FIXED POINT FOR EXPANSION MAPPINGS IN CONE b-METRIC SPACE

ROHIT KUMAR VERMA

Abstract


Abstract. In this paper, we first prove a common fixed point theorem in cone b-metric space of a pair of weakly compatible mappings satisfying an extension mapping. Then we introduce T-extension condition in cone b- metric space. There are some mappings, say S:X→X, which have a fixed point but which can’t be obtained by an extension condition, i.e., which do not satisfy an extension condition. If another mapping T: X→X is introduced, then extension condition holds. In this case S is said to satisfy T-extension condition. We prove our second fixed point theorem using T-extension condition in cone b-metric space.


Full Text:

PDF

References


C. D. Aliprantis and R. Tourky, Cones and Duality, Graduate Studies in Mathematics, Vol. 84, American Mathematical Society,Providence, Rhode Island, 2007.

Anil Kumar and Savita Rathee, Fixed point and common fixed point results in cone metric space and application to invariant approximation, Fixed Point Theory and Applications (2015) 2015:45 DOI 10.1186/s13663-015-0290-9

M. Abbas, G. Jungck, Common fixed point results for non-commuting mappings without continuity in cone metric spaces, J. Math. Anal. Appl., 341 (2008) 416-420.

M. Abbas, B. E. Rhoades, Fixed and periodic point results in cone metric spaces, Appl. Math. Lett. 22 (2009) 511-515.

A. G. B. Ahmad, Z. M. Fadail, M. Abbas, Z. Kadelburg, and S. Radenovic, Some Fixed and Periodic Points in Abstract Metric Spaces, Abstract Apppl. Anal., Vol. 2012, Article ID 908423.

M.A. Ahmed, Common fixed point theorems for set valued and single-valued functions, Demonstratio Math., 36(2012), 471 - 481.

M.A. Alghamdi et al., Fixed point and couple fixed point theorems in b-metric like space, Jour. Inequality & Appl., (2013), 2013:402.

A. Alghamdi Mohammad, H. A. Shahrazad, S. Radenovic and N. Shahzad, Fixed point theorems for convex contraction mappings on cone metric spaces, Math. Comput. Modell., 54 (2011), no. 9-10, 2022-2026.

Azam and Arsad, Common fixed points of generalized contractive maps in cone metric spaces, Bulletin of the Iranian Mathematical Society 35(2)(2009), 255-264.

S. Banach, Sur les op´erations dans les ensembles abstraits et leur application aux ´equations int´egrales., Fund. Math. 3(1922), 133–181.

I. A. Bakhtin, The contraction mapping principle in almost metric spaces, Funct.Anal., Gos. Ped. Inst. Unianowsk, 30(1989), 26-37.

K. P. Chi, E. Karapinar, T. D. Thanh, A generalization of the Meir-Keeler type contraction, Arab Journal of Mathematical Sciences, 18(2012), 141-148.

K.P. Chi, On a fixed point theorem for certain class of maps satisfying a contractive condition depended on an another function, Lobachevskii J. Math. 30(4) (2009), 289–291.

K.P. Chi, H.T. Thuy, A fixed point theorem in 2-metric spaces for a class of maps that satisfy a contractive condition dependent on another function, Lobachevskii J. Math. 31(4) (2010), 338–346.

S. H. Cho and J. S. Bae, Common fixed point theorems for mappings satisfying property (E.A) on cone metric space, Math. Comput. Model. 53(2011), 945-951.

S. Czerwik, Contraction mapping in b-metric spaces, Acta Math. Inform. Univ. Ostrav. 1(1993), 5-11.

A. S. Cvetkovic, M. P. Stanic, S. Dimitrijevic, and S. Simic, Common fixed point theorems for four mappings on cone metric type space, Fixed Point Theory Appl., Vol. 2011, Article ID 589725.

A. K. Dubey, Rohit Verma and R.P.Dubey , On generalization of some coincidence and common fixed point theorems for three self mappings in cone metric spaces, Inter. Jour. of Math. Trends and Tech., 11(2), Jul-2014

A. K. Dubey, Rita Shukla and R. P. Dubey, Cone metric spaces and fixed point theorems of generalized T-Zamfirescu mappings, International Journal of Applied Mathematical Research, 2(1)(2013), 151-156.

