Convergence and fixed point theorems in convex metric spaces : a survey
-
2014-04-13 https://doi.org/10.14419/ijamr.v3i2.2204 -
Abstract
The aim of this paper is to provide a survey of the fixed point theorems, convergence theorems and stability results of iterative schemes that have been studied by many authors in convex metric spaces. This paper should be a useful reference for those persons wishing to become better acquainted with the area.
Keywords: Convex Metric Spaces, Fixed Point, Hyperbolic Spaces, Iterative Schemes, Stability.
-
References
- A. M. Harder and T.L. Hicks, Stability results for fixed point iteration procedures, Math. Japonica, vol. 33 no. 5, (1988), 693-706.
- A. Liepins, A cradle-song for a little tiger on fixed points, Topological Spaces and their mappings. Riga, (1983), 61-69.
- A. Rafiq and S. Zafar, On the convergence of implicit Ishikawa iterations with errors to a common fixed point of two mappings in convex metric spaces, General Mathematics, vol. 14, no. 2, (2006), 95-108.
- A. Rafiq, Fixed point of Ciric quasi-contractive operators in generalized convex metric spaces, General Mathematics, vol. 14, no. 3, (2006), 79-90.
- A. R. Khan, H. Fukha-ud-din and M. A. A. Khan, An implicit algorithm for two finite families of nonexpansive maps in hyperbolic spaces, Fixed Point Theory and Applications, vol. 2012, (2012), 1-12.
- A. M. Harder and T. L. Hicks, A stable iteration procedure for nonexpansive mappings, Math. Japon., vol.33 no. 5, (1988), 687-692.
- B. E. Rhoades, Fixed point theorems and stability results for fixed point iteration procedures II, Indian J. Pure Appl. Math., vol. 24, no. 11, (1993), 691-703.
- B. E. Rhoades, Some theorems on weakly contractive maps, Nonlinear Analysis, vol. 47, (2001), 2683-2693.
- C. Wang and L.W. Liu, Convergence theorems of fixed points of uniformly quasi-Lipschitzian mappings in convex metric spaces, Nonlinear Anal, TMA, vol. 70, (2009), 2067-2071.
- D. Ariza-Ruiz, Convergence and stability of some iterative processes for a class of quasi nonexpansive type mappings, Journal of Nonlinear Science and Applications, vol. 5, (2012), 93-103.
- F. E. Browder, Non expansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. USA, vol. 54, (1965), 1041-1044.
- G.S Saluja, Three-step iteration process for a finite family of Asymptotically quasi-nonexpansive mappings in the intermediate sense in convex metric spaces, Bulletin of Mathematical Analysis and Applications, vol. 3, (2011), 89-101.
- G. S. Saluja and H. K. Nashine, Convergence of implicit iteration process for a finite family of asymptotically Quasi-nonexpansive mappings in convex metric spaces, vol. 30, no. 3, (2010), 331-340.
- G. S. Saluja, Approximating common fixed points for asymptotically quasi-non-expansive mappings in the intermediate sense in convex metric spaces, Functional Analysis, Approximation and Computation, vol. 3, no. 1, (2011), 33-44.
- G. Jungck, Compatible mappings and common fixed points, Internat J. Math. Math. Sci., vol. 9, (1986), 771-779.
- G. Jungck and B.E. Rhoades, Some fixed point theorems for compatible maps, Internat J. Math. Math. Sci., vol. 16, (1993), 417-428.
- H. Fukhar-ud-din, A.R. Khan A. Kalsoom and M.A.A. Khan, One step implicit algorithm for two finite families of nonexpansive maps in Hyperbolic spaces, J. Adv. Math. Stud., vol. 6 no. 1, (2013), 73-81.
- H. Fukhar-ud-din, A. R. Khan and M. Ubaid-Ur-Rehman, Ishikawa type algorithm of multi-valued Quasi-nonexpansive maps on nonlinear domains, Annals of Functional. Analysis, vol. 4 no. 2, (2013), 97-110.
- H. F. Senter and W. G. Doston, Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc., vol. 44, (1974), 375-380.
