In the present paper, we extend the concept of contraction in a new manner by introducing D-contraction defined on a family F of bounded functions. We also introduce a new notion of a fixed function on a metric space. Some fixed function theorems along with illustrative examples and application are also given to verify the effectiveness of our results.
1. Aggarwal, R. P., Meehan, M., and O’Regan, D. Fixed Point Theory and Applications. Cambridge University Press, Cambridge, UK, 2001.
https://doi.org/10.1017/CBO9780511543005
2. Banach, S. Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fund. Math., 1922, 3, 133–181.
https://doi.org/10.4064/fm-3-1-133-181
3. Border, K. C. Fixed Point Theorems with Applications to Economics and Game Theory. Cambridge University Press, UK, 1985.
https://doi.org/10.1017/CBO9780511625756
4. Bortfeld, T. Optimized planning using physical objectives and constraints. Semin. Radiat. Oncol., 1999, 9, 20–34.
https://doi.org/10.1016/S1053-4296(99)80052-6
5. Chandok, S. and Narang, T. D. Some common fixed point theorems for Banach operator pairs with applications in best approx-imation. Nonlinear Anal.: Theory, Methods Appl., 2010, 73, 105–109.
6. Gupta, V., Shatanawi, W., and Mani, N. Fixed point theorems for (y;b )-Geraghty contraction type maps in ordered metric spaces and some applications to integral and ordinary differential equations. JFPTA, 2017, 19, 1251–1267.
https://doi.org/10.1007/s11784-016-0303-2
7. Gupta, V. and Kanwar, A. 2016. V-Fuzzy metric space and related fixed point theorems. Fixed Point Theory Appl., 2016, 2016: 51.
8. Kannan, R. Some results on fixed points. Bull. Calcutta Math. Soc., 1968, 25, 71–76.
9. Reich, S. Some remarks concerning contraction mappings. Canadian Math. Bull., 1971, 14, 121–124.
https://doi.org/10.4153/CMB-1971-024-9
10. Rhoades, B. E. A comparison of various definitions of contractive mappings. Trans. Amer. Math. Soc., 1977, 25, 257–290.
https://doi.org/10.1090/S0002-9947-1977-0433430-4
11. Samet, B., Vetro, C., and Vetro, P. Fixed point theorems for α – ѱ contractive type mappings. Nonlinear Anal., 2012, 75, 2154–2165.
https://doi.org/10.1016/j.na.2011.10.014
12. Shahi, P., Kaur, J., and Bhatia, S. S. Fixed point theorems for (ξ ,α)-expansive mappings in complete metric space. Fixed Point Theory Appl., 2012, 2012: 157.
13. Shepard, D. M., Olivera, G. H., Reckwerdt, P. J., and Mackie, T. R. Iterative approaches to dose optimization in tomotherapy. Phys. Med. Biol., 2000, 45, 69–90.
https://doi.org/10.1088/0031-9155/45/1/306
14. Tian, Z., Zarepisheh, M., Jia, X., and Jiang, S. B. 2013. The fixed-point iteration method for IMRT optimization with truncated dose deposition coefficient matrix. arXiv: 1303.3504 [physics.med-ph].