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September  2021, 17(5): 2733-2759. doi: 10.3934/jimo.2020092

Viscosity approximation method for solving the multiple-set split equality common fixed-point problems for quasi-pseudocontractive mappings in Hilbert spaces

Adeolu Taiwo , Lateef Olakunle Jolaoso and  Oluwatosin Temitope Mewomo ,

School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa

* Corresponding author: Oluwatosin Temitope Mewomo

Received  May 2019 Revised  February 2020 Published  September 2021 Early access  May 2020

We propose a parallel iterative scheme with viscosity approximation method which converges strongly to a solution of the multiple-set split equality common fixed point problem for quasi-pseudocontractive mappings in real Hilbert spaces. We also give an application of our result to approximation of minimization problem from intensity-modulated radiation therapy. Finally, we present numerical examples to demonstrate the behaviour of our algorithm. This result improves and generalizes many existing results in literature in this direction.

Keywords: Split equality common fixed point problem, Lipschitzian quasi-pseudocontractive mapping, viscosity approximation.
Mathematics Subject Classification: Primary: 47H10, 47J25; Secondary: 65J15.
Citation: Adeolu Taiwo, Lateef Olakunle Jolaoso, Oluwatosin Temitope Mewomo. Viscosity approximation method for solving the multiple-set split equality common fixed-point problems for quasi-pseudocontractive mappings in Hilbert spaces. Journal of Industrial & Management Optimization, 2021, 17 (5) : 2733-2759. doi: 10.3934/jimo.2020092
References:
[1]

T. O. Alakoya, L. O. Jolaoso and O. T. Mewomo, Modified inertia subgradient extragradient method with self adaptive stepsize for solving monotone variational inequality and fixed point problems, Optimization, (2020), 1–30. Google Scholar

[2]

Q. H. Ansari and A. Rehan, Split feasibility and fixed point problems, in Nonlinear Analysis, Trends Math., Birkhäuser/Springer, New Delhi, 2014,281–322.  Google Scholar

[3]

C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Problems, 18 (2002), 441-453.  doi: 10.1088/0266-5611/18/2/310.  Google Scholar

[4]

Y. Censor, T. Bortfeld, B. Martin and A. Trofimov, A unified approach for inversion problems in intensity-modulated radiation therapy, Phys. Med. Biol., 51 (2006), 2353-2365.  doi: 10.1088/0031-9155/51/10/001.  Google Scholar

[5]

Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms, 8 (1994), 221-239.  doi: 10.1007/BF02142692.  Google Scholar

[6]

Y. Censor, T. Elfving, N. Kopf and T. Bortfeld, The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Problems, 21 (2005), 2071-2084.  doi: 10.1088/0266-5611/21/6/017.  Google Scholar

[7]

S. Chang, L. Wang and L.-J. Qin, Split equality fixed point problem for quasi-pseudo-contractive mappings with applications, Fixed Point Theory Appl., 208 (2015), 12 pp. doi: 10.1186/s13663-015-0458-3.  Google Scholar

[8]

H. Che and M. Li, A simultaneous iterative method for split equality problems of two finite families of strictly pseudononspreading mappings without prior knowledge of operator norms, Fixed Point Theory Appl., 1 (2015), 14 pp. doi: 10.1186/1687-1812-2015-1.  Google Scholar

[9]

W.-Z. Chen, Y. Xiao and J. Li, Impact of dose calculation algorithm on radiation theraphy, World J. Radiol., 6 (2014), 874-880.   Google Scholar

[10]

A. Hanjing and S. Suantai, The split common fixed point problem for infinite families of demicontractive mappings, Fixed Point Theory Appl., (2018), Paper No. 14, 21 pp. doi: 10.1186/s13663-018-0639-y.  Google Scholar

[11]

C. Izuchukwu, G. C. Ugwunnadi, O. T. Mewomo, A. R. Khan and M. Abbas, Proximal-type algorithms for split minimization problem in $p$-uniformly convex metric spaces, Numer. Algorithms, 82 (2019), 909-935.  doi: 10.1007/s11075-018-0633-9.  Google Scholar

[12]

L. O. Jolaoso, T. O. Alakoya, A. Taiwo and O. T. Mewomo, A parallel combination extragradient method with Armijo line searching for finding common solutions of finite families of equilibrium and fixed point problems, Rend. Circ. Mat. Palermo II. Ser, 2 (2019), 1-25.   Google Scholar

[13]

