GEORGIAN MATHEMATICAL JOURNAL: Vol. 2, No. 5, 1995, 469-490

ON A DARBOUX PROBLEM FOR A THIRD ORDER HYPERBOLIC EQUATION WITH MULTIPLE CHARACTERISTICS

O. JOKHADZE

Abstract. A Darboux type problem for a model hyperbolic equation of the third order with multiple characteristics is considered in the

case of two independent variables. The Banach space C^{∘} _{α}^{2,1}, α ≥ 0, is introduced where the problem under consideration is investigated. The real number α_{0} is found such that for α > α_{0} the problem is solved uniquely and for α < α_{0} it is normally solvable in Hausdorff’s sense. In the class of uniqueness an estimate of the solution of the problem is obtained which ensures stability of the solution.

§ 1. Formulation of the Problem

In the plane of independent variables x, y let us consider a third order hyperbolic equation

u_{xxy} = f,  (1.1)

where f is the given and u the unknown real function.

Straight lines y = const form a double family of characteristics of (1.1), while x = const a single family.

Let γ_{i} : φ = φ_{i}, 0 ≤ r < +∞, i = 1, 2, 0 ≤ φ_{1} < φ_{2} ≤ \frac{π}{2} , be two rays coming out of the origin of the coordinates O(0, 0) written in terms of the polar coordinates r, φ. The angle formed by these rays will be denoted by D : φ_{1} < φ < φ_{2}, 0 < r < +∞. Let P_{1}^{0} and P_{2}^{0} be the points at which γ_{1} and γ_{2} intersect respectively the characteristics L_{1}(P^{0}) : x = x_{0} and L_{2}(P^{0}) : y = y_{0} coming out of an arbitrarily taken point P^{0}(x_{0}, y_{0}) ∈ D. Equation (1.1) will be considered in the rectangular domain D_{0} : 0 < x <

469

1072-947X/95/900-0469$7.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1995 Plenum Publishing Corporation

1991 Mathematics Subject Classification. 35L35.

Key words and phrases. Darboux problem, hyperbolic equation, characteristic line, Banach space, normal solvability, stability of solution, integral representation, regular solution.

x_{0}, 0 < y < y_{0}, bounded by the characteristics x = 0, x = x_{0} and y = 0, y = y_{0}.

Equations of segments OP_{1}^{0}, OP_{2}^{0} of the rays γ_{1}, γ_{2} will be written in terms of rectangular coordinates x, y as OP_{1}^{0} : y = ρ_{1}x, 0 ≤ x ≤ x_{0}; OP_{2}^{0} : x = ρ_{2}y, 0 ≤ y ≤ y_{0}, where ρ_{1} = tg φ_{1}, ρ_{2} = ctg φ_{2} and 0 ≤ ρ_{1}ρ_{2} = \frac{tg φ_{1}}{tg φ_{2}} < 1.

For equation (1.1) we shall consider a Darboux type problem formulated as follows: Find in \overline{D}_{0} a regular solution u : \overline{D}_{0} → R, R ≡ (−∞, +∞), of equation (1.1), satisfying on the segments OP_{1}^{0} and OP_{2}^{0} the conditions

(M_{1}u_{xx} + N_{1}u_{xy} + P_{1}u_{x} + Q_{1}u_{y} + S_{1}u)|_{OP_{1}^{0}} = f_{1},  (1.2)
(M_{2}u_{xx} + N_{2}u_{xy} + P_{2}u_{x} + Q_{2}u_{y} + S_{2}u)|_{OP_{2}^{0}} = f_{2},  (1.3)
(M_{3}u_{xx} + N_{3}u_{xy} + P_{3}u_{x} + Q_{3}u_{y} + S_{3}u)|_{OP_{2}^{0}} = f_{3},  (1.4)

where M_{i}, N_{i}, P_{i}, Q_{i}, S_{i}, f_{i}, i = 1, 2, 3, are given continuous real functions.

A function u : \overline{D}_{0} → R continuous in \overline{D}_{0} together with its partial derivatives D_{x}^{i}D_{y}^{j}u, i = 0, 1, 2, j = 0, 1, i + j > 0, D_{x} ≡ \frac{∂}{∂x}, D_{y} ≡ \frac{∂}{∂y} and satisfying equation (1.1) in \overline{D}_{0} and conditions (1.2)–(1.4) is called a regular solution of problem (1.1)– (1.4).

It should be noted that the boundary value problem (1.1)–(1.4) is the natural development of the well-known classical formulations of Goursat and Darboux problems (see, for example, [1]–[4]) for second-order linear hyperbolic equations. Variants of Goursat and Darboux problems for one hyperbolic equation and systems of second order, and also for systems of first order, are investigated in some papers (see, for example, [5]–[13]). Note that the results obtained in [7]–[9] are new even for one equation and in a certain sense bear a complete character.

Initial-boundary and characteristic problems for a wide class of hyperbolic equations of third and higher order with dominating derivatives are treated in [14]–[19] and other papers.

Remark 1.1. Conditions (1.2)–(1.4) take into account the hyperbolic nature of problem (1.1)–(1.4) as they contain only the derivatives dominated by the derivative D_{x}^{2}D_{y}u.

Remark 1.2. Since the family of characteristics y = const is the double one for the hyperbolic equation (1.1), two conditions (1.3), (1.4) are given on the segment OP_{2}^{0}.

Remark 1.3. The above problem could be formulated also for an angular domain bounded by the rays γ_{1}, γ_{2} and the characteristics L_{1}(P^{0}), L_{2}(P^{0}) of (1.1) under the same boundary conditions (1.2)–(1.4), but, as is well known, the solution u(x, y) of the thus formulated problem continues into the domain D_{0} as the solution of the original problem (1.1)–(1.4). The problem formulated in the form of (1.1)–(1.4) is convenient for further investigations provided that equation (1.1) contains dominated lowest terms and the Riemann function is effectively used.

Let us introduce the functional spaces

C^{∘}_{α} (\overline{D}_{0}) ≡ \{u : u ∈ C(\overline{D}_{0}), u(0) = 0, sup_{z≠0,z∈\overline{D}_{0}} |z|^{−α}|u(z)| < ∞\},

α ≥ 0, z = x + iy,

C^{∘}_{α} [0, d] ≡ {φ : φ ∈ C[0, d], φ(0) = 0, sup_{0<t≤d} t^{−α}|φ(t)| < ∞},

α ≥ 0, d > 0.

For α = 0 the above classes will be denoted by C^{∘} (\overline{D}_{0}) and C^{∘} [0, d], respectively.

Obviously, C^{∘}_{α} (\overline{D}_{0}) and C^{∘}_{α} [0, d] will be the Banach spaces with respect to the norms

∥u∥_{C^{∘}_{α}(\overline{D}_{0})} = sup_{z≠0,z∈\overline{D}_{0}} |z|^{−α}|u(z)|, ∥φ∥_{C^{∘}_{α}[0,d]} = sup_{0<t≤d} t^{−α}|φ(t)|,

respectively.

It is easy to see that the belonging of the functions u ∈C^{∘} (\overline{D}_{0}) and

φ ∈C^{∘} [0, d] to the spaces C^{∘}_{α} (\overline{D}_{0}) and C^{∘}_{α} [0, d], respectively, is equivalent to the fulfillment of the following inequalities:

|u(z)| ≤ c_{1}|z|^{α}, z ∈ \overline{D}_{0},  (1.5)
|φ(t)| ≤ c_{2}t^{α}, t ∈ [0, d], c_{i} ≡ const > 0, i = 1, 2.  (1.6)

The boundary value problem (1.1)–(1.4) will be investigated in the space

C^{∘} _{α}^{2,1}(\overline{D}_{0}) ≡ \{u : D_{x}^{i}D_{y}^{j}u ∈C^{∘}_{α} (\overline{D}_{0}), i = 0, 1, 2, j = 0, 1\},

which is Banach with respect to the norm

∥u∥_{C^{∘}_{α}^{2,1}(\overline{D}_{0})} = ∑_{i=0}^{2} ∑_{j=0}^{1} ∥D_{x}^{i}D_{y}^{j}u∥_{C^{∘}_{α}(\overline{D}_{0})}.

To consider (1.1)–(1.4) in the class C^{∘} _{α}^{2,1}(\overline{D}_{0}) it is required that

f ∈C^{∘}_{α} (\overline{D}_{0}), M_{1}, N_{1}, P_{1}, Q_{1}, S_{1} ∈ C[0, x_{0}], M_{i}, N_{i}, P_{i}, Q_{i}, S_{i} ∈ C[0, y_{0}],

i = 2, 3, f_{1} ∈C^{∘}_{α} [0, x_{0}], f_{i} ∈C^{∘}_{α} [0, y_{0}], i = 2, 3.

