Nonlinear Functional Analysis and Applications

  • 제26권3호
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  • Pages.451-474
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  • 2021
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  • 1229-1595(pISSN)
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  • 2466-0973(eISSN)
경남대학교 수학교육과 (Kyungnam University, Department of Mathematics Eduaction)
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OUTER APPROXIMATION METHOD FOR ZEROS OF SUM OF MONOTONE OPERATORS AND FIXED POINT PROBLEMS IN BANACH SPACES

  • 투고 : 2020.08.27
  • 심사 : 2021.02.19
  • 발행 : 2021.09.15
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⟨ 이전 논문 다음 논문 ⟩

초록

In this paper, we investigate a hybrid algorithm for finding zeros of the sum of maximal monotone operators and Lipschitz continuous monotone operators which is also a common fixed point problem for finite family of relatively quasi-nonexpansive mappings and split feasibility problem in uniformly convex real Banach spaces which are also uniformly smooth. The iterative algorithm employed in this paper is design in such a way that it does not require prior knowledge of operator norm. We prove a strong convergence result for approximating the solutions of the aforementioned problems and give applications of our main result to minimization problem and convexly constrained linear inverse problem.

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과제정보

The first and second authors acknowledge with thanks the bursary and financial support from Department of Science and Technology and National Research Foundation, Republic of South Africa, Centre of Excellence in Mathematical and Statistical Sciences (DST-NRF COE-MaSS) Post Doctoral Bursary. Opinions expressed and conclusions arrived are those of the authors and are not necessarily to be attributed to the CoE-MaSS. And the fourth author was supported by the Basic Science Research Program through the National Research Foundation(NRF) Grant funded by Ministry of Education of the republic of Korea (2018R1D1A1B07045427).