M. Demmaa, Reza Saadati, P. Vetro, Fixed point results on b-metric space via picard sequences and b-simulation functions, Iranian Journal of Mathematical Sciences and Informatics, 11(1)(2016), 123–136 DOI: 10.7508/ ijmsi.2016.01.011

H. Huang and Shaoyuan Xu, Fixed point theorems of contractive mappings in cone b-metric spaces and applications, Fixed Point Theory and Applications 2013, 2013:112, http://www.fixedpointtheoryandapplications.com/content/2013/1/112

H. Huang, S. Xu, Fixed point theorems of contractive mappings in cone b-metric spaces and applications, Fixed Point Theory & Appl., 2012, 2012:220.

L. G. Huang and X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332(2)(2007), 1468-1476.

X. Huang, Chuanxi Zhu and Xi Wen, Fixed point theorems for expanding mappings in cone metric spaces, math. reports 14(64), 2(2012), 141–148.

N. Hussain and M. H. Shah, KKM mappings in cone b-metric spaces, Computers and Mathematics with Applications 62(2011), 1677-1684, www.elsevier.com/locate/camwa

S. Jankovi´c, Z. Kadelburg, and S. Radenovi´c, On cone metric spaces: a survey, Nonlinear Anal. 4(7)(2011), 2591-2601.

M. Jovanovic, Z.Kadelburg and S. Redenovic, Common Fixed Point Results in Metric-Type Space, Hindawi Publishing Corp. Fixed Point Theorey and Applications, (2010), Article ID 978121.

G. Jungck, Compatible mappings and common fixed points, Int. Journal of Mathematics & Mathematical Sciences 9(4)(1986), 771-779.

G. Jungck, Common fixed points for non-continuous non-self mappings on a non-numeric spaces, Far East J. Math. Sci. 4(2)(1996), 199-212.

M. A. Khamsi, Remarks on cone metric spaces and fixed point theorems of contractive mappings, Fixed Point Theory and Applications, Volume 2010, Article ID 315398, 7 pages, doi:10.1155/2010/315398.

M.A. Khamsi, N. Hussain, KKM mappings in metric type spaces, Nonlinear Anal. 73 (2010) 3123–3129.

D. Ilic, V. Rakocevic, Common fixed points for maps on cone metric space, J. Math. Anal. Appl., 341 (2008) 876-882.

M. Kir, H. Kizitune, On some well known fixed point theorems in b-metric space, Turkish Journal of Analysis and Number Theory, 1(2013), 13-16.

R. Maitra, Caccioppoli type fixed point result in cone b-metric space, IJISET - International Journal of Innovative Science, Engineering & Technology, 1(4), June 2014, www.ijiset.com, ISSN 2348–7968.

S. K. Mohanta, coincidence points and common fixed points for expansive type mappings in b-metric spaces, Iranian Journal of Mathematical Sciences and Informatics, 11(1)(2016), 101-113, DOI: 10.7508/ ijmsi.2016.01.009

H. Mohebi, Topical functions and their properties in a class of ordered Banach spaces, Part II of the book "Continuous Optimization, Curent Trends and Modern Applications", Springer, 2005 343-361.

S. Moradi, M. Omid, A fixed-point theorem for integral type inequality depending on another function, Int. J. Math. Anal. (Russ.) 4(29–32) (2010), 1491–1499.

M. Öztürk, N. Kaplan, Common fixed points of f-contraction mappings in complex valued metric spaces, Math. Sci. 8(2014)129, doi: 10.1007/s40096-014-0129-2.

M. P. Stanic, A. S. Cvetkovic, S. Simic, and S. Dimitrijevic, Common fixed point under contractive condition of Ciric's type on cone metric type spaces, Fixed Point Theory Appl.,2012, 2012:35.

B. Popovic, S. Radenovic and S. Shukla, fixed point results to tvs-cone b-metric spaces, Gulf Journal of Mathematics, vol 1 (2013) 51-64

S. Radenovic and Z. Kadelburg, Quasi-contractions on symmetric and cone symmetric spaces, Banach J. Math. Anal., 5(1) (2011), 38-50.

