Skip to content

Advertisement

  • Research
  • Open Access

New aspects of Opial-type integral inequalities

Advances in Difference Equations20182018:452

https://doi.org/10.1186/s13662-018-1912-4

  • Received: 9 August 2018
  • Accepted: 2 December 2018
  • Published:

Abstract

In this manuscript, we prove new aspects for several Opial-type integral inequalities for the left and right Caputo–Fabrizio operators with nonsingular kernel. For this purpose we use the inequalities obtained by Andrić et al. (Integral Transforms Spec. Funct. 25(4):324–335, 2014), which is the generalization of an inequality of Agarwal and Pang (Opial Inequalities with Applications in Differential and Difference Equations, 1995). Besides, examples are presented to validate the reported results.

Keywords

  • Opial-type inequality
  • The left operator
  • The right operator
  • The left integral
  • The right integral
  • Caputo–Fabrizio fractional derivative

MSC

  • 26D10
  • 26D15
  • 26A33
  • 26A40
  • 26A42
  • 26A51

1 Introduction and preliminaries

Since the discovery of Opial’s inequality, it has found interesting applications. Really, Opial’s inequality and its generalizations, extensions, and discretizations have been playing an important role in the study of the existence and uniqueness of initial and boundary value problems for ordinary and partial differential equations besides difference equations [3, 4, 21, 39, 42].

In 1960, Opial [43] obtained the following integral inequality:

Theorem 1.1

Let \(x(t)\in C^{1}[0,h]\) be such that \(x(0)=x(h)=0\) and \(x(t)>0\) in \((0,h)\). Then the following integral inequality holds:
$$ \int^{h}_{0} \bigl\vert x (t )x' (t ) \bigr\vert \,dt \leq\frac {h}{4} \int^{h}_{0} \bigl(x (t ) \bigr)^{2} \,dt. $$
(1.1)
Here \(\frac{h}{4}\) is a constant best possibility.

From that time, Opial’s inequality [43] has been studied extensively by many mathematicians. This inequality has been extended, generalized in different ways, see [2, 5, 6, 22, 2428, 39, 40, 4447, 54]. Also, various mathematicians studied Opial-type integral inequalities for different types of fractional derivative and integral operators involving Caputo, Canavati, Riemann–Liouville, and so on, see [9, 14, 1619, 2931] and the references therein.

In 2000, Anastassiou [7] obtained Opial-type inequalities involving functions and their ordinary and fractional derivatives. In 2002, Anastassiou and Goldstein [12] presented the Opial-type inequalities involving fractional derivatives of different orders. The same year, Anastassiou et al. [13] studied a class of \(L_{p}\)-type Opial inequalities for generalized fractional derivatives for integrable functions based on the results obtained earlier by the first author in 1998. In 2004, Anastassiou [8] established the Opial-type inequalities including fractional derivatives of two functions in different order and power. In 2008, he presented Opial-type inequalities involving Riemann–Liouville fractional derivatives of two functions with applications, see [9]. Also, in 2009, he presented fractional Opial-type inequalities subject to high order boundary conditions in \(L_{p}\) for \(p > 1\), and in 2012, he extended Opial’s integral inequality using the right and left Caputo as well as Riemann–Liouville fractional derivatives, respectively, see [10, 11].

In 2013, Andrić et al. [17] obtained several Opial-type inequalities including Caputo, Canavati, and Riemann–Liouville fractional derivatives. The same year, they presented developments of composition identities for the Caputo fractional derivatives. They gave applications to Opial-type inequalities in [18]. Also, the same year, they studied some Opial-type inequalities for Riemann–Liouville fractional derivatives obtained by Fink in [34] and Pang and Agarwal in [48], see [19].

In 2014, Andrić et al. [15] gave expansions of the Opial-type integral inequalities. Also, they presented a generalization of an inequality obtained by Agarwal and Pang [4].

In 2015, Farid et al. [32] studied the Opial-type inequalities by using generalized fractional integral operator including the Mittag-Leffler function in the kernel. One year later, they presented Opial-type integral inequalities for Hilfer differential and fractional integral operators involving a generalized Mittag-Leffler function in the kernel, see [33].

In 2017, Tomovski et al. [53] gave the generalization of weighted Opial-type inequalities for fractional integral and differential operators involving generalized Mittag-Leffler functions by using Hölder’s integral inequality motivated by the work of Koliha and Pečarić [38].

In 2017, Sarıkaya and Budak [50] obtained new inequalities of Opial-type for conformable integrals.

Recently, researchers have proposed different fractional-time operators from the well-known Riemann–Liouville operator, see [20, 3537, 52]. They are defined by nonsingular memory kernels. Also, they used these new operators to generalize the usual diffusion equation. In fact, these new operators can describe better the evolution of some dynamics of complex systems which cannot be done within the standard fractional calculus operators (for more details, see Refs. [3537] and the references therein).

The purpose of this paper is to establish some Opial-type integral inequalities for the left and right operators with nonsingular kernel. The organization of this paper is given below. The introduction is given in Sect. 1. In Sect. 2, basic definitions and theorems are introduced. Motivated by [4] and [15], we establish several Opial-type inequalities in Sect. 3. Several examples are given for our results in Sect. 4.

2 Basic definitions and theorems

In this section, we present the following theorems, corollaries, and definitions which are useful in the proofs of our results.

Let \(U_{1} (u,K_{1} )\) denote the class of functions \(v: [a_{1},b_{1} ]\rightarrow\mathbb{R}\) with the representation
$$v(t)= \int^{t}_{a_{1}} K_{1}(t,s)u(s)\,ds. $$
Here, the function u is continuous and \(K_{1}\) is an arbitrary nonnegative kernel function such that \(u(t)>0\) implies \(v(t)>0\) for all \(t\in[a_{1},b_{1}]\). Similarly, let \(U_{2} (u,K_{1} )\) denote the class of functions \(v: [a_{1},b_{1} ]\rightarrow\mathbb{R}\) with the representation
$$v(t)= \int^{b_{1}}_{t} K_{1}(t,s)u(s)\,ds. $$
We suppose that all integrals exist. Also, they are finite.

