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Viscosity implicit midpoint scheme for enriched nonexpansive mappings

Year 2024, , 160 - 179, 15.12.2024
https://doi.org/10.33205/cma.1540982

Abstract

This article proposes and analyses a viscosity scheme for an enriched nonexpansive mapping. The scheme is incorporated with the implicit midpoint rule of stiff differential equations. We deduce some convergence properties of the scheme and establish that a sequence generated therefrom converges strongly to a fixed point of an enriched nonexpansive mapping provided such a point exists. Furthermore, we provide some examples of the implementation of the schemes with respect to certain enriched mappings and show the numerical pattern of the scheme.

References

  • M. A. Alghamdi, M. A. Alghamdi, N. Shahzad and H.-K. Xu: The implicit midpoint rule for nonexpansive mappings, Fixed Point Theory Appl., 2014 (2014), Article ID: 96.
  • H. Attouch: Viscosity solutions of minimization problems, SIAM J. Optim., 6 (3) (1996), 769–806.
  • W. Auzinger, R. Frank: Asymptotic error expansions for stiff equations: an analysis for the implicit midpoint and trapezoidal rules in the strongly stiff case, Numer. Math., 56 (5) (1989), 469–499.
  • G. Bader, P. Deuflhard: A semi-implicit mid-point rule for stiff systems of ordinary differential equations Numer. Math., 41 (3) (1983), 373–398.
  • V. Berinde: Approximating fixed points of enriched nonexpansive mappings by Krasnoselskij iteration in Hilbert spaces, Carpathian J. Math., 35 (3) (2019), 293–304.
  • V. Berinde: Approximating fixed points of enriched nonexpansive mappings in banach spaces by using a retractiondisplacement condition, Carpathian J. Math., 36 (1) (2020), 27–34.
  • V. Berinde: A modified krasnosel’skiˇı–mann iterative algorithm for approximating fixed points of enriched nonexpansive mappings, Symmetry, 14 (1) (2022), Article ID: 123.
  • V. Berinde, M. P˘acurar: Recent developments in the fixed point theory of enriched contractive mappings. A survey, Creat. Math. Inform., 33 (2024), 137–159.
  • C. Izuchukwu, C. C. Okeke and F. O. Isiogugu: A viscosity iterative technique for split variational inclusion and fixed point problems between a hilbert space and a banach space, J. Fixed Point Theory Appl., 20 (4) (2018), 1–25.
  • A. Moudafi: Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 241 (1) (2000), 46–55.
  • S. Salisu, V. Berinde, S. Sriwongsa and P. Kumam: On approximating fixed points of strictly pseudocontractive mappings in metric spaces, Carpathian J. Math., 40 (2) (2024), 419–430.
  • S. Salisu, L. Hashim, A. Y. Inuwa and A. U. Saje: Implicit midpoint scheme for enriched nonexpansive mappings, Nonlinear Convex Anal. Opt., 1 (2) (2022), 211–225.
  • S. Salisu, P. Kumam and S. Sriwongsa: Strong convergence theorems for fixed point of multi-valued mappings in Hadamard spaces, J. Inequal. Appl., 2022 (2022), Article ID: 143.
  • S. Salisu, P. Kumam and S. Sriwongsa: On fixed points of enriched contractions and enriched nonexpansive mappings, Carpathian J. Math., 39 (1) (2023), 237–254.
  • S. Salisu, P. Kumam, S. Sriwongsa and V. Berinde: Viscosity scheme with enriched mappings for hierarchical variational inequalities in certain geodesic spaces, Fixed Point Theory, (in press), 2023.
  • S. Salisu, P. Kumam, S. Sriwongsa and A. Y. Inuwa: Enriched multi-valued nonexpansive mappings in geodesic spaces, Rend. Circ. Mat. Palermo (2), 73 (4) (2024), 1435–1451.
  • S. Salisu, M. S. Minjibir, P. Kumam and S. Sriwongsa: Convergence theorems for fixed points in catp(0) spaces, J. Appl. Math. Comput., 69 (2023), 631–650.
  • S. Salisu, S. Sriwongsa, P. Kumam and V. Berinde: Variational inequality and proximal scheme for enriched nonexpansive mappings in cat(0) spaces, J. Nonlinear Convex Anal., 25 (7) (2024), 1759–1776.
  • C. Schneider: Analysis of the linearly implicit mid-point rule for differential-algebraic equations, Electron. Trans. Numer. Anal., 1 (1993), 1–10.
  • S. Somali: Implicit midpoint rule to the nonlinear degenerate boundary value problems, Int. J. Comput. Math., 79 (3) (2002), 327–332.
  • S. Somali, S. Davulcu: Implicit midpoint rule and extrapolation to singularly perturbed boundary value problems, Int. J. Comput. Math., 75 (1) (2000), 117–127.
  • Y. Song, X. Liu: Convergence comparison of several iteration algorithms for the common fixed point problems, Fixed Point Theory Appl., 2009 (2009), Article ID: 824374.
  • H.-K. Xu: Iterative algorithms for nonlinear operators, J. Lond. Math. Soc., 66 (1) (2002), 240–256.
  • H.-K. Xu: Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 298 (1) (2004), 279–291.
  • H.-K. Xu, M. A. Alghamdi and N. Shahzad: The viscosity technique for the implicit midpoint rule of nonexpansive mappings in Hilbert spaces, Fixed Point Theory Appl., 2015 (2015), Article ID: 41.
  • H.-K. Xu, R. G. Ori: An implicit iteration process for nonexpansive mappings, Numer. Funct. Anal. Optim., 22 (5-6) (2001), 767–773.
  • Y. Yao, H. Zhou and Y.-C. Liou: Strong convergence of a modified Krasnoselski-Mann iterative algorithm for non-expansive mappings, J. Appl. Math. Comput., 29 (1-2) (2009), 383–389.
Year 2024, , 160 - 179, 15.12.2024
https://doi.org/10.33205/cma.1540982

