THE D'ALEMBERT AND LOBACZEVSKI DIFFERENCE OPERATORS IN X_lambda SPACES

Abstract

In this paper, by simple calculations we find norms of the d'Alembert and Lobaczevski difference operators connected with the d'Alembert and Lobaczevski functional equations. Because of nonlinearity of these operators, we use a norm for a quadratic operator and introduce a new class of operators, which are a sum of a linear and a quadratic operator, and provide a norm for this class. As an example, we find norms of these operators in X_lambda spaces. Then, we study Pexider type generalizations of the d'Alembert and Lobaczevski difference operators in X^4_lambda spaces. This paper is based on the article "Cauchy and Pexider operators in some function spaces" by S. Czerwik and K. Dlutek [2] and its continuation in a certains sense. The aim of the paper is drawing a reader's attention to a problem of a boundedness of a quadratic operator.