Existence of Iterative Cauchy Fractional Differential Equation

Fractional calculus and its applications (that is the theory of derivatives and integrals of any arbitrary real or complex order) are important in several widely diverse areas of mathematical, physical, and engineering sciences. It generalized the ideas of integer order differentiation and n-fold integration. Fractional derivatives introduce an excellent instrument for the description of general properties of various materials and processes.This is themain advantage of fractional derivatives in comparison with classical integer-order models, in which such effects are in fact neglected.The advantages of fractional derivatives become apparent in modeling mechanical and electrical properties of real materials, as well as in the description of properties of gases, liquids, rocks, and in many other fields. Our aim in this paper is to consider the existence and uniqueness of nonlinear Cauchy problems of fractional order in sense of Riemann-Liouville operators. Also, two theorems in the analytic continuation of solutions are studied. In the fractional Cauchy problems, we replace the first-order time derivative by a fractional derivative. Fractional Cauchy problems are useful in physics. Recently, the author studied the the fractional Cauchy problems in complex domain [1]. One of the most frequently used tools in the theory of fractional calculus is furnished by the Riemann-Liouville operators. The Riemann-Liouville fractional derivative could hardly pose the physical interpretation of the initial conditions required for the initial value problems involving fractional differential equations. Moreover, this operator possesses advantages of fast convergence, higher stability, and higher accuracy to derive different types of numerical algorithms (see [2]).


Introduction
Fractional calculus and its applications (that is the theory of derivatives and integrals of any arbitrary real or complex order) are important in several widely diverse areas of mathematical, physical, and engineering sciences.It generalized the ideas of integer order differentiation and -fold integration.Fractional derivatives introduce an excellent instrument for the description of general properties of various materials and processes.This is the main advantage of fractional derivatives in comparison with classical integer-order models, in which such effects are in fact neglected.The advantages of fractional derivatives become apparent in modeling mechanical and electrical properties of real materials, as well as in the description of properties of gases, liquids, rocks, and in many other fields.
Our aim in this paper is to consider the existence and uniqueness of nonlinear Cauchy problems of fractional order in sense of Riemann-Liouville operators.Also, two theorems in the analytic continuation of solutions are studied.In the fractional Cauchy problems, we replace the first-order time derivative by a fractional derivative.Fractional Cauchy problems are useful in physics.Recently, the author studied the the fractional Cauchy problems in complex domain [1].
One of the most frequently used tools in the theory of fractional calculus is furnished by the Riemann-Liouville operators.The Riemann-Liouville fractional derivative could hardly pose the physical interpretation of the initial conditions required for the initial value problems involving fractional differential equations.Moreover, this operator possesses advantages of fast convergence, higher stability, and higher accuracy to derive different types of numerical algorithms (see [2]).
Definition 2. The fractional (arbitrary) order derivative of the function  of order 0 ≤  < 1 is defined by Remark 3. From Definitions 1 and 2, we have + ,  > −1,  > 0. ( Definition 4. The Caputo fractional derivative of order  > 0 is defined, for a smooth function (), by where  = [] + 1, (the notation [] stands for the largest integer not greater than ).
Note that there is a relationship between Riemann-Liouville differential operator and the Caputo operator and they are equivalent in a physical problem (i.e., a problem which specifies the initial conditions, that is, if (0) = 0, then the Riemann-Liouville derivative and the Caputo derivative of order  ∈ (0, 1) coincide).

Preliminaries
We extract here the basic theory of nonexpansive mappings in order to offer the notions and results that will be needed in the next sections of the paper.Let (, ) be a metric space.A mapping  :  →  is said to be an ]-contraction if there exists ] ∈ [0, 1) such that  (, ) ≤ ] (, ) , ∀,  ∈ .
In the case where ] = 1, the mapping  is said to be nonexpansive.Let  be a nonempty subset of a real normed linear space  and  :  →  be a map.
The following result is a fixed point theorem for non expansive mappings, according to Berinde; see for example [4].Theorem 5. Let  be a nonempty closed convex and bounded subset of a uniformly Banach space .Then any nonexpansive mapping  :  →  has at least a fixed point.Definition 6.Let  be a convex subset of a normed linear space  and let  :  →  be a self-mapping.Given an  0 ∈  and a real number  ∈ [0, 1], the sequence   defined by the formula is usually called Krasnoselskij iteration or Krasnoselskij-Mann iteration.
Definition 7. Let  be a convex subset of a normed linear space  and let  :  →  be a self-mapping.Given an  0 ∈  and a real number   ∈ [0, 1], the sequence   defined by the formula is usually called Mann iteration.
Edelstein [5] proved that strict convexity of  suffices for the Krasnoselskij iteration converge to a fixed point of .While, Egri and Rus [6] proved that for any subset of , the Mann iteration converge to a fixed point of  when  is a nonexpansive mapping.
We need the following results, which can be found in [7].
Lemma 8. Let  be a convex and compact subset of a Banach space  and let  :  →  be a non-expansive mapping.If the Mann iteration process   satisfies the assumptions (a)   ∈  for all positive integers , Then   converges strongly to a fixed point of .

Existence Theorems and Approximation of Solutions
For most of the differential and integral equations with deviating arguments that appear in recent literature, the deviation of the argument usually involves only the time itself.However, another case, in which the deviating arguments depend on both the state variable  and the time , is of importance in theory and practice.Equations of the form are called iterative differential equations.These equations are important in the study of infection models and are related to the study of the motion of charged particles with retarded interaction (see [3,8,9]).
then there exists at least one solution of problem (11) in  , which can be approximated by the Krasnoselskij iteration where  ∈ (0, 1) and  1 ∈  , are arbitrary.
Our aim is to show that  has a fixed point in  , .We proceed to apply Schauder fixed point theorem or Banach fixed point theorem.
First we show that  , is invariant set with respect to , that is, ( , ) ⊂  , .In virtue of condition (A4(a)) and for all  ∈ ,  ∈  , , we have where Now, by taking the supremum in the last assertion, we get If ( L+1)  ℓ/Γ(+1) < 1, then  is a contraction mapping and hence in view of Banach fixed point theorem, (11) has a unique solution.Now if then  is non-expansive and, hence, continuous; thus Schauder fixed point theorem implies that (11) has a solution in  , .Finally, in view of Lemmas 8 and 9, we obtain the second part of the theorem.
Proof.We assume the Banach space [] endowed with Bielecki's norm is given by the formula Let  be defined as in the proof of Theorem 10.By assumptions (A2), (A4), and (A6), it follows that  ( ,, ) ⊂  ,, . ( Now we prove that  ,, is an invariant set with respect to the operator .Indeed, if  ∈  ,, and  ∈  then in view of (A5) and (A6), we have that is,  ∈  ,, .Let , V ∈  ,, and  ∈ , we have Thus we have which shows that  is Lipschitzian, hence continuous.By Schauder's fixed point theorem, it follows that  has at least one fixed point which is actually a solution of the initial value problem (11).We proceed to show that  is nonexpansive function.The function is strictly increasing on  and ( 0 ) = 0; furthermore, Similarly for the function Now the function is strictly decreasing on ; hence, For  ∈ (0, 1) and  ̸ =  0 then by the assumption (A7), there exists a  > 0 such that for sufficient , , , and  0 .Moreover, we have Consequently, we receive This shows that  is non-expansive.Similar argument holds when  =  0 , in (38) we have () =  > 0, hence ℎ is strictly increasing on .Finally, one can use Lemmas 8 and 9 to obtain the second part of the theorem.This completes the proof.

Example 12 .
Consider the following initial value problem associated to an fractional iterative differential equation  0.5  ()