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.
Similar content being viewed by others
1 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 [1–4]. 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 [5–10] 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 [11–20] 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.
2 Preliminaries
In this section, we recall several definitions needed for the discussion.
Definition 2.1
The Riemann–Liouville integral of a function \(f(x)\) of order \(\alpha >0\) is defined as
where Γ is the gamma function.
Definition 2.2
The Riemann–Liouville derivative of a function \(f(x)\) of order α is defined as
where \(n-1<\alpha <n\in \mathbb{Z}^{+}\).
Definition 2.3
The Caputo derivative of a function \(f(x)\) of order α is defined as
where \(n-1<\alpha <n\in \mathbb{Z}^{+}\).
Definition 2.4
A function \(f:[a, b]\subset \mathbb{R}\mapsto \mathbb{R}\) is said to be convex on \([a, b]\) if
for all \(x,y\in [a, b]\) and \(\lambda \in [0, 1]\).
Definition 2.5
([11])
A function \(f:[a, b]\subset \mathbb{R}\mapsto \mathbb{R}\) is said to be \((s, m)\)-convex with modulus \(\mu \geq 0\) (in the second sense) if
for all \(x,y\in [a, b]\), \(\lambda \in [0, 1]\), and \(s,m\in [0, 1]\).
Definition 2.6
A function \(f:[a, b]\subset \mathbb{R}\mapsto \mathbb{R}\) is said to be s-convex (in the second sense) if
for all \(x, y\in [a, b]\), \(\lambda \in [0, 1]\), and \(s\in [0, 1]\).
Definition 2.7
([25])
Let \(f:[a, b]\subset \mathbb{R}\mapsto \mathbb{R}\) be a convex function. Then the Hermite–Hadamard inequality is given by
3 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
-
1.
\(x(t), f(t), c(t), g_{1}(t), g_{2}(t),r(t)\in C([a, b], \mathbb{R}\mathbbm{_{+}})\) with \(r(t)\leq t\),
-
2.
\(h(t)\in C(\mathbb{R}\mathbbm{_{+}}, \mathbb{R}\mathbbm{_{+}})\) is a nondecreasing function with \(h(t)\geq 1\),
-
3.
\(Q(v)=\int _{0}^{v}\frac{\mathrm{d}r}{h(r)}\) for \(v\geq 0\).
However, for fractional order α, we only consider the case \(1<\alpha <2\).
Theorem 3.1
If
then
Proof
From (3.1) it follows that
For \(t\in [a,b]\), defining the function \(y(t)\) by the right-hand side of (3.3), we have
which implies
Multiplying both sides of (3.4) by \((b-a)^{\alpha -1}\), 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. □
Theorem 3.2
If \(x(t)\) satisfies
then
Proof
Simplifying (3.7), we can easily get that
It is known that \((u-v)^{\alpha -1}\leq 2^{2-\alpha } u^{\alpha -1}-v^{\alpha -1}\). Therefore
For the convenience of calculation, denote the right-hand side of (3.10) as \(y(t)\). Then
Hence
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. □
Theorem 3.3
Let \(S(t)=Q(2t+c(b)-2c(a))-Q(t)\) be a nondecreasing function. If there is a function \(x(t)\) such that
then
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
According to (3.17), we have
Thus
Since Q and S are nondecreasing, we have
Using (3.17) and (3.21), it follows that
From (3.16) and (3.22) the expected result follows. □
Similarly to the above case with single-variable functions, we will consider bivariate functions.
Let \(I_{1}=[u_{0}, u_{T}]\) and \(I_{2}=[v_{0}, v_{T}]\) with \(u_{0}, v_{0} \geq 0\). Assume that:
-
1.
\(x(u, v), f(u, v), c(u, v), g_{1}(u, v), g_{2} (u, v)\in C(I_{1} \times I_{2}, \mathbb{R}\mathbbm{_{+}})\),
-
2.
\(r_{1}(u)\in C(I_{1}, \mathbb{R}\mathbbm{_{+}})\) and \(r_{2}(v)\in C(I_{2}, \mathbb{R}\mathbbm{_{+}})\) with \(r_{1}(u)\leq u\), \(r_{2}(v)\leq v\),
-
3.
\(h(t)\in C(\mathbb{R}\mathbbm{_{+}}, \mathbb{R}\mathbbm{_{+}})\) is a nondecreasing function with \(h(t)\geq 1\),
-
4.
\(Q(v)=\int _{0}^{v}\frac{\mathrm{d}r}{h(r)}\) for \(v\geq 0\).
