Fixed Point, Coincidence Point and Common Fixed Point Theorems under Various Expansive Conditions in Parametric Metric Spaces and Parametric b-Metric Spaces

Yıl 2016, Cilt: 29 Sayı: 1, 95 - 107, 21.03.2016

Rashmi Jain , R. D. Daheriya Manoj Ughade

Öz

In this article, we establish some fixed, common fixed and coincidence point theorems for expansive type mappings in parametric metric spaces and parametric b-metric spaces. The presented theorems extend, generalize and improve many existing results in the literature. Also, we introduce some examples the support the validity of our results.

Anahtar Kelimeler

Coincidence point, fixed point, common fixed point, parametric metric space, parametric b-metric space, weakly compatible maps.

Kaynakça

• L. A. Zadeh, “Fuzzy sets,” Information and Computation, vol. 8, pp. 338–353, 1965.
• I. Kramosil and J. Michalek, “Fuzzy metrics and statistical metric spaces,” Kybernetika, vol. 11, no. 5, pp. 336–344, 1975.
• M. Grabiec, “Fixed points in fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 27, no. 3, pp. 385–389, 1988.
• A. George and P. Veeramani, “On some results in fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 64, no. 3, pp. 395–399, 1994.
• C. Di Bari and C. Vetro, “Fixed points, attractors and weak fuzzy contractive mappings in a fuzzy metric space,” Journal of Fuzzy Mathematics, vol. 13, no. 4, pp. 973–982, 2005.
• D. Gopal, M. Imdad, C. Vetro, and M. Hasan, “Fixed point theory for cyclic weak ϕ-contraction in fuzzy metric spaces,” Journal of Nonlinear Analysis and Application, vol. 2012, Article ID jnaa-00110, 11 pages, 2012.
• P. Salimi, C. Vetro, and P. Vetro, “Some new fixed point results in non-Archimedean fuzzy metric spaces,” Nonlinear Analysis: Modelling and Control, vol. 18, no. 3, pp. 344–358, 2013.
• Y. Shen, D. Qiu, and W. Chen, “Fixed point theorems in fuzzy metric spaces,” Applied Mathematics Letters, vol. 25, no. 2, pp. 138–141, 2012.
• C. Vetro, “Fixed points in weak non-Archimedean fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 162, pp. 84–90, 2011.
• C. Vetro, D. Gopal, and M. Imdad, “Common fixed point theorems for (ϕ,ψ)-weak contractions in fuzzy metric spaces,” Indian Journal of Mathematics, vol. 52, no. 3, pp. 573–590, 2010.
• C. Vetro and P. Vetro, “Common fixed points for discontinuous mappings in fuzzy metric spaces,” Rendiconti del Circolo Matematico di Palermo, vol. 57, no. 2, pp. 295–303, 2008.
• J. H. Park, “Intuitionistic fuzzy metric spaces,” Chaos, Solitons and Fractals, vol. 22, no. 5, pp. 1039–1046, 2004.
• C. Alaca, D. Turkoglu, and C. Yildiz, “Fixed points in intuitionistic fuzzy metric spaces,” Chaos, Solitons & Fractals, vol.29, no. 5, pp.1073–1078, 2006.
• D. Coker, “An introduction to intuitionistic fuzzy topological spaces,” Fuzzy Sets and Systems, vol. 88, no. 1, pp. 81–89, 1997.
• J. S. Park, Y. C. Kwun, and J. H. Park, “A fixed point theorem in the intuitionistic fuzzy metric spaces,” Far East Journal of Mathematical Sciences, vol. 16, no. 2, pp. 137–149, 2005.
• M. Rafi and M. S. M. Noorani, “Fixed point theorem on intuitionistic fuzzy metric spaces,” Iranian Journal of Fuzzy Systems, vol. 3, no. 1, pp. 23–29, 2006.
• B. Schweizer and A. Sklar, “Statistical metric spaces,” Pacific Journal of Mathematics, vol. 10, pp. 313–334, 1960.
• S. Manro, S. Kumar and S.S. Bhatia, Common fixed point theorems for weakly compatible maps satisfying common (E.A) like property in intuitionistic fuzzy metric spaces using implicit relation, Journal of the Indian Math. Soc., 81(1-2) (2014), 123-133.
