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October  2021, 14(10): 3529-3539. doi: 10.3934/dcdss.2020432

On the fuzzy stability results for fractional stochastic Volterra integral equation

Reza Chaharpashlou a, , Abdon Atangana b, and  Reza Saadati a,,

a. 

School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran
b. 

Institute for Groundwater Studies (IGS) Faculty: Natural and Agricultural Sciences, University of the Free State, Bloemfontein 9300, South Africa

* Corresponding author: Reza Saadati (email:rsaadati@eml.cc), https://orcid.org/0000-0002-6770-6951

Received  October 2019 Revised  February 2020 Published  October 2021 Early access  November 2020

By a fuzzy controller function, we stable a random operator associated with a type of fractional stochastic Volterra integral equations. Using the fixed point technique, we get an approximation for the mentioned random operator by a solution of the fractional stochastic Volterra integral equation.

Keywords: Fractional stochastic integral equations, fuzzy controller function, stability, random operator, fuzzy normed space.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3529-3539. doi: 10.3934/dcdss.2020432
References:
[1]

R. P. Agarwal, R. Saadati and A. Salamati, Approximation of the multiplicatives on random multi-normed space, Journal of inequalities and applications, 204 (2017), 204. doi: 10.1186/s13660-017-1478-9.  Google Scholar

[2]

T. Bag and S. K. Samanta, Finite dimensional intuitionistic fuzzy normed linear spaces, Ann. Fuzzy Math. Inform, 6 (2013), 45-57.   Google Scholar

[3]

Y. J. Cho, T. M. Rassias and R. Saadati, Stability of functional equations in random normed spaces, Springer Optimization and Its Applications, 86, Springer, New York, 2013. doi: 10.1007/978-1-4614-8477-6.  Google Scholar

[4]

C. D. Constantinescu, J. M. Ramirez and W. R. Zhu, An application of fractional differential equations to risk theory, Finance and Stochastics, 23 (2019), 1001-1024.  doi: 10.1007/s00780-019-00400-8.  Google Scholar

[5]

L. Cădariu and V. Radu, Fixed points and the stability of Jensen's functional equation, J. Inequal. Pure Appl. Math, 4 (2003), 4.  Google Scholar

[6]

J. B. Diaz and B. Margolis, A fixed point theorem of the alternative, for contractions on a generalized complete metric space, Bulletin of the American Mathematical Society, 74 (1968), 305-309.  doi: 10.1090/S0002-9904-1968-11933-0.  Google Scholar

[7]

M. A. El-Moneam, F. Tarek Ibrahim and S. Elamody, Stability of a fractional difference equation of high order, Journal of Nonlinear Sciences and Applications, 12 (2019), 65-74.  doi: 10.22436/jnsa.012.02.01.  Google Scholar

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A. M. A. El-Sayed and F.M. Gaafar, Positive solutions of singular Hadamard-type fractional differential equations with infinite-point boundary conditions or integral boundary conditions, Advances in Difference Equations, 2019 (2019), 382. doi: 10.1186/s13662-019-2315-x.  Google Scholar

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O. Hadžić and E. Pap, Fixed Point Theory in Probabilistic Metric Spaces, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001. doi: 10.1007/978-94-017-1560-7.  Google Scholar

[10]

J. Jiang, D. O'Regan, J. Xu and Z. Fu, Positive solutions for a system of nonlinear Hadamard fractional differential equations involving coupled integral boundary conditions, Journal of Inequalities and Applications, 2019 (2019), 1-18.  doi: 10.1186/s13660-019-2156-x.  Google Scholar

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S. Jung, A fixed point approach to the stability of differential equations $y^{'} = F (x, y)$, Bulletin of the Malaysian Mathematical Sciences Society, 33 (2010).  Google Scholar

[12]

S. M. Jung, A fixed point approach to the stability of an integral equation related to the wave equation, in Abstract and Applied Analysis, 2013, Hindawi, 2013. doi: 10.1155/2013/612576.  Google Scholar

[13]

H. Khan, T. Abdeljawad, M. Aslam, R. A. Khan and A. Khan, Existence of positive solution and Hyers–Ulam stability for a nonlinear singular-delay-fractional differential equation, Advances in Difference Equations, 2019 (2019), 1-13.  doi: 10.1186/s13662-019-2054-z.  Google Scholar

