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Year 2021, Volume: 5 Issue: 3, 421 - 432, 30.09.2021

Kalyani K Seshagiri Rao Namana Belay Mitiku

https://doi.org/10.31197/atnaa.758962
Cited By: 2

Abstract

References

  • [1] R.P. Agarwal, M.A. El-Gebeily and D. O'Regan, Generalized contractions in partially ordered metric spaces, Appl. Anal. 87(2008), 1-8. doi:10.1080/00036810701714164
  • [2] A. Aghajani and R. Arab, Fixed points of (ψ,φ,θ)-contractive mappings in partially ordered b-metric spaces and application to quadratic integral equations, Fixed Point Theory Appl. 2013:245(2013), 1-20.
  • [3] A. Aghajani, M. Abbas and J.R. Roshan, Common ?xed point of generalized weak contractive mappings in partially ordered b-metric spaces, Math. Slovaca 64(4)(2014), 941-960.
  • [4] I.A. Bakhtin, The contraction principle in quasimetric spaces, Func. An., Ulianowsk, Gos. Fed. Ins. 30(1989), 26-37.
  • [5] Belay Mitiku, K. Kalyani and N. Seshagiri Rao, Some fixed point results of generalized (φ,ψ)-contractive mappings in ordered b-metric spaces, BMC Research Notes 13:537(2020). doi:https://doi.org/10.1186/s13104-020-05354-1
  • [6] T.G. Bhaskar and V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65(2006), 1379-1393. doi:10.1016/j.na.2005.10.017
  • [7] I. Cabrera, J. Harjani and K. Sadarangani, A fixed point theorem for contractions of rational type in partially ordered metric spaces, Ann. Univ. 59(2013), 251-258.
  • [8] S. Chandok and E. Karapinar, Common fixed point of generalized rational type contraction mappings in partially ordered metric spaces, Thai J. Math. 11(2)(2013), 251-260.
  • [9] S. Chandok, B.S. Choudhury and N. Metiya, Fixed point results in ordered metric spaces for rational type expressions with auxiliary functions, Journal of the Egyptian Mathematical Society 23(2015), 95-101.
  • [10] Y.J. Cho, M.H. Shah and N. Hussain, Coupled fixed points of weakly F-contractive mappings in topological spaces, Appl. Math. Lett. 24(2011), 1185-1190. doi:10.1016/j.aml.2011.02.004
  • [11] L.J. Ciric, N. Cakic, M. Rajovic and J.S. Ume, Monotone generalized nonlinear contractions in partially ordered metric spaces, Fixed Point Theory Appl. 2008:11(2008). Article ID 131294
  • [12] Cristian Chifu, Common fixed point results in extended b-metric spaces endowed with a directed graph, Results in Nonlinear Analysis 2(1)(2019), 18-24.
  • [13] S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Univ. Ostrav. 1 (1993), 5-11.
  • [14] S. Czerwik, Nonlinear set-valued contraction mappings in b-metric spaces, Atti. Semin. Mat. Fis. Univ. 46(2)(1998), 263-276.
  • [15] B.K. Dass and S. Gupta, An extension of Banach contraction principle through rational expressions, Inidan J. Pure Appl. Math. 6(1975), 1455-1458.
  • [16] Erdal Karapinar, Farshid Khojasteh and Zoran D. Mitrovi¢, A Proposal for Revisiting Banach and Caristi Type Theorems in b-Metric Spaces, Mathematics 7(2019), 308. doi:10.3390/math7040308
  • [17] E. Graily, S.M. Vaezpour, R. Saadati and Y.J. Cho, Generalization of fixed point theorems in ordered metric spaces concerning generalized distance, Fixed Point Theory Appl. 2011, 30(2011). doi:10.1186/1687-1812-2011-30
  • [18] R. H. Haghi, S. Rezapour and N. Shahzad, Some fixed point generalizations are not real generalizations, Nonlinear Anal. 74(2011), 1799-1803.
  • [19] J. Harjani, B. Lopez and K. Sadarangani, A fixed point theorem for mappings satisfying a contractive condition of rational type on a partially ordered metric space, Abstr. Appl. Anal. 2010(2010), 1-8.
  • [20] N. Hussain, V. Parvaneh, J. R. Roshan and 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. 2013:256(2013), 1-18.
  • [21] D. S. Jaggi, Some unique fixed point theorems, Indian J. Pure Appl. Math. 8(1977), 223-230.
  • [22] E. Karapinar, Couple fixed point theorems for nonlinear contractions in cone metric spaces, Comput. Math. Appl. 59(12)(2010), 3656?3668. doi.org/10.1016/j.camwa.2010.03.062
  • [23] M.S. Khan, M. Swaleh and S. Sessa, Fixed points theorems by altering distances between the points, Bull. Austral. Math. Soc. 30(1984) 1-9.
