nach oben

Journal of Applied Mathematics and Computing

Erschienen in:

01.06.2015 | Original Research

Convergence analysis for proximal split feasibility problems and fixed point problems

verfasst von: Yekini Shehu, Ferdinard U. Ogbuisi

Erschienen in: Journal of Applied Mathematics and Computing | Ausgabe 1-2/2015

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

Our aim in this paper is to introduce iterative algorithms and prove their strong convergence for solving proximal split feasibility problems and fixed point problems for \(k\)-strictly pseudocontractive mappings in Hilbert spaces. The sequence generated by our first iterative scheme converges strongly to an approximate common solution of convex minimization feasibility problem and fixed point problem. Furthermore, our second algorithm generates a strongly convergent sequence to an approximate common solution of non-convex minimization feasibility problem and fixed point problem. In all our results in this work, our iterative schemes are proposed with a way of selecting the step-sizes such that their implementation does not need any prior information about the operator norm because the calculation or at least an estimate of the operator norm \(||A||\) is very difficult, if not an impossible task. Our result complements many recent and important results in this direction.

Vorheriger Artikel Nonlinear modelling of ethanol inhibition with the state feedback control

Nächster Artikel Parallel and sequential hybrid methods for a finite family of asymptotically quasi -nonexpansive mappings

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Acedo, G.L., Xu, H.-K.: Iterative methods for strict pseudo-contractions in Hilbert spaces. Nonlinear Anal. 67, 2258–2271 (2007)CrossRefMATHMathSciNet

Brézis, H.: Analyse Fonctionnelle, Thorie et Applications. Masson, Paris (1983)

Browder, F.E., Petryshyn, W.V.: Construction of fixed points of nonlinear mappings in Hilbert space. J. Math. Anal. Appl. 20, 197–228 (1967)CrossRefMATHMathSciNet

Byrne, C.: Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Probl. 18(2), 441–453 (2002)CrossRefMATHMathSciNet

Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 20(1), 103–120 (2004)CrossRefMATHMathSciNet

Byrne, C., Censor, Y., Gibali, A., Reich, S.: The split common null point problem. J. Nonlinear Convex Anal. 13(4), 759–775 (2012)MATHMathSciNet

Ceng, L.C., Al-Homidan, S., Ansari, Q.H., Yao, J.C.: An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings. J. Comput. Appl. Math. 223, 967–974 (2009)CrossRefMATHMathSciNet

Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 8(2–4), 221–239 (1994)CrossRefMATHMathSciNet

Chidume, C.E., Abbas, M., Ali, B.: Convergence of the Mann iteration algorithm for a class of pseudo-contractive mappings. Appl. Math. Comput. 194, 1–6 (2007)CrossRefMATHMathSciNet

10.

Cholamjiak, P., Suantai, S.: Strong convergence for a countable family of strict pseudocontractions in \(q\)-uniformly smooth Banach spaces. Comput. Math. Appl. 62, 787–796 (2011)CrossRefMATHMathSciNet

11.

Clarke, Z.F.H., Ledyaev, Y.S., Stern, R.J., Wolenski, P.R.: Nonsmooth Analysis and Control Theory. Springer, New York (1998)MATH

12.

Dang, Y., Gao, Y.: The strong convergence of a KM-CQ-like algorithm for a split feasibility problem. Inverse Probl. 27(1), Article ID 015007 (2011)

13.

Hao, Y., Cho, S. Y.: Some results on strictly pseudocontractive nonself mapping and equilibrium problems in Hilbert spaces. Abst. Appl. Anal. 2012, Article ID 543040

14.

Jaiboon, C., Kumam, P.: Strong convergence theorems for solving equilibrium problems and fixed point problems of strictly pseudocontraction mappings by two hybrid projection methods. J. Comput. Appl. Math. 234(3), 722–732 (2010)CrossRefMATHMathSciNet

15.

Jung, J.S.: Iterative methods for mixed equilibrium problems and psedocontractive mappings. Fixed Point Theory Appl. 2012, 184 (2012)CrossRef

16.

Liu, Y.: A general iterative method for equilibrium problems and strict pseudo-contractions in Hilbert spaces. Nonlinear Anal. 71, 4852–4861 (2009)CrossRefMATHMathSciNet

17.

Luke, D.R.: Finding best approximation pairs relative to a convex and prox-regular set in Hilbert space. SIAM J. Optim. 19(2), 714–729 (2008)CrossRefMATHMathSciNet

18.

Luke, D.R., Burke, J.V., Lyon, R.G.: Optical wavefront reconstruction: theory and numerical methods. SIAM Rev. 44, 169–224 (2002)CrossRefMATHMathSciNet

19.

Lopez, G., Martin-Marquez, V., Wang, F., Xu, H.K.: Solving the split feasibility problem without prior knowledge of matrix norms. Inverse Probl. 28, 085004 (2012)CrossRefMathSciNet

20.

Marino, G., Xu, H.-K.: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl. 329(1), 336–346 (2007)CrossRefMATHMathSciNet

21.

