Abstract
We obtain a Meir–Keeler type fixed-point theorem which gives a new solution to the Rhoades’ problem on the existence of contractive mappings that admit discontinuity at the fixed point. Meir–Keeler type solutions of the Rhoades’ problem have not been reported in literature before this. Presenting a new approach, we give another solution to this problem using the set of simulation functions. To emphasize the importance of our theoretical results, we obtain two applications of the main results with some illustrative examples.
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References
Bisht, R.K., Pant, R.P.: A remark on discontinuity at fixed points. J. Math. Anal. Appl. 445, 1239–1242 (2017)
Bisht, R.K., Pant, R.P.: Contractive definitions and discontinuity at fixed point. Appl. Gen. Topol. 18(1), 173–182 (2017)
Bisht, R.K., Rakočević, V.: Generalized Meir–Keeler type contractions and discontinuity at fixed point. Fixed Point Theory 19(1), 57–64 (2018)
Bisht, R.K., Rakočević, V.: Fixed points of convex and generalized convex contractions. Rend. Circ. Mat. Palermo Ser. 2 (2018). https://doi.org/10.1007/s12215-018-0386-2
Ćirić, Lj.B.: On contraction type mapping. Math. Balkanica 1, 52–57 (1971)
Ćirić, Lj.B.: A generalization of Banach’s contraction principle. Proc. Am. Math. Soc. 45(2), 267–273 (1974)
Cromme, L.J., Diener, I.: Fixed point theorems for discontinuous mappings. Math. Program. 51, 257–267 (1991)
Cromme, L.J.: Fixed point theorems for discontinuous functions and applications. Nonlinear Anal. 30(3), 1527–1534 (1997)
Ding, X., Cao, J., Zhao, X., Alsaadi, F.E.: Mittag–Leffler synchronization of delayed fractional-order bidirectional associative memory neural networks with discontinuous activations: state feedback control and impulsive control schemes. Proc. R. Soc. A Math. Eng. Phys. Sci. (2017). https://doi.org/10.1098/rspa.2017.0322.
Forti, M., Nistri, P.: Global convergence of neural networks with discontinuous neuron activations. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 50(11), 1421–1435 (2003)
Kannan, R.: Some results on fixed points. Bull. Calcutta Math. Soc. 60, 71–76 (1968)
Kannan, R.: Some results on fixed points-II. Am. Math. Mon. 76, 405–408 (1969)
Khojasteh, F., Shukla, S., Radenović, S.: A new approach to the study of fixed point theory for simulation functions. Filomat 29(6), 1189–1194 (2015)
Meir, A., Keeler, E.: A theorem on contraction mappings. J. Math. Anal. Appl. 28, 326–329 (1969)
Nie, X., Zheng, W. X.: On multistability of competitive neural networks with discontinuous activation functions. In: 4th Australian Control Conference (AUCC), pp. 245–250 (2014)
Nie, X., Zheng, W.X.: Multistability of neural networks with discontinuous non-monotonic piecewise linear activation functions and time-varying delays. Neural Netw. 65, 65–79 (2015)
Nie, X., Zheng, W.X.: Dynamical behaviors of multiple equilibria in competitive neural networks with discontinuous nonmonotonic piecewise linear activation functions. IEEE Trans. Cybern. 46(3), 679–693 (2015)
Özgür, N.Y., Taş, N.: Some fixed-circle theorems and discontinuity at fixed circle. AIP Conf. Proc. 1926, 020048 (2018)
Özgür, N.Y., Taş, N.: Some fixed-circle theorems on metric spaces. Bull. Malays. Math. Sci. Soc. 42(4), 1433–1449 (2019)
Özgür, N.: Fixed-disc results via simulation functions. Turk. J. Math. 43(6), 2794–2805 (2019)
Pant, A., Pant, R.P.: Fixed points and continuity of contractive maps. Filomat 31(11), 3501–3506 (2017)
Pant, A., Pant, R.P., Joshi, M.C.: Caristi type and Meir–Keeler type fixed point theorems. Filomat 33(12), 3711–3721 (2019)
Pant, R.P.: Discontinuity and fixed points. J. Math. Anal. Appl. 240, 284–289 (1999)
Pant, R.P., Özgür, N.Y., Taş, N.: On discontinuity problem at fixed point. Bull. Malays. Math. Sci. Soc. 43(1), 499–517 (2020)
Pant, R.P., Özgür, N.Y., Ta ş, N.: Discontinuity at fixed points with applications. Bull. Belg. Math. Soc. Simon Stevin 25(4), 571–589 (2019)
Rashid, M., Batool, I., Mehmood, N.: Discontinuous mappings at their fixed points and common fixed points with applications. J. Math. Anal. 9(1), 90–104 (2018)
Rhoades, B.E.: Contractive definitions and continuity. Contemp. Math. 72, 233–245 (1988)
Roldán-López-de-Hierro, A.F., Karapınar, E., Roldán-López-de-Hierro, C., Martínez-Moreno, J.: Coincidence point theorems on metric spaces via simulation functions. J. Comput. Appl. Math. 275, 345–355 (2015)
Roldán López de Hierro, A.F., Shahzad, N.: New fixed point theorem under \(R\)-contractions. Fixed Point Theory Appl. 2015, 98 (2015)
Taş, N., Özgür, N.Y.: A new contribution to discontinuity at fixed point. Fixed Point Theory 20(2), 715–728 (2019)
Taş, N., Özgür, N.Y., Mlaiki, N.: New types of \(F_{c}\)-contractions and the fixed-circle problem. Mathematics 6, 188 (2018)
Wardowski, D.: Solving existence problems via \(F\)-contractions. Proc. Am. Math. Soc. 146(4), 1585–1598 (2018)
Wu, H., Shan, C.: Stability analysis for periodic solution of BAM neural networks with discontinuous neuron activations and impulses. Appl. Math. Model. 33(6), 2564–2574 (2017)
Zheng, D., Wang, P.: Weak \(\theta \)-\(\phi \)-contractions and discontinuity. J. Nonlinear Sci. Appl. 10, 2318–2323 (2017)
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Pant, R.P., Özgür, N., Taş, N. et al. New results on discontinuity at fixed point. J. Fixed Point Theory Appl. 22, 39 (2020). https://doi.org/10.1007/s11784-020-0765-0
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DOI: https://doi.org/10.1007/s11784-020-0765-0