Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-22T06:33:06.257Z Has data issue: false hasContentIssue false
  • English
  • Français

MONOTONE LIPSCHITZIAN SEMIGROUPS IN BANACH SPACES

Part of: Nonlinear operators and their properties

Published online by Cambridge University Press:  18 June 2018

WOJCIECH M. KOZLOWSKI
WOJCIECH M. KOZLOWSKI*
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia email [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove the existence of common fixed points for monotone contractive and monotone nonexpansive semigroups of nonlinear mappings acting in Banach spaces equipped with partial order. We also discuss some applications to differential equations and dynamical systems.

Keywords

fixed pointcommon fixed pointmonotone nonexpansive mappingnonexpansive mappingcontractionsemigroup of mappingsmonotone nonexpansive semigroupdifferential equationsdynamical systems

MSC classification

Primary: 47H10: Fixed-point theorems
Secondary: 47H09: Contraction-type mappings, nonexpansive mappings, $A$-proper mappings, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 105 , Issue 3 , December 2018 , pp. 417 - 428
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

References

Alfuraidan, M. R. and Khamsi, M. A., ‘Fixed points of monotone nonexpansive mappings on a hyperbolic metric space with a graph’, Fixed Point Theory Appl. 2015(44) (2015); doi:10.1186/s13663-015-0294-5.Google Scholar
Alfuraidan, M. R. and Khamsi, M. A., ‘Fibonacci–Mann iteration for monotone asymptotically nonexpansive mappings’, Bull. Aust. Math. Soc. 96(2) (2017), 307–316.Google Scholar
Bachar, M. and Khamsi, M. A., ‘Recent contributions to fixed point theory of monotone mappings’, J. Fixed Point Theory Appl. 19(3) (2015), 1953–1976.Google Scholar
Bachar, M. and Khamsi, M. A., ‘Fixed points of monotone mappings and application to integral equations’, Fixed Point Theory Appl. 2015(110) (2015); doi:10.1186/s13663-015-0362-x.Google Scholar
Bachar, M. and Khamsi, M. A., ‘On common approximate fixed points of monotone nonexpansive semigroups in Banach spaces’, Fixed Point Theory Appl. 2015(160) (2016); doi:10.1186/s13663-015-0405-3.Google Scholar
Bin Dehaish, B. A. and Khamsi, M. A., ‘Browder and Göhde fixed point theorem for monotone nonexpansive mappings’, Fixed Point Theory Appl. 2016(20) (2016); doi:10.1186/s13663-016-0505-8.Google Scholar
El-Sayed, S. M. and Ran, A. C. M., ‘On an iteration method for solving a class of nonlinear matrix equations’, SIAM J. Matrix Anal. Appl. 23 (2002), 632–645.Google Scholar
Jachymski, J., ‘The contraction principle for mappings on a metric space with a graph’, Proc. Amer. Math. Soc. 136 (2007), 1359–1373.Google Scholar
Khamsi, M. A. and Khan, A. R., ‘On monotone nonexpansive mappings in L 1([0, 1])’, Fixed Point Theory Appl. 2015(94) (2015); doi:10.1186/s13663-015-0346-x.Google Scholar
Kirk, W. A. and Xu, H. K., ‘Asymptotic pointwise contractions’, Nonlinear Anal. 69 (2008), 4706–4712.Google Scholar
Kozlowski, W. M., ‘Common fixed points for semigroups of pointwise Lipschitzian mappings in Banach spaces’, Bull. Aust. Math. Soc. 84 (2011), 353–361.Google Scholar
Kozlowski, W. M., ‘On nonlinear differential equations in generalized Musielak–Orlicz spaces’, Comment. Math. 53(2) (2013), 113–133.Google Scholar
Kozlowski, W. M., ‘On the Cauchy problem for the nonlinear differential equations with values in modular function spaces’, in: Differential Geometry, Functional Analysis and Applications (Narosa Publishing House, New Delhi, 2015), 75–105.Google Scholar
Kozlowski, W. M., ‘On the construction algorithms for the common fixed points of the monotone nonexpansive semigroups’, in: Proceedings of ICFPTA 2017, 24–28 July 2017, Newcastle, Australia, to appear.Google Scholar
Kozlowski, W. M. and Sims, B., ‘On the convergence of iteration processes for semigroups of nonlinear mappings in Banach spaces’, in: Proc. Math. Statist., Computational and Analytical Mathematics, 50 (Springer, New York–Heidelberg–Dordrecht–London, 2013), 463–484.Google Scholar
Nieto, J. J. and Rodriguez-Lopez, R., ‘Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations’, Order 22(3) (2005), 223–239.Google Scholar
Ran, A. C. M. and Reurings, M. C. B., ‘A fixed point theorem in partially ordered sets and some applications to matrix equations’, Proc. Amer. Math. Soc. 132(5) (2004), 1435–1443.Google Scholar
Xu, H.-K., ‘Inequalities in Banach spaces with applications’, Nonlinear Anal. 16 (1991), 1127–1138.Google Scholar