Hyers-Ulam Stability of $n^{th}$ Order Non-Linear Differential Equations with Initial Conditions

Abstract

In this paper, we investigate the Hyers - Ulam stability of a Generalized $n^{th}$ order Non Linear Differential Equation of the form $x^{(n)}(t) - F(t, x(t)) = 0$ with initial conditions $x(a) = x^{'}(a) = x^{''}(a) = ... = x^{(n-1)}(a) = 0$, where $x \in C^n (I), \ I = [a, b], \ -\infty<a<b<\infty$ and="" $\left|{f(t,="" x(t))}\right|="" \leq="" \="" l="" \left|{x^{(n-2)}(t)}\right|^{\alpha}$,="" $\alpha=""> 0, -\infty< x <\infty,$ with $F(t, 0) = 0$. Moreover, we prove the Hyers - Ulam stability of the Emden - Fowler type differential equation of $n^{th}$ order $x^{(n)}(t) - h(t) \ \left|x(t)\right|^{\alpha} \ sgn \ x(t) = 0$, with the initial conditions $x(a) = x^{'}(a) = x^{''}(a) = ... = x^{(n-1)}(a) = 0$. Where $x \in C^n (I), \ I = [a, b], \ -\infty<a<b<\infty$, $\alpha="">0$, $\alpha \neq 1$ and $h(t)$ is bounded in $\mathbb{R}$.</a<b<\infty$,></a<b<\infty$>