Weakly Picard mappings: Retraction-displacement condition, quasicontraction notion and weakly Picard admissible perturbation

Ioan A. Rus

Abstract


Let \((X,d)\) be a metric space, \(f:X\to X\) be a mapping and \(G(\cdot, f(\cdot))\) be an admissible perturbation of \(f\). In this paper we study the following problems: In which conditions imposed on \(f\) and \(G\) we have the following:\((DDE)\) data dependence estimate for the mapping \(f\) perturbation;\((UH)\) Ulam-Hyers stability for the equation, \(x=f(x)\);\((WP)\) well-posedness of the fixed point problem for \(f\);\((OP)\) Ostrowski property of the mapping \(f\). Some research directions are suggested.

Keywords


metric space; fixed point equation; Picard mapping; weakly Picard mapping; admissible perturbation; retraction-displacement condition; data dependence estimate; Ulam-Hyers stability; well-posedness; Ostrowski property; quasicontraction

Full Text:

PDF

References


Agarwal, P., Jleli, M., Samet, B., Fixed Point Theory in Metric Spaces, Springer, 2018.

Alghamdi, M.A., Shahzad, N., Valero, O., Fixed point theorems in generalized metric spaces with applications to computer science, Fixed Point Theory Appl., 118(2013), 20

pp. DOI: https://doi.org/10.1186/1687-1812-2013-118

Berinde, V., Iterative Approximation of Fixed Points, Springer Berlin, Heidelberg, 2007. DOI: https://doi.org/10.1007/978-3-540-72234-2

Berinde, V., Approximating fixed points of enriched nonexpansive mappings by Krasnoselskij iteration in Hilbert spaces, Carpathian J. Math., 35(2019), no. 3, 293-304. DOI:

https://doi.org/10.37193/cjm.2019.03.04

Berinde, V., Choban, M., Generalized distances and their associate metrics. Impact on fixed point theory, Creat. Math. Inform., 22(2013), no. 1, 23-32.

Berinde, V., M aru ster, S t., Rus, I.A., An abstract point of view on iterative approximation of fixed points of nonself operators, J. Nonlinear Convex Anal., 15(2014), no. 5,

-865.

Berinde, V., P acurar, M., Approximating fixed points of enriched contractions in Banach spaces, J. Fixed Point Theory Appl., 2020, 22-38. DOI: https://doi.org/10.1007/s11784-

-0769-9

Berinde, V., Petru sel, A., Rus, I.A., Remarks on the terminology of the mappings in fixed point iterative methods in metric spaces, Fixed Point Theory, 24(2023), no. 2, 525-540.

Berinde, V., Petru sel, A., Rus, I.A., S erban, M.A., The retraction-displacement condition in the theory of fixed point equation with a convergent iterative algorithm, In: Rassias,

T.M., Gupta, V. (eds.), Mathematical Analysis, Approximation Theory and Their Applications, Springer, 2016, 75-106. DOI: https://doi.org/10.1007/978-3-319-31281-1 4

Berinde, V., Rus, I.A., Asymptotic regularity, fixed point and successive approximations, Filomat, 34(2020), no. 3, 965-981. DOI: https://doi.org/10.2298/ l2003965b

Browder, F.E., Convergence of approximants to fixed points of nonexpansive nonlinear mapping in Banach spaces, Arch. Rat. Mech. Anal., 24(1967), no. 1, 82-90. DOI:

https://doi.org/10.1007/bf00251595

Browder, F.E., Petryshyn, W.V., Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20(1967), no. 2, 197-228. DOI:

https://doi.org/10.1016/0022-247x(67)90085-6

Bruck, R.E., Random products of contractions in metric and Banach spaces, J. Math. Anal. Appl., 88(1982), 319-332. DOI: https://doi.org/10.1016/0022-247x(82)90195-0

Bruck, R.E., Asymptotic behavior of nonexpansive mappings, Contemporary Mathematics, 18(1983), 1-47. DOI: https://doi.org/10.1090/conm/018/728592

Buica, A., Principii de coincidenta si aplicatii, Presa Univ. Clujean a, Cluj-Napoca, 2001.

Buica, A., Rus, I.A., Serban, M.A., Zero point principle of ball-near identity operators

and applications to implicit operator problem, Fixed Point Theory, 21(2020), no. 1, 79-92. DOI: https://doi.org/10.24193/fpt-ro.2020.1.06

Chidume, C.E., Maruster, S t., Iterative methods for the computation of fixed points of demicontractive mappings, J. Comput. Appl. Math., 234(2010), 861-882. DOI:

https://doi.org/10.1016/j.cam.2010.01.050

Chis-Novac, A., Precup, R., Rus, I.A., Data dependence of fixed points for non-self generalized contractions, Fixed Point Theory, 10(2009), no. 1, 73-87.

