Existence, error estimation, rate of convergence, Ulam-Hyers stability, well-posedness and limit shadowing property related to a fixed point problem
Resumen
In this paper we consider a fixed point problem where the mapping is supposed to satisfy a generalized contractive inequality involving rational terms. We first prove the existence of a fixed point of such mappings. Then we show that the fixed point is unique under some additional assumptions. We investigate four aspects of the problem, namely, error estimation and rate of convergence of the fixed point iteration, Ulam-Hyers stability, well-psoedness and limit shadowing property. In the existence theorem we use an admissibility condition. Two illustration are given. The research is in the line with developing fixed point approaches relevant to applied mathematics.
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Amiri, P., Rezapour, Sh., Shahzad, N., Fixed points of generalized α − ψ-contractions, RACSAM 108 (2), 519-526, (2014). https://doi.org/10.1007/s13398-013-0123-9
Berinde, V., Error estimates for approximating fixed points of quasi contractions, Gen. Math. 13(2), 23-34, (2005).
Carl, S., Heikkila, S., Fixed Point Theory in Ordered Sets and Applications, Springer-Verlag New York, 2011, https://doi.org/10.1007/978-1-4419-7585-0
Chifu, C., Petru¸sel, G., Coupled fixed point results for (ϕ, G)-contractions of type (b) in b-metric spaces endowed with a graph, J. Nonlinear Sci. Appl. 10, 671-683, (2017). https://doi.org/10.22436/jnsa.010.02.29
Cieplinski, K., Applications of fixed point theorems to the Hyers-Ulam stability of functional equations - a suvey, Ann. Funct. Anal. 3(1), 151-164, (2012). https://doi.org/10.15352/afa/1399900032
Elqorachi, E., Rassias, T. M. , Generalized Hyers-Ulam stability of trigonometric functional equations, P Mathematics 6, (2018). https://doi.org/10.3390/math6050083
Hussain, N., Rafiq, A., Damjanovic, B., Lazovic, R., On rate of convergence of various iterative schemes, Fixed Point Theory Appl. 2011: 45, (2011). https://doi.org/10.1186/1687-1812-2011-45
Hussain, N., Karapinar, E., Salimi, P., Akbar, F., α-admissible mappings and related fixed point theorems, J. Inequal. Appl. 2013 : 114, (2013). https://doi.org/10.1186/1029-242X-2013-114
Hyers, D. H., On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27(4), 222-224, (1941). https://doi.org/10.1073/pnas.27.4.222
Kutbi, M. A., Sintunavarat, W., Ulam-Hyers stability and well-posedness of fixed point problems for α − λ-contraction mapping in metric spaces, Abstr. Appl. Anal. 2014, Article ID 268230, 6 pages, (2014). https://doi.org/10.1155/2014/268230
Lahiri, B. K., Das, P., Well-posedness and porosity of a certain class of operators, Demonstratio Math. XXXVIII(1), 169-176, (2005). https://doi.org/10.1515/dema-2005-0119
Murali, R., Antony Raj, A., Deboral, M., Hyers-Ulam stability of the isometric Cauchy-Jenson mapping in generalized quasi-banach spaces, Int. J. Adv. Appl. Math. Mech. 3(4), 16-21, (2016).
Phiangsungnoen, S., Kumam, P., Generalized Ulam-Hyers stability and well-posedness for fixed point equation via α-admissibility, J. Inequal. Appl. 2014: 418, (2014). https://doi.org/10.1186/1029-242X-2014-418
Phuengrattana, W., Suantai, S., On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval, J. Comput. Appl. Math. 235, 3006-3014, (2011). https://doi.org/10.1016/j.cam.2010.12.022
Popa, V., Well-posedness of fixed point problem in orbitally complete metric spaces, Stud. Cercet. Stiint., Ser. Mat. 16, 209-214, (2006).
Rassias, T. M., On the stability of the linear mappings in Banach spaces, Proc. Amer. Math. Soc. 72, 297-300, (1978). https://doi.org/10.1090/S0002-9939-1978-0507327-1
Rassias, T. M., Isometries and approximate isometries, IJMMS 25:2, 73-91, (2001). https://doi.org/10.1155/S0161171201004392
Rus, I. A., Remarks on Ulam stability of the operatorial equations, Fixed Point Theory 10(2), 305-320, (2009).
Samet, B., Vetro, C., Vetro, P., Fixed point theorem for α − ψ- contractive type mappings, Nonlinear Anal. 75, 2154-2165, (2012). https://doi.org/10.1016/j.na.2011.10.014
Samet, B., Fixed points for α − ψ- contractive mappings with an application to quadratic integral equations, Electron. J. Differential Equations 2014 ( No.152), 1-18, (2014).
Sen, M., Saha, D., Agarwal, R. P., A Darbo fixed point theory approach towards the existence of a functional integral equation in a Banach algebra, Appl. Math. Comput. 358, 111-118, (2019). https://doi.org/10.1016/j.amc.2019.04.021
Sintunavarat, W., Generalized Ulam-Hyers stability, well-posedness and limit shadowing of fixed point problems for α − β− contraction mapping in metric spaces, The Scientific World Journal 2014, Article ID 569174, 7 pages, (2014). https://doi.org/10.1155/2014/569174
Salimi, P., Latif, A., Hussain, N. Modified α − ψ- contractive mappings with applications, Fixed Point Theory Appl. 2013 : 151, (2013). https://doi.org/10.1186/1687-1812-2013-151
Ulam, S. M., Problems in Modern Mathematics, Wiley, New York (1964).
Yildirim, I., Abbas, M. Convergence rate of implicit iteration process and a data dependence result, Eur. J. Pure Appl. Math. 11 (1), 189-201, (2018). https://doi.org/10.29020/nybg.ejpam.v11i1.2911
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