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Year 2021, Volume: 9 Issue: 3, 95 - 107, 30.09.2021
https://doi.org/10.36753/mathenot.592227

Abstract

References

  • [1] L. Ambrosio, N. Gigli, G. Savaré, Gradient flows in metric spaces and in the space of probability measures, Second edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, (2008).
  • [2] Berinde,V.,Pcurar,M.,TheroleofthePompeiu-Hausdorffmetricinfixedpointtheory.Creat.Math.Inform.22 (2013), no. 2, 143-150.
  • [3] F.S. Blasi, J. Myjak, S. Reich, A.J Zaslavski, Generic existence and approximation of fixed points for nonexpansive set-valued maps, Set-Valued Var. Anal. 17(1), 97-112 (2009).
  • [4] F.E.Browder,ConvergengetheoremforsequenceofnonlinearoperatorinBanachspaces,Math.Z.100(1967).201-225. Vol. EVIII, part 2, 1976.
  • [5] F. E. Browder, and W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert spaces, J. Math. Anal. Appl. 20 (1967) 197- 228.
  • [6] C. E. Chidume, Geometric Properties of Banach spaces and Nonlinear Iterations, Springer Verlag Series: Lecture Notes in Mathematics, Vol. 1965,(2009), ISBN 978-1-84882-189-7.
  • [7] C.E. Chidume, N. Djitte, Iterative algorithm for zeros of bounded m-Accretive nonlinear operators, to appear, J. Nonlinear and convex analysis.
  • [8] C.E. Chidume, N. Djitte, Strong convergence theorems for zeros of bounded maximal monotone nonlinear operators, J. Abstract and Applied Analysis, Volume 2012, Article ID 681348, 19 pages, doi:10.1155/2012/681348.
  • [9] C. E. Chidume, N. Djitté, M. Sène, Iterative algorithm for zeros of multi-valued accretive operators in certain Banach spaces, Afr. Mat. 26 (2015), no. 3-4: 357-368.
  • [10] C. E. Chidume, C. O. Chidume, N. Djitte, and M. S. Minjibir, Krasnoselskii-type algorithm for fixed points of multi-valued strictly pseudo-contractive, Fixed Point Theory and Applications 2013, 2013:58.
  • [11] J.Jost,Convexfunctionalsandgeneralizedharmonicmapsintospacesofnonpositivecurvature,Comment. Math. Helv., 70 (1995), 659-673.
  • [12] O.Güler,Ontheconvergenceoftheproximalpointalgorithmforconvexminimization,SIAMJ.ControlOptim., 29 (1991), 403-419.
  • [13] S.Kakutani,AgeneralizationofBrouwer’sfixedpointtheorem,DukeMathematicalJournal8(1941),no.3,457-459.
  • [14] S. Kamimura, W. Takahashi, Strong convergence of a proximal-type algorithm in a Banach space, SIAM J. of Optimization 13(3) (5003), 938-945.
  • [15] N. Lehdili, A. Moudafi, Combining the proximal algorithm and Tikhonov regularization, Optimization 37(1996), 239-252.
  • [16] W.R.Mann,Meanvaluemethodsiniteration,Proc.Amer.Math.Soc.,4(1953)506-510.
  • [17] P. E. Mainge, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Analysis, 16, 899-912 (2008).
  • [18] G.MarinoandH.K.Xu,AgeneraliterativemethodfornonexpansivemappingsinHibertspaces,J.Math.Anal.Appl. 318 (2006), 43-52.
  • [19] G.Marino,H.K.Xu,Weakandstrongconvergencetheoremsforstrictpseudo-contractionsinHilbertspaces,J.Math. Math. Appl., 329(2007), 336-346.
  • [20] Moudafi,A:Viscosityapproximationmethodsforfixedpointproblems.J.Math.Anal.Appl.241,46-55(2000).
  • [21] B.Martinet,Régularisationd’inéquationsvariationnellesparapproximationssuccessives,(French)Rev.Franaise Informat. Recherche Opérationnelle, 4 (1970), 154-158.
  • [22] A.A.Mebawondu,Proximalpointalgorithmsforfindingcommonfixedpointsofafinitefamilyofnonexpan- sive mapping of nonexpansive multivalued mappings in real Hilbert spaces, Khayyam J. Math. 5 (2019) no. 2, 113-123.
  • [23] I.Miyadera,Nonlinearsemigroups,TranslationsofMathematicalMonographs,AmericanMathematicalSociety, Providence, (1992).
  • [24] J.F.Nash,Equilibriumpointsinn-persongames,ProceedingsoftheNationalAcademyofSciencesoftheUnited States of America, 36 (1950), no1, 48-49.
  • [25] J.F.Nash,Non-coperativegames,AnnalsofMathematics,Secondseries54(1951),286-295.
  • [26] B. Panyanak, Ishikawa iteration processes for multi-valued mappings in Banach Spaces, Comput. Math. Appl. 54 (2007), 872-877.
  • [27] J.Garcia-Falset,E.Lorens-Fuster,andT.Suzuki,Fixedpointtheoryforaclasssofgeneralisednonexpansivemappings, J. Math. Anal. Appl. 375 (2011), 185-195.
  • [28] S. Reich, Strong Convergence theorems for resolvents of accretive operators in Banach spaces, in J. Math. Anal. Appl. 183 (1994), 118-120.
  • [29] R.T.Rockafellar,Onthemaximalityofsumsofnonlinearmonotoneoperators.Trans.Am.Math.Soc.149,7588 (1970).
  • [30] M.Sene,P.FayeandN.Djitté,AKrasnoselskiitypeAlgorithmapproximatingacommonFixedPointofafinitefamily of multivalued strictly pseudo-contractive mappings in Hilbert spaces ,J. Maths. Sci. Adv. Appl.,Volume 27, 2014, Pages 59-80.
  • [31] M.V. Solodov, B.F. Svaiter, Forcing strong convergence of proximal point iterations in a Hilber space, Math. Program., Ser. A 87 (5000) 189-202.
  • [32] T.M.M.Sow,AnewiterativemethodformultivaluednonexpansivemappingsinBanachspaceswithapplicationJournal Nonlinear Analysis and Application 2018 No.2 (2018) 212-222.
  • [33] T.M.M.Sow,N.Djitté,andC.E.Chidume,Apathconvergencetheoremandconstructionoffixedpointsfornonexpan- sive mappings in certain Banach spaces, Carpathian J.Math.,32(2016),No.2,217-226,2016.
  • [34] T. M. M. Sow, M. Sène, N. Djitté, Strong convergence theorems for a common fixed point of a finite family of multi-valued Mappings in certain Banach Spaces, Int. J. Math. Anal., Vol. 9, 2015, no. 9, 437-452.
  • [35] H.K.Xu,Aniterativeapproachtoquadraticoptimization,J.Optim.TheoryAppl.116(2003)659-678.
  • [36] H.K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. 66 (2002), no. 2, 240 - 256.
  • [37] Y.Yao,H.Zhou,Y.C.Liou,StrongconvergenceofmodifiedKrasnoselskii-Manniterativealgorithmfornonexpansive mappings, J. Math. Anal. Appl. Comput. 29 (2009) 383-389.
  • [38] S. Wang, A general iterative method for an infinite family of strictly pseudo-contractive mappings in Hilbert spaces, Applied Mathematics Letters, 24(2011): 901-907.

