Stability and convergence of new random approximation algorithms for random contractive-type operators in separable Hilbert spaces

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DOI:

https://doi.org/10.26637/mjm0903/009

Abstract

In this paper, new iterative schemes called Jungck-DI-Noor random iterative scheme and Jungck-DI-SP random iterative scheme are introduced and studied. Also, stability and convergence results were obtained without necessarily imposing sum conditions on the countably finite family of the control sequences and injectivity condition on the operators, which makes our schemes to be more desirable in applications than the ones studied in this paper and several others currently in the literature.

Keywords:

Strong convergence, Jungck-DI-Noor random iterative scheme, Jungck-DI-SP random iterative scheme, Stability, Contractive-type operator, fixed point, separable Hilbert space

Mathematics Subject Classification:

47H09, 47H10 , 47J05, 65J15
  • Imo Kalu Agwu Department of Mathematics, Micheal Okpara University of Agriculture, Umudike, Umuahia Abia State, Nigeria.
  • Donatus Ikechi Igbokwe Department of Mathematics, Micheal Okpara University of Agriculture, Umudike, Umuahia Abia State, Nigeria.
  • Pages: 148-167
  • Date Published: 01-07-2021
  • Vol. 9 No. 03 (2021): Malaya Journal of Matematik (MJM)

I. K. AGWU AND D. I. IGBOKWE, New iteration algorithm for equilibrium problems and fixed point problems of two finite families of asymptotically demicontractive multivalued mappings (in press.)

H. Akewe and A. Mogbademu, Common fixed point of Jungck-Kirk-type iteration for nonself operators in normed linear spaces, Fasciculi Mathemathici, 2016(2016), 29-41.

H. AKEWe and H. OlaOluWa, On the convergence of modified iteration process for generalise contractive-like operators, Bull. Math. Anal. Appl., 4(3)(2012), 78-86.

H. Akewe, G. A. Okeeke and A. Olayiwola, Strong convergence and stability of Kirk-multistep-type iterative schemes for contractive-type operators. Fixed Point Theory Appl., 2014(2014), 45.

H. AKEWE, Approximation of fixed and common fixed points of generalised contractive-like operators, PhD Thesis, University of Lagos, Nigeria, (2010).

A.T. BhARUCHA-ReID, Fixed point theorems in probabilistic analysis, Bull. Amer. Math. Soc., 82(1976), 641-657.

S. IтоH, Random fixed point theorems with an application to random differential equations in Banach spaces, J. Math. Anal. Appl., 67(1979), 261-273.

V. BERInde, On the stability of some fixed point problems. Bull. Stint. Univ. Bala Mare, Ser. B Fasc. Mat-inform. $X V I I I(1), mathbf{1 4}(2002), 7-14$.

V. BERINDE, Iterative approximation of fixed points. Editorial Efemerede, Bala Mare. (2012).

A. D. Bosede, H. AKewe, A. S. WusU And O. F. BAKre, Random hybrid iterative algorithms of Jungck-type and common random fixed point theorems with stability results, Int'l J. of Research and Innovation in AppL. Sci.,IV(IX), (2019), 2454-6494.

R. Chugh and V. Kummar, Stability of hybrid fixed point iterative algorithm of Kirk-Noor-type in nonlinear spaces for self and nonself operators, Intl. J. Contemp. Math. Sci., 7(24)(2012), 1165-1184.

R. Chugh and V. Kummar, Strong convergence of SP iterative scheme for quasi-contractive operators. Intl. J. Comput. Appl., 31(5)(2011), 21-27.

A. M. Harder AND T. L. Hicks, Stability results for fixed point iterative procedures. Math. Jpn, 33(5)(1988), 693706.

O. Hans, Reduzierende zufallige transformationen, Czechoslov. Math. J., 7(1957), 154-158.

N. Hussain, R. Chugh, V. Kummar and A. Rafig, On the convergence of Kirk-type iterative schemes. J. Appl. Math., 2012 (2012), Article ID 526503, 22 pages.

S. IshIKAWA, Fixed points by a new iteration methods. Proc. Am. Math. Soc., 44(1974), 147-150.

C.O. Imoru and M.O. Olatinwo, On the stability of Picard's and Mann's iteration. Carpath. J. Math., 19(2003), $155-160$.

