Convergence analysis of general parallel $ S $-iteration process for system of mixed generalized Cayley variational inclusions

Iqbal Ahmad; Faizan Ahmad Khan; Arvind Kumar Rajpoot; Mohammed Ahmed Osman Tom; Rais Ahmad; Iqbal Ahmad; Faizan Ahmad Khan; Arvind Kumar Rajpoot; Mohammed Ahmed Osman Tom; Rais Ahmad

doi:10.3934/math.20221109

AIMS Mathematics

2022, Volume 7, Issue 11: 20259-20274. doi: 10.3934/math.20221109

Previous Article Next Article

Research article

Convergence analysis of general parallel $ S $-iteration process for system of mixed generalized Cayley variational inclusions

1.
Department of Mechanical Engineering, College of Engineering, Qassim University, Buraidah 51452, Al-Qassim, Saudi Arabia
2.
Computational & Analytical Mathematics and Their Applications Research Group, Faculty of Science, University of Tabuk, Tabuk-71491, Saudi Arabia
3.
Department of Mathematics, Aligarh Muslim University, Aligarh-202002, India
4.
Department of Mathematics, University of Bahri, Khartoum 11111, Sudan

Received: 17 July 2022 Revised: 29 August 2022 Accepted: 08 September 2022 Published: 16 September 2022
MSC : 47H09, 90C33

This work is concentrated on the study of a system of mixed generalized Cayley variational inclusions. Parallel Mann iteration process is defined in order to achieve the solution. We define an altering point problem which is equivalent to our system and then we construct general parallel $ S $-iteration process. Finally, we discuss convergence criteria and provide an example.
- convergence,
- Lipschitz,
- process,
- Cayley,
- solution
Citation: Iqbal Ahmad, Faizan Ahmad Khan, Arvind Kumar Rajpoot, Mohammed Ahmed Osman Tom, Rais Ahmad. Convergence analysis of general parallel $ S $-iteration process for system of mixed generalized Cayley variational inclusions[J]. AIMS Mathematics, 2022, 7(11): 20259-20274. doi: 10.3934/math.20221109

Related Papers:

Abstract

This work is concentrated on the study of a system of mixed generalized Cayley variational inclusions. Parallel Mann iteration process is defined in order to achieve the solution. We define an altering point problem which is equivalent to our system and then we construct general parallel $ S $-iteration process. Finally, we discuss convergence criteria and provide an example.

