A Mixed Monotone Operator Method for the Existence and Uniqueness of Positive Solutions to Impulsive Caputo Fractional Differential Equations

We establish some sufficient conditions for the existence and uniqueness of positive solutions to a class of initial value problem for impulsive fractional differential equations involving the Caputo fractional derivative. Our analysis relies on a fixed point theorem for mixed monotone operators. Our result can not only guarantee the existence of a unique positive solution but also be applied to construct an iterative scheme for approximating it. An example is given to illustrate our main result.

In [7], Ahmad and Sivasundaram considered the following impulsive hybrid boundary value problem for nonlinear fractional differential equations: where Based on contraction mapping principle and the Krasnoselskii fixed point theorem, they discussed some existence results for (2).
In [24], Benchohra and Slimani concerned the existence and uniqueness of solutions for the following initial value problem for Caputo fractional-order differential equations: where  = 1, . . ., , 0 , and ( +  ) = lim ℎ → 0 + (  + ℎ) and ( −  ) = lim ℎ → 0 − (  + ℎ) represent the right and left limits of () at  =   .They gave an existence and uniqueness result for the IVP (3) which was based on the Banach fixed point theorem and also obtained two existence and uniqueness results; the first one was based on the Schaefer fixed point theorem and the second one was based on the nonlinear alternative of the Leray-Schauder type.
Different from the above works [7,24], in this paper, we will use a fixed point theorem for mixed monotone operators to study the existence and uniqueness of positive solutions for the IVP (1).Our result can not only guarantee the existence of unique positive solution but also be applied to construct iterative scheme for approximating it.
With this context in mind, the outline of this paper is as follows.In Section 2, we will recall certain results from the theory of fractional calculus and some definitions, notations, and results of mixed monotone operators.In Section 3, we will provide some conditions under which the IVP (1) will have a unique positive solution.Finally, in Section 4, we will provide one example, which explicates the applicability of our main result.

Preliminaries
For the convenience of the reader, we present here some definitions, lemmas, and basic results that will be used in the proof of our main theorem.
In the sequel, we present some basic concepts in the ordered Banach spaces for completeness and one fixed point theorem which will be used later.For convenience of readers, we suggest that one refers to [23,25,26] for details.
Suppose that (, ‖ ⋅ ‖) is a real Banach space which is partially ordered by a cone  ⊂ .That is,  ≤  if and only if  −  ∈ .If  ≤  and  ̸ = , then we denote that  <  or  > .By  we denote the zero element of .Recall that a nonempty closed convex set is called normal if there exists a constant  > 0 such that, for all ,  ∈ ,  ≤  ≤  implies that ‖‖ ≤ ‖‖; in this case  is called the normality constant of .

Main Result
In this section, we apply Lemma 4 to study the IVP (1) and then we obtain a new result on the existence and uniqueness of positive solutions.The existence and uniqueness result is relatively new to the fractional differential equations in this literature.
From the very recent paper [21], the approach for finding the solution of impulsive fractional differential equations in [18] is inappropriate and some arguments like Lemma 5 are plausible.
Proof.To begin with, from Lemma 5, the IVP (1) has an integral formulation given by ( Define an operator  :   ×   → [0, ] by It is easy to prove that  is the solution of the IVP (1) if and only if  is the fixed point of .From ( 1 ), ( 4 ), and ( 5 ), we know that for any , V ∈   ,  ∈ [0, ].So,  :   ×   →   .In the sequel we check that  satisfies all assumptions of Lemma 4. Firstly, we prove that  is a mixed monotone operator.In fact, for , and  ∈ [0, ], and by ( 2 ) and Definition 3, we have that That is, Next, we show that  satisfies the condition () of Lemma 4. Now, we set V() ≡ ( ∈ [0, ]).Then, V > .