  • [24] V. Lakshmikantham and L.J. iric, Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal. 70(2009), 4341-4349. doi:10.1016/j.na.2008.09.020
  • [25] N. V. Luong and N. X. Thuan, Fixed point theorem for generalized weak contractions satisfying rational expressions in ordered metric spaces, Fixed Point Theory Appl. 2011:46 (2011), 1-10.
  • [26] Mitrovic Zoran D., Vahid Parvaneh, Nabil Mlaiki, Nawab Hussain and Stojan Radenovic, On some new gen- eralizations of Nadler contraction in b-metric spaces, Cogent Mathematics & Statistics 7:1(2020), 1760189. DOI: 10.1080/25742558.2020.1760189
  • [27] J.R. Morales and A. Vizcaya, Common fixed points for ψ -Geraghty-Jungck contraction type mappings in Branciari b-metric spaces, Results in Nonlinear Analysis 3(3)(2020), 128-136.
  • [28] J.J. Nieto and R.R. Lopez, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22(2005), 223-239. doi:10.1007/s11083-005-9018-5
  • [29] J.J. Nieto and R.R. Lopez, Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta Math Sinica Engl. Ser. 23(12)(2007), 2205?2212. doi:10.1007/s10114-005-0769-0
  • [30] Nguyen T. Hieu and Nguyen V. Dung, Some ?xed point results for generalized rational type contraction mappings in partially ordered b-metric space, Facta Univ. Ser. Math. Inf. 30(1)(2015), 49-66.
  • [31] D. O'Regan and A. Petrusel, Fixed point theorems for generalized contractions in ordered metric spaces, J. Math. Anal. Appl. 341(2008), 1241-1252.
  • [32] V. Parvaneh, J. R. Roshan and S. Radenovic, Existence of tripled coincidence points in ordered b-metric spaces and an application to a system of integral equations, Fixed Point Theory Appl. 2013:130(2013), 1-19.
  • [33] A.C.M. Ran and M.C.B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Am. Math. Soc. 132(2004), 1435-1443. doi:10.1090/S0002-9939-03-07220-4
  • [34] G.V. Ravindranadh Babu and D. Ratna Babu, Fixed points of Suzuki Z-contraction type maps in b-metric spaces, Advances in the Theory of Nonlinear Analysis and its Applications 4(1)(2020), 14-28. https://doi.org/10.31197/atnaa.632075
  • [35] J.R. Roshan, V. Parvaneh, S. Sedghi, N. Shobkolaei and W. Shatanawi, Common fixed points of almost generalized (ψ,φ) s -contractive mappings in ordered b-metric spaces, Fixed Point Theory Appl. 2013:159(2013), 1-23.
  • [36] J.R. Roshan, V.Parvaneh and Z. Kadelburg, Common fixed point theorems for weakly isotone increasing mappings in ordered b-metric spaces, J. Nonlinear Sci. Appl. 7(2014), 229-245.
  • [37] J.R. Roshan, V. Parvaneh and I. Altun, Some coincidence point results in ordered b-metric spaces and applications in a system of integral equations, Appl. Math. Comput. 226(2014), 725-737.
  • [38] N. Seshagiri Rao, K. Kalyani and Kejal Khatri, Contractive mapping theorems in partially ordered metric spaces, CUBO 22(2)(2020), 203-214.
  • [39] N. Seshagiri Rao and K. Kalyani, Generalized Contractions to Coupled Fixed Point Theorems in Partially Ordered Metric Spaces, Journal of Siberian Federal University. Mathematics & Physics 13(4)(2020), 492-502. DOI: 10.17516/1997-1397- 2020-13-4-492-502
  • [40] N. Seshagiri Rao and K. Kalyani, Coupled fixed point theorems with rational expressions in partially ordered metric spaces, The Journal of Analysis 28(4)(2020), 1085-1095. https://doi.org/10.1007/s41478-020-00236-y
  • [41] N. Seshagiri Rao, K. Kalyani and Bely Mituku, Fixed point theorems for nonlinear contractive mappings in ordered b-metric space with auxiliary function, BMC Research Notes 13:451(2020). doi.org/10.1186/s13104-020-05273-1
  • [42] W. Sintunavarat, Y.J. Cho and P. Kumam, Common fixed point theorems for c-distance in ordered cone metric spaces, Comput. Math. Appl. 62(2011), 1969-1978. doi:10.1016/j.camwa.2011.06.040
  • [43] Tawseef Rashid, Mohammed M. M. Jaradat, Qamrul Haq Khan, Zoran D. Mitrovi¢, Hassen Aydi and Zead Mustafa, A new approach in the context of ordered incomplete partial b-metric spaces, Open Mathematics 18(2020), 996-1005. https://doi.org/10.1515/math-2020-0054
  • [44] Z.D. Mitrovic, Fixed point results in b-metric space, Fixed Point Theory, 20(2)(2019), 559-566. DOI: 10.24193/fpt- ro.2019.2.36

