GEORGIAN MATHEMATICAL JOURNAL: Vol. 5, No. 2, 1998, 121-138

ON THE BOUNDARY VALUE PROBLEM IN A DIHEDRAL ANGLE FOR NORMALLY HYPERBOLIC SYSTEMS OF FIRST ORDER

O. JOKHADZE

Abstract. Some structural properties as well as a general threedimensional boundary value problem for normally hyperbolic systems of partial differential equations of first order are studied. A condition is given which enables one to reduce the system under consideration to a first-order system with the spliced principal part. It is shown that the initial problem is correct in a certain class of functions if some conditions are fulfilled.

§ 1. Some Structural Properties of Normally Hyperbolic Systems of First Order

In the Euclidean space ℝ^{n+1}, n ≥ 2, of independent variables (x, t), x = (x_{1}, …, x_{n}), we consider the system of partial differential equations of first order

A_{0}u_{t} + ∑_{i=1}^{n} A_{i}u_{x_{i}} + Bu = F,  (1.1)

where A_{i}, i = 0, 1, …, n, B are the given real m × m matrix-functions, m ≥ 2, F is the given and u is the unknown m-dimensional real vectorfunction. It is assumed that det A_{0} ≠ 0.

Denote by p(x, t; λ, ξ) the characteristic determinant of system (1.1), i.e., p(x, t; λ, ξ) ≡ det Q(x, t; λ, ξ), where

Q(x, t; λ, ξ) ≡ A_{0}λ + ∑_{i=1}^{n} A_{i}ξ_{i}, λ ∈ ℝ, ξ = (ξ_{1}, …, ξ_{n}) ∈ ℝ^{n}.

121 Since det A_{0} ≠ 0, we have the representation

1991 Mathematics Subject Classification. 35L50.

Key words and phrases. Normally hyperbolic systems, dihedral angle, reduction of a boundary value problem to a spliced system, bicharacteristic.

1072-947X/98/0300-0121$15.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1998 Plenum Publishing Corporation

p(x, t; λ, ξ) = det A_{0} ∏_{i=1}^{l} (λ − λ_{i}(x, t; ξ))^{k_{i}}, ∑_{i=1}^{l} k_{i} = m,

l = l(x, t; ξ), k_{i} = k_{i}(x, t; ξ), i = 1, …, l.

System (1.1) is said to be hyperbolic at the point (x, t) if all roots λ_{1}(x, t; ξ), …, λ_{l}(x, t; ξ) of the polynomial p(x, t; λ, ξ) are real numbers.

One can easily verify that

k_{i}(x, t; ξ) ≥ m − rank Q(x, t; λ_{i}(x, t; ξ), ξ), i = 1, …, l.

The hyperbolic system (1.1) is said be normally hyperbolic at the point (x, t), if the equalities

k_{i}(x, t; ξ) = m − rank Q(x, t; λ_{i}(x, t; ξ), ξ), i = 1, …, l,

are fulfilled (see, e.g., [1], [2]).

Note that strictly hyperbolic systems, i.e., when l = m, k_{i} = 1, i = 1, …, m, form a subclass of normally hyperbolic systems.

Since det A_{0} ≠ 0, it can be assumed without loss of generality that A_{0} = E, where E is the m×m unit matrix. For simplicity, we shall always assume that (i) n = 2, x_{1} = x, x_{2} = y; (ii) the matrices A_{1} and A_{2} are constant; (iii) system (1.1) is normally hyperbolic.

In our assumptions, in the space of independent variables x, y and t, system (1.1) is rewritten as

u_{t} + A_{1}u_{x} + A_{2}u_{y} + Bu = F.  (1.2)

It is easy to show that since system (1.2) is normally hyperbolic, each of the matrices A_{i}, i = 1, 2, has only real characteristic roots so that the corresponding eigenvectors of the operator A_{i}, 1 ≤ i ≤ 2, form a complete system, i.e., a basis in the space ℝ^{m}. Therefore the matrices A_{i}, i = 1, 2, are diagonalizable, i.e., there exist real nondegenerate matrices C_{i}, i = 1, 2, such that the matrices C_{i}^{−1}A_{i}C_{i}, i = 1, 2, are diagonal.

The normally hyperbolic system (1.2) will be said to be diagonalizable if there exists a real nondegenerate matrix C such that the matrices C^{−1}A_{i}C, i = 1, 2, are diagonal. We have

Lemma 1.1. The normally hyperbolic system (1.2) is diagonalizable if and only if the matrices A_{1} and A_{2} are commutative, i.e., A_{1}A_{2} = A_{2}A_{1}.

Proof. The necessity readily follows from the fact that the diagonal matrices C^{−1}A_{1}C and C^{−1}A_{2}C are commutative, C^{−1}A_{1}CC^{−1}A_{2}C = C^{−1}A_{2}CC^{−1}A_{1}C, i.e., C^{−1}A_{1}A_{2}C = C^{−1}A_{2}A_{1}C. This immediately implies A_{1}A_{2} = A_{2}A_{1}.

To prove sufficiency note that since system (1.2) is normally hyperbolic, we have dim ℝ_{λ_{i}} = k_{i}, where ℝ_{λ_{i}} ≡ Ker (A_{1} − λ_{i}E), 1 ≤ i ≤ l. Clearly, Â_{1}(ℝ_{λ_{i}}) ⊂ ℝ_{λ_{i}}, 1 ≤ i ≤ l, where Â_{1} stands for the linear transform corresponding to the matrix A_{1}.

Let {ν_{ij}}_{j=1}^{k_{i}} be an arbitrary basis of the space ℝ_{λ_{i}}, 1 ≤ i ≤ l. By the definition of the space ℝ_{λ_{i}}, the vectors ν_{i1}, …, ν_{ik_{i}} are the eigenvectors for the transform Â_{1} and correspond to the eigenvalue λ_{i}, 1 ≤ i ≤ l. Therefore, the matrix of the transform Â_{1} in the basis {ν_{ij}}_{j=1}^{k_{i}} of the space ℝ_{λ_{i}} will be diagonal of order (k_{i} × k_{i}) and written as diag [\underbrace{λ_{i}, …, λ_{i}}_{k_{i}−times}], 1 ≤ i ≤ l.

Hence, recalling that the decomposition of the space ℝ^{m} as the direct sum of subspaces ℝ_{λ_{i}}, i = 1, …, l, i.e., ℝ^{m} = ℝ_{λ_{1}} ⊕ ⋯ ⊕ ℝ_{λ_{l}} is unique, we can write the matrix D_{1} of the transform Â_{1} in the basis {ν_{ij}; i = 1, …, l; j =

1, …, k_{i}\} as D_{1} = diag [\underbrace{λ_{1}, …, λ_{1}}_{k_{1}−times}, …, \underbrace{λ_{l}, …, λ_{l}}_{k_{l}−times}].

Let Ã_{2i} be the matrix corresponding to the linear transform Â_{2} of the subspace ℝ_{λ_{i}}, 1 ≤ i ≤ l in the basis {ν_{ij}}_{j=1}^{k_{i}}. Since the matrices A_{1} and A_{2} are commutative, the subspace ℝ_{λ_{i}} is invariant with respect to the linear transform Â_{2}, i.e., Â_{2}(ℝ_{λ_{i}}) ⊂ ℝ_{λ_{i}}, 1 ≤ i ≤ l (see, e.g., [3]). Therefore, in the basis {ν_{ij}; i = 1, …, l; j = 1, …, k_{i}} of the space ℝ^{m}, the matrix Ã_{2} corresponding to Â_{2} will be block-diagonal and have, on its principal diagonal, matrices Ã_{2i}, i = 1, …, l. It is well known that matrices giving the same linear transform in different bases are similar. At the same time, similar matrices have the same characteristic equation. Therefore we have

det(A_{2} − λE) = det(Ã_{2} − λE) = det(Ã_{21} − λE_{k1}) × ⋯ × det(Ã_{2l} − λE_{kl}).

