Iterative Algorithm for Solution of m-accretive Operator in Uniformly Smooth Banach Spaces
Keywords:
m-accretive operato, uniformly convex Banach space, uniformly smooth, composite iteration, resolvent, retraction
Abstract
The purpose of this paper is to study a composite iterative scheme for approximating solution of m-accretive operator in a uniformly convex and uniformly smooth Banach space using the resolvent and retraction technique. The result presented in this paper thus improve and extend the corresponding results of Kim and Xu [9], Qin and Su [12] and the references therein, to a better iterative scheme and that of Marino and Xu [10], Takahashi [16] and the references therein, to a more general Banach space.
References
V.Barbu, Nonlinear semigroups and differential equations in Banach space, Noordhoff, (1976).
T.D.Benavides, G.L.Acedo and H.K.Xu, Iterative solutions for zeros of accretive operators, Math. Nachr., 248/249(2003), 62-71.
F.E.Browder, Nonlinear mappings of nonexpansive and accretive type in Banach space, Bull. Amer. Math. Soc., 73(1967), 875-882.
F.E.Browder and W.V.Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20(1967), 197-228.
R.E.Bruck, Nonexpansive projections on subsets of Banach spaces, Pacific J. Math., 47(1973), 341-355.
I.Cioranescu, Geometry of Banach spaces, Duality mappings and Nonlinear problems, Kluwer academic Publishers, Dordrecht, (1990).
K.Goebel and S.Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, NewYork, (1984).
S.Kamimura and W.Takahashi, Approximating solutions of maximal monotone operators in Hilbert space, J. Approx. Theory, 106(2006), 226-240.
T.H.Kim and H.K.Xu, Strong convergence of modified Mann iterations, Nonlinear Anal., 61(2005), 51-60.
G.Marino and H.K.Xu, Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl., 329(2007), 336346.
W.V.Petryshyn, A characterization of strict convexity of Banach spaces and other uses of duality mappings, J. Funct. Anal., 6(1970), 282-291.
X.Qin and Y.Su, Approximation of a zero point of accretive operator in Banach spaces, J. Math. Anal. Appl., 329(1)(2007), 415-424.
S.Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl., 75(1980), 287-292.
R.T.Rockafellar, Monotone operators and the proximal point algorithm, SIAM. J. Control. Optim., 14(1976), 877-898.
W.Takahashi and Y.Ueda, On Reich’s strong convergence theorems for resolvents of accretive operators, J. Math. Anal. Appl., 104(1984), 546-553.
W.Takahashi, Y.Takeuchi and R.Kubota, Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 341(2008), 276286.
Z.B.Xu and G.F.Roach, Characterstic inequalities for uniformly convex and uniformly smooth Banach spaces, J. Math. Anal. Appl., 157(1991), 189-210.
E.Zeidler, Nonlinear Functional Analysis and its Applications, Part II: Monotone Operators, Springer-Verlag, Berlin (1985).
T.D.Benavides, G.L.Acedo and H.K.Xu, Iterative solutions for zeros of accretive operators, Math. Nachr., 248/249(2003), 62-71.
F.E.Browder, Nonlinear mappings of nonexpansive and accretive type in Banach space, Bull. Amer. Math. Soc., 73(1967), 875-882.
F.E.Browder and W.V.Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20(1967), 197-228.
R.E.Bruck, Nonexpansive projections on subsets of Banach spaces, Pacific J. Math., 47(1973), 341-355.
I.Cioranescu, Geometry of Banach spaces, Duality mappings and Nonlinear problems, Kluwer academic Publishers, Dordrecht, (1990).
K.Goebel and S.Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, NewYork, (1984).
S.Kamimura and W.Takahashi, Approximating solutions of maximal monotone operators in Hilbert space, J. Approx. Theory, 106(2006), 226-240.
T.H.Kim and H.K.Xu, Strong convergence of modified Mann iterations, Nonlinear Anal., 61(2005), 51-60.
G.Marino and H.K.Xu, Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl., 329(2007), 336346.
W.V.Petryshyn, A characterization of strict convexity of Banach spaces and other uses of duality mappings, J. Funct. Anal., 6(1970), 282-291.
X.Qin and Y.Su, Approximation of a zero point of accretive operator in Banach spaces, J. Math. Anal. Appl., 329(1)(2007), 415-424.
S.Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl., 75(1980), 287-292.
R.T.Rockafellar, Monotone operators and the proximal point algorithm, SIAM. J. Control. Optim., 14(1976), 877-898.
W.Takahashi and Y.Ueda, On Reich’s strong convergence theorems for resolvents of accretive operators, J. Math. Anal. Appl., 104(1984), 546-553.
W.Takahashi, Y.Takeuchi and R.Kubota, Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 341(2008), 276286.
Z.B.Xu and G.F.Roach, Characterstic inequalities for uniformly convex and uniformly smooth Banach spaces, J. Math. Anal. Appl., 157(1991), 189-210.
E.Zeidler, Nonlinear Functional Analysis and its Applications, Part II: Monotone Operators, Springer-Verlag, Berlin (1985).
How to Cite
Niyati Gurudwan. (2016). Iterative Algorithm for Solution of m-accretive Operator in Uniformly Smooth Banach Spaces. International Journal of Current Research in Science and Technology, 2(1), 55-60. Retrieved from https://crst.gfer.org/index.php/crst/article/view/60
Issue
Section
Articles
