Initial Value Problem For Nonlinear Fractional Differential Equations With -Caputo Derivative via Monotone Iterative Technique

In this article, we discuss the existence and uniqueness of extremal solutions for nonlinear initial value problems of fractional differential equations involving the ψ-Caputo derivative. Moreover, some uniqueness results are obtained. Our results rely on the standard tools of functional analysis. More precisely we apply the monotone iterative technique combined with the method of upper and lower solutions to establish sufficient conditions for existence as well as the uniqueness of extremal solutions to the initial value problem. An illustrative example is presented to point out the applicability of our main results.

At the present day, different kinds of fixed point theorems are widely used as fundamental tools in order to prove the existence and uniqueness of solutions for various classes of nonlinear fractional differential equations for details, we refer the reader to a series of papers [24][25][26][27][28][29][30] and the references therein, but here we focus on those using the monotone iterative technique, coupled with the method of upper and lower solutions. This method is a very useful tool for proving the existence and approximation of solutions to many applied problems of nonlinear differential equations and integral equations (see [31][32][33][34][35][36][37][38][39][40][41][42]). However, as far as we know, there is no work yet reported on the existence of extremal solutions for the Cauchy problem with ψ-Caputo fractional derivative. Motivated by this fact, in this paper we deal with the existence and uniqueness of extremal solutions for the following initial value problem of fractional differential equations involving the ψ-Caputo derivative: where c D α;ψ a + is the ψ-Caputo fractional derivative of order α ∈ (0, 1], f : [a, b] × R −→ R is a given continuous function and a * ∈ R.
The rest of the paper is organized as follows: in Section 2, we give some necessary definitions and lemmas. The main results are given in Section 3. Finally, an example is presented to illustrate the applicability of the results developed.

Preliminaries
In this section, we introduce some notations and definitions of fractional calculus and present preliminary results needed in our proofs later.
We begin by defining ψ-Riemann-Liouville fractional integrals and derivatives. In what follows, 8,17]). For α > 0, the left-sided ψ-Riemann-Liouville fractional integral of order α for an integrable function x : J −→ R with respect to another function ψ : J −→ R that is an increasing differentiable function such that ψ (t) = 0, for all t ∈ J is defined as follows where Γ is the classical Euler Gamma function.

Definition 2 ([17]
). Let n ∈ N and let ψ, x ∈ C n (J, R) be two functions such that ψ is increasing and ψ (t) = 0, for all t ∈ J. The left-sided ψ-Riemann-Liouville fractional derivative of a function x of order α is defined by where n = [α] + 1.

Definition 3 ([17]
). Let n ∈ N and let ψ, x ∈ C n (J, R) be two functions such that ψ is increasing and ψ (t) = 0, for all t ∈ J. The left-sided ψ-Caputo fractional derivative of x of order α is defined by where n = [α] + 1 for α / ∈ N, n = α for α ∈ N. To simplify notation, we will use the abbreviated symbol From the definition, it is clear that We note that if x ∈ C n (J, R) the ψ-Caputo fractional derivative of order α of x is determined as (see, for instance, [17], Theorem 3).
Theorem 1 (Weissinger's fixed point theorem [44]). Assume (E, d) to be a non empty complete metric space and let β j ≥ 0 for every j ∈ N such that ∑ n−1 j=0 β j converges. Furthermore, let the mapping T : for every j ∈ N and every u, v ∈ E. Then, T has a unique fixed point u * . Moreover, for any v 0 ∈ E, the sequence {T j v 0 } ∞ j=1 converges to this fixed point u * .

Main Results
Let us recall the definition and lemma of a solution for problem (1). First of all, we define what we mean by a solution for the boundary value problem (1).
, for each t ∈ J and the condition For the existence of solutions for problem (1) we need the following lemma for a general linear equation of α > 0, that generalizes expression (3.1.34) in [8].
Lemma 4. For a given h ∈ C(J, R) and α ∈ (n − 1, n], with n ∈ N, the linear fractional initial value problem has a unique solution given by Moreover, the explicit solution of the Volterra integral equation (6) can be represented by where E α,β (·) is the two-parametric Mittag-Leffer function defined in (4).
Proof. Since α ∈ (n − 1, n], from Lemma 2 we know that the Cauchy problem (5) is equivalent to the following Volterra integral equation Note that the above equation can be written in the following form where the operator T is defined by Let n ∈ N and x, y ∈ C(J, R). Then, we have . . .
Hence, we have it follows that the mapping T n is a contraction. Hence, by Weissinger's fixed point theorem, T has a unique fixed point. That is (5) has a unique solution. Now we apply the method of successive approximations to prove that the integral Equation (6) can be expressed by For this, we set It follows from Equation (8) and Lemma 3 that Similarly, Equations (8) and (9) and Lemmas 1 and 3 yield Continuing this process, we derive the following relation Taking the limit as n → ∞, we obtain the following explicit solution x(t) to the integral Equation (6): Taking into account (4), we get Then the proof is completed.