Since system (1.2) is normally hyperbolic, the matrix A_{2} has only real characteristic roots. Thus for the linear transform Â_{2} : ℝ_{λ_{i}} → ℝ_{λ_{i}} there exists a basis {μ_{ij}}_{j=1}^{k_{i}} which consists of the real vectors of the subspace ℝ_{λ_{i}}, and where the matrix A_{2i}^{∗} of the above-mentioned transform is of Jordan form, 1 ≤ i ≤ l. Therefore, in the basis {μ_{ij}; i = 1, …, l; j = 1, …, k_{i}} of the space ℝ^{m}, the matrix A_{2}^{∗} of the transform Â_{2} will also be of Jordan form. But, since system (1.2) is normally hyperbolic, in the space ℝ^{m} there exists a basis {σ_{i}}_{i=1}^{m} in which the matrix of the transform Â_{2} is diagonal. Further, as is well known, a Jordan matrix similar to the diagonal one is diagonal too. Therefore in the basis {μ_{ij}; i = 1, …, l; j = 1, …, k_{i}} of the space ℝ^{m} the matrix A_{2}^{∗} is diagonal, but the matrix of the transform Â_{1} is diagonal in any basis of the transform ℝ_{λ_{i}}, in particular, in the basis {μ_{ij}}_{j=1}^{k_{i}}, 1 ≤ i ≤ l. Therefore the matrices of the transforms Â_{1} and Â_{2} will be diagonal in the basis {μ_{ij}; i = 1, …, l; j = 1, …, k_{i}} of the space ℝ^{m}.

§ 2. Statement of the Problem and Some Notations

In the discussion below the matrices A_{1} and A_{2} will always be assumed to be commutative, i.e., the equality

A_{1}A_{2} = A_{2}A_{1}  (2.1)

is valid.

After introducing a new unknown function v by the formula u = Cv with the nondegenerate matrix C whose existence was proved by condition (2.1) in §1, system (1.2) takes the form

v_{t} + D_{1}v_{x} + D_{2}v_{y} + B_{0}v = F_{0},  (2.2)

where by virtue of Lemma 1.1 the matrices D_{i} = C^{−1}A_{i}C, i = 1, 2, are diagonal, i.e., D_{1} = diag[ν_{1}, …, ν_{m}], D_{2} = diag[μ_{1}, …, μ_{m}], B_{0} = C^{−1}BC, F_{0} = C^{−1}F.

It is obvious that the directions defined by the vectors l_{i} = (ν_{i}, μ_{i}, 1), i = 1, …, m, are bicharacteristic.

Let Â_{j} be the linear transform corresponding to he matrix A_{j}, 1 ≤ j ≤ 2. Denote by Λ_{i} an m-dimensional vector which is the eigenvalue of the transform Â_{1}, corresponding to the eigenvalue ν_{i}, 1 ≤ i ≤ m. By virtue of (2.1) the vector Λ_{i} is also the eigenvector of the transform Â_{2} corresponding to the eigenvalue μ_{i}, 1 ≤ i ≤ m. By Lemma 1.1, the vectors Λ_{i}, i = 1, …, m, can be chosen such that the (m × m) matrix C = [Λ_{1}, …, Λ_{m}], whose columns consist of these vectors, will reduce the matrices A_{1} and A_{2} to the diagonal form, namely, to D_{i} = C^{−1}A_{i}C, i = 1, 2.

Obviously, the vectors l_{i} = (ν_{i}, μ_{i}, 1) and l_{j} = (ν_{j}, μ_{j}, 1) define the same bicharacteristic direction if the equalities ν_{i} = ν_{j}, μ_{i} = μ_{j}, 1 ≤ i ≠ j ≤ m, are fulfilled. In this context, we divide the set of vectors {l_{1}, …, l_{m}} into nonintersecting classes {l_{11}, …, l_{1s_{1}}}, …, {l_{m_{0}1}, …, l_{m_{0}s_{m_{0}}} } whose representatives with respective “multiplicities” s_{1}, …, s_{m_{0}}, will be denoted by l̃_{1}, …, l̃_{m_{0}}, m_{0} ≤ m. Now the matrix C = [Λ_{1}, …, Λ_{m}] can be represented as

C = [Λ_{11}, …, Λ_{1s_{1}}; …; Λ_{m_{0}1}, …, Λ_{m_{0}s_{m_{0}}} ],

or as C = (C̃_{1}, C̃_{2}), where

C̃_{1} = [Λ_{11}, …, Λ_{1s_{1}}; …; Λ_{q1}, …, Λ_{qs_{q}}],
C̃_{2} = [Λ_{q+11}, …, Λ_{q+1s_{q+1}}; …; Λ_{m_{0}1}, …, Λ_{m_{0}s_{m_{0}}} ]

and q will be defined below.

Denote by D^{∗} the dihedral angle

D^{∗} ≡ {(x, y, t) ∈ ℝ^{3}, t − y > 0, t + y > 0}.

For bicharacteristic directions of system (1.2) we make the following assumption: bicharacteristics passing through any point of the edge Γ^{∗} ≡ {(x, y, t) ∈ ℝ^{3} : y = t = 0, x ∈ ℝ} of the angle D^{∗} have no common points with the set \overline{D^{∗}}\Γ^{∗}. This is equivalent to the fulfillment of the inequalities

|μ_{i}| > 1, i = 1, …, m.  (2.3)

Let P_{0} = P_{0}(x_{0}, y_{0}, t_{0}) be an arbitrary fixed point of the set \overline{D^{∗}}\Γ^{∗}, and let S_{1} ⊃ Γ^{∗} and S_{2} ⊃ Γ^{∗} be the two-dimensional edges of D^{∗}, i.e., ∂D^{∗} = S_{1} ∪ S_{2}, S_{1} ≡ \{(x, y, t) ∈ ℝ^{3} : x ∈ ℝ, y = t, t ∈ \overline{ℝ}_{+}\}, S_{2} ≡ \{(x, y, t) ∈ ℝ^{3} : x ∈ ℝ, y = −t, t ∈ \overline{ℝ}_{+}\}, ℝ_{+} ≡ (0, ∞). From the point P_{0} we draw the bicharacteristic beam L̃_{i}(P_{0}) of system (2.2) which corresponds to the vector l̃_{i}, is directed towards the decreasing values of the t-coordinate of a moving point L̃_{i}(P_{0}), and intersects one of the edges S_{1} or S_{2} at a point P̃_{0}^{i}, 1 ≤ i ≤ m_{0}. It can be assumed without loss of generality that bicharacteristic beams defined by the vectors l̃_{1}, …, l̃_{q} and passing through the point P_{0} intersect the edge S_{1}, while those defined by l̃_{q+1}, …, l̃_{m_{0}} intersect the edge S_{2}.

Below, it will be assumed for simplicity that q = 2 and m_{0} = 3, l̃_{i} = (ν̃_{i}, μ̃_{i}, 1), i = 1, 2, 3, and also rank (l̃_{1}, l̃_{2}, l̃_{3}) = 3.

Through the point P_{0} draw a plane P_{_{l_{i}},_{l_{j}}}, parallel to the vectors l̃_{i} and

l̃_{j}, 1 ≤ i < j ≤ 3. We introduce the following notation:

P_{_{l_{2}},_{l_{3}}} and P_{_{l_{1}},_{l_{3}}} are respectively the intersection points of the planes Q_{1} and Q_{2} with the edge Γ^{∗};

D is a the domain forming a pentahedron with the vertices at the points P_{0}, P̃_{0}^{2}, P̃_{0}^{1}, Q_{2}, P̃_{0}^{3}, Q_{1};

Δ_{1} and Δ_{2} are respectively a triangle and rectangle with the vertices at the points Q_{2}, P̃_{0}^{3}, Q_{1} and Q_{1}, P̃_{0}^{2}, P̃_{0}^{1}, Q_{2}, respectively.

