = f̂_{2},  (36)
(û_{x_{1}} − v̂_{2})|_{l_{1}∪l_{2}} = 0,  (37)
(v̂_{4} − iξ_{1}û)|_{l_{1}∪l_{2}} = 0,  (38)
(v̂_{3} − iξ_{2}û)|_{l_{1}∪l_{2}} = 0,  (39)

where û, v̂_{j}, j = 1, …, 4, F̂, f̂_{1}, f̂_{2} are respectively the Fourier transforms of the functions u, v_{j}, j = 1, …, 4, F, f_{1}, f_{2} with respect to the variables x_{2} and y_{2}; l_{1} : x_{1} − k_{1}y_{1} = 0, 0 ≤ y_{1} < +∞, l_{2} : x_{1} + k_{2}y_{1} = 0, 0 ≤ y_{1} < +∞, are the beams lying in the plane of the variables x_{1}, y_{1} and outgoing from the origin O(0, 0). Here i = \sqrt{−1}.

Thus, after the Fourier transformation with respect to the variables x_{2} and y_{2}, the spatial problem (20)–(29) is reduced to the plane problem (30)– (39) with the parameters ξ_{1} and ξ_{2} lying in the domain Ω_{1} : −k_{2}y_{1} < x_{1} < k_{1}y_{1}, 0 < y_{1} < +∞, of the plane of variables x_{1}, y_{1}. It is easy to verify that in the class Φ^{∘}_{α}^{k}(\overline{Ω}_{1}) of functions defined by inequalities (19) this reduction is equivalent.

Assuming v = (v̂_{1}, v̂_{2}) and v_{5} = (v̂_{3}, v̂_{4}), we rewrite system (31), (32) in the vector form

v_{y_{1}} − Av_{x_{1}} + A_{1}v_{5} = F_{1},  (40)

where

A = (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) , A_{1} = (\begin{matrix}iξ_{2} & −iξ_{1} \\ 0 & 0 \end{matrix}) , F = (F̂ ; 0) .

We easily see that with respect to v system (40) is strictly hyperbolic and its characteristics L_{1}(x_{1}^{0}, y_{1}^{0}) and L_{2}(x_{1}^{0}, y_{1}^{0}) passing through the point (x_{1}^{0}, y_{1}^{0}) are the straight lines defined by the equations

L_{1}(x_{1}^{0}, y_{1}^{0}) : x_{1} − y_{1} = x_{1}^{0} − y_{1}^{0}, L_{2}(x_{1}^{0}, y_{1}^{0}) : x_{1} + y_{1} = x_{1}^{0} + y_{1}^{0}.

If

B = (\begin{matrix}1 & −1 \\ 1 & 1 \end{matrix}) , Λ = (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}) ,  (41)

then, as is easy to verify,

B^{−1} = \frac{1}{2} (\begin{matrix}1 & 1 \\ −1 & 1 \end{matrix}) , B^{−1}AB = Λ.  (42)

Therefore by replacing v = Bw we can rewrite system (40) as

w_{y_{1}} − Λw_{x_{1}} + B^{−1}A_{1}v_{5} = B^{−1}F_{1}.  (43)

By virtue of (41)–(43), after replacing v = Bw, problem (30)–(39) is rewritten as

û_{y_{1}} = w_{1} − w_{2},  (44)
w_{1y_{1}} − w_{1x_{1}} + \frac{1}{2}iξ_{2}v̂_{3} − \frac{1}{2}ξ_{1}v̂_{4} = F_{2},  (45)
w_{1y_{1}} + w_{1x_{1}} − \frac{1}{2}iξ_{2}v̂_{3} + \frac{1}{2}ξ_{1}v̂_{4} = F_{3},  (46)
v̂_{3y_{1}} − iξ_{2}(w_{1} − w_{2}) = 0,  (47)
v̂_{4y_{1}} − iξ_{1}(w_{1} − w_{2}) = 0,  (48)
[(M_{1} + N_{1})w_{1} + (M_{1} − N_{1})w_{2} + M̃_{1}v̂_{4} + Ñ_{1}v̂_{3} + S̃_{1}û]|_{l_{1}} = f̂_{1}, (49)
[(M_{2} + N_{2})w_{1} + (M_{2} − N_{2})w_{2} + M̃_{2}v̂_{4} + Ñ_{1}v̂_{3} + S̃_{2}û]|_{l_{2}} = f̂_{2}, (50)
(û_{x_{1}} − (w_{1} + w_{2}))|_{l_{1}∪l_{2}} = 0,  (51)
(v̂_{4} − iξ_{1}û)|_{l_{1}∪l_{2}} = 0,  (52)
(v̂_{3} − iξ_{2}û)|_{l_{1}∪l_{2}} = 0,  (53)

where (F_{2}, F_{3}) = B^{−1}F_{1}.

