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Weak c-Comparison Function and a Fixed Point Theorem in Cone Metric Spaces over Banach Algebras

Year 2019, Volume: 2 Issue: 1, 25 - 33, 23.07.2019

Muttalip Özavşar , Faruk Develi

Abstract

In this paper, we first give the notion of weak c-comparison function, and prove some associated fixed point theorems containing classes of many well known theorems in the setting of cone metric spaces over Banach algebras with solid cones.

Keywords

Cone Metric Space Banach Algebras

References

  • S. Banach, \textit{Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales}. Fundam. Math., 1922 \textbf{(3)}: 133–181.
  • R. Kannan, \textit{Some results on fixed points}. Bull.Calcultta Math.Soc., 1968 \textbf{(10)}: 71-76.
  • S.K. Chatterjea, \textit{Fixed Point Theorems}. C.R.Acad.Bulgare Sci., 1972 \textbf{(25)}: 727-730.
  • T. Zamfirescu, \textit{Fixed Point theorems in metric spaces}. Arch.Math.(Basel), 1972 \textbf{(23)}: 292-298.
  • V. Berinde, \textit{Approximating fixed points of weak contractions using the Picard iterations}. Nonlinear Anal. Forum, 2004 \textbf{(9)}: 43-53.
  • M.A Alghamdi, V. Berinde and N. Shahzad, \textit{Fixed points of non-self almost contractions}. Carp. J.Math., 2014 \textbf{(30)}: 7-14.
  • V. Berinde and M. Pacurar, \textit{Fixed points and continuity of almost contractions}. Fixed Point Theory, 2008 \textbf{(9)}: 23-34.
  • T. Suzuki, \textit{Fixed point theorems for Berinde mappings}. Bull. Kyushu Inst. Tech. Pure Appl. Math., 2011 \textbf{(58)}: 13-1
  • J. Tiammee, Y.J. Cho and S. Suanta, \textit{Fixed point theorems for nonself G-almost contractive mappings in Banach spaces endowed with graphs}. Carpathian J.Math., 2016 \textbf{(32)}: 375-382.
  • L.G. Huang and X. Zhang, \textit{Cone metric spaces and fixed point theorems of contractive mappings}. J. Math. Anal. Appl.,2007 \textbf{(332)}: 1468-1476.
  • M. Abbas, P. Vetro and S.H. Khan, \textit{On fixed points of Berinde's contractive mappings in cone metric spaces}. Carp.J.Math., 2010 \textbf{(26)}: 121-133.
  • W.S. Du, \textit{A note on cone metric fixed point theory and its equivalence}. Nonlinear Anal., 2010 \textbf{(72)}: 2259-2261.
  • S. Radenovic, S. Simic, N. Cakic and Z. Golubovic, \textit{A note on tvs-cone metric fixed point theory}. Math. Comp. Mod., 2011 \textbf{(54)}: 2418-2422.
  • H. Liu and S. Xu \textit{Cone metric spaces with Banach algebras and fixed point theorems of generalized Lipschitz mappings}. Fix. P. Theory Appl., 2013, 10 pages.
  • H. Huang and S. Radenovic, \textit{Common fixed point theorems of generalized Lipschitz mappings in cone b-metricspaces over Banach algebras and applications}. J. Nonlinear Sci. Appl., 2015 \textbf{(8)}: 787-799.
  • S. Shukla, S. Balasubramanian and M. Pavlovic, \textit{A Generalized Banach Fixed Point Theorem}. Bull. Malays. Math. Sci. Soc., 2016 \textbf{(39)}: 1529-1539.
  • S. Xu and S. Radenovic, \textit{Fixed point theorems of generalized Lipschitz mappings on cone metric spaces over Banach algebras without assumption of normality}. Fix. P. Theory Appl., 2014, 12 pages.
  • M. Özavşar, \textit{Fixed point theorems for (k,l)-almost contractions in cone metric spaces over Banach algebras}. Math. Adv. Pur. Appl. Sci., 2018 \textbf{(1)}: 46-51.
  • W. Rudin, \textit{Functional Analysis}, 2nd edn. McGraw-Hill, New York 1991.
  • Sh. Rezapour and R. Hamlbarani, \textit{Some notes on the paper cone metric spaces and fixed point theorems of contractive mappings}. J. Math. Anal. Appl., 2008 \textbf{(345)}: 719–724.
  • S. Radenovic and B.E. Rhoades, \textit{Fixed point theorem for two non-self mappings in cone metric spaces}. Comput. Math.Appl., 2009 \textbf{(57)}: 1701-1707. Z.
  • Kadelburg and S. Radenovic, \textit{A note on various types of cones and fixed point results in cone metric spaces}. Asian J. Math. Appl., 2013, Article ID ama0104, 7 pages.
  • S. Jankovic, Z. Kadelburg and S. Radenovic, \textit{On cone metric spaces: a survey}. Nonlinear Anal., 2011 \textbf{(74)}: 2591-2601.
  • V. Berinde, \textit{Iterative Approximation of Fixed Points}. Editura Efemeride, Baia Mare, 2002.
  • V. Berinde, \textit{Approximating fixed points of weak $\varphi$-contractions using the Picard iterations}. Fixed Point Theory, 2003, 131-147.
  • B. Li and H. Huang, \textit{Fixed point results for weak $\varphi$-contractions in cone metric spaces over Banach algebras and applications}. Journal of Function Spaces, 2017, 6 pages.
  • H. Huang, G. Deng and S. Radenovic, \textit{Some topological properties and fixed point results in cone metric spaces over Banach algebras}. Positivity, 2019.

