GEORGIAN MATHEMATICAL JOURNAL: Vol. 3, No. 6, 1996, 547-564

SPATIAL PROBLEM OF DARBOUX TYPE FOR ONE MODEL EQUATION OF THIRD ORDER

O. JOKHADZE

Abstract. For a hyperbolic type model equation of third order a Darboux type problem is investigated in a dihedral angle. It is shown that there exists a real number ρ_{0} such that for α > ρ_{0} the problem under consideration is uniquely solvable in the Frechet space. In the case where the coefficients are constants, Bochner’s method is developed in multidimensional domains, and used to prove the uniquely solvability of the problem both in Frechet and in Banach spaces.

§ 1. Statement of the Problem

Let us consider a partial differential equation of hyperbolic type

u_{xyz} = F  (1.1)

in ℝ^{3}, where F is a given function and u is an unknown real function.

For equation (1.1) the family of planes x = const, y = const, z = const is characteristic, while the directions determined by the unit vectors e_{1}, e_{2}, e_{3} of the coordinate axes are bicharacteristic.

In the space ℝ^{3} let S_{i}^{0} : p_{i}(x, y, z) ≡ α_{i}^{0}x + β_{i}^{0}y + γ_{i}^{0}z = 0, i = 1, 2, be arbitrarily given planes passing through the origin. Assume that ν_{1}^{0} ∦ ν_{2}^{0}, |ν_{i}^{0}| ≠ 0, where ν_{i}^{0} ≡ (α_{i}^{0}, β_{i}^{0}, γ_{i}^{0}), i = 1, 2. The space ℝ^{3} is partitioned by the planes S_{i}^{0}, i = 1, 2, into four dihedral angles. We consider equation (1.1) in one of these angles D_{0} which, without loss of generality, is assumed to be given in the form

D_{0} ≡ {(x, y, z) ∈ ℝ^{3} : α_{i}^{0}x + β_{i}^{0}y + γ_{i}^{0}z > 0, i = 1, 2}.

For the domain D_{0} we make the following assumptions:

547

1991 Mathematics Subject Classification. 35L35.

Key words and phrases. Hyperbolic equation, Darboux type problem for the hyperbolic equation in the dihedral angle, characteristic plane, bicharacteristic line.

1072-947X/96/1100-0547$09.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1996 Plenum Publishing Corporation

(a) the edge Γ_{0} ≡ {(x, y, z) ∈ ℝ^{3} : α_{i}^{0}x + β_{i}^{0}y + γ_{i}^{0}z = 0, i = 1, 2} of D_{0} lies in none of the coordinate planes, which is equivalent to

|\begin{matrix}α_{1}^{0} & β_{1}^{0} \\ α_{2}^{0} & β_{2}^{0} \end{matrix}| ≠ 0, |\begin{matrix}α_{1}^{0} & γ_{1}^{0} \\ α_{2}^{0} & γ_{2}^{0} \end{matrix}| ≠ 0, |\begin{matrix}β_{1}^{0} & γ_{1}^{0} \\ β_{2}^{0} & γ_{2}^{0} \end{matrix}| ≠ 0;  (1.2)

this implies that Γ_{0} has no bicharacteristic direction, i.e., ν^{0} ∦ e_{j}, j = 1, 2, 3, where ν^{0} ≡ ν_{1}^{0} × ν_{2}^{0} is the vector product of the vectors ν_{1}^{0} and ν_{2}^{0};

(b) the bicharacteristics passing through the edge Γ_{0} do not pass into the domain D_{0}, and this is equivalent to the fulfillment of the inequalities α_{1}^{0}α_{2}^{0} < 0, β_{1}^{0}β_{2}^{0} < 0, γ_{1}^{0}γ_{2}^{0} < 0.

Let P_{0}(x_{0}, y_{0}, z_{0}) be an arbitrary point of the domain D_{0}, and let S_{1} ⊃ Γ_{0} and S_{2} ⊃ Γ_{0} be the plane sides of the angle D_{0}, ∂D_{0} = S_{1} ∪ S_{2}. Let the bicharacteristic beams L_{i}(P_{0}), i = 1, 2, 3, of equation (1.1) radiate from the point P_{0}, in the direction of decreasing values of the z-coordinate of the moving points L_{i}(P_{0}) to the intersection with one of the sides S_{1} or S_{2} at the points P_{i}, i = 1, 2, 3. Assume that these three points do not lie on the same side. Without loss of generality the points P_{1} and P_{2} are assumed to lie on S_{1}, while the point P_{3} lies on S_{2}.

In the domain D_{0} let us consider the following Darboux type problem: Find in D_{0} a regular solution u(x, y, z) of equation (1.1) satisfying the boundary conditions (to shorten the formulas here and below we assume that S_{0} ≡ S_{1})

(M_{i}u_{xy} + N_{i}u_{xz} + Q_{i}u_{yz})|_{S_{i−1}} = f_{i}, i = 1, 2, 3.  (1.3)

For convenience we transform the domain D_{0} into the domain D : z_{1} − y_{1} > 0, z_{1} + y_{1} > 0 of the space of variables x_{1}, y_{1}, z_{1}. To this end let us introduce new independent variables defined by the equalities

\begin{matrix}x_{1} = x, y_{1} = \frac{1}{2}(p_{1}(x, y, z) − p_{2}(x, y, z)), \\ z_{1} = \frac{1}{2}(p_{1}(x, y, z) + p_{2}(x, y, z)). \end{matrix}  (1.4)

Owing to (1.2), the linear transform (1.4) is obviously nondegenerate, it establishes the one-to-one correspondence between the domains D_{0} and D.

Retaining, the previous notation for u, F, M_{i}, N_{i}, Q_{i}, f_{i} and S_{j}, i = 1, 2, 3, j = 1, 2, in the domain D for the variables x_{1}, y_{1}, z_{1} we rewrite the problem (1.1), (1.2) as

\begin{matrix}\frac{∂^{3}u}{∂μ_{1}∂μ_{2}∂μ_{3}} = F, \\ (M_{i} \frac{∂^{2}u}{∂μ_{1}∂μ_{2}} + N_{i} \frac{∂^{2}u}{∂μ_{1}∂μ_{3}} + Q_{i} \frac{∂^{2}u}{∂μ_{2}∂μ_{3}} )|_{S_{i−1}} = f_{i}, i = 1, 2, 3, \end{matrix} (1.5)

where \frac{∂}{∂μ_{i}} , i = 1, 2, 3, are well defined derivatives with respect to various directions expressed by the values α_{i}^{0}, β_{i}^{0}, γ_{i}^{0}, i = 1, 2, 3,

S_{1} = \{(x, y, z) ∈ ℝ^{3} : x ∈ ℝ, y = z, z ∈ \overline{ℝ}_{+}\},
S_{2} = \{(x, y, z) ∈ ℝ^{3} : x ∈ ℝ, y = −z, z ∈ \overline{ℝ}_{+}\}, \overline{ℝ}_{+} ≡ [0, ∞).

