Fejér monotonicity and fixed point theorems with applications to a nonlinear integral equation in complex valued Banach spaces

Godwin Amechi Okeke, Mujahid Abbas


It is our purpose in this paper to prove some fixed point results and Fej´er monotonicity of some faster fixed point iterative sequences generated by some nonlinear operators satisfying rational inequality in complex valued Banach spaces. We prove that results in complex valued Banach spaces are valid in cone metric spaces with Banach algebras. Furthermore, we apply our results in solving certain mixed type VolterraFredholm functional nonlinear integral equation in complex valued Banach spaces.


Complex valued Banach spaces; fixed point theorems; Féjer monotonicity; iterative processes; cone metric spaces with Banach algebras; mixed type Volterra-Fredholm functional nonlinear integral equation

Subject classification

47H09; 47H10; 49M05; 54H25

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M. Abbas and B. E. Rhoades, Fixed and periodic point results in cone metric spaces, Appl. Math. Lett. 22 (2009), 511-515. https://doi.org/10.1016/j.aml.2008.07.001

M. Abbas, V. C. Rajic, T. Nazir and S. Radenovic, Common fixed point of mappings satisfying rational inequalities in ordered complex valued generalized metric spaces, Afr. Mat. 2013, 14 pages. https://doi.org/10.1007/s13370-013-0185-z

M. Abbas, M. Arshad and A. Azam, Fixed points of asymptotically regular mappings in complex-valued metric spaces, Georgian Math. J. 20 (2013), 213-221. https://doi.org/10.1515/gmj-2013-0013

M. Abbas and T. Nazir, A new faster iteration process applied to constrained minimization and feasibility problems, Mat. Vesn. 66 (2014), 223-234.

M. Abbas, M. De la Sen and T. Nazir, Common fixed points of generalized cocyclic mappings in complex valued metric spaces, Discrete Dynamics in Nature and Society 2015, Article ID: 147303, 2015, 11 pages. https://doi.org/10.1155/2015/147303

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, no. 4 (2020), 9-29. https://doi.org/10.5269/bspm.v38i4.37148

R. P. Agarwal, D. O'Regan and D.R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, Journal of Nonlinear and Convex Analysis 8, no. 1 (2007), 61-79.

J. Ahmad, N. Hussain, A. Azam and M. Arshad, Common fixed point results in complex valued metric space with applications to system of integral equations, Journal of Nonlinear and Convex Analysis 29, no. 5 (2015), 855-871.

H. Akewe, G. A. Okeke and A. F. Olayiwola, Strong convergence and stability of Kirk-multistep-type iterative schemes for contractive-type operators, Fixed Point Theory and Applications 2014, 2014:46, 24 pages. https://doi.org/10.1186/1687-1812-2014-45

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 and Applications (2015) 2015:66, 8 pages. https://doi.org/10.1186/s13663-015-0315-4

A. Azam, B. Fisher and M. Khan, Common fixed point theorems in complex valued metric spaces, Numerical Functional Analysis and Optimization 32, no. 3 (2011), 243-253. https://doi.org/10.1080/01630563.2011.533046

S. Banach, Sur les opérations dans les ensembles abstraits et leurs applications aux équations intégrales, Fund. Math. 3, (1922), 133-181. https://doi.org/10.4064/fm-3-1-133-181

H. H. Bauschke and P. L. Combettes, Convex analysis and monotone operator theory in Hilbert spaces, CMS Books in Mathematics, Second Edition, Springer International Publishing AG, 2017. https://doi.org/10.1007/978-3-319-48311-5_2

V. Berinde, Summable almost stability of fixed point iteration procedures, Carpathian J. Math. 19, no. 2 (2003), 81-88.

V. Berinde, Iterative approximation of fixed points, Lecture Notes in Mathematics, Springer-Verlag Berlin Heidelberg, 2007. https://doi.org/10.1109/SYNASC.2007.49

A. Cegielski, Iterative methods for fixed point problems in Hilbert spaces, Lecture Notes in Mathematics, Springer Heidelberg New York Dordrecht London, 2012. https://doi.org/10.1007/978-3-642-30901-4

R. Chugh, V. Kumar and S. Kumar, Strong convergence of a new three step iterative scheme in Banach spaces, American Journal of Computational Mathematics 2 (2012), 345-357. https://doi.org/10.4236/ajcm.2012.24048

C. Craciun and M.-A. Serban, A nonlinear integral equation via Picard operators, Fixed Point Theory 12, no. 1 (2011), 57-70.

F. Gürsoy, Applications of normal S-iterative method to a nonlinear integral equation, The Scientific World Journal 2014, Article ID 943127, 2014, 5 pages. https://doi.org/10.1155/2014/943127

F. Gürsoy and V. Karakaya, A Picard-S hybrid type iteration method for solving a differential equation with retarded argument, arXiv:1403.2546v2 [math.FA] 2014.