A New Hybrid Iterative Method for Solving Fixed Points Problems for a Finite Family of Multivalued Strictly Pseudo-Contractive Mappings and Convex Minimization Problems in Real Hilbert Spaces

Year 2021, Volume: 9 Issue: 3, 95 - 107, 30.09.2021
https://doi.org/10.36753/mathenot.592227

Abstract

In the present work, we introduce a new hybrid iterative  process  which is a combination of proximal point algorithms and a modified Krasnoselskii-Mann algorithm for approximating a common element of the set of minimizers of a convex function and the set of common fixed points of a finite family of multivalued strictly pseudo-contractive mappings in the framework of Hilbert spaces. We then prove strong convergence of the proposed iterative process without imposing any compactness condition on the mapping or the space. The results we obtain extend and improve some recent known results.

References

  • [1] L. Ambrosio, N. Gigli, G. Savaré, Gradient flows in metric spaces and in the space of probability measures, Second edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, (2008).
  • [2] Berinde,V.,Pcurar,M.,TheroleofthePompeiu-Hausdorffmetricinfixedpointtheory.Creat.Math.Inform.22 (2013), no. 2, 143-150.
  • [3] F.S. Blasi, J. Myjak, S. Reich, A.J Zaslavski, Generic existence and approximation of fixed points for nonexpansive set-valued maps, Set-Valued Var. Anal. 17(1), 97-112 (2009).
  • [4] F.E.Browder,ConvergengetheoremforsequenceofnonlinearoperatorinBanachspaces,Math.Z.100(1967).201-225. Vol. EVIII, part 2, 1976.
  • [5] F. E. Browder, and W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert spaces, J. Math. Anal. Appl. 20 (1967) 197- 228.
  • [6] C. E. Chidume, Geometric Properties of Banach spaces and Nonlinear Iterations, Springer Verlag Series: Lecture Notes in Mathematics, Vol. 1965,(2009), ISBN 978-1-84882-189-7.
  • [7] C.E. Chidume, N. Djitte, Iterative algorithm for zeros of bounded m-Accretive nonlinear operators, to appear, J. Nonlinear and convex analysis.
  • [8] C.E. Chidume, N. Djitte, Strong convergence theorems for zeros of bounded maximal monotone nonlinear operators, J. Abstract and Applied Analysis, Volume 2012, Article ID 681348, 19 pages, doi:10.1155/2012/681348.
  • [9] C. E. Chidume, N. Djitté, M. Sène, Iterative algorithm for zeros of multi-valued accretive operators in certain Banach spaces, Afr. Mat. 26 (2015), no. 3-4: 357-368.
  • [10] C. E. Chidume, C. O. Chidume, N. Djitte, and M. S. Minjibir, Krasnoselskii-type algorithm for fixed points of multi-valued strictly pseudo-contractive, Fixed Point Theory and Applications 2013, 2013:58.
  • [11] J.Jost,Convexfunctionalsandgeneralizedharmonicmapsintospacesofnonpositivecurvature,Comment. Math. Helv., 70 (1995), 659-673.
  • [12] O.Güler,Ontheconvergenceoftheproximalpointalgorithmforconvexminimization,SIAMJ.ControlOptim., 29 (1991), 403-419.
  • [13] S.Kakutani,AgeneralizationofBrouwer’sfixedpointtheorem,DukeMathematicalJournal8(1941),no.3,457-459.
  • [14] S. Kamimura, W. Takahashi, Strong convergence of a proximal-type algorithm in a Banach space, SIAM J. of Optimization 13(3) (5003), 938-945.
  • [15] N. Lehdili, A. Moudafi, Combining the proximal algorithm and Tikhonov regularization, Optimization 37(1996), 239-252.
  • [16] W.R.Mann,Meanvaluemethodsiniteration,Proc.Amer.Math.Soc.,4(1953)506-510.