In the domain D let us consider, instead of the problem (1.5), the following boundary value problem in more general terms. Find in the domain D a regular solution u(x, y, z) of the equation

\frac{∂^{3}u}{∂l_{1}∂l_{2}∂l_{3}} = F  (1.6)

satisfying the boundary conditions

(M_{i} \frac{∂^{2}u}{∂l_{1}∂l_{2}} + N_{i} \frac{∂^{2}u}{∂l_{1}∂l_{3}} + Q_{i} \frac{∂^{2}u}{∂l_{2}∂l_{3}} )|_{S_{i−1}} = f_{i}, i = 1, 2, 3. (1.7)

Here for the variables x_{1}, y_{1}, z_{1} we use the previous notation: x, y, z;

\frac{∂}{∂l_{i}} ≡ α_{i} \frac{∂}{∂x} + β_{i} \frac{∂}{∂y} + γ_{i} \frac{∂}{∂z}, |α_{i}| + |β_{i}| + |γ_{i}| ≠ 0, l_{i} ≡ (α_{i}, β_{i}, γ_{i}), i = 1, 2, 3,

is the derivative with respect to the direction, M_{i}, N_{i}, Q_{i}, f_{i}, F, i = 1, 2, 3, are the given functions, and u is the unknown real function. Moreover, the bicharacteristics of equation (1.6) and the domain D will be assumed to satisfy conditions (a) and (b) formulated above for equation (1.1) in the domain D_{0}.

A regular solution of equation (1.6) is said to be a function u(x, y, z)

which is continuous in D together with its partial derivatives \frac{∂^{i+j+k}u}{∂l_{1}^{i}∂l_{2}^{j}∂l_{3}^{k}},

i, j, k = 0, 1, and satisfying equation (1.6) in D.

It should be noted that the boundary value problem (1.6), (1.7) is a natural continuation of the known classical statements of the Goursat and Darboux problems (see, e.g., [1]–[3]) for linear hyperbolic equations of second order with two independent variables on a plane. The multidimensional analogues of the Goursat and Darboux problems for one hyperbolic equation of second order in a dihedral angle were studied by Beudon and many other authors (see, e.g., [2], [4]–[7]).

Many works are devoted to the initial boundary value and characteristic problems for a wide class of hyperbolic equations of third and higher orders in multidimensional domains with dominating derivatives (see, e.g., [8], [9]).

Remark 1.1. Note that the hyperbolicity of problem (1.6), (1.7) is taken into account in conditions (1.7) because of the presence of dominating

derivatives of second order with respect to \frac{∂^{3}u}{∂l_{1}∂l_{2}∂l_{3}} .

Remark 1.2. Since the bicharacteristic beams of equation (1.6) radiating from an arbitrary point of the domain D in the direction of decreasing values of the z-coordinate of moving points of these beams intersect the side S_{1} twice, while the side S_{2} only once, we take in the boundary conditions (1.7) two conditions on S_{1} and one condition on S_{2}, respectively.

In the domains D and Π_{+} ≡ {(x, z) ∈ ℝ^{2} : x ∈ ℝ, z ∈ ℝ_{+}}, ℝ_{+} ≡ (0, ∞), let us introduce into consideration the functional spaces

C^{0}_{α}(\overline{D}) ≡ \{v ∈ C(\overline{D}) : v|_{Γ} = 0, sup_{\begin{matrix}(x,y,z)∈\overline{D}\Γ \\ z≤N \end{matrix}} ρ^{−α}|v(x, y, z)| < ∞, ∀N ∈ ℕ\},

C^{0}_{α,β}(\overline{D}) ≡ \{v ∈ C^{0}_{α}(\overline{D}) : sup_{(x,y,z)∈\overline{D},z>1} ρ^{−β}|v(x, y, z)| < ∞\},

where Γ ≡ {(x, y, z) ∈ ℝ^{3} : x ∈ ℝ, y = z = 0}, ρ is the distance from the point (x, y, z) ∈ \overline{D} to the edge Γ of D, and the parameters α = const ≥ 0, β = const ≥ 0.

Similarly, we introduce the spaces

C^{0}_{α}(\overline{Π}_{+}) ≡ \{φ ∈ C(\overline{Π}_{+}) : φ|_{Γ_{1}} = 0, sup_{\begin{matrix}(x,z)∈Π_{+} \\ z≤N \end{matrix}} z^{−α}|φ(x, z)| < ∞, ∀N ∈ ℕ\},

C^{0}_{α,β}(\overline{Π}_{+}) ≡ \{φ ∈ C^{0}_{α}(\overline{Π}_{+}) : sup_{(x,z)∈Π_{+},z>1} z^{−β}|φ(x, z)| < ∞\},

Γ_{1} ≡ {(x, z) ∈ ℝ^{2} : x ∈ ℝ, z = 0}.

Obviously, for the semi-norms

∥v∥_{C^{0}_{α}(\overline{D}_{N})} = sup_{(x,y,z)∈\overline{D}_{N} \Γ} ρ^{−α}|v(x, y, z)|,

∥φ∥_{C^{0}_{α}(\overline{Π}_{+,N})} = sup_{(x,z)∈\overline{Π}_{+,N} \Γ_{1}} z^{−α}|φ(x, z)|,

where D_{N} ≡ D ∩ {z < N}, Π_{+,N} ≡ Π_{+} ∩ {z < N}, N ∈ ℕ, the spaces

C^{0}_{α}(\overline{D}) and C^{0}_{α}(\overline{Π}_{+}) are the countable normed Frechet spaces.

The spaces C^{0}_{α,β}(\overline{D}) and C^{0}_{α,β}(\overline{Π}_{+}) are the Banach spaces with the norms

∥v∥_{C^{0}_{α,β}(\overline{D})} =
= max \{ sup_{(x,y,z)∈\overline{D}\Γ,z≤1} ρ^{−α}|v(x, y, z)|, sup_{(x,y,z)∈\overline{D},z>1} ρ^{−β}|v(x, y, z)|\},

SPATIAL PROBLEM OF DARBOUX TYPE 551

∥φ∥_{C^{0}_{α,β}(\overline{Π}_{+})} =

= max { sup_{(x,z)∈Π_{+},z≤1} z^{−α}|φ(x, z)|, sup_{(x,z)∈Π_{+},z>1} z^{−β}|φ(x, z)|}.

Remark 1.3. Because of the uniform estimate 1 ≤ \frac{ρ}{z} ≤ \sqrt{2}, (x, y, z) ∈ D, we can replace the value ρ in the definition of the spaces C^{0}_{α}(\overline{D}), C^{0}_{α}(\overline{D}_{N}), C^{0}_{α,β}(\overline{D}) by the variable z which will be used below.

Throughout this paper we denote by c a positive constant whose particular value is not of principal interest for our investigation.