[14]

H. Khan, F. Jarad, T. Abdeljawad and A. Khan, A singular ABC-fractional differential equation with $p$-Laplacian operator, Chaos, Solitons & Fractals, 129 (2019), 56-61.  doi: 10.1016/j.chaos.2019.08.017.  Google Scholar

[15]

H. Khan, A. Khan, T. Abdeljawad and A. Alkhazzan, Existence results in Banach space for a nonlinear impulsive system, Advances in Difference Equations, 2019 (2019), 18. doi: 10.1186/s13662-019-1965-z.  Google Scholar

[16]

A. Khan, H. Khan, J. F. Gómez-Aguilar and T. Abdeljawad, Existence and Hyers-Ulam stability for a nonlinear singular fractional differential equations with Mittag-Leffler kernel, Chaos, Solitons & Fractals, 127 (2019), 422-427.  doi: 10.1016/j.chaos.2019.07.026.  Google Scholar

[17]

H. Khan, A. Khan, F. Jarad and A. Shah, Existence and data dependence theorems for solutions of an ABC-fractional order impulsive system, Chaos, Solitons & Fractals, (2019), 109477. doi: 10.1016/j.chaos.2019.109477.  Google Scholar

[18]

Y. Ma and W. Li, Application and research of fractional differential equations in dynamic analysis of supply chain financial chaotic system, Chaos, Solitons & Fractals, 130 (2020), 109417. doi: 10.1016/j.chaos.2019.109417.  Google Scholar

[19]

D. Miheţ and V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, Journal of mathematical Analysis and Applications, 343 (2008), 567-572.  doi: 10.1016/j.jmaa.2008.01.100.  Google Scholar

[20]

D. Miheţ and R. Saadati, On the stability of the additive Cauchy functional equation in random normed spaces, Applied mathematics letters, 24 (2011), 2005-2009.  doi: 10.1016/j.aml.2011.05.033.  Google Scholar

[21]

A. K. Mirmostafaee, Perturbation of generalized derivations in fuzzy Menger normed algebras, Fuzzy sets and systems, 195 (2012), 109-117.  doi: 10.1016/j.fss.2011.10.015.  Google Scholar

[22]

H. K. Nashine and R. W. Ibrahim, Symmetric solutions of nonlinear fractional integral equations via a new fixed point theorem under FG-contractive condition, Numerical Functional Analysis and Optimization, 40 (2019), 1448-1466.  doi: 10.1080/01630563.2019.1602779.  Google Scholar

[23]

S. Nadaban, T. Binzar and F. Pater, Some fixed point theorems for $\varphi$-contractive mappings in fuzzy normed linear spaces, Journal of Nonlinear Sciences and Applications, 10 (2017), 5668-5676.  doi: 10.22436/jnsa.010.11.05.  Google Scholar

[24]

R. Naeem and M. Anwar, Jessen type functionals and exponential convexity, J. Math. Comput. Sci, 17 (2017), 429-436.  doi: 10.22436/jmcs.017.03.08.  Google Scholar

[25]

R. Naeem and M. Anwar, Weighted Jessen's functionals and exponential convexity, J. Math. Comput. Sci, 19 (2019), 171-180.  doi: 10.22436/jmcs.019.03.04.  Google Scholar

[26]

C. Park, D. Y. Shin, R. Saadati and J. R. Lee, A fixed point approach to the fuzzy stability of an AQCQ-functional equation, Filomat, 30 (2016), 1833-1851.  doi: 10.2298/FIL1607833P.  Google Scholar

[27]

C. Park, S. O. Kim and C. Alaca, Stability of additive-quadratic rho-functional equations in Banach spaces: A fixed point approach, J. Nonlin. Sci. Appl., 10 (2017), 1252-1262.  doi: 10.22436/jnsa.010.03.34.  Google Scholar

[28]

G. Sadeghi, M. Nazarianpoor and J. M. Rassias, Solution and stability of quattuorvigintic functional equation in intuitionistic fuzzy normed spaces, Iranian Journal of Fuzzy Systems, 15 (2018), 13-30.   Google Scholar

[29]

R. Saadati and S. M. Vaezpour, Some results on fuzzy Banach spaces, Journal of Applied Mathematics and Computing, 17 (2005), 475-484.  doi: 10.1007/BF02936069.  Google Scholar

