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

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.

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 and 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.

[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.

[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.

[4]

Y. CensorT. BortfeldB. 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.

[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.

[6]

Y. CensorT. ElfvingN. 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.

[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.

[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.

[9]

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

[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.

[11]

C. IzuchukwuG. C. UgwunnadiO. T. MewomoA. 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.

[12]

L. O. JolaosoT. O. AlakoyaA. 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. 

[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.

[14]

L. O. JolaosoK. O. OyewoleC. 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.

[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.

[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.

[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.

[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.

[19]

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

[20]

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

[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.

[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.

[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.

[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.

[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.

[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.

[27]

A. TaiwoL. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[4]

Y. CensorT. BortfeldB. 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.

[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.

[6]

Y. CensorT. ElfvingN. 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.

[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.

[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.

[9]

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

[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.

[11]

C. IzuchukwuG. C. UgwunnadiO. T. MewomoA. 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.

[12]

L. O. JolaosoT. O. AlakoyaA. 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. 

[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.

[14]

L. O. JolaosoK. O. OyewoleC. 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.

[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.

[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.

[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.

[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.

[19]

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

[20]

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

[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.

[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.

[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.

[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.

[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.

[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.

[27]

A. TaiwoL. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

Figure 1.  Example 4.3. Top left: Case Ⅰ; Top right: Case Ⅱ; Bottom left: Case Ⅲ; Bottom right: Case Ⅳ
Figure 2.  Top left: Case IIa; Top right: Case IIb; Bottom left: Case IIc; Bottom right: Case IId
Figure 3.  Left : Case IIa* Right: Case IIc*
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
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
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
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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