On the Nonlinear Fractional Differential Equations with Caputo Sequential Fractional Derivative

0 y(0) = b 1 , where cDα 0 , cDα 0 are Caputo fractional derivatives, 0 < α 1 , α 2 ≤ 1, p > 1, and b 0 , b 1 ∈ R. Local existence of solutions is established by employing Schauder fixed point theorem. Then a growth condition imposed to f guarantees not only the global existence of solutions on the interval [0, +∞), but also the fact that the intervals of existence of solutions with any fixed initial value can be extended to [0, +∞). Three illustrative examples are also presented. Existence results for initial value problems of ordinary differential equations with p-Laplacian on the half-axis follow as a special case of our results.


Introduction
This paper deals with the following initial value problem for nonlinear fractional differential equation involving Caputo sequential fractional derivative: where    1 0 ,    2 0 are Caputo fractional derivatives, 0 <  1 ,  2 ≤ 1,  > 1, and  0 ,  1 ∈ R, and   (, ) is continuous on [0, +∞) × R, 0 ≤  <  2 .When  = 2, the equation in (1) becomes a sequential fractional differential equation.Here, we follow the definition of sequential fractional derivative presented by Podlubny [1]: where the symbol  ]  ( = 1, 2, . . ., ) means the Riemann-Liouville derivative or the Caputo derivative.It is easy to see that (2) is a generalized expression presented by Miller and Ross in [2].Fractional differential equations have been of great interest for the past three decades; see the monographs [1][2][3][4] and the papers of [5][6][7][8].This is due to the intensive development of the theory of fractional calculus itself as well as its applications.Apart from diverse areas of pure mathematics, fractional differential equations can be used in modeling of various fields of science and engineering such as rheology, self-similar dynamical processes, porous media, fluid flows, viscoelasticity, electrochemistry, control, electromagnetic, and many other branches of science.For details, see [9][10][11][12] and the references therein.
Recently, we note that the investigation for the existence of solutions of sequential fractional differential equations associated with a variety of initial and boundary value conditions has attracted the considerable attention of researchers.Here, we mention some works on them.In [13,14], the authors investigated a class of Riemann-Liouville sequential fractional differential equation (  0 )  +  ()  = 0,  ≥ 0, and obtained existence, nonexistence, and asymptotic property of the solutions by some nonlinear analysis methods.In 2 Advances in Mathematical Physics [15,16], the authors considered the existence and uniqueness of the solutions for initial value problems involving Riemann-Liouville sequential fractional derivative by using monotone iterative method and fixed point method.As for sequential fractional differential equations associated with boundary value conditions, we refer the reader to [17][18][19][20][21][22].For example, Chen and Tang [21] considered the existence of the solutions for the following Caputo sequential fractional differential equation: with -point boundary value conditions by the coincidence degree continuation theorem.After that, in 2015, Jiang [22] investigated the following Riemann-Liouville sequential fractional differential equations with -Laplacian at resonance: where By the extension of the continuous theorem in [23] and constructing suitable operators, they obtained the existence of solutions satisfied integral boundary value conditions.
In view of the facts that the Laplace transform of the Caputo derivative allows utilization of initial values of classical integer-order derivatives and that the Caputo derivative of a constant is 0, sequential fractional differential equations involving the Caputo fractional derivative have more clear physical interpretations than those involving the Riemann-Liouville fractional derivative (see [3,4]).
To the best of our knowledge, there is no paper dealing with the existence of solutions of Caputo sequential fractional differential equations with initial value conditions.In our latest paper [24], by virtue of uniform Lipschitz continuity of   (, ⋅) on R for every  > 0, we proved the existence and uniqueness of solutions of problem (1).Now, in this paper, we are concerned with the initial value problem (1) without uniform Lipschitz continuity of   (, ⋅).By fractional Taylor expansion theorem, we first obtain an integral equation equivalent to the initial value problem (1), to which local existence of solutions is established utilizing Schauder fixed point theorem.Then a growth condition imposed to  guarantees not only the global existence of solutions on the interval [0, +∞), but also the fact that the intervals of existence of solutions with any fixed initial value can be extended to [0, +∞).In addition, existence results for initial value problems of ordinary differential equations with -Laplacian on the half-axis follow as a special case of our results.
The paper is organized as follows.In Section 2, we present some necessary definitions and preliminary results that will be used in our discussions.The main results and their proofs are given in Section 3. In Section 4, we will give three examples to illustrate our results.

