Rate of convergence of Euler approximation of time-dependent mixed SDEs driven by Brownian motions and fractional Brownian motions

1.   Introduction

The study of integral inequality is an interesting area for research in mathematical analysis [1,2]. The fundamental integral inequalities can be instrumental in cultivating the subjective properties of convexity. The existence of massive literature surrounding integral inequalities for convex functions [3-7] depicts the importance of this topic. The most beautiful fact about convex function is that, it has a very elegant representation based on an inequality presented when the functional value of a linear combination of two points in its domain does not exceed the linear combination of the functional values at those two points. Fractional calculus owes its starting point to whether or not the importance of a derivative to an integer order could be generalized to a fractional order which is not an integer. Following this unique conversation between L'Hopital and Leibniz, the concept of fractional calculus grabbed the eye of some extraordinary researchers like Euler, Laplace, Fourier, Lacroix, Abel, Riemann, and Liouville. Over time, fractional operators have been differentiated with their singularity, locality and having general forms with the improvements made in their kernel structures. In this sense, based on the basic concepts of fractional analysis, Riemann-Liouville(R-L) and Caputo operators, various new trends have been successful. Fractional integral inequalities are marvelous tools for building up the qualitative and quantitative properties of preinvex functions. There has been a ceaseless development of intrigue in such a region of research so as to address the issues of different utilizations of these variants. In $\rm 1938$, Ostrowski inequality established the following useful and interesting integral inequality, (see [12] and [13]). This review assumed a vital part in growing and getting varieties of well-known integral inequalities with the assistance of fractional integral operators. Then again, by characterizing various forms of Riemann-Liouville(R-L) fractional operator somewhat recently, new forms and refinements of integral inequalities involving differentiable functions have been presented. Studies in the field of fractional calculus have carried another point of view and direction in different fields of applied sciences. It has revealed insight into numerous real-life issues with the utilizations of recently characterized fractional operators.

For recent result and their related some generalizations, variants and extensions concerning Ostrowski inequality (see [9,10,14-17]).

The aim of this paper is to establish some integral inequalities for functions whose derivatives in absolute value are preinvex. Now we recall some notions in invexity analysis which will be used through the paper (see [20,21,24,26,28]) and references therein.

Let $g:\mathcal K \rightarrow \Re$ and $\eta : \mathcal K \times \mathcal K \rightarrow \Re$, where $\mathcal K$ is a nonempty set in $\Re^n,$ be continuous functions.

Definition [19] A function $g:\mathcal K\subseteq \Re = \left( -\infty , \infty \right) \rightarrow \Re$ is said to be convex, if we have

for all $\mathfrak{c}, \mathfrak{e}\in \mathcal K\;$ and $v\in \left[0, 1\right].$

Definition [25] The set $\mathcal K \subset \Re^n$ is said to be invex with respect to $\eta(., .)$, if for every $\mathfrak{c}, \mathfrak{e}\in \mathcal K$ and $v\in[0, 1]$

The above set $\mathcal K$ is also called $\eta$-connected set.

It is obvious that every convex set is invex with respect to $\eta(\mathfrak{e}, \mathfrak{c}) = \mathfrak{e}-\mathfrak{c}$ but there exist invex sets which are not convex [20].

Definition The function $g$ on the invex set $\mathcal K$ is said to be preinvex with respect to $\eta$ if

The function $-g$ is said to be preconcave if and only if $g$ is preinvex .

The important note that every convex function is a preinvex function but the converse is not true [21]. For example $g(v) = -|v|, \ \ \forall \ v\in \Re$, is not convex function but it is preinvex function with respect to

We also want the following hypothesis regarding the function $\eta$ which is due to Mohan et al. [22]. Condition-$\mathcal C$: Let $\mathcal K \subset \Re^n$ be an open invex subset with respect to $\eta : \mathcal K \times \mathcal K \rightarrow \Re$. For any $\mathfrak{c}, \mathfrak{e} \in \mathcal K$ and $v \in [0, 1]$

For any $\mathfrak{c}, \mathfrak{e}\in \mathcal K$ and $v_{1}, v_{2} \in [0, 1]$ from condition $\mathcal C$, we have

