International Journal of Nonlinear Analysis and Applications
A nonunique common fixed point theorem of Rhoades type in b-metric spaces with applications
Document Type : Research Paper
Authors
System Dynamics and Control Laboratory , Department of mathematics and informatics, OEB University, Algeria
Abstract
The aim of this paper is to prove a nonunique common fixed point theorem of Rhoades type for two self-mappings in complete b-metric spaces. This theorem extends the results of [17] and [49]. Examples are furnished to illustrate the validity of our results. We apply our theorem to establish the existence of common solutions of a system of two nonlinear integral equations and a system of two functional equations arising in dynamic programming.
Keywords
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[2] S. Aleksic, H. Huang, Z. D. Mitrovic and S. Radenovic, Remarks on some fixed point results in b-metric spaces, J. Fixed Point Theory Appl. 20(4) (2018) 1–17.
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[29] Z. Kadelburg and S. Radenovic, Notes on some recent papers concerning F-contractions in b-metric spaces, Construct. Math. Anal. 1(2) (2018) 108–112.
[30] A. K. Kalinde, S. N. Mishra and H. K. Pathak, Some results on common fixed points with applications, Fixed Point Theory, 6(2) (2005) 285–301.
[31] S. Khalehoghli, H. Rahimi and M. E. Gordji, Fixed point theorems in R-metric spaces with applications, AIMS Mathematics, 5(4) (2020) 3125-3137.
[32] S. Khalehoghli, H. Rahimi and M. E. Gordji, R-topological spaces and SR-topological spaces with their applications, Math Sci. 14 (2020) 249–255.
[2] S. Aleksic, H. Huang, Z. D. Mitrovic and S. Radenovic, Remarks on some fixed point results in b-metric spaces, J. Fixed Point Theory Appl. 20(4) (2018) 1–17.
[3] S. Aleksic, Z. D. Mitrovic and S. Radenovic, On some recent fixed point results for single and multivalued mappings in b-metric spaces, Fasc. Math. 61 (2018) 5–16.[4] T. V. An, L. Q. Tuyen and N. V. Dung, Stone-type theorem on b-metric spaces and applications, Topology Appl.185 (2015) 50–64.
[5] H. Aydi, M. Bota, E. Karapinar and S. Mitrovic, A fixed point theorem for set-valued quasi-contractions in b-metric spaces, Fixed Point Theory Appl. 2012(1) (2012) 1–8.
[6] G. V. R. Babu and T. M. Dula, Fixed points of almost generalized (α, β)-(ψ, ϕ) contractive mappings in b-metric spaces, Facta Univ. Ser. Math. Inform. 33(2) (2018) 177–196.
[7] H. Baghani, M. E. Gordji and M. Ramezani, Orthogonal sets: The axiom of choice and proof of a fixed point theorem, J. Fixed Point Theory Appl. 18(3) (2016) 465–477.
[8] I. A. Bakhtin, The contraction mapping principle in almost metric spaces, Funct. Anal. Unianowsk, Gos. Ped. Inst. 30 (1989) 26–37.
[9] S. Banach, Sur les op´erations dans les ensembles abstraits et leur application aux quations int´egrales, Fund. Math. 3 (1922) 133–181.
[10] R. Bellman and E. S. Lee, Functional equations arising in dynamic programming, Aequationes Math. 17 (1978)1–18.
[11] V. Berinde, Generalized contractions in quasi-metric spaces, Seminar on Fixed Point Theory, (1993) 3–9.
[12] P. C. Bhakta and S. Mitra, Some existence theorems for functional equations arising in dynamic programming, J. Math. Anal. Appl. 98 (1984) 348–362.
[13] M. Boriceanu, Fixed point theory for multivalued generalized contraction on a set with two b-metrics, Studia Universitatis Babes-Bolyai, Series Mathematica, 54(3) (2009) 3–14.
[14] S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostrav. 1 (1993) 5–11.
[15] S. Czerwik, Nonlinear set-valued contraction mappings in b-metric spaces, Atti. Sem. Mat. Univ. Modena, 46 (1998) 263-276.
[16] M. Demma and P. Vetro, Picard sequence and fixed point results on b-metric spaces, J. Function Spaces, Vol. 2015, Article ID 189861, 6 pages.
[17] N. V. Dung, V. T. Le. Hang and S. Sedghi, Remarks on metric-type spaces and applications, Southeast Asian Bull. Math. 39 (2015) 755–768.
[18] N. V. Dung and V. T. Le Hang, On relaxations of contraction constants and Caristi’s theorem in b-metric spaces, J. Fixed Point Theory Appl. 18 (2016) 267–284.
[19] N. V. Dung and V. T. Le Hang, Remarks on cyclic contractions in b-metric spaces and applications to integral equations, R. Acad. Cienc. Exactas Fs. Nat. Ser. A Math. 111 (2017) 247–255.
[20] M. E. Gordji, H. Habibi, Fixed point theory in generalized orthogonal metric space, J - Linear Topolog. Algebra,2017.
[21] M. E. Gordji, M. Rameani, M. De La Sen and Yeol Je Cho, On orthogonal sets and Banach fixed point theorem, Fixed Point Theory, 18(2) (2017) 569-578.