L. O. Jolaoso, T. O. Alakoya, A. Taiwo and O. T. Mewomo, Inertial extragradient method via viscosity approximation approach for solving equilibrium problem in Hilbert space, Optimization, (2020), 1–26. doi: 10.1080/02331934.2020.1716752.  Google Scholar

[14]

L. O. Jolaoso, K. O. Oyewole, C. C. Okeke and O. T. Mewomo, A unified algorithm for solving split generalized mixed equilibrium problem and fixed point of nonspreading mapping in Hilbert space, Demonstr. Math., 51 (2018), 211-232.  doi: 10.1515/dema-2018-0015.  Google Scholar

[15]

L. O. Jolaoso, A. Taiwo, T. O. Alakoya and O. T. Mewomo, A unified algorithm for solving variational inequality and fixed point problems with application to the split equality problem, Comput. Appl. Math., 39 (2020), 28 pp. doi: 10.1007/s40314-019-1014-2.  Google Scholar

[16]

P.-E. Maingé, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 16 (2008), 899-912.  doi: 10.1007/s11228-008-0102-z.  Google Scholar

[17]

G. Marino and H.-K. Xu, Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl., 329 (2007), 336-346.  doi: 10.1016/j.jmaa.2006.06.055.  Google Scholar

[18]

A. Moudafi, A relaxed alternating CQ-algorithm for convex feasibility problems, Nonlinear Anal., 79 (2013), 117-121.  doi: 10.1016/j.na.2012.11.013.  Google Scholar

[19]

A. Moudafi, Alternating CQ-algorithms for convex feasibility and fixed-point problems, J. Nonlinear Convex Anal., 15 (2014), 809-818.   Google Scholar

[20]

A. Moudafi and E. Al-Shemas, Simultaneous iterative methods for split equality problems and applications, Trans. Math. Program. Appl., 1 (2013), 1-11.   Google Scholar

[21]

S. A. Naimpally and K. L. Singh, Extensions of some fixed point theorems of Rhoades, J. Math. Anal. Appl., 96 (1983), 437-446.  doi: 10.1016/0022-247X(83)90052-5.  Google Scholar

[22]

F. U. Ogbuisi and O. T. Mewomo, On split generalised mixed equilibrium problems and fixed-point problems with no prior knowledge of operator norm, J. Fixed Point Theory Appl., 19 (2017), 2109-2128.  doi: 10.1007/s11784-016-0397-6.  Google Scholar

[23]

F. U. Ogbuisi and O. T. Mewomo, Iterative solution of split variational inclusion problem in a real Banach spaces, Afr. Mat., 28 (2017), 295-309.  doi: 10.1007/s13370-016-0450-z.  Google Scholar

[24]

F. U. Ogbuisi and O. T. Mewomo, Convergence analysis of common solution of certain nonlinear problems, Fixed Point Theory, 19 (2018), 335-358.  doi: 10.24193/fpt-ro.2018.1.26.  Google Scholar

[25]

Y. Shehu and O. T. Mewomo, Further investigation into split common fixed point problem for demicontractive operators, Acta Math. Sin. (Engl. Ser.), 32 (2016), 1357-1376.  doi: 10.1007/s10114-016-5548-6.  Google Scholar

[26]

A. Taiwo, L. O. Jolaoso and O. T. Mewomo, A modified Halpern algorithm for approximating a common solution of split equality convex minimization problem and fixed point problem in uniformly convex Banach spaces, Comput. Appl. Math., 38 (2019), 28 pp. doi: 10.1007/s40314-019-0841-5.  Google Scholar

[27]

A. Taiwo, L. O. Jolaoso and O. T. Mewomo, Parallel hybrid algorithm for solving pseudomonotone equilibrium and split common fixed point problems, Bull. Malays. Math. Sci. Soc., 43 (2020), 1893-1918.  doi: 10.1007/s40840-019-00781-1.  Google Scholar

[28]

A. Taiwo, L. O. Jolaoso and O. T. Mewomo, General alternative regularization method for solving split equality common fixed point problem for quasi-pseudocontractive mappings in Hilbert spaces, Ricerche Mat., (2019), 1–25. doi: 10.1007/s11587-019-00460-0.  Google Scholar

[29]

Y. Wang and X. Fang, Viscosity approximation methods for the multiple-set split equality common fixed-point problems of demicontractive mappings, J. Nonlinear Sci. Appl., 10 (2017), 4254-4268.  doi: 10.22436/jnsa.010.08.20.  Google Scholar

[30]

H.-K. Xu, Another control condition in an iterative method for nonexpansive mappings, Bull. Austral. Math. Soc., 65 (2002), 109-113.  doi: 10.1017/S0004972700020116.  Google Scholar