Year 2019, Volume: 2 Issue: 1, 25 - 33, 23.07.2019

Muttalip Özavşar , Faruk Develi

Abstract

References

  • S. Banach, \textit{Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales}. Fundam. Math., 1922 \textbf{(3)}: 133–181.
  • R. Kannan, \textit{Some results on fixed points}. Bull.Calcultta Math.Soc., 1968 \textbf{(10)}: 71-76.
  • S.K. Chatterjea, \textit{Fixed Point Theorems}. C.R.Acad.Bulgare Sci., 1972 \textbf{(25)}: 727-730.
  • T. Zamfirescu, \textit{Fixed Point theorems in metric spaces}. Arch.Math.(Basel), 1972 \textbf{(23)}: 292-298.
  • V. Berinde, \textit{Approximating fixed points of weak contractions using the Picard iterations}. Nonlinear Anal. Forum, 2004 \textbf{(9)}: 43-53.
  • M.A Alghamdi, V. Berinde and N. Shahzad, \textit{Fixed points of non-self almost contractions}. Carp. J.Math., 2014 \textbf{(30)}: 7-14.
  • V. Berinde and M. Pacurar, \textit{Fixed points and continuity of almost contractions}. Fixed Point Theory, 2008 \textbf{(9)}: 23-34.
  • T. Suzuki, \textit{Fixed point theorems for Berinde mappings}. Bull. Kyushu Inst. Tech. Pure Appl. Math., 2011 \textbf{(58)}: 13-1
  • J. Tiammee, Y.J. Cho and S. Suanta, \textit{Fixed point theorems for nonself G-almost contractive mappings in Banach spaces endowed with graphs}. Carpathian J.Math., 2016 \textbf{(32)}: 375-382.
  • L.G. Huang and X. Zhang, \textit{Cone metric spaces and fixed point theorems of contractive mappings}. J. Math. Anal. Appl.,2007 \textbf{(332)}: 1468-1476.
  • M. Abbas, P. Vetro and S.H. Khan, \textit{On fixed points of Berinde's contractive mappings in cone metric spaces}. Carp.J.Math., 2010 \textbf{(26)}: 121-133.
  • W.S. Du, \textit{A note on cone metric fixed point theory and its equivalence}. Nonlinear Anal., 2010 \textbf{(72)}: 2259-2261.
  • S. Radenovic, S. Simic, N. Cakic and Z. Golubovic, \textit{A note on tvs-cone metric fixed point theory}. Math. Comp. Mod., 2011 \textbf{(54)}: 2418-2422.
  • H. Liu and S. Xu \textit{Cone metric spaces with Banach algebras and fixed point theorems of generalized Lipschitz mappings}. Fix. P. Theory Appl., 2013, 10 pages.
  • H. Huang and S. Radenovic, \textit{Common fixed point theorems of generalized Lipschitz mappings in cone b-metricspaces over Banach algebras and applications}. J. Nonlinear Sci. Appl., 2015 \textbf{(8)}: 787-799.
  • S. Shukla, S. Balasubramanian and M. Pavlovic, \textit{A Generalized Banach Fixed Point Theorem}. Bull. Malays. Math. Sci. Soc., 2016 \textbf{(39)}: 1529-1539.
  • S. Xu and S. Radenovic, \textit{Fixed point theorems of generalized Lipschitz mappings on cone metric spaces over Banach algebras without assumption of normality}. Fix. P. Theory Appl., 2014, 12 pages.
  • M. Özavşar, \textit{Fixed point theorems for (k,l)-almost contractions in cone metric spaces over Banach algebras}. Math. Adv. Pur. Appl. Sci., 2018 \textbf{(1)}: 46-51.
  • W. Rudin, \textit{Functional Analysis}, 2nd edn. McGraw-Hill, New York 1991.
  • Sh. Rezapour and R. Hamlbarani, \textit{Some notes on the paper cone metric spaces and fixed point theorems of contractive mappings}. J. Math. Anal. Appl., 2008 \textbf{(345)}: 719–724.
  • S. Radenovic and B.E. Rhoades, \textit{Fixed point theorem for two non-self mappings in cone metric spaces}. Comput. Math.Appl., 2009 \textbf{(57)}: 1701-1707. Z.
  • Kadelburg and S. Radenovic, \textit{A note on various types of cones and fixed point results in cone metric spaces}. Asian J. Math. Appl., 2013, Article ID ama0104, 7 pages.
  • S. Jankovic, Z. Kadelburg and S. Radenovic, \textit{On cone metric spaces: a survey}. Nonlinear Anal., 2011 \textbf{(74)}: 2591-2601.
  • V. Berinde, \textit{Iterative Approximation of Fixed Points}. Editura Efemeride, Baia Mare, 2002.
  • V. Berinde, \textit{Approximating fixed points of weak $\varphi$-contractions using the Picard iterations}. Fixed Point Theory, 2003, 131-147.
  • B. Li and H. Huang, \textit{Fixed point results for weak $\varphi$-contractions in cone metric spaces over Banach algebras and applications}. Journal of Function Spaces, 2017, 6 pages.
  • H. Huang, G. Deng and S. Radenovic, \textit{Some topological properties and fixed point results in cone metric spaces over Banach algebras}. Positivity, 2019.
There are 27 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Muttalip Özavşar

