THE EQUIVALENCE BETWEEN THE CONVERGENCES OF MANN AND ISHIKAWA ITERATION METHODS WITH ERRORS FOR DEMICONTINUOUS φ-STRONGLY ACCRETIVE OPERATORS IN UNIFORMLY SMOOTH BANACH SPACES

We investigate the equivalence between the convergences of the Mann iteration method and the Ishikawa iteration method with errors for demicontinuous φ-strongly accretive operators in uniformly smooth Banach spaces. A related result deals with the equivalence of the Mann iteration method and the Ishikawa iteration method for φ-pseudocontractive operators in nonempty closed convex subsets of uniformly smooth Banach spaces. The results presented in this paper extend and improve the corresponding results in the literature.


Introduction and preliminaries
For a Banach space X we will denote by J the normalized duality mapping from X into 2 X * given by J(x) = f * ∈ X * : Re x, f * = f * 2 = x 2 , x ∈ X, (1.1) where X * denotes the dual space of X and •, • denotes the generalized duality pairing.It is known that X is a uniformly smooth Banach space if and only if J is single-valued and uniformly continuous on any bounded subset of X.Let I denote the identity operator in X.
An operator T with domain D(T) and range R(T) in X is said to be strongly accretive if there exists a constant k > 0 such that for any x, y ∈ D(T), there exists j(x − y) ∈ J(x − y) satisfying Re Tx − T y, j(x − y) ≥ k x − y 2 . (1.2) Without loss of generality we may assume that k ∈ (0,1).It is known that T is accretive if and only if for any x, y ∈ D(T), there exists j(x − y) ∈ J(x − y) such that Re Tx − T y, j(x − y) ≥ 0.
(1.3) Furthermore, T is called φ-strongly accretive if there exists a strictly increasing function φ : [0,∞) → [0,∞) with φ(0) = 0 such that for any x, y ∈ D(T) there exists j(x − y) ∈ J(x − y) satisfying Re Tx − T y, j(x − y) ≥ φ x − y x − y .(1.4) Closely related to the class of strongly accretive operators is the class of strongly pseudocontractive operators where an operator T is called strongly pseudocontractive if there exists t > 1 such that for any x, y ∈ D(T), there exists j(x − y) ∈ J(x − y) satisfying Re Tx − T y, j(x − y) ≤ 1 t x − y 2 . (1.5) T is called φ-strongly pseudocontractive if there exists a strictly increasing function φ : [0,∞] → [0,∞] with φ(0) = 0 such that for any x, y ∈ D(T) there exists j( An operator A : X → X * is said to be demicontinuous on X if {Ax n } n≥1 converges weakly to Ax 0 for any x 0 ∈ X and {x n } n≥1 ⊂ X with lim n→∞ x n = x 0 .It is well known that if X is a finite-dimensional space, then A is demicontinuous if and only if it is continuous. Within the past 20 years or so, various authors have applied the Mann iteration method, the Mann iteration method with errors, the Ishikawa iteration method, and the Ishikawa iteration method with errors to approximate fixed points of pseudocontractive, strongly pseudocontractive, φ-strongly pseudocontractive, and to approximate solutions of nonlinear equations Tx = f and x + Tx = f in the case when T is accretive, strongly accretive, and φ-strongly accretive (see, e.g., [1,[3][4][5][6][7][8][9][10][11][12][13][14][15]20] and the references therein).Recently, the equivalence of the Mann iteration method and the Ishikawa iteration method for various nonlinear operators and nonlinear equations has been established in Banach spaces or uniformly smooth Banach spaces.For details, we refer to [2,[16][17][18][19]. Especially, Rhoades and Soltuz [18] obtained the equivalence of the Mann iteration method and the Ishikawa iteration method for strongly pseudocontractive operators, strongly accretive operators, and accretive operators, respectively, in uniformly smooth Banach spaces.
It is our purpose in this paper to show the equivalence of both the Mann iteration method with errors and the Ishikawa iteration method with errors for φ-strongly accretive operators in uniformly smooth Banach spaces, and the Mann iteration method and the Ishikawa iteration method for φ-strongly pseudocontractive operators in nonempty closed and convex subsets of uniformly smooth Banach spaces.The results presented in this paper extend, improve, and unify the corresponding results due to Chang [1], Chang et al. cite3, Chidume [4][5][6], Chidume and Osilike [7], Osilike [14], Rhoades and Soltuz [18], Zhou [20], and others.Two examples which dwell upon the importance of our results are given.
The following lemmas play crucial roles in the proofs of our main results.
Lemma 1.1 [11].Suppose that X is a uniformly smooth Banach space and T : X → X is a demicontinuous φ-strongly accretive operator.Then the equation Tx = f has a unique solution for any f ∈ X.

