## Regular solutions of the Shabat equation.(English)Zbl 0929.34054

The author discusses the Shabat equation $f'(t)+q^2f'(qt)+f^2(t)-q^2f^2(qt)=\mu,\tag{1}$ where $$q$$ and $$\mu$$ are real parameters, $$q\in (-1,0)\cup (0,1)$$. A function $$f(t)$$ is called a global solution to (1) if $$f(t)\in C^1(\mathbb{R})$$ and it satisfies (1) for every $$t\in\mathbb{R}$$. In the case $$\mu >0$$, a function $$f(t)$$ is called a regular solution to (1) if it is a global solution to (1) and it satisfies the asymptotic expansions $f(t)=\pm f^*+O(t^{-2})\qquad\text{and}\qquad f'(t)=O(t^{-3})\quad\text{as }t\to\pm\infty,\text{ with }f^*=\sqrt{\mu\big /(1-q^2)}.$ Considering the initial condition (2) $$f(0)=f_0$$ the author proves that the initial value problem (1), (2) has a unique solution in a neighborhood of the origin and set up conditions under which the solution is regular, resp. it is not global. It is shown that for $$\mu >0$$ any global solution $$f(t)$$ to (1) satisfying $$f(t)\to\pm f^*$$ as $$t\to\pm\infty$$ is a regular solution.
The author discusses also some related topics, including $$q$$-difference equations, neutral equations and Dirichlet series.

### MSC:

 34K12 Growth, boundedness, comparison of solutions to functional-differential equations 34K05 General theory of functional-differential equations 34K40 Neutral functional-differential equations
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### References:

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