It can be easily seen that the belonging of the functions v ∈ C^{0}(\overline{D}) and φ ∈ C^{0}(\overline{Π}_{+}), respectively, to the spaces C^{0}_{α}(\overline{D}) and C^{0}_{α}(\overline{Π}_{+}) is equivalent to the fulfillment of the inequalities

\begin{matrix}|v(x, y, z)| ≤ cz^{α}, (x, y, z) ∈ \overline{D}, z ≤ N, \\ |φ(x, z)| ≤ cz^{α}, (x, z) ∈ \overline{Π}_{+}, z ≤ N, N ∈ ℕ. \end{matrix}  (1.8)

We investigate problem (1.6), (1.7) in the Frechet space

C^{0}_{α}^{l}(\overline{D}) ≡ \{u : \frac{∂^{i+j+k}u}{∂l_{1}^{i}∂l_{2}^{j}∂l_{3}^{k}} ∈ C^{0}_{α}(\overline{D}), i, j, k = 0, 1\}, l ≡ (l_{1}, l_{2}, l_{3}),

with respect to the semi-norms

∥u∥_{C^{0}_{α}^{l}(\overline{D}_{N})} = ∑_{i,j,k=0}^{1} ∥\frac{∂^{i+j+k}u}{∂l_{1}^{i}∂l_{2}^{j}∂l_{3}^{k}} ∥_{C^{0}_{α}(\overline{D}_{N})}, N ∈ ℕ,

and in the Banach space

C^{0}_{α,β}^{l}(\overline{D}) ≡ \{u : \frac{∂^{i+j+k}u}{∂l_{1}^{i}∂l_{2}^{j}∂l_{3}^{k}} ∈ C^{0}_{α,β}(\overline{D}), i, j, k = 0, 1\}

with the norm

∥u∥_{C^{0}_{α,β}^{l}(\overline{D})} = ∑_{i,j,k=0}^{1} ∥\frac{∂^{i+j+k}u}{∂l_{1}^{i}∂l_{2}^{j}∂l_{3}^{k}} ∥_{C^{0}_{α,β}(\overline{D})}.

In considering the problem (1.6), (1.7) in the class C^{0}_{α}^{l}(\overline{D}) (C^{0}_{α,β}^{l}(\overline{D})), we require that F ∈ C^{0}_{α}(\overline{D}) (C^{0}_{α,β}(\overline{D})), M_{i}, N_{i}, Q_{i} ∈ C(\overline{Π}_{+}) (M_{i} ≡ const, N_{i} ≡ const, Q_{i} ≡ const), f_{i} ∈ C^{0}_{α}(\overline{Π}_{+}) (C^{0}_{α,β}(\overline{Π}_{+})), i = 1, 2, 3.

§ 2. Equivalent Reduction of Problem (1.6), (1.7) to a Functional Equation

Using the notation \frac{∂^{2}u}{∂l_{1}∂l_{2}} ≡ v_{1}, \frac{∂^{2}u}{∂l_{1}∂l_{3}} ≡ v_{2}, \frac{∂^{2}u}{∂l_{2}∂l_{3}} ≡ v_{3}, problem

(1.6), (1.7), in the domain D, can be rewritten equivalently as a boundary value problem for a system of partial differential equations of first order with respect to the unknown functions v_{1}, v_{2}, v_{3}:

\frac{∂v_{1}}{∂l_{3}} = F, \frac{∂v_{2}}{∂l_{2}} = F, \frac{∂v_{3}}{∂l_{1}} = F,  (2.1)
(M_{i}v_{1} + N_{i}v_{2} + Q_{i}v_{3})|_{S_{i−1}} = f_{i}, i = 1, 2, 3.  (2.2)

The equivalence of the initial problem (1.6), (1.7) and problem (2.1), (2.2) is an obvious consequence of

Lemma 2.1. In the closed domain \overline{D}_{0} there exists a unique function u ∈ \{u : D_{x}^{i}D_{y}^{j}D_{z}^{k}u ∈ C(\overline{D}_{0}), i, j, k = 0, 1\}, satisfying both the redefined system of partial differential equations of second order

u_{xy} = v_{1}, u_{xz} = v_{2}, u_{yz} = v_{3}  (2.3)

and the conditions

u(P^{0}) = c_{0}, u_{x}|_{Γ_{0}} = ω_{1}, u_{y}|_{Γ_{0}} = ω_{2}, u_{z}|_{Γ_{0}} = ω_{3}.  (2.4)

Here v_{1}, v_{2}, v_{3} are given functions such that v_{i}, \frac{∂v_{1}}{∂z} , \frac{∂v_{2}}{∂y} , \frac{∂v_{3}}{∂x} ∈ C(\overline{D}_{0}), i = 1, 2, 3; \frac{∂v_{1}}{∂z} (x, y, z) = \frac{∂v_{2}}{∂y} (x, y, z) = \frac{∂v_{3}}{∂x} (x, y, z), (x, y, z) ∈ \overline{D}_{0}; c_{0} and ω_{i} ∈ C(Γ_{0}), i = 1, 2, 3, are, respectively, the given constant and functions on Γ_{0}; P^{0} = P^{0}(x^{0}, y^{0}, z^{0}) is an arbitrarily fixed point of Γ_{0}.

Proof. Let P_{0}(x_{0}, y_{0}, z_{0}) be an arbitrary point of \overline{D}_{0}. It is obvious that owing to the requirement (a) on Γ_{0} in §1, the plane x = x_{0} has the unique point of intersection of P_{0}^{∗}(x_{0}, y^{∗}(x_{0}), z^{∗}(x_{0})) with the edge Γ_{0}. Since (u_{x}(x_{0}, y, z))_{y} = v_{1}(x_{0}, y, z), (u_{x}(x_{0}, y, z))_{z} = v_{2}(x_{0}, y, z) and u_{x}(P_{0}^{∗}) = ω_{1}(P_{0}^{∗}), the function u_{x}(x_{0}, y, z) is defined uniquely at the point P_{0}(x_{0}, y_{0}, z_{0}) by the formula

u_{x}(P_{0}) = ω_{1}(P_{0}^{∗}) +  ∫_{(y^{∗}(x_{0}),z^{∗}(x_{0}))}^{(y_{0},z_{0})} v_{1}(x_{0}, y, z)dy + v_{2}(x_{0}, y, z)dz. (2.5)

Here the curvilinear integral is taken along any simple smooth curve connecting the points (y^{∗}(x_{0}), z^{∗}(x_{0})) and (y_{0}, z_{0}) of the plane x = x_{0} and lying wholly in \overline{D}_{0}. Since the point P_{0} is chosen arbitrarily, in the closed domain \overline{D}_{0} formula (2.5) gives the representation of the function u_{x}(P), P ≡ P (x, y, z), which is written in terms of the given functions v_{1} and v_{2}.

Analogously, the representation formulas for the functions u_{y}(P) and u_{z}(P) in \overline{D}_{0} are given by the known functions v_{1}, v_{3} and v_{2}, v_{3}, respectively. It remains only to note that the function u(P) defined by the formula

u(P) = c_{0} + ∫_{P^{0}P} u_{x}dx + u_{y}dy + u_{z}dz = c_{0} + ∫_{P^{0}P} {ω_{1}(x, y^{∗}(x), z^{∗}(x)) +

+ ∫_{P^{∗}P^{1}} v_{1}(x, η, ζ)dη + v_{2}(x, η, ζ)dζ}dx + {ω_{2}(x^{∗∗}(y), y, z^{∗∗}(y)) +

+ ∫_{P^{∗∗}P^{2}} v_{1}(ξ, y, ζ)dξ + v_{3}(ξ, y, ζ)dζ}dy + {ω_{3}(x^{∗∗∗}(z), y^{∗∗∗}(z), z) +

+ ∫_{P^{∗∗∗}P^{3}} v_{2}(ξ, η, z)dξ + v_{3}(ξ, η, z)dη}dz  (2.6)

defines actually the unique solution of the problem (2.3), (2.4). Here P^{1} ≡ P^{1}(y, z), P^{2} ≡ P^{2}(x, z), P^{3} ≡ P^{3}(x, y), P^{∗} ≡ P^{∗}(y^{∗}(x), z^{∗}(x)), P^{∗∗} ≡ P^{∗∗}(x^{∗∗}(y), z^{∗∗}(y)), P^{∗∗∗} ≡P^{∗∗∗}(x^{∗∗∗}(z), y^{∗∗∗}(z)).

