Document Type : Research Paper
Authors
Department of Mathematics, College of Education for Pure Sciences (Ibn AlHaitham), University of Baghdad, Baghdad - Iraq
Abstract
In this paper, we established weak convergence theorems by using appropriate conditions for approximating common fixed points and equivalence between the convergence of the Picard-Mann iteration scheme and Liu et al iteration scheme in Banach spaces. As well as, numerical examples are given to show that Picard-Mann is faster than Liu et al iteration schemes.
Keywords
Main Subjects
[1] J.B. Diaz, F.T. Metcalf, “On the structure of the set
of subsequential limit points of successive
approximation,” Bull. Am. Math. Sco. 73, 516-519,
1967.
[2] T. Suzuki, “Fixed point theorems and convergence
theorems for som generalized- nonexpansive
mappings,” J. Math. Anal. Appl. 340, 1088–1095, 2008.
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Edelstein’s, “Method and Fixed point theorems for
some generalized nonexpansive mappings,” J. Math.
Anal. Appl. 350, 12-17, 2009.
[4] W. Phuengrattana, “Approximating fixed points of
suzuki-generalized nonexpansive mappings,”
Nonlinerar Anal. Hybrid. Syst. 5, 3, 583-590, 2011.
[5] S.H. Khan, T. Suzuki, “A Reich-type convergence
theorem for generalized nonexpansive mappings in
uniformly convex Banach spaces,” Nonlinear Anal. 80,
211-215, 2013.
[6] J. Garcial-Falset, E. Llorens-Fuster, T. Suzuki,
“Fixed point theory for a class of generalized
nonexpansive mappings,” J. Math. Anal. Appl. 375,
185-195, 2011.
[7] W. Takahashi, T. Tamura, T “Convergence theorem
for a pair of nonexpansive mappings in Banach spaces,”
J. convex Analysis, 5, I, 45-58.
[8] S.H. Khan, “A Picard-Mann hybrid iteration
process,” Fixed Point Theory. Appl., 2013:69, 2013.
[9] Z. Liu, C. Feng, J.S. Ume, S.M. Kang, “Weak and
strong convergence for common fixed points of a pair
of nonexpansive and asymptotically nonexpansive
mappings,” Taiwanese Journal of Mathematics, 11, I,
27-42, 2007.
[10] V.K. Sahu, “Convergence results of implicit
iteration scheme for two asymptotically quasi-Inonexpansive mappings in Banach spaces,” Global
Journal of pure and Applied Mathematices, Vol. 12, No.
2, pp. 1723-1742, 2016.
[11] B. Gunduz, “A new two step iterative scheme for
a finite family of nonself I-asymptoically nonexpansive
mappings in Banach space,” NTMSCI 5, No. 2, 16-28,
2017.
[12] E.L. Fuster, E.M. Galvez, “The fixed point theory
for some generalized nonexpansive mapping,” Abstract
. Appl. Anal. Vol 2011, Article ID 435686, 15 page,
2011.
[13] A. Sharma, M. Imdad, “Approximating fixed
points of generalized nonexpansive mappings Via
faster iteration schemes,” Fixed point theory, 4, no.4,
605-623, 2014.
[14] F.E. Browder,”Semicontractive and semiaccretive
nonlinear mappings in Banach spaces,” Bull. Amer.
Math. Soc. 74, 660-665, 1968.
[15] D.R, Sahu, D.O. Regan, R.P. Agarwal, “Fixed
applications, Topological fixed point theory and its
applications,” doi:10.1007/978-387-75818-3-1.
[16] E. Zeidler, “Nonlinear Functional analysis and
applications,” Fixed point theorems, Springer Verilage,
New York Inc. 1986.
[17] I. Yildirim, M. Abbas, N. Karaca, “On the
convergence and data dependence results for multistep
Picard-Mann iteration process in the class of
contractive-like operators,” J. Nonlinear. Sci. Appl. 9,
3773-3786, 2016.
[18] G.S. Saluja, “Weak convergence theorems for
Asymptotically Nonexpansive Mappings and Total
Asymptotically Non-self Mappings,” Sohag J. Math. 4,
No.2, 49-57, 2017.
[19] B.E. Kashem, “Partition method for solving
Boundary value problem using B-Spline functions,”
Eng & Tech. Journal, Vol. 27, No. 11, 2009.
