Browse > Article
http://dx.doi.org/10.22771/nfaa.2021.26.01.09

FIXED POINT THEOREMS IN COMPLEX VALUED CONVEX METRIC SPACES  

Okeke, G.A. (Department of Mathematics, School of Physical Sciences Federal University of Technology Owerri)
Khan, S.H. (Department of Mathematics, Statistics and Physics Qatar University)
Kim, J.K. (Department of Mathematics, Statistics and Physics Qatar University)
Publication Information
Nonlinear Functional Analysis and Applications / v.26, no.1, 2021 , pp. 117-135 More about this Journal
Abstract
Our purpose in this paper is to introduce the concept of complex valued convex metric spaces and introduce an analogue of the Picard-Ishikawa hybrid iterative scheme, recently proposed by Okeke [24] in this new setting. We approximate (common) fixed points of certain contractive conditions through these two new concepts and obtain several corollaries. We prove that the Picard-Ishikawa hybrid iterative scheme [24] converges faster than all of Mann, Ishikawa and Noor [23] iterative schemes in complex valued convex metric spaces. Also, we give some numerical examples to validate our results.
Keywords
Complex valued convex metric spaces; iterative schemes; fixed point; rate of convergence;
Citations & Related Records
연도 인용수 순위
  • Reference
1 H. Akewe, G.A. Okeke and A.F. Olayiwola, Strong convergence and stability of Kirkmultistep-type iterative schemes for contractive-type operators, Fixed Point Theory Appl., 2014:46 (2014), 24 pages.   DOI
2 H. Akewe and G.A. Okeke, Convergence and stability theorems for the Picard-Mann hybrid iterative scheme for a general class of contractive-like operators, Fixed Point Theory Appl., 2015:66 (2015), 8 pages.   DOI
3 C.D. Alecsa, On new faster fixed point iterative schemes for contraction operators and comparison of their rate of convergence in convex metric spaces, Int. J. Nonlinear Anal. Appl., 8(1) (2017), 353-388.
4 W.M. Alfaqih, M. Imdad and F. Rouzkard, Unified common fixed point theorems in complex valued metric spaces via an implicit relation with applications, Bol. Soc. Paran. Mat. (3s), 38(4) (2020), 9-29.
5 A. Azam, B. Fisher and M. Khan, Common fixed point theorems in complex valued metric spaces, Numer. Funct. Anal. Optimi., 32(3) (2011), 243-253.   DOI
6 I. Beg, M. Abbas and J.K. Kim, Convergence theorems of the iterative schemes in convex metric spaces, Nonlinear Funct. Anal. Appl., 11(3) (2006), 421-436.
7 V. Berinde, Iterative approximation of fixed points, Lecture Notes in Mathematics, Springer-Verlag Berlin Heidelberg, 2007.
8 S.S. Chang and J.K. Kim, Convergence theorems of the Ishikawa type iterarive sequences with errors for generalized quasi-contractive mappings in convex metric spaces, Appl. Math. Letters, 16(4) (2003), 535-542   DOI
9 B.K. Dass and S. Gupta, An extension of Banach contraction principle through rational expression, Indian J. Pure Appl. Math., 6 (1975), 1455-1458.
10 O. Ege, Complex valued rectangular b-metric spaces and an application to linear equations, J. Nonlinear Sci. Appl., 8(6) (2015), 1014-1021.   DOI
11 O. Ege, Complex valued Gb-metric spaces, J. Comput. Anal. Appl., 21(2) (2016), 363-368.
12 O. Ege, Some fixed point theorems in complex valued Gb-metric spaces, J. Nonlinear Convex Anal., 18(11) (2017), 1997-2005.
13 O. Ege and I. Karaca, Common fixed point results on complex valued Gb-metric spaces, Thai J. Math., 16(3) (2018), 775-787.
14 O. Ege and I. Karaca, Complex valued dislocated metric spaces, Korean J. Math., 26(4) (2018), 809-822.   DOI
15 H. Fukhar-ud-din and V. Berinde, Iterative methods for the class of quasi-contractive type operators and comparison of their rate of convergence in convex metric spaces, Filomat, 30(1) (2016), 223-230.   DOI
16 K. Goebel and W.A. Kirk, Topics in metric fixed point theory, Cambridge Stud. Adv. Math., 28, Cambridge University Press, London, 1990.