[22] T. Hamaizia and A. Aliouche, A fixed point theorem in b-metric spaces and its application in linear integral equation, Asian J. Math. Computer Research, 15(1) (2017) 23-29.
[23] H. Hosseini and M. Eshaghi, Fixed Point Results in Orthogonal Modular Metric Spaces, Int. J. Nonlinear Anal. Appl. 11 (2020) 425-436.
[24] H. Huang, T. Dosenovic and S. Radenovic, Some fixed point results in b-metric spaces approach to the existence of a solution to nonlinear integral equations, J. Fixed Point Theory Appl. 20(3) (2018) 1-19.
[25] H. Huang, G. Deng and S. Radenovic, Fixed point theorems in b-metric spaces with applications to differential equations, J. Fixed Point Theory Appl. 20(1) (2018) 1:24.
[26] N. Hussain, V. Parvaneh, J. R. Roshan and Z. Kadelburg, Fixed points of cyclic (ψ, ϕ, L, A, B)-contractive mappings in ordered b-metric spaces with applications, Fixed Point Theory Appl. 2013 (2013), 256, 18 pages.
[27] M. Jleli, B Samet, C. Vetro and F. Vetro, Fixed points for multivalued mappings in b-metric spaces, Abstract Appl. Anal. Vol. 2015, Article ID 718074, 7 pages.
[28] Z. Kadelburg, S. Radenovic and M. Rajovic, A note on fixed point theorems for rational Geraghty contractive mappings in ordered b-metric spaces, Kragujevac J. Math. 39(2) (2015) 183–191.
[29] Z. Kadelburg and S. Radenovic, Notes on some recent papers concerning F-contractions in b-metric spaces, Construct. Math. Anal. 1(2) (2018) 108–112.
[30] A. K. Kalinde, S. N. Mishra and H. K. Pathak, Some results on common fixed points with applications, Fixed Point Theory, 6(2) (2005) 285–301.
[31] S. Khalehoghli, H. Rahimi and M. E. Gordji, Fixed point theorems in R-metric spaces with applications, AIMS Mathematics, 5(4) (2020) 3125-3137.
[32] S. Khalehoghli, H. Rahimi and M. E. Gordji, R-topological spaces and SR-topological spaces with their applications, Math Sci. 14 (2020) 249–255.
[33] M. A. Khamsi, Remarks on cone metric spaces and fixed point theorems of contractive mappings, Fixed Point Theory Appl. 2010 (2010) 1–7.
[34] M. A. Khamsi and N. Hussain, KKM mappings in metric type spaces. Nonlinear Anal. 7 (2010) 3123–3129.
[35] F. Khojasteh, M. Abbas and S. Costache, Two new types of fixed point theorems in complete metric spaces, Abstract Appl. Anal., Article ID 3258040 (2014), 5 pages.
[36] A. Latif, V. Parvaneh, P. Salimi and A. E. Al-Mazrooei, Various Suzuki type theorems in b-metric spaces. J. Nonlinear Sci. Appl. 8(4) (2015) 363–377.
[37] J. Li, M. Fu, Z. Liu and S. M. Kang, A common fixed point theorem and its application in dynamic programming, Appl. Math. Sci. 2(17) (2008) 829–842.
[38] Z. Liu, Compatible mappings and fixed points, Acta Sci. Math. 65 (1999) 371–383.
[39] Z. Liu, R. P. Agarwal and S. M. Kang, On solvability of functional equations and system of functional equations arising in dynamic programming, J. Math. Anal. Appl. 297 (2004) 111–130.
[40] N. Lu, F. He and W. S. Du, Fundamental questions and new counter-examples for b-metric spaces and Fatou property, Mathematics 7 (2019) 1107.
[41] R. Miculescu and A. Mihail, New fixed point theorems for set-valued contractions in b-metric spaces, J. Fixed Point Theory Appl. 19 (2017) 2153–2163.
[42] Z. D. Mitrovic, S. Radenovic, F. Vetro, and J. Vujakovic, Some remarks on TAC-contractive mappings in b-metric spaces, Mat. Vesnik, 70(2) (2018) 167–175.
[43] H. K. Pathak,Y. J. Cho, S. M. Kang, B. S. Lee, Fixed point theorems for compatible mappings of type (P) and applications to dynamic programming, Le Matematiche, 50 (1995) 15–33.
[44] S. Radenovic, T. Dosenovic, V. Ozturk and C. Dolicanin, A note on the paper: “Nonlinear integral equations with new admissibility types in b-metric spaces”, J. Fixed Point Theory Appl. 19 (2017) 2287–2295.
[45] M. Ramezani, H. Baghani, Contractive gauge functions in strongly orthogonal metric spaces, Int. J. Nonlinear Anal. Appl. 8 (2017) 23–28.
[46] B. E. Rhoades, Two New Fixed Point Theorems, Gen. Math. Notes, 27(2) (2015) 123–132.
[47] B. Samet, The class of (α, ψ)-type contractions in b-metric spaces and fixed point theorems, Fixed Point Theory Appl. 2015 (2015) 1–15.
