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Modified fractional Cauchy problem in a complex domain
Advances in Difference Equations volume 2013, Article number: 149 (2013)
In this paper, we make an extension to the Srivastava-Owa fractional operators in the space , where U is the open unit disk and H is a complex Hilbert space. Some recurrent relations are imposed on these extended operators. Moreover, by employing the theory of sums of accretive operators, the existence and uniqueness of the solution of the fractional Cauchy problem (in the sense of extended Srivastava-Owa fractional operators) is studied in a complex Hilbert space. Applications are illustrated.
Fractional derivatives can express the properties of memory and heredity of materials, which is the chief benefit of fractional derivatives compared with integer-order derivatives. Practical problems require definitions of fractional derivatives allowing the use of physically interpretable initial conditions. Fractional time derivatives are linked with irregular sub-diffusion, where a darken of particles spreads more slowly than a classical diffusion. While the fractional space derivatives are used to model irregular diffusion or dispersion, where a particle follow spreads at a rate not in agreement with the classical Brownian motion model, and the follow can be asymmetric [1–3].
Fractional differential and integro-differential equations occur from different real processes, and phenomena arise in physics such as signal processing and image processing, optics, engineering, control system, computer science (such as real neural network, complex neural network, information technology), statistics and probability, astronomy, geophysics, hydrology, chemical technology, materials, robots, earthquake analysis, electric fractal network, statistical mechanics, biotechnology, medicine, and economics [4–10].
Fractional Cauchy problems restore the integer time derivative by its fractional complement. Nigmatullin  posed a physical derivation of the fractional Cauchy problem; Kochubei  introduced the mathematical study of fractional Cauchy problems; Meerschaert et al.  constructed the stochastic solutions for fractional Cauchy problems in a bounded domain; Zaslavsky , falsified fractional Cauchy problems as a model for Hamiltonian chaos; Kexue and Jigen , concerned with fractional abstract Cauchy problems with order , proposed the sufficient conditions for the existence and uniqueness of mild solutions and strong solutions of the inhomogeneous fractional Cauchy problem; Li et al.  established an existence theorem for mild solutions to the nonlocal Cauchy problem by virtue of measure of noncompactness and the fixed point theorem for condensing maps; Zhong et al.  studied the Cauchy problem for some local fractional abstract differential equation with fractal conditions; Yang  considered the problem for local fractional derivatives from local fractional functional analysis theory; finally, local fractional Cauchy formula within fractal complex domain was investigated in [19–21]. Recently, the author studied the fractional Cauchy problem in a complex domain [22–26].
In this article, we shall make an extension to the Srivastava-Owa fractional operators in the space , where U is the open unit disk and H is a complex Hilbert space. Some properties are discussed such as the recurrent relations. Moreover, by applying the theory of sums of accretive operators, the existence and uniqueness of the solution of the fractional Cauchy problem (in the sense of extended Srivastava-Owa fractional operators) is established in a complex Hilbert space. Applications are introduced.
2 Fractional calculus
In , Srivastava and Owa provided the definitions for fractional operators (derivative and integral) in the complex z-plane ℂ as follows.
Definition 2.1 The fractional derivative of order α is defined for a function by
where the function is analytic in a simply-connected region of the complex z-plane ℂ containing the origin, and the multiplicity of is removed by requiring to be real when .
Definition 2.2 The fractional integral of order α is defined, for a function , by
where the function is analytic in a simply-connected region of the complex z-plane (ℂ) containing the origin, and the multiplicity of is removed by requiring to be real when .
Remark 2.1 From Definitions 2.1 and 2.2, we have
In this note, we are concerned about the following fractional Cauchy problem (in the sense of the Srivastava-Owa operator):
where is a closed densely defined linear operator on a complex Hilbert space H, is a bounded operator defined everywhere in , and , , . Denote by and . For complex Hilbert spaces and H with the inner product and respectively, let be the space of all bounded linear operators from to H; if , we write . Recall that the operator P is called accretive if , , and m-accretive if , . Denoted by , the resolvent set of the operator A. Note that the resolvent set of a bounded linear operator A is an open set. Moreover, the space is a Hilbert space with the inner product
Throughout the paper, we consider , and .
Definition 2.3 Equation (1) has maximal regularity in if for every , such that
where is the Sobolev space defined by
By employing the concept of sums of accretive operators, we shall prove the maximal regularity of problem (1).
We proceed to extend the fractional integral operator to the space . Define the fractional integral operator by
where . We have the following property.
Lemma 2.1 .
Proof By making use of the Young inequality, it follows that
Similarly, we extend the fractional integral operator to the space by the operator
Furthermore, we define the space as follows:
Lemma 2.2 Let , then
Proof For a function f, using the Dirichlet technique yields
Let , we impose
Thus we have
Lemma 2.3 .