For system (1.2) we consider the boundary value problem formulated as follows: Find, in the domain D, a regular solution u(x, y, t) of system (1.2) satisfying the boundary conditions

B_{i}u|\overline{Δ_{i}} = f_{i},  (2.4)

where B_{i} are the given (m_{i} × m) matrix-functions and f_{i} are the given m_{i}- dimensional vector-functions, i = 1, 2, m_{1} = s_{1} + ⋯ + s_{q}, m_{2} = s_{q+1} + ⋯ + s_{m_{0}}. It is obvious that m_{1} + m_{2} = m though we do not exclude the cases with m_{1} = 0 or m_{2} = 0, which correspond to the Cauchy problem. Below it will always be assumed that 0 < m_{i} < m, i = 1, 2.

A function u(x, y, t) which, together with its partial derivatives u_{x}, u_{y}, u_{t}, is continuous in D and satisfies system (1.2) is called a regular solution of system (1.2).

Some analogs of the Goursat problem for hyperbolic systems of first order with two independent variables have been studied in [4]–[8]. A lot of papers are devoted to general boundary value problems of the Darboux type for normally hyperbolic systems of second order on a plane (see, e.g., [1], [2]). Some multidimensional problems of the Goursat and Darboux type are considered in several papers (see, e.g., [9]–[11]) both for a hyperbolic equation and for a system of equations in a dihedral angle. For hyperbolic equations of third order, a boundary value problem in a dihedral angle is investigated in [12].

Denote by Δ_{i}^{∗} the orthogonal projection of the polygons Δ_{i}^{∗}, i = 1, 2, onto the plane x, t. The restrictions of B_{i} and f_{i} on the sets \overline{Δ_{i}^{∗}}, i = 1, 2, will be denoted as before.

In the domains D and Δ_{i}^{∗}, we introduce the following functional spaces

C^{0}_{α}(\overline{D}) ≡ \{w ∈ C(\overline{D}) : w|_{Γ} = 0, sup_{(x,y,t)∈\overline{D}\Γ^{∗}} ρ^{−α}||w(x, y, t)||_{ℝ^{m}} < ∞\},

C^{0}_{α}(\overline{Δ_{i}^{∗}}) ≡ \{ψ ∈ C(\overline{Δ_{i}^{∗}}) : ψ|_{Γ_{1}} = 0, sup_{(x,t)∈\overline{Δ_{i}^{∗}}\Γ_{1}^{∗}} t^{−α}||ψ(x, t)||_{ℝ^{m}} < ∞\},

where Γ ≡ \overline{D}∩Γ^{∗}, Γ_{1} ≡ \overline{Δ_{i}^{∗}} ∩Γ_{1}^{∗}, i = 1, 2, Γ_{1}^{∗} ≡ \{(x, t) ∈ ℝ^{2} : x ∈ ℝ, t = 0\}, ρ is the distance from the point (x, y, t) ∈ \overline{D}\Γ^{∗} to the edge Γ^{∗} of the domain D^{∗}, the real parameter α = const ≥ 0. For a = (a_{1}, …, a_{m}) ∈ ℝ^{m}, m ≥ 2, denote ∥a∥_{ℝ_{m}} = |a_{1}| + ⋯ + |a_{m}|.

Obviously, the spaces C^{0}_{α}(\overline{D}) and C^{0}_{α}(\overline{Δ_{i}^{∗}}), i = 1, 2, are Banach ones with the norms

∥w∥_{C^{0}_{α}(\overline{D})} = sup_{(x,y,t)∈\overline{D}\Γ^{∗}} ρ^{−α}||w(x, y, t)||_{ℝ^{m}},

∥ψ∥_{C^{0}_{α}(\overline{Δ_{i}^{∗}})} = sup_{(x,t)∈\overline{Δ_{i}^{∗}}\Γ_{1}^{∗}} t^{−α}||ψ(x, t)||_{ℝ^{m}}.

Remark 2.1. Since the estimate 1 ≤ ρ/t ≤ \sqrt{2}, (x, y, t) ∈ D^{∗}, is uniform,

the value ρ in the definition of the space C^{0}_{α}(\overline{D}) below will be replaced by the variable t.

It is easy to verify that the fact that w ∈ C^{0}(\overline{D}) and ψ ∈ C^{0}(\overline{Δ_{i}^{∗}}) belong to

the spaces C^{0}_{α}(\overline{D}) and C^{0}_{α}(\overline{Δ_{i}^{∗}}), respectively, is equivalent to the fulfillment of the inequalities

\begin{matrix}||w(x, y, t)||_{ℝ^{m}} ≤ ct^{α}, (x, y, t) ∈ \overline{D}, \\ ||ψ(x, t)||_{ℝ^{m}} ≤ ct^{α}, (x, t) ∈ \overline{Δ_{i}^{∗}}, i = 1, 2. \end{matrix}  (2.5)

We shall investigate the boundary value problem (1.2), (2.4) in the Banach space

C^{0}_{α}^{1,1,1}(\overline{D}) ≡ \{u : \frac{∂^{|i|}u}{∂x^{i_{1}}∂y^{i_{2}}∂t^{i_{3}}} ∈ C^{0}_{α}(\overline{D}), |i| ≤ 1, |i| = ∑_{j=1}^{3} i_{j}\},

with respect to the norm

∥u∥_{C^{0}_{α}^{1,1,1}(\overline{D})} = ∑_{|i|≤1} ∥\frac{∂^{|i|}u}{∂x^{i_{1}}∂y^{i_{2}}∂t^{i_{3}}} ∥_{C^{0}_{α}(\overline{D})}

assuming that the matrix-functions B ∈ C(\overline{D}), B_{i} ∈ C(\overline{Δ_{i}^{∗}}) and the vector-

functions F ∈ C^{0}_{α}(\overline{D}), f_{i} ∈ C^{0}_{α}(\overline{Δ_{i}^{∗}}), i = 1, 2.

§ 3. Equivalent Reduction of Problem (1.2), (2.4) to a System of Integro-Differential Equations

From an arbitrary point P (x, y, t) ∈ \overline{D}\Γ we draw the bicharacteristic beam L̃_{i}(P) of system (2.2) which corresponds to the vector l̃_{i} and is directed towards the decreasing values of the t-coordinate of a moving point of L̃_{i}(P), 1 ≤ i ≤ 3. The points of intersection of beams L̃_{i}(P), i = 1, 2, 3, with the faces S_{1} and S_{2} are P̃^{i} ∈ S_{1}, i = 1, 2, and P̃^{3} ∈ S_{2}. Denote by (ω_{i}^{1}(x, y, t), ω_{i}^{2}(x, y, t)) the coordinates of orthogonal projection of the point P̃^{i} onto the plane (x, t), 1 ≤ i ≤ 3. A simple calculation yields

\begin{matrix}ω_{i}^{1}(x, y, t) = x + ν̃_{i}(1 − μ̃_{i})^{−1}(y − t), \\ ω_{i}^{2}(x, y, t) = t + (1 − μ̃_{i})^{−1}(y − t), \end{matrix} i = 1, 2,

ω_{3}^{1}(x, y, t) = x − ν̃_{3}(1 + μ̃_{3})^{−1}(y + t), ω_{3}^{2}(x, y, t) = t − (1 + μ̃_{3})^{−1}(y + t).

Let ξ = x_{i}(x, y, t; τ), η = y_{i}(x, y, t; τ), ζ = τ be the parametrization of a segment L̃_{i}(P) ∩ \overline{D}, where ω_{i}^{2}(x, y, t) ≤ τ ≤ t, 1 ≤ i ≤ 3.

