Our main aim in this paper is to use the technique of nonexpansive operators in more general iterative and noniterative fractional differential equations (Cauchy type). The noninteger case is taken in sense of the Riemann-Liouville fractional operators. Applications are illustrated.

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

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 [

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 [

The fractional (arbitrary) order integral of the function

The fractional (arbitrary) order derivative of the function

From Definitions

The Caputo fractional derivative of order

Note that there is a relationship between Riemann-Liouville differential operator and the Caputo operator

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

The following result is a fixed point theorem for non expansive mappings, according to Berinde; see for example [

Let

Let

Let

Edelstein [

We need the following results, which can be found in [

Let

Then

Let

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

In this section, we establish the existence and uniqueness results for the fractional differential equation

Assume that the following conditions are satisfied for the initial value problem (

if

one of the following conditions holds:

If

Consider the integral operator

Our aim is to show that

First we show that

Thus

Hence

where

Now, by taking the supremum in the last assertion, we get

If

then

Next we establish the solution of (

It is clear that

Assume that the following conditions are satisfied.

If

There exists a

If (A2), (A4) hold then there exists at least one solution of problem (

We assume the Banach space

Let

Now we prove that

that is,

Let

This yields

where

Thus we have

which shows that

We proceed to show that

is strictly increasing on

Similarly for the function

then

Now the function

is strictly decreasing on

For

which implies that

But since

for sufficient

Consequently, we receive

This shows that

Similar argument holds when

Consider the following initial value problem associated to an fractional iterative differential equation

To satisfy (A4(a)), we have

Hence (A4(a)) is satisfied. The function

Therefore, by Theorem

If we consider the function

Therefore, again by Theorem

Again, we consider the problem (

Hence in view of Theorem

We can observe that problem (

Finally, problem (

As such iterative fractional differential equations are used to generalize the model infective disease processes, pattern formation in the plane, and are important in investigations of dynamical systems, future works will be also devoted to them.

The author is thankful to the anonymous referee for his/her helpful suggestions for the improvement of this paper.