- I. Beg, Inequalities in metric spaces with applications, Topological Methods in Nonlinear Analysis, vol. 17, (2001), 183-190.
- I. Beg, An iteration scheme for asymptotically nonexpansive mappings on uniformly convex metric spaces, Nonlinear Analysis Forum, vol. 6, no. 1, (2001), 27-34.
- I. Bula, Strictly convex metric spaces and fixed points, Mathematica Moravica, vol. 3, (1999), 5-16.
- I. Yildirim, S. H. Khan and M. Ozdemir, Some fixed point results for uniformly Quasi-Lipschitzian mappings in convex metric spaces, Journal of Nonlinear Analysis and Optimization, vol. 4, (2013), 143-148.
- I. Beg, M. Abbas and J. K. Kim, Convergence theorems of the iterative schemes in convex metric spaces: Nonlinear Funct. Anal. & Appl., vol. 11, no. 3, 421-436.
- I. Beg and M. Abbas, Common fixed points and best approximation in convex metric spaces, Soochow Journal of Mathematics, vol. 33, no. 4, (2007), 729-738.
- I. Beg, Structure of the set of fixed points of nonexpansive mappings in convex metric spaces, Ann. Univ. Curie-Sklodowska Sec. vol. 52, (1998), 7-14.
- J. B. Baillon, Nonexpansive mappings and Hyperconvex spaces, Contemp. Math., vol. 72, (1988), 11-19.
- J. P. Penot, Fixed point theorems without convexity, Bull. Soc. Math. France, vol. 60, (1979), 129-152.
- K. Geobel and W.A. Kirk, Iteration processes for nonexpansive mapping in topological methods in nonlinear functional Analysis (Toronto, Canada, 1982), American Mathematical Society of contemporary Mathematics, Providence, RI, USA, and vol. 21, (1983), 115-123.
- K. P. R. Sastry, G. V. R. Babu and Ch. Srinivasa Rao, Convergence of Ishikawa iteration scheme for a nonlinear quasi-contractive pair of selfmaps of convex metric spaces, Indian J. pure appl. Math., vol. 33, no. 2, (2002), 203-214.
- L.P. Belluce and W.A. Kirk, Fixed-point theorems for families of contraction mappings, Pacific J. Math., vol. 18, (1966), 213-218.
- Lj. B. Ciric, A generalizations of Banach’s contraction principle. Proc. Amer. Math.,vol. 45, (1974), 267-273.
- Lj. B. Ciric, on some discontinuous fixed point theorems on convex metric spaces, Czechoslovak Math. J., vol. 43, (1993), 319-326.
- Lj. B. Ciric et al, on the convergence of the Ishikawa iterates to a common fixed point of two mappings, Archivum Mathematicum (Brno), Tomus, vol. 39, (2003), 123-127.
- M. Edelstein, on nonexpansive mapping of Banach spaces, Proc. Camb. Phil. Soc., vol. 60, (1964), 439-447.
- M.A. Khamsi, M. Lin and R. Sine: On the fixed points of commuting nonexpansive maps in hyperconvex spaces. J. Math. Anal. Appl., vol. 168, (1992), 372-380.
- M. O. Olatinwo, Convergence results for Jungck-type iterative processes in convex metric spaces. Acta Univ. Palacki. Olomuc. Fac. rer. nat., Mathematica, vol. 51, no. 1, (2012), 79-87.
- M.O. Olatinwo, Stability results for some fixed point iterative processes in convex metric spaces, Int. journal of engineering, vol. 3, (2011), 103-106.
- M. R. Taskovic, General convex topological spaces and fixed points, Math. Moravica, vol. 1, (1997), 127-134.
- M. Lin and R. Sine, on the fixed point set of order preserving maps, Math. Zeit. vol. 203, (1990), 227-234.
- M. O. Olatinwo, Convergence and stability results for some iterative schemes, Acta Universities Apulensis, vol. 26, (2011), 225-236.
- M. D. Guay, K. L. Singh and J. H. M. Whitfield, Fixed point theorems for nonexpansive mappings in convex metric spaces, In: S. P.Singh J. H. Barry (eds.) Proceedings of Conference on Nonlinear Analysis, vol. 80, (1982), 179-189.