§ 2. Integral Representation of a Regular Solution of the Class C^{∘} _{α}^{2,1}(\overline{D}_{0}) of Equation (1.1)

The integration of equation (1.1) along the characteristics enables us to prove that the following lemma is valid.

Lemma 2.1. The formula

u(x, y) = ∫_{0}^{x}(x − ξ)φ(ξ)dξ + ∫_{0}^{y} ψ(η)dη + x ∫_{0}^{y} ν(η)dη +

+ ∫_{0}^{x} ∫_{0}^{y}(x − ξ)f(ξ, η)dξdη, (x, y) ∈ \overline{D}_{0},  (2.1)

establishes the one-to-one correspondence between regular solutions u(x, y)

of the class C^{∘} _{α}^{2,1}(\overline{D}_{0}) of equation (1.1) and values φ ∈C^{∘}_{α} [0, x_{0}], ψ, ν ∈

C^{∘}_{α} [0, y_{0}]. Note that φ(x) = u_{xx}(x, 0), 0 ≤ x ≤ x_{0}, ψ(y) = u_{y}(0, y), ν(y) = u_{xy}(0, y), 0 ≤ y ≤ y_{0}.

Proof. Introduce the notation u(x, 0) ≡ φ_{1}(x), 0 ≤ x ≤ x_{0}, u(0, y) ≡ ψ_{1}(y), u_{x}(0, y) ≡ ν_{1}(y), 0 ≤ y ≤ y_{0}. It is obvious that φ_{1}(0) = φ′_{1}(0) = ψ_{1}(0) = ν_{1}(0) = 0. Further, by integrating (1.1) twice and one time along the characteristics y = const and x = const, respectively, we obtain

u(x, y) = φ_{1}(x) + ψ_{1}(y) + xν_{1}(y) +
+ ∫_{0}^{x} ∫_{0}^{y}(x − ξ)f(ξ, η)dξdη, (x, y) ∈ \overline{D}_{0}.  (2.2)

Let φ′′_{1}(x) ≡ φ(x), 0 ≤ x ≤ x_{0}, ψ′_{1}(y) ≡ ψ(y), ν′_{1}(y) ≡ ν(y), 0 ≤ y ≤ y_{0}. Then (2.2) takes the form of (2.1).

From u ∈C^{∘} _{α}^{2,1}(\overline{D}_{0}) it follows that φ ∈C^{∘}_{α} [0, x_{0}], ψ, ν ∈C^{∘}_{α} [0, y_{0}]. To prove the converse statement note that by (1.5), (1.6) we have the estimates

|φ(x)| ≤ c_{3}x^{α}, x ∈ [0, x_{0}], |ψ(y)| ≤ c_{4}y^{α}, |ν(y)| ≤ c_{5}y^{α}, y ∈ [0, y_{0}],
|f(x, y)| ≤ c_{6}|z|^{α}, z ≡ (x, y) ∈ \overline{D}_{0}, c_{2+i} ≡ const > 0, i = 1, 2, 3.

ON A DARBOUX PROBLEM 473

Setting ζ = (ξ, η), from formula (2.1) we have

|u(x, y)| ≤ c_{3}x_{0} ∫_{0}^{x} ξ^{α}dξ + c_{4} ∫_{0}^{y} η^{α}dη + c_{5}x_{0} ∫_{0}^{y} η^{α}dη +

+c_{6}x_{0} ∫_{0}^{x} ∫_{0}^{y} |ζ|^{α}dξdη ≤ c_{7}|z|^{α},

where c_{7} ≡ \frac{1}{α+1}(c_{3}x_{0}^{2} + c_{4}y_{0} + c_{5}x_{0}y_{0}) + c_{6}x_{0}^{2}y_{0} > 0, z ∈ \overline{D}_{0}.

Hence u ∈C^{∘}_{α} (\overline{D}_{0}). In a similar but relatively simpler manner it is proved that D_{x}^{i}D_{y}^{j}u ∈C^{∘}_{α} (\overline{D}_{0}), i = 0, 1, 2, j = 0, 1, i + j > 0, and therefore u ∈

C^{∘} _{α}^{2,1}(\overline{D}_{0}).

§ 3. Reducing Problem (1.1)–(1.4) to a System of Integro-Functional Equations

Substituting (2.1) into conditions (1.2)–(1.4) gives us

\begin{cases}M_{1}(x)φ(x) + ∫_{0}^{x}[P_{1}(x) + S_{1}(x)(x − ξ)]φ(ξ)dξ + Q_{1}(x)ψ(ρ_{1}x)+ \\ +S_{1}(x) ∫_{0}^{ρ_{1}x} ψ(η)dη + [N_{1}(x) + xQ_{1}(x)]ν(ρ_{1}x) + [P_{1}(x)+ \\ +xS_{1}(x)] ∫_{0}^{ρ_{1}x} ν(η)dη = f̃_{1}(x), 0 ≤ x ≤ x_{0}, \\ Q_{i}(y)ψ(y) + S_{i}(y) ∫_{0}^{y} ψ(η)dη + [N_{i}(y) + ρ_{2}yQ_{i}(y)]ν(y)+ \\ +[P_{i}(y) + ρ_{2}yS_{i}(y)] ∫_{0}^{y} ν(η)dη + M_{i}(y)φ(ρ_{2}y) + ∫_{0}^{ρ_{2}y}[P_{i}(y)+ \\ +S_{i}(y)(ρ_{2}y − ξ)]φ(ξ)dξ = f̃_{i}(y), i = 2, 3, 0 ≤ y ≤ y_{0}, \end{cases} (3.1)

where

\begin{cases}f̃_{1}(x) ≡ f_{1}(x) − M_{1}(x) ∫_{0}^{ρ_{1}x} f(x, η)dη − N_{1}(x) ∫_{0}^{x} f(ξ, ρ_{1}x)dξ− \\ −P_{1}(x) ∫_{0}^{x} ∫_{0}^{ρ_{1}x} f(ξ, η)dξ dη − Q_{1}(x) ∫_{0}^{x}(x − ξ)f(ξ, ρ_{1}x)dξ− \\ −S_{1}(x) ∫_{0}^{x} ∫_{0}^{ρ_{1}x}(x − ξ)f(ξ, η)dξ dη, 0 ≤ x ≤ x_{0}, \\ f̃_{i}(y) ≡ f_{i}(y) − M_{i}(y) ∫_{0}^{y} f(ρ_{2}y, η)dη − N_{i}(y) ∫_{0}^{ρ_{2}y} f(ξ, y)dξ− \\ −P_{i}(y) ∫_{0}^{ρ_{2}y} ∫_{0}^{y} f(ξ, η)dξ dη − Q_{i}(y) ∫_{0}^{ρ_{2}y}(ρ_{2}y − ξ)f(ξ, y)dξ− \\ −S_{i}(y) ∫_{0}^{ρ_{2}y} ∫_{0}^{y}(ρ_{2}y − ξ)f(ξ, η)dξ dη, i = 2, 3, 0 ≤ y ≤ y_{0}. \end{cases} (3.2)

Transferring all the integral terms contained in system (3.1) to the righthand side, we have

M_{1}(x)φ(x) + Q_{1}(x)ψ(ρ_{1}x) + R_{1}(x)ν(ρ_{1}x) = F_{1}(x), 0 ≤ x ≤ x_{0}, (3.3)
Q_{2}(y)ψ(y) + R_{2}(y)ν(y) + M_{2}(y)φ(ρ_{2}y) = F_{2}(y), 0 ≤ y ≤ y_{0}, (3.4)
Q_{3}(y)ψ(y) + R_{3}(y)ν(y) + M_{3}(y)φ(ρ_{2}y) = F_{3}(y), 0 ≤ y ≤ y_{0}, (3.5)

where

F_{1}(x) ≡ f̃_{1}(x) − ∫_{0}^{x}[P_{1}(x) + S_{1}(x)(x − ξ)]φ(ξ)dξ − S_{1}(x) ∫_{0}^{ρ_{1}x}ψ(η)dη −

−[P_{1}(x) + xS_{1}(x)] ∫_{0}^{ρ_{1}x}ν(η)dη, 0 ≤ x ≤ x_{0},

F_{i}(y) ≡ f̃_{i}(y) − S_{i}(y) ∫_{0}^{y} ψ(η)dη − [P_{i}(y) + ρ_{2}yS_{i}(y)] ∫_{0}^{y} ν(η)dη −

− ∫_{0}^{ρ_{2}y}[P_{i}(y) + S_{i}(y)(ρ_{2}y − ξ)]φ(ξ)dξ, i = 2, 3, 0 ≤ y ≤ y_{0},

R_{1}(x) ≡ N_{1}(x) + xQ_{1}(x), 0 ≤ x ≤ x_{0}, R_{i}(y) ≡ N_{i}(y) + ρ_{2}yQ_{i}(y), i = 2, 3, 0 ≤ y ≤ y_{0}.