Under such conditions, we state the following theorem.
Theorem 3.4
Suppose that \(x(u,v)\) satisfies the following inequality:
for \(u\in I_{1}\), \(v\in I_{2}\). Then
where \(l(w,z,s,\tau )=(2^{2-\alpha }s^{\alpha -1}-w^{\alpha -1})(2^{2-\alpha } \tau ^{\alpha -1}- z^{\alpha -1})\). The constants \(k_{0}\) and k are defined by \(k_{0}=(u-r_{1}(u_{0}))^{2-\alpha }(v -r_{2}(v_{0}))^{2-\alpha }\) and \(k=(u_{T}-r_{1}( u_{0}))^{\alpha -1}(v_{T}-r_{2}(v_{0}))^{\alpha -1 }\), respectively.
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
where
Integrating both sides of (3.28) with respect to t over \([u_{0}, u]\) yields the relation
where
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:
and
where \(G(s,\tau )=\int _{u _{0}}^{s}\int _{v_{0}}^{ \tau }(s-w)^{\alpha -1}( \tau -z)^{\alpha -1}x^{p} (w,z)\,\mathrm{d}w\,\mathrm{d}z\), \(0< p<1\).
Corollary 3.1
Suppose that \(x(t)\) satisfies
for \(c(b)\leq 2c(a)+1\). Then we can get an explicit estimation of \(x(t)\) in (3.30):
where
Proof
Since \(x(t)\leq x(t)+1\), using \(h(t)=t+1\) in Theorem 3.3, we can get that \(Q(v)=\log (v+1)\), \(Q^{-1}(t)=\exp (t)-1\), \(S(t)=\log (2t+c(b)-2c(a)+1)- \log (t+1)\), and \(S^{-1}(t)=\frac{c(b)-(2c(a)+1)}{\exp (t)- 2}-1\). So (3.34) can be easily proved. □
Corollary 3.2
If \(r(t)\), a, b, and \(x(t)\) in (3.31) meet the \(r(t)\leq t\), \(1\leq a\), \(b \leq \log (2)+2c(a)-c(b)\), and
then
where
and
Proof
Note that
We take \(h(t)=e^{t}\). Then \(Q(t)=1-e^{-t}\), \(Q^{-1} (t)=\log ( \frac{1}{1-t})\), \(S(t)=\exp (-t)-\exp (-2t-c(b) +2c(a))\), \(S^{-1}(t)= \log [\exp (\frac{-c(b)+ 2c(a)}{2} ) (\exp (c(b)-2c(a))-4t )^{\frac{1}{2}}+1 ]-\log (2t)\), and \(Q [S^{-1}(t) ] =1- \frac{2\exp [\frac{c(b)t-2c(a)t}{2} ]}{[\exp (c(b)-2c(a))-4t]^{\frac{1}{2}}+1}\). By applying Theorem 3.3 we deduce the corollary. □
Corollary 3.3
If \(F(y_{1}, y_{2}, y_{3}, y_{4})\leq A(y_{1}, y_{2}) (y_{3}^{p}+y_{4} )\) and \(C(u, v)\leq c(u, v)(u-u_{0}) (v-v_{0})\) for \(u\in [u_{0}, u_{T}]\), \(v\in [v_{0}, v_{T}]\), then \(x(u,v)\) defined by (3.32) satisfies
where
and \(k_{0}\) and k are as in Theorem 3.4.
Proof
From the assumptions of the corollary we can deduce that
Applying Theorem 3.4 to \(h(t)=t^{p}\) completes the proof. □
4 Hermite–Hadamard-type fractional integral inequalities
In this section, we present some Hermite–Hadamard-type fractional integral inequalities by integration, differentiation, and convex functions.
Lemma 4.1
Let \(c, \alpha \in (0, 1)\), and let \(f\in C^{3}([a, b])\). Then
Furthermore, \(Q(t)\) and \(T_{f, c}(a, b)\) can be expressed as follows:
and
Here \(b_{0}=\frac{\alpha c}{2(c-1)}\), \(b_{1}=\frac{\alpha }{1-c}\), \(b_{2}=- \frac{\alpha (c-2)}{2(c-1)}-1\), and \(2(c^{\alpha +1}-1)=(\alpha +2)(c-1)\).
