Ulam's problem for approximate homomorphisms in connection with Bernoulli's differential equation

Abstract

Ulam's problem for approximate homomorphisms and its application to certain types of differential equations was first investigated by Alsina and Ger. They proved in [C. Alsina, R. Ger, On some inequalities and stability results related to the exponential function, J. Inequal. Appl. 2 (1998) 373-380] that if a differentiable function f : I -> R satisfies the differential inequality vertical bar y'(t) - y(t)vertical bar <= epsilon, where I is an open subinterval of R, then there exists a solution f(0) : I R -> of the equation y'(t) = y(t) such that vertical bar(t) - fo(t)vertical bar <= 3 epsilon for any t epsilon I. In this paper, we investigate the Ulam's problem concerning the Bernoulli's differential equation of the form y(t)(-x)y'(t) + g(t)y(t)(1-x) + h(t) = 0. (C) 2006 Elsevier Inc. All rights reserved.