[20] A.J. Kadhim, “Expansion method for solving
Linear integral equations with Multiple Lags using BSpline and Orthogonal functions,” Eng & Tech.
Journal,Vol.29, No.9, 2011.
of subsequential limit points of successive
approximation,” Bull. Am. Math. Sco. 73, 516-519,
1967.
[2] T. Suzuki, “Fixed point theorems and convergence
theorems for som generalized- nonexpansive
mappings,” J. Math. Anal. Appl. 340, 1088–1095, 2008.
[3] S. Dhompongsa, W. Inthakon, A. Kaewkhao,
Edelstein’s, “Method and Fixed point theorems for
some generalized nonexpansive mappings,” J. Math.
Anal. Appl. 350, 12-17, 2009.
[4] W. Phuengrattana, “Approximating fixed points of
suzuki-generalized nonexpansive mappings,”
Nonlinerar Anal. Hybrid. Syst. 5, 3, 583-590, 2011.
[5] S.H. Khan, T. Suzuki, “A Reich-type convergence
theorem for generalized nonexpansive mappings in
uniformly convex Banach spaces,” Nonlinear Anal. 80,
211-215, 2013.
[6] J. Garcial-Falset, E. Llorens-Fuster, T. Suzuki,
“Fixed point theory for a class of generalized
nonexpansive mappings,” J. Math. Anal. Appl. 375,
185-195, 2011.
[7] W. Takahashi, T. Tamura, T “Convergence theorem
for a pair of nonexpansive mappings in Banach spaces,”
J. convex Analysis, 5, I, 45-58.
[8] S.H. Khan, “A Picard-Mann hybrid iteration
process,” Fixed Point Theory. Appl., 2013:69, 2013.
[9] Z. Liu, C. Feng, J.S. Ume, S.M. Kang, “Weak and
strong convergence for common fixed points of a pair
of nonexpansive and asymptotically nonexpansive
mappings,” Taiwanese Journal of Mathematics, 11, I,
27-42, 2007.
[10] V.K. Sahu, “Convergence results of implicit
iteration scheme for two asymptotically quasi-Inonexpansive mappings in Banach spaces,” Global
Journal of pure and Applied Mathematices, Vol. 12, No.
2, pp. 1723-1742, 2016.
[11] B. Gunduz, “A new two step iterative scheme for
a finite family of nonself I-asymptoically nonexpansive
mappings in Banach space,” NTMSCI 5, No. 2, 16-28,
2017.
[12] E.L. Fuster, E.M. Galvez, “The fixed point theory
for some generalized nonexpansive mapping,” Abstract
. Appl. Anal. Vol 2011, Article ID 435686, 15 page,
2011.
[13] A. Sharma, M. Imdad, “Approximating fixed
points of generalized nonexpansive mappings Via
faster iteration schemes,” Fixed point theory, 4, no.4,
605-623, 2014.
[14] F.E. Browder,”Semicontractive and semiaccretive
nonlinear mappings in Banach spaces,” Bull. Amer.
Math. Soc. 74, 660-665, 1968.
[15] D.R, Sahu, D.O. Regan, R.P. Agarwal, “Fixed
applications, Topological fixed point theory and its
applications,” doi:10.1007/978-387-75818-3-1.
[16] E. Zeidler, “Nonlinear Functional analysis and
applications,” Fixed point theorems, Springer Verilage,
New York Inc. 1986.
[17] I. Yildirim, M. Abbas, N. Karaca, “On the
convergence and data dependence results for multistep
Picard-Mann iteration process in the class of
contractive-like operators,” J. Nonlinear. Sci. Appl. 9,
3773-3786, 2016.
[18] G.S. Saluja, “Weak convergence theorems for
Asymptotically Nonexpansive Mappings and Total
Asymptotically Non-self Mappings,” Sohag J. Math. 4,
No.2, 49-57, 2017.
[19] B.E. Kashem, “Partition method for solving
Boundary value problem using B-Spline functions,”
Eng & Tech. Journal, Vol. 27, No. 11, 2009.
[20] A.J. Kadhim, “Expansion method for solving
Linear integral equations with Multiple Lags using BSpline and Orthogonal functions,” Eng & Tech.
Journal,Vol.29, No.9, 2011.
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