Proof
We represent A as
First, we estimate \(A_{1}\). It is clear that
Using integration by parts and the facts \(Q_{1}(0)=Q_{1}(c)=0\), \(Q'_{1}(0)=0\), and \(Q'_{1}(c)=\alpha c^{\alpha +1}\), we have
In a similar manner, we find that
Thus A can be written as
which yields the desired result
with
The proof is completed. □
Corollary 4.1
Let \(c, \alpha \in (0, 1)\), and let \(f\in C^{4}([a, b])\). Under the assumptions of Lemma 4.1for \(1-\alpha \), we have
Theorem 4.1
Let \(f\in C^{3}([a, b])\). Denote by d a division of the interval \([a, b]\), i.e., \(d: a=t_{0} < t_{1}<\cdots <t_{n-1}<t_{n}=b\), \(h=t_{i+1} -t_{i}\), \(i=0,\dots ,n-1\). Then for \(c, \alpha \in (0, 1)\),
where \(F_{i}(u)= (\frac{t_{i+1}-u}{t_{n}-u} )^{1-\alpha }f(u)\), \(i=0,\dots ,n-2\).
Proof
From Lemma 4.1 we have
This proves the theorem. □
Theorem 4.2
Let \(f\in C^{4}([a, b])\) and suppose that \(|f^{(3)} |\) is an \((s, m)\)-convex function on \([a, b]\). Under the assumptions of Lemma 4.1, we have the following inequality for \(c, \alpha \in (0, 1)\):
Proof
Equation (4.1) yields the inequality
Since \(| f^{(3)} |\) is \((s,m)\)-convex, \(I_{1}\) can be calculated as
Furthermore, we have
Hence it is easily shown that
Next, using \(c^{\alpha }-t^{\alpha }\leq \alpha (c-t)t^{\alpha -1}\), we have
As for \(|Q_{2}(1-t) |\), we have the inequality
in which \(m(t)=(1-t)^{\alpha +2}+(1-t)^{2}\), and \((1-t)^{r}+t^{r}\leq 1\) for \(r\geq 1\).
Thus
and
By (4.15), (4.18), and (4.19) we get the inequality
Finally, the proof can be fulfilled by (4.14) and (4.20). □
Lemma 4.2
Let \(f\in C([a, b])\). Then for \(c, \alpha \in (0, 1)\),
where
Proof
Let \(J=\int _{0}^{1}p(t)f'(ta+(1-t)b)\,\mathrm{d}t\). We have
Multiplying both sides of (4.23) by \(\frac{1}{\Gamma (\alpha )}\), we obtain (4.21). The lemma is proved. □
Theorem 4.3
Let \(c, \alpha \in (0, 1)\), let \(f:[a, b]\mapsto \mathbb{R}\) be a differentiable function on \([a, b]\), and let \(p\geq 1\). If \(\vert f'\vert ^{\frac{p}{p-1}}\) is convex, then
Proof
By (4.21) we have
Using the Hölder inequality for \(q, p\geq 1\) such that \(\frac{1}{p}+\frac{1}{q}=1\), we can prove that
where we used the inequalities \(A^{r}+B^{r}\leq 2^{1-r}(A+B)^{r}\) for \(A, B\geq 0\) and \(0\leq r \leq 1\), and \((1-t^{\alpha })^{p}+t^{\alpha p}\leq 1\) for \(0\leq t \leq 1\). The proof is completed. □
Proposition 4.1
Let \(p, \beta \geq 0, c\), \(\alpha \in (0,1)\), and \(0\leq a\leq b\). Then
where \(\mathrm{B} (\alpha ,\beta )=\int _{0} ^{1}u^{\alpha -1}(1-u)^{ \beta -1}\,\mathrm{d}u\).
Proof
The proposition comes directly by applying Theorem 4.3 to \(f(t)=t^{\beta }\). □
Proposition 4.2
Let \(\beta \geq 3\) and \(c, \alpha \in (0,1)\) with \((\alpha +2)(c-1)=2(c^{\alpha +1}-1)\). Then
Proof
Applying Lemma 4.1 to \(f(t)=t^{\beta }\) and \(\beta \geq 3\), we arrive at (4.28). □
5 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.
Availability of data and materials
Not applicable.
