Zeitschrift für Analysis und ihre Anwendungen

Journal for Analysis and its Applications Volume 18 (1999), No. 2, 287–295

Nonlinear Equations with Operators Satisfying Generalized Lipschitz Conditions in Scales

J. Janno

Dedicated to my good collaborator Prof. L. von Wolfersdorf

Abstract. By means of the contraction principle we prove existence, uniqueness and stability of solutions for nonlinear equations u + G_{0}[D, u] + L(G_{1}[D, u], G_{2}[D, u]) = f in a Banach space E, where G_{0}, G_{1}, G_{2} satisfy Lipschitz conditions in scales of norms, L is a bilinear operator and D is a data parameter. The theory is applicable for inverse problems of memory identification and generalized convolution equations of the second kind.

Keywords: Nonlinear operator equations, nonlinear convolution equations, scales of norms, fixed point theorems, existence, uniqueness and stability of solutions of nonlinear equations

AMS subject classification: 47 H 15, 45 G 10, 45 D 05

0. Introduction

Recently by the author and L.v. Wolfersdorf global solvability and stability with respect to free terms has been proved for nonlinear convolution equations of the second kind [11, 12]. These results were generalized to abstract equations containing operators which are Lipschitz-continuous in scales of norms [9].

On the other hand, in the most important applications of this theory (see [7, 8, 10]) the operators involved are dependent on the data. In the present paper we generalize the results of the work [9] to the case when the Lipschitz operators depend on a data parameter D and derive stability estimates with respect to D and the free term f. In contrast to [9] the stability results obtained in this paper are global. As in [11, 12] we will use the Banach fixed point theorem in scales of norms.

J. Janno: Estonian Academy of Sciences, Institute of Cybernetics, Akadeemia teer 21, EE-0026 Tallinn, Estonia. The paper was written during a stay of the author at the Freiberg University of Mining and Technology with a grant of DFG from September 1997 to December 1997. The author acknowledges also the support of the Estonian Science Foundation.

ISSN 0232-2064 / $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag Berlin

1. Equations with Lipschitz operators in scales

We will study the operator equation

u + G_{0}[D, u] + L(G_{1}[D, u], G_{2}[D, u]) = f  (1)

in a Banach space E, where G_{0} ∈ (Ê × E → E) and G_{i} ∈ (Ê × E → E_{i}) (i = 1, 2), with Ê a linear space, E_{1} and E_{2} are normed spaces, and L is a bilinear operator from E_{1} × E_{2} into E.

We suppose that the spaces E and E_{i} are endowed with scales of norms ∥ · ∥_{σ} and ∥ · ∥_{i,σ} (σ ≥ 0), respectively, which satisfy the conditions

κ(σ)∥u∥_{0} ≤ ∥u∥_{σ} ≤ ∥u∥_{0}  (u ∈ E)  (2)

and

∥v_{i}∥_{i,σ} ≤ ∥v_{i}∥_{i,0}  (v_{i} ∈ E_{i})  (3)

where

κ ∈ C(ℝ_{+} → ℝ_{+})  (κ > 0),  (4)

the linear space Ê is endowed with a semi-norm | · | and for the operators G_{0}, G_{i} and L the assumptions

∥G_{0}[D_{1}, u_{1}] − G_{0}[D_{2}, u_{2}]∥_{σ}
≤ M_{0}(|D_{1}|, |D_{2}|, ∥u_{1}∥_{σ}, ∥u_{2}∥_{σ}, σ)(|D_{1} − D_{2}| + ∥u_{1} − u_{2}∥_{σ})  (5)
∥G_{i}[D_{1}, u_{1}] − G_{i}[D_{2}, u_{2}]∥_{i,σ}
≤ M_{i}(|D_{1}|, |D_{2}|, ∥u_{1}∥_{σ}, ∥u_{2}∥_{σ})(|D_{1} − D_{2}| + ∥u_{1} − u_{2}∥_{σ})  (6)

for D_{1}, D_{2} ∈ Ê and u_{1}, u_{2} ∈ E and

∥L(v_{1}, v_{2})∥_{σ} ≤ N∥v_{1}∥_{1,σ}∥v_{2}∥_{2,σ}  (7)
∥L(v_{1}, v_{2})∥_{σ} ≤ λ(σ) min {∥v_{1}∥_{1,0}∥v_{2}∥_{2,σ}; ∥v_{1}∥_{1,σ}∥v_{2}∥_{2,0}}  (8)

for v_{i} ∈ E_{i} (i = 1, 2) hold. Here N ≥ 0 and the functions M_{0}, M_{i} and λ satisfy the conditions

