doi: 10.3934/dcdss.2020432

On the fuzzy stability results for fractional stochastic Volterra integral equation

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  November 2020

By a fuzzy controller function, we stabilize 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.

Citation: Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 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. ConstantinescuJ. 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-MoneamF. 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

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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. JiangD. O'ReganJ. 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. KhanT. AbdeljawadM. AslamR. 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. KhanF. JaradT. 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. KhanH. KhanJ. 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. NadabanT. 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. ParkD. Y. ShinR. 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. ParkS. 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. SadeghiM. 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. WeiX. 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. ConstantinescuJ. 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-MoneamF. 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. JiangD. O'ReganJ. 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. KhanT. AbdeljawadM. AslamR. 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. KhanF. JaradT. 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. KhanH. KhanJ. 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. NadabanT. 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. ParkD. Y. ShinR. 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. ParkS. 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. SadeghiM. 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. WeiX. 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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