References
Li, C.P., Li, Z.Q., Wang, Z.: Mathematical analysis and the local discontinuous Galerkin method for Caputo–Hadamard fractional partial differential equation. J. Sci. Comput. 85(2), 41 (2020)
Li, C.P., Li, Z.Q.: Stability and logarithmic decay of the solution to Hadamard-type fractional differential equation. J. Nonlinear Sci. 31(2), 31 (2021)
Li, C.P., Li, Z.Q.: The blow-up and global existence of solution to Caputo–Hadamard fractional partial differential equation with fractional Laplacian. J. Nonlinear Sci. 31(5), 80 (2021)
Fan, E.Y., Li, C.P., Li, Z.Q.: Numerical approaches to Caputo–Hadamard fractional derivatives with application to long-term integration of fractional systems. Commun. Nonlinear Sci. Numer. Simul. 106, 106096 (2022)
Abdeldaim, A., El-Deeb, A.A.: On generalized of certain retarded nonlinear integral inequalities and its applications in retarded integro-differential equations. Appl. Math. Comput. 256, 375–380 (2015)
Boudeliou, A.: On certain new nonlinear retarded integral inequalities in two independent variables and applications. Appl. Math. Comput. 335, 103–111 (2018)
El-Deeb, A.A., Ahmed, R.G.: On some generalizations of certain nonlinear retarded integral inequalities for Volterra–Fredholm integral equations and their applications in delay differential equations. J. Egypt. Math. Soc. 25, 279–285 (2017)
Gu, J., Meng, F.W.: Some new nonlinear Volterra–Fredholm type dynamic integral inequalities on time scales. Appl. Math. Comput. 245, 235–242 (2014)
Liu, H.D.: A class of retarded Volterra–Fredholm type integral inequalities on time scales and their applications. J. Inequal. Appl. 2017(1), 293 (2017)
Xu, R., Ma, X.T.: Some new retarded nonlinear Volterra–Fredholm type integral inequalities with maxima in two variables and their applications. J. Inequal. Appl. 2017(1), 187 (2017)
Bracamonte, M., Giménez, J., Vivas-Cortez, M.: Hermite–Hadamard–Fejér type inequalities for \((s, m)\)-strongly convex functions with module c, in the second sense. Appl. Math. Inf. Sci. 10(6), 2015–2053 (2016)
Alabdali, O., Guessab, A., Schmeisser, G.: Characterizations of uniform convexity for differentiable functions. Appl. Anal. Discrete Math. 13(3), 721–732 (2019)
Guessab, A., Moncayo, M., Schmeisser, G.: A class of nonlinear four-point subdivision schemes. Adv. Comput. Math. 37(2), 151–190 (2012)
Guessab, A.: Direct and converse results for generalized multivariate Jensen-type inequalities. J. Nonlinear Convex Anal. 13(4), 777–797 (2012)
Guessab, A.: Sharp Approximations Based on Delaunay Triangulations and Voronoi Diagrams: Textbook, Novosibirsk State University. NSU Publishing and Printing Center, Novosibirsk (2022)
Guessab, A.: Generalized barycentric coordinates and approximations of convex functions on arbitrary convex polytopes. Comput. Math. Appl. 66(6), 1120–1136 (2013)
Deng, J.H., Wang, J.R.: Fractional Hermite–Hadamard inequalities for \((\alpha , m)\)-logarithmically convex functions. J. Inequal. Appl. 2013, 364 (2013)
Hudzik, H., Maligranda, L.: Some remarks on s-convex functions. Aequ. Math. 48, 100–111 (1994)
Iscan, I.: Hermite–Hadamard’s inequalities for prequasiinvex functions via fractional integrals. Konuralp J. Math. 2(2), 76–84 (2014)
Set, E., Özdemir, M.E., Dragomir, S.S.: On Hadamard-type inequalities involving several kinds of convexity. J. Inequal. Appl. 2010, Article ID 286845 (2010)
Li, C.P., Cai, M.: Theory and Numerical Approximations of Fractional Integrals and Derivatives. SIAM, Philadelphia (2019)
Li, C.P., Zeng, F.H.: Numerical Methods for Fractional Calculus. Chapman & Hall/CRC, Boca Raton (2015)
Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)
Alomari, M., Darus, M.: The Hadamard’s inequality for s-convex function. Int. J. Math. Anal. 2(13), 639–646 (2008)
Hadamard, J.: Étude sur les propriétés des fonctions entières et en particulier d’une fonction considérée par Riemann. J. Math. Pures Appl. 58, 171–215 (1893)
Acknowledgements
The authors would like to express their sincere thanks to the editor and the anonymous reviewers for their helpful comments and suggestions.
Funding
Not applicable.
Author information
Authors and Affiliations
Contributions
MDB made the major analysis and the original draft preparation. JT contributed significantly in writing this paper by analyzing the results, reviewing and editing. Both authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Doubbi Bounoua, M., Tang, J. On some Volterra–Fredholm and Hermite–Hadamard-type fractional integral inequalities. J Inequal Appl 2022, 36 (2022). https://doi.org/10.1186/s13660-022-02772-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-022-02772-6