참고문헌

  1. H.A. Abass, F.U. Ogbuisi and O.T. Mewomo, Common solution of split equilibrium problem with no prior knowledge of operator norm, UPB Sci. Bull., Series A, 80 (2018), 175-190.
  2. H.A. Abass, C. Izuchukwu, O.T. Mewomon and Q.L. Dong, Strong convergence of an inertial forward-backward splitting method for accretive operators in real Banach space, Fixed Point Theory, 21 (2020), 397-412. https://doi.org/10.24193/fpt-ro.2020.2.28
  3. H.A. Abass, K.O. Aremu, L.O. Jolaoso and O.T. Mewomo, An inertial forward-backward splitting method for approximating solutions of certain optimization problem, J. Nonlinear Funct. Anal., 2020 (2020), Article ID 6.
  4. Y.I. Alber, Metric and generalized projection operators in Banach spaces: properties and applications in: Kartsatos, A.G (Ed). Theory and Applications of Nonlinear Operators and Accretive and Monotone Type. Lecture Notes Pure Appl. Math., 178, Dekker, New York (1996), 15-50.
  5. Y.I. Alber and S. Reich, An iterative method for solving a class of nonlinear operator equations in Banach spaces, PanAmer. Math. J., 4 (1994), 39-54.
  6. Y. Alber and L. Ryazantseva, Nonlinear ill-posed problems of monotone type, Springer, Dordrecht (2006). xiv+410 pp. ISBN:978-1-4020-4395-6, 1-4020-4395-3.
  7. K. Aoyama and F. Koshaka, Strongly relatively nonexpansive sequences generated by firmly nonexpansive-like mappings, Fixed Point Theory Appl., 95 (2014), pp.13.
  8. K. Avetisyan, O. Djordjevic and M. Pavlovic, Littlewood-Paley inequalities in uniformly convex and uniformly smooth Banach spaces, J. Math. Anal. Appl., 336 (2007), 31-43. https://doi.org/10.1016/j.jmaa.2007.02.056
  9. V. Barbu, Nonlinear semigroups and differential equations in Banach spaces, Editura Academiei, R.S.R, Bucharest, English transl. Noordhof, Leyden, 1976.
  10. Y. Censor and T. Elfving,A multiprojection algorithm using Bregman projection in a product space, Numer. Algor., 8 (1994), 221-239. https://doi.org/10.1007/BF02142692
  11. Y. Censor, T. Elfving, N. Kopf and T. Bortfield, The multiple-sets split feasibilty problem and its applications for inverse problems, Inverse Prob., 21 (2005), 2071-2084. https://doi.org/10.1088/0266-5611/21/6/017
  12. Q.L. Dong, D. Jiang, P. Cholmjiak and Y. Shehu, A strong convergence result involving an inertial forward-backward splitting algorithm for monotone inclusions, J. Fixed Theory Appl., 19 (2017), 3097-3118. https://doi.org/10.1007/s11784-017-0472-7
  13. B. Eicke, Iteration methods for convexly constrained ill-posed problems in Hilbert space, Numer. Funct. Anal. Optim., 13 (1992), 413-429. https://doi.org/10.1080/01630569208816489
  14. H.W. Engl, M. Hanke and A. Neubauer, Regularization of inverse problems, Kluwer Academic Publishers Group, Dordrecht, 1996.
  15. J.N. Ezeora, H.A. Abass and C. Izuchukwu, Strong convergence of an inertial-type algorithm to a common solution of minimization and fixed point problems, Mathematik Vesnik, 71 (2019), 338-350.
  16. Z. Jouymandi and F. Moradlou, Retraction algorithms for solving variational inequalities, pseudomonotone equilibrium problems and fixed point problems in Banach spaces, Numer. Algor., 78 (2018), 1153-1182. https://doi.org/10.1007/s11075-017-0417-7
  17. S. Kamimura and W. Takahashi, Strong convergence of a proximal-type algorithm in Banach space, SIAM J. Optim., 13 (2002), 938-945. https://doi.org/10.1137/S105262340139611X
  18. J.K. Kim, A.H. Dar and Salahuddin, Existence theorems for the generalized relaxed pseudomonotone variational inequalities, Nonlinear Funct. Anal. and Appl., 25(1) (2020), 25-34. doi.org/10.22771/nfaa.2020.25.01.03
  19. J.K. Kim, T.M. Tuyen and M.T. Ngoc Ha, Two projection methods for solving the split common fixed point problem with multiple output sets in Hilbert spaces, Numer. Funct. Anal. Optimiz., 42(8) (2021), 973-988, https://doi.org/10.1080/01630563.2021.1933528
  20. J.K. Kim and T.M. Tuyen, A parallel iterative method for a finite family of Bregman strongly nonexpansive mappings in reflexive Banach spaces, J. Korean Math. Soc., 57(3) (2020), 617-640. https://doi.org/10.4134/JKMS.j190268
  21. J.K. Kim and T.M. Tuyen,Parallel iterative method for solving the split common null point problem in Banach spaces, J. Nonlinear and Convex Anal., 20(10) (2019), 2075-2093.
  22. P.L. Lions and B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM J. Numer. Anal., 16 (1979), 964-979. https://doi.org/10.1137/0716071
  23. Z. Ma, L. Wang and S.S. Chang, On the split feasibility problem and fixed point problem of quasi-φ-nonexpansive in Banach spaces, Numer. Algor., 80(4) (2019), 1203-1218. https://doi.org/10.1007/s11075-018-0523-1.
  24. S. Matsushita and W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in Banach spaces, J. Approx. Theory. 134 (2005), 257-266. https://doi.org/10.1016/j.jat.2005.02.007
  25. A.A. Mebawondu, Proximal Point Algorithms for Finding Common Fixed Points of a Finite Family of Nonexpansive Multivalued Mappings in Real Hilbert Spaces, Khayyam Journal of Mathematics, 5(2), 113-123.
  26. A. Moudafi and B.S. Thakur, Solving proximal split feasibility problems without prior knowledge of operator norms, Optim. Letter, 8 (2014), 2099-2110. https://doi.org/10.1007/s11590-013-0708-4
  27. D.H. Peaceman and H.H. Rashford, The numerical solution of parabolic and elliptic differential equations, J. Soc. Ind. Appl. Math., 3 (1995), 267-275.
  28. J. Peypouquet, Convex optimization in Normed spaces. Theory, Methods and Examples. With a foreword by H. Attouch. Springer briefs in optimization. Springer, Cham., 2015. xiv+124 pp. ISBN: 978-3-319-13709-4, 978-3-319-13710-0.
  29. X. Qin, Y.J. Cho and S.M. Kang, Convergence theorems of common elements for equilibrium problem and fixed point problems in Banach spaces, J. Comput. Appl. Math., 225 (2009), 20-30. https://doi.org/10.1016/j.cam.2008.06.011
  30. R.T. Rockfellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1977), 877-808. https://doi.org/10.1137/0314056
  31. Y. Shehu, Convergence results of forward-backward algorithms for sum of monotone operators in Banach spaces, Results Math., 74 (2019), 138. https://doi.org/10.1007/s00025-019-1061-4
  32. Y. Shehu, Iterative approximation for zeros of sum of accretive operators in Banach spaces, J. Funct. Spaces, (2015), Article ID 5973468, 9 pages.
  33. Y. Shehu, Iterative approximation method for finite family of relatively quasinonexpansive mapping and systems of equilibrium problem, J. Glob. Optim., (2014), DOI.10.1007/s10898-010-9619-4.
  34. Y. Shehu and O.S. Iyiola, Convergence analysis for the proximal split feasibility using an inertial extrapolation term method, J. Fixed Point Theory Appl., 19 (2017), 2483-2510. https://doi.org/10.1007/s11784-017-0435-z
  35. W. Takahashi,Nonlinear Functional Analysis, Fixed Theory Appl., Yokohama-Publishers, 2000.
  36. Ng.T.T. Thuy, P.T.T. Hoai and Ng.T.T. Hoa, Explicit iterative methods for maximal monotone operators in Hilbert spaces, Nonlinear Funct. Anal. Appl., 25(4) (2020), 753-767. doi.org/10.22771/nfaa.2020.25.04.09.
  37. P.T. Vuong, J.J. Stroduot and V.H. Nguyen, A gradient projection method for solving split equality and split feasibility problems in Hilbert space, Optimization, 64 (2015), 2321-2341. https://doi.org/10.1080/02331934.2014.967237
  38. K. Wattanawitoon and P. Kuman, Strong convergence theorems by a new hybrid projection algorithm for fixed point problem and equilibrium problems of two relatively quasinonexpansive mappings, Nonlinear Anal. Hybrid Syst., 3 (2009), 11-20. https://doi.org/10.1016/j.nahs.2008.10.002
  39. H.K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal. Theory Methods Appl., 16 (1991), 1127-1138. https://doi.org/10.1016/0362-546X(91)90200-K
  40. J.C. Yao, Variational inequalities with generalized monotone operators, Math. Oper. Res., 19 (1994), 691-705. https://doi.org/10.1287/moor.19.3.691
  41. H. Zhang and L. Ceng, Projection splitting methods for sums of maximal monotone operators with applications, J. Math. Anal. Appl., 406 (2013), 323-334. https://doi.org/10.1016/j.jmaa.2013.04.072
  42. J. Zhang and N. Jiang, Hybrid algorithm for common solution of monotone inclusion problem and fixed point problem and applications to variational inequalities, Springer Plus, 5: 803 (2016). https://doi.org/10.1186/s40064-016-2389-9