Ulam Stability for Fractional Differential Equation in Complex Domain

and Applied Analysis 3 Theorem 2.1. Consider the problem 2.1 with a2 < α, 1 < α ≤ 2, and there exists a constant δ > 0 satisfying the condition |an 2| ≤ Γ n α 1 ( n α − a2Γ n 1 δΓ n 1 |cn|, 2.2 for all sufficiently large integers n, where cn ⎧ ⎪⎨ ⎪⎩ a0 Γ α 1 − a2 , n 0 an − cn−2 Γ n α 1 /Γ n 1 n α − a2 , n ≥ 1, 2.3 where c−1 0. Then every solution u : U → C of the fractional differential equation 2.1 can be expressed by


Introduction
Fractional calculus is a rapidly growing subject of interest for physicists and mathematicians. The reason for this is that problems may be discussed in a much more stringent and elegant way than using traditional methods. Fractional differential equations have emerged as a new branch of applied mathematics which has been used for many mathematical models in science and engineering. In fact, fractional differential equations are considered as an alternative model to nonlinear differential equations 1-7 .
The class of fractional differential equations of various types plays important roles and tools not only in mathematics but also in physics, control systems, dynamical systems, and engineering to create the mathematical modeling of many physical phenomena. Naturally, such equations required to be solved. Many studies on fractional calculus and fractional differential equations, involving different operators such as Riemann-Liouville operators, Erdlyi-Kober operators, Weyl-Riesz operators, Caputo operators, and Grünwald-Letnikov operators, have appeared during the past three decades with its applications in other field. Moreover, the existence and uniqueness of holomorphic solutions for nonlinear fractional differential equations such as Cauchy problems and diffusion problems in complex domain are established and posed 8-15 . The present article deals with a nonhomogeneous fractional differential equation. The nonhomogeneous fractional differential equations involving the Bessel differential equation where c −1 0. Then every solution u : U → C of the fractional differential equation 2.1 can be expressed by where u h z is a solution of the homogeneous Proof. We assume that u : U → C is a function given in the form 2.4 , and we define that u p z u z − u h z ∞ n 0 c n z n α . Then, it follows from 2.2 and 2.3 that That is, the power series for u p z converges for all z ∈ U. Hence, we see that the domain of u z is well defined. We now prove that the function u p z satisfies the nonhomogeneous equation 2.1 . Indeed, it follows from 2.3 that  In this section, we consider the Hyers-Ulam stability for fractional differential equation 2.1 . Let H be the space of all analytic functions on U.

Ulam Stability
We need the following results. Proof. In virtue of Theorem 2.1, 2.2 has a holomorphic solution in the U. According to Lemma 3.2, we have 3.4 Let > 0 and w ∈ U be such that ∞ n 0 a n w n α ≤ ∞ n 0 |a n | p 2 n . 3.5 We will show that there exists a constant K independent of such that |w m − u m | ≤ K, w ∈ U, u ∈ U 3.6 and satisfies 3.1 . We put the function f w −1 λa m ∞ n 0, n / m a n w n α , a m / 0, 0 < λ < 1, 3.7 thus, for w ∈ ∂U, we obtain ∞ n 0, n / m a n w n α λ|w m − u m | 1 |a m | ∞ n 0 a n w n α λ|w m − u m |.

6 Abstract and Applied Analysis
Without loss of the generality, we consider |a m | max n≥1 |a n | yielding

3.9
This completes the proof.

3.12
Since |f z | < 1 for |z| 1, hence for |f z | | − z| and by Rouche's theorem, we observe that f z − z has exactly one zero in U, which yields that f has a unique fixed point in U.
Abstract and Applied Analysis 7

Conclusion
From above, we conclude that fractional differential equations of Bessel type have holomorphic solutions in the unit disk. The uniqueness imposed by employing the Rouche's theorem. Furthermore, this solution satisfied the generalized Ulam stability for infinite series of fractional power.