It is easy to see that the characteristic L_{1}(x_{1}^{0}, y_{1}^{0}) : x_{1} + y_{1} = x_{1}^{0} + y_{1}^{0} of equation (45) passing through the point (x_{1}^{0}, y_{1}^{0}) ∈ \overline{Ω}_{1} intersects the beam l_{1} at a single point whose ordinate is denoted by ω_{1}(x_{1}^{0}, y_{1}^{0}). In a similar manner, the characteristic L_{2}(x_{1}^{0}, y_{1}^{0}) : x_{1} − y_{1} = x_{1}^{0} − y_{1}^{0}, (x_{1}^{0}, y_{1}^{0}) ∈ \overline{Ω}_{1}, of equation (46) intersects l_{2} at a single point with ordinate ω_{2}(x_{1}^{0}, y_{1}^{0}). Clearly,

ω_{i}(x_{1}^{0}, y_{1}^{0}) ∈ C^{∞}(\overline{Ω}_{1}), ω_{i}(x_{1}^{0}, y_{1}^{0}) ≤ y_{1}^{0}, i = 1, 2.  (54)

It should also be noted that a segment of each characteristic L_{i}(x_{1}^{0}, y_{1}^{0}), i = 1, 2, (x_{1}^{0}, y_{1}^{0}) ∈ \overline{Ω}_{1}, drawn from the point (x_{1}^{0}, y_{1}^{0}) towards decreasing ordinate values to the intersection with one of the beams l_{1} or l_{2}, admits a parametrization of the form

x_{1} = z_{i}(x_{1}^{0}, y_{1}^{0}; t), y_{1} = t, ω_{i}(x_{1}^{0}, y_{1}^{0}) ≤ t ≤ y_{1}^{0},  (55)

where z_{1}(x_{1}^{0}, y_{1}^{0}; t) = x_{1}^{0} + y_{1}^{0} − t ∈ C^{∞}[ω_{1}(x_{1}^{0}, y_{1}^{0}), y_{1}^{0}], z_{2}(x_{1}^{0}, y_{1}^{0}; t) = x_{1}^{0} − y_{1}^{0} + t ∈ C^{∞}[ω_{2}(x_{1}^{0}, y_{1}^{0}), y_{1}^{0}].

Remark. The functions û, w_{1}, w_{2}, v̂_{3}, v̂_{4}, F_{2}, F_{3}, f̂_{1}, f̂_{2} depend not only on the independent variables x_{1} and x_{2}, but also on the parameters ξ_{1} and ξ_{2}. For simplicity, these parameters will be omitted below. For example, instead of û(x_{1}, y_{1}, ξ_{1}, ξ_{2}) we shall write û(x_{1}, y_{1}).

Using (44), (51) and the fact that û(0, 0) = 0, for u ∈ Φ^{∘}_{α}^{k}(\overline{D}_{1}) we obtain

û(x_{1}, y_{1}) = ∫_{0}^{y_{1}}(k_{1}u_{x_{1}} + u_{y_{1}})(k_{1}t, t)dt = ∫_{0}^{y_{1}}[(1 + k_{1})w_{1} +
+ (1 − k_{1})w_{2}](k_{1}t, t)dt, (x_{1}, y_{1}) ∈ l_{1},  (56)
û(x_{1}, y_{1}) = ∫_{0}^{y_{1}}(−k_{2}u_{x_{1}} + u_{y_{1}})(−k_{2}t, t)dt = ∫_{0}^{y_{1}}[(1 − k_{2})w_{1} +
+ (1 + k_{2})w_{2}](−k_{2}t, t)dt, (x_{1}, y_{1}) ∈ l_{2}.  (57)

For (x_{1}, y_{1}) ∈ \overline{Ω}_{1} and x_{1} > 0, by integrating equations (44), (47), (48) with respect to the variable y_{1} and taking into account the boundary conditions (52), (53) and equalities (56), (57) we obtain

û(x_{1}, y_{1}) = ∫_{0}^{k_{1}^{−1}x_{1}}[(1 + k_{1})w_{1} + (1 − k_{1})w_{2}](k_{1}t, t)dt +

+ ∫_{k_{1}^{−1}x_{1}}^{y_{1}} (w_{1} − w_{2})(x_{1}, t)dt,  (58)

v̂_{3}(x_{1}, y_{1}) = iξ_{2} ∫_{0}^{k_{1}^{−1}x_{1}}[(1 + k_{1})w_{1} + (1 − k_{1})w_{2}](k_{1}t, t)dt +

+ iξ_{2} ∫_{k_{1}^{−1}x_{1}}^{y_{1}} (w_{1} − w_{2})(x_{1}, t)dt,  (59)

v̂_{4}(x_{1}, y_{1}) = iξ_{1} ∫_{0}^{k_{1}^{−1}x_{1}}[(1 + k_{1})w_{1} + (1 − k_{1})w_{2}](k_{1}t, t)dt +

+ iξ_{1} ∫_{k_{1}^{−1}x_{1}}^{y_{1}} (w_{1} − w_{2})(x_{1}, t)dt,  (60)

and for (x_{1}, y_{1}) ∈ \overline{Ω}_{1} and x_{1} ≤ 0 we have

û(x_{1}, y_{1}) = ∫_{0}^{−k_{1}^{−1}x_{1}}[(1 − k_{2})w_{1} + (1 + k_{2})w_{2}](−k_{2}t, t)dt +

+ ∫_{−k_{2}^{−1}x_{1}}^{y_{1}} (w_{1} − w_{2})(x_{1}, t)dt,  (61)

v̂_{3}(x_{1}, y_{1}) = iξ_{2} ∫_{0}^{−k_{2}^{−1}x_{1}}[(1 − k_{2})w_{1} + (1 + k_{2})w_{2}](−k_{2}t, t)dt +

+ iξ_{2} ∫_{−k_{2}^{−1}x_{1}}^{y_{1}} (w_{1} − w_{2})(x_{1}, t)dt,  (62)

v̂_{4}(x_{1}, y_{1}) = iξ_{1} ∫_{0}^{−k_{2}^{−1}x_{1}}[(1 − k_{2})w_{1} + (1 + k_{2})w_{2}](−k_{2}t, t)dt +