M_{0} ∈ C(ℝ_{+}^{5} → ℝ_{+}), M_{0}(x_{1}, ..., x_{4}, σ) is increasing in x_{1}, ..., x_{4}  (9)
lim_{σ→∞} M_{0}(x_{1}, ..., x_{4}, σ) = 0 for any (x_{1}, ..., x_{4}) ∈ ℝ^{4}  (10)
M_{i} ∈ C(ℝ_{+}^{4} → ℝ_{+}), M_{i}(x_{1}, ..., x_{4}) is increasing in x_{1}, ..., x_{4}  (11)
λ ∈ C(ℝ_{+} → ℝ_{+}), lim_{σ→∞} λ(σ) = 0.  (12)

where as usual ℝ_{+} = [0, ∞).

Theorem. Let assumptions (2) − (12) be fulfilled. Then equation (1) has a unique solution u ∈ E for any D ∈ Ê and f ∈ E. Moreover, for solutions u_{1} and u_{2} corresponding to data D_{1}, f_{1} and D_{2}, f_{2}, respectively, the stability estimate

∥u_{1} − u_{2}∥_{0} ≤ Λ(Q_{1}, Q_{2})(|D_{1} − D_{2}| + ∥f_{1} − f_{2}∥_{0})  (13)

holds where

Q_{i} = (|D_{i}|, ∥f_{i}∥_{0}, ∥G_{0}[D_{i}, f_{i}]∥_{0}, ∥G_{1}[D_{i}, f_{i}]∥_{1,0}, ∥G_{2}[D_{i}, f_{i}]∥_{2,0}) (i = 1, 2)

and Λ ∈ C(ℝ_{+}^{10} → ℝ_{+}), Λ > 0 and Λ – increasing in x_{1}, ..., x_{10}. If the operators G_{i} (i = 0, 1, 2) satisfy the conditions G_{0}[0, 0] = G_{1}[0, 0] = G_{2}[0, 0] = 0, then (13) has the simplified form

∥u_{1} − u_{2}∥_{0} ≤ Λ_{1}(|D_{1}|, ∥f_{1}∥_{0}, |D_{2}|, ∥f_{2}∥_{0})(|D_{1} − D_{2}| + ∥f_{1} − f_{2}∥_{0}),  (14)

where Λ_{1} ∈ C(ℝ_{+}^{4} → ℝ_{+}), Λ_{1} > 0 and Λ_{1} is increasing in x_{1}, ..., x_{4}.

The proof of Theorem is given in the next section.

The main area of applications of equation (1) and the related Theorem are inverse problems for determining memory kernels in heat flow [7], viscoelasticity [5, 8] and thermo- and poroviscoelasticity [10]. All these problems admit reductions to integral equations or systems of integral equations of the form

m(t) + G_{0}[D, m](t) + K[D, m] ∗ m(t) = f(t) (t ∈ [0, T])

in a Banach space E = X^{n}, where X a functional space over the interval [0, T] and n ≥ 1. Here m is the memory kernel, or a vector of independent memory kernels, and

G_{0}[D, ·] ∈ (E → E)  and  K[D, ·] ∈ (E → X^{n×n})

are operators of m depending on the data vector D of the inverse problem. The bilinear operator L in these cases is the convolution operator

L(v_{1}, v_{2})(t) ≡ v_{1} ∗ v_{2}(t) = ∫_{0}^{t} v_{1}(t − τ)v_{2}(τ) dτ

and the scales of norms are defined using exponential weights of the form e^{−σt} (σ ≥ 0).

The technique of scales of norms enables to formulate statements about global existence, uniqueness and stability of solutions of these nonlinear integral equations of the second kind.

It is remarkable that the method of weighted norms applies to inverse problems of memory identification also in the case if one makes use of an approach different from the reduction to integral equations (e.g., a priori estimates [2, 3], the theory of semigroups [1], etc.). This is due to the fact that these problems, if they are constructed from linear constitutive laws, contain only nonlinearities of convolution type.

Other areas of application of the theory of the present paper are equations of autoconvolution type [1, 6, 11] arising in stochastics and spectroscopy as well as more theoretical examples of equations involving various types of generalized convolutions. Concerning the latter examples we refer the reader to the previous papers of the author and L. v. Wolfersdorf [9, 11, 12].