Maingé, P.E.: The viscosity approximation process for quasi-nonexpansive mappings in Hilbert spaces. Comput. Math. Appl. 59(1), 74–79 (2010)CrossRefMATHMathSciNet

22.

Moudafi, A.: Split monotone variational inclusions. J. Optim. Theory Appl. 150(2), 275–283 (2011)CrossRefMATHMathSciNet

23.

Moudafi, A., Thakur, B. S.: Solving proximal split feasibility problems without prior knowledge of operator norms. Optim. Lett. doi:10.1007/s11590-013-0708-4

24.

Osilike, M.O., Shehu, Y.: Cyclic algorithm for common fixed points of finite family of strictly pseudocontractive mappings of Browder–Petryshyn type. Nonlinear Anal. 70, 3575–3583 (2009)CrossRefMATHMathSciNet

25.

Qin, L.-J., Wang, L.: An iteration method for solving equilibrium problems, common fixed point problems of pseudocontractive mappings of Browder-Petryshyn type in Hilbert spaces. Int. Math. Forum 6(2), 63–74 (2011)MATHMathSciNet

26.

Qu, B., Xiu, N.: A note on the CQ algorithm for the split feasibility problem. Inverse Probl. 21(5), 1655–1665 (2005)CrossRefMATHMathSciNet

27.

Rockafellar, R.T., Wets, R.: Variational Analysis. Springer, Berlin (1988)

28.

Shehu, Y.: Iterative methods for family of strictly pseudocontractive mappings and system of generalized mixed equilibrium problems and variational inequality problems. Fixed Point Theory Appl. 2011, Article ID 852789 (2011)

29.

Xu, H.-K.: Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces. Inverse Probl. 26(10), Article ID 105018 (2010)

30.

Xu, H.-K.: A variable Krasnoselskii–Mann algorithm and the multiple-set split feasibility problem. Inverse Probl. 22(6), 2021–2034 (2006)CrossRefMATH

31.

Xu, H.K.: Iterative algorithm for nonlinear operators. J. London Math. Soc. 66(2), 1–17 (2002)MathSciNet

32.

Yang, Q.: The relaxed CQ algorithm solving the split feasibility problem. Inverse Probl. 20(4), 1261–1266 (2004)CrossRefMATH

33.

Yang, Q., Zhao, J.: Generalized KM theorems and their applications. Inverse Probl. 22(3), 833–844 (2006)CrossRef

34.

Yao, Y., Chen, R., Liou, Y.-C.: A unified implicit algorithm for solving the triple-hierarchical constrained optimization problem. Math. Comput. Model. 55(3–4), 1506–1515 (2012)CrossRefMATHMathSciNet

35.

Yao, Y., Cho, Y.-J., Liou, Y.-C.: Hierarchical convergence of an implicit doublenet algorithm for nonexpansive semigroups and variational inequalities. Fixed Point Theory Appl. 2011, article 101 (2011)

36.

Yao, Y., Jigang, W., Liou, Y.-C.: Regularized methods for the split feasibility problem. Abstr. Appl Anal. 2012, Article ID 140679 (2012)

37.

Yao, Y., Liou, Y.-C., Kang, S.M.: Two-step projection methods for a system of variational inequality problems in Banach spaces. J. Global Optim. 55(4), 801–811 (2013)CrossRefMATHMathSciNet

38.

Yao, Y.H., Zhou, H.Y., Liou, Y.C.: Strong convergence of a modified Krasnoselski–Mann iterative algorithm for nin-expansive mappings. J. Math. Comput. 29, 383–389 (2009)

39.

Zhao, J., Yang, Q.: Several solution methods for the split feasibility problem. Inverse Probl. 21(5), 1791–1799 (2005)CrossRefMATH

40.

Zhou, H.Y.: Convergence theorems of fixed points for strict pseudo-contractions in Hilbert spaces. Nonlinear Anal. 69(2), 456–462 (2008)CrossRefMATHMathSciNet

Titel: Convergence analysis for proximal split feasibility problems and fixed point problems
verfasst von: Yekini Shehu
Ferdinard U. Ogbuisi
Publikationsdatum: 01.06.2015
Verlag: Springer Berlin Heidelberg
Erschienen in: Journal of Applied Mathematics and Computing / Ausgabe 1-2/2015
Print ISSN: 1598-5865
Elektronische ISSN: 1865-2085
DOI: https://doi.org/10.1007/s12190-014-0800-7

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Weitere Artikel der Ausgabe 1-2/2015

Deteriorating jobs scheduling on a single machine with release dates, rejection and a fixed non-availability interval

Existence result for nonlinear fractional differential equation with -Laplacian operator at resonance

Minimal cyclic codes of length over , where is prime power of the form

Higher-order symmetric duality with higher-order generalized invexity

A finite iterative algorithm for Hermitian reflexive and skew-Hermitian solution groups of the general coupled linear matrix equations

Single machine scheduling problem with multiple due windows assignment in a group technology

Premium Partner