Some fixed point results in ordered $b$-metric space with auxiliary functions

Year 2021, Volume: 5 Issue: 3, 421 - 432, 30.09.2021

Kalyani K Seshagiri Rao Namana Belay Mitiku

https://doi.org/10.31197/atnaa.758962
Cited By: 2

Abstract

The purpose of this paper is to establish some fixed point results for a class of generalized $(\phi, \psi)$-weak contraction mapping in a partially ordered $b$-metric space. This mapping necessarily have a unique fixed point under ordered relation in a space. Also, the results for common and coincidence fixed points of the self mappings are presented. These results generalize and extend an existing results in the literature. Some illustrations are given at the end to support the results.

Keywords

Ordered $b$-metric space Common fixed point Coincidence point

Thanks

We are very thankful to the Editor-in-Chief of the Journal for providing an opportunity to publish our research article.

References

  • [1] R.P. Agarwal, M.A. El-Gebeily and D. O'Regan, Generalized contractions in partially ordered metric spaces, Appl. Anal. 87(2008), 1-8. doi:10.1080/00036810701714164
  • [2] A. Aghajani and R. Arab, Fixed points of (ψ,φ,θ)-contractive mappings in partially ordered b-metric spaces and application to quadratic integral equations, Fixed Point Theory Appl. 2013:245(2013), 1-20.
  • [3] A. Aghajani, M. Abbas and J.R. Roshan, Common ?xed point of generalized weak contractive mappings in partially ordered b-metric spaces, Math. Slovaca 64(4)(2014), 941-960.
  • [4] I.A. Bakhtin, The contraction principle in quasimetric spaces, Func. An., Ulianowsk, Gos. Fed. Ins. 30(1989), 26-37.
  • [5] Belay Mitiku, K. Kalyani and N. Seshagiri Rao, Some fixed point results of generalized (φ,ψ)-contractive mappings in ordered b-metric spaces, BMC Research Notes 13:537(2020). doi:https://doi.org/10.1186/s13104-020-05354-1
  • [6] T.G. Bhaskar and V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65(2006), 1379-1393. doi:10.1016/j.na.2005.10.017
  • [7] I. Cabrera, J. Harjani and K. Sadarangani, A fixed point theorem for contractions of rational type in partially ordered metric spaces, Ann. Univ. 59(2013), 251-258.
  • [8] S. Chandok and E. Karapinar, Common fixed point of generalized rational type contraction mappings in partially ordered metric spaces, Thai J. Math. 11(2)(2013), 251-260.
  • [9] S. Chandok, B.S. Choudhury and N. Metiya, Fixed point results in ordered metric spaces for rational type expressions with auxiliary functions, Journal of the Egyptian Mathematical Society 23(2015), 95-101.
  • [10] Y.J. Cho, M.H. Shah and N. Hussain, Coupled fixed points of weakly F-contractive mappings in topological spaces, Appl. Math. Lett. 24(2011), 1185-1190. doi:10.1016/j.aml.2011.02.004
  • [11] L.J. Ciric, N. Cakic, M. Rajovic and J.S. Ume, Monotone generalized nonlinear contractions in partially ordered metric spaces, Fixed Point Theory Appl. 2008:11(2008). Article ID 131294
  • [12] Cristian Chifu, Common fixed point results in extended b-metric spaces endowed with a directed graph, Results in Nonlinear Analysis 2(1)(2019), 18-24.