- M. A. Noor, New approximation schemes for general variational inequalities, Journal of Mathematical Analysis and applications, vol. 251, no. 1, (2000), 217-229.
- N. Aronszajn and P. Panitchpakdi, Extensions of uniformly continuous transformations and hyperconvex metric spaces, Pac. J. Math., vol. 6, (1956), 405-439.
- O. Hadzic, A common fixed point theorem for a family of mappings in convex metric spaces, Univ. u Novom Sadu, vol. 20, (1990), 89-95.
- O. Hadzic, Some common fixed point theorems in convex metric spaces, Univ. u Novom Sadu, Zb. Prirod.- Mat. Fak. Ser. Mat., vol. 15, no. 2, (1985), 1-13.
- P. Kuhfitting, Fixed points of several classes of non-linear mapping in Banach space, Proc. Amer. Math. Soc., vol. 44, (1974), 300-306.
- P. Soardi, Existence of fixed points of nonexpansive mappings in certain Banach lattices, Proc. Am. Math. Soc., vol. 73, (1979), 25-29.
- Q. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings, Journal of Mathematical Analysis and applications, vol. 207, no. 1, (1997), 96-103.
- Q. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings with error member, Journal of Mathematical Analysis and Applications, vol. 259, no. 1, (2001), 18-24.
- Q. Liu, Iterative sequences for asymptotically quasi-nonexpansive mapping with an error member of uniform convex Banach space. Journal of Mathematical Analysis and Applications, vol. 266, no. 2, (2002), 468-471.
- Q. Li-Hua and Y. Si-Sheng, Convergence of the Projection type Ishikawa iteration process with errors for a finite family of I-Asymptotically non-expansive mappings in Generalized Convex metric spaces, International Mathematical Forum, vol. 5, no. 43, 2010, 2103-2109.
- R. Chugh., V. Kumar and S. Kumar, Strong convergence of a new three step Iterative scheme in Banach spaces. American Journal of Computational Mathematics, vol. 2, (2012), 345-357.
- R. Sine, Hyperconvexity and approximate fixed points, Nonlinear Anal., vol. 13, (1989), 863-869.
- R. Sine, Hyperconvexity and nonexpansive multifunctions, Trans. Am. Math. Soc., vol. 315, (1989), 755-767.
- R. Sine, on nonlinear contraction semigroups in sup norm spaces, Nonlinear Anal., vol. 3, (1979), 885-890.
- R. P. Agarwal, D. O’Regan and D. R. Sahu, Fixed point theory for Lipschitzian-type mappings with applications, Topological Fixed Point Theory and Its Applications, vol. 6, (2009).
- R. P. Agarwal, D. O’Regan and D. R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mapping, Journal of Nonlinear and convex Analysis, vol.8, no. 1, (2007), 61-79.
- R. Chugh, Preety and M. Aggarwal, Some convergence results for modified S-iterative scheme in Hyperbolic spaces, International Journal of Computer Applications, vol. 80, no. 6, (2013), 20-23.
- S. Sharma and B. Deshpande, Common fixed point theorems of Gregus type in convex metric spaces, Mathematica Moravica, vol. 6, (2002), 77-85.
- S. Sharma and B. Deshpande, Discontinuity and weak compatibility in fixed point consideration of Gregus type in convex metric spaces, Fasciculi Mathematici, vol. 36, (2005), 91-100.
- S. A. Naimpally and K. L. Singh, Extensions of some fixed point theorems of Rhoades, J. Math. Anal. Appl., vol. 96, (1983), 437-446.
- S. Ishikawa, Fixed point by a new iteration method, Proc. Amer. Math. Soc., vol. 44, no. 1, (1974), 147-150.
- S. S. Chang, J. K. Kim and D. S. Jin, Iterative sequences with errors for asymptotically quasi-nonexpansive type mappings in convex metric spaces, Archives of inequality and Applications, vol. 2, (2004), 365-374.
- T. Shimizu and W. Takahashi, Fixed point theorems in certain convex metric spaces, Math. Japon., vol. 37, (1992), 855-859.