Coman, Gh., Pavel, G., Rus, I., Rus, I.A., Introducere in teoria ecuatiilor operatoriale, Editura Dacia, Cluj-Napoca, 1976.

Edelstein, M., A remark on a theorem of M.A. Krasnoselski, Amer. Math. Monthly, 73(1966), 509-510. DOI: https://doi.org/10.2307/2315474

Edelstein, M., O'Brien, R.C., Nonexpansive mappings, asymptotic regularity and successive approximations, J. London Math. Soc., 17(1978), 547-554. DOI:

https://doi.org/10.1112/jlms/s2-17.3.547

Eirola, T., Nevanlinna, O., Pilyugin, S.Yu., Limit shadowing property, Numer. Funct. Anal. Optim., 18(1997), no. 1-2, 75-92.

DOI: https://doi.org/10.1080/01630569708816748

Ey, K., Potzsche, C., Asymptotic behavior of recursions via fixed point theory, J. Math. Anal. Appl., 337(2008), 1125-1141. DOI: https://doi.org/10.1016/j.jmaa.2007.04.052

Filip, A.D., Fixed Point Theory in Kasahara Spaces, Casa Cartii de Stiinta, Cluj-Napoca, 2015.

Filip, A.D., Conversions between generalized metric spaces and standard metric spaces with applications in fixed point theory, Carpathian J. Math., 37(2021), no. 2, 345-354.

DOI: https://doi.org/10.37193/cjm.2021.02.19

Filip, A.D., Rus, I.A., Fixed point theory for nonself generalized contractions in Kasahara spaces, An. Univ. Vest, Timi soara, Mat.-Inform., 57(2019), no. 1, 66-76. DOI:

https://doi.org/10.2478/awutm-2019-0007

Frigon, M., Fixed point and continuation results for contractions in metric and gauge spaces, Banach Center Publications, 77(2007), 89-114.

DOI: https://doi.org/10.4064/bc77-0-8

Goebel, K., Kirk, W.A., Topics in Metric Fixed Point Theory, Cambridge Univ. Press, 1990.

Hitzler, P., Generalized Metrics and Topology in Logic Programming Semantics, Dissertation for Doctor in Philosophy, National Univ. of Ireland, 2001.

Jleli, M., Nashine, H.K., Samet, B., Vetro, C., On multivalued weakly Picard operators in partial Hausdor metric spaces, Fixed Point Theory Appl., 2015:52. DOI:

https://doi.org/10.1186/s13663-015-0293-6

Kirk, W.A., Shahzad, N., Fixed Point Theory in Distance Spaces, Springer, 2014. DOI: https://doi.org/10.1007/978-3-319-10927-5

Lee, K., Sakai, K., Various shadowing properties and their equivalence, Disc. Contin. Dynamical Systems, 13(2005), no. 2, 533-539.

DOI: https://doi.org/10.3934/dcds.2005.13.533

Lemair, B., Well-posedness, conditioning and regularization of minimization, inclusion and fixed-point problems, Pliska Stud. Math. Bulgar, 12(1998), 71-84.

Ortega, J.M., Rheinboldt, W.C., On a class of approximate iterative processes, Arch. Rat. Mech. Anal., 23(1967), 352-365. DOI: https://doi.org/10.1007/bf00276778

Ortega, J.M., Rheinboldt, W.C., Iterative Solution of Nonlinear Equations in Several Variables, Acad. Press, New York, 1970. DOI: https://doi.org/10.1016/c2013-0-11263-9

Park, S., Almost all about Rus-Hicks-Rhoades maps in quasi-metric spaces, Adv. Theory of Nonlinear Anal. Appl., 7(2023), no. 1, 455-472.

Park, S., Relatives of a Theorem of Rus-Hicks-Rhoades, Letters in Nonlinear Analysis and its Application, 1(2023), no. 2, 57-63.

Pacurar, M., Rus, I.A., Some remarks on the notions and terminology in the ordered set theory, Creat. Math. Inform., 27(2018), no. 2, 191-195.

Pacurar, M., Rus, I.A., Fixed point theory of cyclic operators, J. Fixed Point Appl., 2022, 24:79. DOI: https://doi.org/10.1007/s11784-022-00996-z

Petrusel, A., Rus, I.A., An abstract point of view on iterative approximation schemes of fixed points for multivalued operators, J. Nonlinear Sci. Appl., 6(2013), 97-107. DOI:

https://doi.org/10.22436/jnsa.006.02.05

Petrusel, A., Rus, I.A., Fixed Point Theory in terms of a metric and of an order relation, Fixed Point Theory, 20(2019), no. 2, 601-622. DOI: https://doi.org/10.24193/fptro.