Abstract

References

  • M. A. Alghamdi, M. A. Alghamdi, N. Shahzad and H.-K. Xu: The implicit midpoint rule for nonexpansive mappings, Fixed Point Theory Appl., 2014 (2014), Article ID: 96.
  • H. Attouch: Viscosity solutions of minimization problems, SIAM J. Optim., 6 (3) (1996), 769–806.
  • W. Auzinger, R. Frank: Asymptotic error expansions for stiff equations: an analysis for the implicit midpoint and trapezoidal rules in the strongly stiff case, Numer. Math., 56 (5) (1989), 469–499.
  • G. Bader, P. Deuflhard: A semi-implicit mid-point rule for stiff systems of ordinary differential equations Numer. Math., 41 (3) (1983), 373–398.
  • V. Berinde: Approximating fixed points of enriched nonexpansive mappings by Krasnoselskij iteration in Hilbert spaces, Carpathian J. Math., 35 (3) (2019), 293–304.
  • V. Berinde: Approximating fixed points of enriched nonexpansive mappings in banach spaces by using a retractiondisplacement condition, Carpathian J. Math., 36 (1) (2020), 27–34.
  • V. Berinde: A modified krasnosel’skiˇı–mann iterative algorithm for approximating fixed points of enriched nonexpansive mappings, Symmetry, 14 (1) (2022), Article ID: 123.
  • V. Berinde, M. P˘acurar: Recent developments in the fixed point theory of enriched contractive mappings. A survey, Creat. Math. Inform., 33 (2024), 137–159.
  • C. Izuchukwu, C. C. Okeke and F. O. Isiogugu: A viscosity iterative technique for split variational inclusion and fixed point problems between a hilbert space and a banach space, J. Fixed Point Theory Appl., 20 (4) (2018), 1–25.
  • A. Moudafi: Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 241 (1) (2000), 46–55.
  • S. Salisu, V. Berinde, S. Sriwongsa and P. Kumam: On approximating fixed points of strictly pseudocontractive mappings in metric spaces, Carpathian J. Math., 40 (2) (2024), 419–430.
  • S. Salisu, L. Hashim, A. Y. Inuwa and A. U. Saje: Implicit midpoint scheme for enriched nonexpansive mappings, Nonlinear Convex Anal. Opt., 1 (2) (2022), 211–225.
  • S. Salisu, P. Kumam and S. Sriwongsa: Strong convergence theorems for fixed point of multi-valued mappings in Hadamard spaces, J. Inequal. Appl., 2022 (2022), Article ID: 143.
  • S. Salisu, P. Kumam and S. Sriwongsa: On fixed points of enriched contractions and enriched nonexpansive mappings, Carpathian J. Math., 39 (1) (2023), 237–254.
  • S. Salisu, P. Kumam, S. Sriwongsa and V. Berinde: Viscosity scheme with enriched mappings for hierarchical variational inequalities in certain geodesic spaces, Fixed Point Theory, (in press), 2023.
  • S. Salisu, P. Kumam, S. Sriwongsa and A. Y. Inuwa: Enriched multi-valued nonexpansive mappings in geodesic spaces, Rend. Circ. Mat. Palermo (2), 73 (4) (2024), 1435–1451.
  • S. Salisu, M. S. Minjibir, P. Kumam and S. Sriwongsa: Convergence theorems for fixed points in catp(0) spaces, J. Appl. Math. Comput., 69 (2023), 631–650.
  • S. Salisu, S. Sriwongsa, P. Kumam and V. Berinde: Variational inequality and proximal scheme for enriched nonexpansive mappings in cat(0) spaces, J. Nonlinear Convex Anal., 25 (7) (2024), 1759–1776.
  • C. Schneider: Analysis of the linearly implicit mid-point rule for differential-algebraic equations, Electron. Trans. Numer. Anal., 1 (1993), 1–10.
  • S. Somali: Implicit midpoint rule to the nonlinear degenerate boundary value problems, Int. J. Comput. Math., 79 (3) (2002), 327–332.
  • S. Somali, S. Davulcu: Implicit midpoint rule and extrapolation to singularly perturbed boundary value problems, Int. J. Comput. Math., 75 (1) (2000), 117–127.
  • Y. Song, X. Liu: Convergence comparison of several iteration algorithms for the common fixed point problems, Fixed Point Theory Appl., 2009 (2009), Article ID: 824374.
  • H.-K. Xu: Iterative algorithms for nonlinear operators, J. Lond. Math. Soc., 66 (1) (2002), 240–256.
  • H.-K. Xu: Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 298 (1) (2004), 279–291.
  • H.-K. Xu, M. A. Alghamdi and N. Shahzad: The viscosity technique for the implicit midpoint rule of nonexpansive mappings in Hilbert spaces, Fixed Point Theory Appl., 2015 (2015), Article ID: 41.
  • H.-K. Xu, R. G. Ori: An implicit iteration process for nonexpansive mappings, Numer. Funct. Anal. Optim., 22 (5-6) (2001), 767–773.
  • Y. Yao, H. Zhou and Y.-C. Liou: Strong convergence of a modified Krasnoselski-Mann iterative algorithm for non-expansive mappings, J. Appl. Math. Comput., 29 (1-2) (2009), 383–389.
There are 27 citations in total.

