In the present manuscript, notions of $ C^* $-algebra valued $ \mathcal{R} $-metric space and $ C^* $-algebra valued $ \mathcal{R} $-contractive map are introduced along with some fixed point results which in turn generalizes and unifies certain well known results in the existing literature. Further, in support of the obtained results some illustrative examples have been provided.
Citation: Astha Malhotra, Deepak Kumar, Choonkil Park. $ C^* $-algebra valued $ \mathcal{R} $-metric space and fixed point theorems[J]. AIMS Mathematics, 2022, 7(4): 6550-6564. doi: 10.3934/math.2022365
In the present manuscript, notions of $ C^* $-algebra valued $ \mathcal{R} $-metric space and $ C^* $-algebra valued $ \mathcal{R} $-contractive map are introduced along with some fixed point results which in turn generalizes and unifies certain well known results in the existing literature. Further, in support of the obtained results some illustrative examples have been provided.
| [1] |
A. Alam, M. Imdad, Relation-theoretic contraction principle, J. Fixed Point Theory Appl., 17 (2015), 693–702. https://doi.org/10.1007/s11784-015-0247-y doi: 10.1007/s11784-015-0247-y
|
| [2] |
S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math., 3 (1922), 133–181. https://doi.org/10.4064/fm-3-1-133-181 doi: 10.4064/fm-3-1-133-181
|
| [3] | V. Berinde, Approximating fixed points of weak contractions using the Picard iteration, Nonlinear Anal. Forum, 9 (2004), 43–53. |
| [4] | S. K. Chatterjea, Fixed-point theorems, C. R. Acad. Bulgare Sci., 25 (1972), 727–730. |
| [5] |
L. B. Ćirić, A generalization of Banach's contraction principle, Proc. Amer. Math. Soc., 45 (1974), 267–273. https://doi.org/10.1090/S0002-9939-1974-0356011-2 doi: 10.1090/S0002-9939-1974-0356011-2
|
| [6] |
S. Chandok, D. Kumar, C. Park, $C^*$-algebra-valued partial metric space and fixed point theorems, Indian Acad. Sci. Math. Sci., 129 (2019), 37. https://doi.org/10.1007/s12044-019-0481-0 doi: 10.1007/s12044-019-0481-0
|
| [7] | R. G. Douglas, Banach algebra techniques in operator theory, 2 Eds., New York: Springer-Verlag, 1998. https://doi.org/10.1007/978-1-4612-1656-8 |
| [8] |
D. J. Guo, V. Lakshmikantham, Coupled fixed points of nonlinear operators with applications, Nonlinear Anal., 11 (1987), 623–632. https://doi.org/10.1016/0362-546X(87)90077-0 doi: 10.1016/0362-546X(87)90077-0
|
| [9] |
A. Ghanifard, H. P. Masiha, M. De La Sen, Approximation of fixed points of $C^*$-algebra-multi-valued contractive mappings by the Mann and Ishikawa processes in convex $C^*$-algebra-valued metric spaces, Mathematics, 8 (2020), 392. https://doi.org/10.3390/math8030392 doi: 10.3390/math8030392
|
| [10] |
G. E. Hardy, T. D. Rogers, A generalization of a fixed point theorem of Reich, Canad. Math. Bull., 16 (1973), 201–206. https://doi.org/10.4153/CMB-1973-036-0 doi: 10.4153/CMB-1973-036-0
|
| [11] |
L. G. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 332 (2007), 1468–1476. https://doi.org/10.1016/j.jmaa.2005.03.087 doi: 10.1016/j.jmaa.2005.03.087
|
| [12] | N. Hussain, J. Ahmad, New Suzuki-Berinde type fixed point results, Carpathian J. Math., 33 (2017), 59–72. |
| [13] |
S. Hussain, Fixed point and common fixed point theorems on ordered cone b-metric space over Banach algebra, J. Nonlinear Sci. Appl., 13 (2020), 22–33. http://dx.doi.org/10.22436/jnsa.013.01.03 doi: 10.22436/jnsa.013.01.03
|
| [14] | R. Kannan, Some results on fixed points, Bull. Cal. Math. Soc., 60 (1968), 71–76. |
| [15] | B. Kolman, R. C. Busby, S. C. Ross, Discrete mathematical structures, Prentice-Hall, 1996. |
| [16] |
S. Khalehoghli, H. Rahimi, M. E. Gordji, Fixed point theorems in $\mathcal{R}$-metric spaces with applications, AIMS Math., 5 (2020), 3125–3137. https://doi.org/10.3934/math.2020201 doi: 10.3934/math.2020201
|
| [17] | S. Lipschutz, Schaum's outline of theory and problems of set theory and related topics, New York: McGraw-Hill, 1964. |
| [18] |
A. Meir, E. Keeler, A theorem on contraction mappings, J. Math. Anal. Appl., 28 (1969), 326–329. https://doi.org/10.1016/0022-247X(69)90031-6 doi: 10.1016/0022-247X(69)90031-6
|
| [19] | G. J. Murphy, $C^*$-algebras and operator theory, San Diego: Academic Press, 1990. https://doi.org/10.1016/C2009-0-22289-6 |
| [20] |
Z. H. Ma, L. N. Jiang, H. K. Sun, $C^*$-algebra-valued metric spaces and related fixed point theorems, Fixed Point Theory Appl., 2014 (2014), 206. https://doi.org/10.1186/1687-1812-2014-206 doi: 10.1186/1687-1812-2014-206
|
| [21] |
Z. H. Ma, L. N. Jiang, $ C^{\ast} $-algebra-valued b-metric spaces and related fixed point theorems, Fixed Point Theory Appl., 2015 (2015), 222. https://doi.org/10.1186/s13663-015-0471-6 doi: 10.1186/s13663-015-0471-6
|
| [22] |
G. Prasad, Fixed points of Kannan contractive mappings in relational metric spaces, J. Anal., 29 (2021), 669–684. https://doi.org/10.1007/s41478-020-00273-7 doi: 10.1007/s41478-020-00273-7
|
| [23] |
S. Reich, Some remarks concerning contraction mappings, Canad. Math. Bull., 14 (1971), 121–124. https://doi.org/10.4153/cmb-1971-024-9 doi: 10.4153/cmb-1971-024-9
|
| [24] |
C. C. Shen, L. N. Jiang, Z. H. Ma, $C^*$-algebra-valued $G$-metric spaces and related fixed-point theorems, J. Funct. Space., 2018 (2018), 3257189. https://doi.org/10.1155/2018/3257189 doi: 10.1155/2018/3257189
|
| [25] |
D. Wardowski, Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory Appl., 2012 (2012), 94. https://doi.org/10.1186/1687-1812-2012-94 doi: 10.1186/1687-1812-2012-94
|
| [26] |
T. Zamfirescu, Fix point theorems in metric spaces, Arch. Math., 23 (1972), 292–298. https://doi.org/10.1007/BF01304884 doi: 10.1007/BF01304884
|
Astha Malhotra, Deepak Kumar, Choonkil Park. $ C^* $-algebra valued $ \mathcal{R} $-metric space and fixed point theorems[J]. AIMS Mathematics, 2022, 7(4): 6550-6564. doi: 10.3934/math.2022365
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