Nonlinear Functional Analysis and Applications

Kyungnam University, Department of Mathematics Eduaction (경남대학교 수학교육과)
권호 표지
DOI QR코드

DOI QR Code

ALTERNATED INERTIAL RELAXED TSENG METHOD FOR SOLVING FIXED POINT AND QUASI-MONOTONE VARIATIONAL INEQUALITY PROBLEMS

  • A. E. Ofem (School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal) ;
  • A. A. Mebawondu (Department of Computer Science and Mathematics, Mountain Top University) ;
  • C. Agbonkhese (Department of Computer Science and Mathematics, Mountain Top University) ;
  • G. C. Ugwunnadi (Department of Mathematics, University of Eswatini) ;
  • O. K. Narain (School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal)
  • Received : 2023.06.22
  • Accepted : 2023.07.16
  • Published : 2024.03.15
Download PDF
⟨ Previous Next ⟩

Abstract

In this research, we study a modified relaxed Tseng method with a single projection approach for solving common solution to a fixed point problem involving finite family of τ-demimetric operators and a quasi-monotone variational inequalities in real Hilbert spaces with alternating inertial extrapolation steps and adaptive non-monotonic step sizes. Under some appropriate conditions that are imposed on the parameters, the weak and linear convergence results of the proposed iterative scheme are established. Furthermore, we present some numerical examples and application of our proposed methods in comparison with other existing iterative methods. In order to show the practical applicability of our method to real word problems, we show that our algorithm has better restoration efficiency than many well known methods in image restoration problem. Our proposed iterative method generalizes and extends many existing methods in the literature.