  • [13] S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Univ. Ostrav. 1 (1993), 5-11.
  • [14] S. Czerwik, Nonlinear set-valued contraction mappings in b-metric spaces, Atti. Semin. Mat. Fis. Univ. 46(2)(1998), 263-276.
  • [15] B.K. Dass and S. Gupta, An extension of Banach contraction principle through rational expressions, Inidan J. Pure Appl. Math. 6(1975), 1455-1458.
  • [16] Erdal Karapinar, Farshid Khojasteh and Zoran D. Mitrovi¢, A Proposal for Revisiting Banach and Caristi Type Theorems in b-Metric Spaces, Mathematics 7(2019), 308. doi:10.3390/math7040308
  • [17] E. Graily, S.M. Vaezpour, R. Saadati and Y.J. Cho, Generalization of fixed point theorems in ordered metric spaces concerning generalized distance, Fixed Point Theory Appl. 2011, 30(2011). doi:10.1186/1687-1812-2011-30
  • [18] R. H. Haghi, S. Rezapour and N. Shahzad, Some fixed point generalizations are not real generalizations, Nonlinear Anal. 74(2011), 1799-1803.
  • [19] J. Harjani, B. Lopez and K. Sadarangani, A fixed point theorem for mappings satisfying a contractive condition of rational type on a partially ordered metric space, Abstr. Appl. Anal. 2010(2010), 1-8.
  • [20] N. Hussain, V. Parvaneh, J. R. Roshan and 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. 2013:256(2013), 1-18.
  • [21] D. S. Jaggi, Some unique fixed point theorems, Indian J. Pure Appl. Math. 8(1977), 223-230.
  • [22] E. Karapinar, Couple fixed point theorems for nonlinear contractions in cone metric spaces, Comput. Math. Appl. 59(12)(2010), 3656?3668. doi.org/10.1016/j.camwa.2010.03.062
  • [23] M.S. Khan, M. Swaleh and S. Sessa, Fixed points theorems by altering distances between the points, Bull. Austral. Math. Soc. 30(1984) 1-9.
  • [24] V. Lakshmikantham and L.J. iric, Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal. 70(2009), 4341-4349. doi:10.1016/j.na.2008.09.020
  • [25] N. V. Luong and N. X. Thuan, Fixed point theorem for generalized weak contractions satisfying rational expressions in ordered metric spaces, Fixed Point Theory Appl. 2011:46 (2011), 1-10.
  • [26] Mitrovic Zoran D., Vahid Parvaneh, Nabil Mlaiki, Nawab Hussain and Stojan Radenovic, On some new gen- eralizations of Nadler contraction in b-metric spaces, Cogent Mathematics & Statistics 7:1(2020), 1760189. DOI: 10.1080/25742558.2020.1760189
  • [27] J.R. Morales and A. Vizcaya, Common fixed points for ψ -Geraghty-Jungck contraction type mappings in Branciari b-metric spaces, Results in Nonlinear Analysis 3(3)(2020), 128-136.
  • [28] J.J. Nieto and R.R. Lopez, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22(2005), 223-239. doi:10.1007/s11083-005-9018-5
  • [29] J.J. Nieto and R.R. Lopez, Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta Math Sinica Engl. Ser. 23(12)(2007), 2205?2212. doi:10.1007/s10114-005-0769-0
  • [30] Nguyen T. Hieu and Nguyen V. Dung, Some ?xed point results for generalized rational type contraction mappings in partially ordered b-metric space, Facta Univ. Ser. Math. Inf. 30(1)(2015), 49-66.
  • [31] D. O'Regan and A. Petrusel, Fixed point theorems for generalized contractions in ordered metric spaces, J. Math. Anal. Appl. 341(2008), 1241-1252.