If $g$ is a preinvex function on $[\mathfrak{c}, \mathfrak{c}+\eta(\mathfrak{e}, \mathfrak{c})]$ and the mapping $\eta$ satisfies condition $\mathcal C$, then for every $v \in [0, 1]$, from Eq (1.1), it yields that

and

There are many vector functions that satisfy the condition $\mathcal C$ in [25], which trivial case $\eta \left( \mathfrak{c}, \mathfrak{e}\right) = \mathfrak{c}-\mathfrak{e}$. For example suppose $\mathcal K = \Re\backslash \{0\}$ and

The set $\mathcal K$ is invex set and $\eta$ satisfies the condition $\mathcal C$.

Noor et al. [23], proved the following Hermite-Hadamard type inequalities.

Theorem 1.1. Let $g:\mathcal K = [\mathfrak{c}, \mathfrak{c}+\eta(\mathfrak{e}, \mathfrak{c})]\to (0, \infty)$ be a preinvex function on the interval of real numbers $\mathcal K^0$ with $\eta(\mathfrak{e}, \mathfrak{c})>0$, then the following inequalities hold:

Then Riemann-Liouville(R-L) fractional integrals of order $\varepsilon >0$ with $\mathfrak{c} \geq 0$ are defined as follows:

and

In [30], Sarikaya et al. also described the inequality in fractional integral version. In this study, considering the above mentioned theoretical framework, firstly, an integral identity which is candidate to produce Ostrowski type inequalities has been proved. With the help of such identity like Hölder, Power mean, Young's inequalities, Hölder-İşcan, Improved power means inequality and convexity, a new type of inequality, Ostrowski type inequalities, has been obtained.

2.   Main results

In this section, we give Ostrowski inequalities for Riemann-Liouville(R-L) fractional integrals operator are obtained for a differentiable functions on $(\mathfrak{c}, \mathfrak{c}+\eta \left( \mathfrak{e}, \mathfrak{c}\right))$. For this purpose, we give a new identity that involve Riemann-Liouville(R-L) fractional integrals operator whose second derivatives are preinvex functions.

Lemma 2.1. Suppose that a mapping $g:[\mathfrak{c}, \mathfrak{c}+\eta \left( \mathfrak{e}, \mathfrak{c}\right)]\rightarrow \Re$ istwice differentiable with $\mathfrak{c}< \mathfrak{c}+\eta \left( \mathfrak{e}, \mathfrak{c}\right)$. If $g^{{\prime \prime }}\in \mathcal L_{1}[\mathfrak{c}, \mathfrak{c}+\eta \left( \mathfrak{e}, \mathfrak{c}\right)],$ then for all $\mathfrak{z}\in\lbrack \mathfrak{c}, \mathfrak{c}+\eta \left( \mathfrak{e}, \mathfrak{c}\right)]$ and $\varepsilon >0,$ the following equality

satisfies for $v\in [0, 1]$ .

Proof. Let us assume that

where

and similarly

Combining $I_{1}$ and $I_{2}$ with (2.2), we get (2.3).

Remark 2.1. If we set $\varepsilon = 1$ and $\eta\left( \mathfrak{c}, \mathfrak{e}\right) = \mathfrak{c}-\mathfrak{e}$ in Lemma 2.1, we get (Lemma 1 in [11]).

Theorem 2.1. Assume that all the assumptions as defined in Lemma 2.1 and $|g^{{\prime \prime }}|$ is preinvex function on $[\mathfrak{c}, \mathfrak{c}+\eta \left( \mathfrak{e}, \mathfrak{c}\right)],$ then for all$\varepsilon >0,$ the following inequality

satisfies for $v\in \lbrack 0, 1].$

Proof. From Lemma 2.1 and since $|g^{{\prime \prime}}|$ is preinvex function on $[\mathfrak{c}, \mathfrak{c}+\eta \left( \mathfrak{e}, \mathfrak{c}\right)],$ we obtain

This completes the proof.

Remark 2.2. If we set $\varepsilon = 1$ and $\eta\left( \mathfrak{c}, \mathfrak{e}\right) = \mathfrak{c}-\mathfrak{e},$ then from Theorem 2.1, we get (Theorem 4 in [11]) with $s = 1$.