Theorem 2.1

([15])

Let \(\psi: [0,\infty )\rightarrow \mathbb{R}\) be a differentiable function such that, for \(q_{1}>1\), \(\psi (t^{1/q_{1}} )\) is a convex function and \(\psi (0 )=0\). Also, let \(v\in U_{1} (u,K_{1} )\) such that \((\int ^{t}_{a_{1}} (K_{1} (t,s ) )^{p_{1}}\,ds )^{1/p_{1}}\leq C\) and \(\frac{1}{p_{1}}+\frac{1}{q_{1}}=1\). Then
$$\begin{aligned} & \int^{b_{1}}_{a_{1}} \bigl\vert v (t ) \bigr\vert ^{1-q_{1}}\psi^{\prime} \bigl( \bigl\vert v (t ) \bigr\vert \bigr) \bigl\vert u (t ) \bigr\vert ^{q_{1}}\,dt \\ &\quad \leq\frac{q_{1}}{C^{q_{1}}}\psi \biggl(C \biggl( \int^{b_{1}}_{a_{1}} \bigl\vert u (t ) \bigr\vert ^{q_{1}}\,dt \biggr)^{1/q_{1}} \biggr) \end{aligned}$$
(2.1)
$$\begin{aligned} &\quad \leq\frac{q_{1}}{C^{q_{1}} (b_{1}-a_{1} )} \int^{b_{1}}_{a_{1}} \psi \bigl( (b_{1}-a_{1} )^{1/q_{1}}C \bigl\vert u (t ) \bigr\vert \bigr)\,dt. \end{aligned}$$
(2.2)
If \(\psi (t^{1/q_{1}} )\) is a concave function, then reverse inequalities are valid.

When \(\psi (x )=t^{p_{1}+q_{1}}\), the following corollary is obtained.

Corollary 2.1

([15])

Let \(v\in U_{1} (u,K_{1} )\) where \((\int ^{t}_{a_{1}} (K_{1} (t,s ) )^{p_{1}}\,ds )^{1/p_{1}}\leq C\) and \(\frac{1}{p_{1}}+\frac{1}{q_{1}}=1\). Then
$$\begin{aligned} \int^{b_{1}}_{a_{1}} \bigl\vert v (t ) \bigr\vert ^{p_{1}} \bigl\vert u (t ) \bigr\vert ^{q_{1}}\,dt &\leq \frac{q_{1} C^{p_{1}}}{p_{1}+q_{1}} \biggl( \int^{b_{1}}_{a_{1}} \bigl\vert u (t ) \bigr\vert ^{q_{1}}\,dt \biggr)^{ (p_{1}+q_{1} )/q_{1}} \\ &\leq\frac{q_{1} C^{p_{1}} (b_{1}-a_{1} )^{p_{1}/q_{1}}}{p_{1}+q_{1}} \int^{b_{1}}_{a_{1}} \bigl\vert u (t ) \bigr\vert ^{p_{1}+q_{1}}\,dt. \end{aligned}$$
(2.3)

Theorem 2.2

([15])

Let the function \(\psi: [0,\infty )\rightarrow\mathbb{R}\) be differentiable such that, for \(q_{1}>1\), \(\psi (t^{1/q_{1}} )\) is a convex function and \(\psi (0 )=0\). Let \(v\in U_{2} (u,K_{1} )\) such that \((\int ^{b_{1}}_{t} (K_{1} (t, s ) )^{p_{1}} \,ds )^{1/p_{1}}\leq C\) and \(\frac{1}{p_{1}}+\frac{1}{q_{1}}=1\). Then
$$\begin{aligned} &\int^{b_{1}}_{a_{1}} \bigl\vert v (t ) \bigr\vert ^{1-q_{1}}\psi^{\prime} \bigl( \bigl\vert v (t ) \bigr\vert \bigr) \bigl\vert u (t ) \bigr\vert ^{q_{1}}\,dt \\ &\quad \leq\frac{q_{1}}{C^{q_{1}}}\psi \biggl(C \biggl( \int^{b_{1}}_{a_{1}} \bigl\vert u (t ) \bigr\vert ^{q_{1}}\,dt \biggr)^{1/q_{1}} \biggr) \\ &\quad \leq\frac{q_{1}}{C^{q_{1}} (b_{1}-a_{1} )} \int^{b_{1}}_{a_{1}} \psi \bigl( (b_{1}-a_{1} )^{1/q_{1}}C \bigl\vert u (t ) \bigr\vert \bigr)\,dt. \end{aligned}$$
(2.4)
If \(\psi (t^{1/q_{1}} )\) is a concave function, then reverse inequalities are valid.

When \(\psi (x )=t^{p_{1}+q_{1}}\), the following corollary is obtained.

Corollary 2.2

([15])