Details

Primary Language English
Subjects Numerical Analysis, Operator Algebras and Functional Analysis
Journal Section Articles
Authors

Sani Salisu 0000-0003-3387-4188

Songpon Sriwongsa 0000-0002-5137-8113

Poom Kumam 0000-0002-5463-4581

Cho Yeolb Je 0000-0002-1250-2214

Early Pub Date December 4, 2024
Publication Date December 15, 2024
Submission Date August 31, 2024
Acceptance Date December 2, 2024
Published in Issue Year 2024

Cite

APA Salisu, S., Sriwongsa, S., Kumam, P., Yeolb Je, C. (2024). Viscosity implicit midpoint scheme for enriched nonexpansive mappings. Constructive Mathematical Analysis, 7(4), 160-179. https://doi.org/10.33205/cma.1540982
AMA Salisu S, Sriwongsa S, Kumam P, Yeolb Je C. Viscosity implicit midpoint scheme for enriched nonexpansive mappings. CMA. December 2024;7(4):160-179. doi:10.33205/cma.1540982
Chicago Salisu, Sani, Songpon Sriwongsa, Poom Kumam, and Cho Yeolb Je. “Viscosity Implicit Midpoint Scheme for Enriched Nonexpansive Mappings”. Constructive Mathematical Analysis 7, no. 4 (December 2024): 160-79. https://doi.org/10.33205/cma.1540982.
EndNote Salisu S, Sriwongsa S, Kumam P, Yeolb Je C (December 1, 2024) Viscosity implicit midpoint scheme for enriched nonexpansive mappings. Constructive Mathematical Analysis 7 4 160–179.
IEEE S. Salisu, S. Sriwongsa, P. Kumam, and C. Yeolb Je, “Viscosity implicit midpoint scheme for enriched nonexpansive mappings”, CMA, vol. 7, no. 4, pp. 160–179, 2024, doi: 10.33205/cma.1540982.
ISNAD Salisu, Sani et al. “Viscosity Implicit Midpoint Scheme for Enriched Nonexpansive Mappings”. Constructive Mathematical Analysis 7/4 (December 2024), 160-179. https://doi.org/10.33205/cma.1540982.
JAMA Salisu S, Sriwongsa S, Kumam P, Yeolb Je C. Viscosity implicit midpoint scheme for enriched nonexpansive mappings. CMA. 2024;7:160–179.
MLA Salisu, Sani et al. “Viscosity Implicit Midpoint Scheme for Enriched Nonexpansive Mappings”. Constructive Mathematical Analysis, vol. 7, no. 4, 2024, pp. 160-79, doi:10.33205/cma.1540982.
Vancouver Salisu S, Sriwongsa S, Kumam P, Yeolb Je C. Viscosity implicit midpoint scheme for enriched nonexpansive mappings. CMA. 2024;7(4):160-79.