Solving nonlinear \(\phi\) -strongly accretive operator equations by a one-step-two-mappings iterative scheme

Volume 8, Issue 5, pp 837--846 http://dx.doi.org/10.22436/jnsa.008.05.33

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Authors

Safeer Hussain Khan - Department of Mathematics, Statistics and Physics, Qatar University, Doha 2713, Qatar. Birol Gunduz - Department of Mathematics, Faculty of Science and Art, Erzincan University, Erzincan, 24000, Turkey. Sezgin Akbulut - Department of Mathematics, Faculty of Science, Ataturk University, Erzurum, 25240, Turkey.

Abstract

A solution of nonlinear \(\phi\)-strongly accretive operator equations is found in this paper by using a one-step- two-mappings iterative scheme in arbitrary real Banach spaces. We give an example to validate our main theorem. Our results are different from those of Khan et. al., [S. H. Khan, A. Rafiq, N. Hussain, Fixed Point Theory Appl., 2012 (2012), 10 pages] in view of different and independent iterative schemes in the sense that none reduces to the other but extend and improve the results of Ding [X. P. Ding, Computers Math. Appl., 33 (1997), 75-82] and many others.

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Keywords

• One-step-two-mappings iterative scheme
• \(\phi\)-strongly accretive operator
• \(\phi\)-hemicontractive operator.

MSC

•  47H06
•  47H10

References

• [1] C. E. Chidume , Iterative approximation of fixed points of Lipschitz strictly pseudo-contractive mappings, Proc. Amer. Math. Soc., 99 (1987), 283-288.


• [2] C. E. Chidume, M. O. Osilike, Fixed point iterations for strictly hemicontractive maps in uniformly smooth Banach spaces, Numer. Funct. Anal. Optimiz., 15 (1994), 779-790.


• [3] C. E. Chidume, M. O. Osilike, Nonlinear accretive and pseudo-contractive operator equations in Banach spaces, Nonlinear Anal., 31 (1998), 779-789.


• [4] R. Chugh, V. Kumar, Convergence of SP iterative scheme with mixed errors for accretive Lipschitzian and strongly accretive Lipschitzian operators in Banach space, Int. J. Comput. Math., 90 (2013), 1865-1880.


• [5] X. P. Ding, Iterative process with errors to nonlinear \(\phi\)-strongly accretive operator equations in arbitrary Banach spaces, Comput. Math. Appl., 33 (1997), 75-82.


• [6] N. Gurudwan, B. K. Sharma, Approximating solutions for the system of \(\phi\)-strongly accretive operator equations in reflexive Banach space, Bull. Math. Anal. Appl., 2 (2010), 32-39.


• [7] S. Kamimura, S. H. Khan, W. Takahashi, Iterative schemes for approximating solutions of relations involving accretive operators in Banach spaces, Fixed Point Theory Appl., 5 (2003), 41-52.


• [8] T. Kato , Nonlinear semigroups and evolution equations, J. Math. Soc. Japan., 19 (1967), 508-520.


• [9] S. H. Khan, N. Hussain , Convergence theorems for nonself-asymptotically nonexpansive mappings, Comput. Math. Appl., 55 (2008), 2544-2553.


• [10] A. R. Khan, V. Kumar, N. Hussain, Analytical and Numerical Treatment of Jungck-Type Iterative Schemes, Appl. Math. Comput., 231 (2014), 521-535.


• [11] S. H. Khan, A. Rafiq, N. Hussain, A three-step iterative scheme for solving nonlinear \(\phi\)-strongly accretive operator equations in Banach spaces, Fixed Point Theory Appl., 2012 (2012), 10 pages.


• [12] S. H. Khan, I. Yildirim, M. Ozdemir , Convergence of a generalized iteration process for two finite families of Lipschitzian pseudocontractive mappings, Math. Comput. Model., 53 (2011), 707-715.


• [13] J. K. Kim, Z. Liu, S. M. Kang, Almost stability of Ishikawa iterative schemes with errors for \(\phi\)-strongly quasi-accretive and \(\phi\)-hemicontractive operators, Commun. Korean Math. Soc., 19 (2004), 267-281.


• [14] Z. Liu, Z. An, Y. Li, S. M. Kang, Iterative approximation of fixed points for \(\phi\)-hemicontractive operators in Banach spaces, Commun. Korean Math. Soc., 19 (2004), 63-74.


• [15] Z. Liu, M. Bounias, S. M. Kang, Iterative approximation of solution to nonlinear equations of \(\phi\)-strongly accretive operators in Banach spaces, Rocky Mountain J. Math., 32 (2002), 981-997.


• [16] Z. Liu, S. M. Kang , Convergence and stability of the Ishikawa iteration procedures with errors for nonlinear equations of the \(\phi\)-strongly accretive type, Neural Parallel Sci. Comput., 9 (2001), 103-118.


• [17] Y. Miao , S. H. Khan, Strong Convergence of an implicit iterative algorithm in Hilbert spaces, Commun. Math. Anal., 4 (2008), 54-60.


• [18] M. O. Osilike, Iterative solution of nonlinear equations of the \(\phi\)-strongly accretive type, J. Math. Anal. Appl, 200 (1996), 259-271.


• [19] M. O. Osilike, Iterative solution of nonlinear \(\phi\)-strongly accretive operator equations in arbitrary Banach spaces, Nonlinear Anal., 36 (1999), 1-9.


• [20] A. Rafiq , Iterative solution of nonlinear equations involving generalized \(\phi\)-hemicontractive mappings, Indian J. Math., 50 (2008), 365-380.


• [21] A. Rafiq, On iterations for families of asymptotically pseudocontractive mappings, Appl. Math. Lett., 24 (2011), 33-38.


• [22] K. K. Tan, H. K. Xu, Iterative solutions to nonlinear equations of strongly accretive operators in Banach spaces, J. Math. Anal. Appl., 178 (1993), 9-21.


• [23] K. K. Tan, H. K. Xu , Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl., 178 (1993), 301-308.