After integrating the (q_{i} + j)-th equation of system (2.2), where q_{1} = 0, q_{i} = s_{1} + ⋯ + s_{i−1}, i ≥ 2, j = 1, …, s_{i}, along the i-th bicharacteristic L̃_{i}(P) drawn from an arbitrary point P (x, y, t) ∈ \overline{D}\Γ and lying between the point P (x, y, t) and the point of intersection of L̃_{i}(P) with the face S_{1} or S_{2} (depending on the index i of L̃_{i}(P)), we obtain

v_{q_{i}+j}(x, y, t) = v_{q_{i}+j}(ω_{i}^{1}(x, y, t), ω_{i}^{2}(x, y, t)) +

+ ∫_{ω_{i}^{2}(x,y,t)}^{t} ( ∑_{p′=1}^{m} b_{ijp′}v_{p′})(x_{i}(x, y, t; τ), y_{i}(x, y, t; τ), τ)dτ +

+F_{ij}(x, y, t), 1 ≤ i ≤ 3, 1 ≤ j ≤ s_{i},  (3.1)

where v_{q_{i}+j} are the components of the vector v, b_{ijp′} and F_{ij} are the welldefined functions depending only on the coefficients and the right-hand side of system (2.2).

We set

\begin{matrix}φ_{q_{i}+j}^{1}(x, t)≡v_{q_{i}+j}|\overline{Δ_{1}} ≡v_{q_{i}+j}(x, t, t), (x, t)∈\overline{Δ_{1}^{∗}}, i = 1, 2; j = 1, …, s_{i}, \\ φ_{q_{i}+j}^{2}(x, t)≡v_{q_{i}+j}|\overline{Δ_{2}} ≡v_{q_{i}+j}(x, −t, t), (x, t)∈\overline{Δ_{2}^{∗}}, i = 3; j = 1, …, s_{i}. \end{matrix} (3.2)

It is obvious that the number of components of the vectors

φ^{1}(x, t) ≡ (φ_{q_{i}+j}^{1}(x, t)), (x, t) ∈ \overline{Δ_{1}^{∗}}, i = 1, 2; j = 1, …, s_{i},
φ^{2}(x, t) ≡ (φ_{q_{i}+j}^{2}(x, t)), (x, t) ∈ \overline{Δ_{2}^{∗}}, i = 3; j = 1, …, s_{i},

is equal to the numbers m_{1} and m_{2}, respectively.

By substituting the expressions of v from equality (3.1) into the boundary conditions (2.4) and taking into account (3.2) we have

Q_{0}^{1}(x, t)φ^{1}(x, t)+Q_{3}^{1}(x, t)φ^{2}(σ_{3}(x, t))+(T_{1}v)(x, t)=f^{1}(x, t), (x, t) ∈ \overline{Δ_{1}^{∗}},

Q_{0}^{2}(x, t)φ^{2}(x, t) + ∑_{i=1}^{2} Q_{i}^{2}(x, t)φ^{1}(σ_{i}(x, t)) + (T_{2}v)(x, t) =  (3.3)
= f^{2}(x, t), (x, t) ∈ \overline{Δ_{2}^{∗}},

where

\begin{matrix}(T_{1}v)(x, t) ≡ ∫_{ω_{3}^{2}(x,t,t)}^{t} (Ã_{3}v)(x_{3}(x, t, t; τ), y_{3}(x, t, t; τ), τ)dτ, \\ (T_{2}v)(x, t) ≡ ∑_{i=1}^{2} ∫_{ω_{i}^{2}(x,−t,t)}^{t} (Ã_{i}v)(x_{i}(x, −t, t; τ), y_{i}(x, −t, t; τ), τ)dτ, \end{matrix} (3.4)

and Q_{3}^{1}, Ã_{3}, Q_{i}^{2}, Ã_{i}, f^{i}, i = 1, 2, are respectively the well-defined matrices and vectors.

It is obvious that Q_{0}^{i} from (3.3) are matrices of order (m_{i} × m_{i}) which can be represented as the product

Q_{0}^{i} = B_{i} × C̃_{i}, i = 1, 2,  (3.5)

and the functions σ_{i} are defined by the equalities

σ_{i} : (x, t) → (ω_{i}^{1}(x, −t, t), ω_{i}^{2}(x, −t, t)), i = 1, 2,
σ_{i} : (x, t) → (ω_{i}^{1}(x, t, t), ω_{i}^{2}(x, t, t)), i = 3.

Assuming that

det Q_{0}^{i}|\overline{Δ_{i}^{∗}} ≠ 0, i = 1, 2,  (3.6)

where the matrices Q_{0}^{i} are given by (3.5), we rewrite system (3.3) as

\begin{matrix}φ^{1}(x, t)−∑_{i=1}^{2} G_{i}^{1}(x, t)φ^{1}(J_{i}^{1}(x, t))+(T_{3}v)(x, t)=f^{3}(x, t), (x, t)∈\overline{Δ_{1}^{∗}}, \\ φ^{2}(x, t)−∑_{i=1}^{2} G_{i}^{2}(x, t)φ^{2}(J_{i}^{2}(x, t))+(T_{4}v)(x, t)=f^{4}(x, t), (x, t)∈\overline{Δ_{2}^{∗}}, \end{matrix} (3.7)

where G_{i}^{p} are the known matrix-functions of order (m_{p} ×m_{p}), p = 1, 2, f^{i+2} are the known vector-functions,

J_{i}^{1}(x, t) ≡ σ_{i}(σ_{3}(x, t)), (x, t) ∈ \overline{Δ_{1}^{∗}},
J_{i}^{2}(x, t) ≡ σ_{3}(σ_{i}(x, t)), (x, t) ∈ \overline{Δ_{2}^{∗}}, i = 1, 2.

It is easy to verify that by virtue of equalities (3.4) the linear integral operators T_{3} and T_{4} can be represented as

(T_{3}v)(x, t) = ∑_{i=1}^{2} ∫_{ω_{i}^{2}(x,t)}^{ω_{3}^{2}(x,t,t)}(B̃_{i}v)(x_{i}(ω_{3}^{1}(x, t, t), −ω_{3}^{2}(x, t, t), ω_{3}^{2}(x, t, t); τ),

y_{i}(ω_{3}^{1}(x, t, t), −ω_{3}^{2}(x, t, t), ω_{3}^{2}(x, t, t); τ), τ)dτ +

+ ∫_{ω_{3}^{2}(x,t,t)}^{t} (B̃_{3}v)(x_{3}(x, t, t; τ), y_{3}(x, t, t; τ), τ)dτ,

(T_{4}v)(x, t)=∑_{i=1}^{2} ∫_{ω_{3}^{2}(x,t)}^{ω_{i}^{2}(x,−t,t)}(E_{i}v)(x_{3}(ω_{i}^{1}(x, −t, t), ω_{i}^{2}(x, −t, t), ω_{i}^{2}(x, −t, t); τ),

y_{3}(ω_{i}^{1}(x, −t, t), ω_{i}^{2}(x, −t, t), ω_{i}^{2}(x, −t, t); τ), τ)dτ +

+ ∑_{i=1}^{2} ∫_{ω_{i}^{2}(x,−t,t)}^{t} (H_{i}v)(x_{i}(x, −t, t; τ), y_{i}(x, −t, t; τ), τ)dτ,

where ω_{i}^{2}(x, t) ≡ ω_{i}^{2}(ω_{3}^{1}(x, t, t), −ω_{3}^{2}(x, t, t), ω_{3}^{2}(x, t, t)), ω_{3}^{2}(x, t) ≡ ω_{3}^{2}(ω_{i}^{1}(x, −t, t), ω_{i}^{2}(x, −t, t), ω_{i}^{2}(x, −t, t)), and E_{i}, H_{i}, i = 1, 2, B̃_{j}, j = 1, 2, 3, are the well-defined matrices.

For the functions J_{i}^{k} : \overline{Δ_{k}^{∗}} → \overline{Δ_{k}^{∗}} we have the formulas

J_{i}^{k} : (x, t) → (x + δ_{i}^{k}t, τ_{i}t), (x, t) ∈ \overline{Δ_{k}^{∗}},

where δ_{i}^{k}, τ_{i}, i, k = 1, 2, are the well-defined constants written in terms of ν̃_{i}, μ̃_{i}, i = 1, 2, 3.

Remark 3.1. Note that by virtue of condition (2.3) it is easy to establish that 0 < τ_{i} < 1, i = 1, 2.