Remark 2.1. If instead of system (2.3) we consider the system

\frac{∂^{2}u}{∂l_{1}∂l_{2}} = v_{1}, \frac{∂^{2}u}{∂l_{1}∂l_{3}} = v_{2}, \frac{∂^{2}u}{∂l_{2}∂l_{3}} = v_{3},  (2.7)

in the dihedral angle \overline{D}, then similarly to item (a) for Γ_{0} one should require that the edge Γ of D lie in none of the three planes passing through the origin and spanned to the pairs of vectors (l_{1}, l_{2}), (l_{1}, l_{3}) and (l_{2}, l_{3}).

Note that system (2.7) reduces to system (2.3) by means of the nondegenerate transform of variables x, y, z,

x = α_{1}ξ + α_{2}η + α_{3}ζ, y = β_{1}ξ + β_{2}η + β_{3}ζ, z = γ_{1}ξ + γ_{2}η + γ_{3}ζ,

under the assumption that the vectors l_{1}, l_{2} and l_{3} are linearly independent.

Let the bicharacteristic beams L_{i}(P), i = 1, 2, 3, of equation (1.6) rediate from an arbitrary point P (x, y, z) ∈ D in the direction of decreasing values of the z-coordinate of moving points L_{i}(P) to the intersection with the sides S_{1} and S_{2} at the points P_{i}, i = 1, 2, 3.

Denoting for (x, z) ∈ \overline{Π}_{+} v_{1}|_{S_{2}} ≡ φ_{1}(x, z), v_{2}|_{S_{1}} ≡ φ_{2}(x, z), v_{3}|_{S_{1}} ≡ φ_{3}(x, z), and integrating the equations of system (2.1) along the corresponding bicharacteristics, for (x, y, z) ∈ D we get

\begin{cases}v_{1}(x, y, z) = φ_{1}(σ(x, y + z, z; α̃_{1}, α̃_{2})) + F_{1}(x, y, z), \\ v_{2}(x, y, z) = φ_{2}(σ(x, y − z, z; α̃_{3}, α̃_{4})) + F_{2}(x, y, z), \\ v_{3}(x, y, z) = φ_{3}(σ(x, y − z, z; α̃_{5}, α̃_{6})) + F_{3}(x, y, z), \end{cases}  (2.8)

where F_{i}, i = 1, 2, 3, are known functions, σ(x, y, z; λ_{1}, λ_{2}) ≡ (x + λ_{1}y, z + λ_{2}y), σ_{1}(x, y; λ) ≡ x + λy, and the superscript −1 here and below denotes the inverse value.

Substituting the expressions for v_{1}, v_{2}, and v_{3} from equalities (2.8) into the boundary conditions (2.2), for (x, z) ∈ \overline{Π}_{+} we obtain

\{ \begin{matrix}M_{1}(x, z) & φ_{1}(σ(x, z, 0; 2α̃_{1}, α̃_{7})) + N_{1}(x, z)φ_{2}(x, z) + \\ + Q_{1}(x, z)φ_{3}(x, z) = f_{4}(x, z), \\ M_{2}(x, z) & φ_{1}(σ(x, z, 0; 2α̃_{1}, α̃_{7})) + N_{2}(x, z)φ_{2}(x, z) + \\ + Q_{2}(x, z)φ_{3}(x, z) = f_{5}(x, z), \\ M_{3}(x, z) & φ_{1}(x, z) + N_{3}(x, z)φ_{2}(σ(x, z, 0; −2α̃_{3}, α̃_{8})) + \\ + Q_{3}(x, z)φ_{3}(σ(x, z, 0; −2α̃_{5}, α̃_{9})) = f_{6}(x, z). \end{matrix}  \begin{matrix}(2.9) \\ (2.10) \\ (2.11) \end{matrix}

In equalities (2.8)–(2.11) the constants α̃_{i}, i = 1, …, 9, are well defined and written in terms of α_{i}, β_{i}, γ_{i}, i = 1, 2, 3.

Condition I. The functions M_{i}, N_{j}, Q_{j}, i = 1, 2, j = 1, 2, 3, are bounded and uniformly continuous in \overline{Π}_{+}.

Note that by Condition I the known functions f_{i}, i = 4, 5, 6, belong to

the class C^{0}_{α}(\overline{Π}_{+}). We rewrite equations (2.9) and (2.10) as follows:

\begin{cases}N_{1}(x, z)φ_{2}(x, z) + Q_{1}(x, z)φ_{3}(x, z) = f_{4}(x, z)− \\ −M_{1}(x, z)φ_{1}(σ(x, z, 0; 2α̃_{1}, α̃_{7})), \\ N_{2}(x, z)φ_{2}(x, z) + Q_{2}(x, z)φ_{3}(x, z) = f_{5}(x, z)− \\ −M_{2}(x, z)φ_{1}(σ(x, z, 0; 2α̃_{1}, α̃_{7})). \end{cases} (x, z) ∈ \overline{Π}_{+}, (2.12)

Condition II. The inequality

|Δ_{0}(x, z)| ≥ c, (x, z) ∈ \overline{Π}_{+},  (2.13)

holds for the determinant Δ_{0}(x, z) ≡ (N_{1}Q_{2} − N_{2}Q_{1})(x, z).

On account of (2.12) and (2.13) we find that for (x, z) ∈ \overline{Π}_{+}

φ_{i+1}(x, z) = a_{i}(x, z) − b_{i}(x, z)φ_{1}(σ(x, z, 0; 2α̃_{1}, α̃_{7})), i = 1, 2, (2.14)

where a_{i}, b_{i}, i = 1, 2, are given functions. Bearing in mind Conditions I

and II, we find that a_{i} ∈ C^{0}_{α}(\overline{Π}_{+}), i = 1, 2, while the continuous functions b_{i}, i = 1, 2, are bounded in \overline{Π}_{+}.

Condition III. The function M_{3} satisfies the inequality

|M_{3}(x, z)| ≥ c, (x, z) ∈ \overline{Π}_{+},  (2.15)

and the function M_{3}^{−1} is uniformly continuous in \overline{Π}_{+}.

Taking into account (2.15) from (2.11) and (2.14), we obtain the functional equation

φ_{1}(x, z) − G_{2}(x, z)φ_{1}(J_{2}(x, z)) − G_{3}(x, z)φ_{1}(J_{3}(x, z)) = g(x, z) (2.16)

with respect to φ_{1} : \overline{Π}_{+} → ℝ.

Due to Conditions I–III, the functions G_{2}, G_{3} are known uniformly continuous and bounded in \overline{Π}_{+}, while the function g is expressed by the known

functions and belongs to the class C^{0}_{α}(\overline{Π}_{+}). The functions J_{i} : \overline{Π}_{+} → \overline{Π}_{+}, i = 2, 3, act by the formulas

J_{i} : (x, z) → (x + δ_{i}z, τ_{i}z), (x, z) ∈ \overline{Π}_{+}, i = 2, 3,  (2.17)

where δ_{i}, τ_{i}, i = 2, 3 are well-defined constants written in terms of α_{i}, β_{i}, γ_{i}, i = 1, 2, 3.