References

[1]	G. Stampacchia, Formes bilineaires coercivities sur les ensembles convexes, C. R. Acad. Sci. Paris, 258 (1964), 4413–4416.
[2]	Q. H. Ansari, Topics in nonlinear analysis and optimization, World Education, Delhi, 2012.
[3]	C. L. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Probl., 20 (2004), 103–120. https://doi.org/10.1088/0266-5611/20/1/006 doi: 10.1088/0266-5611/20/1/006
[4]	A. Gibali, Two simple relaxed perturbed extragradient methods for solving variational inequalities in Euclidean spaces, J. Nonlinear Var. Anal., 2 (2018), 49–61. https://doi.org/10.23952/jnva.2.2018.1.05 doi: 10.23952/jnva.2.2018.1.05
[5]	D. Kinderlehrer, G. Stampacchia, An introduction to variational inequalities and their applications, Academic Press, New York, 1980.
[6]	Y. Liu, A modified hybrid method for solving variational inequality problems in Banach spaces, J. Nonlinear Funct. Anal., 2017 (2017). https://doi.org/10.23952/jnfa.2017.31
[7]	L. C. Zeng, S. M. Guu, J. C. Yao, Characterization of $H$-monotone operators with applications to variational inclusions, Comput. Math. Appl., 50 (2011), 329–337. https://doi.org/10.1016/j.camwa.2005.06.001 doi: 10.1016/j.camwa.2005.06.001
[8]	A. Hassouni, A. Moudafi, A perturbed algorithm for variational inclusions, J. Math. Anal. Appl., 185 (1994), 706–712. https://doi.org/10.1006/jmaa.1994.1277 doi: 10.1006/jmaa.1994.1277
[9]	Q. H. Ansari, J. C. Yao, A fixed point theorem and its applications to a system of variational inequalities, Bull. Aust. Math. Soc., 59 (1999), 433–442. https://doi.org/10.1017/S0004972700033116 doi: 10.1017/S0004972700033116
[10]	G. Cohen, F. Chaplais, Nested monotony for variational inequalities over a product of spaces and convergence of iterative algorithms, J. Optimiz. Theory Appl., 59 (1988), 360–390. https://doi.org/10.1007/BF00940305 doi: 10.1007/BF00940305
[11]	H. Piri, R. Yavarimehr, Solving systems of monotone variational inequalities on fixed point sets of strictly pseudo-contractive mappings, J. Nonlinear Funct. Anal., 19 (2016).
[12]	R. U. Verma, Projection methods, algorithms and a new system of nonlinear variational inequalities, Comput. Math. Appl., 41 (2001), 1025–1031. https://doi.org/10.1016/S0898-1221(00)00336-9 doi: 10.1016/S0898-1221(00)00336-9
[13]	J. S. Pang, Asymmetric variational inequality problems over product sets: Applications and iterative methods, Math. Program., 31 (1985), 206–219. https://doi.org/10.1007/BF02591749 doi: 10.1007/BF02591749
[14]	A. Cayley, Sur quelques propriétés des déterminants gauches, J. Rein. Angew. Math., 32 (1846), 119–123.
[15]	G. Helmberg, Introduction to spectral theory in Hilbert space: The Cayley transform, Applied Mathematics and Mechanics, North Holland, Amsterdam, 6 (1969).
[16]	R. P. Agarwal, D. O'Regan, D. R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex Anal., 8 (2007), 61–79.
[17]	W. R. Mann, Mean value methods in iteration, Proc. Am. Math. Soc., 4 (1953), 506–610. https://doi.org/10.1090/S0002-9939-1953-0054846-3 doi: 10.1090/S0002-9939-1953-0054846-3
[18]	S. Ishikawa, Fixed points by a new iteration method, Proc. Am. Math. Soc. 44 (1974), 147–150. https://doi.org/10.1090/S0002-9939-1974-0336469-5
[19]	D. R. Sahu, Applications of $S$-iteration process to contrained minimization problem and split feasibility problems, Fixed Point Theor., 12 (2011), 187–204.
[20]	R. P. Agarwal, D. O'Regan, D. R. Sahu, Fixed point theory for Lipschitzian-type mappings with Applications, Springer, New York, 2009. http://dx.doi.org/10.1016/B978-0-12-775850-3.50017-0
[21]	C. E. Chidume, C. E. Chidume, Geometric properties of Banach spaces and nonlinear iterations, Lecture Notes in Mathematics, Springer Verlag, 2009.
[22]	E. Hacioğlu, F. Gürsoy, S. Maldar, Y. Atalan, G. V. Milovanović, Iterative approximation of fixed points and applications to two-point second-order boundary value problems and to machine learning, Appl. Numer. Math., 167 (2021), 143–172. https://doi.org/10.1016/j.apnum.2021.04.020 doi: 10.1016/j.apnum.2021.04.020
[23]	X. Ju, D. Hu, C. Li, X. He, G. Feng, A novel fixed-time converging neurodynamic approach to mixed variational inequalities and applications, IEEE T. Cybernetics, 2021. http://dx.doi.org/10.1109/TCYB.2021.3093076
[24]	X. Ju, C. Li, Y. H. Dai, J. Chen, A new dynamical system with self-adaptive dynamical stepsize for pseudomonotone mixed variational inequalities, Optimization, 2022. http://dx.doi.org/10.1080/02331934.2022.2094795
[25]	W. Kumam, K. Khammahawong, P. Kumam, Error estimate of data dependence for discontinuous operators by new iteration process with convergence analysis, Numer. Funct. Anal. Optim., 40 (2019), 1644–1677. https://doi.org/10.1080/01630563.2019.1610437 doi: 10.1080/01630563.2019.1610437
[26]	G. Maniu, On a three-step iteration process for Suzuki mappings with Qualitative study, Numer. Funct. Anal. Optim., 41 (2020), 929–949. https://doi.org/10.1080/01630563.2020.1719415 doi: 10.1080/01630563.2020.1719415
[27]	D. R. Sahu, Q. H. Ansari, J. C. Yao, Convergence of inexact Mann iterations generated by nearly nonexpansive sequences and applications, Numer. Funct. Anal. Optim., 37 (2016), 1312–1338. https://doi.org/10.1080/01630563.2016.1206566 doi: 10.1080/01630563.2016.1206566
[28]	D. R. Sahu, Q. H. Ansari, J. C. Yao, The prox-Tiknonov-like forward-backward method and applications, Taiwanese J. Math., 19 (2015), 481–503. https://doi.org/10.11650/tjm.19.2015.4972 doi: 10.11650/tjm.19.2015.4972
[29]	D. R. Sahu, S. M. Kang, A. Kumar, Convergence analysis of parallel $S$-iteration process for system of generalized variational inequalities, J. Funct. Space., 2017 (2017). https://doi.org/10.1155/2017/5847096
[30]	D. R. Sahu, J. C. Yao, V. K. Sing, S. Kumar, Semilocal convergence analysis of $S$-iteration process of Newton-Kantorovich like in Banach spaces, J. Optim. Theory App., 172 (2016), 102–127. https://doi.org/10.1007/s10957-016-1031-x doi: 10.1007/s10957-016-1031-x
[31]	B. Tan, S. Y. Cho, Inertial extragradient algorithms with non-monotone stepsizes for pseudomonotone variational inequalities and applications, Comput. Appl. Math., 41 (2022), 121. http://dx.doi.org/10.1007/s40314-022-01819-0 doi: 10.1007/s40314-022-01819-0
[32]	B. Tan, X. Qin, J. C. Yao, Strong convergence of inertial projection and contraction methods for pseudomonotone variational inequalities with applications to optimal control problems, J. Global Optim., 82 (2022), 523–557. https://doi.org/10.1007/s10898-021-01095-y doi: 10.1007/s10898-021-01095-y
[33]	H. K. Xu, D. R. Sahu, Parallel normal $S$-iteration methods with applications to optimization problems, Numer. Func. Anal. Opt., 42 (2021), 1925–1953. https://doi.org/10.1080/01630563.2021.1950761 doi: 10.1080/01630563.2021.1950761
[34]	D. R. Sahu, Altering points and applications, Nonlinear Stud., 21 (2014), 349–365.
[35]	R. Ahmad, I. Ali, M. Rahaman, M. Ishtyak, J. C. Yao, Cayley inclusion problem with its corresponding generalized resolvent equation problem in uniformly smooth Banach spaces, Appl. Anal., 101 (2022), 1354–1368. https://doi.org/10.1080/00036811.2020.1781822 doi: 10.1080/00036811.2020.1781822
[36]	W. V. Petryshyn, A characterization of strict convexity of Banach spaces and other uses of duality mappings, J. Funct. Anal., 6 (1970), 282–291. https://doi.org/10.1016/0022-1236(70)90061-3 doi: 10.1016/0022-1236(70)90061-3

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)