2. Proof of Theorem

The proof uses the contraction principle in balls

B_{ρ,σ}(w) = {u ∈ E : ∥u − w∥_{σ} ≤ ρ}  (ρ > 0, σ ≥ 0, w ∈ E).  (15)

Step 1. At first we show that the auxiliary equation

g + G_{0}[D, g] = f  (16)

has a solution in B_{R,σ}(f), where R = 2∥G_{0}[D, f]∥_{0} and σ is chosen large enough. We define the operator A_{1}g = f − G_{0}[D, g]. Then equation (16) reads g = A_{1}g. In view of (2), (5) and the monotonicity of M_{0} we have

\begin{matrix}∥A_{1}g − f∥_{σ} \\ ≤ ∥G_{0}[D, g] − G_{0}[D, f]∥_{σ} + ∥G_{0}[D, f]∥_{σ} \\ ≤ M_{0}(|D|, |D|, ∥g − f∥_{σ} + ∥f∥_{0}, ∥f∥_{0}, σ) ∥g − f∥_{σ} + ∥G_{0}[D, f]∥_{0} \end{matrix}  (17)

and

\begin{matrix}∥A_{1}g_{1} − A_{1}g_{2}∥_{σ} \\ = ∥G_{0}[D, g_{1}] − G_{0}[D, g_{2}]∥_{σ} \\ ≤ M_{0}(|D|, |D|, ∥g_{1} − f∥_{σ} + ∥f∥_{0}, ∥g_{2} − f∥_{σ} + ∥f∥_{0}, σ) ∥g_{1} − g_{2}∥_{σ}. \end{matrix}  (18)

In the case g, g_{1}, g_{2} ∈ B_{R,σ}(f) inequalities (17) and (18) yield

\begin{aligned}∥A_{1}g − f∥_{σ} & ≤ M_{0}(|D|, |D|, R + ∥f∥_{0}, ∥f∥_{0}, σ)R + \frac{R}{2} \\ ∥A_{1}g_{1} − A_{1}g_{2}∥_{σ} & ≤ M_{0}(|D|, |D|, R + ∥f∥_{0}, R + ∥f∥_{0}, σ) ∥g_{1} − g_{2}∥_{σ}. \end{aligned}

By the continuity of M_{0} and the limit condition (10) there exists a quite large σ = σ_{0} depending continuously on ∥f∥_{0}, |D| and R, so that

A_{1}B_{R,σ_{0}}(f) ⊆ B_{R,σ_{0}}(f), A_{1} is a contraction in B_{R,σ_{0}}(f).

Therefore, equation (16) has for every f ∈ E a unique solution g in the ball B_{R,σ_{0}}(f).

Next we derive some estimates for the solution g of equation (16). To this end we introduce the vector

Q = (|D|, ∥f∥_{0}, ∥G_{0}[D, f]∥_{0}, ∥G_{1}[D, f]∥_{1,0}, ∥G_{2}[D, f]∥_{2,0}) ∈ ℝ_{+}^{5}.

By definition (15) of the ball B_{R,σ_{0}}(f) the solution g satisfies the inequality

∥g − f∥_{σ_{0}} ≤ R = 2∥G_{0}[D, f]∥_{0}.

Thus, by means of (2) and (4) we obtain

∥g∥_{0} ≤ μ_{0}(Q)  (19)

where

μ_{0}(Q) = 2∥G_{0}[D, f]∥_{0} (κ(σ_{0}(Q)))^{−1} + ∥f∥_{0}

is a positive continuous function of Q. Further, using (2) and (6) we can estimate

\begin{aligned}∥G_{i}[D, g]∥_{i,0} & ≤ ∥G_{i}[D, g] − G_{i}[D, f]∥_{i,0} + ∥G_{i}[D, f]∥_{i,0} \\ & ≤ M_{i}(|D|, |D|, ∥g∥_{0}, ∥f∥_{0}) ∥g − f∥_{0} + ∥G_{i}[D, f]∥_{i,0}. \end{aligned}

Applying here (11) and (19) we obtain

∥G_{i}[D, g]∥_{i,0} ≤ μ_{i}(Q)  (i = 1, 2)  (20)

where

μ_{i}(Q) = 2M_{i}(|D|, |D|, μ_{0}(Q), ∥f∥_{0})\frac{∥G_{0}[D, f]∥_{0}}{κ(σ_{0}(Q))} + ∥G_{i}[D, f]∥_{i,0}

are again positive continuous functions of Q.

