Regularization of a three-element functional equation

Abstract

In this paper we study the three-element functional equation

$$ (V\Phi )(z) \equiv \Phi (iz) + \Phi ( - iz) + G(z)\Phi \left( {\frac{1} {z}} \right) = g(z), z \in R, $$

, subject to

$$ R: = \{ z:\left| z \right| < 1, \left| {\arg z} \right| < \frac{\pi } {4}\} . $$

We assume that the coefficients G(z) and g(z) are holomorphic in R and their boundary values G ⁺(t) and g ⁺(t) belong to H(Γ), G(t)G(t ⁻¹) = 1. We seek for solutions Φ(z) in the class of functions holomorphic outside of $\bar R$ such that they vanish at infinity and their boundary values Φ⁻(t) also belong to H(Γ). Using the method of equivalent regularization, we reduce the problem to the 2nd kind integral Fredholm equation.