Remark 2.2. Note that under the assumptions with respect to the coefficients α_{i}, β_{i}, γ_{i}, i = 1, 2, 3, we can easily see that 0 < τ_{i} < 1, i = 2, 3.

Remark 2.3. It is obvious that when Conditions I–III are fulfilled, prob-

lem (1.6), (1.7) in the class C^{0}_{α}^{l}(\overline{D}) is equivalently reduced to (2.16) with

respect to the unknown function φ_{1} of the class C^{0}_{α}(\overline{Π}_{+}). Furthermore, if

u ∈ C^{0}_{α}^{l}(\overline{D}), then φ_{1} ∈ C^{0}_{α}(\overline{Π}_{+}), and vice versa: if φ_{1} ∈ C^{0}_{α}(\overline{Π}_{+}), then taking into account the inequalities (1.8), we find from equalities (2.14), (2.8),

(2.6) that u ∈ C^{0}_{α}^{l}(\overline{D}).

§ 3. Investigation of the Functional Equation (2.16)

Let us introduce the notation

(T φ_{1})(x, z) ≡ φ_{1}(x, z) − ∑_{i=2}^{3} G_{i}(x, z)φ_{1}(J_{i}(x, z)), (x, z) ∈ \overline{Π}_{+}, (3.1)

h(ρ) ≡ ∑_{i=2}^{3} η_{i}τ_{i}^{ρ}, η_{1} ≡ max_{2≤i≤3} sup_{(x,z)∈\overline{Π}_{+}} |G_{i}(x, z)|,

η_{i} ≡ sup_{x∈ℝ} |G_{i}(x, 0)|, i = 2, 3, ρ ∈ ℝ.

(3.2)

Let for some value of the index i the number η_{i} be different from zero. In that case, owing to (2.2), the function h : ℝ → \overline{ℝ}_{+} is continuous and strictly monotonically decreasing on ℝ; moreover, lim_{ρ→−∞} h(ρ) = +∞ and lim_{ρ→+∞} h(ρ) = 0. Therefore there exists a unique real number ρ_{0} such that h(ρ_{0}) = 1. For η_{2}^{2} + η_{3}^{2} = 0 we assume ρ_{0} = −∞.

Lemma 3.1. If α > ρ_{0}, then (2.16) is uniquely solvable in C^{0}_{α}(\overline{Π}_{+,A}), Π_{+,A} ≡ ℝ × (0, A), ∀A > 0, and for the solution φ_{1} = T^{−1}g the estimate

|(T^{−1}g)(x, z)| ≤ Cz^{α}∥g∥_{C^{0}_{α}(\overline{Π}_{+,z})}, x ∈ ℝ, 0 ≤ z ≤ A,  (3.3)

holds, where C is a positive constant not depending on the function g.

Proof. By the condition α > ρ_{0} and the definition of the function h it follows from (3.2) that

h(α) = ∑_{i=2}^{3} η_{i}τ_{i}^{α} < 1.  (3.4)

Because of (3.4) and the uniform continuity of the functions G_{2}, G_{3} there are positive numbers ε (ε < A) and δ such that the inequalities

|G_{i}(x, z)| ≤ η_{i} + δ, i = 2, 3, x ∈ ℝ,  (3.5)
∑_{i=2}^{3}(η_{i} + δ)τ_{i}^{α} ≡ β < 1  (3.6)

are valid for 0 ≤ z ≤ ε.

According to Remark 2.2, there exists a natural number q_{0}(A) such that for q ≥ q_{0}

τ_{i_{q}}τ_{i_{q−1}} ⋯ τ_{i_{1}}z ≤ ε, 0 ≤ z ≤ A,  (3.7)

where 2 ≤ i_{s} ≤ 3, s = 1, …, q.

Let us introduce into consideration the operators Λ and T^{−1} defined by

(Λφ_{1})(x, z) = ∑_{i=2}^{3} G_{i}(x, z)φ_{1}(J_{i}(x, z)), T^{−1} = I + ∑_{q=1}^{∞} Λ^{q},

where (x, z) ∈ \overline{Π}_{+,A} and I is an identical operator. The operator T^{−1} is still formally inverse to the operator T defined by equality (3.1). To prove that the operator T^{−1} is really inverse to the operator T, it suffices to establish

its continuity in the space C^{0}_{α}(\overline{Π}_{+,A}).

Indeed, it is easily seen that the expression Λ^{q}g represents the sum of summands of the type

I_{i_{1}···i_{q}}(x, z) = G_{i_{1}}(x, z)G_{i_{2}}(J_{i_{1}}(x, z))G_{i_{3}}(J_{i_{2}}(J_{i_{1}}(x, z))) ⋯

⋯ G_{i_{q}}(J_{i_{q−1}}(J_{i_{q−2}}(⋯ (J_{i_{1}}(x, z)) ⋯)))g(J_{i_{q}}(J_{i_{q−1}}(⋯ (J_{i_{1}}(x, z)) ⋯))),

where 2 ≤ i_{s} ≤ 3, s = 1, …, q.

By virtue of (3.2), (3.5), (3.7) and Remark 2.2, for q > q_{0}, g ∈ C^{0}_{α}(\overline{Π}_{+,A}) we have

|I_{i_{1}···i_{q}}(x, z)| ≤ |G_{i_{1}}(x, z)| ⋯ |G_{i_{q_{0}}} (J_{i_{q_{0}−1}}(J_{i_{q_{0}−2}}(⋯ (J_{i_{1}}(x, z)) ⋯)))| × ×|G_{i_{q_{0}+1}}(J_{i_{q_{0}}} (J_{i_{q_{0}−1}}(⋯ (J_{i_{1}}(x, z)) ⋯)))| ⋯

⋯ |G_{i_{q}}(J_{i_{q−1}}(J_{i_{q−2}}(⋯ (J_{i_{1}}(x, z)) ⋯)))| ×

×|g(J_{i_{q}}(J_{i_{q−1}}(⋯ (J_{i_{1}}(x, z)) ⋯)))| ≤ η_{1}^{q_{0}}(η_{i_{q_{0}+1}} + δ) ⋯ (η_{i_{q}} + δ) ×

×(τ_{i_{q}}τ_{i_{q−1}} ⋯ τ_{i_{1}}z)^{α}∥g∥_{C^{0}_{α}(\overline{Π}_{+,z})} ≤ η_{1}^{q_{0}}( ∏_{s=q_{0}+1}^{q}(η_{i_{s}} + δ))( ∏_{s=q_{0}+1}^{q} τ_{i_{s}}^{α}) ×

×z^{α}∥g∥_{C^{0}_{α}(\overline{Π}_{+,z})} = η_{1}^{q_{0}}( ∏_{s=q_{0}+1}^{q}(η_{i_{s}} + δ)τ_{i_{s}}^{α})z^{α}∥g∥_{C^{0}_{α}(\overline{Π}_{+,z})}, (3.8)

while for 1 ≤ q ≤ q_{0}

|I_{i_{1}···i_{q}}(x, z)| ≤ η_{1}^{q}(τ_{i_{q}}τ_{i_{q−1}} ⋯ τ_{i_{1}}z)^{α}∥g∥_{C^{0}_{α}(\overline{Π}_{+,z})} ≤

≤ η_{1}^{q}z^{α}∥g∥_{C^{0}_{α}(\overline{Π}_{+,z})}.  (3.9)

