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On fractional continuous weighted OWA (FCWOWA) operator with applications

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Abstract

In order statistics, CWOWA operator in real line based on Choquet integral was recently introduced by Narukawa et al. (Ann Oper Res 244(2):571–581, 2016). Recently, the relation of Choquet integral with fractional integral on a continuous support was recently discussed by Sugeno (IEEE Trans Fuzzy Syst 23:1439–1457, 2015). As an open problem, Sugeno emphasized that “it is necessary to consider fractional Choquet integral with respect to any monotone measures”. This reason permit us to consider this problem and introduce the fractional Choquet integral with respect to any monotone measures. We also introduce the fractional CWOWA operator (FCWOWA) which includes CWOWA and WOWA operators as special cases. As an application, we also present some important inequalities for FCWOWA operator.

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Acknowledgements

The author is very grateful to the anonymous reviewers for their suggestions. This paper was supported by Babol Noshirvani University of Technology with Grant Program No. BNUT/392100/98.

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  1. Department of Mathematics, Faculty of Basic Science, Babol Noshirvani University of Technology, Shariati Ave., Babol, 47148-71167, Iran

    Hamzeh Agahi

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Appendix A

Appendix A

Proof of Theorem 24

Assume \({ FCWOWA}_{\mu }^{ \mathbf {k}}\left( \left| X\right| ^{p}\right) \left( \omega _{1}\right) , { FCWOWA}_{\mu }^{\mathbf {k}}\left( \left| Y\right| ^{q}\right) \left( \omega _{1}\right) \ne 0\) (in the other cases the thesis immediately follows). Then

