Existence Results for a Class of Semilinear Fractional Partial Differential Equations with Delay in Banach Spaces

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 0 < k < 1. Therefore, under some general conditions, we obtain our main results.


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][2][3][4][5][6][7][8][9] and the references therein.In [7,10], the authors studied the existence results of the fractional integrodifferential equations of order 1 <  ≤ 2. In [11,12], the authors considered a class of fractional differential equations, where the fractional derivative operator is  0    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.
For the Riemann-Liouville fractional integral operator and the Caputo fractional derivative operator, we have Definition (see [23]).Let  be a closed linear operator with dense domain () in a Banach space ;  > 0. A family {  ()} ≥0 ⊂ () 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 R + and   (0) = .
We call  the infinitesimal generator of   () or say that  generates   ().
Remark .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.
where () =  * ∑     ().The following proof of the remainder is similar to the proof of Theorem 6.Therefore, for fixed constant  > 0, there exists a positive integer  0 such that, for any , V ∈   , we have By general Banach contraction mapping principle, for operator Q 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.

An Application
Using the main results of this paper, we can solve the following time fractional partial differential equation with delay: Therefore, all the conditions of Theorems 6 and 8 are satisfied; for problem (36), there exists a unique mild solution.

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 0 <  < 1.In this paper, we successfully overcome this condition.We need not extra conditions to ensure the contraction constant 0 <  < 1.Therefore, under some general conditions, we obtain the main results of this paper.Our results generalize and improve many classical results [18][19][20].