Keywords

References

  1. J.A. Abuchu, A.E. Ofem, G.C. Ugwunnadi, O.K. Narain and A. Hussain, Hybrid Alternated Inertial Projection and Contraction Algorithm for Solving Bilevel Variational Inequality Problems, J. Math., 2023, https://doi.org/10.1155/2023/3185746.
  2. F. Akutsah, A.A. Mebawondu, H.A. Abass and O.K. Narain, A self adaptive method for solving a class of bilevel variational inequalities with split variational inequality and composed fixed point problem constraints in Hilbert spaces, Numer. Algebra Control Optim., 122 (2023), DOI: 10.3934/naco.2021046.
  3. F. Akutsah, A.A. Mebawondu, G.C. Ugwunnadi and O.K. Narain, Inertial extrapolation method with regularization for solving monotone bilevel variation inequalities and fixed point problems in real Hilbert space, J. Nonlinear Funct. Anal., 2022, Article ID 5,
  4. F. Akutsah, A.A. Mebawondu, G.C. Ugwunnadi, P. Pillay and O.K. Narain, Inertial extrapolation method with regularization for solving a new class of bilevel problem in real Hilbert spaces, SeMA J., 80 (2023), 503-524, https://doi.org/10.1007/s40324-022-00293-2.
  5. F. Alvares and H. Attouch, An inertial proximal monotone operators via discretization of a nonlinear oscillator with damping, Set Valued Anal., 9 (2001), 3-11. https://doi.org/10.1023/A:1011253113155
  6. P.K. Anh, T. Anh and L.D. Muun, On bilevel split pseudomonotone variational inequality problems with applications, Acta. Math. Vietnam., 42 (2017), 413-429. https://doi.org/10.1007/s40306-016-0178-8
  7. R.Y. Apostol, A.A. Grynenko and V.V. Semenov, Terative algorithms for monotone bilevel variational inequalities, J. Comput. Appl. Math., 107 (2012), 3-14.
  8. L.C. Ceng, A. Gibali and S. Reich, The subgradient extragradient method for solving variational inequalities in Hilbert space, J. Optim. Theory Appl., 148 (2011), 318-335. https://doi.org/10.1007/s10957-010-9757-3
  9. L.C. Ceng, N. Hadjisavvas and N.C. Wong, Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems, Glob. Optim., 46 (2020), 635-646. https://doi.org/10.1007/s10898-009-9454-7
  10. E.O. Effanga, J.A. Ugboh, E.I. Enang and B.E.A. Eno, A tool for constructing pair-wise balanced incomplete block design, Journal of Modern Mathematics and Statistics, 3(4), 69-72.
  11. I.M. Esuabana, U.A. Abasiekwere, J.A. Ugboh and Z. Lipcsey, Equivalent Construction of Ordinary Differential Equations from Impulsive System, Academic Journal of Applied Mathematical Sciences, 4(8), 77-89.
  12. G. Ficher, Sur problem elastostatico di signorini con ambigue condizioni al contorno, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 34 (1963), 138-142.
  13. G. Ficher, Problemi elastostatici con vincoli unilaterali: il problema di Signorini con ambigue condizioni al contorno, Atti Accad. Naz. Lincci, Cl. Sci. Fis. Mat. Nat., Sez., 7 (1964), 91-140.
  14. N. Hadjisavvas and S. Schaible, Quasimonotone variational inequalities in Banach spaces, J. Optim. Theory Appl., 90 (1996), 95-111. https://doi.org/10.1007/BF02192248
  15. Y. He, A new double projection algorithm for variational inequalities, J. Comput. Appl. Math., 185(1) (2006), 166-173. https://doi.org/10.1016/j.cam.2005.01.031
  16. B.S. He and L.Z. Liao, Improvements of some projection methods for monotone nonlinear variational inequalities, Optim. Theory Appl., 112 (2002), 111-128. https://doi.org/10.1023/A:1013096613105
  17. H. He, C. Ling and H.K. Xu, A relaxed projection method for split variational inequalities, J. Optim. Theory Appl., 166 (2015), 213-233. https://doi.org/10.1007/s10957-014-0598-3
  18. G.M. Korpelevich, An extragradient method for finding saddle points and for other problems, Ekon. Mat. Metody., 12 (1976), 747-756.
  19. N. Nadezhkina and W. Takahashi, Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 128 (2006), 191-201. https://doi.org/10.1007/s10957-005-7564-z
  20. A.E. Ofem, J.A. Abuchu, H.A. Nabwey, G.C. Ugwunnadi and O.K. Narain, On Bilevel Monotone Inclusion and Variational Inequality Problems, Mathematics, 11(22) (2023), 4643, https://0-doi-org.brum.beds.ac.uk/10.3390/math11224643.
  21. A.E. Ofem, H. I,sik, F. Ali and J. Ahmad, A new iterative approximation scheme for Reich-Suzuki type nonexpansive operators with an application, J. Inequal. Appl., 28 (2022), doi.org/101186/s13660-022-02762-8. 101186/s13660-022-02762-8
  22. A.E. Ofem, A.A. Mebawondu, G.C. Ugwunnadi, P. Cholamjiak and O.K. Narain, Relaxed Tseng splitting method with double inertial steps for solving monotone inclusions and fixed point problems, Numer. Algorithms, (2023), https://doi.org/10.1007/s11075-023-01674-y.
  23. A.E. Ofem, A.A. Mebawondu, G.C. Ugwunnadi, H. I,sik and O.K. Narainl, A modified subgradient extragradient algorithm-type for solving quasimonotone variational inequality problems with applications, J. Inequal. Appl. 2023(73) (2023), https://doi.org/10.1186/s13660-023-02981-7.
  24. F. Ogbuisi, Yekini Shehu and Jen-Chih Yao, An alternated inertial method for pseudomonotone variational inequalities in Hilbert spaces, Optim. Eng., 23(1) (2022), 1-29, DOI:10.1007/s11081-021-09615-1.
  25. G.A. Okeke, A.E. Ofem, T. Abdeljawad, M.A. Alqudah and A. Khan, A solution of a nonlinear Volterra integral equation with delay via a faster iteration method, AIMS Mathematics, 8(1) (2023), 102-124. https://doi.org/10.3934/math.2023005
  26. M.O. Osilike and S.C. Aniagbosor, Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings, Math. Comput. Model., 32(2000), 1181-1191. https://doi.org/10.1016/S0895-7177(00)00199-0
  27. B.T. Polyak, Some methods of speeding up the convergence of iteration methods, Politehn Univ Buchar Sci Bull Ser A Appl Math Phys., 4(5) (1964), 1-17. https://doi.org/10.1016/0041-5553(64)90137-5
  28. Y. Shehu and O.S. Iyiola, Projection methods with alternating inertial steps for variational inequalities: Weak and linear convergence, Appl. Numer. Math., 57 (2020), 315-337. https://doi.org/10.1016/j.apnum.2020.06.009
  29. Y. Shehu, O.S. Iyiola and F.U. Ogbuisi, Iterative method with inertial terms for nonexpansive mappings, Applications to compressed sensing, Numer. Algor., 83 (2020), 1321-1347. https://doi.org/10.1007/s11075-019-00727-5
  30. Y. Shehu and P.T. Vuong and P. Cholamjiak, A self-adaptive projection method with an inertial technique for split feasibility problems in Banach spaces with applications to image restoration problems, J. Fixed Point Theory Appl., 21 ( 2019), 1-24. https://doi.org/10.1007/s11784-018-0638-y
  31. Y. Shehu, P.T. Vuong and A. Zemkoho, An inertial extrapolation method for convex simple bilevel optimization, Optim. Meth. Softw., (2019), https://doi.org/10.1080/10556788.2019.1619729.
  32. G. Stampacchia, Formes bilineaires coercitives sur les ensembles convexes, C. R. Math. Acad. Sci., 258 (1964), 4413-4416.
  33. W. Takahashi, The split common fixed point problem and the shrinking projection method in Banach spaces, J. Conv. Anal., 24 (2017), 1015-1028.
  34. W. Takahashi, C.F. Wen and J.C. Yao., The shrinking projection method for a finite family of demimetric mappings with variational inequality problems in a Hilbert space, Fixed Point Theory, 19 (2018), 407-419. https://doi.org/10.24193/fpt-ro.2018.1.32
  35. D.V. Thong and D.V. Hieu, Inertial subgradient extragradient algorithms with line- search process for solving variational inequality problems and fixed point problems, Numer. Algor., 80(4) (209), 1283-1307, DOI: 10.1007/s11075-018-0527-x.
  36. M. Tian and M. Tong, Self-adaptive subgradient extragradient method with inertial modification for solving monotone variational inequality problems and quasinonexpansive fixed point problems, J. Inequ. Appl., 2019(7) (2019).
  37. P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control. Optim., 38 (2000), 431-446. https://doi.org/10.1137/S0363012998338806
  38. H.K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66(2002), 240-256. https://doi.org/10.1112/S0024610702003332
  39. L. Zheng, A double projection algorithm for quasimonotone variational inequalities in Banach spaces, Inequal. Appl., 123 (2018), 1-20.