Corollary 2.1. By using Theorem 2.1 with $|g^{{\prime \prime }}|\leq \mathcal M,$ we get the following inequality

Remark 2.3. If we set $\varepsilon = 1$ and $\eta\left( \mathfrak{c}, \mathfrak{e}\right) = \mathfrak{c}-\mathfrak{e},$ then from Corollary 2.1, we recapture (Theorem 2.1, [32]).

Corollary 2.2. If we set $\eta\left( \mathfrak{c}, \mathfrak{e}\right) = \mathfrak{c}-\mathfrak{e}$ and $\mathfrak{z} = \frac{\mathfrak{c}+\mathfrak{e}}{2},$ in Corollary 2.1, we get the mid-point inequality

Theorem 2.2. Assume that all the assumptions as defined in Lemma 2.1 and $|g^{{\prime \prime }}|^q,$ $q>1$ is preinvex function on $[\mathfrak{c}, \mathfrak{c}+\eta \left( \mathfrak{e}, \mathfrak{c}\right)],$ then for all$\varepsilon >0,$ the following inequality

satisfies for $v\in \lbrack 0, 1],$ where $q^{-1}+p^{-1} = 1.$

Proof. Suppose that $p>1.$ From Lemma 2.1, by using the well-known Hölder integral inequality and the preinvexity of $|g^{{\prime \prime}}|^q$, we obtain

Since $|g^{{\prime \prime}}|^q$ is preinvexity on $[\mathfrak{c}, \mathfrak{c}+\eta \left( \mathfrak{e}, \mathfrak{c}\right)],$ we obtain

and

By using (2.8) and (2.9) with (2.7), we obtain

This completes the proof.

Remark 2.4. If we set $\varepsilon = 1$ and $\eta\left( \mathfrak{c}, \mathfrak{e}\right) = \mathfrak{c}-\mathfrak{e},$ then from Theorem 2.2, we get ( Theorem 5, [11]) with $s = 1$.

Corollary 2.3. Using Theorem 2.2 with $|g^{{\prime \prime }}|\leq \mathcal M,$ we get

Corollary 2.4. If in Corollary 2.3, we set $\eta\left( \mathfrak{c}, \mathfrak{e}\right) = \mathfrak{c}-\mathfrak{e}$ and $\mathfrak{z} = \frac{\mathfrak{c}+\mathfrak{e}}{2},$ then we get the mid-point inequality

Theorem 2.3. Assume that all the assumptions as defined in Lemma 2.1 and $|g^{{\prime \prime }}|^q,$ $q\geq1$ is preinvex function on $[\mathfrak{c}, \mathfrak{c}+\eta \left( \mathfrak{e}, \mathfrak{c}\right)],$ then for all$\varepsilon >0,$ the following inequality

satisfies for $v\in \lbrack 0, 1].$

Proof. Suppose that $q\geq1.$ From Lemma 2.1, by using the power-mean integral inequality and preinvexity of $|g^{{\prime \prime }}|^q$, we obtain

Since $|g^{{\prime \prime}}|^q$ is preinvexity on $[\mathfrak{c}, \mathfrak{c}+\eta \left( \mathfrak{e}, \mathfrak{c}\right)],$ we obtain

and

By using (2.12) and (2.13) with (2.11), we obtain

This completes the proof.

Remark 2.5. If we set $\varepsilon = 1$ and $\eta\left( \mathfrak{c}, \mathfrak{e}\right) = \mathfrak{c}-\mathfrak{e},$ then from Theorem 2.3, we get (Theorem 6, [11])with $s = 1$.

Corollary 2.5. Under the same assumptions in Theorem 2.3 with $|g^{{\prime \prime }}|\leq \mathcal M,$ we get the following inequality

Corollary 2.6. If in Corollary 2.5, we set $\eta\left( \mathfrak{c}, \mathfrak{e}\right) = \mathfrak{c}-\mathfrak{e}$ and $\mathfrak{z} = \frac{\mathfrak{c}+\mathfrak{e}}{2},$ then we get the mid-point inequality

Theorem 2.4. Assume that all the assumptions as defined in Lemma 2.1 and $|g^{{\prime \prime }}|^q,$ $q>1$ is preinvex function on $[\mathfrak{c}, \mathfrak{c}+\eta \left( \mathfrak{e}, \mathfrak{c}\right)],$ then for all$\varepsilon >0,$ the following inequality

satisfies for $v\in \lbrack 0, 1].$

Proof. From Lemma 2.1, we obtain

By using the Young's inequality as

This completes the proof.