Rewrite equations (3.4) and (3.5) as follows:

Q_{i}(y)ψ(y) + R_{i}(y)ν(y) = F_{i}(y) − M_{i}(y)φ(ρ_{2}y),

i = 2, 3, 0 ≤ y ≤ y_{0}.  (3.6)

Assuming that

Δ(y) ≡ | \begin{matrix}Q_{2}(y) & N_{2}(y) \\ Q_{3}(y) & N_{3}(y) \end{matrix} | ≠ 0, 0 ≤ y ≤ y_{0},  (3.7)

we find from system (3.6) that

ψ(y) = a_{1}(y) − b_{1}(y)φ(ρ_{2}y),

ν(y) = a_{2}(y) − b_{2}(y)φ(ρ_{2}y), 0 ≤ y ≤ y_{0},  (3.8)

where a_{1}(y) ≡ Δ^{−1}(y)[F_{2}(y)R_{3}(y) − F_{3}(y)R_{2}(y)], b_{1}(y) ≡ Δ^{−1}(y)[M_{2}(y)R_{3}(y) − M_{3}(y)R_{2}(y)], a_{2}(y) ≡ Δ^{−1}(y)[F_{3}(y)Q_{2}(y) − F_{2}(y)Q_{3}(y)], b_{2}(y) ≡ Δ^{−1}(y)[M_{3}(y)Q_{2}(y) − M_{2}(y)Q_{3}(y)], 0 ≤ y ≤ y_{0}.

Note that here and in what follows the upper index −1 means the inverse value.

Let

M_{1}(x) ≠ 0, 0 ≤ x ≤ x_{0}.  (3.9)

If the obtained expressions for the functions ψ(y), ν(y), 0 ≤ y ≤ y_{0}, are substituted from (3.8) into equality (3.3), then we shall have

φ(x) − a(x)φ(τ_{0}x) = F (x), 0 ≤ x ≤ x_{0},  (3.10)

where a(x) ≡ M_{1}^{−1}(x)[Q_{1}(x)b_{1}(ρ_{1}x)+R_{1}(x)b_{2}(ρ_{1}x)], F (x) ≡ M_{1}^{−1}(x)[F_{1}(x) − Q_{1}(x)a_{1}(ρ_{1}x) − R_{1}(x)a_{2}(ρ_{1}x)], 0 ≤ x ≤ x_{0}, τ_{0} ≡ ρ_{1}ρ_{2}.

Simple calculations lead to

\begin{cases}F (x) = ∫_{0}^{x} K_{1}(x, ξ)φ(ξ)dξ + ∫_{0}^{τ_{0}x} K_{2}(x, ξ)φ(ξ)dξ+ \\ +K_{3}(x) ∫_{0}^{ρ_{1}x} ψ(η)dη + K_{4}(x) ∫_{0}^{ρ_{1}x} ν(η)dη + F_{4}(x), 0 ≤ x ≤ x_{0}, \\ a_{1}(y) = ∫_{0}^{ρ_{2}y} K_{5}(ξ, y)φ(ξ)dξ + K_{6}(y) ∫_{0}^{y} ψ(η)dη+ \\ +K_{7}(y) ∫_{0}^{y} ν(η)dη + F_{5}(y), 0 ≤ y ≤ y_{0}, \\ a_{2}(y) = ∫_{0}^{ρ_{2}y} K_{8}(ξ, y)φ(ξ)dξ + K_{9}(y) ∫_{0}^{y} ψ(η)dη+ \\ +K_{10}(y) ∫_{0}^{y} ν(η)dη + F_{6}(y), 0 ≤ y ≤ y_{0}, \end{cases} (3.11)

where K_{1}(x, ξ), 0 ≤ x ≤ x_{0}, 0 ≤ ξ ≤ x, K_{2}(x, ξ), 0 ≤ x ≤ x_{0}, 0 ≤ ξ ≤ τ_{0}x, K_{i}(x), 0 ≤ x ≤ x_{0}, i = 3, 4, K_{i}(ξ, y), 0 ≤ ξ ≤ ρ_{2}y, 0 ≤ y ≤ y_{0}, i = 5, 8, K_{i}(y), 0 ≤ y ≤ y_{0}, i = 6, 7, 9, 10, expressed in terms of the M_{1}, N_{i}, P_{i}, Q_{i}, S_{i}, ρ_{1}, ρ_{2}, i = 1, 2, 3, are continuous kernels of the integral terms contained on the right-hand sides of system (3.11), while the functions F_{i}, i = 4, 5, 6, denote the following values:

\begin{cases}F_{4}(x) ≡ M_{1}^{−1}(x)f̃_{1}(x) + [M_{1}^{−1}(x)R_{1}(x)Δ^{−1}(ρ_{1}x)Q_{3}(ρ_{1}x)− \\ −M_{1}^{−1}(x)Q_{1}(x)Δ^{−1}(ρ_{1}x)R_{3}(ρ_{1}x)]f̃_{2}(ρ_{1}x)+ \\ +[M_{1}^{−1}(x)Q_{1}(x)Δ^{−1}(ρ_{1}x)R_{2}(ρ_{1}x)− \\ −M_{1}^{−1}(x)R_{1}(x)Δ^{−1}(ρ_{1}x)Q_{2}(ρ_{1}x)]f̃_{3}(ρ_{1}x), 0 ≤ x ≤ x_{0}, \\ F_{5}(y) ≡ Δ^{−1}(y)R_{3}(y)f̃_{2}(y) − Δ^{−1}(y)R_{2}(y)f̃_{3}(y), 0 ≤ y ≤ y_{0}, \\ F_{6}(y) ≡ Δ^{−1}(y)Q_{2}(y)f̃_{3}(y) − Δ^{−1}(y)Q_{3}(y)f̃_{2}(y), 0 ≤ y ≤ y_{0}. \end{cases} (3.12)

Introducing the notation

(Kφ)(x) ≡ φ(x) − a(x)φ(τ_{0}x), 0 ≤ x ≤ x_{0},  (3.13)

equalities (3.8), (3.10) due to (3.11) we can write as

\begin{cases}(Kφ)(x) = ∫_{0}^{x} K_{1}(x, ξ)φ(ξ)dξ + ∫_{0}^{τ_{0}x} K_{2}(x, ξ)φ(ξ)dξ+ \\ +K_{3}(x) ∫_{0}^{ρ_{1}x} ψ(η)dη + K_{4}(x) ∫_{0}^{ρ_{1}x} ν(η)dη + F_{4}(x), 0 ≤ x ≤ x_{0}, \\ ψ(y) = ∫_{0}^{ρ_{2}y} K_{5}(ξ, y)φ(ξ)dξ + K_{6}(y) ∫_{0}^{y} ψ(η)dη+ \\ +K_{7}(y) ∫_{0}^{y} ν(η)dη − b_{1}(y)φ(ρ_{2}y) + F_{5}(y), 0 ≤ y ≤ y_{0}, \\ ν(y) = ∫_{0}^{ρ_{2}y} K_{8}(ξ, y)φ(ξ)dξ + K_{9}(y) ∫_{0}^{y} ψ(η)dη+ \\ +K_{10}(y) ∫_{0}^{y} ν(η)dη − b_{2}(y)φ(ρ_{2}y) + F_{6}(y), 0 ≤ y ≤ y_{0}. \end{cases} (3.14)

Remark 3.1. It is obvious that if conditions (3.7), (3.9) are fulfilled, then

in the class C^{∘} _{α}^{2,1}(\overline{D}_{0}) problem (1.1)–(1.4) is equivalent to the system

of equations (3.14) with respect to the unknowns φ ∈C^{∘}_{α} [0, x_{0}], ψ, ν ∈

C^{∘}_{α} [0, y_{0}].

§ 4. Invertibility of the Functional Operator K Defined by Equality (3.13)

Assume that conditions (3.7), (3.9) are fulfilled. Set σ ≡ a(0) and α_{0} ≡ − log |σ|/ log τ_{0} (σ ≠ 0).

