EXISTENCE AND UNIQUENESS OF ANALYTIC SOLUTIONS OF THE SHABAT EQUATION

Sufficient conditions are given so that the initial value problem for the Shabat equation has a unique analytic solution, which, together with its first derivative, converges absolutely for z ∈ C : |z| < T , T > 0. Moreover, a bound of this solution is given. The sufficient conditions involve only the initial condition, the parameters of the equation, and T . Furthermore, from these conditions, one can obtain an upper bound for T . Our results are in consistence with some recently found results.

The analytic solutions of the initial value problem (1.1)-(1.2),for complex or real parameters (q, µ, f 0 ), have been studied in [7,13].More precisely, it was proved in [13, pages 63-65] that there exists a unique solution of the form which converges in the disc |z| < R q for every fixed value of q ∈ C, |q| < 1.For the radius of convergence R q , the following estimate was given in the same paper: (1.20) In order to prove this result, S. Skorik and V. Spiridonov used Taylor series, which is a method not applicable when |q| = 1, as they state.However, they were able to give exact solutions of (1.1) in the cases where q 3 = 1 and q 4 = 1.
Recently, it was proved in [7] that if |q| = 1 and for some positive constant ν depending on q only, there exists a unique solution of the form (1.19) of the initial value problem (1.1)-(1.2) (for q,µ, f 0 ∈ C), which converges for |z| < ρ/2 7ν+1 , where For the proof of this result Liu followed Siegel's approach [12].It is mentioned in [7] that "when q is on the unit circle but not a root of unity, the analysis of the convergence of (1.19) is nontrivial since the coefficient 1 + q n+1 can be arbitrarily small and a straightforward estimation of the coefficient f n is not enough."It is also mentioned there that in order to prove the convergence of (1.19), good estimates of δ n , n = 1,2,..., are needed, where Finally in [6], the regular solutions of the initial value problem (1.1)-(1.2) for q,µ, f 0 ∈ R were studied.The following were proved among other things.
(1) If q ∈ (0,1), then the initial value problem (1.1)-(1.2) has one and only one solution in a neighborhood of the origin with open maximal interval of existence (T min ,T max ), for which Remark 1.8.(i) For µ, q, and f 0 ∈ R, our interval of existence is (−T,T), which is a subset of (T min ,T max ).Thus we have T min ≤ −T < T ≤ T max .Therefore T max > 0 and T min < 0, which is consistent with the second result of [6] mentioned above.
(ii) For µ = 0 and q ∈ (0,1), we obtain the following from (1.7): It can be easily proved that Thus we have which is consistent with what is already mentioned.
The method we use is a functional analytic method developed by Ifantis [3] for differential and functional differential equations and used also in [8,9,11] for functional and functional differential equations.The basic idea of the method is the equivalent transformation of the functional differential equation under consideration into an operator equation.By use of this method and due to the space H 1 (∆) where we work, the convergence of the established solution (in H 1 (∆)) of the functional differential equation under consideration is immediately proved.In this way, we avoid the use of the method of majorizing series which is often used for proving the convergence of series and which was also used in [7].
This functional analytic method is briefly presented in Section 2. The proof of our main result (Theorem 1.1) is given in Section 3.

The functional analytic method
Denote by H an abstract separable Hilbert space over the complex field with the orthonormal base {e n }, n = 1,2,3,..., and by (•,•) and • , the scalar product and the norm in H, respectively.Consider now those elements f ∈ H, which satisfy the condition ∞ n=1 |( f ,e n )| < +∞.These elements form a Banach space We also define the shift operators V and V * as follows: V : Ve n = e n+1 , n = 1,2,..., It is proved [2, page 3139] that the mapping is a one-to-one mapping from H 1 onto H 1 (∆) which preserves the norm, where f z = ∞ n=1 z n−1 e n , f 0 = e 1 , |z| < 1 are the eigenelements of V * , which form a complete system in H, in the sense that ( f z ,h) = 0, for all z, |z| < 1, implies that h = 0.The element f ∈ H 1 defined by (2.2) is called the abstract form of φ(z).In general, the abstract form of a function G(φ(z)) : H 1 (∆) → H 1 (∆) is a mapping N( f ) : H 1 → H 1 for which the following relation holds: where Q * is the adjoint of the diagonal operator Qe n = q n−1 e n , |q| ≤ 1, n = 1,2,.... (ii) The abstract form of f (z) is the element C 0 V * f , where C 0 is the diagonal operator for which it was proved in [3, Proposition 2] that it has a selfadjoint extension with discrete spectrum, that is, the domain of C 0 can be extended to the range of the bounded diagonal operator B 0 : In [3,Proposition 3], it was also proved that the range of B 0 , that is, the definition domain of C 0 , is isomorphic with the linear manifold of H 2 (∆) which consists of all functions f (z) with f (z) ∈ H 2 (∆), where H 2 (∆) is the following Hilbert space of analytic functions: where ∆ = {z ∈ C : |z| < 1}.(iii) The abstract form of [ f (z)] 2 is the element f (V ) f , where (2.7) It was also proved in [4, page 355], [5, page 386] that the operator N( f ) is Frechét differentiable at every point Due to the above known results, we can easily prove that (iv) the abstract form of f (qz) is the element

Proof of Theorem 1.1
Proof.First of all we set x = z/T, f (z According to what is mentioned in Section 2, the abstract form of (3.1) in H 1 is Eugenia N. Petropoulou 861 where F is the abstract form in H 1 of F(x) and F(V ) is defined by (2.7).Equation (3.3) can also be written as where B 0 is defined by (2.5).Since |q| < 1, it follows that qQ * 1 = |q| < 1.Thus (I + qQ * ) −1 exists, is uniquely determined on all H 1 , and is bounded by Thus from (3.4), we obtain We would like to apply to (3.6) the fixed point theorem of Earle and Hamilton [1] which states that if g : X → X is holomorphic, that is, its Fréchet derivative exists, and g(X) lies strictly inside X, then g has a unique fixed point in X, where X is a bounded, connected, and open subset of a Banach space B. According to what is mentioned in Section 2, it is obvious that the mapping φ 1 (F) is Frechét differentiable.
Let F 1 ≤ R, R sufficiently large but finite.Then we find from (3.6) (3.