Let \(v\in U_{2} (u,K_{1} )\) where \((\int ^{b_{1}}_{t} (K_{1} (t,s ) )^{p_{1}}\,ds )^{1/p_{1}}\leq C\) and \(\frac{1}{p_{1}}+\frac{1}{q_{1}}=1\). Then
$$\begin{aligned} \int^{b_{1}}_{a_{1}} \bigl\vert v (t ) \bigr\vert ^{p_{1}} \bigl\vert u (t ) \bigr\vert ^{q_{1}}\,dt &\leq \frac{q_{1} C^{p_{1}}}{p_{1}+q_{1}} \biggl( \int^{b_{1}}_{a_{1}} \bigl\vert u (t ) \bigr\vert ^{q_{1}}\,dt \biggr)^{ (p_{1}+q_{1} )/q_{1}} \\ &\leq\frac{q_{1} C^{p_{1}} (b_{1}-a_{1} )^{p_{1}/q_{1}}}{p_{1}+q_{1}} \int^{b_{1}}_{a_{1}} \bigl\vert u (t ) \bigr\vert ^{p_{1}+q_{1}}\,dt. \end{aligned}$$
(2.5)
Below, we show the definitions of the left and right operators with nonsingular kernel introduced in [23]. According to [1, 23], if \(g\in H^{1}(a_{1},b_{1})\), \(0< a_{1}< b_{1}\leq\infty\), \(\alpha\in(0,1)\), the left operator \({}^{CFR}_{\ \ \, a_{1}}D^{\alpha}\) is defined by
$$ \bigl({}^{CFR}_{\ \ \, a_{1}}D^{\alpha} g \bigr) (t )= \frac{M(\alpha )}{1-\alpha}\frac{d}{dt} \int_{a_{1}} ^{t} g(s)\exp \bigl(\lambda (t-s ) \bigr) \,ds $$
(2.6)
and the right operator \({}^{CFR} D^{\alpha}_{b_{1}}\) is defined by
$$ \bigl({}^{CFR} D^{\alpha}_{b_{1}} g \bigr) (t )=- \frac{M(\alpha )}{1-\alpha}\frac{d}{dt} \int_{t} ^{b_{1}} g(s)\exp \bigl(\lambda (s-t ) \bigr) \,ds, $$
(2.7)
with \(\lambda=-\frac{\alpha}{1-\alpha}\) and \(t\geq a_{1}\). Here \(M(\alpha )\) is a normalization function depending on α. Also, the left integral operator is defined as
$$ \bigl({}^{CF} _{\ a_{1}} I^{\alpha} g \bigr) (t )= \frac {1-\alpha}{B (\alpha )}g (t )+\frac{\alpha}{B (\alpha )} \int^{t}_{a_{1}}g (s )\,ds $$
(2.8)
and the right integral operator is defined as
$$ \bigl({}^{CF} I^{\alpha} _{b_{1}} g \bigr) (t )= \frac {1-\alpha}{B (\alpha )}g (t )+\frac{\alpha}{B (\alpha )} \int^{b_{1}}_{t}g (s )\,ds. $$
(2.9)

Definition 2.1

([49])

Let f and g be two functions that are piecewise continuous on every finite closed interval \(0 \leq t \leq b\) and of exponential order. The function denoted by \(f*g\) and defined by
$$f(t)*g(t)= \int^{t}_{0} f(s)g(t-s)\,ds $$
is called the convolution of the functions f and g.

Definition 2.2

([51])

Let \(f(x)\) and \(g(x)\) be positive and be in \(L^{1}\). Moreover, they are differentiable and their derivative is integrable. Then the derivative of a convolution is
$$(f*g )^{\prime}=f^{\prime}*g=f*g^{\prime}. $$

3 Main results

In this section, we give the Opial-type integral inequalities for the left and right of the operator using the inequalities obtained by Andrić et al. [15], which is the generalization of an inequality of Agarwal and Pang [4].

The following result is obtained by using Theorem 2.1 and the left operator.

Theorem 3.1

Let \(\psi: [0,\infty )\rightarrow\mathbb{R}\) be a differentiable function such that, for \(q_{1}>1\), \(\psi (t^{1/q_{1}} )\) is a convex function and \(\psi (0 )=0\). Also, let \(0<\alpha<1\), \(g\in H^{1}(a_{1},b_{1})\), and let \({}^{CFR} _{\ \ \, a_{1}}D^{\alpha}\) be defined by (2.6). If \(\frac{1}{p_{1}}+\frac {1}{q_{1}}=1\), then the following inequalities hold:
$$\begin{aligned}& \int^{b_{1}}_{a_{1}} \bigl\vert \bigl({}^{CFR}_{\ \ \, a_{1}}D^{\alpha} g \bigr) (t ) \bigr\vert ^{1-q_{1}}\psi^{\prime} \bigl( \bigl\vert \bigl({}^{CFR}_{\ \ \, a_{1}}D^{\alpha} g \bigr) (t ) \bigr\vert \bigr) \bigl\vert g' (t ) \bigr\vert ^{q_{1}}\,dt \\& \quad \leq\frac{q_{1}}{C^{q_{1}}}\psi \biggl(C \biggl( \int^{b_{1}}_{a_{1}} \bigl\vert g' (t ) \bigr\vert ^{q_{1}}\,dt \biggr)^{1/q_{1}} \biggr) \\& \quad \leq\frac{q_{1}}{C^{q_{1}} (b_{1}-a_{1} )} \int^{b_{1}}_{a_{1}} \psi \bigl( (b_{1}-a_{1} )^{1/q_{1}}C \bigl\vert g' (t ) \bigr\vert \bigr)\,dt, \end{aligned}$$
(3.1)
where
$$ C=\frac{M (\alpha )}{1-\alpha} \biggl(\frac{1-\exp (p_{1}\lambda (b_{1}-a_{1} ) )}{-p_{1}\lambda} \biggr)^{1/p_{1}}. $$
(3.2)
If \(\psi (t^{1/q_{1}} )\) is a concave function, then reverse inequalities hold.

Proof

For \(t \in [a_{1},b_{1} ]\), let
$$\begin{aligned}& v (t )= \bigl({}^{CFR}_{\ \ \, a_{1}}D^{\alpha} g \bigr) (t )= \frac{M(\alpha)}{1-\alpha}\frac{d}{dt} \int_{a_{1}} ^{t} g(s)\exp \bigl(\lambda (t-s ) \bigr) \,ds \\& \hphantom{v (t )}=\frac{M (\alpha )}{1-\alpha}\frac{d}{dt} \bigl(g(t)*\exp (\lambda t ) \bigr) \\& \hphantom{v (t )}=\frac{M (\alpha )}{1-\alpha} \biggl(\frac{dg}{dt}(t)*\exp (\lambda t ) \biggr) \\& \hphantom{v (t )}=\frac{M (\alpha )}{1-\alpha} \int_{a_{1}}^{t} g'(s)\exp \bigl(\lambda (t-s ) \bigr)\,ds, \\& K_{1}(t,s)=\textstyle\begin{cases} \frac{M (\alpha )}{1-\alpha}\exp (\lambda (t-s ) ), & a_{1}\leq s\leq t; \\ 0 ,& t\leq s\leq b_{1}, \end{cases}\displaystyle \end{aligned}$$
(3.3)
and
$$\phi (t )= \biggl( \int^{t}_{a_{1}} \bigl(K_{1} (t,s ) \bigr)^{p_{1}} \,ds \biggr)^{1/p_{1}}=\frac{M (\alpha )}{1-\alpha} \biggl( \frac{1-\exp (p_{1}\lambda (t-a_{1} ) )}{-p_{1}\lambda } \biggr)^{1/p_{1}}. $$
From \(\lambda<0\), the function ϕ is increasing on \([a_{1},b_{1} ]\). Thus, we can write
$$\max_{t\in [a_{1},b_{1} ]}\phi (t )=\frac{M (\alpha )}{1-\alpha} \biggl( \frac{1-\exp (p_{1}\lambda (b_{1}-a_{1} ) )}{-p_{1}\lambda} \biggr)^{1/p_{1}}=C. $$
Then \((\int^{t}_{a_{1}} (K_{1} (t,s ) )^{p_{1}} \,ds )^{1/p_{1}}\leq C\). Also, if it is taken as \(u=g'\) and v as in (3.3), then from Theorem 2.1 it gives us (3.1) in Theorem 3.1. This completes the proof. □

When \(\psi (t )=t^{p_{1}+q_{1}}\) in Theorem 3.1, the following corollary is obtained.