Lemma 4.1. Let either γ_{1} or γ_{2} be the characteristic of equation (1.1) (i.e., τ_{0} = 0).Then the equation

(Kφ)(x) = g(x), 0 ≤ x ≤ x_{0},  (4.1)

has a unique solution in the space C^{∘}_{α} [0, x_{0}] for all α ≥ 0.

The proof follows from the fact that under the assumption of the lemma K is the identity operator in the space C^{∘}_{α} [0, x_{0}].

Lemma 4.2. Let the straight lines γ_{1}, γ_{2} be not the characteristics of equation (1.1) (i.e., 0 < τ_{0} < 1) and σ ≠ 0. Then for α > α_{0} equation (4.1)

has a unique solution in the space C^{∘}_{α} [0, x_{0}] and for the inverse operator K^{−1} we have the estimate

|(K^{−1}g)(x)| ≤ cx^{α}∥g∥_{C^{∘}_{α}[0,x]},  (4.2)

where the positive constant c does not depend on the function g.

Proof. We introduce into consideration the operators

(Γφ)(x) = a(x)φ(τ_{0}x), 0 ≤ x ≤ x_{0}, K^{−1} = I + ∑_{j=1}^{∞} Γ^{j},  (4.3)

where I is the identity operator. It is easy to see that the operator K^{−1} is formally inverse to the operator K. Thus it enough for us to prove that the Neuman series I + ∑_{j=1}^{∞} Γ^{j} converges in the space C^{∘}_{α} [0, x_{0}].

By the definition of the operator Γ from (4.3) we have (Γ^{j}φ)(x) = a(x)a(τ_{0}x) … a(τ_{0}^{j−1}x)φ(τ_{0}^{j}x), 0 ≤ x ≤ x_{0}. The condition α > α_{0} is equivalent to the inequality τ_{0}^{α}|σ| < 1. Therefore by virtue of the continuity of the function a and the equality a(0) = σ there are positive numbers ε (ε < x_{0}), δ and q such that the inequalities

|a(x)| ≤ |σ| + δ, τ_{0}^{α}(|σ| + δ) ≡ q < 1  (4.4)

will hold for 0 ≤ x ≤ ε.

It is obvious that the sequence {τ_{0}^{j}x}_{j=0}^{∞} uniformly converges to zero as j → ∞ on the segment [0, x_{0}]. Therefore there is a natural number j_{0} such that

τ_{0}^{j}x ≤ ε, for 0 ≤ x ≤ x_{0}, j ≥ j_{0}.  (4.5)

We can take as j_{0}, say, j_{0} = [\frac{log εx_{0}^{−1}}{log τ_{0}} ] + 1, where [p] denotes the integral part of the number p.

Let max_{0≤x≤x_{0}} |a(x)| ≡ β. By virtue of (4.4), (4.5) the following estimates

hold for j > j_{0}, g ∈C^{∘}_{α} [0, x_{0}]:

\begin{matrix}|(Γ^{j}g)(x)| = |a(x)a(τ_{0}x) … a(τ_{0}^{j_{0}−1}x)| · |a(τ_{0}^{j_{0}}x) … a(τ_{0}^{j−1}x)| · |g(τ_{0}^{j}x)| ≤ \\ ≤ β^{j_{0}}(|σ| + δ)^{j−j_{0}}(τ_{0}^{j}x)^{α}∥g∥_{C^{∘}_{α}[0,x]} ≤ \\ ≤ β^{j_{0}}(|σ| + δ)^{−j_{0}}(τ_{0}^{α}(|σ| + δ))^{j}x^{α}∥g∥_{C^{∘}_{α}[0,x]} = c_{0}q^{j}x^{α}∥g∥_{C^{∘}_{α}[0,x]}, \end{matrix} (4.6)

where c_{0} ≡ β^{j_{0}}(|σ| + δ)^{−j_{0}}.

For 1 ≤ j ≤ j_{0} we have

|(Γ^{j}g)(x)| ≤ β^{j}(τ_{0}^{j}x)^{α}∥g∥_{C^{∘}_{α}[0,x]} ≤ β^{j}x^{α}∥g∥_{C^{∘}_{α}[0,x]}.  (4.7)

Now by (4.6) and (4.7) we eventually have

|φ(x)| = |(K^{−1}g)(x)| ≤ |g(x)| + | ∑_{j=1}^{j_{0}}(Γ^{j}g)(x)| + | ∑_{j=j_{0}+1}^{∞}(Γ^{j}g)(x)| ≤

≤ (1 + ∑_{j=1}^{j_{0}} β^{j} + c_{0} ∑_{j=j_{0}+1}^{∞} q^{j})x^{α}∥g∥_{C^{∘}_{α}[0,x]} =

= (1 + ∑_{j=1}^{j_{0}} β^{j} + c_{0} \frac{q^{j_{0}+1}}{1 − q} )x^{α}∥g∥_{C^{∘}_{α}[0,x]},

from which we obtain the continuity of the operator K^{−1} in the space C^{∘}_{α} [0, x_{0}] and the validity of estimate (4.2).

Remark 4.1. If σ = 0, then the inequality τ_{0}^{α}|σ| < 1 is fulfilled for any α ≥ 0 and, as seen from the proof, in that case Lemma 4.2 holds for all α ≥ 0.

Lemma 4.3. Let the straight lines γ_{1}, γ_{2} be not the characteristics of equation (1.1) (i.e., 0 < τ_{0} < 1) and σ ≠ 0. Then equation (4.1) is solvable

in the space C^{∘}_{α} [0, x_{0}] for α < α_{0} and the homogeneous equation corresponding to (4.1) has in the said space an infinite number of linearly independent solutions, i.e., dim Ker K = ∞.

Proof. The condition α < α_{0} is equivalent to the inequality τ_{0}^{α}|σ| > 1. Therefore, as in proving Lemma 4.2, there are positive numbers ε_{1} (ε_{1} < x_{0}), δ_{1} and q_{1} such that the inequalities

|a^{−1}(x)| ≤ (|σ| − δ_{1})^{−1}, |σ| − δ_{1} > 0, τ_{0}^{α}(|σ| − δ_{1}) ≡ q_{1}^{−1} > 1 (4.8)

will hold for 0 ≤ x ≤ ε_{1}.

It is easy to see that the operator Γ from (4.3) is invertible and

(Γ^{−1}φ)(x) = a^{−1}(τ_{0}^{−1}x)φ(τ_{0}^{−1}x), 0 ≤ x ≤ τ_{0}ε_{1}.

Rewrite (4.3) in the equivalent form

φ(x) − (Γ^{−1}φ)(x) = −(Γ^{−1}g)(x), 0 ≤ x ≤ τ_{0}ε_{1}.  (4.9)

Obviously, for any x from the interval 0 < x < τ_{0}ε_{1} there exists a unique natural number n_{1} = n_{1}(x) satisfying the inequalities

τ_{0}ε_{1} < τ_{0}^{−n_{1}}x ≤ ε_{1}.

It is easy to verify that

n_{1}(x) = [\frac{log ε_{1}^{−1}x}{log τ_{0}}] ≥ \frac{log ε_{1}^{−1}x}{log τ_{0}} − 1.  (4.10)

Similarly, for ε_{1} < x ≤ x_{0} there exists a unique natural number n_{2} = n_{2}(x) satisfying the inequalities

τ_{0}ε_{1} ≤ τ_{0}^{n_{2}}x < ε_{1}.

Clearly, n_{2}(x) = [1 − \frac{log ε_{1}^{−1}x}{log τ_{0}}].

One can easily verify that any continuous solution on the half-interval 0 < x ≤ x_{0} of equation (4.1) or (4.9) is given by the formula

φ(x) = \begin{cases}φ^{0}(x), τ_{0}ε_{1} ≤ x ≤ ε_{1}, \\ (Γ^{−n_{1}(x)}φ^{0})(x) − ∑_{j=1}^{n_{1}(x)}(Γ^{−j}g)(x), 0 < x < τ_{0}ε_{1}, \\ (Γ^{n_{2}(x)}φ^{0})(x) − ∑_{j=0}^{n_{2}(x)−1}(Γ^{j}g)(x), x > ε_{1}, \end{cases} (4.11)

where φ^{0} is an arbitrary function from the class C[τ_{0}ε_{1}, ε_{1}], satisfying the condition φ^{0}(ε_{1}) − a(ε_{1})φ^{0}(τ_{0}ε_{1}) = g(ε_{1}).

Let us show that the function φ given by (4.11) belongs to the class

C^{∘}_{α} [0, x_{0}] for g ∈C^{∘}_{α} [0, x_{0}]. The arbitrariness of φ^{0} implies that Lemma 4.3 holds for equation (4.1).