[30]

R. Saadati and C. Park, Approximation of derivations and the superstability in random Banach $\ast$-algebras, Advances in Difference Equations, 2018 (2018), 1-12.  doi: 10.1186/s13662-018-1882-6.  Google Scholar

[31]

W. Wei, X. Li and X. Li, New stability results for fractional integral equation, Computers & Mathematics with Applications, 64 (2012), 3468-3476.  doi: 10.1016/j.camwa.2012.02.057.  Google Scholar

show all references

References:
[1]

R. P. Agarwal, R. Saadati and A. Salamati, Approximation of the multiplicatives on random multi-normed space, Journal of inequalities and applications, 204 (2017), 204. doi: 10.1186/s13660-017-1478-9.  Google Scholar

[2]

T. Bag and S. K. Samanta, Finite dimensional intuitionistic fuzzy normed linear spaces, Ann. Fuzzy Math. Inform, 6 (2013), 45-57.   Google Scholar

[3]

Y. J. Cho, T. M. Rassias and R. Saadati, Stability of functional equations in random normed spaces, Springer Optimization and Its Applications, 86, Springer, New York, 2013. doi: 10.1007/978-1-4614-8477-6.  Google Scholar

[4]

C. D. Constantinescu, J. M. Ramirez and W. R. Zhu, An application of fractional differential equations to risk theory, Finance and Stochastics, 23 (2019), 1001-1024.  doi: 10.1007/s00780-019-00400-8.  Google Scholar

[5]

L. Cădariu and V. Radu, Fixed points and the stability of Jensen's functional equation, J. Inequal. Pure Appl. Math, 4 (2003), 4.  Google Scholar

[6]

J. B. Diaz and B. Margolis, A fixed point theorem of the alternative, for contractions on a generalized complete metric space, Bulletin of the American Mathematical Society, 74 (1968), 305-309.  doi: 10.1090/S0002-9904-1968-11933-0.  Google Scholar

[7]

M. A. El-Moneam, F. Tarek Ibrahim and S. Elamody, Stability of a fractional difference equation of high order, Journal of Nonlinear Sciences and Applications, 12 (2019), 65-74.  doi: 10.22436/jnsa.012.02.01.  Google Scholar

[8]

A. M. A. El-Sayed and F.M. Gaafar, Positive solutions of singular Hadamard-type fractional differential equations with infinite-point boundary conditions or integral boundary conditions, Advances in Difference Equations, 2019 (2019), 382. doi: 10.1186/s13662-019-2315-x.  Google Scholar

[9]

O. Hadžić and E. Pap, Fixed Point Theory in Probabilistic Metric Spaces, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001. doi: 10.1007/978-94-017-1560-7.  Google Scholar

[10]

J. Jiang, D. O'Regan, J. Xu and Z. Fu, Positive solutions for a system of nonlinear Hadamard fractional differential equations involving coupled integral boundary conditions, Journal of Inequalities and Applications, 2019 (2019), 1-18.  doi: 10.1186/s13660-019-2156-x.  Google Scholar

[11]

S. Jung, A fixed point approach to the stability of differential equations $y^{'} = F (x, y)$, Bulletin of the Malaysian Mathematical Sciences Society, 33 (2010).  Google Scholar

[12]

S. M. Jung, A fixed point approach to the stability of an integral equation related to the wave equation, in Abstract and Applied Analysis, 2013, Hindawi, 2013. doi: 10.1155/2013/612576.  Google Scholar

[13]

H. Khan, T. Abdeljawad, M. Aslam, R. A. Khan and A. Khan, Existence of positive solution and Hyers–Ulam stability for a nonlinear singular-delay-fractional differential equation, Advances in Difference Equations, 2019 (2019), 1-13.  doi: 10.1186/s13662-019-2054-z.  Google Scholar

[14]

H. Khan, F. Jarad, T. Abdeljawad and A. Khan, A singular ABC-fractional differential equation with $p$-Laplacian operator, Chaos, Solitons & Fractals, 129 (2019), 56-61.  doi: 10.1016/j.chaos.2019.08.017.  Google Scholar

[15]

H. Khan, A. Khan, T. Abdeljawad and A. Alkhazzan, Existence results in Banach space for a nonlinear impulsive system, Advances in Difference Equations, 2019 (2019), 18. doi: 10.1186/s13662-019-1965-z.  Google Scholar