Corollary 3.1

Let \(0<\alpha<1\), \(g\in H^{1}(a_{1},b_{1})\), and let \({}^{CFR} _{a_{1}}D^{\alpha}\) be defined by (2.6). Also let \(\frac {1}{p_{1}}+\frac{1}{q_{1}}=1\). Then the following inequalities hold:
$$\begin{aligned} \int^{b_{1}}_{a_{1}} \bigl\vert \bigl({}^{CFR}_{\ \ \, a_{1}}D^{\alpha} g \bigr) (t ) \bigr\vert ^{p_{1}} \bigl\vert g' (t ) \bigr\vert ^{q_{1}}\,dt &\leq\frac{q_{1} C^{p_{1}}}{p_{1}+q_{1}} \biggl( \int^{b_{1}}_{a_{1}} \bigl\vert g' (t ) \bigr\vert ^{q_{1}}\,dt \biggr)^{ (p_{1}+q_{1} )/q_{1}} \\ &\leq\frac{q_{1} C^{p_{1}} (b_{1}-a_{1} )^{p_{1}/q_{1}}}{p_{1}+q_{1}} \int^{b_{1}}_{a_{1}} \bigl\vert g' (t ) \bigr\vert ^{p_{1}+q_{1}}\,dt, \end{aligned}$$
(3.4)
where C is defined as in (3.2).

Theorem 3.2

Let the function \(\psi: [0,\infty )\rightarrow \mathbb{R}\) be differentiable such that, for \(q_{1}>1\), \(\psi (t^{1/q_{1}} )\) is a convex function and \(\psi (0 )=0\). Also, let \(0<\alpha<1\), \(g\in H^{1}(a_{1},b_{1})\), and let \({}^{CFR} D^{\alpha }_{b_{1}}\) be defined by (2.7). If \(\frac{1}{p_{1}}+\frac{1}{q_{1}}=1\), then the following inequalities hold:
$$\begin{aligned}& \int^{b_{1}}_{a_{1}} \bigl\vert \bigl({}^{CFR} D^{\alpha}_{b_{1}} g \bigr) (t ) \bigr\vert ^{1-q_{1}} \psi^{\prime} \bigl( \bigl\vert \bigl({}^{CFR} D^{\alpha }_{b_{1}} g \bigr) (t ) \bigr\vert \bigr) \bigl\vert g' (t ) \bigr\vert ^{q_{1}}\,dt \\& \quad \leq\frac{q_{1}}{C^{q_{1}}}\psi \biggl(C \biggl( \int^{b_{1}}_{a_{1}} \bigl\vert g' (t ) \bigr\vert ^{q_{1}}\,dt \biggr)^{1/q_{1}} \biggr) \\& \quad \leq\frac{q_{1}}{C^{q_{1}} (b_{1}-a_{1} )} \int^{b_{1}}_{a_{1}} \psi \bigl( (b_{1}-a_{1} )^{1/q_{1}}C \bigl\vert g' (t ) \bigr\vert \bigr)\,dt, \end{aligned}$$
(3.5)
where C is defined as in (3.2). If \(\psi (t^{1/q_{1}} )\) is a concave function, then reverse inequalities hold.

Proof

Using the same method as the proof of Theorem 3.1, inequalities follow from Theorem 2.2. □

When \(\psi (t )=t^{p_{1}+q_{1}}\) in Theorem 3.2, the following corollary is obtained.

Corollary 3.2

Let \(0<\alpha<1\), \(g\in H^{1}(a_{1},b_{1})\), and let \({}^{CFR} D^{\alpha}_{b_{1}}\) be defined by (2.7). Also let \(\frac{1}{p_{1}}+\frac {1}{q_{1}}=1\). Then the following inequalities hold:
$$\begin{aligned} \int^{b_{1}}_{a_{1}} \bigl\vert \bigl({}^{CFR} D^{\alpha}_{b_{1}} g \bigr) (t ) \bigr\vert ^{p_{1}} \bigl\vert g' (t ) \bigr\vert ^{q_{1}}\,dt &\leq \frac{q_{1} C^{p_{1}}}{p_{1}+q_{1}} \biggl( \int^{b_{1}}_{a_{1}} \bigl\vert g' (t ) \bigr\vert ^{q_{1}}\,dt \biggr)^{ (p_{1}+q_{1} )/q_{1}} \\ &\leq\frac{q_{1} C^{p_{1}} (b_{1}-a_{1} )^{p_{1}/q_{1}}}{p_{1}+q_{1}} \int^{b_{1}}_{a_{1}} \bigl\vert g' (t ) \bigr\vert ^{p_{1}+q_{1}}\,dt, \end{aligned}$$
(3.6)
where C is defined as in (3.2).

The next result is obtained by using Theorem 2.1 and the left integral operator, see for more details [41].