By (4.8), (4.10) the estimates

|(Γ^{−n_{1}(x)}φ(x)| = |a^{−1}(τ_{0}^{−1}x)a^{−1}(τ_{0}^{−2}x) … a^{−1}(τ_{0}^{−n_{1}(x)}x)φ(τ_{0}^{−n_{1}(x)}x)| ≤

≤ (|σ| − δ_{1})^{−n_{1}(x)}∥φ^{0}∥_{C[τ_{0}ε_{1},ε_{1}]} ≤ τ_{0}^{αn_{1}(x)}∥φ^{0}∥_{C[τ_{0}ε_{1},ε_{1}]} ≤
≤ τ_{0}^{(\frac{log ε{−1 \atop 1}x}{log τ_{0}} −1)α}∥φ^{0}∥_{C[τ_{0}ε_{1},ε_{1}]} = τ_{0}^{−α}ε_{1}^{−α}x^{α}∥φ^{0}∥_{C[τ_{0}ε_{1},ε_{1}]} (4.12)

hold for 0 < x < τ_{0}ε_{1}.

In a similar manner for 0 < x < τ_{0}ε_{1} and 1 ≤ j ≤ n_{1}(x) we have

|(Γ^{−j}g)(x)| ≤ (|σ| − δ_{1})^{−j}(τ_{0}^{−j}x)^{α}∥g∥_{C^{∘}_{α}[0,x_{0}]} =

= [τ_{0}^{α}(|σ| − δ_{1})]^{−j}x^{α}∥g∥_{C^{∘}_{α}[0,x_{0}]} = q_{1}^{j}x^{α}∥g∥_{C^{∘}_{α}[0,x_{0}]}.

Hence it follows that

| ∑_{j=1}^{n_{1}(x)}(Γ^{−j}g)(x)| ≤ ( ∑_{j=1}^{n_{1}(x)} q_{1}^{j})x^{α}∥g∥_{C^{∘}_{α}[0,x_{0}]} ≤

≤ \frac{q_{1}}{1 − q_{1}} x^{α}∥g∥_{C^{∘}_{α}[0,x_{0}]}.  (4.13)

By (4.12) and (4.13) we conclude that the function φ given by (4.11) and

being a solution of equation (4.1) belongs to the class C^{∘}_{α} [0, x_{0}].

§ 5. Proof of the Main Results

Theorem 5.1. If at least either γ_{1} or γ_{2} is the characteristic of equation (1.1) (i.e., τ_{0} = 0) and conditions (3.7), (3.9) are fulfilled, then problem

(1.1)–(1.4) has a unique solution in the class C^{∘} _{α}^{2,1}(\overline{D}_{0}) for all α ≥ 0.

Theorem 5.2. Let conditions (3.7), (3.9) be fulfilled and the straight lines γ_{1}, γ_{2} be not the characteristics of equation (1.1) (i.e., 0 < τ_{0} < 1). If the equality σ = 0 holds, then problem (1.1)–(1.4) is uniquely solvable in the

class C^{∘} _{α}^{2,1}(\overline{D}_{0}) for all α ≥ 0. If however σ ≠ 0, then problem (1.1)–(1.4)

is uniquely solvable in the class C^{∘} _{α}^{2,1}(\overline{D}_{0}) for α > α_{0}, while for α < α_{0} problem (1.1)–(1.4) is normally solvable in Hausdorff’s sense in the class

C^{∘} _{α}^{2,1}(\overline{D}_{0}) and its index κ = +∞. In particular, the homogeneous problem corresponding to (1.1)–(1.4) has an infinite number of linearly independent solutions.

Proof. Rewrite system (3.14) in terms of the new unknown functions

ψ(y) + b_{1}(y)φ(ρ_{2}y) ≡ ω(y), ν(y) + b_{2}(y)φ(ρ_{2}y) ≡ λ(y), 0 ≤ y ≤ y_{0},

as

\begin{cases}(Kφ)(x) = ∫_{0}^{x} K_{11}(x, ξ)φ(ξ)dξ + K_{3}(x) ∫_{0}^{ρ_{1}x} ω(η)dη+ \\ +K_{4}(x) ∫_{0}^{ρ_{1}x} λ(η)dη + F_{4}(x), 0 ≤ x ≤ x_{0}, \\ ω(y) = ∫_{0}^{ρ_{2}y} K_{12}(ξ, y)φ(ξ)dξ + K_{6}(y) ∫_{0}^{y} ω(η)dη+ \\ +K_{7}(y) ∫_{0}^{y} λ(η)dη + F_{5}(y), 0 ≤ y ≤ y_{0}, \\ λ(y) = ∫_{0}^{ρ_{2}y} K_{13}(ξ, y)φ(ξ)dξ + K_{9}(y) ∫_{0}^{y} ω(η)dη+ \\ +K_{10}(y) ∫_{0}^{y} λ(η)dη + F_{6}(y), 0 ≤ y ≤ y_{0}, \end{cases}  (5.1)

where

K_{11}(x, ξ) ≡ K_{1}(x, ξ) + K_{1}^{∗}(x, ξ),

K_{1}^{∗}(x, ξ)≡\begin{cases}K_{2}(x, ξ)+ρ_{2}^{−1}[K_{3}(x)b_{1}(ρ_{2}^{−1}ξ)+K_{4}(x)b_{2}(ρ_{2}^{−1}ξ)], 0≤ξ ≤τ_{0}x, \\ 0, τ_{0}x < ξ ≤ x, \end{cases}

K_{12}(ξ, y) ≡ K_{5}(ξ, y) + ρ_{2}^{−1}[K_{6}(y)b_{1}(ρ_{2}^{−1}ξ) + K_{7}(y)b_{2}(ρ_{2}^{−1}ξ)],

K_{13}(ξ, y) ≡ K_{8}(ξ, y) + ρ_{2}^{−1}[K_{9}(y)b_{1}(ρ_{2}^{−1}ξ) + K_{10}(y)b_{2}(ρ_{2}^{−1}ξ)].

Remark 5.1. If ρ_{2} = 0, then ψ(y) ≡ ω(y), ν(y) ≡ λ(y), 0 ≤ y ≤ y_{0}, and the introduction of the new unknown functions ω and λ is superfluous.

We can rewrite system (5.1) in terms of the new independent variables x = x_{0}t, y = y_{0}t, ξ = x_{0}τ, η = y_{0}τ, 0 ≤ t, τ ≤ 1, as

\begin{cases}(Kφ̃)(t) = ∫_{0}^{t} K̃_{11}(t, τ)φ̃(τ)dτ + K̃_{3}(t) ∫_{0}^{τ_{1}t} ω̃(τ)dτ+ \\ +K̃_{4}(t) ∫_{0}^{τ_{1}t} λ̃(τ)dτ + F̃_{4}(t), 0 ≤ t ≤ 1, 0 < \frac{ρ_{1}x_{0}}{y_{0}} ≡ τ_{1} < 1, \\ ω̃(t) = ∫_{0}^{τ_{2}t} K̃_{12}(τ, t)φ̃(τ)dτ + K̃_{6}(t) ∫_{0}^{t} ω̃(τ)dτ+ \\ +K̃_{7}(t) ∫_{0}^{t} λ̃(τ)dτ + F̃_{5}(t), 0 ≤ t ≤ 1, 0 < \frac{ρ_{2}y_{0}}{x_{0}} ≡ τ_{2} < 1, \\ λ̃(t) = ∫_{0}^{τ_{2}t} K̃_{13}(τ, t)φ̃(τ)dτ + K̃_{9}(t) ∫_{0}^{t} ω̃(τ)dτ+ \\ +K̃_{10}(t) ∫_{0}^{t} λ̃(τ)dτ + F̃_{6}(t), 0 ≤ t ≤ 1, \end{cases} (5.2)

where the functions with waves are expressions of the corresponding functions in terms of the variables t and τ, for example, φ(x) = φ(x_{0}t) ≡ φ̃(t), ω(y) = ω(y_{0}t) ≡ ω̃(t), K̃_{11}(t, τ) ≡ x_{0}K_{11}(x_{0}t, x_{0}τ), K̃_{12}(τ, t) ≡ x_{0}K_{12}(x_{0}τ, y_{0}t), K̃_{3}(t) ≡ y_{0}K_{3}(x_{0}t), K̃_{6}(t) ≡ y_{0}K_{6}(y_{0}t), 0 ≤ t, τ ≤ 1.

