Fixed point results of enriched interpolative Kannan type operators with applications

Authors

  • Mujahid Abbas Government College University ; University of Pretoria
  • Rizwan Anjum Riphah International University
  • Shakeela Riasat Government College University

DOI:

https://doi.org/10.4995/agt.2022.16701

Keywords:

Fixed point, enriched Kannan operators, interpolative Kannan type contraction, Krasnoselskij iteration, well-posedness, periodic point, Ulam-Hyers stability, variational inequality problem

Abstract

The purpose of this paper is to introduce the class of enriched interpolative Kannan type operators on Banach space that contains the
classes of enriched Kannan operators, interpolative Kannan type contraction operators and some other classes of nonlinear operators. Some examples are presented to support the concepts introduced herein. A convergence theorem for the Krasnoselskij iteration method to approximate fixed point of the enriched interpolative Kannan type operators is proved. We study well-posedness, Ulam-Hyers stability and periodic point property of operators introduced herein. As an application of the main result, variational inequality problems is solved.

Downloads

Download data is not yet available.

Author Biographies

Mujahid Abbas, Government College University ; University of Pretoria

Department of Mathematics, Government College University (Pakistan) ; Department of Mathematics and Applied Mathematics, University of Pretoria (South Africa)

Rizwan Anjum, Riphah International University

Department of Mathematics, Riphah Institute of Computing and Applied Sciences

Shakeela Riasat, Government College University

Abdus Salam School of Mathematical Sciences

References

R. Anjum and M. Abbas, Fixed point property of a nonempty set relative to the class offriendly mappings, Rev. R. Acad. Cienc. Exactas Fıs. Nat. Ser. A Mat. RACSAM 116,no. 1 (2022), Paper no. 32. https://doi.org/10.1007/s13398-021-01158-5

R. Anjum and M. Abbas, Common fixed point theorem for modified Kannan enrichedcontraction pair in Banach spaces and its applications, Filomat 35, no. 8 (2021) 2485-2495. https://doi.org/10.2298/FIL2108485A

M. Abbas, R. Anjum and V. Berinde, Enriched multivalued contractions with applica-tions to differential inclusions and dynamic programming, Symmetry 13, no. 8 (2021),Paper no. 1350. https://doi.org/10.3390/sym13081350

M. Abbas, R. Anjum and V. Berinde, Equivalence of certain iteration processes obtainedby two new classes of operators, Mathematics 9, no. 18 (2021), Paper no. 2292. https://doi.org/10.3390/math9182292

M. Abbas, R. Anjum and H. Iqbal, Generalized enriched cyclic contractions with appli-cation to generalized iterated function system, Chaos, Solitons and Fractals 154 (2022),Paper no. 111591. https://doi.org/10.1016/j.chaos.2021.111591

R. P. Agarwal and E. Karapinar, Interpolative Rus-Reich- Ciric type contractions viasimulation functions, An. St. Univ. Ovidius Constanta, Ser. Mat. 27, no. 3 (2019), 137-152. https://doi.org/10.2478/auom-2019-0038

R. P. Agarwal, E. Karapinar, D. O'Regan and A. F. Roldan Lopez de Hierro, Fixedpoint theory in metric type spaces, Springer International Publishing, 2015.

H. Aydi, E. Karapinar and A. F. Roldan Lopez de Hierro, w-Interpolative Ciric-Reich-Rus-type contractions, Mathematics 7, no. 57 (2019), Paper no. 57. https://doi.org/10.3390/math7010057

H. Aydi, C. M. Chen and E. Karapinar, Interpolative Ciric-Reich-Rustypes via the Branciari distance, Mathematics 7, no. 1 (2019), Paper no. 84. https://doi.org/10.3390/math7010084

F. S. D. Blasi and J. Myjak, Sur la porosité de l'ensemble des contractions sans point fixe, C. R. Acad. Sci. Paris. 308 (1989), 51-54.

V. Berinde and M. Păcurar, Approximating fixed points of enriched contractions in Banach spaces, Journal of Fixed Point Theory and Applications 22, no. 2 (2020), 1-10. https://doi.org/10.1007/s11784-020-0769-9

V. Berinde and M. Păcurar, Kannan's fixed point approximation for solving split feasibility and variational inequality problems, Journal of Computational and Applied Mathematics 386 (2021), Paper no. 113217. https://doi.org/10.1016/j.cam.2020.113217

V. Berinde, Approximating fixed points of enriched nonexpansive mappings by Krasnoselskii iteration in Hilbert spaces, Carpathian J. Math. 35, no. 3 (2019), 293-304. https://doi.org/10.37193/CJM.2019.03.04

V. Berinde and M. Păcurar, Approximating fixed points of enriched Chatterjea contractions by Krasnoselskii iterative algorithm in Banach spaces, Journal of Fixed Point Theory and Applications 23, no. 4 (2021), 1-16. https://doi.org/10.1007/s11784-021-00904-x

V. Berinde and M. Păcurar, Fixed point theorems for enriched Ćirić-Reich-Rus contractions in Banach spaces and convex metric spaces, Carpathian J. Math. 37 (2021), 173-184. https://doi.org/10.37193/CJM.2021.02.03

F. E. Browder and W. V. Petryshyn, The solution by iteration of nonlinear functional equations in Banach spaces, Bull. Am. Math. Soc. 72 (1966), 571-575. https://doi.org/10.1090/S0002-9904-1966-11544-6

