Ulam Stability for Fractional Differential Equation in Complex Domain
The present paper deles with a fractional differential equation , , where in sense of Srivastava-Owa fractional operators. The existence and uniqueness of holomorphic solutions are established. Ulam stability for the approximation and holomorphic solutions are suggested.
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 which appears frequently in practical problems and applications. These equations have proved useful in many branches of physics and engineering. They have been used in problems of treating the boundary value problems exhibiting cylindrical symmetries.
In , Srivastava and Owa gave definitions for fractional operators (derivative and integral) in the complex -plane as follows.
Definition 1.1. The fractional derivative of order is defined, for a function by where the function is analytic in simply-connected region of the complex -plane containing the origin, and the multiplicity of is removed by requiring to be real when .
Definition 1.2. The fractional integral of order is defined, for a function , by where the function is analytic in simply-connected region of the complex -plane containing the origin, and the multiplicity of is removed by requiring to be real when .
2. Nonhomogeneous Fractional Differential Equation
In this section, we will express the solution for the nonhomogeneous problem where the radius of convergence of power series is at least , for all and . Note that for , (2.1) reduces to the Bessel differential equation.
Theorem 2.1. Consider the problem (2.1) with , and there exists a constant satisfying the condition for all sufficiently large integers , where where . Then every solution of the fractional differential equation (2.1) can be expressed by where is a solution of the homogeneous
Proof. We assume that is a function given in the form (2.4), and we define that . Then, it follows from (2.2) and (2.3) that
That is, the power series for converges for all . Hence, we see that the domain of is well defined. We now prove that the function satisfies the nonhomogeneous equation (2.1). Indeed, it follows from (2.3) that
Hence is a particular solution of the nonhomogeneous equation (2.1). On the other hand, since every solution to (2.1) can be expressed as a sum of a solution of the homogeneous equation and a particular solution of the nonhomogeneous equation, every solution of (2.1) is certainly of the form (2.4).
3. Ulam Stability
A classical problem in the theory of functional equations is that if a function approximately satisfies functional equation when does, there exists an exact solution of which approximates. In 1960, Ulam  imposed the question of the stability of Cauchy equation, and, in 1941, Hyers solved it . In 1978, Rassias  provided a generalization of Hyers, theorem by proving the existence of unique linear mappings near approximate additive mappings. The problem has been considered for many different types of spaces (see [19–21]). Recently, Li and Hua  have discussed and proved the Hyers-Ulam stability of spacial type of finite polynomial equation, and Bidkham et al.  have introduced the Hyers-Ulam stability of generalized finite polynomial equation.
In this section, we consider the Hyers-Ulam stability for fractional differential equation (2.1). Let be the space of all analytic functions on .
Definition 3.1. Let be a real number. We say that has the generalized Hyers-Ulam stability if there exists a constant with the following property: for every , if then there exists some that satisfies (2.1) such that
We need the following results.
Lemma 3.2 (see ). If ( is a complex Banach space) is holomorphic, then is a subharmonic of . It follows that can have no maximum in unless is of constant value throughout .
Theorem 3.3. Let be holomorphic in the unit disk and then (2.1) has the generalized Hyers-Ulam stability.
Proof. In virtue of Theorem 2.1, (2.2) has a holomorphic solution in the . According to Lemma 3.2, we have
Let and be such that
We will show that there exists a constant independent of such that
and satisfies (3.1). We put the function
thus, for , we obtain
Without loss of the generality, we consider yielding This completes the proof.
Theorem 3.4. If then (2.1) has a unique solution in the unit disk.
Proof. By setting
for , we pose
Since for , hence for and by Rouche’s theorem, we observe that has exactly one zero in , which yields that has a unique fixed point in .
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.
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