- T. Shimizu and W. Takahashi, Fixed points of multivalued mappings in certain convex metric spaces, Topological Methods in Nonlinear Analysis, vol. 8, (1996), 197-203.
- T. Zamfirescu, Fix point theorems in metric spaces, Arch. Math., vol. 23, (1972), 292-298.
- U. Kohlenbach, Some logical metatheorems with applications in functional analysis, Trans. Amer. Math. Soc., vol. 357, (2005), 89-128.
- V. Berinde, Some remarks on a fixed point theorem for Ciric-type almost contractions, Carpathian J. Math., vol. 25, no. 2, (2009), 157-162.
- V. Berinde, On the convergence of Mann iteration for a class of quasi-contractive operators, North University of Baia Mare,Preprint, (2003).
- V. Berinde, on the convergence of the Ishikawa iteration in the class of quasicontractive operators, Acta Math. Univ. Comenianae, vol. 73, no. 1, (2004), 119-126.
- V. Berinde, A convergence theorem for Mann iteration in the class of Zamfirescu operators, Analele Universitatic de vest, Timisoara, Seria Mathematica-Informatica, vol. 45, no. 1, (2007), 33-44.
- V. Berinde, Iterative approximation of fixed points, Springer-Verlog Berlin Heidelberg, (2007).
- W.G. Doston, Fixed points of quasi-nonexpansive mapping, J. Austral. Math. Soc., vol. 13, (1972), 167-170.
- W.A. Kirk, Krasnoselskij’s iteration processes in hyperbolic space, Numerical functional Analysis and optimization, vol. 4, (1982), 371-381.
- W.A. Kirk, An abstract fixed point theorem for nonexpansive mappings, Proc. Amer. Math. Soc., vol. 82, (1981), 640-642.
- W.A. Kirk, Fixed point theory for nonexpansive mappings II. Contemporary Math., vol. 18, (1983), 121-140.
- W. Takahashi, A convexity in metric space and nonexpansive mappings, I, Kodai Math. Sem. Rep., vol. 22, (1970), 142-149.
- W. Kirk and B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal., vol. 68, (2008), 3689-3696.
- W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., vol. 44, (1953), 506-510.
- W. Phuegrattana and S. Suantai, On the rate of convergence of Mann, Ishikawa, Noor and SP iterations for continuous function on an arbitrary interval, Journal of Computational and Applied Mathematics, vol. 235, (2011), 3006-3014.
- X. Zhiqun,L.Guiwen and B. E. Rhoades, On Equivalence of some iterations convergence for Quasi-Contraction maps in convex metric spaces, Fixed Point Theory and Applications, vol. 2010, (2010), 1-10.
- Y. X. Tian and Chun-de Yang, Convergence theorems of three-step iterative scheme for a finite family of uniformly Quasi-Lipschitzian mappings in Convex metric spaces, Fixed Point Theory and Applications, vol. 2009, (2009), 1-12.
- Y. X. Tian, Convergence of an Ishikawa type Iterative scheme for asymptotically quasi- nonexpansive mapping. Computer and Mathematics with Applications, vol. 49, (2005), 1905-1912.
- Ya. I. Alber and S. Guwrre-Delabriere, Principles of weakly contractive maps in Hilbert spaces, new results in operator theory, Advances and Appl. Birkhauser Verlag, Basel, vol. 98, (1976), 7-22.
- Y. Xu, Ishikawa and Mann iteration process with errors for nonlinear strongly accretive operator equations, J. Math. Anal. Appl., vol. 224, (1998), 91-101.
- Z. Opial, Weak convergence of the sequence of successive approximation for nonexpansive mapping, Bull. Amer. Math. Soc., vol. 73, (1967), 591-597.
-
Downloads
-
How to Cite
Chugh, R., & Malik, P. (2014). Convergence and fixed point theorems in convex metric spaces : a survey. International Journal of Applied Mathematical Research, 3(2), 133-160. https://doi.org/10.14419/ijamr.v3i2.2204Received date: 2014-03-13
Accepted date: 2014-04-05
Published date: 2014-04-13