Abstract
In this paper, we consider a class of nonlinear time fractional partial differential equations with delay. We obtain the existence and uniqueness of the mild solutions for the problem by the theory of solution operator and the general Banach contraction mapping principle. We need not extra conditions to ensure the contraction constant . Therefore, under some general conditions, we obtain our main results.
1. Introduction
Fractional derivatives can describe the property of the memory, and they have more advantages than integer-order derivatives. Therefore, fractional differential equations have been successfully applied in many fields, such as engineering and physics. About the fractional differential equations, we refer to these papers [1–9] and the references therein. In [7, 10], the authors studied the existence results of the fractional integrodifferential equations of order In [11, 12], the authors considered a class of fractional differential equations, where the fractional derivative operator is with fractional order and is a closed densely defined operator in a Banach space. Goufo [13] studied the existence results for a class of fractional fragmentation model by theory of strongly continuous solution operators. In [14, 15], the authors investigated a class of space-time fractional diffusion equations, while in [16] the authors studied a class of linear fractional differential equations by the variational iteration method and the Adomian decomposition method. In [17–19], the authors studied the following fractional partial differential equations:Ouyang [17] studied the existence of the local mild solutions for such problem by Leray-Schauder’s fixed theorem. Zhu et al. [18, 19] also studied the existence of the mild solutions by strict set contraction and Banach contraction mapping theorem of the problem. Li et al. [20] investigated the following fractional differential equations:where In [20], the Lipschitz coefficient of the nonlinear function is a constant.
Inspired by the above said work, we investigate the following nonlinear fractional partial differential equations with delay:where , is Caputo’s fractional derivative of order , is the Riemann-Liouville fractional integral of order , is a nature number, is a bounded domain with regular boundary , are continuous functions, and these functions satisfy ,
In this paper, we consider the existence results of the mild solutions of problem (3) by general Banach contraction mapping theorem. We need not extra conditions to ensure the contraction constant . Under some general conditions, we obtain our main results. Therefore, our results presented in this paper improve many classical results.
2. Preliminaries
Let be a Banach space and let is be a Banach space with norm .
Definition 1 (see [21, 22]). The Riemann-Liouville fractional integral of a function of order is defined aswhere .
For convenience, we let and then
Definition 2 (see [21, 22]). The Riemann-Liouville fractional derivative of a function of order is defined as where .
Definition 3 (see [21, 22]). The Caputo fractional derivative of function of order is defined as where .
For the Riemann-Liouville fractional integral operator and the Caputo fractional derivative operator, we have
Definition 4 (see [23]). Let be a closed linear operator with dense domain in a Banach space ; . A family of bounded linear operators in is called a solution operator for the integral equation if the following conditions are satisfied:(i) is strongly continuous on and (ii) and for all and (iii) is a solution offor all
We call the infinitesimal generator of or say that generates .
Let ; then the fractional partial differential equation (3) can be rewritten in the following abstract form:where and is defined by for and
It is easy to see that generates a semigroup on . Theorem 3.1 in [23] means that is the infinitesimal generator of a solution operator .
Definition 5 (see [24]). If is a mild solution of problem (12), then satisfies the following integral equations:
3. Main Results
We define the operator by
Theorem 6. Assume that the conditions and hold.
There exists a real number such that
The function is continuous in on , and there exist nonnegative Lebesgue integrable functions such thatwhere .
Then problem (12) has a unique mild solution , which means that (3) has a unique mild solution.
Proof. For any , by the property of the Lebesgue integrable function, there exists a continuous function such that , where . From conditions - and (15), for any , we obtainwhere Next, for any nature number , we will prove the following inequality:Obviously, for , (18) holds. Assume that and (18) holds; that is,By (15), -, and formula , we have By mathematical induction, we obtain that (18) holds for . Therefore, for any , we havewhere . It is easy to see that Therefore, we choose sufficiently large nature number such thatFor any nature number , such that . Obviously, . For any sufficiently large positive integer ; by the Stirling formula and by (23), we haveOn the other hand, without loss of generality, we assume that . By Stirling formula and , we getwhere . By (21), (25), and (26), we get Therefore, for fixed constant , there exists a positive integer such that, for any , we have By general Banach contraction mapping principle, for operator there exists a unique fixed point , which means that problem (3) has a unique mild solution.
Remark 7. In Theorem 6, we obtain the existence and uniqueness of the global mild solution of problem (3) under the uniform Lipschitz condition of the function . In next Theorem 8, we assume that the function satisfies the local Lipschitz condition.
Theorem 8. Let be continuous in for , and satisfies the following local Lipschitz contition: for any , there exist nonnegative Lebesgue integrable functions such that where with .
Then there exists a such that problem (3) has a unique mild solution on
Proof. For all fixed , there exists such that as . Let such that , where . Take . LetObviously, is a closed subset of ; thus is a Banach space. Now we consider the mapping byNext, we will prove that maps into .
Let ; we haveTherefore, maps into .
Let and ; we havewhere The following proof of the remainder is similar to the proof of Theorem 6. Therefore, for fixed constant , there exists a positive integer such that, for any , we have By general Banach contraction mapping principle, for operator there exists a unique fixed point , which means that is the unique mild solution of problem (12). That is, problem (3) has a unique mild solution.
4. An Application
Using the main results of this paper, we can solve the following time fractional partial differential equation with delay:where . Similar to Section 2, we can rewrite the time fractional partial differential equation (36) as the following abstract fractional evolution equation:Therefore, all the conditions of Theorems 6 and 8 are satisfied; for problem (36), there exists a unique mild solution.
5. Conclusion
This paper considers the existence and uniqueness of the mild solutions for a class of nonlinear fractional partial differential equations with delay by general Banach contraction mapping principle. We know that the Banach contraction mapping principle needs the special conditions to ensure the contraction constant . In this paper, we successfully overcome this condition. We need not extra conditions to ensure the contraction constant . Therefore, under some general conditions, we obtain the main results of this paper. Our results generalize and improve many classical results [18–20].
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Acknowledgments
This work was supported by the Project of Shandong Province Higher Educational Science and Technology Program (no. J16LI14) and by the National Natural Science Foundation of China (no. 11871302).