Taking into account (3.8), (3.9), and (3.6) for q > q_{0}, we get

|(Λ^{q}g)(x, z)| = | ∑_{i_{1},...,i_{q}} I_{i_{1}···i_{q}}(x, z)| ≤ ( ∑_{i_{1},...,i_{q_{0}}} 1)^{q_{0}}η_{1}^{q_{0}} ×

×[ ∑_{i=2}^{3}(η_{i} + δ)τ_{i}^{α}]^{q−q_{0}}z^{α}∥g∥_{C^{0}_{α}(\overline{Π}_{+,z})} ≤ c_{1}β^{q}z^{α}∥g∥_{C^{0}_{α}(\overline{Π}_{+,z})}, (3.10)

while for 1 ≤ q ≤ q_{0}

|(Λ^{q}g)(x, z)| ≤ c_{2}z^{α}∥g∥_{C^{0}_{α}(\overline{Π}_{+,z})},  (3.11)

where c_{1} = η_{1}^{q_{0}}β^{−q_{0}}( ∑_{i_{1},...,i_{q_{0}}} 1)^{q_{0}}, c_{2} = η_{1}^{q}( ∑_{i_{1},...,i_{q}} 1).

From (3.10) and (3.11) we finally find

|(T^{−1}g)(x, z)| ≤ |g(x, z)| +∑_{q=1}^{q_{0}} |(Λ^{q}g)(x, z)|+ ∑_{q=q_{0}+1}^{∞} |(Λ^{q}g)(x, z)|≤

≤ (1 + c_{2}q_{0} + c_{1}β^{q_{0}+1}(1 − β)^{−1})z^{α}∥g∥_{C^{0}_{α}(\overline{Π}_{+,z})} = Cz^{α}∥g∥_{C^{0}_{α}(\overline{Π}_{+,z})},

where C ≡ 1 + c_{2}q_{0} + c_{1}β^{q_{0}+1}(1 − β)^{−1}. This implies that the operator T^{−1}

is continuous in the space C^{0}_{α}(\overline{Π}_{+,A}) and estimate (3.3) is valid. Thus the unique solvability of equation (2.16) on \overline{Π}_{+,A} is proved for any A > 0.

The unique solvability of this equation on the whole \overline{Π}_{+} in the class

C^{0}_{α}(\overline{Π}_{+}) follows from

Lemma 3.2. If the equation

(T φ_{1})(x, z) = g(x, z), (x, z) ∈ \overline{Π}_{+},  (3.12)

is uniquely solvable on \overline{Π}_{+,A} for any A > 0, then equation (3.12) is uniquely solvable on the whole \overline{Π}_{+}.

Proof. Indeed, let φ_{1,n}(x, z) be that unique solution of equation (3.12) on \overline{Π}_{+,n} whose existence has been proved above. Owing to the aboveestablished uniqueness of the solution, we have φ_{1,n}(x, z) = φ_{1,m}(x, z) if (x, z) ∈ \overline{Π}_{+,n} and m > n. Then it is obvious that φ_{1}(x, z) = φ_{1,n}(x, z) is the unique solution of equation (3.12) for (x, z) ∈ \overline{Π}_{+,n}.

Finally, by Lemmas 3.1 and 3.2, the lemma below is valid.

Lemma 3.3. If α > ρ_{0}, then equation (2.16) is uniquely solvable in the

space C^{0}_{α}(\overline{Π}_{+}) for any g ∈ C^{0}_{α}(\overline{Π}_{+}).

Since problem (1.6), (1.7) in Conditions I–III in the class C^{0}_{α}^{l}(\overline{D}) has been

equivalently reduced to (2.16) in C^{0}_{α}(\overline{Π}_{+}), from Lemmas 3.1–3.3 we have

Theorem 3.1. If α > ρ_{0}, then the boundary value problem (1.6), (1.7)

is uniquely solvable in the class C^{0}_{α}^{l}(\overline{D}).

Consider now the case where coefficients of the problem (1.6), (1.7) are constant, i.e., M_{i} ≡ const, N_{i} ≡ const, Q_{i} ≡ const, i = 1, 2, 3. It is evident that in this case Condition I is fulfilled automatically. As for Conditions II and III, they can respectively be replaced by relatively weaker conditions:

Condition II′. Δ_{0} ≠ 0.

Condition III′. M_{3} ≠ 0.

By developing Bochner’s method (see, e.g., [11]) in multidimensional do-

mains, we investigate the functional equation (2.16) in the classes C^{0}_{α}(\overline{Π}_{+})

and C^{0}_{α,β}(\overline{Π}_{+}).

Let Z ≡ {σ_{0}, σ_{1}, …, σ_{i}, … } be a set of all numbers representable in the form ∑_{k=2}^{3} n_{k} log τ_{k}, where n_{k} are arbitrary integers, and σ_{0} = 0, σ_{i} ≠ σ_{j}, for i ≠ j. Denote by ℂ a set of all complex numbers. Let us consider the entire function Δ(s), s ∈ ℂ, corresponding to the operator T_{3} from (3.1):

\begin{matrix}Δ(s) = ∑_{p=1}^{3} G̃_{p}e^{s log τ_{p}}, G̃_{1} ≡ 1, \\ G̃_{p} ≡ −G_{p}, p = 2, 3, τ_{1} = 1, s ∈ ℂ. \end{matrix}  (3.13)

Obviously, Δ(s) ≢ 0, s ∈ ℂ, since G̃_{1} ≡ 1. Denote by H a set of real parts of all zeros of the entire function Δ(s), s ∈ ℂ. The set H is either finite or closed countably-bounded [10].

Since the set H is finite or countably-bounded, the complement H on the real axis consists of a finite or a countable set of intervals two of which are half-lines. Let H_{0} = (−∞, b_{0}) and H_{1} = (a, ∞) be respectively the left and the right half-line, and let H_{i}, i = 2, 3, …, be the rest of the intervals.

It is shown in [11] and [12] that the analytic almost-periodic function \frac{1}{Δ(s)} decomposes in the strip Π_{i} ≡ \{s : Re s ∈ H_{i}\}, i ≥ 0, and takes the form of an absolutely convergent series

\frac{1}{Δ(s)} = ∑_{j=0}^{∞} γ_{ij}e^{σ_{j}s}, σ_{j} ∈ Z, s ∈ Π_{i},  (3.14)

with uniquely determined coefficients γ_{ij}, j = 0, 1, . . ..

On acount of (3.13) and (3.14) it can be easily seen that for Re s ∈ H_{i_{0}}, i_{0} ≥ 0, we have the equality

∑_{q=0}^{∞} ∑_{p=1}^{3} G̃_{p}γ_{i_{0}q}e^{(log τ_{p}+σ_{q})s} = ∑_{ν=0}^{∞} ( ∑_{(q,p)∈I_{ν}} G̃_{p}γ_{i_{0}q})e^{σ_{ν}s} = 1, (3.15)

where I_{ν} is the set of all elements (q, p) for which log τ_{p} + σ_{q} = σ_{ν}.

