Approximation of common fixed points in 2-Banach spaces with applications

[EN] The purpose of this paper is to establish the existence and uniqueness of common fixed points of a family of self-mappings satisfying generalized rational contractive condition in 2-Banach spaces. An example is included to justify our results. We approximate the common fixed point by Mann and Picard type iteration schemes. Further, an application to well-posedness of the common fixed point problem is given. The presented results generalize many known results on 2-Banach spaces.The authors thank the reviewers for valuable comments. The first author D. Ramesh Kumar would like to thank the University Grants Commission, New Delhi, India for providing the financial support in preparation of this manuscript.Kumar, DR.; Pitchaimani, M. (2019). Approximation of common fixed points in 2-Banach spaces with applications. Applied General Topology. 20(1):43-55. https://doi.org/10.4995/agt.2019.9168SWORD4355201M. Abbas, B. E. Rhoades and T. Nazir, Common fixed points for four maps in cone metric spaces, Applied Mathematics and Computation 216 (2010), 80-86. https://doi.org/10.1016/j.amc.2010.01.003M. Abbas and G. Jungck, Common fixed point results for noncommuting mappings without continuity in cone metric spaces, J. Math. Anal. Appl. 341 (2008), 416-420. https://doi.org/10.1016/j.jmaa.2007.09.070M. Arshad, E. Karapinar and J. Ahmad, Some unique fixed point theorems for rational contractions in partially ordered metric spaces, J. Inequal. Appl. 2013(1) (2013), 1-16. https://doi.org/10.1186/1029-242x-2013-248A. Azam, B. Fisher and M. Khan, Common fixed point theorems in complex valued metric spaces, Numerical Functional Analysis and optimization 32, no. 3 (2011), 243-253. https://doi.org/10.1080/01630563.2011.533046S. Banach, Sur les opérations dans les ensembles abstraits et leur application auxéquations intégrales, Fund. Math. 3 (1922), 133-181. https://doi.org/10.4064/fm-3-1-133-181I. Beg and A. R. Butt, Fixed point for set-valued mappings satisfying an implicit relation in partially ordered metric spaces, Nonlinear Anal. 71, no.9 (2009), 3699-3704. https://doi.org/10.1016/j.na.2009.02.027K. Cieplinski, Approximate multi-additive mappings in 2-Banach spaces, Bull. Iranian Math. Soc. 41, no. 3 (2015), 785-792.B. K. Dass and S. Gupta, An extension of Banach's contraction principle through rational expression, Indian J. Pure appl. Math. 6, no. 4 (1975), 1445-1458.F. S. De Blasi and J. Myjak, Sur la porosité de l'ensemble des contractions sans point fixe, C. R. Acad. Sci. Paris 308 (1989), 51-54.S. Gähler, 2-metrische Räume and ihre topologische strucktur, Math. Nachr. 26 (1963), 115-148. https://doi.org/10.1002/mana.19630260109S. Gähler, Uber die unifromisieberkeit 2-metrischer Räume, Math. Nachr. 28 (1965), 235-244. https://doi.org/10.1002/mana.19640280309S. Gähler, Über 2-Banach-Räume, Math. Nachr. 42 (1969), 335-347. https://doi.org/10.1002/mana.19690420414L.-G. Huang and X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332, no. 2 (2007), 1468-1476. https://doi.org/10.1016/j.jmaa.2005.03.087W. A. Kirk and N. Shahzad, Some fixed point results in ultrametric spaces, Topology and its Applications 159 (2012), 3327-3334. https://doi.org/10.1016/j.topol.2012.07.016K. Iseki, Fixed point theorems in 2-metric space, Math.Seminar. Notes, Kobe Univ. 3(1975), 133-136.E. Matouskova, S. Reich and A. J. Zaslavski, Genericity in nonexpansive mapping theory, Advanced Courses of Mathematical Analysis I, World Scientific Hackensack (2004), 81-98. https://doi.org/10.1142/9789812702371_0004S. B. Nadler, Sequence of contraction and fixed points, Pacific J.Math. 27 (1968), 579-585.H. K. Nashinea, M. Imdadb and M. Hasan, Common fixed point theorems under rational contractions in complex valued metric spaces, J. Nonlinear Sci. Appl. 7 (2014), 42-50. https://doi.org/10.22436/jnsa.007.01.05B. G. Pachpatte, Common fixed point theorems for mappings satisfying rational inequalities, Indian J. Pure appl. Math. 10, no. 11 (1979), 1362-1368.A.-D. Filip and A. Petrusel, Fixed point theorems for operators in generalized Kasahara spaces, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas 109, no. 1 (2015), 15-26. https://doi.org/10.1007/s13398-014-0163-9M. Pitchaimani and D. Ramesh Kumar, Some common fixed point theorems using implicit relation in 2-Banach spaces, Surv. Math. Appl. 10 (2015), 159-168.M. Pitchaimani and D. Ramesh Kumar, Common and coincidence fixed point theorems for asymptotically regular mappings in 2-Banach Spaces, Nonlinear Funct. Anal. Appl.21, no. 1 (2016), 131-144.M. Pitchaimani and D. Ramesh Kumar, On construction of fixed point theory under implicit relation in Hilbert spaces, Nonlinear Funct. Anal. Appl. 21, no. 3 (2016), 513-522.M. Pitchaimani and D. Ramesh Kumar, On Nadler type results in ultrametric spaces with application to well-posedness, Asian-European Journal of Mathematics 10, no. 4(2017), 1750073(1-15). https://doi.org/10.1142/s1793557117500735M. Pitchaimani and D. Ramesh Kumar, Generalized Nadler type results in ultrametric spaces with application to well-posedness, Afr. Mat. 28 (2017), 957-970. https://doi.org/10.1007/s13370-017-0496-6V. Popa, Well-Posedness of fixed problem in compact metric space, Bull. Univ. Petrol-Gaze, Ploicsti, sec. Mat Inform. Fiz. 60, no. 1 (2008), 1-4.D. Ramesh Kumar and M. Pitchaimani, Set-valued contraction mappings of Presic-Reichtype in ultrametric spaces, Asian-European Journal of Mathematics 10, no. 4 (2017), 1750065 (1-15). https://doi.org/10.1142/s1793557117500656D. Ramesh Kumar and M. Pitchaimani, A generalization ofset-valued Presic-Reich type contractions in ultrametric spaces with applications, J. Fixed Point Theory Appl. 19,no. 3 (2017), 1871-1887. https://doi.org/10.1007/s11784-016-0338-4D. Ramesh Kumar and M. Pitchaimani, Approximation and stability of common fixed points of Presic type mappings in ultrametric spaces, J. Fixed Point Theory Appl. 20:4(2018). https://doi.org/10.1007/s11784-018-0504-yD. Ramesh Kumar and M. Pitchaimani, New coupled fixed point theorems in cone metric spaces with applications to integral equations and Markov process, Transactions of A. Razmadze Mathematical Institute, to appear. https://doi.org/10.1016/j.trmi.2018.01.006S. Reich and A. T. Zaslawski, Well- Posedness of fixed point problems, Far East J. Math.sci, Special volume part III (2011), 393-401.W. Sintunavarat and P. Kumam, Generalized common fixed point theorems in complex valued metric spaces and applications, J. Inequal. Appl. 2012, no. 1 (2012), 1-12. https://doi.org/10.1186/1029-242x-2012-84R. J. Shahkoohi and A. Razani, Some fixed point theorems for rational Geraghty contractive mappings in ordered b-metric spaces, J. Inequal. Appl. 2014, no. 1 (2014), 1-23. https://doi.org/10.1186/1029-242x-2014-373S. Shukla, Presic type results in 2-Banach spaces Afr. Mat. 25, no. 4 (2014), 1043-1051. https://doi.org/10.1007/s13370-013-0174-2A. White, 2-Banach spaces, Math. Nachr. 42 (1969), 43-60

