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AIMS Mathematics

2023, Volume 8, Issue 5: 11037-11056. doi: 10.3934/math.2023559
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A generalized Halpern-type forward-backward splitting algorithm for solving variational inclusion problems

  • 1.
    Mathematics, Department of Learning Management, Faculty of Education, Burapha University, Chonburi Campus, 169 Long Haad Bangsaen Road, Saen Suk, Mueang, Chonburi 20131, Thailand
  • 2.
    Center of Excellence in Theoretical and Computational Science, SCL 802 Fixed Point Laboratory, Science Laboratory Building, King Mongkut's University of Technology Thonburi, 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand
  • 3.
    Mathematics Institute, African University of Science and Technology, Abuja 900107, Nigeria
  • 4.
    Operational Research Center in Healthcare, Near East University, Nicosia, TRNC, Turkey
  • 5.
    KMUTTFixed Point Research Laboratory, KMUTT-Fixed Point Theory and Applications Research Group, Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi, 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand
  • 6.
    Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
  • Received: 15 August 2022 Revised: 18 January 2023 Accepted: 02 February 2023 Published: 08 March 2023

  • MSC : 46N10, 90C25

  • In this paper, we investigate the problem of finding a zero of sum of two accretive operators in the setting of uniformly convex and $ q $-uniformly smooth real Banach spaces ($ q > 1 $). We incorporate the inertial and relaxation parameters in a Halpern-type forward-backward splitting algorithm to accelerate the convergence of its sequence to a zero of sum of two accretive operators. Furthermore, we prove strong convergence of the sequence generated by our proposed iterative algorithm. Finally, we provide a numerical example in the setting of the classical Banach space $ l_4(\mathbb{R}) $ to study the effect of the relaxation and inertial parameters in our proposed algorithm.

    Citation: Premyuda Dechboon, Abubakar Adamu, Poom Kumam. A generalized Halpern-type forward-backward splitting algorithm for solving variational inclusion problems[J]. AIMS Mathematics, 2023, 8(5): 11037-11056. doi: 10.3934/math.2023559

    Related Papers:

  • In this paper, we investigate the problem of finding a zero of sum of two accretive operators in the setting of uniformly convex and $ q $-uniformly smooth real Banach spaces ($ q > 1 $). We incorporate the inertial and relaxation parameters in a Halpern-type forward-backward splitting algorithm to accelerate the convergence of its sequence to a zero of sum of two accretive operators. Furthermore, we prove strong convergence of the sequence generated by our proposed iterative algorithm. Finally, we provide a numerical example in the setting of the classical Banach space $ l_4(\mathbb{R}) $ to study the effect of the relaxation and inertial parameters in our proposed algorithm.