F. O. Isiogugu, C. IzUChUKwu AND C. C. OKEKE, New iteration scheme for approximating a common fixed point of a finite family of mappings. Hindawi J. Math., 2020(2020), Article ID 3287968.

G. JungCK, Commuting mappings and fixed points, Amer. Math. Monthly, 83(4)(1976), 261-263.

W. A. KICK, On successive approximations for nonexpansive mappings in Banach spaces, Glasg. Math. J., 12(1971), $6-9$.

W. R. MAnN, Mean value method in iteration, Proc. Am. Math. Soc., 44(2000), 506-510.

M. A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl., 251(2000), 217229.

M. O. Olatinwo, A generalization of some convergence results using a Jungck-Noor three-step iteration process in arbitrary Banach space, Fasciculi Mathemathici, 40(2008), 37-43.

J. O. Olaleru And H. AKEwE, On the convergence of Jungck-type iterative schemes for generalized contractive-like operators, Fasciculi Mathemathici, 45(2010), 87-98.

M. O. Olatinwo, Stability results for Jungck-kirk-Mann and Jungck-kirk hybrid iterative algorithms, Anal. Theory Appl., 29(2013), 12-20.

M. O. Olatinwo, Convergence results for Jungck-type iterative process in convex metric spaces,, Acta Univ. Palacki Olomue, Fac. Rer. Nat. Math., 51(2012), 79-87.

M. O. Olatinwo, Some stability and strong convergence results for the Jungck-Ishikawa iteration process, Creative Math. and Inf., 17(2008), 33-42.

M. O. Olaeru, H. Akewe, An extension of Gregus fixed point theorem, Fixed Point Theory Appl., 2007(2007), Article ID 78628 .

M. O. Olutinwo, Some stability results for two hybrid fixed point iterative algorithms in normed linear space. Mat. vesn, 61(4)(2009), 247-256.

M. O. OSILIKE AND A. UDOEMENE, A short proof of stability resultsfor fixed point iteration procedures for a class of contractive-type mappings, Indian J. Pure Appl. Math., 30(1999), 1229-1234.

M. O. OsILIKE, Stability results for lshikawa fixed point iteration procedure, Indian J. Pure Appl. Math., 26(10)(1996), 937-941.

A. M. Ostrowski, The round off stability of iterations, Z. Angew Math. Mech., 47(1967), 77-81.

B. E. RHOAdE, Fixed point theorems and stability results for fixed point iteration procedures. Indian J. Pure Appl. Math., 24(11)(1993), 691-03.

B. E. RhoAde, Fixed point theorems and stability results for fixed point iteration procedures. Indian J. Pure Appl. Math., 21(1990), 1-9.

A. RATIQ, A convergence theprem for Mann's iteration procedure, Appl. Math. E-Note, 6(2006), 289-293.

B. E. RHOADE, A comparison of various definitions of contractive mappings, Trans. Am. Math. Soc., 266(1977), 257290.

B. E. RhOAde, Comments on two fixed point iteration methods, Trans. Am. Math. Soc., 56(1976), 741-750.

B. E. RHOADE, Fixed point iteration using infinite matrices, Trans. Am. Math. Soc., 196(1974), 161-176.

A. RATIQ, On the convergence of the three step iteration process in the class of quasi-contractive operators, Acta. Math. Acad. Paedagag Nayhazi, 22(2006), 300-309.

R. A. RashWan and H. A. Hammad, Stability and strong convergence results for random Jungck-Kirk-Noor iterative scheme, Fasciculi Mathemathici, (2017), 167-182.

S. L. Singh, C. Bhatnagar And S.N. Mishra, Stability of Jungck-type iteration procedures, Int. J. Math. Math. Sci., 19(2005), 3035-3043.

A. SpACEK, Zufallige Gleichungen, Czechoslovak, Math. J., 5(1955), 462-466.

H. K. Xu, Some random fixed point theorems for condensing and nonexpansive operators, Proc. Ame. Math. Soc., 110(2)(1990), 395-400.

T. Z AMFIRESCU , Fixed point theorems in metric spaces, Arch. Math., 23(1972), 292–298.

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Published

01-07-2021

How to Cite

Agwu , I. K. ., and D. I. Igbokwe. “Stability and Convergence of New Random Approximation Algorithms for Random Contractive-Type Operators in Separable Hilbert Spaces”. Malaya Journal of Matematik, vol. 9, no. 03, July 2021, pp. 148-67, doi:10.26637/mjm0903/009.