[16]

A. Khan, H. Khan, J. F. Gómez-Aguilar and T. Abdeljawad, Existence and Hyers-Ulam stability for a nonlinear singular fractional differential equations with Mittag-Leffler kernel, Chaos, Solitons & Fractals, 127 (2019), 422-427.  doi: 10.1016/j.chaos.2019.07.026.  Google Scholar

[17]

H. Khan, A. Khan, F. Jarad and A. Shah, Existence and data dependence theorems for solutions of an ABC-fractional order impulsive system, Chaos, Solitons & Fractals, (2019), 109477. doi: 10.1016/j.chaos.2019.109477.  Google Scholar

[18]

Y. Ma and W. Li, Application and research of fractional differential equations in dynamic analysis of supply chain financial chaotic system, Chaos, Solitons & Fractals, 130 (2020), 109417. doi: 10.1016/j.chaos.2019.109417.  Google Scholar

[19]

D. Miheţ and V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, Journal of mathematical Analysis and Applications, 343 (2008), 567-572.  doi: 10.1016/j.jmaa.2008.01.100.  Google Scholar

[20]

D. Miheţ and R. Saadati, On the stability of the additive Cauchy functional equation in random normed spaces, Applied mathematics letters, 24 (2011), 2005-2009.  doi: 10.1016/j.aml.2011.05.033.  Google Scholar

[21]

A. K. Mirmostafaee, Perturbation of generalized derivations in fuzzy Menger normed algebras, Fuzzy sets and systems, 195 (2012), 109-117.  doi: 10.1016/j.fss.2011.10.015.  Google Scholar

[22]

H. K. Nashine and R. W. Ibrahim, Symmetric solutions of nonlinear fractional integral equations via a new fixed point theorem under FG-contractive condition, Numerical Functional Analysis and Optimization, 40 (2019), 1448-1466.  doi: 10.1080/01630563.2019.1602779.  Google Scholar

[23]

S. Nadaban, T. Binzar and F. Pater, Some fixed point theorems for $\varphi$-contractive mappings in fuzzy normed linear spaces, Journal of Nonlinear Sciences and Applications, 10 (2017), 5668-5676.  doi: 10.22436/jnsa.010.11.05.  Google Scholar

[24]

R. Naeem and M. Anwar, Jessen type functionals and exponential convexity, J. Math. Comput. Sci, 17 (2017), 429-436.  doi: 10.22436/jmcs.017.03.08.  Google Scholar

[25]

R. Naeem and M. Anwar, Weighted Jessen's functionals and exponential convexity, J. Math. Comput. Sci, 19 (2019), 171-180.  doi: 10.22436/jmcs.019.03.04.  Google Scholar

[26]

C. Park, D. Y. Shin, R. Saadati and J. R. Lee, A fixed point approach to the fuzzy stability of an AQCQ-functional equation, Filomat, 30 (2016), 1833-1851.  doi: 10.2298/FIL1607833P.  Google Scholar

[27]

C. Park, S. O. Kim and C. Alaca, Stability of additive-quadratic rho-functional equations in Banach spaces: A fixed point approach, J. Nonlin. Sci. Appl., 10 (2017), 1252-1262.  doi: 10.22436/jnsa.010.03.34.  Google Scholar

[28]

G. Sadeghi, M. Nazarianpoor and J. M. Rassias, Solution and stability of quattuorvigintic functional equation in intuitionistic fuzzy normed spaces, Iranian Journal of Fuzzy Systems, 15 (2018), 13-30.   Google Scholar

[29]

R. Saadati and S. M. Vaezpour, Some results on fuzzy Banach spaces, Journal of Applied Mathematics and Computing, 17 (2005), 475-484.  doi: 10.1007/BF02936069.  Google Scholar

[30]

R. Saadati and C. Park, Approximation of derivations and the superstability in random Banach $\ast$-algebras, Advances in Difference Equations, 2018 (2018), 1-12.  doi: 10.1186/s13662-018-1882-6.  Google Scholar

[31]

W. Wei, X. Li and X. Li, New stability results for fractional integral equation, Computers & Mathematics with Applications, 64 (2012), 3468-3476.  doi: 10.1016/j.camwa.2012.02.057.  Google Scholar

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