+ iξ_{1} ∫_{−k_{2}^{−1}x_{1}}^{y_{1}} (w_{1} − w_{2})(x_{1}, t)dt.  (63)

Now, by integrating equations (45) and (46) along the respective characteristics from the point (x_{1}, y_{1}) ∈ \overline{Ω}_{1} to the intersection points of these characteristics with the beams l_{1} and l_{2} we obtain

w_{1}(x_{1}, y_{1}) = w_{1}(k_{1}ω_{1}(x_{1}, y_{1}), ω_{1}(x_{1}, y_{1})) +
+ \frac{1}{2}i ∫_{ω_{1}(x_{1},y_{1})}^{y_{1}} (ξ_{1}v̂_{4}−ξ_{2}v̂_{3})(z_{1}(x_{1}, y_{1}; t), t)dt+F̃_{2}(x_{1}, y_{1}),  (64)
w_{2}(x_{1}, y_{1}) = w_{2}(−k_{2}ω_{2}(x_{1}, y_{1}), ω_{2}(x_{1}, y_{1})) +

+ \frac{1}{2}i ∫_{ω_{2}(x_{1},y_{1})}^{y_{1}} (ξ_{2}v̂_{3}−ξ_{1}v̂_{4})(z_{2}(x_{1}, y_{1}; t), t)dt+F̃_{3}(x_{1}, y_{1}),  (65)

where

F̃_{2}(x_{1}, y_{1}) = ∫_{ω_{1}(x_{1},y_{1})}^{y_{1}} F_{2}(z_{1}(x_{1}, y_{1}; t), t)dt,

F̃_{3}(x_{1}, y_{1}) = ∫_{ω_{2}(x_{1},y_{1})}^{y_{1}} F_{3}(z_{2}(x_{1}, y_{1}; t), t)dt.

We set

φ = w_{1}|_{l_{1}}, ψ = w_{2}|_{l_{2}}.

On substituting the obtained expressions for û, w_{1}, w_{2}, v̂_{3}, v̂_{4} from (58)– (65) into the boundary conditions (49), (50) and taking into account that

ω_{1}(x_{1}, y_{1})|_{l_{1}} = y_{1}, ω_{1}(x_{1}, y_{1}) = \frac{1 − k_{2}}{1 + k_{1}} y_{1},

ω_{2}(x_{1}, y_{1}) = \frac{1 − k_{1}}{1 + k_{2}} y_{1}, ω_{2}(x_{1}, y_{1})|_{l_{2}} = y_{1}

we obtain

(M_{1} + N_{1})φ(y_{1}) + (M_{1} − N_{1})ψ(τ_{2}y_{1}) + (M_{1} − N_{1})\frac{1}{2}i ×

× ∫_{τ_{2}y_{1}}^{y_{1}} (ξ_{2}v̂_{3} − ξ_{1}v̂_{4})(z_{2}(k_{1}y_{1}, y_{1}; t), t)dt + (M̃_{1}iξ_{1} + Ñ_{1}iξ_{2} + S̃_{1}) ×

× ∫_{0}^{y_{1}}[(1 + k_{1})w_{1} + (1 − k_{1})w_{2}](k_{1}t, t)dt = f̂_{1}(y_{1}) − (M_{1} + N_{1})(y_{1}) ×

×F̃_{2}(k_{1}y_{1}, y_{1}) − (M_{1} − N_{1})(y_{1})F̃_{3}(k_{1}y_{1}, y_{1}),  (66)

(M_{2} + N_{2})φ(τ_{1}y_{1}) + (M_{2} − N_{2})ψ(y_{1}) + (M_{2} + N_{2})\frac{1}{2}i ×

× ∫_{τ_{1}y_{1}}^{y_{1}} (ξ_{1}v̂_{4} − ξ_{2}v̂_{3})(z_{1}(−k_{2}y_{1}, y_{1}; t), t)dt + (M̃_{2}iξ_{1} + Ñ_{2}iξ_{2} + S̃_{2}) ×

× ∫_{0}^{y_{1}}[(1 − k_{2})w_{1} + (1 + k_{2})w_{2}](−k_{2}t, t)dt = f̂_{2}(y_{1}) −

−(M_{2} + N_{2})(y_{1})F̃_{2}(−k_{2}y_{1}, y_{1}) − (M_{2} − N_{2})(y_{1})F̃_{3}(−k_{2}y_{1}, y_{1}). (67)

Now, since 0 < k_{i} < 1, i = 1, 2, we have

0 < τ_{1} = \frac{1 − k_{2}}{1 + k_{1}} < 1, 0 < τ_{2} = \frac{1 − k_{1}}{1 + k_{2}} < 1, 0 < τ = τ_{1}τ_{2} < 1. (68)

It will assumed below that

(M_{1} + N_{1})|_{l_{1}} ≠ 0, (M_{2} − N_{2})|_{l_{2}} ≠ 0.