Remark 3.2. It is obvious that when conditions (3.6) are fulfilled, prob-

lem (1.2), (2.4) in the class C^{0}_{α}^{1,1,1}(\overline{D}) is equivalently reduced to system

(3.7) for the unknown vector-function φ^{i} of the class C^{0}_{α}(\overline{Δ_{i}^{∗}}), i = 1, 2.

Furthermore, if u ∈ C^{0}_{α}^{1,1,1}(\overline{D}), then φ^{i} ∈ C^{0}_{α}(\overline{Δ_{i}^{∗}}), i = 1, 2. Vice versa, if

φ^{i} ∈ C^{0}_{α}(\overline{Δ_{i}^{∗}}), i = 1, 2, then with regard to inequality (2.5) equalities (3.1),

(3.2) and u = Cv readily imply that u ∈ C^{0}_{α}^{1,1,1}(\overline{D}).

§ 4. Investigation of the System of Integro-Functional Equations (3.1), (3.7) and the Proof of the Main Result

Let us consider the system of functional equations

(K_{p}φ^{p})(x, t)≡φ^{p}(x, t)−∑_{i=1}^{2} G_{i}^{p}(x, t)φ^{p}(J_{i}^{p}(x, t))=g_{p}(x, t), (x, t)∈\overline{Δ_{p}^{∗}}, (4.1)

and introduce the notation

h_{p}(ρ) ≡ ∑_{i=1}^{2} η_{ip}τ_{i}^{ρ}, η_{p} ≡ max_{1≤i≤2} sup_{(x,t)∈\overline{Δ_{p}^{∗}}} ∥G_{i}^{p}(x, t)∥,

η_{ip} ≡ sup_{x∈[Q_{1},Q_{2}]} ∥G_{i}^{p}(x, 0)∥, i, p = 1, 2, ρ ∈ ℝ.

(4.2)

Here and in what follows by ∥.∥ we understand the norm of a matrix operator acting from one Euclidean space into another.

If all values η_{ip} = 0, then it is assumed that ρ_{p} = −∞, i = 1, 2, 1 ≤ p ≤ 2. Let now for some value of the index i the value η_{ip}, 1 ≤ p ≤ 2, be different from zero. In that case, by Remark 3.1, the function h_{p} : ℝ → \overline{ℝ}_{+} is continuous and strictly decreasing on ℝ; also, lim_{ρ→−∞} h_{p}(ρ) = +∞ and lim_{ρ→+∞} h_{p}(ρ) = 0, 1 ≤ p ≤ 2. Therefore there exists a unique real number ρ_{p} such that h_{p}(ρ_{p}) = 1, 1 ≤ p ≤ 2. It is assumed that ρ_{0} ≡ max (ρ_{1}, ρ_{2}).

Lemma 4.1. If α > ρ_{0}, then equation (4.1) is uniquely solvable in the

space C^{0}α(\overline{Δ_{p}^{∗}}) and for the solution φ^{p} = K_{p}^{−1}g^{p} the estimate

∥φ^{p}(x, t)∥_{ℝ^{mp}} = ∥(K_{p}^{−1}g_{p})(x, t)∥_{ℝ^{mp}} ≤
≤ C_{2+p}t^{α}∥g_{p}∥_{C^{0} α(\overline{Δ_{p}^{∗}}∩\{t_{1}≤t\})}, (x, t) ∈ \overline{Δ_{p}^{∗}},  (4.3)

holds, where C_{2+p} is a positive constant not depending on the function g_{p}, 1 ≤ p ≤ 2.

Proof. We shall consider the case p = 1, since the case p = 2 is considered analogously. The condition α > ρ_{0} and the definition of the function h_{1} from (4.2) imply

h_{1}(α) = ∑_{i=1}^{2} η_{i1}τ_{i}^{α} < 1.  (4.4)

By inequality (4.4) and the continuity of the functions G_{i}^{1}, i = 1, 2, there exist positive numbers ε_{1} (ε_{1} < t^{0}) and δ_{1} such that the inequalities

∥G_{i}^{1}(x, t)∥ ≤ η_{i1} + δ_{1}, i = 1, 2,  (4.5)
∑_{i=1}^{2}(η_{i1} + δ_{1})τ_{i}^{α} ≡ γ_{1} < 1  (4.6)

hold for (x, t) ∈ \overline{Δ_{1}^{∗}} ∩ \{0 ≤ t ≤ ε_{1}\}.

By Remark 3.1 there is a natural number r_{0} such that for r ≥ r_{0}

τ_{i_{r}}τ_{i_{r−1}} ⋯ τ_{i_{1}}t ≤ ε_{1}, 0 ≤ t ≤ t^{0},  (4.7)

where 1 ≤ i_{s} ≤ 2, s = 1, …, r.

We introduce the operators Λ_{1} and K_{1}^{−1} acting by the formulas

(Λ_{1}φ^{1})(x, t) = ∑_{i=1}^{2} G_{i}^{1}(x, t)φ^{1}(J_{i}^{1}(x, t)), (x, t) ∈ \overline{Δ_{1}^{∗}}, K_{1}^{−1} = I + ∑_{r=1}^{∞} Λ_{1}^{r},

where I is the identical operator. Obviously, the operator K_{1}^{−1} is the formally inverse operator to the operator K_{1} defined by equality (4.1). Hence

it is sufficient for us to show that K_{1}^{−1} is continuous in the space C^{0}_{α}(\overline{Δ_{1}^{∗}}).

As is easily seen, the expression Λ_{1}^{r}g_{1} is the sum consisting of terms of the form

I_{i_{1}···i_{r}}(x, t) = G_{i_{1}}^{1}(x, t)G_{i_{2}}^{1}(J_{i_{1}}^{1}(x, t))G_{i_{3}}^{1}(J_{i_{2}}^{1}(J_{i_{1}}^{1}(x, t))) ⋯

⋯ G_{i_{r}}^{1}(J_{i_{r−1}}^{1}(J_{i_{r−2}}^{1}(⋯ (J_{i_{1}}^{1}(x, t)) ⋯)))g_{1}(J_{i_{r}}^{1}(J_{i_{r−1}}^{1}(⋯ (J_{i_{1}}^{1}(x, t)) ⋯))),

where 1 ≤ i_{s} ≤ 2, s = 1, …, r.

132 O. JOKHADZE

Hence, using (4.2), (4.5), (3.7) and Remark 3.1, we obtain: for r > r_{0},

g_{1} ∈ C^{0}_{α}(\overline{Δ_{1}^{∗}})

∥I_{i_{1}···i_{r}}(x, t)∥_{ℝ^{m_{1}}} ≤
≤ ∥G_{i_{1}}^{1}(x, t)∥ ⋯ ∥G_{i_{r_{0}}}^{1} (J_{i_{r_{0}−1}}^{1}(J_{i_{r_{0}−2}}^{1}(⋯ (J_{i_{1}}^{1}(x, t)) ⋯)))∥ ×
×∥G_{i_{r_{0}+1}}^{1}(J_{i_{r_{0}}}^{1} (J_{i_{r_{0}−1}}^{1}(⋯ (J_{i_{1}}^{1}(x, t)) ⋯)))∥ ⋯
⋯ ∥G_{i_{r}}^{1}(J_{i_{r−1}}^{1}(J_{i_{r−2}}^{1}(⋯ (J_{i_{1}}^{1}(x, t)) ⋯)))∥ ×
×∥g_{1}(J_{i_{r}}^{1}(J_{i_{r−1}}^{1}(⋯ (J_{i_{1}}^{1}(x, t)) ⋯)))∥_{ℝ^{m_{1}}} ≤

≤ η_{1}^{r_{0}}(η_{i_{r_{0}+11}} + δ_{1}) ⋯ (η_{i_{r}1} + δ_{1})(τ_{i_{r}}τ_{i_{r−1}} ⋯ τ_{i_{1}}t)^{α}∥g_{1}∥_{C^{0}_{α}(\overline{Δ_{1}^{∗}}∩\{t_{1}≤t\})} ≤