Step 2. Let us return to equation (1). In view of (16) we write it in the operator form u = Au, where

Au = g − L(G_{1}[D, u], G_{2}[D, u]) + G_{0}[D, g] − G_{0}[D, u].

We are going to show that equation (1) has a solution in the ball B_{ρ,σ}(g), where ρ is small enough and σ is large enough.

Observing estimates (2), (3) and (5) - (8), the bilinearity of L as well as the monotonicity of the functions M_{0}, M_{1} and M_{2} we obtain

\begin{matrix}∥Au − g∥_{σ} \\ ≤ ∥L(G_{1}[D, u] − G_{1}[D, g], G_{2}[D, u] − G_{2}[D, g])∥_{σ} \\ + ∥L(G_{1}[D, u] − G_{1}[D, g], G_{2}[D, g])∥_{σ} \\ + ∥L(G_{1}[D, g], G_{2}[D, u] − G_{2}[D, g])∥_{σ} \\ + ∥L(G_{1}[D, g], G_{2}[D, g])∥_{σ} \\ + ∥G_{0}[D, g] − G_{0}[D, u]∥_{σ} \\ ≤ NM_{1}(|D|, |D|, ∥u − g∥_{σ} + ∥g∥_{0}, ∥g∥_{0}) \\ × M_{2}(|D|, |D|, ∥u − g∥_{σ} + ∥g∥_{0}, ∥g∥_{0}) ∥u − g∥_{σ}^{2} \\ + λ(σ) ∑_{i=1}^{2} M_{i}(|D|, |D|, ∥u − g∥_{σ} + ∥g∥_{0}, ∥g∥_{0}) ∥G_{j_{i}}[D, g]∥_{j_{i},0} ∥u − g∥_{σ} \\ + λ(σ)∥G_{1}[D, g]∥_{1,0}∥G_{2}[D, g]∥_{2,0} \\ + M_{0}(|D|, |D|, ∥u − g∥_{σ} + ∥g∥_{0}, ∥g∥_{0}, σ) ∥u − g∥_{σ} \end{matrix} (21)

and

∥Au_{1} − Au_{2}∥_{σ}

≤ ∥L(G_{1}[D, u_{1}] − G_{1}[D, u_{2}], G_{2}[D, u_{1}] − G_{2}[D, g] + G_{2}[D, g])∥_{σ}

+ ∥L(G_{1}[D, u_{2}] − G_{1}[D, g] + G_{1}[D, g], G_{2}[D, u_{1}] − G_{2}[D, u_{2}])∥_{σ}

+ ∥G_{0}[D, u_{1}] − G_{0}[D, u_{2}]∥_{σ}  (22)

≤ { ∑_{i=1}^{2} M_{i}(|D|, |D|, ∥u_{1} − g∥_{σ} + ∥g∥_{0}, ∥u_{2} − g∥_{σ} + ∥g∥_{0})

× [NM_{j_{i}}(|D|, |D|, ∥u_{i} − g∥_{σ} + ∥g∥_{0}, ∥g∥_{0}) ∥u_{i} − g∥_{σ} + λ(σ)∥G_{j_{i}}[D, g]∥_{j_{i},0}]

+ M_{0}(|D|, |D|, ∥u_{1} − g∥_{σ} + ∥g∥_{0}, ∥u_{2} − g∥_{σ} + ∥g∥_{0}, σ)}∥u_{1} − u_{2}∥_{σ}

where j_{i} is defined so that j_{1} = 2 and j_{2} = 1. Further, assuming that u, u_{1}, u_{2} ∈ B_{ρ,σ}(g) and applying estimates (19) and (20) in (21) and (22) we have

\begin{matrix}∥Au − g∥_{σ} \\ ≤ ρ^{2}NM_{1}(|D|, |D|, ρ + μ_{0}(Q), μ_{0}(Q))M_{2}(|D|, |D|, ρ + μ_{0}(Q), μ_{0}(Q)) \\ + ρλ(σ) ∑_{i=1}^{2} M_{i}(|D|, |D|, ρ + μ_{0}(Q), μ_{0}(Q))μ_{j_{i}}(Q) \\ + λ(σ)μ_{1}(Q)μ_{2}(Q) + ρM_{0}(|D|, |D|, ρ + μ_{0}(Q), μ_{0}(Q), σ) \end{matrix}  (23)

and

\begin{matrix}∥Au_{1} − Au_{2}∥_{σ} \\ ≤ \{ ∑_{i=1}^{2} M_{i}(|D|, |D|, ρ + μ_{0}(Q), ρ + μ_{0}(Q)) \\ × [ρN M_{j_{i}}(|D|, |D|, ρ + μ_{0}(Q), μ_{0}(Q)) + λ(σ)μ_{j_{i}}(Q)] \\ + M_{0}(|D|, |D|, ρ + μ_{0}(Q), ρ + μ_{0}(Q), σ)\}∥u_{1} − u_{2}∥_{σ}. \end{matrix}  (24)