Proof By Lemma 2.2, we have
From the last assertion, we conclude that . □
Lemma 2.4 Let , then .
then, by using integration by parts, we get
Combining the last two assertions, we end the proof. □
Remark 2.2 For a special case , we have the relation
Note that the initial condition of problem (1) implies that of the form
(this class of analytic functions has wide applications in the geometric function theory and the univalent function theory when (see )); hence we obtain
By virtue of the last discussion, we have the following result.
Lemma 2.5 Let , then , is an accretive operator.
Proof To prove that is an accretive operator, it suffices to show that , where u is in the domain of . By the definition of , we receive that ; consequently, this implies that
where . Hence, by Remark 2.2, we have
but u is in the domain of , so, consequently, is an accretive operator. □
Lemma 2.6 Let , then is an m-accretive operator.
Proof To prove that , it suffices to show that the function is well defined for all , and bounded in . A simple computation shows that
is a Mittag-Leffler function. Therefore, by applying the Young inequality, we conclude that
where . Thus is well defined for all , and bounded in . This implies that is an m-accretive operator. □
3 Existence and uniqueness
In this section, we study the maximal regularity of the fractional Cauchy problem (1)
under the following assumptions.
(H1) A is a linear m-accretive operator in H, , where H is a complex Hilbert space.
(H2) C is a bounded operator in and there exists with
and , .
We provide the main result of this section. We need the following result.
Lemma 3.1 
Let P be m-accretive and let Q be accretive Lipschitz continuous in a Hilbert space H. Then is m-accretive with .
Theorem 3.1 Let the assumptions (H1)-(H3) hold. Then there exists a unique function satisfying (1) and the inequality
Proof Our aim is to rewrite Eq. (1) as an operator equation in . For this purpose, we define the following operators in :
thus Eq. (1) becomes
Since (H1), therefore, we let , then we pose
Our point is to prove that (9) has a unique solution. It suffices to show that is m-accretive. For the accretivity, we must show that
According to Lemma 2.5 and Lemma 2.6, we obtain that is m-accretive. Moreover, by (H2) it follows that the operator is a bounded accretive operator. By virtue of Lemma 3.1, it follows that , , is m-accretive. Together with (H3), we obtain that Eq. (9) has a unique solution . □
Next, we proceed to prove the inequality (7). By taking the inner product of (9) with w and using the accretivity of the operators and , we observe that
and by applying the Cauchy-Schwarz inequality, we obtain
Moreover, by the boundedness of the operator ℭ, there exists a constant such that
Consequently, we have
Since is bounded in (Lemma 2.1) and by the fact that , it follows
Hence, from the last two inequalities, we conclude the inequality (9). This completes the proof.
Corollary 3.1 Let A be accretive and self-adjoint in H, and let the hypotheses (H2) and (H3) hold. Then there exists a unique function satisfying (1) and the inequality (7).
Proof It is well known that an accretive and self-adjoint operator implies a symmetric and m-accretive one . Hence, by virtue of Theorem 3.1, there exists a unique function satisfying (1) and the inequality (7). □
Corollary 3.2 Let C be accretive and Lipschitz continuous in H and let the hypotheses (H1) and (H3) hold. Then there exists a unique function satisfying (1) and the inequality (7).
Proof Since C is Lipschitz continuous in H, then for all and we have
Consequently, we get
where and are positive; thus in view of Theorem 3.1, there exists a unique function satisfying (1) and the inequality (7). □
Example 3.1 Consider the problem
where , , , , and . Obviously, by employing a fixed point argument to the linearized equation, we receive
thus we can define
It follows that A and C satisfy the conditions of Corollary 3.1 for some λ, and therefore (10) has a unique function .
Note that self-adjoint operators on a Hilbert space are applied in quantum mechanics to describe a physical observation such as the position, momentum, angular momentum and spin. The differential operators corresponding to the Legendre differential equation and the harmonic motion equation are self-adjoint, while those corresponding to the Laguerre differential equation and Hermite differential equation are not. A nonself-adjoint second-order linear differential operator can be viewed as a self-adjoint by using Sturm-Liouville theory.
Example 3.2 Consider the problem
such that and that . Let and
It is clear that A is a self-adjoint operator and C is bounded; thus it follows that the problem (11) has a unique solution.
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The author is thankful to the referees for helpful suggestions for the improvement of this article. This research has been funded by the university of Malaya, under Grant No. RG208-11AFR.
The author declares that she has no competing interests.
The author read and approved the final manuscript.
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Ibrahim, R.W. Modified fractional Cauchy problem in a complex domain. Adv Differ Equ 2013, 149 (2013). https://doi.org/10.1186/1687-1847-2013-149
- fractional calculus
- fractional differential equations
- Srivastava-Owa fractional operators
- unit disk
- analytic function
- Cauchy problem