Theorem 3.3

Let the function \(\psi: [0,\infty )\rightarrow \mathbb{R}\) be differentiable such that, for \(q_{1}>1\), \(\psi (t^{1/q_{1}} )\) is a convex function and \(\psi (0 )=0\). Also, let \(0<\alpha<1\), \(g\in H^{1}(a_{1},b_{1})\), and let \({}^{CF} _{\ a_{1}}I^{\alpha}\) be defined by (2.8). If \(\frac{1}{p_{1}}+\frac {1}{q_{1}}=1\), then the following inequalities hold:
$$\begin{aligned}& \int^{b_{1}}_{a_{1}} \biggl\vert \bigl({}^{CF}_{\ a_{1}}I^{\alpha} g \bigr) (t )-\frac{1-\alpha}{B (\alpha )}g (t ) \biggr\vert ^{1-q_{1}} \psi^{\prime} \biggl( \biggl\vert \bigl({}^{CF}_{\ a_{1}}I^{\alpha} g \bigr) (t )-\frac{1-\alpha}{B (\alpha )}g (t ) \biggr\vert \biggr) \bigl\vert g (t ) \bigr\vert ^{q_{1}}\,dt \\& \quad \leq\frac{q_{1}}{C_{1}^{{q}_{1}}}\psi \biggl(C_{1} \biggl( \int^{b_{1}}_{a_{1}} \bigl\vert g (t ) \bigr\vert ^{q_{1}}\,dt \biggr)^{1/q_{1}} \biggr) \\& \quad \leq\frac{q_{1}}{C_{1}^{{q_{1}}} (b_{1}-a_{1} )} \int^{b_{1}}_{a_{1}} \psi \bigl( (b_{1}-a_{1} )^{1/q_{1}}C_{1} \bigl\vert g (t ) \bigr\vert \bigr)\,dt, \end{aligned}$$
(3.7)
where
$$ C_{1}=\frac{\alpha}{B (\alpha )} (b_{1}-a_{1} )^{1/p_{1}}. $$
(3.8)
If \(\psi (t^{1/q_{1}} )\) is a concave function, then reverse inequalities hold.

Proof

For \(t \in [a_{1},b_{1} ]\), let
$$\begin{aligned}& v (t )= \bigl({}^{CF}_{\ a_{1}}I^{\alpha} g \bigr) (t )- \frac{1-\alpha}{B (\alpha )}g (t ), \\& K_{1}(t,s)= \textstyle\begin{cases} \frac{\alpha}{B (\alpha )} ,& a_{1}\leq s\leq t; \\ 0 ,& t\leq s\leq b_{1}, \end{cases}\displaystyle \end{aligned}$$
(3.9)
and
$$\phi (t )= \biggl( \int^{t}_{a_{1}} \bigl(K_{1} (t,s ) \bigr)^{p_{1}} \,ds \biggr)^{1/p_{1}}=\frac{\alpha}{B (\alpha )} (t-a_{1} )^{1/p_{1}}. $$
From \(\lambda<0\), the function ϕ is increasing on \([a_{1},b_{1} ]\). Thus, we can write
$$\max_{t\in [a_{1},b_{1} ]}\phi (t )=\frac{\alpha }{B (\alpha )} (b_{1}-a_{1} )^{1/p_{1}}=C_{1}. $$
Then \((\int^{t}_{a_{1}} (K_{1} (t,s ) )^{p_{1}} \,ds )^{1/p_{1}}\leq C_{1}\). Also, if it is taken as \(u=g\) and v as in (3.9), then from Theorem 2.1 it gives us (3.7) in Theorem 3.3. This completes the proof. □

When \(\psi (t )=t^{p_{1}+q_{1}}\) in Theorem 3.3, we obtain the following corollary.

Corollary 3.3

Let \(0<\alpha<1\), \(g\in H^{1}(a_{1},b_{1})\), and let \({}^{CF} _{\ a_{1}}I^{\alpha}\) be defined by (2.8). Also let \(\frac {1}{p_{1}}+\frac{1}{q_{1}}=1\). Then the following inequalities hold:
$$\begin{aligned}& \int^{b_{1}}_{a_{1}} \biggl\vert \bigl({}^{CF}_{\ a_{1}}I^{\alpha} g \bigr) (t )-\frac{1-\alpha}{B (\alpha )}g (t ) \biggr\vert ^{p_{1}} \bigl\vert g (t ) \bigr\vert ^{q_{1}}\,dt \\& \qquad \qquad \leq\frac{q_{1} C_{1}^{{p_{1}}}}{p_{1}+q_{1}} \biggl( \int^{b_{1}}_{a_{1}} \bigl\vert g (t ) \bigr\vert ^{q_{1}}\,dt \biggr)^{ (p_{1}+q_{1} )/q_{1}} \\& \qquad \qquad \leq\frac{q_{1} C_{1}^{{p_{1}}} (b_{1}-a_{1} )^{p_{1}/q_{1}}}{p_{1}+q_{1}} \int^{b_{1}}_{a_{1}} \bigl\vert g (t ) \bigr\vert ^{p_{1}+q_{1}}\,dt, \end{aligned}$$
(3.10)
where \(C_{1}\) is defined as in (3.8).

Theorem 3.4

Let the function \(\psi: [0,\infty )\rightarrow \mathbb{R}\) be differentiable such that, for \(q_{1}>1\), \(\psi (t^{1/q_{1}} )\) is a convex function and \(\psi (0 )=0\). Also, let \(0<\alpha<1\), \(g\in H^{1}(a_{1},b_{1})\), and let \({}^{CF} I^{\alpha }_{b_{1}}\) be defined by (2.9). If \(\frac{1}{p_{1}}+\frac{1}{q_{1}}=1\), then the following inequalities hold:
$$\begin{aligned}& \int^{b_{1}}_{a_{1}} \biggl\vert \bigl({}^{CF} I^{\alpha}_{b_{1}} g \bigr) (t )-\frac{1-\alpha}{B (\alpha )}g (t ) \biggr\vert ^{1-q_{1}}\psi^{\prime} \biggl( \biggl\vert \bigl({}^{CF} I^{\alpha}_{b_{1}} g \bigr) (t )- \frac{1-\alpha}{B (\alpha )}g (t ) \biggr\vert \biggr) \bigl\vert g (t ) \bigr\vert ^{q_{1}}\,dt \\& \quad \leq\frac{q_{1}}{C_{1}^{{q}_{1}}}\psi \biggl(C_{1} \biggl( \int^{b_{1}}_{a_{1}} \bigl\vert g (t ) \bigr\vert ^{q_{1}}\,dt \biggr)^{1/q_{1}} \biggr) \\& \quad \leq\frac{q_{1}}{C_{1}^{{q}_{1}} (b_{1}-a_{1} )} \int^{b_{1}}_{a_{1}} \psi \bigl( (b_{1}-a_{1} )^{1/q_{1}}C_{1} \bigl\vert g (t ) \bigr\vert \bigr)\,dt, \end{aligned}$$
(3.11)
where \(C_{1}\) is defined as in (3.8). If \(\psi (t^{1/q_{1}} )\) is a concave function, then reverse inequalities hold.