Let T_{i}(φ̃, ω̃, λ̃), i = 1, 2, 3, be the linear integral operators acting by the formulas

\begin{cases}T_{1}(φ̃, ω̃, λ̃)(t) ≡ ∫_{0}^{t} K̃_{11}(t, τ)φ̃(τ)dτ + K̃_{3}(t) ∫_{0}^{τ_{1}t} ω̃(τ)dτ+ \\ +K̃_{4}(t) ∫_{0}^{τ_{1}t} λ̃(τ)dτ, 0 ≤ t ≤ 1, 0 < τ_{1} < 1, \\ T_{2}(φ̃, ω̃, λ̃)(t) ≡ ∫_{0}^{τ_{2}t} K̃_{12}(τ, t)φ̃(τ)dτ + K̃_{6}(t) ∫_{0}^{t} ω̃(τ)dτ+ \\ +K̃_{7}(t) ∫_{0}^{t} λ̃(τ)dτ, 0 ≤ t ≤ 1, 0 < τ_{2} < 1, \\ T_{3}(φ̃, ω̃, λ̃)(t) ≡ ∫_{0}^{τ_{2}t} K̃_{13}(τ, t)φ̃(τ)dτ + K̃_{9}(t) ∫_{0}^{t} ω̃(τ)dτ+ \\ +K̃_{10}(t) ∫_{0}^{t} λ̃(τ)dτ, 0 ≤ t ≤ 1. \end{cases} (5.3)

Remark 5.2. The integral operators T_{i}, i = 1, 2, 3, acting by formulas (5.3) are Volterra type operators.

To prove Theorems 5.1 and 5.2 we shall solve system (5.2) for the un-

known functions φ̃ ∈C^{∘}_{α}, ω̃ ∈C^{∘}_{α}, λ̃ ∈C^{∘}_{α}, using the method of successive approximations.

Set φ̃_{0}(t) ≡ 0, ω̃_{0}(t) ≡ 0, λ̃_{0}(t) ≡ 0, 0 ≤ t ≤ 1, and for n ≥ 1,

(Kφ̃_{n})(t) = ∫_{0}^{t} K̃_{11}(t, τ)φ̃_{n−1}(τ)dτ + K̃_{3}(t) ∫_{0}^{τ_{1}t}ω̃_{n−1}(τ)dτ +

+K̃_{4}(t) ∫_{0}^{τ_{1}t}λ̃_{n−1}(τ)dτ + F̃_{4}(t), 0 ≤ t ≤ 1,  (5.4)

ω̃_{n}(t) = ∫_{0}^{τ_{2}t}K̃_{12}(τ, t)φ̃_{n−1}(τ)dτ + K̃_{6}(t) ∫_{0}^{t} ω̃_{n−1}(τ)dτ +

+ K̃_{7}(t) ∫_{0}^{t} λ̃_{n−1}(τ)dτ + F̃_{5}(t), 0 ≤ t ≤ 1,  (5.5)

λ̃_{n}(t) = ∫_{0}^{τ_{2}t}K̃_{13}(τ, t)φ̃_{n−1}(τ)dτ + K̃_{9}(t) ∫_{0}^{t} ω̃_{n−1}(τ)dτ +

+ K̃_{10}(t) ∫_{0}^{t} λ̃_{n−1}(τ)dτ + F̃_{6}(t), 0 ≤ t ≤ 1,  (5.6)

where the operator K acts by (3.13).

Using estimate (4.2) and taking into account Remark 5.2, we shall prove below that under the assumptions of Lemma 4.1 or Lemma 4.2 we have the estimates

|φ̃_{n+1}(t) − φ̃_{n}(t)| ≤ M \frac{L^{n}}{n!} t^{n+α},  (5.7)
|ω̃_{n+1}(t) − ω̃_{n}(t)| ≤ M \frac{L^{n}}{n!} t^{n+α},  (5.8)
|λ̃_{n+1}(t) − λ̃_{n}(t)| ≤ M \frac{L^{n}}{n!} t^{n+α},  (5.9)

where M = M(M_{i}, N_{i}, P_{i}, Q_{i}, S_{i}, f_{i}, i = 1, 2, 3, f, c, ρ_{1}, ρ_{2}) > 0, L = L(M_{i}, N_{i}, P_{i}, Q_{i}, S_{i}, i = 1, 2, 3, c, ρ_{1}, ρ_{2}) > 0 are sufficiently large positive numbers which do not depend on n and which are to be defined, while c is the constant from (4.2). Proof of Estimates (5.7)–(5.9). Since a restriction is imposed on f, f_{i}, i =

1, 2, 3, we have F̃_{3+i} ∈C^{∘}_{α}, i = 1, 2, 3. Indeed, by (1.6) we have |f_{1}(x)| ≤ k_{1}x^{α}, k_{1} > 0, α ≥ 0, x ∈ [0, x_{0}]. Further, the first of equalities (3.2) gives

|f̃_{1}(x)| ≤ k_{1}x^{α} + k_{4}c_{6} ∫_{0}^{ρ_{1}x}(x^{2} + η^{2})^{α/2}dη + k_{4}c_{6} ∫_{0}^{x}(ξ^{2} + ρ_{1}^{2}x^{2})^{α/2}dξ +

+k_{4}c_{6} ∫_{0}^{x} ∫_{0}^{ρ_{1}x}(ξ^{2} + η^{2})^{α/2}dξdη ≤ k_{5}x^{α}, x ∈ [0, x_{0}], α ≥ 0,

where k_{4} ≡ k_{4}(M_{1}, N_{1}, P_{1}, Q_{1}, S_{1}), k_{5} ≡ k_{5}(k_{1}, k_{4}, c_{6}, x_{0}, α, ρ_{1}) are the completely defined positive numbers.

Hence we conclude that f̃_{1} ∈C^{∘}_{α}. The case f̃_{2}, f̃_{3} ∈C^{∘}_{α} is proved similarly. Taking into account the expressions of the functions F_{3+i}, i = 1, 2, 3, from

(3.12), it is now easy to establish that F̃_{3+i} ∈C^{∘}_{α}, i = 1, 2, 3. Therefore by (1.6) the following estimates hold: |F̃_{3+i}(t)| ≤ θ_{3+i}t^{α} or t^{−α}|F̃_{3+i}(t)| ≤ θ_{3+i}, i = 1, 2, 3, α ≥ 0, t ∈ [0, 1]. If in this inequality t is replaced by s ∈ [0, t],

then by the definition of a norm in the space C^{∘}_{α} [0, t] we shall have

∥F̃_{3+i}∥_{C^{∘}_{α}[0,t]} ≤ θ_{3+i} i = 1, 2, 3, ∀t ∈ [0, 1].  (5.10)

Since φ̃_{0}(t) ≡ ω̃_{0}(t) ≡ λ̃_{0}(t) ≡ 0, 0 ≤ t ≤ 1, and under the assumptions of Lemma 4.2 estimate (4.2) holds, from (5.4), (5.10) we shall have

|φ̃_{1}(t) − φ̃_{0}(t)| = |φ̃_{1}(t)| = |(K^{−1}F̃_{4})(t)| ≤
≤ ct^{α}∥F̃_{4}∥_{C^{∘}_{α}[0,t]} ≤ cθ_{4}t^{α}.  (5.11)

(5.5), (5.10) in turn imply

|ω̃_{1}(t) − ω̃_{0}(t)| = |ω̃_{1}(t)| = |F̃_{5}(t)| ≤ θ_{5}t^{α}.  (5.12)

Similarly, (5.6), (5.10) give

|λ̃_{1}(t) − λ̃_{0}(t)| = |λ̃_{1}(t)| = |F̃_{6}(t)| ≤ θ_{6}t^{α}.  (5.13)

Assuming that estimates (5.7)–(5.9) hold for n, n > 0, let us prove that they are valid for n + 1 for sufficiently large M and L.

Denote by K̃ the largest of the numbers sup_{(t,τ)∈[0,1]×[0,1]} |K̃_{1i}(t, τ)|,

i = 1, 2, 3, sup_{t∈[0,1]} |K̃_{i}(t)|, i = 3, 4, 6, 7, 9, 10.

From (5.4) we have

K(φ̃_{n+2} − φ̃_{n+1})(t) = T (φ̃_{n+1} − φ̃_{n}, ω̃_{n+1} − ω̃_{n}, λ̃_{n+1} − λ̃_{n})(t), (5.14)

where

T (φ̃_{n+1} − φ̃_{n}, ω̃_{n+1} − ω̃_{n}, λ̃_{n+1} − λ̃_{n})(t) ≡ ∫_{0}^{t} K̃_{11}(t, τ)(φ̃_{n+1} − φ̃_{n})(τ)dτ +

+K̃_{3}(t) ∫_{0}^{τ_{1}t}(ω̃_{n+1} − ω̃_{n})(τ)dτ + K̃_{4}(t) ∫_{0}^{τ_{1}t}(λ̃_{n+1} − λ̃_{n})(τ)dτ.

