Some fixed point results on double controlled cone metric spaces
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Abstract
In this text, we investigate some fixed point results in double-controlled cone metric spaces using several contraction mappings such as the B-contraction, the Hardy-Rogers contraction, and so on. Additionally, we prove the same fixed point results by using rational type contraction mappings, which were discussed by the authors Dass. B. K and Gupta. S. Also, a few examples are included to illustrate the results. Finally, we discuss some applications that support our main results in the field of applied mathematics.
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References
[1] Abbas, M., Rhoades, B. E., Fixed and periodic point results in cone metric space, Appl. Math. Lett. 22 (4) (2009), 511–515. https://doi.org/10.1016/j.aml.2008.07.001 Google Scholar
[2] Abdeljawad, T., Mlaiki, N., Aydi, H., Souayah, N. Double controlled metric type spaces and some fixed point results, Mathematics. 6 (2018), 320. https://doi.org/10.3390/math6120320 Google Scholar
[3] Bakhtin, I. A., The contraction mapping principle in almost metric spaces, Funct. Anal., 30 (1989), 26–37. Google Scholar
[4] Banach, S., Surles operations dans les ensembles abstract et leur application aux equation inte-grals, Fund.Math., 3(1922), 133–181. Google Scholar
[5] Bianchini, R. M. T., Su un problema di S. Reich riguardante la teoria dei punti fissi, Bolletino U.M.I., 4 (5) (1972), 103–106. Google Scholar
[6] Brouwer, F., The fixed point theory of multiplicative mappings in topological vector spaces, Math-ematische Annalen., 177 (1968), 283–301. Google Scholar
[7] Chatterjea, S. K., Fixed point theorems , C. R. Acad. Bulg. Sci., 25 (1972), 727–730. Google Scholar
[8] Ciric, L. B., Generalized contractions and fixed point theorems, Publ. Inst. Math. (Bulgr). 12 (26) (1971), 19–26. Google Scholar
[9] Dass, B. K., Gupta, S., An extension of Banach contraction principle through rational expression, Communicated by F.C. Auluck, FNA.,1975. Google Scholar
[10] Hardy, G. E., Rogers, T.D., A generalization of fixed point theorem of Reich, Can. Math. Bull., 16 (1973), 201–206. Google Scholar
[11] Haung, L. G., Zhang, X., Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (4) (2007), 1468–1476. https://doi.org/10.1016/j.jmaa.2005.03.087 Google Scholar
[12] Huaping Huang, Stojan Radenovi ́c, Guantie Deng, A sharp generalization on cone b-metric space over Banach algebra, J. Nonlinear Sci. Appl. 10 (2017), 429–435. http://dx.doi.org/10.22436/jnsa.010.02.09 Google Scholar
[13] Jaggi, D. S., Some unique fixed point theorems, Indian Journal of Pure and Applied Mathematics, 8 (1977), 223–230. Google Scholar
[14] Kannan, R., Some results on fixed points, Bull Calcutta Math.Soc, 60 (1968), 71–76. Google Scholar
[15] Kannan, R., Some results on fixed points II, Am.Math.Mon. 76 (1969), 405–408. Google Scholar
[16] Karapinar, E., A new non-unique fixed point theorem, Ann. Funct. Annals. 2 (1) (2011), 51–58. Google Scholar
[17] Khan, M. S., A fixed point theorems for metric spaces, Rendiconti Dell ’istituto di mathematica dell’ Universtia di tresti, 8 (1976), 69–72. Google Scholar
[18] Khuri, S. A., Louhichi, I., A novel Ishikawa-Green’s fixed point scheme for the solution of BVPs, Appl. Math. Lett. 82 (2018), 50–57. https://doi.org/10.1016/j.aml.2018.02.016 Google Scholar
[19] Kumar. K, Rathour. L, Sharma. M. K, Mishra V. N. Fixed point approximation for suzuki generalized nonexpansive mapping using B(δ,μ) condition, Applied Mathematics 13 (2) (2022), 215–227. https://doi.org/10.4236/am.2022.132017 Google Scholar
[20] Marudai, M., Bright V. S., Unique fixed point theorem weakly B-contractive mappings, Far East journal of Mathematical Sciences (FJMS), 98 (7) (2015), 897–914. Google Scholar
[21] Mishra. L. N, Dewangan. V, Mishra. V. N, Karateke. S, Best proximity points of admissible almost generalized weakly contractive mappings with rational expressions on b-metric spaces, J. Math. Computer Sci. 22 (2) (2021), 97—109. https://doi.org/10.22436/jmcs.022.02.01 Google Scholar
[22] Mishra. L. N, Dewangan. V, Mishra. V. N, Amrulloh. H, Coupled best proximity point theorems for mixed g-monotone mappings in partially ordered metric spaces, J. Math. Comput. Sci. 11 (5) (2021), 6168–6192. https://doi.org/10.28919/jmcs/6164 Google Scholar
[23] Mishra. L. N, Mishra. V. N, Gautam. P, Negi. K, Fixed point Theorems for Cyclic-C ́iri ́c-Reich-Rus contraction mapping in Quasi-Partial b-metric spaces, Scientific Publications of the State University of Novi Pazar Ser. A: Appl. Math. Inform. and Mech. 12 (1) (2020), 47–56. http://dx.doi.org/10.5937/SPSUNP2001047M Google Scholar
[24] Mishra L. N, Tiwari. S. K, Mishra. V. N, Fixed point theorems for generalized weakly S-contractive mappings in partial metric spaces, Journal of Applied Analysis and Computation 5 (4) (2015), 600–612. https://doi.org/10.11948/2015047 Google Scholar