Faruk Develi

Publication Date July 23, 2019
Published in Issue Year 2019 Volume: 2 Issue: 1

Cite

APA Özavşar, M., & Develi, F. (2019). Weak c-Comparison Function and a Fixed Point Theorem in Cone Metric Spaces over Banach Algebras. Mathematical Advances in Pure and Applied Sciences, 2(1), 25-33.
AMA Özavşar M, Develi F. Weak c-Comparison Function and a Fixed Point Theorem in Cone Metric Spaces over Banach Algebras. MAPAS. July 2019;2(1):25-33.
Chicago Özavşar, Muttalip, and Faruk Develi. “Weak C-Comparison Function and a Fixed Point Theorem in Cone Metric Spaces over Banach Algebras”. Mathematical Advances in Pure and Applied Sciences 2, no. 1 (July 2019): 25-33.
EndNote Özavşar M, Develi F (July 1, 2019) Weak c-Comparison Function and a Fixed Point Theorem in Cone Metric Spaces over Banach Algebras. Mathematical Advances in Pure and Applied Sciences 2 1 25–33.
IEEE M. Özavşar and F. Develi, “Weak c-Comparison Function and a Fixed Point Theorem in Cone Metric Spaces over Banach Algebras”, MAPAS, vol. 2, no. 1, pp. 25–33, 2019.
ISNAD Özavşar, Muttalip - Develi, Faruk. “Weak C-Comparison Function and a Fixed Point Theorem in Cone Metric Spaces over Banach Algebras”. Mathematical Advances in Pure and Applied Sciences 2/1 (July 2019), 25-33.
JAMA Özavşar M, Develi F. Weak c-Comparison Function and a Fixed Point Theorem in Cone Metric Spaces over Banach Algebras. MAPAS. 2019;2:25–33.
MLA Özavşar, Muttalip and Faruk Develi. “Weak C-Comparison Function and a Fixed Point Theorem in Cone Metric Spaces over Banach Algebras”. Mathematical Advances in Pure and Applied Sciences, vol. 2, no. 1, 2019, pp. 25-33.
Vancouver Özavşar M, Develi F. Weak c-Comparison Function and a Fixed Point Theorem in Cone Metric Spaces over Banach Algebras. MAPAS. 2019;2(1):25-33.