Further, for the right-hand side of (5.14) we have the estimate

T (φ̃_{n+1} − φ̃_{n}, ω̃_{n+1} − ω̃_{n}, λ̃_{n+1} − λ̃_{n})(t) ≤ K̃M \frac{L^{n}}{n!} ∫_{0}^{t} τ^{n+α}dτ +

+K̃M \frac{L^{n}}{n!} ∫_{0}^{τ_{1}t}τ^{n+α}dτ + K̃M \frac{L^{n}}{n!} ∫_{0}^{τ_{1}t}τ^{n+α}dτ ≤

≤ 3K̃M \frac{L^{n}}{(n + 1)!}t^{n+1+α}.  (5.15)

As in deriving inequality (5.10), we shall have

∥T (φ̃_{n+1} − φ̃_{n}, ω̃_{n+1} − ω̃_{n}, λ̃_{n+1} − λ̃_{n})∥_{C^{∘}_{α}[0,t]} ≤ 3K̃M \frac{L^{n}}{(n + 1)!}t^{n+1}.

Now (4.2), (5.14), and (5.15) imply

|(φ̃_{n+2} − φ̃_{n+1})(t)| = |{K^{−1}T (φ̃_{n+1} − φ̃_{n}, ω̃_{n+1} − ω̃_{n}, λ̃_{n+1} − λ̃_{n})}(t)| ≤

≤ ct^{α}∥T (φ̃_{n+1} − φ̃_{n}, ω̃_{n+1} − ω̃_{n}, λ̃_{n+1} − λ̃_{n})∥_{C^{∘}_{α}[0,t]} ≤

≤ 3cK̃M \frac{L^{n}}{(n + 1)!}t^{n+1+α}.  (5.16)

Similarly, from (5.5) and (5.6) we find

\begin{matrix}|(ω̃_{n+2} − ω̃_{n+1})(t)| ≤ 3cK̃M \frac{L^{n}}{(n + 1)!}t^{n+1+α}. \\ |(λ̃_{n+2} − λ̃_{n+1})(t)| ≤ 3cK̃M \frac{L^{n}}{(n + 1)!}t^{n+1+α}. \end{matrix}  (5.17)

From (5.11)–(5.13), (5.16), and (5.17) it immediately follows that if we set

M = max{cθ_{4}, θ_{5}, θ_{6}}, L = max{3cK̃, 3K̃},  (5.18)

then estimates (5.7)–(5.9) shall be valid for any integer n ≥ 0.

(5.7)–(5.9) imply that the series

\begin{cases}φ̃(t) = lim_{n→∞} φ̃_{n}(t) = ∑_{n=0}^{∞}(φ̃_{n+1}(t) − φ̃_{n}(t)), \\ ω̃(t) = lim_{n→∞} ω̃_{n}(t) = ∑_{n=0}^{∞}(ω̃_{n+1}(t) − ω̃_{n}(t)), \\ λ̃(t) = lim_{n→∞} λ̃_{n}(t) = ∑_{n=0}^{∞}(λ̃_{n+1}(t) − λ̃_{n}(t)), 0 ≤ t ≤ 1, \end{cases} (5.19)

converge in the space C^{∘}_{α} [0, 1] and by virtue of (5.4)–(5.6) the limit functions φ̃, ω̃, λ̃ satisfy system (5.2). Returning to the previous variables x, y and the functions φ, ψ, ν, we thus conclude that these values satisfy system (3.14). Further, by Lemma 2.1 the function u represented by formula (2.1)

belongs to the class C^{∘} _{α}^{2,1}(\overline{D}_{0}). It is thereby shown that in the plane of the variables x, y the function u is a solution of problem (1.1)–(1.4) belonging

to the class C^{∘} _{α}^{2,1}(\overline{D}_{0}).

We shall now show that problem (1.1)–(1.4) has no other solutions in

the class C^{∘} _{α}^{2,1}(\overline{D}_{0}). Indeed, let the function u^{0} be a solution of the homogeneous problem corresponding to (1.1)–(1.4) belonging to the class

C^{∘} _{α}^{2,1}(\overline{D}_{0}). Then the functions φ^{0}(t) ≡ φ^{0}(x_{0}t) = φ^{0}(x) ≡ u_{xx}^{0}(x, 0), ω^{0}(t)≡ω^{0}(y_{0}t)=ω^{0}(y)≡ψ^{0}(y) + b_{1}(y)φ^{0}(ρ_{2}y)≡u_{y}^{0}(0, y) + b_{1}(y)u_{xx}^{0}(ρ_{2}y, 0), λ̃^{0}(t)≡λ^{0}(y_{0}t)=λ^{0}(y)≡ν^{0}(y) + b_{2}(y)φ^{0}(ρ_{2}y)≡u_{xy}^{0}(0, y) + b_{2}(y)u_{xx}^{0}(ρ_{2}y, 0) satisfy the homogeneous system of equations

\begin{cases}(Kφ^{0})(t) = ∫_{0}^{t} K̃_{11}(t, τ)φ^{0}(τ)dτ + K̃_{3}(t) ∫_{0}^{τ_{1}t} ω^{0}(τ)dτ+ \\ +K̃_{4}(t) ∫_{0}^{τ_{1}t} λ̃^{0}(τ)dτ, 0 ≤ t ≤ 1, \\ ω^{0}(t) = ∫_{0}^{τ_{2}t} K̃_{12}(τ, t)φ^{0}(τ)dτ + K̃_{6}(t) ∫_{0}^{t} ω^{0}(τ)dτ+ \\ +K̃_{7}(t) ∫_{0}^{t} λ̃^{0}(τ)dτ, 0 ≤ t ≤ 1, \\ λ̃^{0}(t) = ∫_{0}^{τ_{2}t} K̃_{13}(τ, t)φ^{0}(τ)dτ + K̃_{9}(t) ∫_{0}^{t} ω^{0}(τ)dτ+ \\ +K̃_{10}(t) ∫_{0}^{t} λ̃^{0}(τ)dτ, 0 ≤ t ≤ 1. \end{cases} (5.20)

Apply the method of successive approximations to system (5.20), taking the functions φ^{0}, ω^{0}, λ̃^{0} themselves as zero approximations. Since these functions satisfy system (5.20), each next aprroximation will coincide with the latter, i.e., φ_{n}^{0}(t) ≡ φ^{0}(t), ω_{n}^{0}(t) ≡ ω^{0}(t), λ̃_{n}^{0}(t) ≡ λ̃^{0}(t), 0 ≤ t ≤ 1. Recalling that these functions satisfy estimates of form (1.6), by a reasoning similar to that used in deriving inequalities (5.7)–(5.9) we obtain

|φ^{0}(t)| = |φ_{n+1}^{0}(t)| ≤ M_{0} \frac{L_{0}^{n}}{n!} t^{n+α},

|ω^{0}(t)| = |ω_{n+1}^{0}(t)| ≤ M_{0} \frac{L_{0}^{n}}{n!} t^{n+α},

|λ̃^{0}(t)| = |λ̃_{n+1}^{0}(t)| ≤ M_{0} \frac{L_{0}^{n}}{n!} t^{n+α},

where M_{0} and L_{0} are positive constants defined as M and L. When n → ∞ we obtain φ^{0} ≡ ω^{0} ≡ λ̃^{0} ≡ 0 or, which is the same, φ^{0} ≡ ψ̃^{0} ≡ ν^{0} ≡ 0. Finally, by (2.1) we have u^{0}(x, y) ≡ 0 everywhere in \overline{D}_{0}.

We have thus proved Theorem 5.1 and also the first part of Theorem 5.2. To prove the second part of Theorem 5.2, rewrite system (5.2) as a single equation

K_{1}χ̃ + T_{4}χ̃ = F̃,  (5.21)

where χ̃ ≡ (φ̃, ω̃, λ̃) ∈C^{∘}_{α} × C^{∘}_{α} × C^{∘}_{α}, K_{1}χ̃ ≡ (Kφ̃, Iω̃, Iλ̃), T_{4}χ̃ ≡

(T_{1}(φ̃, ω̃, λ̃), T_{2}(φ̃, ω̃, λ̃), T_{3}(φ̃, ω̃, λ̃)), F̃ ≡ (F̃_{4}, F̃_{5}, F̃_{6}), and the operators T_{i}, i = 1, 2, 3, are defined by (5.3).

