Semi-strongly asymptotically non-expansive mappings and their applications on fixed point theory

Year 2017, Volume: 46 Issue: 4, 613 - 620, 01.08.2017

Chris Lennard , Veysel Nezir

Abstract

We study fixed point theory for semi-strongly asymptotically nonexpansive and strongly asymptotically nonexpansive mappings. We consider these mappings for renormings of $l^1$ and $c_0$, and show that $l^1$ cannot be equivalently renormed to have the fixed point property for semi-strongly asymptotically nonexpansive mappings, while $c_0$ cannot be equivalently renormed to have the fixed point property for strongly asymptotically nonexpansive mappings Next and more importantly, we show reflexivity is equivalent to the fixed point property for affine semi-strongly asymptotically nonexpansive mappings in Banach lattices. Finally, we give an application of our results in Lorentz-Marcinkiewicz spaces $l_{w,\infty}^0$, and some examples of these new types of mappings associated with a large class of $c_0$-summing basic sequences in $c_0$.

Keywords

nonexpansive mapping, reflexive Banach space, fixed point property, weak fixed point property, closed bounded convex set, asymptotically isometric $c_0$-summing basic sequence, Lorentz-Marcinkiewicz spaces, semi-strongly asymptotically nonexpansive mapping, strongly asymptotically nonexpansive mapping

References

• Alspach D. E., A fixed point free nonexpansive map, Proceedings of the American Mathematical Society 82 , no. 3, 423424, 1981.
• Beauzamy B., Introduction to Banach spaces and their Geometry, 11 (Elsevier Science Pub. Co., 1982).
• Domínguez Benavides T., A renorming of some nonseparable Banach spaces with the Fixed Point Property, J. Math. Anal. Appl. 350 (2), 525530, 2009.
• Dowling P. N. and Lennard C. J., Every nonreflexive subspace of L1[0; 1] fails the fixed point property, Proceedings of the American Mathematical Society 125 (2), 443446, 1997.
• Dowling P. N., Lennard C. J., and Turett B., Some fixed point results in l1 and c0, Nonlinear Analysis 39, 929936, 2000.
• Dowling P. N., Lennard C. J., and Turett B., Weak compactness is equivalent to the fixed point property in c0, Proc. Amer. Math. Soc. 132 (6) 16591666, 2004.
• van Dulst D., Equivalent norms and the fixed point property for nonexpansive mappings, J. London Math. Soc. 25 (2), 139144, 1982.
• James R. C., Uniformly non-square Banach spaces, Ann. of Math. 80, 542550, 1964.
• Lennard C. J. and Nezir V., The closed, convex hull of every ai c0-summing basic sequence fails the fpp for ane nonexpansive mappings, J. Math. Anal. Appl. 381, 678688, 2011.
• Lennard C. J. and Nezir V., Reflexivity is equivalent to the perturbed fixed point property for cascading nonexpansive maps in Banach lattices, Nonlinear Analysis: Theory, Methods & Applications 95, 414420, 2014.
• Lin P. K., There is an equivalent norm on $\ell_1$ that has the fixed point property, Nonlinear Analysis 68, 23032308, 2008.
• Lindenstrauss J. and Tzafriri L., Classical Banach spaces I: Sequence Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, 92 (Springer-Verlag, 1977).
• Lindenstrauss J. and Tzafriri L., Classical Banach spaces II: Function Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, 97 (Springer-Verlag, 1979).
• Mil'man D. P. and Mil'man V. D., Some properties of non-reflexive Banach spaces, Mat. Sb. (N.S.) 65, 486497, 1964 (in Russian).
• Nezir V., Fixed Point Properties for c0-like Spaces, Ph.D. thesis, University of Pittsburgh, 2012.