Proof

Using the same method as the proof of Theorem 3.1, inequalities follow from Theorem 2.2. □

When \(\psi (t )=t^{p_{1}+q_{1}}\) in Theorem 3.4, we obtain the following corollary.

Corollary 3.4

Let \(0<\alpha<1\), \(g\in H^{1}(a_{1},b_{1})\), and let \({}^{CF} I^{\alpha}_{b_{1}}\) be defined by (2.9). Also let \(\frac{1}{p_{1}}+\frac {1}{q_{1}}=1\). Then the following inequalities hold:
$$\begin{aligned} &\int^{b_{1}}_{a_{1}} \biggl\vert \bigl({}^{CF} I^{\alpha}_{b_{1}} g \bigr) (t )-\frac{1-\alpha}{B (\alpha )}g (t ) \biggr\vert ^{p_{1}} \bigl\vert g (t ) \bigr\vert ^{q_{1}}\,dt \\ &\quad \leq \frac{q_{1} C_{1}^{p_{1}}}{p_{1}+q_{1}} \biggl( \int^{b_{1}}_{a_{1}} \bigl\vert g (t ) \bigr\vert ^{q_{1}}\,dt \biggr)^{ (p_{1}+q_{1} )/q_{1}} \\ &\quad \leq\frac{q_{1} C_{1}^{p_{1}} (b_{1}-a_{1} )^{p_{1}/q_{1}}}{p_{1}+q_{1}} \int^{b_{1}}_{a_{1}} \bigl\vert g (t ) \bigr\vert ^{p_{1}+q_{1}}\,dt, \end{aligned}$$
(3.12)
where \(C_{1}\) is defined as in (3.8).

4 Examples

Below, we will show the application of our main results with two examples.

Example 4.1

In Corollary 3.1, let \(g(t)=e^{t}\), \(\alpha=\frac{1}{2}\), \(p_{1}=q_{1}=2\), \(M(\alpha)=\alpha\), and \(t\in[a_{1},b_{1}]=[1,3]\). Then \(\lambda=-1\), \(M(\frac{1}{2})=\frac{1}{2}\), and \(C=\sqrt{\frac {1-e^{-4}}{2}}\). So, we obtain
$$\begin{aligned} \bigl({}^{CFR}_{\ \ \ \, 1}D^{\frac{1}{2}} g \bigr) (t )&= \bigl({}^{CFR}_{\ \ \ \, 1}D^{\frac{1}{2}} g \bigr) (t )= \frac{d}{dt} \int ^{t}_{1} e^{s} e^{- (t-s )}\,ds = \int^{t}_{1} \frac{\partial}{\partial t} \bigl(e^{-t+2s} \bigr)\,ds+ e^{t} \\ &=- \int^{t}_{1} e^{-t+2s} \,ds+ e^{t}= \frac {e^{t}+e^{-t+2}}{2}. \end{aligned}$$
Then we apply Corollary 3.1 to obtain the following inequalities:
$$\begin{aligned} \int^{3}_{1} \bigl\vert \bigl({}^{CFR}_{\ \ \ \, 1}D^{\frac{1}{2}} g \bigr) (t ) \bigr\vert ^{2} \bigl\vert g' (t ) \bigr\vert ^{2}\,dt &= \int^{3}_{1} \biggl\vert \frac{e^{t}+e^{-t+2}}{2} \biggr\vert ^{2} e^{2t}\,dt \\ &\leq\frac {1-e^{-4}}{4} \biggl( \int^{3}_{1} e^{2t} \,dt \biggr)^{2} \leq\frac{1-e^{-4}}{2} \int^{3}_{1} e^{4t} \,dt. \end{aligned}$$

Example 4.2

In Corollary 3.3, let \(g(t)=\sin t\), \(\alpha=\frac {1}{2}\), \(p_{1}=q_{1}=2\), \(B(\alpha)=1-\alpha\), and \(t\in[a_{1},b_{1}]=[\frac {\pi}{2},\pi]\). Then \(\lambda=-1\), \(B(\frac{1}{2})=\frac{1}{2}\), and \(C_{1}=\sqrt{\pi-\frac{\pi}{2}}\). So, we obtain
$$\bigl({}^{CF} _{\ \frac{\pi}{2}} I^{\frac{1}{2}} g \bigr) (t )= \bigl({}^{CF} _{\ \frac{\pi}{2}} I^{\frac{1}{2}} \sin \bigr) (t ) =\sin t+ \int^{t}_{\frac{\pi}{2}} \sin (s )\,ds=\sin t-\cos t. $$
Then we apply Corollary 3.3 to obtain the following inequalities:
$$\begin{aligned} & \int^{\pi}_{\frac{\pi}{2}} \biggl\vert \bigl({}^{CF}_{\ \frac{\pi }{2}}I^{\frac{1}{2}} g \bigr) (t )-\frac{1-\alpha}{B (\alpha )}g (t ) \biggr\vert ^{2} \bigl\vert g (t ) \bigr\vert ^{2}\,dt \\ &\quad = \int^{\pi}_{\frac{\pi}{2}} \vert \sin t-\cos t-\sin t \vert ^{2} \vert \sin t \vert ^{2}\,dt \\ &\quad = \int^{\pi}_{\frac{\pi}{2}} \cos^{2} t \sin^{2} t \,dt \leq \frac{\pi-\frac{\pi}{2}}{2} \biggl( \int^{\pi}_{\frac{\pi}{2}} \sin^{2} t \,dt \biggr)^{2} \\ &\quad \leq\frac{\pi}{4} \biggl(\pi-\frac{\pi}{2} \biggr) \int^{\pi}_{\frac{\pi }{2}} \sin^{4} t \,dt. \end{aligned}$$

5 Conclusion

Caputo–Fabrizio operator has recently started to play an important role in modeling of a class of real world dissipative phenomena [35]. In fact some real data have confirmed that this operator is important for describing the dynamics of specific classes of real world problems. At the same time new mathematical generalizations of this operator were developed. In our manuscript, with the help of inequalities obtained by Andrić et al. [15], we proposed, within four theorems and their related corollaries, several Opial-type integral inequalities for Caputo–Fabrizio operators. Finally, we analyzed two illustrative examples carefully. The results reported in this manuscript can find applications within the evaluation of the existence and uniqueness of initial and boundary value problems related to diffusion process within the Caputo–Fabrizio operators.