≤ η_{1}^{r_{0}}( ∏_{s=r_{0}+1}^{r}(η_{i_{s}1} + δ_{1}))( ∏_{s=r_{0}+1}^{r} τ_{i_{s}}^{α})t^{α}∥g_{1}∥_{C^{0} α(\overline{Δ_{1}^{∗}}∩\{t_{1}≤t\})} =

= η_{1}^{r_{0}}( ∏_{s=r_{0}+1}^{r}(η_{i_{s}1} + δ_{1})τ_{i_{s}}^{α})t^{α}∥g_{1}∥_{C^{0} α(\overline{Δ_{1}^{∗}}∩\{t_{1}≤t\})},  (4.8)

and for 1 ≤ r ≤ r_{0}

∥I_{i_{1}···i_{r}}(x, t)∥_{ℝ^{m_{1}}} ≤ η_{1}^{r}(τ_{i_{r}}τ_{i_{r−1}} ⋯ τ_{i_{1}}t)^{α}∥g_{1}∥_{C^{0} α(\overline{Δ_{1}^{∗}}∩\{t_{1}≤t\})} ≤

≤ η_{1}^{r}t^{α}∥g_{1}∥_{C^{0} α(\overline{Δ_{1}^{∗}}∩\{t_{1}≤t\})}.  (4.9)

By (4.8), (4.9), and (4.6) we have: for r > r_{0}

∥(Λ_{1}^{r}g_{1})(x, t)∥_{ℝ^{m_{1}}} = ∥ ∑_{i_{1},...,i_{r}} I_{i_{1}···i_{r}}(x, t)∥_{ℝ^{m_{1}}} ≤

≤ ( ∑_{i_{1},...,i_{r_{0}}} 1)^{r_{0}}η_{1}^{r_{0}}[ ∑_{i=1}^{2}(η_{i1} + δ_{1})τ_{i}^{α}]^{r−r_{0}}t^{α}∥g_{1}∥_{C^{0}_{α}(\overline{Δ_{1}^{∗}}∩\{t_{1}≤t\})} ≤

≤ C_{5}γ_{1}^{r}t^{α}∥g_{1}∥_{C^{0}_{α}(\overline{Δ_{1}^{∗}}∩\{t_{1}≤t\})},  (4.10)

and for 1 ≤ r ≤ r_{0}

∥(Λ_{1}^{r}g_{1})(x, t)∥_{ℝ^{m_{1}}} ≤ C_{6}t^{α}∥g_{1}∥_{C^{0}_{α}(\overline{Δ_{1}^{∗}}∩\{t_{1}≤t\})},  (4.11)

where C_{5} ≡ η_{1}^{r_{0}}γ_{1}^{−r_{0}}( ∑_{i_{1},...,i_{r_{0}}} 1)^{r_{0}}, C_{6} ≡ η_{1}^{r}( ∑_{i_{1},...,i_{r}} 1).

Inequalities (4.10) and (4.11) finally imply

∥φ^{1}(x, t)∥_{ℝ^{m_{1}}} = ∥(K_{1}^{−1}g_{1})(x, t)∥_{ℝ^{m_{1}}} ≤

≤ ∥g_{1}(x, t)∥_{ℝ^{m_{1}}} + ∑_{r=1}^{r_{0}} ∥(Λ_{1}^{r}g_{1})(x, t)∥_{ℝ^{m_{1}}} + ∑_{r=r_{0}+1}^{∞} ∥(Λ_{1}^{r}g_{1})(x, t)∥_{ℝ^{m_{1}}} ≤

≤ (1 + C_{6}r_{0} + C_{5}γ_{1}^{r_{0}+1}(1 − γ_{1})^{−1})t^{α}∥g_{1}∥_{C^{0}_{α}(\overline{Δ_{1}^{∗}}∩\{t_{1}≤t\})} =

= C_{3}t^{α}∥g_{1}∥_{C^{0}_{α}(\overline{Δ_{1}^{∗}}∩\{t_{1}≤t\})},

where C_{3} ≡ 1 + C_{6}r_{0} + C_{5}γ_{1}^{r_{0}+1}(1 − γ_{1})^{−1}. Hence we conclude that the

operator K_{1}^{−1} is continuous in the space C^{0}_{α}(\overline{Δ_{1}^{∗}}) and therefore Lemma 4.1 is true.

On the basis of this lemma we have

Theorem 4.1. Let conditions (3.6) be fulfilled. If α > ρ_{0}, then problem

(1.2), (2.4) is uniquely solvable in the space C^{0}_{α}^{1,1,1}(\overline{D}).

Proof. First we solve the system of equations (3.1), (3.7) with respect to

the unknown functions v ∈ C^{0}_{α}^{1,1,1}(\overline{D}) and φ^{p} ∈ C^{0}_{α}(\overline{Δ_{p}^{∗}}), p = 1, 2, using the method of successive approximations.

Let

\begin{matrix}v_{0}(x, y, t) ≡ 0, (x, y, t) ∈ \overline{D}; φ_{0}^{p}(x, t) ≡ 0, (x, t) ∈ \overline{Δ_{p}^{∗}}, p = 1, 2; \\ v_{q_{i}+j,k}(x, y, t) = φ_{q_{i}+j,k}(ω_{i}^{1}(x, y, t), ω_{i}^{2}(x, y, t)) + \\ + ∫_{ω_{i}^{2}(x,y,t)}^{t} ( ∑_{p′=1}^{m} b_{ijp′}v_{p′,k−1})(x_{i}(x, y, t; τ), y_{i}(x, y, t; τ), τ)dτ + \\ +F̃_{ij}(x, y, t), 1 ≤ i ≤ 3, 1 ≤ j ≤ s_{i}, \end{matrix}(4.12)

where

φ_{q_{i}+j,k}(ω_{i}^{1}(x, y, t), ω_{i}^{2}(x, y, t)) =

= \begin{cases}φ_{q_{i}+j,k}^{1}(ω_{i}^{1}(x, y, t), ω_{i}^{2}(x, y, t)), & 1 ≤ i ≤ 2, 1 ≤ j ≤ s_{i}, \\ φ_{q_{i}+j,k}^{2}(ω_{i}^{1}(x, y, t), ω_{i}^{2}(x, y, t)), & i = 3, 1 ≤ j ≤ s_{i}, \end{cases} (x, y, t) ∈ \overline{D}.

The values φ_{k}^{p}(x, t) are defined by the equations

(K_{p}φ_{k}^{p})(x, t) + (T_{2+p}v_{k−1})(x, t) = f^{2+p}(x, t),  (4.13)
(x, t) ∈ \overline{Δ_{p}^{∗}}, p = 1, 2, k ≥ 1,

where the operators K_{p}, p = 1, 2, act by (4.1).

For convenience, system (4.12) is rewritten as

v_{k}(x, y, t) = φ_{k}(x, y, t) +

+ ∑_{i=1}^{3} ∫_{ω_{i}^{2}(x,y,t)}^{t} (Ω_{i}v_{k−1})(x_{i}(x, y, t; τ), y_{i}(x, y, t; τ), τ)dτ +

+F̃(x, y, t), (x, y, t) ∈ \overline{D},  (4.14)

where the (q_{i} + j)-th component of the vector φ_{k}(x, y, t) is equal to φ_{q_{i}+j,k}(ω_{i}^{1}(x, y, t), ω_{i}^{2}(x, y, t)), 1 ≤ i ≤ 3, 1 ≤ j ≤ s_{i}, k ≥ 1; Ω_{i}, i = 1, 2, 3, and F̃ are respectively the well-defined matrices and vector-functions.

We shall now show that the following estimates are true:

∥v_{k+1}(x, y, t) − v_{k}(x, y, t)∥_{ℝ^{m}} ≤ M^{∗} \frac{M_{∗}^{k}}{k!} t^{k+α}, (x, y, t) ∈ \overline{D}, (4.15)
∥φ_{k+1}^{p}(x, t) − φ_{k}^{p}(x, t)∥_{ℝ^{mp}} ≤ M^{∗} \frac{M_{∗}^{k}}{k!} t^{k+α}, (x, t) ∈ \overline{Δ_{p}^{∗}}, (4.16)

where M_{∗} and M^{∗} are well-defined sufficiently large numbers not depending on k, k ≥ 1, p = 1, 2.