It is obvious that the operator T_{4} is compact in the space C^{∘} _{α}^{3}[0, 1] ≡

C^{∘}_{α} [0, 1]× C^{∘}_{α} [0, 1]× C^{∘}_{α} [0, 1], since each of the operators T_{i}, i = 1, 2, 3, is represented as the sum of completely defined linear integral operators of the Volterra type in the space C^{∘}_{α} [0, 1].

Now let us show that under the assumptions of the second part of Theorem 5.2, i.e., under

σ ≠ 0, α < α_{0},  (5.22)

the equation

K_{1}χ̃ = Φ̃  (5.23)

is normally solvable in Hausdorff’s sense (see, for example, [6]) in the space

C^{∘} _{α}^{3}[0, 1] and its index κ = +∞, i.e., the image of the space C^{∘} _{α}^{3}[0, 1] at the mapping K_{1} is closed in this very space and dim Ker K_{1} = +∞, dim Ker K_{1}^{∗} < +∞, where K_{1}^{∗} is the conjugate operator of K_{1}. To this end, as is easy to see, it is enough to show that the equation

Kφ = Φ̃_{1}  (4.1′)

possesses the said property in the space C^{∘}_{α} [0, 1].

Lemma 4.3 implies by (5.22) that in the space C^{∘}_{α} [0, 1] equation (4.1′) is normally solvable in Hausdorff’s sense and its index κ = +∞, dim Ker K = +∞, dim Ker K^{∗} = 0. Therefore equation (5.23), too, is also normally sol-

vable in Hausdorff’s sense in Banach space C^{∘} _{α}^{3}[0, 1] and its index κ = +∞. Hence, in turn, it follows that equation (5.21), too, possesses this property

in the space C^{∘} _{α}^{3}[0, 1], since the operator T_{4} is compact and the property of the equation to be normally solvable and to have an index equal to +∞ is stable at compact perturbations (see, for example, [20]).

The latter arguments prove the second part of Theorem 5.2, since in the class C^{∘} _{α}^{2,1}(\overline{D}_{0}) problem (1.1)–(1.4) is equivalently reduced to equation

(5.21) in the space C^{∘} _{α}^{3}[0, 1].

Remark 5.3. It should be noted that the value α_{0} appearing in the solvability conditions of problem (1.1)–(1.4) depends only on the value at the point O(0, 0) of the coefficients M_{i}, N_{i}, Q_{i}, i = 1, 2, 3, and on the value τ_{0} = ρ_{1}ρ_{2}, since a simple verification shows that a(0)=M_{1}^{−1}Δ^{−1}(M_{2}N_{3}Q_{1}−M_{3}N_{2}Q_{1} + M_{3}N_{1}Q_{2} − M_{2}N_{1}Q_{3})(0).

§ 6. Estimation of a Regular Solution of Problem (1.1)–(1.4)

Belonging to the Class C^{∘} _{α}^{2,1}(\overline{D}_{0})

Below it will be shown that if the conditions of Theorems 5.1 and 5.2 guaranteeing the unique solvability of problem (1.1)–(1.4) are fulfilled, then

for the solution u of this problem belonging to the class C^{∘} _{α}^{2,1}(\overline{D}_{0}) we have the estimate

∥u∥_{C^{∘}_{α}^{2,1}(\overline{D}_{0})} ≤ C(∥f_{1}∥_{C^{∘}_{α}[0,x_{0}]} + ∥f_{2}∥_{C^{∘}_{α}[0,y_{0}]} +

+∥f_{3}∥_{C^{∘}_{α}[0,y_{0}]} + ∥f∥_{C^{∘}_{α}(\overline{D}_{0})}) ≡ CC^{∗}(f_{1}, f_{2}, f_{3}, f),  (6.1)

where C is a positive constant not depending on f, f_{i}, i = 1, 2, 3.

The proof of the above statement will be divided into two parts. First we shall prove that for the solution u of the class C^{∘} _{α}^{2,1}(\overline{D}_{0}) of problem (1.1)–(1.4) we have the estimate

∥u∥_{C^{∘}_{α}^{2,1}(\overline{D}_{0})} ≤ C_{1}C^{∗}(φ, ψ, ν, f),  (6.2)

where C_{1} is a positive constant not depending on f, φ(x) = u_{xx}(x, 0), 0 ≤ x ≤ x_{0}, ψ(y) = u_{y}(0, y), ν(y) = u_{xy}(0, y), 0 ≤ y ≤ y_{0}.

Indeed, similarly to the proof of Lemma 2.1, by virtue of the definition of norms in the spaces C^{∘}_{α} [0, d], C^{∘}_{α} (\overline{D}_{0}) formula (2.1) yields

|u(x, y)| ≤ x_{0}∥φ∥_{C^{∘}_{α}[0,x_{0}]} ∫_{0}^{x} ξ^{α}dξ + ∥ψ∥_{C^{∘}_{α}[0,y_{0}]} ∫_{0}^{y} η^{α}dη +

+x_{0}∥ν∥_{C^{∘}_{α}[0,y_{0}]} ∫_{0}^{y} η^{α}dη + x_{0}∥f∥_{C^{∘}_{α}(\overline{D}_{0})} ∫_{0}^{x} ∫_{0}^{y} |ζ|^{α}dξdη ≤

≤ \frac{x_{0}^{2}}{α + 1}∥φ∥_{C^{∘}_{α}[0,x_{0}]}|z|^{α} + \frac{y_{0}^{2}}{α + 1}∥ψ∥_{C^{∘}_{α}[0,y_{0}]}|z|^{α} +

+\frac{x_{0}y_{0}}{α + 1}∥ν∥_{C^{∘}_{α}[0,y_{0}]}|z|^{α} + x_{0}^{2}y_{0}∥f∥_{C^{∘}_{α}(\overline{D}_{0})}|z|^{α},

from which it follows that

∥u∥_{C^{∘}_{α}(\overline{D}_{0})} ≤ C_{0,0}C^{∗}(φ, ψ, ν, f),  (6.3)

where C_{0,0} ≡ max \{\frac{x_{0}^{2}}{α+1}, \frac{y_{0}}{α+1}, \frac{x_{0}y_{0}}{α+1} , x_{0}^{2}y_{0}\}. In a similar manner one can prove the estimates

∥D_{x}^{i}D_{y}^{j}u∥_{C^{∘}_{α}(\overline{D}_{0})} ≤ C_{i,j}C^{∗}(φ, ψ, ν, f),  (6.4)

where C_{i,j}, i = 0, 1, 2, j = 0, 1, i + j > 0, are positive constants not depending on the functions f, φ, ψ, ν.

Estimates (6.3), (6.4) immediately imply (6.2) where C_{1} ≡ ∑_{i=0}^{2} ∑_{j=0}^{1} C_{ij}.

Now let us prove the second part of the statement. From (5.7) and (5.19) we immediately have

|φ(x)| ≤ ∑_{n=0}^{∞} |φ_{n+1}(x) − φ_{n}(x)| ≤ M(\frac{x}{x_{0}} )^{α} ∑_{n=0}^{∞} \frac{L^{n}}{n!} (\frac{x}{x_{0}} )^{n} =

= M(\frac{x}{x_{0}} )^{α}e^{L \frac{x}{x_{0}}} , 0 ≤ x ≤ x_{0}.

Hence we easily obtain

∥φ∥_{C^{∘}_{α}[0,x_{0}]} ≤ Mγ,  (6.5)

where γ ≡ x_{0}^{−α}e^{L}.

Similar estimates hold for the functions ψ, ν as well:

∥ψ∥_{C^{∘}_{α}[0,y_{0}]} ≤ Mγ̃, ∥ν∥_{C^{∘}_{α}[0,y_{0}]} ≤ Mγ̃,  (6.6)

where γ̃ ≡ y_{0}^{−α}L.

By (5.18) and the proof of inequality (5.10) it is easy to see that we can take as M the value

M = C_{2}C^{∗}(f_{1}, f_{2}, f_{3}, f),  (6.7)

where C_{2} is a sufficiently large positive constant not depending on f, f_{i}, i = 1, 2, 3.

Finally, with regard to (6.7) inequalities (6.2), (6.5), (6.6) give estimate (6.1), where the positive constant C is expressed in terms of C_{1}, C_{2}, γ, γ̃.

Estimate (6.1) immediately implies that the solution of problem (1.1)– (1.4) is stable.

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(Received 21.06.94)

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