Properties for ψ -Fractional Integrals Involving a General Function ψ and Applications

: In this paper, we are concerned with the ψ -fractional integrals, which is a generalization of the well-known Riemann–Liouville fractional integrals and the Hadamard fractional integrals, and are useful in the study of various fractional integral equations, fractional differential equations, and fractional integrodifferential equations. Our main goal is to present some new properties for ψ -fractional integrals involving a general function ψ by establishing several new equalities for the ψ -fractional integrals. We also give two applications of our new equalities.


Introduction
Fractional integrals and fractional derivatives are generalizations of classical integer-order integrals and integer-order derivatives, respectively, which have been found to be more adequate in the study of a lot of real world problems. In recent decades, various fractional-order models have been used in plasma physics, automatic control, robotics, and many other branches of science (cf.,  and the references therein).
It is known that the ψ-fractional derivative operator, which was introduced in [22], extends the well-known Riemann-Liouville fractional derivative operator. Moreover, it is also easy to see that the ψ-fractional integral operator [14] extends the well-known Riemann-Liouville fractional integral operator and the Hadamard fractional integral operator (see Remark 1 below). Both the ψ-fractional derivative operator and the ψ-fractional integral operator are useful in the study of various fractional integral equations, fractional differential equations, and fractional integrodifferential equations.
The following known definitions about fractional integrals are used later. Definition 1. [14] Let [t 1 , t 2 ] ∈ R and α > 0. The Riemann-Liouville fractional integrals (left-sided and right-sided) of order α are defined by

Remark 1.
(i) From [14], we know that, for a function f , the right-sided and left-sided Riemann-Liouville fractional integral of order α are defined by respectively. If we take ψ(x) = x, then it follows from (1) that which is the right-sided Riemann-Liouville fractional integral. (ii) From [23], we know that, for a function f , the right-sided and left-sided Hadamard fractional integral of order α are defined by respectively. Hence, taking ψ(x) = lnx in (1), we have which is the right-sided Hadamard fractional integral.
Throughout this paper, we suppose that ψ(µ) is a strictly increasing function on (0, ∞) and ψ (µ) is continuous, 0 ≤ t 1 < t 2 . ζ(µ) is the inverse function of ψ(µ) and The rest of the paper is organized as follows. In Section 2, we give some new equalities for ψ-fractional integrals involving a general function ψ. To illustrate the applicability of our new equalities, we give two examples in Section 3 by introducing the ψ-means and presenting relationships between the arithmetic mean and the ψ-means, and by establishing a prior estimate for a class of fractional differential equations in view of the equalities established in Section 2.
Based on Theorem 1, we can obtain the following Theorems 2 and 3.
The following result involves a point µ between t 1 and t 2 .
Next, we will give two equalities involving function φ.

Theorem 5.
Let the function f : [t 1 , where

Proof. Write
Then, for I 1 , we have For I 2 , we obtain By adding (16) and (17), we get This implies that the equality (15) is true. where

Proof. Write
As we can see from (24) that the ψ-mean M ψ (t 1 , t 2 ) is just the following logarithmic mean [25] when ψ(µ) = ln µ: Moreover, we see that, when ψ(µ) = µ, the ψ-mean M ψ (t 1 , t 2 ) is just the arithmetic mean A(t 1 , t 2 ). The following two results, which are deduced by virtue of our new equalities in the last section, show new relationships between the arithmetic mean A and the two ψ-means above.

Conclusions
In this paper, we present new properties for ψ-fractional integrals involving a general function ψ by establishing several new equalities for the ψ-fractional integrals. The ψ-fractional integrals are generalizations of Riemann-Liouville fractional integrals and Hadamard fractional integrals, and our equalities are more general and new. To illustrate the applicability of our new equalities, we introduce the ψ-means and explore the relationships between the arithmetic mean and the ψ-means with the aid of our equalities. Moreover, we use our equalities to obtain an prior estimate for a class of fractional differential equations. How to study the properties of solutions to fractional equations involving ψ-Caputo fractional derivative? How to reveal other new properties about ψ-fractional integrals? How to find more applications of these properties? We will pay our attention to these problems in our future research.
Author Contributions: All the authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Funding:
The work was supported partly by the NSF of China (11571229).