Since by assumptions the functions M_{j}, N_{j}, j = 1, 2, do not depend on the variables x_{2} and y_{2}, these conditions are evidently equivalent to the conditions

(M_{1} + N_{1})|_{S_{1}^{0}} ≠ 0,  (69)
(M_{2} − N_{2})|_{S_{2}^{0}} ≠ 0.  (70)

Solving system (66), (67) with respect to φ and ψ, we obtain

φ(y_{1}) − a(y_{1})φ(τy_{1}) = (T_{1}(w))(y_{1}) + (T_{2}(v_{5}))(y_{1}) + g_{1}(y_{1}), (71)
ψ(y_{1}) − b(y_{1})ψ(τy_{1}) = (T_{3}(w))(y_{1}) + (T_{5}(v_{5}))(y_{1}) + g_{2}(y_{1}), (72)

where

a(y_{1}) = [(M_{1} − N_{1})(M_{1} + N_{1})^{−1}](y_{1})[(M_{2} + N_{2})(M_{2} − N_{2})^{−1}](τ_{2}y_{1}),

b(y_{1}) = [(M_{2} + N_{2})(M_{2} − N_{2})^{−1}](y_{1})[(M_{1} − N_{1})(M_{1} + N_{1})^{−1}](τ_{1}y_{1}),

and T_{j}, j = 1, …, 4, are linear integral operators of the form

(T_{1}(w))(y_{1}) = (E_{11}ξ_{1} + E_{12}ξ_{2} + E_{13}) ∫_{0}^{y_{1}}w(k_{1}t, t)dt +

+ (E_{14}ξ_{1} + E_{15}ξ_{2} + E_{16}) ∫_{0}^{τ_{2}y_{1}}w(−k_{2}t, t)dt,

(T_{2}(v_{5}))(y_{1}) = (E_{21}ξ_{1} + E_{22}ξ_{2}) ∫_{τ_{2}y_{1}}^{y_{1}} v_{5}(z_{2}(k_{1}y_{1}, y_{1}; t), t)dt +

+ (E_{23}ξ_{1} + E_{24}ξ_{2}) ∫_{τy_{1}}^{τ_{2}y_{1}}v_{5}(z_{1}(−k_{2}τ_{2}y_{1}, τ_{2}y_{1}; t), t)dt,

(T_{3}(w))(y_{1}) = (E_{31}ξ_{1} + E_{32}ξ_{2} + E_{33}) ∫_{0}^{y_{1}}w(−k_{2}t, t)dt +

+ (E_{34}ξ_{1} + E_{35}ξ_{2} + E_{36}) ∫_{0}^{τ_{1}y_{1}}w(k_{1}t, t)dt,

(T_{4}(v_{5}))(y_{1}) = (E_{41}ξ_{1} + E_{42}ξ_{2}) ∫_{τ_{1}y_{1}}^{y_{1}} v_{5}(z_{1}(−k_{2}y_{1}, y_{1}; t), t)dt +

+ (E_{43}ξ_{1} + E_{44}ξ_{2}) ∫_{τy_{1}}^{τ_{1}y_{1}}v_{5}(z_{2}(k_{1}τ_{1}y_{1}, τ_{1}y_{1}; t), t)dt.

Here v_{5} = (v̂_{3}, v̂_{4}), E_{ij} are the completely defined 1 × 2 matrices of the class C^{k−1}, and g_{1}, g_{2} are functions expressed in terms of the well-known functions F and f_{1}, f_{2}.

As mentioned above, in equations (58)–(65), (71), (72) the unknown functions and the right-hand sides depend on the parameter ξ = (ξ_{1}, ξ_{2}), which we omit for simplicity.

Remark. As follows from the above reasoning, if conditions (69), (70) are fulfilled, then in the class Φ^{∘}_{α}^{k}(\overline{D}_{1}) problem (1), (17), (18) is equivalent to the problem of finding a system of functions û, w_{1}, w_{2}, v̂_{3}, v̂_{4}, φ, and ψ from equations (58)–(65), (71), (72), where

û, w_{1}, w_{2}, v̂_{3}, v̂_{4} ∈ Φ^{∘}_{α}^{k−1}(\overline{Ω}_{1}), φ ∈ Φ^{∘}_{α}^{k−1}[0, +∞), ψ ∈ Φ^{∘}_{α}^{k−1}[0, +∞)

and

F̃_{2}, F̃_{3} ∈ Φ^{∘}_{α}^{k−1}(\overline{Ω}_{1}), g_{1} ∈ Φ^{∘}_{α}^{k−1}[0, +∞), g_{2} ∈ Φ^{∘}_{α}^{k−1}[0, +∞).

Let Q(x_{1}^{0}, y_{1}^{0}) ∈ Ω_{1}. Denote by P_{1} and P_{2} the points of intersection of the characteristics L_{1}(x_{1}^{0}, y_{1}^{0}) : x_{1} − y_{1} = x_{1}^{0} − y_{1}^{0} and L_{2}(x_{1}^{0}, y_{1}^{0}) : x_{1} + y_{1} = x_{1}^{0} + y_{1}^{0} from system (40) with the beams l_{1} and l_{2}, respectively. Denote by Ω_{1Q} ⊂ Ω_{1} a rectangle with vertices at the points O(0, 0), P_{1}, P_{2}, and Q.

We set

σ = a(0) = b(0) = [\frac{(M_{1} − N_{1})(M_{2} + N_{2})}{(M_{2} − N_{2})(M_{1} + N_{1})}](0),

where a(y_{1}) and b(y_{1}) are the coefficients in equations (71), (72).

Lemma 1. Let conditions (69), (70) be fulfilled. Then for k + α > log |σ|/ log τ + 1 the boundary value problem (44)–(53) is uniquely solvable in the class Φ^{∘}_{α}^{k−1}(\overline{Ω}_{1}); the domain of the dependence of the solution of this problem for the point Q(x_{1}^{0}, y_{1}^{0}) ∈ Ω_{1} is \overline{Ω}_{1Q}.

