Stability and convergence analysis of hybrid algorithms for Berinde contraction mappings and its applications

Year 2021, , 159 - 168, 30.09.2021

Raweerote Suparatulatorn , Suthep Suantai

https://doi.org/10.53006/rna.950067

Cited By: 1

Abstract

In this paper, we construct a new hybrid iteration, called SR-iteration, and prove its stability and convergence analysis for weak contraction mappings in a Banach space. We compare rate of convergence between the SR-iteration and other iterations. Moreover, we provide numerical comparisons for supporting our main theorem and apply our main result to prove existence problem of mixed type Volterra-Fredholm functional nonlinear integral equation.

Keywords

SR-iteration, rate of convergence, weak contraction, Banach space

References

• [1] W. Chaolamjiak, D. Yambangwai, H.A. Hammad, Modified hybrid projection methods with SP iterations for quasi- nonexpansive multivalued mappings in Hilbert spaces, Bull. Iran. Math. Soc. (2020). https://doi.org/10.1007/s41980-020- 00448-9.
• [2] W. Cholamjiak, D. Yambangwai, H. Dutta, H.A. Hammad, Modi?ed CQ-algorithms for G-nonexpansive mappings in Hilbert spaces involving graphs, New Math. Nat. Comput. 16(1) (2019) 89-103.
• [3] W. Cholamjiak, S. Suantai, R. Suparatulatorn, S. Kesornprom, P. Cholamjiak, Viscosity approximation methods for fixed point problems in Hilbert spaces endowed with graphs, J. Appl. Numer. Optim. 1 (2019) 25-38.
• [4] E. Picard, Mémoire sur la théorie des équations aux dérivées partielles et la méthode des approximations successives, J. Math. Pures Appl. 6(4) (1890) 145-210.
• [5] W.R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953) 506-510.
• [6] S. Ishikawa, Fixed point by a new iteration method, Proc. Amer. Math. Soc. 44 (1974) 147-150.
• [7] M.A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251(1) (2000) 217-229.
• [8] W. Phuengrattana, S. Suantai, On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval, J. Comput. Appl. Math. 235(9) (2011) 3006-3014.
• [9] W. Phuengrattana, 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(1) (2013) 217-226.
• [10] M.O. Osilike, Stability results for Ishikawa fixed point iteration procedure, Indian J. Pure Appl. Math. 26(10) (1995) 937-941.
• [11] B.E. Rhoades, Fixed point theorems and stability results for fixed point iteration procedures, Indian J. Pure Appl. Math. 21 (1990) 1-9.
• [12] R.L. Burden, J.D. Faires, Numerical Analysis, 9th edn. Brooks/Cole Cengage Learning, Boston (2010).
• [13] V. Berinde, Iterative Approximation of Fixed Points. Editura Efemeride, Baia Mare (2002).
• [14] F. Gürsoy, A Picard-S iterative method for approximating fixed point of weak-contraction mappings, Filomat, 30(10) (2016) 2829-2845.
• [15] F. Gürsoy, V. Karakaya, A Picard-S hybrid type iteration method for solving a differential equation with retarded argument, arXiv:1403.2546v2 (2014).
• [16] C. Craciun, M.A. Serban, A nonlinear integral equation via Picard operators, Fixed Point Theory. 12(1) (2011) 57-70.
• [17] H.A. Hammad, M. De la Sen, Solution of nonlinear integral equation via fixed point of cyclic α ψ L -rational contraction mappings in metric-like spaces, Bull. Braz. Math. Soc. New Ser. 51 (2020) 81-105.
• [18] H.A. Hammad, M. De la Sen, Generalized contractive mappings and related results in b-metric like spaces with an application, Symmetry, 11(5) (2019) 667.
• [19] H.A. Hammad, M. De la Sen, A solution of Fredholm integral equation by using the cyclic η q s -rational contractive mappings technique in b-metric-like spaces, Symmetry, 11(9) (2019) 1184.