The convex real projective orbifolds with radial or totally geodesic ends: a survey of some partial results

A real projective orbifold has a radial end if a neighborhood of the end is foliated by projective geodesics that develop into geodesics ending at a common point. It has a totally geodesic end if the end can be completed to have the totally geodesic boundary. The purpose of this paper is to announce some partial results. A real projective structure sometimes admits deformations to parameters of real projective structures. We will prove a homeomorphism between the deformation space of convex real projective structures on an orbifold $\mathcal{O}$ with radial or totally geodesic ends with various conditions with the union of open subspaces of strata of the corresponding subset of $$Hom(\pi_{1}(\mathcal{O}), PGL(n+1, \mathbb{R}))/PGL(n+1, \mathbb{R}).$$ Lastly, we will talk about the openness and closedness of the properly (resp. strictly) convex real projective structures on a class of orbifold with generalized admissible ends.Comment: 36 pages, 2 figure. Corrected a few mistakes including the condition (NA) on page 22, arXiv admin note: text overlap with arXiv:1011.106

Morse actions of discrete groups on symmetric space

We study the geometry and dynamics of discrete infinite covolume subgroups of higher rank semisimple Lie groups. We introduce and prove the equivalence of several conditions, capturing "rank one behavior'' of discrete subgroups of higher rank Lie groups. They are direct generalizations of rank one equivalents to convex cocompactness. We also prove that our notions are equivalent to the notion of Anosov subgroup, for which we provide a closely related, but simplified and more accessible reformulation, avoiding the geodesic flow of the group. We show moreover that the Anosov condition can be relaxed further by requiring only non-uniform unbounded expansion along the (quasi)geodesics in the group. A substantial part of the paper is devoted to the coarse geometry of these discrete subgroups. A key concept which emerges from our analysis is that of Morse quasigeodesics in higher rank symmetric spaces, generalizing the Morse property for quasigeodesics in Gromov hyperbolic spaces. It leads to the notion of Morse actions of word hyperbolic groups on symmetric spaces,i.e. actions for which the orbit maps are Morse quasiisometric embeddings, and thus provides a coarse geometric characterization for the class of subgroups considered in this paper. A basic result is a local-to-global principle for Morse quasigeodesics and actions. As an application of our techniques we show algorithmic recognizability of Morse actions and construct Morse "Schottky subgroups'' of higher rank semisimple Lie groups via arguments not based on Tits' ping-pong. Our argument is purely geometric and proceeds by constructing equivariant Morse quasiisometric embeddings of trees into higher rank symmetric spaces.Comment: 93 page

Relative Hyperbolicity, Trees of Spaces and Cannon-Thurston Maps

We prove the existence of continuous boundary extensions (Cannon-Thurston maps) for the inclusion of a vertex space into a tree of (strongly) relatively hyperbolic spaces satisfying the qi-embedded condition. This implies the same result for inclusion of vertex (or edge) subgroups in finite graphs of (strongly) relatively hyperbolic groups. This generalises a result of Bowditch for punctured surfaces in 3 manifolds and a result of Mitra for trees of hyperbolic metric spaces.Comment: 27pgs No figs, v3: final version, incorporating referee's comments, to appear in Geometriae Dedicat