Declarations

Acknowledgements

The authors would like to thank the referees for their useful comments and remarks.

Funding

No funding.

Authors’ contributions

All authors contributed to each part of this work equally, and they all read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Arts and Sciences, Bolu Abant Izzet Baysal University, Bolu, Turkey
(2)
Department of Mathematics, Faculty of Arts and Sciences, Çankaya University, Ankara, Turkey
(3)
Institutes of Space Sciences, Magurele-Bucharest, Romania

References

  1. Abdeljawad, T., Baleanu, D.: On fractional derivatives with exponential kernel and their discrete versions. Rep. Math. Phys. 80, 1–27 (2017) MathSciNetView ArticleGoogle Scholar
  2. Agarwal, R.P.: Sharp Opial-type inequalities involving r-derivatives and their applications. Tohoku Math. J. 47, 567–593 (1995) MathSciNetView ArticleGoogle Scholar
  3. Agarwal, R.P., Lakshmikantham, V.: Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations. World Scientific, Singapore (1993) View ArticleGoogle Scholar
  4. Agarwal, R.P., Pang, P.Y.H.: Opial Inequalities with Applications in Differential and Difference Equations. Kluwer Academic, Dordrecht (1995) View ArticleGoogle Scholar
  5. Agarwal, R.P., Pang, P.Y.H.: Opial-type inequalities involving higher order derivatives. J. Math. Anal. Appl. 189, 85–103 (1995) MathSciNetView ArticleGoogle Scholar
  6. Agarwal, R.P., Pang, P.Y.H.: Remarks on the generalizations of Opial’s inequality. J. Math. Anal. Appl. 190, 559–577 (1995) MathSciNetView ArticleGoogle Scholar
  7. Anastassiou, G.A.: Opial type inequalities involving functions and their ordinary and fractional derivatives. Commun. Appl. Anal. 4(4), 547–560 (2000) MathSciNetMATHGoogle Scholar
  8. Anastassiou, G.A.: Opial-type inequalities involving fractional derivatives of two functions and applications. Comput. Math. Appl. 48, 1701–1731 (2004) MathSciNetView ArticleGoogle Scholar
  9. Anastassiou, G.A.: Opial type inequalities involving Riemann–Liouville fractional derivatives of two functions with applications. Math. Comput. Model. 48, 344–374 (2008) MathSciNetView ArticleGoogle Scholar
  10. Anastassiou, G.A.: Balanced fractional Opial inequalities. Chaos Solitons Fractals 42, 1523–1528 (2009) MathSciNetView ArticleGoogle Scholar
  11. Anastassiou, G.A.: Opial-type inequalities for functions and their ordinary and balanced fractional derivatives. J. Comput. Anal. Appl. 14(5), 862–879 (2012) MathSciNetMATHGoogle Scholar
  12. Anastassiou, G.A., Goldstein, J.A.: Fractional Opial-type inequalities and fractional differential equations. Results Math. 41, 197–212 (2002) MathSciNetView ArticleGoogle Scholar
  13. Anastassiou, G.A., Koliha, J.J., Pečarić, J.: Opial type \(L_{p}\)-inequalities for fractional derivatives. Int. J. Math. Math. Sci. 31(2), 85–95 (2002) MathSciNetView ArticleGoogle Scholar
  14. Andrić, M., Barbir, A., Farid, G., Pečarić, J.: More on certain Opial-type inequality for fractional derivatives. Nonlinear Funct. Anal. Appl. 19(4), 563–583 (2014) MATHGoogle Scholar
  15. Andrić, M., Barbir, A., Farid, G., Pečarić, J.: Opial-type inequality due to Agarwal–Pang and fractional differential inequalities. Integral Transforms Spec. Funct. 25(4), 324–335 (2014) MathSciNetView ArticleGoogle Scholar
  16. Andrić, M., Pečarić, J., Perić, I.: Improvements of composition rule for the Canavati fractional derivatives and applications to Opial-type inequalities. Dyn. Syst. Appl. 20, 383–394 (2011) MathSciNetMATHGoogle Scholar
  17. Andrić, M., Pečarić, J., Perić, I.: An Opial-type inequality for fractional derivatives of two functions. Fract. Differ. Calc. 3(1), 55–68 (2013) MathSciNetView ArticleGoogle Scholar
  18. Andrić, M., Pečarić, J., Perić, I.: Composition identities for the Caputo fractional derivatives and applications to Opial-type inequalities. Math. Inequal. Appl. 16(3), 657–670 (2013) MathSciNetMATHGoogle Scholar
  19. Andrić, M., Pečarić, J., Perić, I.: A multiple Opial type inequality for the Riemann–Liouville fractional derivatives. J. Math. Inequal. 7(1), 139–150 (2013) MathSciNetView ArticleGoogle Scholar
  20. Atanacković, T.M., Pilipović, S., Zorica, D.: Survey paper properties of the Caputo–Fabrizio fractional derivative and its distributional settings. Fract. Calc. Appl. Anal. 21(1), 29–44 (2018) MathSciNetView ArticleGoogle Scholar
  21. Bainov, D., Simeonov, P.: Integral Inequalities and Applications. Kluwer Academic, Dordrecht (1992) View ArticleGoogle Scholar
  22. Boyd, D.W.: Best constants in inequalities related to Opial’s inequality. J. Math. Anal. Appl. 25, 378–387 (1969) MathSciNetView ArticleGoogle Scholar
  23. Caputo, M., Fabrizio, M.: A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1(2), 73–85 (2015) Google Scholar
  24. Cheung, W.S.: On Opial-type inequalities in two variables. Aequ. Math. 38, 236–244 (1989) MathSciNetView ArticleGoogle Scholar