From (3.15), because of the absolute convergence of the series (3.14) in Π_{i_{0}} and the uniqueness theorem for analytic almost-periodic functions, we get

∑_{(q,p)∈I_{ν}} G̃_{p}γ_{i_{0}q} = \begin{cases}1, for ν = 0, \\ 0, for ν ≥ 1. \end{cases}  (3.16)

When investigating the functional equation (2.16) by Bochner’s method, we assume that the displacement operators defined by equality (2.17) are permutational, i.e., J_{2}(J_{3})(x, z) = J_{3}(J_{2})(x, z), (x, z) ∈ \overline{Π}_{+}, which is equivalent to the fulfillment of the equality

\frac{δ_{3}}{1 − τ_{3}} − \frac{δ_{2}}{1 − τ_{2}} = 0.  (3.17)

Below equality (3.17) is assumed to be fulfilled.

Lemma 3.4. The operator T defined by formula (3.1) is invertible in the

space C^{0}_{α}(\overline{Π}_{+}) if α > sup H.

Proof. Consider the operator T̃_{i_{0}} acting in the space C^{0}_{α}(\overline{Π}_{+}) by the formula

(T̃_{i_{0}}φ)(x, z) = ∑_{q=0}^{∞} γ_{i_{0}q}(T_{2}^{n_{q}}T_{3}^{m_{q}}φ)(x, z), i_{0} = 1,  (3.18)

where (T_{p}φ)(x, z) ≡ φ(x + δ_{p}z, τ_{p}z), (x, z) ∈ \overline{Π}_{+}, φ ∈ C^{0}_{α}(\overline{Π}_{+}), δ_{1} = 0,

σ_{q} = n_{q} log τ_{2} + m_{q} log τ_{3}, p = 1, 2, 3.

The operator T̃_{i_{0}} acting by formula (3.18) is defined correctly according to

Lemma 3.5. The operator T_{2}^{n_{q}}T_{3}^{m_{q}} appearing in formula (3.18) and corresponding to the decomposition σ_{q} = n_{q} log τ_{2} + m_{q} log τ_{3} is correctly defined, i.e., if for σ_{q} another decomposition σ_{q} = n′_{q} log τ_{2} + m′_{q} log τ_{3} takes

place, then the equality T_{2}^{n_{q}}T_{3}^{m_{q}} = T_{2}^{n′_{q}}T_{3}^{m′_{q}} is valid.

Proof. Simple calculations show that

(T_{2}^{n_{q}}T_{3}^{m_{q}}φ)(x, z) = φ[x + (δ_{2} \frac{1 − τ_{2}^{n_{q}}}{1 − τ_{2}} τ_{3}^{m_{q}} + δ_{3} \frac{1 − τ_{3}^{m_{q}}}{1 − τ_{3}} )z, τ_{2}^{n_{q}}τ_{3}^{m_{q}}z],

which, because of (3.17), takes the form

(T_{2}^{n_{q}}T_{3}^{m_{q}}φ)(x, z) = φ[x + \frac{δ_{2}}{1 − τ_{2}} (1 − e^{σ_{q}})z, e^{σ_{q}}z], (x, z) ∈ \overline{Π}_{+}. (3.19)

Equality (3.19) implies that Lemma 3.5 is valid.

By the assumption

c_{3} ≡ ∑_{q=0}^{∞} |γ_{1q}|e^{σ_{q}α} < ∞, where α > sup H.  (3.20)

Let P_{z}(φ) ≡ ∥φ∥_{C^{0}_{α}(\overline{Π}_{+,z})}, φ̃(x, z) ≡ φ[x + \frac{δ_{2}}{1−τ_{2}} (1 − e^{σ_{q}})z, e^{σ_{q}}z], (x, z) ∈

\overline{Π}_{+}. Then

P_{z}(φ̃) = ∥φ̃∥_{C^{0}_{α}(\overline{Π}_{+,z})} ≤ sup_{\begin{matrix}x∈ℝ, \\ 0<τ_{1}≤z_{0}z \end{matrix}} |τ_{1}^{−α}e^{σ_{q}α}φ(x, τ_{1})| ≤ e^{σ_{q}α}P_{z_{0}z}(φ), (3.21)

where z_{0} ≡ e^{max_{q} σ_{q}} = 1 because max_{q} σ_{q} = − max{0, log τ_{2}, log τ_{3}} = 0.

Let us now prove that the operator T̃_{1} is continuous. By virtue of (3.18) and (3.21) we have P_{z}(T̃_{1}φ) ≤ ∑_{q=0}^{∞} |γ_{1q}|P_{z}(φ̃) ≤ ∑_{q=0}^{∞} |γ_{1q}|e^{σ_{q}α}P_{z}(φ), whence, taking into account (3.20), we find P_{z}(T̃_{1}φ) ≤ c_{3}P_{z}(φ), which proves that the operator T̃_{1} is continuous.

Let us prove now that T T̃_{1} = I. According to the definition of the operator T, formulas (3.1), (3.18), and (3.16), we have

((T T̃_{1})φ)(x, z) = ∑_{p=1}^{3} G̃_{p}(T_{p}(T̃_{1}φ))(x, z) = ∑_{q=0}^{∞} ∑_{p=1}^{3} G̃_{p}γ_{1q} ×

×φ\{x + [\frac{δ_{2}}{1 − τ_{2}} (1 − e^{σ_{q}}) + δ_{p}e^{σ_{q}}]z, τ_{p}e^{σ_{q}}z\} = ∑_{ν=0}^{∞} ( ∑_{(q,p)∈I_{ν}} G̃_{p}γ_{1q}) ×

×φ\{x + [\frac{δ_{2}}{1 − τ_{2}} (τ_{p} − e^{σ_{ν}}) + δ_{p}e^{σ_{q}}]τ_{p}^{−1}z, e^{σ_{ν}} z\} =

= φ\{x + [\frac{δ_{2}}{1 − τ_{2}} (τ_{p} − 1) + δ_{p}]τ_{p}^{−1}z, z\} = φ(x, z),

since

\frac{δ_{2}}{1 − τ_{2}} (τ_{p} − 1) + δ_{p} = 0  (3.22)

for condition (3.17) for any p = 1, 2, 3.

Similarly, taking into account (3.1), (3.16), (3.18), and (3.22), we can prove that

((T̃_{1}T)φ)(x, z) = ∑_{q=0}^{∞} γ_{1q}(T_{2}^{n_{q}}T_{3}^{m_{q}}(T φ))(x, z) =

= ∑_{q=0}^{∞} γ_{1q} ∑_{p=1}^{3} G̃_{p}(T_{2}^{n_{q}}T_{3}^{m_{q}}(T_{p}φ))(x, z) =

= ∑_{p=1}^{3} ∑_{q=0}^{∞} γ_{1q}G̃_{p}φ\{x + [\frac{δ_{2}}{1 − τ_{2}} (1 − e^{σ_{q}})τ_{p} + δ_{p}]z, τ_{p}e^{σ_{q}}z\} =

= ∑_{ν=0}^{∞} ( ∑_{(q,p)∈I_{ν}} γ_{1q}G̃_{p})φ\{x + [\frac{δ_{2}}{1 − τ_{2}} (τ_{p} − e^{σ_{ν}}) + δ_{p}]z, τ_{p}e^{σ_{ν}} z\} =

= φ\{x + [\frac{δ_{2}}{1 − τ_{2}} (τ_{p} − 1) + δ_{p}]z, z\} = φ(x, z).

Thus Lemma 3.4 is proved completely.

Let now I_{α,β} ≡ [min(α, β), max(α, β)]. It is obvious that if H ∩I_{α,β} = ∅, then the segment I_{α,β} is wholly contained in the strip Π_{i_{0}}, i_{0} ≥ 0.