Remark. As mentioned above, the boundary value problem (44)–(53) is equivalent to the system of integro-functional equations (58)–(65), (71), (72). When problem (44)–(53) is considered in the domain Ω_{1Q}, it is sufficient to investigate equations (71) and (72) on the segments [0, d_{1}] and [0, d_{2}], where d_{1} and d_{2} are the ordinates of the points P_{1} and P_{2}.

Let us consider the functional equations

(K_{1j}(φ))(y_{1}) = φ(y_{1}) − a_{j}(y_{1})φ(τy_{1}) = χ_{1}(y_{1}), 0 ≤ y_{1} ≤ d_{1}, (73)
(K_{2j}(ψ))(y_{1}) = ψ(y_{1}) − b_{j}(y_{1})ψ(τy_{1}) = χ_{2}(y_{2}), 0 ≤ y_{1} ≤ d_{2}, (74)

where a_{j}(y_{1}) = τ^{j}a(y_{1}), b_{j}(y_{1}) = τ^{j}b(y_{1}), j = 0, 1, …, k − 1. Note that if we differentiate j-times the expression (K_{10}(φ))(y_{1}), which is the left-hand side of equation (71), with respect to y_{1}, then in the obtained expression the sum of the terms which contain the function φ(y_{1}) with its derivative φ^{(j)}(y_{1}) gives (K_{1j}(φ^{(j)}))(y_{1}). A similar statement also holds for K_{2j}.

Let in equations (73), (74) the right-hand sides χ_{p}(y_{1}) ∈ Φ^{∘}_{k−1+α−j}[0, d_{p}],

p = 1, 2. It is assumed that Φ^{∘}_{α}^{k}[0, d_{p}] = Φ^{∘}_{α}[0, d_{p}] for k = 0. In that case, by the definition of the space Φ^{∘}_{k−1+α−j}[0, d_{p}], for any natural number N there exist positive numbers C_{p} = C_{p}(y, N, χ_{p}), B_{p} = B_{p}(y_{1}, N, χ_{p}), not depending on ξ = (ξ_{1}, ξ_{2}), such that for 0 ≤ y_{1} ≤ d_{p} and |ξ| = |ξ_{1}|+|ξ_{2}| > B_{p} the inequality

|χ_{p}(y_{1})| ≤ C_{p}y_{1}^{k−1+α−j}e^{−N|ξ|}

is fulfilled and

C_{p}^{0}(y_{1}) = sup_{0≤y_{1}^{0}≤y_{1}} C_{p}(y_{1}^{0}) < +∞, B_{p}^{0}(y_{1}) = sup_{0≤y_{1}^{0}≤y_{1}} B_{p}(y_{1}^{0}) < +∞.

Lemma 2. Let conditions (69), (70) be fulfilled. Then for k −1 +α > −\frac{log |σ|}{log τ} equations (73) and (74) are uniquely solvable in the spaces Φ^{∘}_{k−1+α−j}

[0, d_{1}] and Φ^{∘}_{k−1+α−j}[0, d_{2}], and for |ξ| > B_{1} and |ξ| > B_{2} the estimates

|(K_{1j}^{−1}(χ_{1}))(y_{1})| = |φ(y_{1})| ≤ σ_{1}C_{1}y_{1}^{k−1+α−j}e^{−N|ξ|},  (75)
|(K_{2j}^{−1}(χ_{2}))(y_{1})| = |ψ(y_{1})| ≤ σ_{2}C_{2}y_{1}^{k−1+α−j}e^{−N|ξ|}  (76)

hold, where the positive constants σ_{1} and σ_{2} do not depend on N, ξ, and the functions χ_{1}, χ_{2}.

The proof of this lemma repeats the reasoning from [9].

To prove Lemma 1, we shall solve the system of integro-functional equations (58)–(65), (71), (72) with respect to the unknowns

û, w_{1}, w_{2}, v̂_{3}, v̂_{4} ∈ Φ^{∘}_{α}^{k−1}(\overline{Ω}_{1Q}), φ ∈ Φ^{∘}_{α}^{k−1}[0, d_{1}], ψ ∈ Φ^{∘}_{α}^{k−1}[0, d_{2}]

by the method of successive approximations.

356 S. KHARIBEGASHVILI

We set

û_{0}(x_{1}, y_{1}) ≡ 0, w_{1,0}(x_{1}, y_{1}) ≡ 0, w_{2,0}(x_{1}, y_{1}) ≡ 0, v̂_{3,0}(x_{1}, y_{1}) ≡ 0, v̂_{4,0}(x_{1}, y_{1}) ≡ 0, φ_{0}(y_{1}) ≡ 0, ψ_{0}(y_{1}) ≡ 0,

û_{n}(x_{1}, y_{1}) = ∫_{0}^{β^{−1}x_{1}}[(1 + β)w_{1,n−1} + (1 − β)w_{2,n−1}](βt, t)dt +

+ ∫_{β^{−1}x_{1}}^{y_{1}} (w_{1,n−1} − w_{2,n−1})(x_{1}, t)dt,  (77)

v̂_{3,n}(x_{1}, y_{1}) = iξ_{2} ∫_{0}^{β^{−1}x_{1}}[(1 + β)w_{1,n−1} + (1 − β)w_{2,n−1}](βt, t)dt +

+iξ_{2} ∫_{β^{−1}x_{1}}^{y_{1}} (w_{1,n−1} − w_{2,n−1})(x_{1}, t)dt,  (78)

v̂_{4,n}(x_{1}, y_{1}) = iξ_{1} ∫_{0}^{β^{−1}x_{1}}[(1 + β)w_{1,n−1} + (1 − β)w_{2,n−1}](βt, t)dt +