  25. Cheung, W.S.: Some new Opial-type inequalities. Mathematika 37, 136–142 (1990) MathSciNetView ArticleGoogle Scholar
  26. Cheung, W.S.: Generalizations of Opial-type inequalities in two variables. Tamkang J. Math. 22(1), 43–50 (1991) MathSciNetMATHGoogle Scholar
  27. Cheung, W.S.: Some generalized Opial-type inequalities. J. Math. Anal. Appl. 162(2), 317–321 (1991) MathSciNetView ArticleGoogle Scholar
  28. Cheung, W.S.: Opial-type inequalities with m functions in n variables. Mathematika 39, 319–326 (1992) MathSciNetView ArticleGoogle Scholar
  29. Farid, G., Pečarić, J.: Opial-type integral inequalities for fractional derivatives. Fract. Differ. Calc. 2(1), 31–54 (2012) MathSciNetView ArticleGoogle Scholar
  30. Farid, G., Pečarić, J.: Opial-type integral inequalities for fractional derivatives II. Fract. Differ. Calc. 2(2), 139–155 (2012) MathSciNetView ArticleGoogle Scholar
  31. Farid, G., Pečarić, J.: Opial-type integral inequalities for Widder derivatives and linear differential operators. Int. J. Anal. Appl. 7(1), 38–49 (2015) MATHGoogle Scholar
  32. Farid, G., Pečarić, J., Tomovski, Z.: Opial-type integral inequalities for fractional integral operator involving Mittag-Leffler function. Fract. Differ. Calc. 5(1), 93–106 (2015) MathSciNetView ArticleGoogle Scholar
  33. Farid, G., Pečarić, J., Tomovski, Z.: Generalized Opial-type inequalities for differential and integral operators with special kernels in fractional calculus. J. Math. Inequal. 10(4), 1019–1040 (2016) MathSciNetView ArticleGoogle Scholar
  34. Fink, A.M.: On Opial’s inequality for \(f^{(n)}\). Proc. Am. Math. Soc. 115, 177–181 (1992) MATHGoogle Scholar
  35. Hristov, J.: Transient heat diffusion with a non-singular fading memory: from the Cattaneo constitutive equation with Jeffrey’s kernel to the Caputo–Fabrizio time-fractional derivative. Therm. Sci. 20, 757–762 (2016) View ArticleGoogle Scholar
  36. Hristov, J.: Derivation of fractional Dodson equation and beyond: transient mass diffusion with a non-singular memory and exponentially fading-out diffusivity. Prog. Fract. Differ. Appl. 3(4), 255–270 (2017) View ArticleGoogle Scholar
  37. Hristov, J.: Derivatives with non-singular kernels from the Caputo–Fabrizio definition and beyond: appraising analysis with emphasis on diffusion models. Front. Fract. Calc. 1, 270–342 (2017) Google Scholar
  38. Koliha, J.J., Pečarić, J.: Weighted Opial inequalities. Tamkang J. Math. 33(1), 83–92 (2002) MathSciNetMATHGoogle Scholar
  39. Li, J.-D.: Opial-type inequalities involving several higher order derivatives. J. Math. Anal. Appl. 167, 98–110 (1992) MathSciNetView ArticleGoogle Scholar
  40. Lin, C.T.: Some generalizations of Opial’s inequality. Tamkang J. Math. 16, 123–129 (1985) MathSciNetGoogle Scholar
  41. Losada, J., Nieto, J.: Properties of a new fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1(2), 87–92 (2015) Google Scholar
  42. Mitrinovič, D.S., Pečarić, J., Fink, A.M.: Inequalities Involving Functions and Their Integrals and Derivatives. Kluwer Academic, Dordrecht (1991) View ArticleGoogle Scholar
  43. Opial, Z.: Sur une inégalité. Ann. Pol. Math. 8, 29–32 (1960) MathSciNetView ArticleGoogle Scholar
  44. Pachpatte, B.G.: On Opial-type integral inequalities. J. Math. Anal. Appl. 120, 547–556 (1986) MathSciNetView ArticleGoogle Scholar
  45. Pachpatte, B.G.: On some new generalizations of Opial inequality. Demonstr. Math. 19, 281–291 (1986) MathSciNetMATHGoogle Scholar
  46. Pachpatte, B.G.: On certain integral inequalities related to Opial’s inequality. Period. Math. Hung. 17, 119–125 (1986) MathSciNetView ArticleGoogle Scholar
  47. Pachpatte, B.G.: On inequalities of Opial type. Demonstr. Math. 25, 35–45 (1992) MATHGoogle Scholar
  48. Pang, P.Y.H., Agarwal, R.P.: On an Opial type inequality due to Fink. J. Math. Anal. Appl. 196(2), 748–753 (1995) MathSciNetView ArticleGoogle Scholar
  49. Ross, S.L.: Differential Equations. Wiley, New York (1984) MATHGoogle Scholar
  50. Sarıkaya, M.Z., Budak, H.: New inequalities of Opial-type for conformable fractional integrals. Turk. J. Math. 41, 1164–1173 (2017) MathSciNetView ArticleGoogle Scholar
  51. Schwartz, L.: Mathematics for the Physical Sciences. Addison-Wesley, Paris (1966) MATHGoogle Scholar
  52. Tateishi, A.A., Ribeiro, H.V., Lenzi, E.K.: The role of fractional time-derivative operators on anomalous diffusion. Front. Phys. (2017). https://doi.org/10.3389/fphy.2017.00052 View ArticleGoogle Scholar
  53. Tomovski, Z., Pecarić, J., Farid, G.: Weighted Opial-type inequalities for fractional integral and differential operators involving generalized Mittag-Leffler function. Eur. J. Pure Appl. Math. 10(3), 419–439 (2017) MathSciNetMATHGoogle Scholar
  54. Yang, G.S.: A note on an inequality similar to Opial inequality. Tamkang J. Math. 18, 101–104 (1987) MathSciNetMATHGoogle Scholar

Copyright

© The Author(s) 2018

Advertisement