B. K. Dass and S. Gupta, An extension of Banach contraction principle through rational expression, Indian J. Pure Appl. Math. 6 (1975), 1455-1458.

B. C. Dhage, Generalized metric spaces with fixed point, Bull. Calcutta Math. Soc. 84 (1992), 329-336.

L.-G. Huang and X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007), 1468-1476. https://doi.org/10.1016/j.jmaa.2005.03.087

N. Hussain, V. Kumar and M. A. Kutbi, On rate of convergence of Jungck-type iterative schemes, Abstract and Applied Analysis 2013, Article ID 132626, 15 pages. https://doi.org/10.1155/2013/132626

H. Humaira, M. Sarwar and P. Kumam, Common fixed point results for fuzzy mappings on complex-valued metric spaces with homotopy results, Symmetry 11, no. 1 (2019),17 pages. https://doi.org/10.3390/sym11010061

S. Ishikawa, Fixed points by a new iteration method, Proc. Am. Math. Soc. 44 (1974), 147-150. https://doi.org/10.1090/S0002-9939-1974-0336469-5

I. Karahan and M. Ozdemir, A general iterative method for approximation of fixed points and their applications, Advances in Fixed Point Theory 3 (2013), 510-526. https://doi.org/10.1186/1687-1812-2013-244

S. H. Khan, A Picard-Mann hybrid iterative process, Fixed Point Theory and Applications 2013, 2013:69, 10 pages. https://doi.org/10.1186/1687-1812-2013-69

H. Liu and S. Xu, Cone metric spaces with Banach algebras and fixed point theorems of generalized Lipschitz mappings, Fixed Point Theory and Appl. 2013, 2013:320, 10 pages. https://doi.org/10.1186/1687-1812-2013-320

W. R. Mann, Mean value methods in iteration, Proc. Am. Math. Soc. 4 (1953), 506-510. https://doi.org/10.1090/S0002-9939-1953-0054846-3

M. A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251 (2000), 217-229. https://doi.org/10.1006/jmaa.2000.7042

G. A. Okeke, Iterative approximation of fixed points of contraction mappings in complex valued Banach spaces, Arab. J. Math. Sci. 25, no. 1 (2019), 83-105. https://doi.org/10.1016/j.ajmsc.2018.11.001

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. https://doi.org/10.1007/s40065-017-0162-8

M. Öztürk and M. Basarir, On some common fixed point theorems with rational expressions on cone metric spaces over a Banach algebra, Hacettepe J. Math. and Stat. 41, no. 2 (2012), 211-222.

W. Phuengrattana and S. Suantai, On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval, Journal of Computational and Applied Mathematics 235 (2011), 3006-3014. https://doi.org/10.1016/j.cam.2010.12.022

F. Rouzkard and M. Imdad, Some common fixed point theorems on complex valued metric spaces, Computers and Mathematics with Applications 64 (2012), 1866-1874. https://doi.org/10.1016/j.camwa.2012.02.063

W. Rudin, Functional analysis, 2nd edn. McGraw-Hill, New York, 1991.

D. R. Sahu and A. Petrusel, Strong convergence of iterative methods by strictly pseudocontractive mappings in Banach spaces, Nonlinear Analysis: Theory, Methods & Applications 74, no. 17 (2011), 6012-6023. https://doi.org/10.1016/j.na.2011.05.078

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. https://doi.org/10.22436/jnsa.006.03.02

S. Shukla, R. Rodríguez-López and M. Abbas, Fixed point results for contractive mappings in complex valued fuzzy metric spaces, Fixed Point Theory 19, no. 2 (2018), 751-774.

N. Singh, D. Singh, A. Badal and V. Joshi, Fixed point theorems in complex valued metric spaces, Journal of the Egyptian Math. Soc. 24 (2016), 402-409. https://doi.org/10.1016/j.joems.2015.04.005

W. Sintunavarat and P. Kumam, Generalized common fixed point theorems in complex valued metric spaces and applications, J. Ineq. Appl. 2012, 2012:84' 12 pages. https://doi.org/10.1186/1029-242X-2012-84

S. M. Soltuz and T. Grosan, Data dependence for Ishikawa iteration when dealing with contractive-like operators, Fixed Point Theory and Applications 2008, Article ID 242916, 2008, 7 pages. https://doi.org/10.1155/2008/242916

B. S. Thakur, D. Thakur and M. Postolache, A new iterative scheme for numerical reckoning fixed points of Suzuki's generalized nonexpansive mappings, App. Math. Comp. 275 (2016), 147-155. https://doi.org/10.1016/j.amc.2015.11.065

K. Ullah and M. Arshad, Numerical reckoning fixed points for Suzuki's generalized nonexpansive mappings via new iteration process, Filomat 32, no. 1 (2018), 187-196. https://doi.org/10.2298/FIL1801187U

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