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    [1] A. Adamu, D. Kitkuan, A. Padcharoen, C. E. Chidume, P. Kumam, Inertial viscosity-type iterative method for solving inclusion problems with applications, Math. Comput. Simulat., 194 (2022), 445–459. https://doi.org/10.1016/j.matcom.2021.12.007 doi: 10.1016/j.matcom.2021.12.007
    [2] A. Adamu, P. Kumam, D. Kitkuan, A. Padcharoen, Relaxed modified Tseng algorithm for solving variational inclusion problems in real Banach spaces with applications, Carpathian J. Math., 39 (2023), 1–26. https://doi.org/10.37193/CJM.2023.01.01 doi: 10.37193/CJM.2023.01.01
    [3] A. Adamu, J. Deepho, A. H. Ibrahim, A. B. Abubakar, Approximation of zeros of sum of monotone mappings with applications to variational inequality problem and image processing, Nonlinear Funct. Anal. Appl., 26 (2021), 411–432. https://doi.org/10.22771/NFAA.2021.26.02.12 doi: 10.22771/NFAA.2021.26.02.12
    [4] A. Adamu, D. Kitkuan, P. Kumam, A. Padcharoen, T. Seangwattana, Approximation method for monotone inclusion problems in real Banach spaces with applications, J. Inequal. Appl., 2022 (2022), 70. https://doi.org/10.1186/s13660-022-02805-0 doi: 10.1186/s13660-022-02805-0
    [5] A. Adamu, A. A. Adam, Approximation of solutions of split equality fixed point problems with applications, Carpathian J. Math., 37 (2021), 381–392. https://doi.org/10.37193/CJM.2021.03.02 doi: 10.37193/CJM.2021.03.02
    [6] F. Alvarez, On the minimizing property of a second order dissipative system in Hilbert spaces, SIAM J. Control Optim., 38 (2000), 1102–1119. https://doi.org/10.1137/S0363012998335802 doi: 10.1137/S0363012998335802
    [7] F. Alvarez, H. Attouch, An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping, Set-Valued Anal., 9 (2001), 3–11. https://doi.org/10.1023/A:1011253113155 doi: 10.1023/A:1011253113155
    [8] H. Attouch, A. Cabot, Convergence of a relaxed inertial forward-backward algorithm for structured monotone inclusions, Appl. Math. Optim., 80 (2019), 547–598. https://doi.org/10.1007/s00245-019-09584-z doi: 10.1007/s00245-019-09584-z
    [9] J. Abubakar, P. Kumam, A. H. Ibrahim, A. Padcharoen, Relaxed inertial Tseng's type method for solving the inclusion problem with application to image restoration, Mathematics, 8 (2020), 818. https://doi.org/10.3390/math8050818 doi: 10.3390/math8050818
    [10] Y. Alber, I. Ryazantseva, Nonlinear ill-posed problems of monotone type, Dordrecht: Springer, 2006. https://doi.org/10.1007/1-4020-4396-1
    [11] A. Beck, M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci., 2 (2009), 183–202. https://doi.org/10.1137/080716542 doi: 10.1137/080716542
    [12] C. Chidume, Geometric properties of Banach spaces and nonlinear iterations, London: Springer, 2009. https://doi.org/10.1007/978-1-84882-190-3
    [13] C. E. Chidume, A. Adamu, P. Kumam, D. Kitkuan, Generalized hybrid viscosity-type forward-backward splitting method with application to convex minimization and image restoration problems, Numer. Func. Anal. Opt., 42 (2021), 1586–1607. https://doi.org/10.1080/01630563.2021.1933525 doi: 10.1080/01630563.2021.1933525
    [14] C. E Chidume, A. Adamu, M. O. Nnakwe, Strong convergence of an inertial algorithm for maximal monotone inclusions with applications, Fixed Point Theory Appl., 2020 (2020), 13. https://doi.org/10.1186/s13663-020-00680-2 doi: 10.1186/s13663-020-00680-2
    [15] C. E Chidume, P. Kumam, A. Adamu, A hybrid inertial algorithm for approximating solution of convex feasibility problems with applications, Fixed Point Theory Appl., 2020 (2020), 12. https://doi.org/10.1186/s13663-020-00678-w doi: 10.1186/s13663-020-00678-w
    [16] C. E. Chidume, A. Adamu, L. O. Chinwendu, Strong convergence theorem for some nonexpansive-type mappings in certain Banach spaces, Thai J. Math., 18 (2020), 1537–1548.
    [17] C. E. Chidume, A. Adamu, M. S. Minjibir, U. V. Nnyaba, On the strong convergence of the proximal point algorithm with an application to Hammerstein euations, J. Fixed Point Theory Appl., 22 (2020), 61. https://doi.org/10.1007/s11784-020-00793-6 doi: 10.1007/s11784-020-00793-6
    [18] P. Cholamjiak, D. Van Hieu, Y. J. Cho, Relaxed forward-backward splitting methods for solving variational inclusions and applications, J. Sci. Comput., 88 (2021), 85. https://doi.org/10.1007/s10915-021-01608-7 doi: 10.1007/s10915-021-01608-7
    [19] P. Cholamjiak, P. Sunthrayuth, A. Singta, K. Muangchoo, Iterative methods for solving the monotone inclusion problem and the fixed point problem in Banach spaces, Thai J. Math., 18 (2020), 1225–1246.
    [20] W. Cholamjiak, P. Cholamjiak, S. Suantai, An inertial forward-backward splitting method for solving inclusion problems in Hilbert spaces, J. Fixed Point Theory Appl., 20 (2018), 42. https://doi.org/10.1007/s11784-018-0526-5 doi: 10.1007/s11784-018-0526-5
