On some Volterra–Fredholm and Hermite–Hadamard-type fractional integral inequalities

*Correspondence: tangjianhualzz@126.com 1Department of Mathematics, Shanghai University, Shanghai 200444, China Abstract The main aim of this paper is establishing some new Volterra–Fredholm and Hermite–Hadamard-type fractional integral inequalities, which can be used as auxiliary tools in the study of solutions to fractional differential equations and fractional integral equations. Applications are also given to explicate the availability of our results.


Introduction
The subject of fractional calculus has gained considerable popularity and importance over the past few decades, mainly due to its validated applications in various fields of science and engineering [1][2][3][4]. Integral inequalities, especially fractional integral inequalities, have been paid more and more attention in recent years. These inequalities play important roles in the study of fractional differential equations and fractional integral equations. At present, many scholars are devoted to studying various integral inequalities, such as Volterra-Fredholm and Hermite-Hadamard-type inequalities. In [5][6][7][8][9][10] the authors generalized and analyzed the Volterra-Fredholm-type and delay integral inequalities. In addition, some applications in fractional differential equations were presented to illustrate the validity of their outcomes. Convex functions have found an important place in modern mathematics, as they can be seen in a large number of research papers and books today. In this context, the Hermite-Hadamard inequality can be regarded as the first fundamental result for convex functions, which is defined over an interval of real numbers with natural geometric interpretation and many applications. In [11][12][13][14][15][16][17][18][19][20] a number of Hermite-Hadamard-type inequalities are deduced involving the classical and Riemann-Liouville fractional integrals for different classes of convex functions such as (s, m)-convex, m-convex, log-convex, and prequasi-invex functions. In this paper, we consider Volterra-Fredholm and Hermite-Hadamard-type inequalities involving fractional integrals.
The structure of this paper is as follows. The first part gives some preliminary results about fractional integrals, derivatives, and convex functions. In the second part, we derive some new nonlinear Volterra-Fredholm-type fractional integral inequalities on time scales for one-and two-variable functions. In the third part, we establish Hermite-Hadamard-type inequalities and some other integral inequalities for the Riemann-Liouville fractional integral. Finally, we give some concluding remarks.

Preliminaries
In this section, we recall several definitions needed for the discussion.
where is the gamma function.

Nonlinear Volterra-Fredholm-type fractional integral inequalities
In this section, we show and prove certain Riemann-Liouville fractional integral inequalities of nonlinear Volterra-Fredholm-type by amplification, differentiation, integration, and inverse functions.
In the following discussion, we assume that for v ≥ 0. However, for fractional order α, we only consider the case 1 < α < 2.
For t ∈ [a, b], defining the function y(t) by the right-hand side of (3.3), we have which implies (3.4) Multiplying both sides of (3.4) by (ba) α-1 , we get Setting t = s and integrating both sides of (3.5) over [a, t], we find In fact, Q -1 is nondecreasing, and y(a) = 0. We can deduce that Using (3.3) and (3.6), we can derive the desired inequality (3.2). This ends the proof.
Proof Simplifying (3.7), we can easily get that (3.10) For the convenience of calculation, denote the right-hand side of (3.10) as y(t). Then Hence (3.12) By the same steps from (3.4)-(3.6), as in the proof of Theorem 3.1, we have Combining (3.10) and (3.13), we can easily find (3.8). The proof is completed.
Proof According to (3.14), we have Denote the right-hand side of (3.16) as y(t). Inspired by (3.9)-(3.11), y(t) has the following estimate: By the definition of y(t) we get (3.18) According to (3.17), we have Since Q and S are nondecreasing, we have Q y(a) (3.21) Using (3.17) and (3.21), it follows that Similarly to the above case with single-variable functions, we will consider bivariate functions. Let for v ≥ 0. Under such conditions, we state the following theorem.
Proof From (3.23) we easily derive that where Taking the partial derivative of y(u, v) with respect to u, we have Through a series of calculations, we get l(w, z, s, τ )g 2 (w, z) dw dz dτ , Integrating both sides of (3.28) with respect to t over [u 0 , u] yields the relation l(w, z, s, τ )g 2 (w, z) dw dz dτ ds, Since Q -1 is an increasing function, in the light of (3.25) and (3.29), we observe that (3.24) holds. The theorem is proved.
To illustrate our results, the following Volterra-Fredholm fractional integral equations for one and two variables are separately considered in Corollaries 3.1-3.3: (3.31) and

Corollary 3.2 If r(t), a, b, and x(t) in
, and .
We take h(t) = e t . Then . By applying Theorem 3.3 we deduce the corollary. (3.39) and k 0 and k are as in Theorem 3.4.
Proof From the assumptions of the corollary we can deduce that

Hermite-Hadamard-type fractional integral inequalities
In this section, we present some Hermite-Hadamard-type fractional integral inequalities by integration, differentiation, and convex functions.
Furthermore, Q(t) and T f ,c (a, b) can be expressed as follows:

Conclusion
In this paper, we established some new Volterra-Fredholm and Hermite-Hadamard-type fractional integral inequalities. They extend some known inequalities and provide a handy tool for deriving bounds of solutions to fractional differential equations and fractional integral equations. In the meantime, we obtain new fractional integral inequalities for convex functions and show their applications. Finally, we present some estimates of the Riemann-Liouville fractional integral of functions whose absolute value is convex and the derivative is raised to a positive real power.