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Hyers-Ulam stability for an nth order differential equation using fixed point approach
CHOONKIL PARK,Murali Ramdoss,Ponmana Selvan Arumugam
Keywords:differential equation; Hyers-Ulam stability; Hyers-Ulam-Rassias stability; fixed point method
Abstract:
      In this paper, we prove the Hyers-Ulam stability and the Hyers-Ulam-Rassias stability of the $n^{th}$ order differential equation of the form $$x^{(n)}(t) = f(t, x(t))$$ and $$x^{(n)}(t) = f \left( t, x(t), x'(t), x'(t),\cdots , x^{(n-1)}(t) \right) $$ with initial conditions $$x(a) = x_0 , x'(a) = x_1 , x'(a) = x_2 , \cdots , x^{(n-1)}(a)= x_{n-1}$$ for all $t \in I = [a, b] \subset \mathbb{R}$ and $x \in C^{(n)}(I)$ by using fixed point method in the sense of Cadariu and Radu.