ITERATIVE SOLUTION OF NONLINEAR EQUATIONS WITH STRONGLY ACCRETIVE OPERATORS IN BANACH SPACES

Let X be a real Banach space with a uniformly convex dual X*. Let T: X a X be a Lipschitzian and strongly accretive mapping with a Lipschitzian constant L greater than or equal to 1 and a strongly accretive constant k epsilon (0,1). Let {alpha(n)} and {beta(n)} be two real sequences in [0,1] satisfying: (i) alpha(n) --> 0 as n --> infinity (ii) beta(n) < k(1 - k)/L(1 + L), for all n greater than or equal to 0; (iii) Pi(infinity) alpha(n) = infinity Set Sx = f - Tx + x, For All x epsilon X. Assume that {u(n)}(n=0)(infinity) and {v(n)}(n=0)(infinity) be two sequences in X satisfying parallel to u(n) parallel to = o(alpha(n)) and nu(n) --> 0 as n --> infinity. For arbitrary x(0) epsilon X, the iteration sequence {x(n)} is defined by (IS)(I) {x(n+1) = (1 - alpha(n))x(n) + alpha(n)Sy(n) + u(n), {y(n) = (1 - beta(n)) x(n) + beta(n)Sx(n) + v(n) (n greater than or equal to 0) then {x(n)} converges strongly to the unique solution of the equation Tx = f. related result deals with iterative approximation of fixed points of phi-hemicontractive mappings.

ITERATIVE PROCESS TO -HEMICONTRACTIVE OPERATOR AND-STRONGLY ACCRETIVE OPERATOR EQUATIONS

Let E be an arbitrary real Banach space and K be a nonempty closed convex subsets of E. Let T:K→K be a uniformly continuous _hemicontractive operator with bounded range and a n,b n,c n,a ′ n,b ′ n,c ′ n be sequences in [0,1] satisfying:ⅰ) a n+b n+c n=a ′ n+b ′ n+c ′ n=1. n≥0; ⅱ) lim b n= lim b ′ n= lim c ′ n= 0; ⅲ)∑∞n=0b n=∞; ⅳ) c n=o(b n). For any given x 0,u 0,v 0∈K, define the Ishikawa type iterative sequence x n as follows: x n+1 =a nx n+b nTy n+c nu n, y n=a ′ nx n+b ′ nTx n+c ′ nv n (n≥0), where u n and v n are bounded sequences in K. Then x n converges strongly to the unique fixed point of T. Related result deals with the convergence of Ishikawa type iterative sequence to the solution of _strongly accretive operator equations.

Iterative Approximation with Errors for Generalized Set-Valued Strongly Accretive Mapping in Banach Spaces

A new concept of generalized set-valued strongly accretive mappings in Banach spaces was given and some strong convergence theorems of Ishikawa and Mann iterative process with errors approximation methods by Huang et al. was proved. The results presented in this paper improve and extend the earlier results obtained by Huang et al.

Three-step Iterations with Errors for Nonlinear Strongly Accretive Operator Equations

In this paper, we suggest and analyse a three-step iterative scheme with errors for solving nonlinear strongly accretive operator equation Tx = f without the Lipshitz condition. The results presented in this paper improve and extend current results in the more general setting.

A Note on Iterative Solutions of Nonlinear Accretive Operator Equations

Suppose that X is a real Banach space, H: X→X is a Lipschitz operator, T: X→X is a uniformly continuous operator with bounded range, and H+T is strongly accretive. Then the Ishikawa iteration process converges strongly to the unique solution of the equation Hx+Tx=f . This conclusion extends the corresponding results in recent papers.

A system of general nonlinear variational inclusions in Banach spaces

A new system of general nonlinear variational inclusions is introduced and studied in Banach spaces. An iterative algorithm is developed and analyzed by use of the resolvent operator techniques to find the approximate solutions of the system of general nonlinear variational inclusions involving different nonlinear operators in uniformly smooth Banach spaces.