17 D.S. Jaggi, Some unique fixed point theorems, Indian J. Pure and Appl. Math., 8(2) (1977), 223-230.
18 D.S. Jaggi and B.K. Dass, An extension of Banach's fixed point theorem through rational expression, Bull. Cal. Math., 72 (1980), 261-266.
19 S.H. Khan, A Picard-Mann hybrid iterative process, Fixed Point Theory Appli., 2013:69 (2013), 10 pages.   DOI
20 L. Khan, Fixed Point Theorem for Weakly Contractive Maps in Metrically Convex Spaces under C-Class Function, Nonlinear Funct. Anal. Appl., 25(1) (2020), 153-160.   DOI
21 G.A. Okeke, Convergence analysis of the Picard-Ishikawa hybrid iterative process with applications, Afrika Matematika, 30 (2019), 817-835.   DOI
22 J.K. Kim, S.A. Chun and Y.M. Nam, Convergence theorems of iterative sequences for generalized p-quasicontractive mappings in p-convex metric spaces, J. Comput. Anal. Appl., 10(2) (2008), 147-162
23 J.K. Kim, K.S. Kim and Y.M. Nam, Convergence and stability of iterative processes for a pair of simultaneously asymptotically quasi-nonexpansive type mappings in convex metric spaces, J. Comput. Anal. Appl., 9(2) (2007), 159-172.
24 M.A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl., 251 (2000), 217-229.   DOI
25 G.A. Okeke, Iterative approximation of fixed points of contraction mappings in complex valued Banach spaces, Arab J. Math. Sci., 25(1) (2019), 83-105.   DOI
26 G.A. Okeke and M. Abbas, A solution of delay differential equations via Picard-Krasnoselskii hybrid iterative process, Arab. J. Math., 6 (2017), 21-29.   DOI
27 G.A. Okeke, Convergence theorems for G-nonexpansive mappings in convex metric spaces with a directed graph, Rendiconti del Circolo Matematico di Palermo Series II, (2020), DOI: 10.1007/s12215-020-00535-0.   DOI
28 G.A. Okeke and M. Abbas, Fejer monotonicity and fixed point theorems with applications to a nonlinear integral equation in complex valued Banach spaces, Appl. Gen. Topol., 21(1) (2020), 135-158.   DOI
29 G.A. Okeke and M. Abbas, Convergence analysis of some faster iterative schemes for G-nonexpansive mappings in convex metric spaces endowed with a graph, Thai J. Math., 18(3) (2020), 1475-1496.
30 G.A. Okeke, J.O. Olaleru and M.O. Olatinwo, Existence and approximation of fixed point of a nonlinear mapping satisfying rational type contractive inequality condition in complex-valued Banach spaces, Inter. J. Math. Anal. Optim.: Theory and Appli., 2020(1) (2020), 707-717.
31 M.O. Olatinwo, Convergence and stability results for some iterative schemes, Acta Universitatis Apulensis, 26 (2011), 225-236.
32 W. Phuengrattana and S. Suantai, Comparison of the rate of convergence of various iterative methods for the class of weak contractions in Banach spaces, Thai J. Math.,11 (2013), 217-226.
33 G.S. Saluja, Fixed point theorems under rational contraction in complex valued metric spaces, Nonlinear Funct. Anal. Appl., 22(1) (2017), 209-216.   DOI
34 G.S. Saluja and J.K. Kim, Convergence analysis for total asymptotically nonexpansive mappings in convex metric spaces with applications, Nonlinear Funct. Anal. Appl., 25(2) (2020), 231-247.   DOI
35 B. Samet, C. Vetro and H. Yazidi, A fixed point theorem for a Meir-Keeler type contraction through rational expression, J. Nonlinear Sci. Appl., 6 (2013), 162-169.   DOI
36 W. Takahashi, A convexity in metric spaces and nonexpansive mapping I, Kodai Math. Sem. Rep. 22 (1970), 142-149.   DOI
37 I. Yidirim, S.H. Khan and M. Ozdemir, Some fixed point results for uniformly quasilipschitzian mappings in convex metric spaces, J. Nonlinear Anal. Optimi., 4(2) (2013), 143148.