+iξ_{1} ∫_{β^{−1}x_{1}}^{y_{1}} (w_{1,n−1} − w_{2,n−1})(x_{1}, t)dt,  (79)

β = \begin{cases}k_{1} for x_{1} > 0, \\ −k_{2} for x_{1} ≤ 0, \end{cases} β^{−1}x_{1} ≤ y_{1} for (x_{1}, y_{1}) ∈ Ω_{1Q},

w_{1,n}(x_{1}, y_{1}) = φ_{n}(ω_{1}(x_{1}, y_{1})) +

+\frac{1}{2}i ∫_{ω_{1}(x_{1},y_{1})}^{y_{1}} (ξ_{1}v̂_{4,n−1} − ξ_{2}v̂_{3,n−1})(z_{1}(x_{1}, y_{1}; t), t)dt + F̃_{2}(x_{1}, y_{1}), (80)

w_{2,n}(x_{1}, y_{1}) = ψ_{n}(ω_{2}(x_{1}, y_{1})) +

+\frac{1}{2}i ∫_{ω_{2}(x_{1},y_{1})}^{y_{1}} (ξ_{2}v̂_{3,n−1} − ξ_{1}v̂_{4,n−1})(z_{2}(x_{1}, y_{1}; t), t)dt + F̃_{3}(x_{1}, y_{1}). (81)

To define the functions φ_{n}(y_{1}) and ψ_{n}(y_{1}), we shall use the equations

(K_{10}(φ_{n}))(y_{1}) = (T_{1}(w_{n−1}))(y_{1}) + (T_{2}(v_{5,n−1}))(y_{1}) + g_{1}(y_{1}), (82)
(K_{20}(ψ_{n}))(y_{1}) = (T_{3}(w_{n−1}))(y_{1}) + (T_{4}(v_{5,n−1}))(y_{1}) + g_{2}(y_{1}), (83)

where v_{5} = (v̂_{3}, v̂_{4}).

If the conditions of Lemma 2 are fulfilled, then using estimate (75), (76) for j = 0 and applying (54), (68), also the inequality β^{−1}x_{1} ≤ y_{1} for (x_{1}, y_{1}) ∈ Ω_{1Q}, Q(x_{1}^{0}, y_{1}^{0}) ∈ Ω_{1}, and taking into account the Volterra structure of the integral operators contained in equalities (77)–(78), for |ξ| > β̃ we obtain by the method of mathematical induction the inequalities:

|û_{n+1}(x_{1}, y_{1}) − û_{n}(x_{1}, y_{1})| ≤ M^{∗} \frac{M_{∗}^{n}}{n!} (1 + |ξ|)^{n}e^{−N|ξ|}y_{1}^{n+k−1+α}, (84)
|w_{p,n+1}(x_{1}, y_{1}) − w_{p,n}(x_{1}, y_{1})| ≤ M^{∗} \frac{M_{∗}^{n}}{n!} (1 + |ξ|)^{n}e^{−N|ξ|}y_{1}^{n+k−1+α}, (85)
p = 1, 2,
|v̂_{q,n+1}(x_{1}, y_{1}) − v̂_{q,n}(x_{1}, y_{1})| ≤ M^{∗} \frac{M_{∗}^{n}}{n!} (1 + |ξ|)^{n}e^{−N|ξ|}y_{1}^{n+k−1+α}, (86)
q = 3, 4,
|φ_{n+1}(y_{1}) − φ_{n}(y_{1})| ≤ M^{∗} \frac{M_{∗}^{n}}{n!} (1 + |ξ|)^{n}e^{−N|ξ|}y_{1}^{n+k−1+α},  (87)
|ψ_{n+1}(y_{1}) − ψ_{n}(y_{1})| ≤ M^{∗} \frac{M_{∗}^{n}}{n!} (1 + |ξ|)^{n}e^{−N|ξ|}y_{1}^{n+k−1+α},  (88)

where the positive numbers B̃=B̃(x_{1}^{0}, y_{1}^{0}, N, f_{1}, f_{2}, F), M_{∗}=M_{∗}(x_{1}^{0}, y_{1}^{0}, σ_{1}, σ_{2}) and M^{∗} = M^{∗}(x_{1}^{0}, y_{1}^{0}, N, f_{1}, f_{2}, F, σ_{1}, σ_{2}) do not depend on ξ; σ_{1} and σ_{2} are the constants from (75) and (76).

Remark. As follows from equalities (77)–(79), for x_{1} > 0 and x_{1} ≤ 0 the functions û_{n}, v̂_{3,n} and v̂_{4,n} are defined by different formulas. This does not cause discontinuities of the functions û_{n}, v̂_{3,n} and v̂_{4,n} and their partial derivatives with respect to x_{1} and y_{1} up to order (k − 1) inclusive along the axis Oy_{1} : x_{1} = 0, since it is assumed that the functions F̃_{2}, F̃_{3}, g_{1}, g_{2} from (72)–(83) and their derivatives up to order (k − 1) inclusive are equal to zero at the point O(0, 0).