2.40

Petrusel, A., Rus, I.A., Graphic contraction principle and applications, In: Rassias, Th. et al. (eds.), Springer, 2019, 395-416. DOI: https://doi.org/10.1007/978-3-030-31339-5 15

Petrusel, A., Rus, I.A., Ulam stability of zero point equations, In: J. Brzdek et al. (eds.), Ulam Type Stability, Springer, 2019. DOI: https://doi.org/10.1007/978-3-030-28972-

16

Petrusel, A., Rus, I.A., Serban, M.A., Basic problems of the metric fixed point theory and the relevance of a metric fixed point theorem for a multivalued operator, J. Nonlinear

Convex Anal., 15(2014), no. 3, 493-513.

Petryshyn, W.V., Construction of fixed points of demicompact mappings in Hilbert space, J. Math. Anal. Appl., 14(1966), 276-284. DOI: https://doi.org/10.1016/0022-

x(66)90027-8

Petryshyn, W.V., Williamson, T.F., Strong and weak convergence of the sequence of successive approximations for quasi-nonexpansive mappings, J. Math. Anal. Appl.,

(1973), 459-497. DOI: https://doi.org/10.1016/0022-247x(73)90087-5

Rus, I.A., Principii si aplicat ii ale teoriei punctului x, Editura Dacia, Cluj-Napoca, 1979.

Rus, I.A., Generalized Contractions and Applications, Cluj Univ. Press, Cluj-Napoca, 2001.

Rus, I.A., Data dependence of the fixed points in a set with two metrics, Fixed Point Theory, 8(2007), no. 1, 115-123.

Rus, I.A., Picard operators and well-posedness of fixed point problems, Stud. Univ. Babes-Bolyai Math., 52(2007), no. 3, 147-156.

Rus, I.A., Fixed point theory in partial metric spaces, An. Univ. Vest Timi soara, Mat-Inform., 46(2008), no. 2, 149-160.

Rus, I.A., Properties of the solution of those equations for which the Krasnoselskii iteration converges, Carpathian J. Math., 28(2012), no. 2, 329-336. DOI:

https://doi.org/10.37193/cjm.2012.02.02

Rus, I.A., An abstract point of view on iterative approximation of fixed points: Impact on the theory of fixed point equations, Fixed Point Theory, 13(2012), no. 1, 179-192.

Rus, I.A., The generalized retraction methods in fixed point theory for nonself operators, Fixed Point Theory, 15(2014), no. 2, 559-578.

Rus, I.A., Results and problems in Ulam stability of operatorial equations and inclusions, In: Th.M. Rassias (ed.), Handbook of Functional Equations: Stability Theory, Springer,

, 323-352. DOI: https://doi.org/10.1007/978-1-4939-1286-5 15

Rus, I.A., Some variants of contraction principle, generalizations and applications, Stud. Univ. Babe s-Bolyai Math., 61(2016), no. 3, 343-358.

Rus, I.A., Relevant classes of weakly Picard operators, An. Univ. Vest Timi soara, Mat.-Inform., 54(2016), no. 2, 3-19. DOI: https://doi.org/10.1515/awutm-2016-0019

Rus, I.A., Convergence results for fixed point iterative algorithms in metric spaces, Carpathian J. Math., 35(2019), no. 2, 209-220.

DOI: https://doi.org/10.37193/cjm.2019.02.09

Rus, I.A., Set-theoretical aspect of the fixed point theory: Some examples, Carpathian J. Math., 37(2021), no. 2, 235-258. DOI: https://doi.org/10.37193/cjm.2021.02.10

Rus, I.A., Around metric coincidence point theory, Stud. Univ. Babe s-Bolyai Math., 68(2023), no. 2, 449-463.

Rus, I.A., Petrusel, A., Petrusel, G., Fixed Point Theory, Cluj Univ. Press, Cluj-Napoca, 2008.

Rus, I.A., Petrusel, A., Santamarian, A., Data dependence of the fixed point set of some multivalued weakly Picard operators, Nonlinear Anal., 52(2003), 1947-1959. DOI:

https://doi.org/10.1016/s0362-546x(02)00288-2

Rus, I.A., Serban, M.A., Some generalizations of a Cauchy lemma and applications, In: Topics in Mathematics, Computer Science and Philosophy, Cluj Univ. Press, 2008,

-181.

Rus, I.A., Serban, M.A., Basic problems of the metric fixed point theory and the relevance of a metric fixed point theorem, Carpathian J. Math., 29(2013), no. 2, 239-258.

Sine, R.C. (ed.), Fixed Points and Nonexpansive mappings, Contemporary Mathematics, 18(1983).

Takahashi, W., Nonlinear Functional Analysis, Yokohama Publishers, 2000.

Tricomi, F., Un teorema sulla convergenza delle successioni formate delle successive iterate di una funzione di una variable reale, Giorn. Mat Bottaglini, 54(1916), 1-9.

Zabrejko, P.P., K-metric and K-normed spaces: Survey, Collect. Math., 48(1997), no. 4-6, 825-859.




DOI: http://dx.doi.org/10.24193/subbmath.2024.1.13

Refbacks

  • There are currently no refbacks.