Indeed, by the assumptions for f_{p} and F we have f^{2+p} ∈C^{0} _{α}(\overline{Δ_{p}^{∗}}), F̃ ∈

C^{0} _{α}(\overline{D}), p = 1, 2. Hence, on account of inequalities (2.5) from §2, we conclude that the estimates

∥F̃(x, y, t)∥_{ℝ^{m}} ≤ Θ_{1}t^{α}, (x, y, t) ∈ \overline{D},  (4.17)
∥f^{2+p}(x, t)∥_{ℝ^{mp}} ≤ Θ_{1+p}t^{α}, (x, t) ∈ \overline{Δ_{p}^{∗}},  (4.18)
p = 1, 2, Θ_{i} = const ≥ 0, i = 1, 2, 3,

are fulfilled.

By v_{0} ≡ 0, φ_{0}^{p} ≡ 0, p = 1, 2 and the conditions of Theorem 4.1 estimate (4.3) is true so that (4.13), (4.18) imply

∥φ_{1}^{p}(x, t) − φ_{0}^{p}(x, t)∥_{ℝ^{mp}} = ∥φ_{1}^{p}(x, t)∥_{ℝ^{mp}} ≤ C_{5}Θ_{4}t^{α}, p = 1, 2,

C_{7} = max (C_{3}, C_{4}), Θ_{4} = max (Θ_{2}, Θ_{3})

(4.19)

which in turn gives rise to

∥φ_{1}(x, y, t) − φ_{0}(x, y, t)∥_{ℝ^{m}} = ∥φ_{1}(x, y, t)∥_{ℝ^{m}} =

= ∑_{1≤i≤3} ∑_{1≤j≤s_{i}} |φ_{q_{i}+j,1}(ω_{i}^{1}(x, y, t), ω_{i}^{2}(x, y, t))| ≤

≤ ∑_{1≤i≤3} ∑_{1≤j≤s_{i}} C_{7}Θ_{4}(ω_{i}^{2}(x, y, t))^{α} ≤ mC_{7}Θ_{4}t^{α},  (4.20)

since ∑_{1≤i≤3} ∑_{1≤j≤s_{i}} 1 = m and, as shown in §3, 0 ≤ ω_{i}^{2}(x, y, t) ≤ t, i = 1, 2, 3.

By virtue of (4.17) and (4.20), from (4.14) we have

∥v_{1}(x, y, t) − v_{0}(x, y, t∥_{ℝ^{m}} = ∥v_{1}(x, y, t)∥_{ℝ^{m}} ≤ ∥φ_{1}(x, y, t)∥_{ℝ^{m}} +

+∥F̃(x, y, t)∥_{ℝ^{m}} ≤ mC_{7}Θ_{4}t^{α} + Θ_{1}t^{α} = (mC_{7}Θ_{4} + Θ_{1})t^{α}. (4.21)

Now, assuming that estimates (4.15), (4.16) are fulfilled for k, k > 0, we shall show that they hold for k + 1 when M_{∗} and M^{∗} are sufficiently large.

Using (4.13), for p = 1 we have

\{K_{1}(φ_{k+2}^{1} − φ_{k+1}^{1})\}(x, t) = −\{T_{3}(v_{k+1} − v_{k})\}(x, t), (x, t) ∈ \overline{Δ_{1}^{∗}}. (4.22)

It is obvious that for the right-hand side of equation (4.22) we have the estimate

∥{T_{3}(v_{k+1} − v_{k})}(x, t)∥_{ℝ^{m_{1}}} ≤

≤ ∑_{i=1}^{2} ∫_{ω_{i}^{2}(x,t)}^{ω_{3}^{2}(x,t,t)}∥B̃_{i}∥∥v_{k+1} − v_{k}∥_{ℝ^{m}}(x_{i}(ω_{3}^{1}(x, t, t), −ω_{3}^{2}(x, t, t),

ω_{3}^{2}(x, t, t); τ), y_{i}(ω_{3}^{1}(x, t, t), −ω_{3}^{2}(x, t, t), ω_{3}^{2}(x, t, t); τ), τ)dτ +

+ ∫_{ω_{3}^{2}(x,t,t)}^{t} ∥B̃_{3}∥∥v_{k+1} − v_{k}∥_{ℝ^{m}}(x_{3}(x, t, t; τ), y_{3}(x, t, t; τ), τ)dτ. (4.23)

Denote by ξ_{1} the largest of the numbers max_{x,t,τ} ∥B̃_{i}(x, t, τ)∥, i = 1, 2, 3. Since

0 ≤ ω_{i}^{2}(x, t) ≤ ω_{3}^{2}(x, t, t) ≤ t, by (4.15) we find from (4.23) that

∥{T_{3}(v_{k+1} − v_{k})}(x, t)∥_{ℝ^{m_{1}}} ≤

≤ ξ_{1}M^{∗} \frac{M_{∗}^{k}}{k!} ( ∑_{i=1}^{2} ∫_{ω_{i}^{2}(x,t)}^{ω_{3}^{2}(x,t,t)}τ^{k+α}dτ + ∫_{ω_{3}^{2}(x,t,t)}^{t} τ^{k+α}dτ) ≤

≤ ξ_{1}M^{∗} \frac{M_{∗}^{k}}{k!} ( ∑_{i=1}^{2} 1 + 1) ∫_{0}^{t} τ^{k+α}dτ ≤

≤ 3ξ_{1}M^{∗} \frac{M_{∗}^{k}}{k!} \frac{1}{k + α + 1}t^{k+α+1} ≤ 3ξ_{1}M^{∗} \frac{M_{∗}^{k}}{(k + 1)!}t^{k+1+α}. (4.24)

Now (4.22), (4.24), and (4.3) (for p = 1) imply

∥φ_{k+2}^{1}(x, t) − φ_{k+1}^{1}(x, t)∥_{ℝ^{m_{1}}} ≤ 3C_{3}ξ_{1}M^{∗} \frac{M_{∗}^{k}}{(k + 1)!}t^{k+1+α}. (4.25)

Similarly, (4.13) (for p = 2), (4.15), and (4.3) (for p = 2) give

∥φ_{k+2}^{2}(x, t) − φ_{k+1}^{2}(x, t)∥_{ℝ^{m_{2}}} ≤ 4C_{4}ξ_{2}M^{∗} \frac{M_{∗}^{k}}{(k + 1)!}t^{k+1+α}, (4.26)

where ξ_{2} denotes the largest of the numbers max_{x,t,τ} ∥E_{i}(x, t, τ)∥,

max_{x,t,τ} ∥H_{i}(x, t, τ)∥, i = 1, 2.

Using the same arguments as in deriving estimate (4.20), from (4.25) and (4.26) we obtain

∥φ_{k+2}(x, y, t) − φ_{k+1}(x, y, t)∥_{ℝ^{m}} ≤ ξ_{4}M^{∗} \frac{M_{∗}^{k}}{(k + 1)!}t^{k+1+α}, (4.27)

where ξ_{4} ≡ 4mC_{7}ξ_{3}, ξ_{3} ≡ max (ξ_{1}, ξ_{2}).

We denote by η the largest of the numbers max_{\overline{D}} ∥Ω_{i}∥, where the matrices

Ω_{i}, i = 1, 2, 3, are defined by (4.14). By (4.27) and (4.15), from system (4.14) we have

∥v_{k+2}(x, y, t) − v_{k+1}(x, y, t)∥_{ℝ^{m}} ≤ ∥φ_{k+2}(x, y, t) − φ_{k+1}(x, y, t)∥_{ℝ^{m}} +

+ ∑_{i=1}^{3} ∫_{ω_{i}^{2}(x,y,t)}^{t} ∥Ω_{i}∥∥v_{k+1} − v_{k}∥_{ℝ^{m}}(x_{i}(x, y, t; τ), y_{i}(x, y, t; τ), τ)dτ ≤

≤ ξ_{4}M^{∗} \frac{M_{∗}^{k}}{(k + 1)!}t^{k+1+α} + 3η ∫_{0}^{t} M^{∗} \frac{M_{∗}^{k}}{k!} τ^{k+α}dτ ≤

≤ (ξ_{4} + 3η)M^{∗} \frac{M_{∗}^{k}}{(k + 1)!}t^{k+1+α}, (x, y, t) ∈ \overline{D},  (4.28)

since 0 ≤ ω_{i}^{2}(x, y, t) ≤ t, i = 1, 2, 3.