By virtue of the limit conditions (10) and (12) and the monotonicity of M_{0}, M_{1} and M_{2} there exist a quite small but positive ρ = ρ_{1} and quite large σ = σ_{1} so that

AB_{ρ_{1},σ}(g) ⊆ B_{ρ_{1},σ}(g), A is a contraction in B_{ρ_{1},σ}(g)

for every σ ≥ σ_{1}.

Therefore, equation (1) has a unique solution in every ball B_{ρ_{1},σ}(g), where σ ≥ σ_{1}. Particularly, this proves the existence assertion of Theorem. Since the functions μ_{0}, μ_{1}, μ_{2}, M_{0}, M_{1}, M_{2} and λ are continuous, the quantities ρ_{1} and σ_{1} depend also continuously on Q.

Step 3. Let us show the uniqueness of the solution of equation (1) in the whole space E. Suppose that u_{1} ∈ E and u_{2} ∈ E are two arbitrary solutions of (1). Then from (1) in view of (2), (3), (5), (8) and (16) we obtain

∥u_{i} − g∥_{σ} ≤ ∥L(G_{1}[D, u_{i}], G_{2}[D, u_{i}])∥_{σ}
+ ∥G_{0}[D, u_{i}] − G_{0}[D, g]∥_{σ}
≤ λ(σ)∥G_{1}[D, u_{i}]∥_{1,0}∥G_{2}[D, u_{i}]∥_{2,0}
+ M_{0}(|D|, |D|, ∥u_{i}∥_{0}, ∥g∥_{0}, σ) ∥u_{i} − g∥_{0}.

Due to the limit conditions (10) and (12) the relations lim_{σ→∞} ∥u_{i} − g∥_{σ} = 0 hold for i = 1, 2. Thus, there exists a quite large σ ≥ σ_{1} so that ∥u_{i} − g∥_{σ} ≤ ρ_{1} for i = 1, 2. Observing definition (15) we see that u_{1} and u_{2} belong to a ball B_{ρ_{1},σ}(g), where the uniqueness has already been show. Consequently, u_{1} = u_{2}.

Step 4. Finally, let us derive the stability estimates (13) and (14). To this end we need a bound for ∥u∥_{0} in terms Q. In the second part of the proof we have shown that the solution of equation (1) belongs to the ball B_{ρ_{1},σ_{1}}(g). Thus, by definition (15) we get the inequality ∥u − g∥_{σ_{1}} ≤ ρ_{1}. Using here (2) and (19) we obtain

∥u∥_{0} ≤ μ(Q)  (25)

where μ(Q) = ρ_{1}(Q)κ(σ_{1}(Q))^{−1} + μ_{0}(Q).

Since ρ_{1}, σ_{1}, κ and μ_{0} are positive continuous functions of Q, μ is also positive and continuous.

Further we denote the solutions of equation (16) coresponding the data D_{i} and f_{i} by g_{i} (i = 1, 2), respectively, and subtract the equations (1) with D_{1}, f_{1} and D_{2}, f_{2}. We get the relation

u_{1} − u_{2} = f_{1} − f_{2} + G_{0}[D_{2}, u_{2}] − G_{0}[D_{1}, u_{1}]
+ L(G_{1}[D_{2}, u_{2}] − G_{1}[D_{1}, u_{1}], G_{2}[D_{2}, u_{2}] − G_{2}[D_{2}, g_{2}] + G_{2}[D_{2}, g_{2}])
+ L(G_{1}[D_{1}, u_{1}] − G_{1}[D_{1}, g_{1}] + G_{1}[D_{1}, g_{1}], G_{2}[D_{2}, u_{2}] − G_{2}[D_{1}, u_{1}]).