• C. Ionescu, S. Rezapour, and M. E. Samei, “Fixed points of some new contractions on intuitionistic fuzzy metric spaces,” Fixed Point Theory and Applications, vol. 2013, article 168, 2013.
• N. Hussain, S. Khaleghizadeh, P. Salimi, and Afrah A. N. Abdou, “A New Approach to Fixed Point Results in Triangular Intuitionistic Fuzzy Metric Spaces”, Abstract and Applied Analysis, Volume 2014, Article ID 690139, 16 pages.
• N. Hussain, P. Salimi, V. Parvaneh, Fixed point results for various contractions in parametric and fuzzy b-metric spaces, J. Nonlinear Sci. Appl. 8 (2015), 719-739.
• Jungck, G., Commuting mappings and fixed points. Amer. Math. Monthly, 83(1976), 261-263.
• G. Jungck, “Compatible mappings and common fixed points,” International Journal of Mathematics and Mathematical Sciences, vol. 9, no.4, pp. 771-779, 1986.
• G. Jungck, “Fixed points for non-continuous non-self mappings on non-metric space,” Far East Journal of Mathematical Sciences, vol. 4, pp. 199-212, 1996.
• S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inf. Univ. Ostravensis, 1 (1993), 5-11.
• S. Czerwik, Nonlinear set-valued contraction mappings in b-metric spaces, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 263-276.
• M. A. Alghamdi, N. Hussain, P. Salimi, Fixed point and coupled fixed point theorems on b-metric-like spaces, J. Inequal. Appl., 2013 (2013), 25 pages.
• N. Hussain, V. Parvaneh, J. R. Roshan, Z. Kadelburg, Fixed points of cyclic weakly (ψ,φ,L,A,B)-contractive mappings in ordered b-metric spaces with applications, Fixed Point Theory Appl., 2013 (2013), 18pages.
• A. S. Saluja, A. K. Dhakda and D. Magarde, Some fixed point theorems for expansive type mapping in dislocated metric space, Mathematical Theorey and Modeling, 3(2013).
• R. D. Daheriya, Rashmi Jain, and Manoj Ughade “Some Fixed Point Theorem for Expansive Type Mapping in Dislocated Metric Space”, ISRN Mathematical Analysis, Volume 2012, Article ID 376832, 5 pages, doi:10.5402/2012/376832.
• Yan Han and Shaoyuan Xu, Some new theorems of expanding mappings without continuity in cone metric spaces, Fixed Point Theory and Applications, 2013, 2013:3.
• Wasfi Shatanawi and Fadi Awawdeh, Some fixed and coincidence point theorems for expansive maps in cone metric spaces, Fixed Point Theory and Applications 2012, 2012:19.
• Xianjiu Huang, Chuanxi Zhu, Xi Wen, Fixed point theorems for expanding mappings in partial metric spaces, An. St. Univ. Ovidius Constant_a Vol. 20(1), 2012, 213-224.
• Z. Mustafa, F. Awawdeh and W. Shatanawi, Fixed Point Theorem for Expansive Mappings in G-Metric Spaces, Int. J. Contemp. Math. Sciences, Vol. 5, 2010, no. 50, 2463-2472.
• A. Muraliraj and R. Jahir Hussain, Coincidence and Fixed Point Theorems for Expansive Maps in d−Metric Spaces, Int. Journal of Math. Analysis, Vol. 7, 2013, no. 44, 2171 – 2179.
• P. Z. Daffer, H. Kaneko, On expansive mappings, Math. Japonica. 37 (1992), 733-735.
• S. Z. Wang, B. Y. Li, Z. M. Gao, K. Iseki, Some fixed point theorems for expansion mappings, Math. Japonica. 29 (1984), 631-636.
• Aage, CT, Salunke, JN: Some fixed point theorems for expansion onto mappings on cone metric spaces. Acta Math. Sin. Engl. Ser. 27(6), 1101-1106 (2011).
• G. Jungck, “Commuting mappings and fixed points,” The American Mathematical Monthly, vol. 83, no.4, pp. 261-263, 1976.
• M. A. Alghamdi, N. Hussain and P. Salimi, Fixed point and coupled fixed point theorems on b- metric-like spaces, Journal of Inequalities and Applications, 2013, 2013:402.
• R. D. Daheriya, Rashmi Jain, and Manoj Ughade, Unified fixed point theorems for non-surjective expansion mappings by using class