After differentiating equalities (77)–(83) with respect to x_{1} and y_{1} and using the above estimates (84)–(88) and (75), (76), for |ξ| > B̃_{j_{1}+j_{2}} we obtain by the method of mathematical induction the inequalities

|[∂^{j_{1},j_{2}}(û_{n+1} − û_{n})](x_{1}, y_{1})| ≤ M_{j_{1}+j_{2}}^{∗} \frac{M_{∗j_{1}+j_{2}}^{n}}{n!} (1 + |ξ|)^{n}e^{−N|ξ|} ×
× y_{1}^{n+k+α−j_{1}−j_{2}−1},  (89)
|[∂^{j_{1},j_{2}}(w_{p,n+1} − w_{p,n})(x_{1}, y_{1})| ≤ M_{j_{1}+j_{2}}^{∗} \frac{M_{∗j_{1}+j_{2}}^{n}}{n!} (1 + |ξ|)^{n}e^{−N|ξ|} ×
× y_{1}^{n+k+α−j_{1}−j_{2}−1}, p = 1, 2,  (90)
|[∂^{j_{1},j_{2}}(v̂_{q,n+1} − v̂_{q,n})(x_{1}, y_{1})| ≤ M_{j_{1}+j_{2}}^{∗} \frac{M_{∗j_{1}+j_{2}}^{n}}{n!} (1 + |ξ|)^{n}e^{−N|ξ|} ×

× y_{1}^{n+k+α−j_{1}−j_{2}−1}, q = 3, 4,  (91)
|[\frac{d^{j_{1}+j_{2}}}{dy_{1}^{j_{1}+j_{2}}} (φ_{n+1} − φ_{n})](y_{1})| ≤ M_{j_{1}+j_{2}}^{∗} \frac{M_{∗j_{1}+j_{2}}^{n}}{n!} (1 + |ξ|)^{n} ×
× e^{−N|ξ|}y_{1}^{n+k+α−j_{1}−j_{2}−1},  (92)
|[\frac{d^{j_{1}+j_{2}}}{dy_{1}^{j_{1}+j_{2}}} (ψ_{n+1} − ψ_{n})](y_{1})| ≤ M_{j_{1}+j_{2}}^{∗} \frac{M_{∗j_{1}+j_{2}}^{n}}{n!} (1 + |ξ|)^{n} ×
× e^{−N|ξ|}y_{1}^{n+k+α−j_{1}−j_{2}−1},  (93)

where ∂^{j_{1},j_{2}} = ∂^{j_{1}+j_{2}}/∂x_{1}^{j_{1}}∂y_{1}^{j_{2}}, 0 ≤ j_{1} + j_{2} ≤ k − 1; B̃_{p}, M_{p}^{∗}, and M_{∗p}, p = 1, …, k − 1, are positive integers not depending on ξ.

By (89) we find that for 0 ≤ j_{1} + j_{2} ≤ k − 1 the series

û_{j_{1},j_{2}}(x_{1}, y_{1}) = lim_{n→∞}[∂^{j_{1},j_{2}} û_{n}](x_{1}, y_{1}) = ∑_{n=1}^{∞} [∂^{j_{1},j_{2}}(û_{n} − û_{n−1})](x_{1}, y_{1})

converges uniformly in \overline{Ω}_{1Q} and for the sum of this series we have the estimate

|û_{j_{1},j_{2}}(x_{1}, y_{1})| ≤ M_{j_{1}+j_{2}}^{∗}e^{M_{∗j_{1}+j_{2}} (1+|ξ|)y_{1}}e^{−N|ξ|}y_{1}^{k+α−j_{1}−j_{2}−1}. (94)

This estimate implies that û_{j_{1},j_{2}} ∈ Φ^{∘}_{k−1+α−j_{1}−j_{2}}(\overline{Ω}_{1Q}), since, as is easy to verify, the operator of multiplication by the function

e^{M_{∗j_{1}+j_{2}} (1+|ξ|)y_{1}}

transforms the space Φ^{∘}_{k−1+α−j_{1}−j_{2}}(\overline{Ω}_{1Q}) into itself. This in turn implies

that u^{1}(x_{1}, y_{1})≡û_{0,0}(x_{1}, y_{1})∈ Φ^{∘}_{α}^{k−1}(\overline{Ω}_{1Q}) and û_{j_{1},j_{2}}(x_{1}, y_{1})≡∂^{j_{1},j_{2}}u^{1}(x_{1}, y_{1}).

By (90)–(93) we find in a similar manner that the series

w_{p}^{1}(x_{1}, y_{1})= lim_{n→∞} w_{p,n}(x_{1}, y_{1})= ∑_{n=1}^{∞} (w_{p,n}(x_{1}, y_{1})−w_{p,n−1}(x_{1}, y_{1})), p=1, 2,

v_{q}^{1}(x_{1}, y_{1})= lim_{n→∞} v̂_{q,n}(x_{1}, y_{1})= ∑_{n=1}^{∞} (v̂_{q,n}(x_{1}, y_{1})−v̂_{q,n−1}(x_{1}, y_{1})), q =3, 4,

converge in the space Φ^{∘}_{α}^{k−1}(\overline{Ω}_{1Q}), and the series

φ^{1}(y_{1}) = lim_{n→∞} φ_{n}(y_{1}) = ∑_{n=1}^{∞} (φ_{n}(y_{1}) − φ_{n−1}(y_{1})),

ψ^{1}(y_{1}) = lim_{n→∞} ψ_{n}(y_{1}) = ∑_{n=1}^{∞} (ψ_{n}(y_{1}) − ψ_{n−1}(y_{1}))

converge in the spaces Φ^{∘}_{α}^{k−1}[0, d_{1}] and Φ^{∘}_{α}^{k−1}[0, d_{2}], respectively. Hence by virtue of (77)–(83) it follows that the limit fnctions u^{1}, w^{1}, v_{3}^{1}, v_{4}^{1}, φ^{1}, and ψ^{1} satisfy the system of equations (58)–(65), (71), (72). Thus to prove Lemma 1 it only remains to show that the system of equations (58)–(65), (71), (72) has no other solutions in the classes considered above. Indeed, let us assume that the functions u^{∗}, w^{∗}, v_{3}^{∗}, v_{4}^{∗}, φ^{∗}, and ψ^{∗} from the classes proved above satisfy the homogeneous system of equations corresponding to (58)–(65), (71), (72), i.e., for