  • [32] V. Parvaneh, J. R. Roshan and S. Radenovic, Existence of tripled coincidence points in ordered b-metric spaces and an application to a system of integral equations, Fixed Point Theory Appl. 2013:130(2013), 1-19.
  • [33] A.C.M. Ran and M.C.B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Am. Math. Soc. 132(2004), 1435-1443. doi:10.1090/S0002-9939-03-07220-4
  • [34] G.V. Ravindranadh Babu and D. Ratna Babu, Fixed points of Suzuki Z-contraction type maps in b-metric spaces, Advances in the Theory of Nonlinear Analysis and its Applications 4(1)(2020), 14-28. https://doi.org/10.31197/atnaa.632075
  • [35] J.R. Roshan, V. Parvaneh, S. Sedghi, N. Shobkolaei and W. Shatanawi, Common fixed points of almost generalized (ψ,φ) s -contractive mappings in ordered b-metric spaces, Fixed Point Theory Appl. 2013:159(2013), 1-23.
  • [36] J.R. Roshan, V.Parvaneh and Z. Kadelburg, Common fixed point theorems for weakly isotone increasing mappings in ordered b-metric spaces, J. Nonlinear Sci. Appl. 7(2014), 229-245.
  • [37] J.R. Roshan, V. Parvaneh and I. Altun, Some coincidence point results in ordered b-metric spaces and applications in a system of integral equations, Appl. Math. Comput. 226(2014), 725-737.
  • [38] N. Seshagiri Rao, K. Kalyani and Kejal Khatri, Contractive mapping theorems in partially ordered metric spaces, CUBO 22(2)(2020), 203-214.
  • [39] N. Seshagiri Rao and K. Kalyani, Generalized Contractions to Coupled Fixed Point Theorems in Partially Ordered Metric Spaces, Journal of Siberian Federal University. Mathematics & Physics 13(4)(2020), 492-502. DOI: 10.17516/1997-1397- 2020-13-4-492-502
  • [40] N. Seshagiri Rao and K. Kalyani, Coupled fixed point theorems with rational expressions in partially ordered metric spaces, The Journal of Analysis 28(4)(2020), 1085-1095. https://doi.org/10.1007/s41478-020-00236-y
  • [41] N. Seshagiri Rao, K. Kalyani and Bely Mituku, Fixed point theorems for nonlinear contractive mappings in ordered b-metric space with auxiliary function, BMC Research Notes 13:451(2020). doi.org/10.1186/s13104-020-05273-1
  • [42] W. Sintunavarat, Y.J. Cho and P. Kumam, Common fixed point theorems for c-distance in ordered cone metric spaces, Comput. Math. Appl. 62(2011), 1969-1978. doi:10.1016/j.camwa.2011.06.040
  • [43] Tawseef Rashid, Mohammed M. M. Jaradat, Qamrul Haq Khan, Zoran D. Mitrovi¢, Hassen Aydi and Zead Mustafa, A new approach in the context of ordered incomplete partial b-metric spaces, Open Mathematics 18(2020), 996-1005. https://doi.org/10.1515/math-2020-0054
  • [44] Z.D. Mitrovic, Fixed point results in b-metric space, Fixed Point Theory, 20(2)(2019), 559-566. DOI: 10.24193/fpt- ro.2019.2.36
There are 44 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Kalyani K This is me 0000-0003-2409-6513

Seshagiri Rao Namana 0000-0003-2409-6513

Belay Mitiku This is me 0000-0003-2409-6513

Publication Date September 30, 2021
Published in Issue Year 2021 Volume: 5 Issue: 3

Cite

Cited By

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https://doi.org/10.1186/s13104-021-05801-7