[31]

I. Yamada, The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. Inherently parallel algorithm for feasibility and optimization and their applications, Stud. Comput. Math., 8 (2001), 473-504.  doi: 10.1016/S1570-579X(01)80028-8.  Google Scholar

[32]

H. Zegeye and N. Shahzad, Convergence of Mann's type iteration method for generalized asymptotically nonexpansive mappings, Comput. Math. Appl., 62 (2011), 4007-4014.  doi: 10.1016/j.camwa.2011.09.018.  Google Scholar

[33]

J. Zhao, Solving split equality fixed-point problem of quasi-nonexpansive mappings without prior knowledge of operators norms, Optimization, 64 (2015), 2619-2630.  doi: 10.1080/02331934.2014.883515.  Google Scholar

[34]

J. Zhao and S. Wang, Mixed iterative algorithms for the multiple-set split equality common fixed-point problems without prior knowledge of operator norms, Optimization, 65 (2016), 1069-1083.  doi: 10.1080/02331934.2015.1072716.  Google Scholar

[35]

J. Zhao and S. Wang, Viscosity approximation methods for the split equality common fixed point problem of quasi-nonexpansive operators, Acta Math. Sci. Ser. B (Engl. Ed.), 36 (2016), 1474-1486.  doi: 10.1016/S0252-9602(16)30083-2.  Google Scholar

[36]

J. Zhao and H. Zong, Solving the multiple-set split equality common fixed-point problem of firmly non-expansive operators, J. Inequal. Appl., (2018), Paper No. 83, 18 pp. doi: 10.1186/s13660-018-1668-0.  Google Scholar

show all references

References:
[1]

T. O. Alakoya, L. O. Jolaoso and O. T. Mewomo, Modified inertia subgradient extragradient method with self adaptive stepsize for solving monotone variational inequality and fixed point problems, Optimization, (2020), 1–30. Google Scholar

[2]

Q. H. Ansari and A. Rehan, Split feasibility and fixed point problems, in Nonlinear Analysis, Trends Math., Birkhäuser/Springer, New Delhi, 2014,281–322.  Google Scholar

[3]

C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Problems, 18 (2002), 441-453.  doi: 10.1088/0266-5611/18/2/310.  Google Scholar

[4]

Y. Censor, T. Bortfeld, B. Martin and A. Trofimov, A unified approach for inversion problems in intensity-modulated radiation therapy, Phys. Med. Biol., 51 (2006), 2353-2365.  doi: 10.1088/0031-9155/51/10/001.  Google Scholar

[5]

Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms, 8 (1994), 221-239.  doi: 10.1007/BF02142692.  Google Scholar

[6]

Y. Censor, T. Elfving, N. Kopf and T. Bortfeld, The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Problems, 21 (2005), 2071-2084.  doi: 10.1088/0266-5611/21/6/017.  Google Scholar

[7]

S. Chang, L. Wang and L.-J. Qin, Split equality fixed point problem for quasi-pseudo-contractive mappings with applications, Fixed Point Theory Appl., 208 (2015), 12 pp. doi: 10.1186/s13663-015-0458-3.  Google Scholar

[8]

H. Che and M. Li, A simultaneous iterative method for split equality problems of two finite families of strictly pseudononspreading mappings without prior knowledge of operator norms, Fixed Point Theory Appl., 1 (2015), 14 pp. doi: 10.1186/1687-1812-2015-1.  Google Scholar

[9]

W.-Z. Chen, Y. Xiao and J. Li, Impact of dose calculation algorithm on radiation theraphy, World J. Radiol., 6 (2014), 874-880.   Google Scholar

[10]

A. Hanjing and S. Suantai, The split common fixed point problem for infinite families of demicontractive mappings, Fixed Point Theory Appl., (2018), Paper No. 14, 21 pp. doi: 10.1186/s13663-018-0639-y.  Google Scholar

[11]

C. Izuchukwu, G. C. Ugwunnadi, O. T. Mewomo, A. R. Khan and M. Abbas, Proximal-type algorithms for split minimization problem in $p$-uniformly convex metric spaces, Numer. Algorithms, 82 (2019), 909-935.  doi: 10.1007/s11075-018-0633-9.  Google Scholar

[12]

L. O. Jolaoso, T. O. Alakoya, A. Taiwo and O. T. Mewomo, A parallel combination extragradient method with Armijo line searching for finding common solutions of finite families of equilibrium and fixed point problems, Rend. Circ. Mat. Palermo II. Ser, 2 (2019), 1-25.   Google Scholar