$$\begin{aligned}&\frac{{ FCWOWA}_{\mu }^{\mathbf {k}}\left( \left| XY\right| \right) \left( \omega _{1}\right) }{\left[ { FCWOWA}_{\mu }^{\mathbf {k}}\left( \left| X\right| ^{p}\right) \left( \omega _{1}\right) \right] ^{ \frac{1}{p}}\left[ { FCWOWA}_{\mu }^{\mathbf {k}}\left( \left| Y\right| ^{q}\right) \left( \omega _{1}\right) \right] ^{\frac{1}{q}}} \\&\quad \underset{}{=}\frac{\left( C\right) \int _{A}\mathbf {k}\left( \omega _{1},\omega _{2}\right) \left| X\left( \omega _{2}\right) Y\left( \omega _{2}\right) \right| {d}\mu _{2}\left( \omega _{2}\right) }{\left( \left( C\right) \int _{A}\mathbf {k}\left( \omega _{1},\omega _{2}\right) \left| X\left( \omega _{2}\right) \right| ^{p}{d}\mu _{2}\left( \omega _{2}\right) \right) ^{\frac{1}{p}}\left( \left( C\right) \int _{A}\mathbf {k} \left( \omega _{1},\omega _{2}\right) \left| Y\left( \omega _{2}\right) \right| ^{q}{d}\mu _{2}\left( \omega _{2}\right) \right) ^{\frac{1}{q}}} \\&\quad \text {(by (8))} \\&\quad \underset{}{=}\left( C\right) \int _{A}\frac{\mathbf {k}^{\frac{1}{p}}\left( \omega _{1},\omega _{2}\right) \left| X\left( \omega _{2}\right) \right| }{\left( \left( C\right) \int _{A}\mathbf {k}\left( \omega _{1},\omega _{2}\right) \left| X\left( \omega _{2}\right) \right| ^{p}{d}\mu _{2}\left( \omega _{2}\right) \right) ^{\frac{1}{p}}}\\&\qquad \frac{\mathbf {k }^{\frac{1}{q}}\left( \omega _{1},\omega _{2}\right) \left| Y\left( \omega _{2}\right) \right| }{\left( \left( C\right) \int _{A}\mathbf {k} \left( \omega _{1},\omega _{2}\right) \left| Y\left( \omega _{2}\right) \right| ^{q}{d}\mu _{2}\left( \omega _{2}\right) \right) ^{\frac{1}{q}}} {d}\mu _{2}\left( \omega _{2}\right) \\&\quad \text {(by positive homogeneity of Choquet integral)}\\&\quad \underset{}{\leqslant }\left( C\right) \int _{A}\left( \left| \frac{\mathbf {k} \left( \omega _{1},\omega _{2}\right) \left| X\left( \omega _{2}\right) \right| ^{p}}{\left( C\right) \int _{A}\mathbf {k}\left( \omega _{1},\omega _{2}\right) \left| X\left( \omega _{2}\right) \right| ^{p}{d}\mu _{2} }\right| \frac{1}{p}\right. \\&\qquad \left. +\,\left| \frac{\mathbf {k}\left( \omega _{1},\omega _{2}\right) \left| Y\left( \omega _{2}\right) \right| ^{q}}{\left( \left( C\right) \int _{A}\mathbf {k}\left( \omega _{1},\omega _{2}\right) \left| Y\left( \omega _{2}\right) \right| ^{q}{d}\mu _{2}\left( \omega _{2}\right) \right) }\right| \frac{1}{q}\right) {d}\mu _{2}\left( \omega _{2}\right) \\&\qquad \text {(by monotonicity of Choquet integral and Yong's inequality }|rs| \overset{}{\le }\frac{|r|^{p}}{p}+\frac{|s|^{q}}{q}\text {)} \\&\quad \underset{}{\leqslant } \frac{1}{p}\left( C\right) \int _{A}\left| \frac{\mathbf {k} \left( \omega _{1},\omega _{2}\right) \left| X\left( \omega _{2}\right) \right| ^{p}}{\left( C\right) \int _{A}\mathbf {k}\left( \omega _{1},\omega _{2}\right) \left| X\left( \omega _{2}\right) \right| ^{p}{d}\mu _{2}\left( \omega _{2}\right) }\right| {d}\mu _{2}\left( \omega _{2}\right) \\&\qquad +\frac{1}{q}\left( C\right) \int _{A}\left| \frac{\mathbf {k}\left( \omega _{1},\omega _{2}\right) \left| Y\left( \omega _{2}\right) \right| ^{q}}{\left( \left( C\right) \int _{A}\mathbf {k}\left( \omega _{1},\omega _{2}\right) \left| Y\left( \omega _{2}\right) \right| ^{q}{d}\mu _{2}\left( \omega _{2}\right) \right) }\right| {d}\mu _{2}\left( \omega _{2}\right) \\&\qquad \text {(by submodularity)} \\&\quad \underset{}{=}\frac{1}{p}\frac{\left( C\right) \int _{A}\mathbf {k}\left( \omega _{1},\omega _{2}\right) \left| X\left( \omega _{2}\right) \right| ^{p}{d}\mu _{2}\left( \omega _{2}\right) }{\left( C\right) \int _{A} \mathbf {k}\left( \omega _{1},\omega _{2}\right) \left| X\left( \omega _{2}\right) \right| ^{p}{d}\mu _{2}\left( \omega _{2}\right) }+\frac{1}{q} \frac{\left( C\right) \int _{A}\mathbf {k}\left( \omega _{1},\omega _{2}\right) \left| Y\left( \omega _{2}\right) \right| ^{q}{d}\mu _{2}\left( \omega _{2}\right) }{\left( C\right) \int _{A}\mathbf {k}\left( \omega _{1},\omega _{2}\right) \left| Y\left( \omega _{2}\right) \right| ^{q}{d}\mu _{2}\left( \omega _{2}\right) } \\&\qquad \text {(by positive homogeneity of Choquet integral)} \\&\quad \underset{}{=}\frac{1}{p}+\frac{1}{q}\underset{}{=}1. \end{aligned}$$

Therefore,

$$\begin{aligned} { FCWOWA}_{\mu }^{\mathbf {k}}\left( \left| XY\right| \right) \left( \omega _{1}\right) \le \left[ { FCWOWA}_{\mu }^{\mathbf {k}}\left( \left| X\right| ^{p}\right) \left( \omega _{1}\right) \right] ^{\frac{1}{p}} \left[ { FCWOWA}_{\mu }^{\mathbf {k}}\left( \left| Y\right| ^{q}\right) \left( \omega _{1}\right) \right] ^{\frac{1}{q}}, \end{aligned}$$

which completes the proof. \(\square \)