[25] Mitrovi ́c, Z. D., Radenovi ́c, S., The Banach and Reich contractions in bv(s)-metric spaces, J. Fixed Point Theory Appl. 19 (2017), 3087–3095. http://dx.doi.org/10.1007/s11784-017-0469-2 Google Scholar
[26] Mlaiki, N., Aydi, H., Souayah, N., Abdeljawad, T., Controlled metric type spaces and related contraction principle, Mathematics 6 (10) (2018), 194. https://doi.org/10.3390/math6100194 Google Scholar
[27] Mlaiki, N., Double controlled metric-like spaces, J. Inequal. Appl. 189 2020. https://doi.org/10.1186/s13660-020-02456-z Google Scholar
[28] Reich, S., Some remarks connecting contraction mappings, Can. Math. Bull. 14 (1971), 121–124. https://doi.org/10.4153/CMB-1971-024-9 Google Scholar
[29] Roshan, J. R., Parvanesh, V., Kadelburg, Z., Hussain, N., New fixed point results in b-rectangular metric spaces, Nonlinear Analalysis: Modelling and control 21 (5) (2016), 614–634. http://dx.doi.org/10.15388/NA.2016.5.4 Google Scholar
[30] Sanatee. A. G, Rathour. L, Mishra. V. N, Dewangan. V Some fixed point theorems in regular modular metric spaces and application to Caratheodory’s type anti-periodic boundary value prob-lem, The Journal of Analysis 31 (2023), 619–632. https://doi.org/10.1007/s41478-022-00469-z Google Scholar
[31] Sanatee. A. G, Ranmanesh. M. Mishra. L. N, Mishra. V. N, Generalized 2−proximal C−contraction mappings in complete ordered 2−metric space and their best proximity points, Scientific Publications of the State University of Novi Pazar Ser. A: Appl. Math. Inform. and Mech, 12 (1) (2020), 1–11. http://dx.doi.org/10.5937/SPSUNP2001001S Google Scholar
[32] Shahi P, Rathour L, Mishra. V. N Expansive Fixed Point Theorems for tri-simulation functions, The Journal of Engineering and Exact Sciences –jCEC 08 (3) (2022), 14303–01e. https://doi.org/10.18540/jcecvl8iss3pp14303-01e Google Scholar
[33] Sharma. N, Mishra. L. N, Mishra. V. N, Almusawa. H, Endpoint approximation of standard three-step multi-valued iteration algorithm for nonexpansive mappings, Applied Mathematics and Information Sciences 15 (1) (2021), 73–81. https://doi.org/10.18576/amis/150109 Google Scholar
[34] Sharma. N, Mishra. L. N, Mishra. V. N, Pandey. S, Solution of Delay Differential equation via N1v iteration algorithm, European J. Pure Appl. Math. 13 (5) (2020), 1110–1130. https://doi.org/10.29020/nybg.ejpam.v13i5.3756 Google Scholar
[35] Sharma. N, Mishra. L. N, Mishra. S. N, Mishra. V. N, Empirical study of new iterative algorithm for generalized nonexpansive operators, Journal of Mathematics and Computer Science 25 (3) (2022), 284–295. https://dx.doi.org/10.22436/jmcs.025.03.07 Google Scholar
[36] Shateri, T. L., Double controlled cone metric spaces and the related fixed point theorems, J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math. 30 (1) (2023), 1–13. https://doi.org/10.48550/arXiv.2208.06812 Google Scholar
[37] Slobodanka Jankovi ́c, Zoran Kadelburg, Stojan Radenovi ́c, On cone metric spaces: A survey, Nonlinear Analysis 74 (2011) 2591–2601. Google Scholar
[38] Stojan Radenovi ́c, Common Fixed Points Under Contractive Condition in Cone Metric Spaces, Computers and Mathematics with applycation 58 (2019),1273–1278. https://doi.org/10.1016/j.camwa.2009.07.035 Google Scholar
[39] Suzana Aleksi ́c, Zoran Kadelburg, Zoran. D. Mitrovi ́c, Stojan Radenovi ́c, A new survey: cone metric spaces, Journal of the international Mathematical Vertiual Institute 9(2019), 93-121. https://api.semanticscholar.org/CorpusID:119572977 Google Scholar
[40] Theivaraman. R, Srinivasan. P. S, Thenmozhi. S, Radenovic. S, Some approximate fixed point results for various contraction type mappings, 13 (9) (2023), 1–20. https://doi.org/10.28919/afpt/8080 Google Scholar
[41] Theivaraman. R, Srinivasan. P. S, Radenovic. S, Choonkil Park, New Approximate Fixed Point Results for Various Cyclic Contraction Operators on E-Metric Space, 27 (3) (2023), 160–179. https://doi.org/10.12941/jksiam.2023.27.160 Google Scholar
[42] Vishnu Narayanan P: B. Deshpande, V.N. Mishra, A. Handa, L.N. Mishra, Coincidence Point Results for Generalized (ψ, θ, φ)-Contraction on Partially Ordered Metric Spaces, Thai J. Math., 19 (1) (2021), 93–112. Google Scholar
[43] Vishnu Narayanan P, Mishra. L. N, Tiwari. S. K, Mishra. V. N, Khan. I. A; Unique Fixed Point Theorems for Generalized Contractive Mappings in Partial Metric Spaces, Journal of Function Spaces, 2015 (2021), Article ID 960827, 1–8. Google Scholar
[44] Zamfirescu, T., Fixed point theorems in metric spaces, Arch. Math. (Basel) 23(1972), 292–298. Google Scholar
[45] Zoran Kadelburg, Stojan Radenovi ́c, Vladimir Rakoˇcevi ́c, A note on the equivalence of some metric and cone metric fixed point results, Applied Mathematics Letters, 24 (2011), 370–374. https://doi.org/10.1016/j.aml.2010.10.030 Google Scholar