F̃_{2} = 0, F̃_{3} = 0, g_{1} = 0, g_{2} = 0.  (95)

To this system with the homogeneous conditions (95) we apply the method of successive approximations, assuming that the functions u^{∗}, w^{∗}, v_{3}^{∗}, v_{4}^{∗}, φ^{∗}, and ψ^{∗} are zero approximations. Since this system of functions satisfy the homogeneous system of equations, each of the following approximations will coincide with it, i.e.,

u_{n}^{∗}(x_{1}, y_{1}) ≡ u^{∗}(x_{1}, y_{1}), w_{n}^{∗}(x_{1}, y_{1}) ≡ w^{∗}(x_{1}, y_{1}), v_{3,n}^{∗}(x_{1}, y_{1}) ≡ v_{3}^{∗}(x_{1}, y_{1}), v_{4,n}^{∗}(x_{1}, y_{1}) ≡ v_{4}^{∗}(x_{1}, y_{1}), φ_{n}^{∗}(y_{1}) ≡ φ^{∗}(y_{1}), ψ_{n}^{∗}(y_{1}) ≡ ψ^{∗}(y_{1}).

By the same reasoning as for estimates (84)–(88), for |ξ| > B̃^{∗} we obtain

|u^{∗}(x_{1}, y_{1})| = |u_{n}^{∗}(x_{1}, y_{1})| ≤ M̃^{∗} \frac{M̃_{∗}^{n}}{n!} (1 + |ξ|)^{n}e^{−N|ξ|}y_{1}^{n+k+α−1},

|w_{p}^{∗}(x_{1}, y_{1})| = |w_{p,n}^{∗}(x_{1}, y_{1})| ≤ M̃^{∗} \frac{M̃_{∗}^{n}}{n!} (1 + |ξ|)^{n}e^{−N|ξ|}y_{1}^{n+k+α−1}, p = 1, 2,

|v_{q}^{∗}(x_{1}, y_{1})| = |v_{q,n}^{∗}(x_{1}, y_{1})| ≤ M̃^{∗} \frac{M̃_{∗}^{n}}{n!} (1 + |ξ|)^{n}e^{−N|ξ|}y_{1}^{n+k+α−1}, q = 3, 4,

|φ^{∗}(y_{1})| = |φ_{n}^{∗}(y_{1})| ≤ M̃^{∗} \frac{M̃_{∗}^{n}}{n!} (1 + |ξ|)^{n}e^{−N|ξ|}y_{1}^{n+k+α−1},

|ψ^{∗}(y_{1})| = |ψ_{n}^{∗}(y_{1})| ≤ M̃^{∗} \frac{M̃_{∗}^{n}}{n!} (1 + |ξ|)^{n}e^{−N|ξ|}y_{1}^{n+k+α−1}.

Hence, passing to the limit as n → ∞, we find that

u^{∗} ≡ 0, w_{1}^{∗} ≡ 0, w_{2}^{∗} ≡ 0, v_{3}^{∗} ≡ 0, v_{4}^{∗} ≡ 0, φ^{∗} ≡ 0, ψ^{∗} ≡ 0,

which completes the proof of the lemma.

Let Q_{0}(x_{1}^{0}, x_{2}^{0}, y_{1}^{0}, y_{2}^{0} ∈ D_{1}. Denote by

D_{1Q_{0}} : −k_{2}y_{1} < x_{1} < k_{1}y_{1}, y_{1} − x_{1} < y_{1}^{0} − x_{1}^{0}, y_{1} + x_{1} < y_{1}^{0} + x_{1}^{0}

a subdomain of the domain D_{1} bounded by the surfaces S_{1}^{0}, S_{2}^{0} and the characteristic hyperplanes S̃_{1} : y_{1} − x_{1} = y_{1}^{0} − x_{1}^{0} and S̃_{2} : y_{1} + x_{1} = y_{1}^{0} + x_{1}^{0} passing through the point Q_{0}.

Since problem (1), (17), (18) is equivalently reduced to problem (44)– (53), Lemma 1 immediately implies

Theorem 2. Let conditions (69), (70) be fulfilled. Then for k + α > −\frac{log |σ|}{log τ} + 1 problem (1), (17), (18) is uniquely solvable in the class Φ^{∘}_{α}^{k}(\overline{D}_{1})

for any F ∈ Φ^{∘}_{α}^{k−1}(\overline{D}_{1}) and f_{i} ∈ Φ^{∘}_{α}^{k−1}(S_{i}^{0}), i = 1, 2; the domain of the dependence of the solution u of this problem for the point Q_{0} ∈ D_{1} is contained in \overline{D}_{1Q_{0}}.

Note that if conditions (69), (70) or the inequality k + α > −\frac{log |σ|}{log τ} + 1 are invalid, then problem (1), (17), (18) may turn out to be formulated incorrectly.

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(Recieved 29.12.1995)

Author’s address:

Department of Theoretical Mechanics (No. 4)

Georgian Technical University

77, Kostava St., Tbilisi 380075

Georgia