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 $f:I\to \mathbb{R}$ satisfies the differential inequality ∣y′(t) − y(t)∣ ⩽ ε, where I is an open subinterval of $\mathbb{R}$, then there exists a solution $f_0:I\to \mathbb{R}$ of the equation y′(t) = y(t) such that ∣f(t) − f₀(t)∣ ⩽ 3ε for any t ∈ I.

In this paper, we investigate the Ulam’s problem concerning the Bernoulli’s differential equation of the form y(t)^−αy′(t) + g(t)y(t)^1−α + h(t) = 0.

Introduction

In 1940, Ulam gave a wide ranging talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of important unsolved problems (Ref. [11]). Among those was the question concerning the stability of homomorphisms: Let G₁ be a group and let G₂ be a metric group with a metric d(·, ·). Given any δ > 0, does there exist an ε > 0 such that if a function h : G₁ → G₂ satisfies the inequality d(h(xy), h(x)h(y)) < ε for all x, y ∈ G₁, then there exists a homomorphism H : G₁ → G₂ with d(h(x), H(x)) < δ for all x ∈ G₁?

In the following year, Hyers affirmatively answered in his paper [2] the question of Ulam for the case where G₁ and G₂ are Banach spaces. Furthermore, the result of Hyers has been generalized by Rassias (Ref. [9]). Since then, the stability problems of various functional equations have been investigated by many authors (see [3], [5]).

Assume that X is a normed space over a scalar field $\mathbb{K}$ and that I is an open interval. Let

$$P(D)=a_n{D}^n+a_{n-1}{D}^{n-1}+\cdots +a_{1}D+a_0$$

be an nth order differential operator, where we denote by D the differentiation with respect to t and $a_i:I\to \mathbb{K}$ is a function for every i ∈ {0, 1, …, n}.

Assume that for a fixed h : I → X and for any n times strongly differentiable function f : I → X satisfying the inequality

$$\Vert P(D)f(t)+h(t)\Vert ⩽\epsilon$$

for all t ∈ I and for some ε ⩾ 0, there exists a solution f₀ : I → X of the differential equation P(D)y(t) + h(t) = 0 such that ∥f(t) − f₀(t)∥ ⩽ K(ε) for any t ∈ I, where K(ε) is an expression for ε only. Then, we say that the above differential equation has the Hyers–Ulam stability.

If the above statement is also true when we replace ε and K(ε) by φ(t) and Φ(t), where φ, Φ : I → [0, ∞) are functions not depending on f and f₀ explicitly, then we say that the corresponding differential equation has the generalized Hyers–Ulam stability (or the Hyers–Ulam–Rassias stability).

We may apply these terminologies for other (linear or nonlinear) differential equations. For more detailed definitions of the Hyers–Ulam stability and the generalized Hyers–Ulam stability (or the Hyers–Ulam–Rassias stability), we refer the reader to [3], [4], [5].

Alsina and Ger were the first authors who investigated the Hyers–Ulam stability of differential equations: They proved in [1] that if a differentiable function $f:I\to \mathbb{R}$ is a solution of the differential inequality ∣y′(t) − y(t)∣ ⩽ ε, where I is an open subinterval of $\mathbb{R}$, then there exists a solution $f_0:I\to \mathbb{R}$ of the differential equation y′(t) = y(t) such that ∣f(t) − f₀(t)∣ ⩽ 3ε for any t ∈ I.

This result of Alsina and Ger has been generalized by Takahasi et al. They proved in [10] that the Hyers–Ulam stability holds true for the Banach space valued differential equation y′(t) = λy(t) (see also [7]).

Recently, the Hyers–Ulam stability of differential equations of the form a(t)y′(t) = y(t) was proved by Jung (see [6]).

Let I be an arbitrary real interval. Assume that $g\text{,}h:I\to \mathbb{C}$ are continuous functions and α ≠ 1 is a fixed real number. The Bernoulli’s differential equation

$$y^{\prime }(t)+g(t)y(t)+h(t)y(t{)}^{\alpha }=0$$

is one of the most well known equations in the theory of differential equations. If $y(t{)}^{-\alpha }\in \mathbb{C}$ holds for any t ∈ I, then the Bernoulli’s equation (1) is equivalent to

$$y(t{)}^{-\alpha }{y}^{\prime }(t)+g(t)y(t{)}^{1-\alpha }+h(t)=0\text{.}$$

In Theorem 1, we will investigate the generalized Hyers–Ulam stability of the differential equation (2). As a corollary to Theorem 1, we obtain the generalized Hyers–Ulam stability of the first order linear differential equations of the form y′(t) + g(t)y(t) + h(t) = 0.