H. Rahimi, P. Vetro, and G. S. Rad, Some common fixed point results for weakly Compatible mappings in cone metric type space, Miskolc Mathematical Notes, 14(1) (2013), 233–243.

M. U.Rahman and M. Sarwar, Dislocated quasi b-metric space and fixed point theorems, electronic journal of mathematical analysis and applications,Vol. 4(2) July 2016, pp. 16-24, ISSN: 2090-729(online), http://fcag-egypt.com/Journals/EJMAA/

S. Rezapour and R. Hamlbarani, Some notes on the paper ‘Cone metric spaces and fixed point theorems of contractive mappings’ J. Math. Anal. Appl. 345(2008), 719-724.

B. E. Rhoades, A comparison of various definitions of contractive mappings, Transac. of the American Math. Soc., 336 (1977) 257-290.

B. Rzepecki, On fixed point theorems of Maia type, Publ. de l’Inst. Math. 28 (42) (1980) 179–186.

A. Saleh Al-Mezel, H. H. Alsulami, E. Karapınar and F. Khojasteh, A note on fixed point results in complex-valued metric spaces, Jour. of Inequalities and Applications (2015) 2015:33 DOI 10.1186/s13660-015-0550-6.

B. Samet, Common fixed point under contractive condition of Ciric's type in cone metric spaces, Appl. Anal. Discrete Math., 5 (2011) 159-164.

M. H. Shah, S. Simic, N. Hussain, A. Sretenovic, S. Radenovic, Common fixed points for occasionally weakly compatible pairs on cone metric type spaces, J. Comput. Anal. Appl.14 (2) (2012) 290-297.

H. A. Shahrazad, S. Radenovic, N. Shahzad, Fixed point theorems for mappings with convex diminishing diameters on cone metric spaces, Appl. Math. Lett., 24 (2011) no. 12, 2162-2166.

W. Shatanawi, Partially ordered cone metric spaces and coupled fixed point results, Comput. Math. Appl., 60 (2010) 2508-2515.

W. Shatanawi, On w-compatible mappings and common coupled coincidence point in cone metric spaces, Appl. Math. Lett., (2011), doi:10.1016/j.aml.2011.10.10.037

S. Simic, A note on Stone's, Baire's, Ky Fan's and Dugundji's theorem in tvs-cone metric spaces, Appl. Math. Lett., 24 (6) (2011) 999-1002.

M. P. Stanic, A. S. Cvetkovic, S. Simic, and S. Dimitrijevic, Common fixed point under contractive condition of Ciric's type on cone metric type spaces, Fixed Point Theory Appl.,2012, 2012:35.

S. K. Tiwari1, R. P. Dubey, A. K. Dubey, Cone metric spaces and common fixed point theorems for generalized multi-valued mappings, International Journal of Innovative Research in Science, Engineering and Technology, Vol. 2, Issue 10, October 2013.

D. Turkoglu, M. Abuloha, Cone metric spaces and fixed point theorems in diametrically contractive mappings, Acta Math. Sin. (Engl. Ser.) 26 (3) (2010), 489–496.

J. Vandergraft, Newton method for convex operators in partially ordered spaces, SIAM J. Numer. Anal., 4 (1947) 406-432.

P. Vetro, Common fixed points in cone metric spaces, Circ. Mat. Palermo 56 (2007) 464-468.

S. Z. Wang, B. Y. Li, Z. M. Gao and K. Iseki, Some fixed point theorems on expansion mappings, Math. Japonica 29 (1984), 631-636.

P. P. Zabreiko, K-metric and K-normed spaces: survey, Collectanea Math., 48, 4-6 (1997), 825-859.

X. Zhang, Fixed point theorem of generalized quasi-contractive mapping in cone metric spaces, Comput. Math. Appl., 62 (2011) 1627-1633.

S. Zhang, Fixed point theory and its applications, Chongqing press, Chongqing China, 1984.


Refbacks

  • There are currently no refbacks.


MAYFEB Journal of Mathematics 
Toronto, Ontario, Canada
MAYFEB TECHNOLOGY DEVELOPMENT
ISSN 2371-6193