  • [17] P. E. Mainge, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Analysis, 16, 899-912 (2008).
  • [18] G.MarinoandH.K.Xu,AgeneraliterativemethodfornonexpansivemappingsinHibertspaces,J.Math.Anal.Appl. 318 (2006), 43-52.
  • [19] G.Marino,H.K.Xu,Weakandstrongconvergencetheoremsforstrictpseudo-contractionsinHilbertspaces,J.Math. Math. Appl., 329(2007), 336-346.
  • [20] Moudafi,A:Viscosityapproximationmethodsforfixedpointproblems.J.Math.Anal.Appl.241,46-55(2000).
  • [21] B.Martinet,Régularisationd’inéquationsvariationnellesparapproximationssuccessives,(French)Rev.Franaise Informat. Recherche Opérationnelle, 4 (1970), 154-158.
  • [22] A.A.Mebawondu,Proximalpointalgorithmsforfindingcommonfixedpointsofafinitefamilyofnonexpan- sive mapping of nonexpansive multivalued mappings in real Hilbert spaces, Khayyam J. Math. 5 (2019) no. 2, 113-123.
  • [23] I.Miyadera,Nonlinearsemigroups,TranslationsofMathematicalMonographs,AmericanMathematicalSociety, Providence, (1992).
  • [24] J.F.Nash,Equilibriumpointsinn-persongames,ProceedingsoftheNationalAcademyofSciencesoftheUnited States of America, 36 (1950), no1, 48-49.
  • [25] J.F.Nash,Non-coperativegames,AnnalsofMathematics,Secondseries54(1951),286-295.
  • [26] B. Panyanak, Ishikawa iteration processes for multi-valued mappings in Banach Spaces, Comput. Math. Appl. 54 (2007), 872-877.
  • [27] J.Garcia-Falset,E.Lorens-Fuster,andT.Suzuki,Fixedpointtheoryforaclasssofgeneralisednonexpansivemappings, J. Math. Anal. Appl. 375 (2011), 185-195.
  • [28] S. Reich, Strong Convergence theorems for resolvents of accretive operators in Banach spaces, in J. Math. Anal. Appl. 183 (1994), 118-120.
  • [29] R.T.Rockafellar,Onthemaximalityofsumsofnonlinearmonotoneoperators.Trans.Am.Math.Soc.149,7588 (1970).
  • [30] M.Sene,P.FayeandN.Djitté,AKrasnoselskiitypeAlgorithmapproximatingacommonFixedPointofafinitefamily of multivalued strictly pseudo-contractive mappings in Hilbert spaces ,J. Maths. Sci. Adv. Appl.,Volume 27, 2014, Pages 59-80.
  • [31] M.V. Solodov, B.F. Svaiter, Forcing strong convergence of proximal point iterations in a Hilber space, Math. Program., Ser. A 87 (5000) 189-202.
  • [32] T.M.M.Sow,AnewiterativemethodformultivaluednonexpansivemappingsinBanachspaceswithapplicationJournal Nonlinear Analysis and Application 2018 No.2 (2018) 212-222.
  • [33] T.M.M.Sow,N.Djitté,andC.E.Chidume,Apathconvergencetheoremandconstructionoffixedpointsfornonexpan- sive mappings in certain Banach spaces, Carpathian J.Math.,32(2016),No.2,217-226,2016.
  • [34] T. M. M. Sow, M. Sène, N. Djitté, Strong convergence theorems for a common fixed point of a finite family of multi-valued Mappings in certain Banach Spaces, Int. J. Math. Anal., Vol. 9, 2015, no. 9, 437-452.
  • [35] H.K.Xu,Aniterativeapproachtoquadraticoptimization,J.Optim.TheoryAppl.116(2003)659-678.
  • [36] H.K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. 66 (2002), no. 2, 240 - 256.
  • [37] Y.Yao,H.Zhou,Y.C.Liou,StrongconvergenceofmodifiedKrasnoselskii-Manniterativealgorithmfornonexpansive mappings, J. Math. Anal. Appl. Comput. 29 (2009) 383-389.
  • [38] S. Wang, A general iterative method for an infinite family of strictly pseudo-contractive mappings in Hilbert spaces, Applied Mathematics Letters, 24(2011): 901-907.
There are 38 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Thierno Sow 0000-0002-9687-839X

