On the Bounded Solutions of Volterra Equations

Abstract

We extend the method of sums of commuting operators to the study of the existence and uniqueness of bounded solutions of Volterra equations of the form with bounded f in the infinite dimensional case. The main results are necessary and sufficient conditions for the above equations to have a unique bounded solution with spectrum not intersecting the spectrum of the equation under consideration. Applications are made to illustrate the main results.

Keywords:

• Volterra equations
• Sums of commuting operators
• Spectrum of functions
• Bounded solutions
• Periodic solutions
• Quasi-periodic solutions
• Almost periodic solutions
• 1991 Mathematics Subject Classifications: Primary 45D05
• 45N05
• Secondary 34G10
• 34K30

Acknowledgement

This article was written when Nguyen van Minh was holding a visiting position at the University of Electro-Communications, Tokyo. He thanks the University for the generous support and the faculty members in the Department of Mathmetics for their warm hospitality.

Notes

^†E-mail: [email protected]

* This article is dedicated to 60th anniversary of the birthday of Professor P.P. Zabreiko

* This article is dedicated to 60th anniversary of the birthday of Professor P.P. Zabreiko

** E-mail: [email protected]

Additional information

Notes on contributors

James H Liu **

†E-mail: [email protected] * This article is dedicated to 60th anniversary of the birthday of Professor P.P. Zabreiko ** E-mail: [email protected]