New fractional integral inequalities via Caputo-Fabrizio operator and an open problem concerning an integral inequality

In this paper, author introduces some new integral inequalities by using the Caputo-Fabrizio (CF) fractional integral and functions with the same sense of variation. Also an open problem concerning an integral inequality is discussed.


Introduction
A fractional derivative is just an operator which generalizes the ordinary derivative, such that if the fractional derivative is represented by the operator symbol D α then, when α = n, it coincides with the usual differential operator D n [16]. Its origin dates back to 1695 when L'Hopital raised by a letter to Leibniz the question of how the expression D n u(t) = d n dt n u(t), should be understood if n was a real number [16]. Since then, the fractional derivative has become popular and useful due to its ability to describe some natural phenomena in numerous fields of engineering such as theory of viscoelasticity [3][4][5], study of the anomalous diffusion phenomenon [19][20][21], circuit theory [22][23][24], image processing [25,26] and optimal control theory [12][13][14][15], among other applications. Various definition of fractional derivatives have been introduced [27][28][29][30][31][32][33]. In fact, the Grunwald-Letnikov fractional derivative, defined as a limit of a fractional order backward difference, is one of the first introduced fractional operators. Other definition which also plays a major role in Fractional Calculus is the Riemann-Liouville fractional derivative. The Caputo fractional derivative has also been defined via a modified Riemann-Liouville fractional derivative. This approach is useful for the formulation and solution of applied problems [29]. In 2015, Caputo and Fabrizio introduced a new fractional approach [31], which was born due to the necessity to describe a class of non-local systems which cannot be well described by classical local theories or by fractional models with singular kernel [31].

holds.
Next, they proposed the following open problems:

Open problem 1. Under what conditions does the inequality
hold for α, β and λ ?.
In literature few results have been obtained on properties of the Caputo-Fabrizio fractional integral [10,18]. Motivated from [1], the main purpose of this paper is to establish some new inequalities using Caputo-Fabrizio fractional integral. Also a solution to the open problem 1 is established. The paper has been organized as follows, in Section 2, we define basic concepts and definitions. In Section 3, we give the main results. The paper finalize with the conclusion in the section 4.

Basic Concepts and definitions
Firstly, we give some necessary definitions and preliminaries of fractional calculus theory which are used further in this paper. Definition 1. Let α > 0. The Riemann-Liouville fractional integral of order α of a function f is defined by [29] J α f (t) = 1 Definition 2. Let 0 < α < 1. The Caputo-Fabrizio fractional integral of order α of a function f is defined by [9,31] I Definition 3. Let 0 < α < 1 The Caputo-Fabrizio fractional derivative of order α of a function f is defined by [9,32] D α at f (t) = Definition 4. We say that two functions f and g have the same sense of variation on [0, ∞) if ( f (τ) − f (ρ))(g(τ) − g(ρ)) ≥ 0, τ, ρ ∈ (0,t),t > 0. because Note 2. Let m > 0, p ≥ 1 and f , g be two positive functions on [0, ∞). The inequality f g ≥ m is equivalent to because

Main Results
In literature few results have been obtained on some fractional integral inequalities using Caputo-Fabrizio fractional integral [10]. The purpose of this section is to establish some new inequalities using the Caputo-Fabrizio fractional integral.
Using the condition f (τ) g(τ) ≤ M, τ ∈ (0,t),t > 0, we can write Multiplying both sides of (4) by α, then integrating resulting identity with respect to τ from 0 to t, we get which is equivalent to By using (1) in (5), follows Hence, we can write On the other hand, from the condition m ≤ f (τ) g(τ) , we obtain .
Now, multiplying both sides of (7) by α, then integrating resulting identity with respect to τ from 0 to t, we have the inequality By using (2) into (8), yields Hence, we can write The inequality (3) follows on adding the inequalities (6) and (9).
Remark. Let m > 0, p > 1, 1 p + 1 q = 1 and f , g be two positive functions on [0, ∞). The inequality f g ≥ m is equivalent to as Remark. In the same way, inequality f g ≤ M is equivalent to Lemma 1. Let 0 < α < 1, p > 1, 1 p + 1 q = 1 and let f and g be two positive and continuous functions on [0, ∞). If then the inequality holds.