Publication Date September 30, 2021
Submission Date July 15, 2019
Acceptance Date October 30, 2020
Published in Issue Year 2021 Volume: 9 Issue: 3

Cite

APA Sow, T. (2021). A New Hybrid Iterative Method for Solving Fixed Points Problems for a Finite Family of Multivalued Strictly Pseudo-Contractive Mappings and Convex Minimization Problems in Real Hilbert Spaces. Mathematical Sciences and Applications E-Notes, 9(3), 95-107. https://doi.org/10.36753/mathenot.592227
AMA Sow T. A New Hybrid Iterative Method for Solving Fixed Points Problems for a Finite Family of Multivalued Strictly Pseudo-Contractive Mappings and Convex Minimization Problems in Real Hilbert Spaces. Math. Sci. Appl. E-Notes. September 2021;9(3):95-107. doi:10.36753/mathenot.592227
Chicago Sow, Thierno. “A New Hybrid Iterative Method for Solving Fixed Points Problems for a Finite Family of Multivalued Strictly Pseudo-Contractive Mappings and Convex Minimization Problems in Real Hilbert Spaces”. Mathematical Sciences and Applications E-Notes 9, no. 3 (September 2021): 95-107. https://doi.org/10.36753/mathenot.592227.
EndNote Sow T (September 1, 2021) A New Hybrid Iterative Method for Solving Fixed Points Problems for a Finite Family of Multivalued Strictly Pseudo-Contractive Mappings and Convex Minimization Problems in Real Hilbert Spaces. Mathematical Sciences and Applications E-Notes 9 3 95–107.
IEEE T. Sow, “A New Hybrid Iterative Method for Solving Fixed Points Problems for a Finite Family of Multivalued Strictly Pseudo-Contractive Mappings and Convex Minimization Problems in Real Hilbert Spaces”, Math. Sci. Appl. E-Notes, vol. 9, no. 3, pp. 95–107, 2021, doi: 10.36753/mathenot.592227.
ISNAD Sow, Thierno. “A New Hybrid Iterative Method for Solving Fixed Points Problems for a Finite Family of Multivalued Strictly Pseudo-Contractive Mappings and Convex Minimization Problems in Real Hilbert Spaces”. Mathematical Sciences and Applications E-Notes 9/3 (September 2021), 95-107. https://doi.org/10.36753/mathenot.592227.
JAMA Sow T. A New Hybrid Iterative Method for Solving Fixed Points Problems for a Finite Family of Multivalued Strictly Pseudo-Contractive Mappings and Convex Minimization Problems in Real Hilbert Spaces. Math. Sci. Appl. E-Notes. 2021;9:95–107.
MLA Sow, Thierno. “A New Hybrid Iterative Method for Solving Fixed Points Problems for a Finite Family of Multivalued Strictly Pseudo-Contractive Mappings and Convex Minimization Problems in Real Hilbert Spaces”. Mathematical Sciences and Applications E-Notes, vol. 9, no. 3, 2021, pp. 95-107, doi:10.36753/mathenot.592227.
Vancouver Sow T. A New Hybrid Iterative Method for Solving Fixed Points Problems for a Finite Family of Multivalued Strictly Pseudo-Contractive Mappings and Convex Minimization Problems in Real Hilbert Spaces. Math. Sci. Appl. E-Notes. 2021;9(3):95-107.

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