By means of assumptions (2), (5), (6), (8) and the monotonicity of M_{0}, M_{1}, M_{2} we derive the estimate

∥u_{1} − u_{2}∥_{σ}

≤ M_{0}(|D_{2}|, |D_{1}|, ∥u_{2}∥_{0}, ∥u_{1}∥_{0}, σ)(|D_{1} − D_{2}| + ∥u_{1} − u_{2}∥_{σ})

+ λ(σ) ∑_{i=1}^{2} M_{i}(|D_{j_{i}}|, |D_{i}|, ∥u_{j_{i}}∥_{0}, ∥u_{i}∥_{0})

× [M_{j_{i}}(|D_{j_{i}}|, |D_{j_{i}}|, ∥u_{j_{i}}∥_{0}, ∥g_{j_{i}}∥_{0}) ∥u_{j_{i}} − g_{j_{i}}∥_{0} + ∥G_{j_{i}}[D_{j_{i}}, g_{j_{i}}∥_{j_{i},0}]

× (|D_{1} − D_{2}| + ∥u_{1} − u_{2}∥_{σ}) + ∥f_{1} − f_{2}∥_{0}.

Applying here inequalities (19), (20) and (25) we have

\begin{matrix}∥u_{1} − u_{2}∥_{σ} \\ ≤ \{M_{0}(|D_{2}|, |D_{1}|, μ(Q_{2}), μ(Q_{1}), σ) \\ + λ(σ) ∑_{i=1}^{2} M_{i}(|D_{j_{i}}|, |D_{i}|, μ(Q_{j_{i}}), μ(Q_{i})) \\ × [M_{j_{i}}(|D_{j_{i}}|, |D_{j_{i}}|, μ(Q_{j_{i}}), μ_{0}(Q_{j_{i}}))(μ(Q_{j_{i}}) + μ_{0}(Q_{j_{i}})) \\ + μ_{j_{i}}(Q_{j_{i}})]\}∥u_{1} − u_{2}∥_{σ} \\ + \{M_{0}(|D_{2}|, |D_{1}|, μ(Q_{2}), μ(Q_{1}), σ) \\ + λ(σ) ∑_{i=1}^{2} M_{i}(|D_{j_{i}}|, |D_{i}|, μ(Q_{j_{i}}), μ(Q_{i})) \\ × [M_{j_{i}}(|D_{j_{i}}|, |D_{j_{i}}|, μ(Q_{j_{i}}), μ(Q_{j_{i}}))(μ(Q_{j_{i}}) + μ_{0}(Q_{j_{i}})) \\ + μ_{j_{i}}(Q_{j_{i}})]\}|D_{1} − D_{2}| + ∥f_{1} − f_{2}∥_{0} \end{matrix}  (26)

where, as before, j_{1} = 2 and j_{2} = 1. Due to the limit conditions (10) and (12) there exists a quite large σ_{3} so that the coefficient of ∥u_{1} − u_{2}∥_{σ} in the right-hand side of (26) becomes less than one if σ = σ_{3}. Hence, there holds the relation

∥u_{1} − u_{2}∥_{σ_{3}} ≤ μ_{3}(Q_{1}, Q_{2})(|D_{1} − D_{2}| + ∥f_{1} − f_{2}∥_{0})  (27)

with some coefficient μ_{3}. Since μ, μ_{0}, μ_{1}, μ_{2}, M_{0}, M_{1}, M_{2} and λ are continuous, the quantities σ_{3} and μ_{3} are positive continuous functions of Q_{1} and Q_{2}. Applying in (27) the left inequality (2) we derive the stability estimate (13) with

Λ(Q_{1}, Q_{2}) = μ_{3}(Q_{1}, Q_{2})κ(σ_{3}(Q_{1}, Q_{2}))^{−1}.

Since μ_{3}, σ_{3} and κ are positive and continuous functions of Q_{1} and Q_{2}, the coefficient Λ is also positive and continuous. Without loss of generality we may assume Λ(x_{1}, ..., x_{10}) to be increasing in each of its arguments.

In order to prove estimate (14) we observe that inequalities (5) and (6) in view of the assumptions G_{i}[0, 0] = 0 (i = 0, 1, 2) yield the following bounds for components of the vectors Q_{j}:

\begin{aligned}∥G_{0}[D_{j}, f_{j}]∥_{0} & ≤ M_{0}(|D_{j}|, 0, ∥f_{j}∥_{0}, 0, 0)(|D_{j}| + ∥f_{j}∥_{0}) \\ ∥G_{i}[D_{j}, f_{j}]∥_{i,0} & ≤ M_{i}(|D_{j}|, 0, ∥f_{j}∥_{0}, 0)(|D_{j}| + ∥f_{j}∥_{0}) (i = 1, 2). \end{aligned}

Using these relations in (13) we obtain assertion (14). Theorem is proved

Nonlinear Equations with Operators in Scales 295

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Received 19.03.1998