[48] W. Sintunavarat, S. Plubtieng and P. Katchang, Fixed point results and applications on a b-metric space endowed with an arbitrary binary relation, Fixed Point Theory Appl. 2013 (2013) 1–13.
[49] W. Sintunavarat, Fixed point results in b-metric spaces approach to the existence of a solution for nonlinear integral equations, R. Acad. Cienc. Exactas Fs. Nat. Ser. A Math. 110 (2016) 585–600.
[50] T. Suzuki, Basic inequality on a b-metric space and its applications, J. Inequal. Appl. 2017, 256 (2017).
[51] T. Suzuki, Fixed point theorems for single and set-valued F-contractions in b-metric spaces, J. Fixed Point Theory Appl. (2018) 20:35.
[52] S. Xu, S. Chen and S. Radenovic, Some notes on ”Common fixed point of two R-weakly commuting mappings in b-metric spaces”, Int. J. Nonlinear Anal. Appl. 9(2) (2018) 161–167.
[53] I. Yildirim and A. H. Ansari, Some new fixed point results in b-metric spaces, Topol. Algebra Appl. 6 (2018) 102–106.
[34] M. A. Khamsi and N. Hussain, KKM mappings in metric type spaces. Nonlinear Anal. 7 (2010) 3123–3129.
[35] F. Khojasteh, M. Abbas and S. Costache, Two new types of fixed point theorems in complete metric spaces, Abstract Appl. Anal., Article ID 3258040 (2014), 5 pages.
[36] A. Latif, V. Parvaneh, P. Salimi and A. E. Al-Mazrooei, Various Suzuki type theorems in b-metric spaces. J. Nonlinear Sci. Appl. 8(4) (2015) 363–377.
[37] J. Li, M. Fu, Z. Liu and S. M. Kang, A common fixed point theorem and its application in dynamic programming, Appl. Math. Sci. 2(17) (2008) 829–842.
[38] Z. Liu, Compatible mappings and fixed points, Acta Sci. Math. 65 (1999) 371–383.
[39] Z. Liu, R. P. Agarwal and S. M. Kang, On solvability of functional equations and system of functional equations arising in dynamic programming, J. Math. Anal. Appl. 297 (2004) 111–130.
[40] N. Lu, F. He and W. S. Du, Fundamental questions and new counter-examples for b-metric spaces and Fatou property, Mathematics 7 (2019) 1107.
[41] R. Miculescu and A. Mihail, New fixed point theorems for set-valued contractions in b-metric spaces, J. Fixed Point Theory Appl. 19 (2017) 2153–2163.
[42] Z. D. Mitrovic, S. Radenovic, F. Vetro, and J. Vujakovic, Some remarks on TAC-contractive mappings in b-metric spaces, Mat. Vesnik, 70(2) (2018) 167–175.
[43] H. K. Pathak,Y. J. Cho, S. M. Kang, B. S. Lee, Fixed point theorems for compatible mappings of type (P) and applications to dynamic programming, Le Matematiche, 50 (1995) 15–33.
[44] S. Radenovic, T. Dosenovic, V. Ozturk and C. Dolicanin, A note on the paper: “Nonlinear integral equations with new admissibility types in b-metric spaces”, J. Fixed Point Theory Appl. 19 (2017) 2287–2295.
[45] M. Ramezani, H. Baghani, Contractive gauge functions in strongly orthogonal metric spaces, Int. J. Nonlinear Anal. Appl. 8 (2017) 23–28.
[46] B. E. Rhoades, Two New Fixed Point Theorems, Gen. Math. Notes, 27(2) (2015) 123–132.
[47] B. Samet, The class of (α, ψ)-type contractions in b-metric spaces and fixed point theorems, Fixed Point Theory Appl. 2015 (2015) 1–15.
[48] W. Sintunavarat, S. Plubtieng and P. Katchang, Fixed point results and applications on a b-metric space endowed with an arbitrary binary relation, Fixed Point Theory Appl. 2013 (2013) 1–13.
[49] W. Sintunavarat, Fixed point results in b-metric spaces approach to the existence of a solution for nonlinear integral equations, R. Acad. Cienc. Exactas Fs. Nat. Ser. A Math. 110 (2016) 585–600.
[50] T. Suzuki, Basic inequality on a b-metric space and its applications, J. Inequal. Appl. 2017, 256 (2017).
[51] T. Suzuki, Fixed point theorems for single and set-valued F-contractions in b-metric spaces, J. Fixed Point Theory Appl. (2018) 20:35.
[52] S. Xu, S. Chen and S. Radenovic, Some notes on ”Common fixed point of two R-weakly commuting mappings in b-metric spaces”, Int. J. Nonlinear Anal. Appl. 9(2) (2018) 161–167.
[53] I. Yildirim and A. H. Ansari, Some new fixed point results in b-metric spaces, Topol. Algebra Appl. 6 (2018) 102–106.
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Volume 12, Issue 2
November 2021Pages 399-413
- Receive Date: 01 January 2021
- Revise Date: 22 January 2021
- Accept Date: 04 March 2021