    [21] Q. Dong, D. Jiang, P. Cholamjiak, Y. Shehu, A strong convergence result involving an inertial forward-backward algorithm for monotone inclusions, J. Fixed Point Theory Appl., 19 (2017), 3097–3118. https://doi.org/10.1007/s11784-017-0472-7 doi: 10.1007/s11784-017-0472-7
    [22] A. H. Ibrahim, P. Kumam, A. B. Abubakar, A. Adamu, Accelerated derivative-free method for nonlinear monotone equations with an application, Numer. Linear Algebr., 29 (2022), e2424. https://doi.org/10.1002/nla.2424 doi: 10.1002/nla.2424
    [23] S. He, C. Yang, Solving the variational inequality problem defined on intersection of finite level sets, Abstr. Appl. Anal., 2013 (2013), 942315. https://doi.org/10.1155/2013/942315 doi: 10.1155/2013/942315
    [24] F. Iutzeler, J. M. Hendrickx, A generic online acceleration scheme for optimization algorithms via relaxation and inertia, Optim. Method. Softw., 34 (2019), 383–405. https://doi.org/10.1080/10556788.2017.1396601 doi: 10.1080/10556788.2017.1396601
    [25] D. Kitkuan, P. Kumam, J. Martinez-Moreno, Generalized Halpern-type forward-backward splitting methods for convex minimization problems with application to image restoration problems, Optimization, 69 (2020), 1557–1581. https://doi.org/10.1080/02331934.2019.1646742 doi: 10.1080/02331934.2019.1646742
    [26] D. Kitkuan, P. Kumam, A. Padcharoen, W. Kumam, P. Thounthong, Algorithms for zeros of two accretive operators for solving convex minimization problems and its application to image restoration problems, J. Comput. Appl. Math., 354 (2019), 471–495. https://doi.org/10.1016/j.cam.2018.04.057 doi: 10.1016/j.cam.2018.04.057
    [27] P. L. Lions, Une methode iterative de resolution d'une inequation variationnelle, Israel J. Math., 31 (1978), 204–208. https://doi.org/10.1007/BF02760552 doi: 10.1007/BF02760552
    [28] G. López, V. Martín-Márquez, F. Wang, H. K. Xu, Forward-backward splitting methods for accretive operators in Banach spaces, Abstr. Appl. Anal., 2012 (2012), 109236. https://doi.org/10.1155/2012/109236 doi: 10.1155/2012/109236
    [29] Y. Luo, Weak and strong convergence results of forward-backward splitting methods for solving inclusion problems in Banach spaces, J. Nonlinear Convex A., 21 (2020), 341–353.
    [30] A. Moudafi, M. Oliny, Convergence of a splitting inertial proximal method for monotone operators, J. Comput. Appl. Math., 155 (2003), 447–454. https://doi.org/10.1016/S0377-0427(02)00906-8 doi: 10.1016/S0377-0427(02)00906-8
    [31] K. Muangchoo, A. Adamu, A. H. Ibrahim, A. B. Abubakar, An inertial Halpern-type algorithm involving monotone operators on real Banach spaces with application to image recovery problems, Comp. Appl. Math., 41 (2022), 364. https://doi.org/10.1007/s40314-022-02064-1 doi: 10.1007/s40314-022-02064-1
    [32] G. B. Passty, Ergodic convergence to a zero of the sum of monotone operators in Hilbert space, J. Math. Anal. Appl., 72 (1979), 383–390. https://doi.org/10.1016/0022-247X(79)90234-8 doi: 10.1016/0022-247X(79)90234-8
    [33] N. Pholasa, P. Cholamjiaka, Y. J. Cho, Modified forward-backward splitting methods for accretive operators in Banach spaces, J. Nonlinear Sci. Appl., 9 (2016), 2766–2778. https://doi.org/10.22436/jnsa.009.05.72 doi: 10.22436/jnsa.009.05.72
    [34] P. Phairatchatniyom, H. Rehman, J. Abubakar, P. Kumam, J. Martínez-Moreno, An inertial iterative scheme for solving split variational inclusion problems in real Hilbert spaces, Bangmod Int. J. Math. Comput. Sci., 7 (2021), 35–52.
    [35] S. Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl., 75 (1980), 287–292. https://doi.org/10.1016/0022-247X(80)90323-6 doi: 10.1016/0022-247X(80)90323-6
    [36] 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 doi: 10.1007/s00025-019-1061-4
    [37] G. H. Taddele, A. G. Gebrie, J. Abubakar, An iterative method with inertial effect for solving multiple-set split feasibility problem, Bangmod Int. J. Math. Comput. Sci., 7 (2021), 53–73.
    [38] K. K. Tan, H. K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl., 178 (1993), 301–308. https://doi.org/10.1006/jmaa.1993.1309 doi: 10.1006/jmaa.1993.1309
    [39] W. Takahashi, N. C. Wong, J. C. Yao, Two generalized strong convergence theorems of Halpern's type in Hilbert spaces and applications, Taiwanese J. Math., 16 (2012), 1151–1172. https://doi.org/10.11650/twjm/1500406684 doi: 10.11650/twjm/1500406684
    [40] G. B. Wega, H. Zegeye Convergence results of forward-backward method for a zero of the sum of maximally monotone mappings in Banach spaces, Comp. Appl. Math., 39 (2020), 223. https://doi.org/10.1007/s40314-020-01246-z doi: 10.1007/s40314-020-01246-z
    [41] Y. Wang, F. Wang, H. Zhang, Strong convergence of viscosity forward-backward algorithm to the sum of two accretive operators in Banach space, 70 (2021), 169–190. https://doi.org/10.1080/02331934.2019.1705299
    [42] H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal. Theor., 16 (1991), 1127–1138. https://doi.org/10.1016/0362-546X(91)90200-K doi: 10.1016/0362-546X(91)90200-K
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