Holomorphic solutions of some functional equations related to nonlinear second order q-difference equations which has only one characteristic value

Abstract

Here we consider the following functional equation, $\mathrm{\Psi }(X(t,x,\mathrm{\Psi }(x)))=Y(t,x,\mathrm{\Psi }(x)),$ where $X(t,x,y)$ and $Y(t,x,y)$ are holomorphic functions in $t<\theta$, $|x|<\theta$, $|y|<\theta$, and x is a function of t such that $x=x(t)$.

If we have a solution of nonlinear second order difference equations, then we can write general solutions by making use of a solution Ψ.

In this present paper, we will prove the existence of a solution of the above functional equation, after we will show a relationship between the above functional equation and nonlinear second order q-difference equations which has only one characteristic value.

Keywords:

• Functional equations
• difference equations
• q-difference equations
• analytic solutions
• general solutions
• fixed point theorem

2010 Mathematics Subject Classifications:

• 39A13
• 39A45

Acknowledgements

The author would like to thank the University of Padova giving me a long period for study. I would like to greatly thank especially Prof. Francesco Baldassarri and Prof. Bruno Chiarellotto.

Disclosure statement

No potential conflict of interest was reported by the author(s).