| 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. |
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