Preliminaries
In this section, we introduce some basic definitions and notations (see the monographs [1,2] for further details) and give several useful preliminary results which are used throughout this paper.Definition 1.Let  > 0. The Riemann-Liouville fractional integral of a function  : (0, +∞) → R of order  is given by provided that the right-hand side is pointwise defined on (0, +∞).Here and in what follows Γ is the Gamma function.
The fractional integration operator    has that the following semigroup property holds for all  > 0,  > 0.
Definition 2. Let  > 0 and let  be the smallest integer that exceeds .The Riemann-Liouville fractional derivative of a continuous function  : (0, +∞) → R of order  is given by provided that the right-hand side is pointwise defined on (0, +∞).
Obviously, the Riemann-Liouville fractional differentiation is the left inverse of the Riemann-Liouville fractional integration for continuous function  in the following sense: for  > 0. However, it is not the right inverse.More precisely, we have the following fractional Taylor expansion theorem.
Definition 4. Let  > 0 and let  be the smallest integer that exceeds .The Caputo fractional derivative of a continuous function  : (0, +∞) → R of order  is given by provided that the right-hand side is pointwise defined on (0, +∞).
The definition of solutions of the initial value problem (1) is given as follows.
In order to study the existence of solutions of (1), we should transform problem (1) into an equivalent integral equation.We need the following two lemmas.Lemma 7. Let ,  > 0; then one has Proof.Let  0 > 0 and take  0 ∈ (0,  0 ).For || <  0 , we have Since  is continuous and bounded in the neighborhood of  0 , we conclude that where  = sup 0≤≤ 0 + 0 |()|.Combining ( 13), ( 14) with (15), we arrive at lim In addition, it is easy to see that is also continuous in [0, +∞) and where Proof.First we prove the necessity.Let  ∈ ([0, ]) be a solution of problem (1) and define and then  ∈ ([0, ]) and (0) =   ( 1 ).According to Definition 4 and Theorem 3, the differential equation of problem ( 1) can be transformed into the following form: Obviously,   ( Applying Definition 4 and Theorem 3 again, we have Obviously, Combining with Theorem 3, we have Therefore,  satisfies the integral equation (19).
Next, we prove the sufficiency.Let  ∈ ([0, ]) be a solution of the integral equation (19).Combining with Definition 1, (19) reduces to From Remark 9, we see that  (29) Similarly, applying the operator    1 to both sides of (29), we arrive at Therefore,  is a solution of (1) on [0, ].Summing up, we complete the proof of Proposition 10.
Since  can be chosen arbitrarily large in Proposition 10, according to Definition 6, we have the following result.
The following fixed point lemma is the main tool in the proofs of our results.

Lemma 14 (Schauder fixed point theorem).
Let Ω be a closed, convex, and nonempty subset of a Banach space , and let  : Ω → Ω be a mapping such that { :  ∈ Ω} is a relatively compact subset in .Then  has at least one fixed point in Ω.

Main Results
In this section, we will give and prove our main results in this paper.
Theorem 15.For any fixed initial values  0 and  1 , there exists a sufficiently small constant  * > 0 such that problem (1) has a solution on [0,  * ].
Proof.For any given positive constant , choose  * > 0 sufficiently small which will be determined later.Let Obviously, Ω 1 is a closed, convex, and nonempty subset of ([0,  * ]).On this set Ω 1 we define the operator : where According to Proposition 10, in what follows, it suffices to show that the operator  has a fixed point in Ω 1 .Firstly, we will show that  ∈ Ω 1 for any  ∈ Ω 1 .To this end we begin by noting that for any  ∈ Ω 1 , Φ  ∈ ([0,  * ]) by Remark 9. Then we obtain that  ∈ ([0,  * ]).Furthermore, for  ∈ [0,  * ] we have where (, )     . ( Now we can choose  * > 0 so small that which means that  ∈ Ω 1 ; that is,  maps the set 1 to itself.Secondly, we will also show that the family of functions Ω 1 = { :  ∈ Ω 1 } is a relatively compact set.That is to say, we need to show that Ω 1 is uniformly bounded and equicontinuous on [0,  * ].The uniform boundedness follows from the definition of Ω 1 .As for the equicontinuity, for 0 ≤  1 ≤  2 ≤  * and  ∈ Ω 1 , we see that where  > 0 is independent of  1 ,  2 , and .Therefore Ω 1 is uniformly bounded and equicontinuous on [0,  * ], and thus Ω 1 is a relatively compact subset of ([0,  * ]).By Lemma 14 there exists  ∈ Ω 1 such that  =  and  is a solution of problem ( 1) on [0,  * ].The proof is completed.
We have proved the local existence of solutions of problem (1) in Theorem 15.However, maximal intervals of existence of those solutions are not necessarily [0, +∞), which change according to initial values  0 and  1 .We give an example to illustrate it.
Example 16.Consider the following initial value problem for second-order ordinary differential equation where  ̸ = 0 is a constant.When  < 0, it is easy to see that the function is a solution of (38) on [0, +∞) and [0, +∞) is the maximum interval of existence.When  > 0, however, the function  is a solution of (38) on [0, ) and [0, ) is the maximum interval of existence.
Under certain growth condition on the nonlinearity  in (1), we will show that any solutions obtained in Theorem 15 can be extended to the interval [0, +∞) as solutions of problem (1).
In a special case, an estimate of increasing rate for solutions as  → +∞ is made.

𝛼 2 0
(0, (0)) = 0. Now we are ready to transform problem (1) into an equivalent integral equation.For the reader's convenience, we list two special notations that will be used in the following paper:  = /( − 1) and   () = || −2  for  ∈ R.Proposition 10.A function  defined in ([0, ]) is a solution of problem (1) if and only if it satisfies the following integral equation on [0, ]: