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Near fixed point, near fixed interval circle and their equivalence classes in a $b-$interval metric space | ||
| International Journal of Nonlinear Analysis and Applications | ||
| مقاله 166، دوره 13، شماره 1، خرداد 2022، صفحه 1999-2014 اصل مقاله (459.5 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22075/ijnaa.2021.21721.2291 | ||
| نویسندگان | ||
| Meena Joshi1؛ Anita Tomar* 2 | ||
| 1S. G. R. R. (P. G.) College Dehradun, India | ||
| 2Sri Dev Suman Uttarakhand Vishwavidyalay, Pt. L. M. S. Campus Rishikesh- 249201, Uttarakhand, India | ||
| تاریخ دریافت: 13 آبان 1399، تاریخ پذیرش: 09 آذر 1400 | ||
| چکیده | ||
| We introduce a novel distance structure called a b−interval metric space to generalize and extend metric interval space. Also, we demonstrate that the collection of open balls, which forms a basis of a b−interval metric space, generates a T0−topology on it. Further, we define topological notions like an open ball, closed ball, b−convergence, b−Cauchy sequence and completeness of the space on a b−interval metric space to create an environment for the survival of a near fixed point and a unique equivalence class of near fixed point. Towards the end, we introduce notions of interval circle, fixed interval circle, its equivalence class and established the existence of a near fixed interval circle and its equivalence interval C−class of near fixed interval circle to study the geometric properties of non-unique equivalence C−classes of nearly fixed interval circles. | ||
| کلیدواژهها | ||
| Continuity؛ b−convergence؛ completeness؛ fixed interval circle؛ null set؛ T0−topology | ||
| مراجع | ||
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[1] I.A. Bakhtin, The contraction mapping principle in almost metric spaces, Funct. Anal. 30 (1989) 26–37. [2] S. Banach, Sur les operations dans les ensembles abstraits et leur application aux ´equation int´egrales, Fund. Math. 3 (1922) 133–181. [3] R.M.T. Bianchini, Su un problema di S. Reich aguardante la teor´ıa dei punti fissi, Boll. Un. Mat. Ital. 5 (1972) 103–108. [4] J. Caristi, Fixed point theorems for mappings satisfying inwardness conditions, Trans. Amer. Math. Soc. 215 (1976) 241–251. [5] S. Czerwik, Contraction mappings in b−metric spaces, Acta Math. Inform. Univ. Ostrav. 1 (1993) 5–11. [6] M. Edelstein, An extension of Banach’s contraction principle, Proc. Amer. Math. Soc. 12 (1960) 7–10. [7] M. Fr´echet, Sur quelques points du calcul fonctionnel, Rendi. Circ. Mate. Palermo 22 (1906) 1—72. [8] R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc. 60 (1968) 71–76. [9] M. Joshi, A. Tomar, H.A. Nabwey and R. George, On unique and non-unique fixed points and fixed circles in Mb v−metric space and application to cantilever beam problem, J. Funct. Spaces 2021 (2021) Article ID 66810. [10] M. Joshi, A. Tomar and S.K. Padaliya, Fixed point to fixed disc and application in partial metric spaces, Chapter in a book "Fixed point theory and its applications to real-world problem, Nova Science Publishers, New York, USA, 391–406, 2021. [11] M. Joshi, A. Tomar and S.K. Padaliya, On geometric properties of non-unique fixed points in b−metric spaces, Chapter in a book "Fixed Point Theory and its Applications to Real World Problem” Nova Science Publishers, New York, USA, 33–50, 2021. [12] M. Joshi, A. Tomar and S.K. Padaliya, Fixed point to fixed ellipse in metric spaces and discontinuous activation function, Appl. Math. E-Notes 21 (2021) 225-237. [13] M. Joshi and A. Tomar, On unique and non-unique fixed points in metric spaces and application to chemical sciences, J. Funct. Spaces 2021 (2021) Article ID 6681044. [14] N. Y. Ozgur, N. Ta¸s, Some fixed-circle theorems on metric spaces, Bull. Malays. Math. Sci. Soc., 2017. [15] E. Rakotch, A note on contractive mappings, Proc. Amer. Math. Soc. 13 (1962) 459–465. [16] S. Reich, Some remarks concerning contraction mappings, Canad. Math. Bull. 14 (1971) 121–124. [17] V.M. Sehgal, On fixed and periodic points for a class of mappings, J. London Math. Soc. 5(2) (1972) 571–576. [18] A. Tomar and M. Joshi, Near fixed point, near fixed interval circle and near fixed interval disc in metric interval space, Chapter in a book “Fixed Point Theory and its Applications to Real World Problem” Nova Science Publishers, New York, USA, 131–150, 2021. [19] A. Tomar, M. Joshi and S.K. Padaliya, Fixed point to fixed circle and activation function in partial metric space, J. Appl. Anal. 28(1) (2021). [20] H.C. Wu, A new concept of fixed point in metric and normed interval spaces, Math. 6(11) (2018) 219. | ||
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