Corollary 2.7. If we set $\eta\left( \mathfrak{c}, \mathfrak{e}\right) = \mathfrak{c}-\mathfrak{e}$ and $\varepsilon = 1$ in Theorem 2.4, we get

Corollary 2.8. If in Theorem 2.4, we set $\eta\left( \mathfrak{c}, \mathfrak{e}\right) = \mathfrak{c}-\mathfrak{e}$ and $\mathfrak{z} = \frac{\mathfrak{c}+\mathfrak{e}}{2},$ then we get the mid-point inequality

Theorem 2.5. Assume that all the assumptions as defined in Lemma 2.1 and $|g^{{\prime \prime }}|^q,$ $q>1$ is preinvex function on $[\mathfrak{c}, \mathfrak{c}+\eta \left( \mathfrak{e}, \mathfrak{c}\right)],$ then for all$\varepsilon >0,$ the following inequality

satisfies for $v\in \lbrack 0, 1],$ where $q^{-1}+p^{-1} = 1.$

Proof. From Lemma 2.1, by using the Hölder-İşcan integral inequality (see in [33]) and the preinvexity of $|g^{{\prime \prime}}|^q$, we obtain

After simplification, we get (2.15). This completes the proof.

Corollary 2.9. Using the same assumptions in Theorem 2.5 with $|g^{{\prime \prime }}|\leq \mathcal M,$ we get

Theorem 2.6. Assume that all the assumptions as defined in Lemma 2.1 and $|g^{{\prime \prime }}|^q,$ $q\geq1$ is preinvex function on $[\mathfrak{c}, \mathfrak{c}+\eta \left( \mathfrak{e}, \mathfrak{c}\right)],$ then for all$\varepsilon >0,$ the following inequality

satisfies for $v\in \lbrack 0, 1],$ where $q^{-1}+p^{-1} = 1.$

Proof. From Lemma 2.1, improved power-mean integral inequality(see in [33]) and the preinvexity of $|g^{{\prime \prime}}|^q$, we obtain

This completes the proof.

Corollary 2.10. Using the same assumption of Theorem 2.6 with $|g^{{\prime \prime }}|\leq \mathcal M,$ we get

3.   Modified bessel function

We recall the first kind modified Bessel function $\Im_m,$ which has the series representation (see [42], p.77)

where $\zeta \in \Re$ and $m>-1,$ while the second kind modified Bessel function $g_m$ (see [42], p.78) is usually defined as

Consider the function $\Omega_m\left( \zeta \right): \Re \rightarrow [1, \infty)$ defined by

where $\Gamma$ is the gamma function.

The first order derivative formula of $\Omega_m\left( \zeta \right)$ is given by [42]:

and the second derivative can be easily calculated from (3.1) as

and the third derivative can be easily calculated from (3.2) as

Proposition 3.1. Suppose that $m>-1$ and $0<\mathfrak{c}<\mathfrak{e}.$ Then we get the inequality

Proof. The assertion follows immediately from Corollary 2.7 using $g\left(\zeta\right) = \Omega^{\prime} _{m}(\zeta),$ $\zeta>0$ and the identities (3.2) and (3.3).

4.   Conclusions

In this paper, we have defined an idea of fractional integral inequalities whose second derivatives are preinvex functions. We also investigated and proved a new lemma for the second derivatives of Riemann-Liouville fractional integral operator. Some new special cases are discovered in the form of corollaries. We hope that the strategies of this paper will motivate the researchers working in functional analysis, information theory and statistical theory. It is quite open to think about Ostrowski variants for generalized integral operators having Atangana-Baleanu operator etc. by applying generalized preinvexity. The results, which we have presented in this article, will potentially motivate researchers to study analogous and more general integral inequalities for various other kinds of fractional integral operators.

Conflict of interest

All authors have no conflict of interest.