Proof of Theorem 27

$$\begin{aligned}&{ FCWOWA}_{\mu }^{\mathbf {k}}\left( \left| X+Y\right| ^{p}\right) \left( \omega _{1}\right) \\&\quad \underset{}{=}\left( C\right) \int \mathbf {k}(\omega _{1},\omega _{2})\left| X(\omega _{2})+Y(\omega _{2})\right| ^{p}{d}\mu _{2}(\omega _{2}) \\&\quad \underset{}{\le }\left( C\right) \int \left( \mathbf {k}(\omega _{1},\omega _{2})\left| X(\omega _{2})\right| \left| X(\omega _{2})+Y(\omega _{2})\right| ^{p-1}\right. \\&\qquad +\left. \mathbf {k}(\omega _{1},\omega _{2})\left| Y(\omega _{2})\right| \left| X(\omega _{2})+Y(\omega _{2})\right| ^{p-1}\right) {d}\mu _{2}(\omega _{2})\\&\quad \text {(by monotonicity of Choquet integral and inequality }|x+y|\overset{}{ \le }|x|+|y|\text {)} \\&\quad \underset{}{\le }\left( C\right) \int \mathbf {k}(\omega _{1},\omega _{2})\left| X(\omega _{2})\right| \left| X(\omega _{2})+Y(\omega _{2})\right| ^{p-1}{d}\mu _{2}(\omega _{2}) \\&\qquad +\left( C\right) \int \mathbf {k}\left( \omega _{1},\omega _{2}\right) \left| Y(\omega _{2})\right| \left| X(\omega _{2})+Y(\omega _{2})\right| ^{p-1}{d}\mu _{2}(\omega _{2}) \\&\quad \text {(by submodularity)} \\&\quad \underset{}{=}{} { FCWOWA}_{\mu }^{\mathbf {k}}\left( \left| X\right| \left| X+Y\right| ^{p-1}\right) \left( \omega _{1}\right) +{ FCWOWA}_{\mu }^{\mathbf {k}}\left( \left| Y\right| \left( X+Y\right) ^{p-1}\right) \left( \omega _{1}\right) \\&\quad \underset{}{\le }\left( { FCWOWA}_{\mu }^{\mathbf {k}}\left( \left| X\right| ^{p}\right) \left( \omega _{1}\right) \right) ^{\frac{1}{p} }\left( { FCWOWA}_{\mu }^{\mathbf {k}}\left( \left| X+Y\right| ^{(p-1)q}\right) (\omega _{1})\right) ^{\frac{1}{q}} \\&\qquad +\left( { FCWOWA}_{\mu }^{\mathbf {k}}\left( \left| Y\right| ^{p}\right) \left( \omega _{1}\right) \right) ^{\frac{1}{p}}\left( { FCWOWA}_{\mu }^{\mathbf {k}}\left( \left| X+Y\right| ^{(p-1)q}\right) (\omega _{1})\right) ^{\frac{1}{q}} \\ \end{aligned}$$

by (10), where \(\frac{1}{p}+\frac{1}{q}=1\). Then

$$\begin{aligned}&\frac{{ FCWOWA}_{\mu }^{\mathbf {k}}\left( \left| X+Y\right| ^{p}\right) \left( \omega _{1}\right) }{\left( { FCWOWA}_{\mu }^{\mathbf {k} }\left( \left| X+Y\right| ^{(p-1)q}\right) (\omega _{1})\right) ^{ \frac{1}{q}}} \\&\quad \underset{}{\le }\left( { FCWOWA}_{\mu }^{\mathbf {k}}\left( \left| X\right| ^{p}\right) \left( \omega _{1}\right) \right) ^{\frac{1}{p} }+\left( { FCWOWA}_{\mu }^{\mathbf {k}}\left( \left| Y\right| ^{p}\right) \left( \omega _{1}\right) \right) ^{\frac{1}{p}}. \end{aligned}$$

So,

$$\begin{aligned}&\left( { FCWOWA}_{\mu }^{\mathbf {k}}\left( \left| X+Y\right| ^{p}\right) \left( \omega _{1}\right) \right) ^{\frac{1}{p}} \\&\quad \underset{}{\le }\left( { FCWOWA}_{\mu }^{\mathbf {k}}\left( \left| X\right| ^{p}\right) \left( \omega _{1}\right) \right) ^{\frac{1}{p} }+\left( { FCWOWA}_{\mu }^{\mathbf {k}}\left( \left| Y\right| ^{p}\right) \left( \omega _{1}\right) \right) ^{\frac{1}{p}}. \end{aligned}$$

This completes the proof. \(\square \)

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Agahi, H. On fractional continuous weighted OWA (FCWOWA) operator with applications. Ann Oper Res 287, 1–10 (2020). https://doi.org/10.1007/s10479-019-03450-5

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