[13]

L. O. Jolaoso, T. O. Alakoya, A. Taiwo and O. T. Mewomo, Inertial extragradient method via viscosity approximation approach for solving equilibrium problem in Hilbert space, Optimization, (2020), 1–26. doi: 10.1080/02331934.2020.1716752.  Google Scholar

[14]

L. O. Jolaoso, K. O. Oyewole, C. C. Okeke and O. T. Mewomo, A unified algorithm for solving split generalized mixed equilibrium problem and fixed point of nonspreading mapping in Hilbert space, Demonstr. Math., 51 (2018), 211-232.  doi: 10.1515/dema-2018-0015.  Google Scholar

[15]

L. O. Jolaoso, A. Taiwo, T. O. Alakoya and O. T. Mewomo, A unified algorithm for solving variational inequality and fixed point problems with application to the split equality problem, Comput. Appl. Math., 39 (2020), 28 pp. doi: 10.1007/s40314-019-1014-2.  Google Scholar

[16]

P.-E. Maingé, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 16 (2008), 899-912.  doi: 10.1007/s11228-008-0102-z.  Google Scholar

[17]

G. Marino and H.-K. Xu, Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl., 329 (2007), 336-346.  doi: 10.1016/j.jmaa.2006.06.055.  Google Scholar

[18]

A. Moudafi, A relaxed alternating CQ-algorithm for convex feasibility problems, Nonlinear Anal., 79 (2013), 117-121.  doi: 10.1016/j.na.2012.11.013.  Google Scholar

[19]

A. Moudafi, Alternating CQ-algorithms for convex feasibility and fixed-point problems, J. Nonlinear Convex Anal., 15 (2014), 809-818.   Google Scholar

[20]

A. Moudafi and E. Al-Shemas, Simultaneous iterative methods for split equality problems and applications, Trans. Math. Program. Appl., 1 (2013), 1-11.   Google Scholar

[21]

S. A. Naimpally and K. L. Singh, Extensions of some fixed point theorems of Rhoades, J. Math. Anal. Appl., 96 (1983), 437-446.  doi: 10.1016/0022-247X(83)90052-5.  Google Scholar

[22]

F. U. Ogbuisi and O. T. Mewomo, On split generalised mixed equilibrium problems and fixed-point problems with no prior knowledge of operator norm, J. Fixed Point Theory Appl., 19 (2017), 2109-2128.  doi: 10.1007/s11784-016-0397-6.  Google Scholar

[23]

F. U. Ogbuisi and O. T. Mewomo, Iterative solution of split variational inclusion problem in a real Banach spaces, Afr. Mat., 28 (2017), 295-309.  doi: 10.1007/s13370-016-0450-z.  Google Scholar

[24]

F. U. Ogbuisi and O. T. Mewomo, Convergence analysis of common solution of certain nonlinear problems, Fixed Point Theory, 19 (2018), 335-358.  doi: 10.24193/fpt-ro.2018.1.26.  Google Scholar

[25]

Y. Shehu and O. T. Mewomo, Further investigation into split common fixed point problem for demicontractive operators, Acta Math. Sin. (Engl. Ser.), 32 (2016), 1357-1376.  doi: 10.1007/s10114-016-5548-6.  Google Scholar

[26]

A. Taiwo, L. O. Jolaoso and O. T. Mewomo, A modified Halpern algorithm for approximating a common solution of split equality convex minimization problem and fixed point problem in uniformly convex Banach spaces, Comput. Appl. Math., 38 (2019), 28 pp. doi: 10.1007/s40314-019-0841-5.  Google Scholar

[27]

A. Taiwo, L. O. Jolaoso and O. T. Mewomo, Parallel hybrid algorithm for solving pseudomonotone equilibrium and split common fixed point problems, Bull. Malays. Math. Sci. Soc., 43 (2020), 1893-1918.  doi: 10.1007/s40840-019-00781-1.  Google Scholar

[28]

A. Taiwo, L. O. Jolaoso and O. T. Mewomo, General alternative regularization method for solving split equality common fixed point problem for quasi-pseudocontractive mappings in Hilbert spaces, Ricerche Mat., (2019), 1–25. doi: 10.1007/s11587-019-00460-0.  Google Scholar

[29]

Y. Wang and X. Fang, Viscosity approximation methods for the multiple-set split equality common fixed-point problems of demicontractive mappings, J. Nonlinear Sci. Appl., 10 (2017), 4254-4268.  doi: 10.22436/jnsa.010.08.20.  Google Scholar