Definition 7.
A function x 0 ∈ C(J, R) is said to be a lower solution of the problem (1), if it satisfies Theorem 2. Let the function f ∈ C(J × R, R). In addition assume that: (H 1 ) There exist x 0 , y 0 ∈ C(J, R) such that x 0 and y 0 are lower and upper solutions of problem (1), respectively, with x 0 (t) ≤ y 0 (t), t ∈ J. (H 2 ) There exists a constant r ∈ R such that Then there exist monotone iterative sequences {x n } and {y n }, which converge uniformly on the interval J to the extremal solutions of (1) in the sector [x 0 , y 0 ], where Proof. First, for any x 0 (t), y 0 (t) ∈ C(J, R), we consider the following linear initial value problems of fractional order: and c D α;ψ a + y n+1 (t) = f (t, y n (t)) − r(y n+1 (t) − y n (t)), t ∈ J, y n+1 (a) = a * .
By Lemma 4, we know that (13) and (14) have unique solutions in C(J, R) which are defined as follows: We will divide the proof into three steps.
Note that the sequence {x n (t)} is monotone nondecreasing and is bounded from above by y 0 (t). Since the sequence {y n (t)} is monotone nonincreasing and is bounded from below by x 0 (t), therefore the pointwise limits exist and these limits are denoted by x * and y * . Moreover, since {x n (t)}, {y n (t)} are sequences of continuous functions defined on the compact set [a, b], hence by Dini's theorem [46], the convergence is uniform. This is lim n→∞ x n (t) = x * (t) and lim n→∞ y n (t) = y * (t), uniformly on t ∈ J and the limit functions x * , y * satisfy problem (1). Furthermore, x * and y * satisfy the relation x 0 ≤ x 1 ≤ · · · ≤ x n ≤ x * ≤ y * ≤ · · · ≤ y n ≤ · · · ≤ y 1 ≤ y 0 .
Step 3: We prove that x * and y * are extremal solutions of problem (1) in [x 0 , y 0 ].
Using the same method, we can show that Hence, we have Therefore, (19) holds on J for all n ∈ N. Taking the limit as n → ∞ on both sides of (19), we get Therefore x * , y * are the extremal solutions of (1) in [x 0 , y 0 ]. This completes the proof. Now, we shall prove the uniqueness of the solution of the system (1) by monotone iterative technique. Theorem 3. Suppose that (H1) and (H2) are satisfied. Furthermore, we impose that: (H3) There exists a constant r * ≥ −r such that for every x 0 ≤ x ≤ y ≤ y 0 , t ∈ J. Then problem (1) has a unique solution between x 0 and y 0 .
Therefore, x * ≡ y * is the unique solution of the Cauchy problem (1) in [x 0 , y 0 ]. This ends the proof of Theorem 3.
As a direct consequence of the previous result, we arrive at the following one Corollary 1. Suppose that (H1) is satisfied and that f ∈ C(E, R), is differentiable with respect to x and ∂ f /∂x ∈ C(E, R), with Then problem (1) has a unique solution between x 0 and y 0 .
Proof. The proof follows immediately from the fact that E is a compact set and, as a consequence,

An Example
Example 1. Consider the following problem: Note that, this problem is a particular case of IVP (1), where and f : J × R −→ R given by Taking x 0 (t) ≡ 0 and y 0 (t) = 1 + t, it is not difficult to verify that x 0 , y 0 are lower and upper solutions of (20), respectively, and x 0 ≤ y 0 . So (H 1 ) of Theorem 2 holds On the other hand, it is clear that the function f is continuous and satisfies for all t ∈ [0, 1] and 0 ≤ x ≤ t + 1.
Hence, by Corollary 1, the initial value problem (20) has a unique solution u * and there exist monotone iterative sequences {x n } and {y n } converging uniformly to u * . Furthermore, we have the following iterative sequences We notice that the sequences are obtained by solving a recurrence formula of the type v n+1 = A v n , with A a suitable integral operator and v 0 given. So, by a simple numerical procedure, it is not difficult to represent some iterates of the recurrence sequence. We plot in Figure 1 the four first iterates of each sequence. We point out that the unique solution is lying within x 3 and y 3 which gives us a good approximation of such a solution.

Conclusions
In previous sections, we have presented the existence and uniqueness of extremal solutions to a Cauchy problem with ψ-Caputo fractional derivative. Moreover, some uniqueness results are obtained. The proof of the existence results is based on the monotone iterative technique combined with the method of upper and lower solutions. Moreover, an example is presented to illustrate the validity of our main results. Our results are not only new in the given configuration but also correspond to some new situations associated with the specific values of the parameters involved in the given problem.