C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Probl. 20, no. 1 (2004), 103-120. https://doi.org/10.1088/0266-5611/20/1/006

E. Connell, Properties of fixed point spaces, Proc. Amer. Math. Soc. 10 (1959), 974-979. https://doi.org/10.1090/S0002-9939-1959-0110093-3

I. C. Chifu and E. Karapinar, Admissible hybrid Z-contractions in b-metric spaces, Axioms. 9, no. 1, (2020), Paper no. 2. https://doi.org/10.3390/axioms9010002

P. Debnath and M. de La Sen, Set-valued interpolative Hardy-Rogers and set-valued Reich-Rus-Ćirić-type contractions in b-metric spaces, Mathematics. 7, no. 9 (2019), Paper no. 849. https://doi.org/10.3390/math7090849

Y. U. Gaba and E. Karapinar, A new approach to the interpolative contractions, Axioms 8, no. 4 (2019), Paper no. 110. https://doi.org/10.3390/axioms8040110

Y. U. Gaba, H. Aydi and N. Mlaik, (ρ,η,μ)-interpolative Kannan contractions I, Axioms. 10, no. 3 (2021), Paper no. 212. https://doi.org/10.3390/axioms10030212

G. S. Jeong and B. E. Rhoades, Maps for which $F(T)=F(T^{n}),$ Fixed Point Thoery Appl. 6 (2005), 87-131.

J. Górnicki and R. K. Bisht, Around averaged mappings, Journal of Fixed Point Theory and Applications 23 (2021), Paper no. 48. https://doi.org/10.1007/s11784-021-00884-y

R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc. 60 (1968), 71-76. https://doi.org/10.2307/2316437

R. Kannan, Some results o fixed points, II, Amer. Math. Monthly 76 (1969), 405-408. https://doi.org/10.1080/00029890.1969.12000228

M. Kikkawa and T. Suzuki, Some similarity between contractions and Kannan mappings II, Bull. Kyushu Inst. Technol. Pure Appl. Math. 55 (2008) 1-13. https://doi.org/10.1155/2008/649749

E. Karapinar, Revisiting the Kannan type contractions via interpolation, Adv. Theory Nonlinear Anal. Appl. 2, no. 2 (2018), 85-87. https://doi.org/10.31197/atnaa.431135

E. Karapinar, O. Alqahtani and H. Aydi, On interpolative Hardy-Rogers type contractions, Symmetry. 11, no. 1 (2019), Paper no. 8. https://doi.org/10.3390/sym11010008

E. Karapinar, R. Agarwal and H. Aydi, Interpolative Reich-Rus-Ćirić type contractions on partial metric spaces, Mathematics 6, no. 11 (2018), Paper no. 256. https://doi.org/10.3390/math6110256

E. Karapinar and A. Fulga, New hybrid contractions on b-metric spaces, Mathematics 7, no. 7 (2019), Paper no. 578. https://doi.org/10.3390/math7070578

E. Karapinar, H. Aydi and Z. D. Mitrovic, On interpolative Boyd-Wong and Matkowski type contractions, TWMS J. Pure Appl. Math. 11, no. 2 (2020), 204-212.

E. Karapinar, A. Fulga and A. F. Roldán López de Hierro, Fixed point theory in the setting of (α,β,ψ,ϕ)-interpolative contractions, Advances in Difference Equations 1, (2021), 1-16. https://doi.org/10.1186/s13662-021-03491-w

E. Karapinar, Interpolative Kannan-Meir-Keeler type contractions, Advances in the Theory of Nonlinear Analysis and its Applications 5, no. 4 (2021), 611-614. https://doi.org/10.31197/atnaa.989389

M. S. Khan, Y. M. Singh and E. Karapinar, On the interpolative (ϕ,ψ)-type Z-contractions, U. P. B. Sci. Bull. Series A. 83, no. 2 (2021), 25-38.

E. Karapinar, A. Fulga and S. S. Yesilkaya, New Results on Perov-interpolative contractions of Suzuki type mappings, Journal of Functional Spaces 2021, Art. ID 9587604. https://doi.org/10.1155/2021/9587604

M. A. Krasnoselskii, Two remarks on the method of successive approximations, Uspekhi Mat. Nauk. 10 (1955), 123-127.

M. Noorwali, Common fixed point for Kannan type via interpolation, J. Math. Anal. 9, no. 6 (2018), 92-94.

S. Reich and A. J. Zaslavski, Well-posedness of fixed point problems, Far East Journal of Mathematical Sciences (FJMS) 2001, Special volume, Part III, 393-401.

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

P. V. Subrahmanyam, Remarks on some fixed point theorems related to Banach's contraction principle, J. Math. Phys. Sci. 8 (1974), 445-457.

W. Sintunavarat, Generalized Ulam-Hyres stability, well-posedness and limit shadowing of fixed point problems for α-β-contraction mapping in metric spaces, The Sci. World J. 2014, Article ID 569174. https://doi.org/10.1155/2014/569174

Downloads

Published

2022-10-03

How to Cite

[1]
M. Abbas, R. Anjum, and S. Riasat, “Fixed point results of enriched interpolative Kannan type operators with applications”, Appl. Gen. Topol., vol. 23, no. 2, pp. 391–404, Oct. 2022.

Issue

Section

Articles