[30]

H.-K. Xu, Another control condition in an iterative method for nonexpansive mappings, Bull. Austral. Math. Soc., 65 (2002), 109-113.  doi: 10.1017/S0004972700020116.  Google Scholar

[31]

I. Yamada, The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. Inherently parallel algorithm for feasibility and optimization and their applications, Stud. Comput. Math., 8 (2001), 473-504.  doi: 10.1016/S1570-579X(01)80028-8.  Google Scholar

[32]

H. Zegeye and N. Shahzad, Convergence of Mann's type iteration method for generalized asymptotically nonexpansive mappings, Comput. Math. Appl., 62 (2011), 4007-4014.  doi: 10.1016/j.camwa.2011.09.018.  Google Scholar

[33]

J. Zhao, Solving split equality fixed-point problem of quasi-nonexpansive mappings without prior knowledge of operators norms, Optimization, 64 (2015), 2619-2630.  doi: 10.1080/02331934.2014.883515.  Google Scholar

[34]

J. Zhao and S. Wang, Mixed iterative algorithms for the multiple-set split equality common fixed-point problems without prior knowledge of operator norms, Optimization, 65 (2016), 1069-1083.  doi: 10.1080/02331934.2015.1072716.  Google Scholar

[35]

J. Zhao and S. Wang, Viscosity approximation methods for the split equality common fixed point problem of quasi-nonexpansive operators, Acta Math. Sci. Ser. B (Engl. Ed.), 36 (2016), 1474-1486.  doi: 10.1016/S0252-9602(16)30083-2.  Google Scholar

[36]

J. Zhao and H. Zong, Solving the multiple-set split equality common fixed-point problem of firmly non-expansive operators, J. Inequal. Appl., (2018), Paper No. 83, 18 pp. doi: 10.1186/s13660-018-1668-0.  Google Scholar

Figure 1.  Example 4.3. Top left: Case Ⅰ; Top right: Case Ⅱ; Bottom left: Case Ⅲ; Bottom right: Case Ⅳ
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Figure 2.  Top left: Case IIa; Top right: Case IIb; Bottom left: Case IIc; Bottom right: Case IId
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Figure 3.  Left : Case IIa* Right: Case IIc*
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Table 1.  Numerical result for Example 4.3
Algorithm 3.2 Algorithm 1.1
Case Ⅰ No of Iter. 18 44
CPU time (sec) 7.7529 9.7706
Case Ⅱ No of Iter. 9 19
CPU time (sec) 5.2116 8.8849
Case Ⅲ No of Iter. 10 26
CPU time (sec) 7.7204 12.7338
Case Ⅳ No of Iter. 9 22
CPU time (sec) 5.6424 7.1538
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Algorithm 3.2 Algorithm 1.1
Case Ⅰ No of Iter. 18 44
CPU time (sec) 7.7529 9.7706
Case Ⅱ No of Iter. 9 19
CPU time (sec) 5.2116 8.8849
Case Ⅲ No of Iter. 10 26
CPU time (sec) 7.7204 12.7338
Case Ⅳ No of Iter. 9 22
CPU time (sec) 5.6424 7.1538
Table 2.  Numerical results
Alg. (8) Alg. Algorithm 3.2
Case IIa CPU time (sec) 0.0019 8.8698e-4
No of Iter. 75 16
Case IIb CPU time (sec) 0.0020 8.6745e-4
No. of Iter. 75 16
Case IIc CPU time (sec) 0.0018 8.5652e-4
No of Iter. 81 17
Case IId CPU time (sec) 0.0019 8.9216e-4
No of Iter. 74 16
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Alg. (8) Alg. Algorithm 3.2
Case IIa CPU time (sec) 0.0019 8.8698e-4
No of Iter. 75 16
Case IIb CPU time (sec) 0.0020 8.6745e-4
No. of Iter. 75 16
Case IIc CPU time (sec) 0.0018 8.5652e-4
No of Iter. 81 17
Case IId CPU time (sec) 0.0019 8.9216e-4
No of Iter. 74 16
Table 3.  Numerical results
Alg. (8) Alg. 3.2
Case IIa* CPU time (sec) 0.0020 8.8533e-4
No of Iter. 95 17
Case IIc* CPU time (sec) 0.0021 8.6149e-4
No of Iter. 103 18
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Alg. (8) Alg. 3.2
Case IIa* CPU time (sec) 0.0020 8.8533e-4
No of Iter. 95 17
Case IIc* CPU time (sec) 0.0021 8.6149e-4
No of Iter. 103 18
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