If we set

M^{∗} = mC_{7}Θ_{4} + Θ_{1}, M_{∗} = max (3C_{3}ξ_{1}, 4C_{4}ξ_{2}, ξ_{4} + 3η),

then by (4.19), (4.21), (4.25), (4.26), (4.28) immediately imply that estimates (4.15), (4.16) hold for any integer k ≥ 0.

It follows from (4.15), (4.16) that the series

v(x, y, t) = lim_{k→∞} v_{k}(x, y, t) = ∑_{k=1}^{∞} (v_{k}(x, y, t) − v_{k−1}(x, y, t)), (x, y, t) ∈ \overline{D},

φ^{p}(x, t) = lim_{k→∞} φ_{k}^{p}(x, t) = ∑_{k=1}^{∞} (φ_{k}^{p}(x, t) − φ_{k−1}^{p}(x, t)), (x, t) ∈ \overline{Δ_{p}^{∗}}, p = 1, 2,

converge in the spaces C^{0} _{α}^{1,1,1}(\overline{D}), C^{0} _{α}(\overline{Δ_{p}^{∗}}), p = 1, 2, and by (4.13), (4.14) the limit functions v, φ^{p}, p = 1, 2, satisfy system (3.1), (3.7). Finally, since problem (1.2), (2.4) is equivalent to system (3.1), (3.7) and the equality u = Cv holds, we conclude that the obtained function u(x, y, t) is really a

solution of problem (1.2), (2.4) in the class C^{0} _{α}^{1,1,1}(\overline{D}), α > ρ_{0}.

Now we shall show that under the conditions of Theorem 4.1 prob-

lem (1.2), (2.4) has no other solutions in the class C^{0} _{α}^{1,1,1}(\overline{D}). Indeed,

if u ∈C^{0} _{α}^{1,1,1}(\overline{D}) is the solution of the homogeneous problem corresponding to (1.2), (2.4), then the corresponding functions v, φ^{p}, p = 1, 2, satisfy the homogeneous system of equations

\begin{matrix}v_{q_{i}+j}(x, y, t) = φ_{q_{i}+j}(ω_{i}^{1}(x, y, t), ω_{i}^{2}(x, y, t)) + \\ + ∫_{ω_{i}^{2}(x,y,t)}^{t} ( ∑_{p′=1}^{m} b_{ijp′}v_{p′})(x_{i}(x, y, t; τ), y_{i}(x, y, t; τ), τ)dτ, (x, y, t) ∈ \overline{D}, \\ 1 ≤ i ≤ 3, 1 ≤ j ≤ s_{i}, \\ (K_{p}φ^{p})(x, t) + (T_{2+p}v)(x, t) = 0, (x, t) ∈ \overline{Δ_{p}^{∗}}, p = 1, 2. \end{matrix} (4.29)

We apply the method of successive approximations to system (4.29), assuming that v, φ^{1}, φ^{2}, are zero approximations. Since these values satisfy system (4.29), each next approximation will coincide with it so that we shall have v_{k}(x, y, t) ≡ v(x, y, t), (x, y, t) ∈ \overline{D}, φ_{k}^{p}(x, t) ≡ φ^{p}(x, t), (x, t) ∈ \overline{Δ_{p}^{∗}} for k ≥ 1 and p = 1, 2. Recalling that these values satisfy estimates of form (4.17), (4.18) and arguing as in the case of deriving estimates (4.15), (4.16), we obtain

∥v(x, y, t)∥_{ℝ^{m}} = ∥v_{k+1}(x, y, t)∥_{ℝ^{m}} ≤ M̃^{∗} \frac{M̃_{∗}^{k}}{k!} t^{k+α}, (x, y, t) ∈ \overline{D},

∥φ^{p}(x, t)∥_{ℝ^{mp}} =∥φ_{k+1}^{p}(x, t)∥_{ℝ^{mp}} ≤M̃^{∗} \frac{M̃_{∗}^{k}}{k!} t^{k+α}, (x, t)∈\overline{Δ_{p}^{∗}}, k ≥ 1, p=1, 2,

whence, as k → ∞, we find in the limit that

v(x, y, t) ≡ 0, (x, y, t) ∈ \overline{D}, φ^{p}(x, t) ≡ 0, (x, t) ∈ \overline{Δ_{p}^{∗}}, p = 1, 2.

Next, using inequality (4.15) and recalling that the value M^{∗} is defined by Θ_{i}, i = 1, 2, 3, which are given by the right-hand sides F and f_{i}, i = 1, 2, of problem (1.2), (2.4), we can readily show that for a regular solution of the considered problem the estimate

∥u∥_{C^{0}_{α}^{1,1,1}(\overline{D})} ≤ c( ∑_{i=1}^{2} ∥f_{i}∥_{C^{0}_{α}(\overline{Δ_{i}})} + ∥F ∥_{C^{0}_{α}(\overline{D})})  (4.30)

holds, where the positive constant c does not depend on f_{i}, i = 1, 2, and F. Estimate (4.30) implies that a regular solution of problem (1.2), (2.4) is

stable in the space C^{0}_{α}^{1,1,1}(\overline{D}), α > ρ_{0}.

References

1. A. V. Bitsadze, Some classes of partial equations. (Russian) Nauka, Moscow, 1981.

2. S. S. Kharibegashvili, On one boundary value problem for normally hyperbolic systems of second order with variable coefficients. (Russian) Differentsial’nye Uravneniya 21(1985), No. 1, 149–155.

3. I. M. Gelfand, Lectures on linear algebra, 4th edition. (Russian) Nauka, Moscow, 1971.

4. S. L. Sobolev, On analytic solutions of systems of partial equations with two independent variables. (Russian) Matem. Sb. 38(1931), Nos. 1, 2, 107–147.

5. V. P. Mikhailov, On analytic solution of the Goursat problem for a system of differential equations. (Russian) Dokl. Akad. Nauk SSSR 115(1957), No. 3, 450–453.

6. V. P. Mikhailov, On non-analytic solutions of the Goursat problem for a system of differential equations with two independent variables. (Russian) Dokl. Akad. Nauk SSSR 117(1957), No. 5, 759–762.

7. L. A. Mel’tser, On non-correct formulation of the Goursat problem. (Russian) Mat. Sb. 18(60)(1946), No. 1, 59–104.

8. Z. O. Melnik, One nonclassical boundary value problem for hyperbolic systems of first order of two independent variables. (Russian) Differentsial’nie Uravneniya 17(1981), No. 6, 1096–1104.

9. J. Hadamard, Lectures on Cauchy’s problem in linear partial differential equations. Yale Univ. Press, New Haven; Oxford Univ. Press, London, 1923.

10. J. Tolen, Probléme de Cauchy sur la deux hypersurfaces caracteristique sécantes. C. R. Acad. Sci. Paris, Ser. A–B 291(1980), No. 1, A 49–A 52.

11. S. Kharibegashvili, On the solvability of a spatial problem of Darboux type for the wave equation. Georgian Math. J. 2(1995), No. 4, 385–394.

12. O. Jokhadze, Spatial problem of Darboux type for one model equation of third order. Georgian Math. J. 3(1996), No. 6, 547–564.

(Received 20.10.1995)

Author’s address:

A. Razmadze Mathematical Institute

Georgian Academy of Sciences

1, M. Aleksidze St., Tbilisi 380093, Georgia