562 O. JOKHADZE

Lemma 3.6. The operator T given by formula (3.1) is invertible in the space C^{0}_{α,β}(\overline{Π}_{+}) if H ∩ I_{α,β} = ∅ and T^{−1} = T̃_{i_{0}} [12]. Proof. Since the function \frac{1}{Δ(s)} in the strip Π_{i_{0}}, i_{0} ≥ 0, decomposes into the absolutely convergent series (3.14), we have

c_{4} ≡ ∑_{q=0}^{∞} |γ_{i_{0}q}|e^{σ_{q}α} < ∞, c_{5} ≡ ∑_{q=0}^{∞} |γ_{i_{0}q}|e^{σ_{q}β} < ∞.  (3.23)

By virtue of φ ∈ C^{0}_{α,β}(\overline{Π}_{+}) we have the inequalities

sup_{\begin{matrix}x∈ℝ, \\ 0<τ≤1 \end{matrix}} |τ^{−α}φ(x, τ)| < ∞, sup_{\begin{matrix}x∈ℝ, \\ τ>1 \end{matrix}} |τ^{−β}φ(x, τ)| < ∞.

Consider the case α < β and assume that

φ^{∗}(x, z) ≡ φ[x + \frac{δ_{2}}{1 − τ_{2}} (1 − e^{σ})z, e^{σ}z, ], (x, z) ∈ \overline{Π}_{+}.

I. Let σ ≥ 0. Obviously, φ^{∗} ∈ C^{0}_{α,β}(\overline{Π}_{+}), and we have the estimates

sup_{\begin{matrix}x∈ℝ, \\ 0<τ≤1 \end{matrix}} |τ^{−α}φ^{∗}(x, τ)| ≤ sup_{\begin{matrix}x∈ℝ, \\ 0<τ_{1}≤e^{σ} \end{matrix}} |τ_{1}^{−α}e^{σα}φ(x, τ_{1})| ≤

≤ e^{σα} max \{ sup_{\begin{matrix}x∈ℝ, \\ 0<τ_{1}≤1 \end{matrix}} |τ_{1}^{−α}φ(x, τ_{1})|, sup_{\begin{matrix}x∈ℝ, \\ 1<τ_{1}≤e^{σ} \end{matrix}} |τ_{1}^{−β}φ(x, τ_{1})τ_{1}^{β−α}|\} ≤

≤ max \{e^{σα} sup_{\begin{matrix}x∈ℝ, \\ 0<τ_{1}≤1 \end{matrix}} |τ_{1}^{−α}φ(x, τ_{1})|, e^{σβ} sup_{\begin{matrix}x∈ℝ, \\ 1<τ_{1}≤e^{σ} \end{matrix}} |τ_{1}^{−β}φ(x, τ_{1})|\} ≤

≤ e^{σβ}∥φ∥_{C^{0}_{α,β}(\overline{Π}_{+})},

sup_{\begin{matrix}x∈ℝ, \\ τ>1 \end{matrix}} |τ^{−β}φ^{∗}(x, τ)| ≤ sup_{\begin{matrix}x∈ℝ, \\ τ_{1}>e^{σ} \end{matrix}} |τ_{1}^{−β}e^{σβ}φ(x, τ_{1})| ≤ e^{σβ}∥φ∥_{C^{0}_{α,β}(\overline{Π}_{+})}.

II. In the case σ < 0 we have

sup_{\begin{matrix}x∈ℝ, \\ 0<τ≤1 \end{matrix}} |τ^{−α}φ^{∗}(x, τ)| ≤ sup_{\begin{matrix}x∈ℝ, \\ 0<τ_{1}≤e^{σ} \end{matrix}} |τ_{1}^{−α}e^{σα}φ(x, τ_{1})| ≤ e^{σα}∥φ∥_{C^{0}_{α,β}(\overline{Π}_{+})},

sup_{\begin{matrix}x∈ℝ, \\ τ>1 \end{matrix}} |τ^{−β}φ^{∗}(x, τ)| ≤ sup_{\begin{matrix}x∈ℝ, \\ τ_{1}>e^{σ} \end{matrix}} |τ_{1}^{−β}e^{σβ}φ(x, τ_{1})| ≤

≤ max \{ sup_{\begin{matrix}x∈ℝ, \\ e^{σ}<τ_{1}≤1 \end{matrix}} |τ_{1}^{−α}τ_{1}^{α−β}e^{σβ}φ(x, τ_{1})|, sup_{\begin{matrix}x∈ℝ, \\ τ_{1}>1 \end{matrix}} |τ_{1}^{−β}e^{σβ}φ(x, τ_{1})|\} ≤

≤ max \{e^{σα} sup_{\begin{matrix}x∈ℝ, \\ e^{σ}<τ_{1}≤1 \end{matrix}} |τ_{1}^{−α}φ(x, τ_{1})|, e^{σβ} sup_{\begin{matrix}x∈ℝ, \\ τ_{1}>1 \end{matrix}} |τ_{1}^{−β}φ(x, τ_{1})|\} ≤

≤ e^{σα}∥φ∥_{C^{0}_{α,β}(\overline{Π}_{+})}.

Hence we obtain ∥φ^{∗}∥_{C^{0}_{α,β}(\overline{Π}_{+})} ≤ max\{e^{σα}, e^{σβ}\}∥φ∥_{C^{0}_{α,β}(\overline{Π}_{+})}, whence it

follows that the operator T̃_{i_{0}} is continuous. Indeed, by virtue of (3.18), (3.23) we have

∥T_{i_{0}}φ∥_{C^{0}_{α,β}(\overline{Π}_{+})} ≤ ∑_{q=0}^{∞} |γ_{i_{0}q}| ∥φ^{∗}∥_{C^{0}_{α,β}(\overline{Π}_{+})} ≤

≤ ( ∑_{σ_{q}<0} |γ_{i_{0}q}|e^{σ_{q}α} + ∑_{σ_{q}≥0} |γ_{i_{0}q}|e^{σ_{q}β})∥φ∥_{C^{0}_{α,β}(\overline{Π}_{+})} ≤ c_{6}∥φ∥_{C^{0}_{α,β}(\overline{Π}_{+})},

where c_{6} ≡ c_{4} + c_{5}.

As in Lemma 3.4 we prove that T T̃_{i_{0}} = T̃_{i_{0}}T = I, and thus T^{−1} = T̃_{i_{0}}. Analogously, we consider the case α ≥ β, which proves Lemma 3.6 com-

pletely.

Since problem (1.6), (1.7) in Conditions I, II in the class C^{0}_{α}^{l}(\overline{D}) (C^{0}_{α,β}^{l}(\overline{D})) has been reduced by equivalent transforms to equation (2.16) in the class

C^{0}_{α}(\overline{Π}_{+}) (C^{0}_{α,β}(\overline{Π}_{+})), from Lemmas 3.4 and 3.6 we have

Theorem 3.2. The boundary value problem (1.6), (1.7) is uniquely solv-

able in C^{0}_{α}^{l}(\overline{D}) for α > sup H and in C^{0}_{α,β}^{l}(\overline{D}) for I_{α,β} ∩ H = ∅.

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(Received 24.11.1994)

Author’s address: A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Alexidze St., Tbilisi 380093 Republic of Georgia