Main results
In the following we establish the equivalence between the convergence of the Mann iteration method with errors and the Ishikawa iteration method with errors for demicontinuous strongly accretive operators in uniformly smooth Banach spaces.
Theorem 2.1.Let X be a uniformly smooth Banach space and T : X → X a demicontinuous φ-strongly accretive operator.For a given f ∈ X, let Sx = f + x − Tx for any x ∈ X. Define the Ishikawa iteration sequence with errors {x n } n≥0 iteratively by and the Mann iteration sequence with errors {u n } n≥0 iteratively by ) Assume that either the sequences x n − Tx n n≥0 and y n − T y n n≥0 or the sequences Tx n n≥0 and T y n n≥0 are bounded. (2.5) Then, for u 0 = x 0 ∈ X, the following assertions are equivalent: (i) the Mann iteration sequence with errors {u n } n≥0 converges strongly to the unique solution of the equation Tx = f ; (ii) the Ishikawa iteration sequence with errors {x n } n≥0 converges strongly to the unique solution of the equation Tx = f .Proof.It follows from Lemma 1.1 that the equation Tx = f has a unique solution q ∈ X.
It is clear that S is demicontinuous and q is a unique fixed point of S. Thus (i) follows from (ii) by setting b n = 0 and δ n = 0 for any n ≥ 0. Next we prove that (i) implies (ii).Since T is φ-strongly accretive, it follows that for any where It follows from the φ-strong accretivicity of T that for any x, y ∈ X, which implies that for all x, y ∈ X.Therefore, d, M, and L are bounded by (2.3)-(2.5).It is evident to verify that Zeqing Liu et al. 5 for any n ≥ 0. In terms of Lemma 1.2, (2.1), and (2.2), we arrive at for n ≥ 0 and some constants , and (2.13), we infer that for n ≥ 0 and some constants D 5 > 0, D > 0, where In light of (2.1), (2.4), (2.7)-(2.12),we know that From the continuity of the function b, (2.4) and (2.16), we deduce that (2.17) Put inf{A( y n − u n ) : n ≥ 0} = t and inf{A( x n − u n ) : n ≥ 0} = r.Suppose that rt > 0. From (2.17) we conclude immediately that there exists an integer m such that d n < 1/2r 2 t, for all n ≥ m.By virtue of (2.14) and (2.17), we get that for each n ≥ m, which implies that 3), (2.4), and (2.14), we conclude easily that for given ε > 0 there exists an integer k satisfying for any n ≥ n k .Next we prove by induction that for any i ≥ 0 In fact, (2.21) implies that (2.22) holds for i = 0. Suppose that (2.22) holds for some i > 0. If (2.23) In view of (2.14), we have (2.25) Hence the Ishikawa iteration sequence with errors {x n } n≥0 converges strongly to the unique solution of Tx = f .This completes the proof.
respectively.It is easy to see that T is continuous and for any u 0 = x 0 > 0, {Tx n } n≥0 and {T y n } n≥0 are bounded, where {x n } n≥0 and {y n } n≥0 are as in (2.1).In order to prove that T is φ-strongly accretive, for any x, y ∈ X with x ≥ y, we consider the following cases.
Next we establish the equivalence between the Mann iteration method with errors and the Ishikawa iteration methods with errors for demicontinuous accretive operators in uniformly smooth Banach spaces.
Zeqing Liu et al. 9 Theorem 2.4.Let X be a uniformly smooth Banach space and T : X → X a demicontinuous accretive operator.For a given f ∈ X, define S : X → X by Sx = f − Tx for x ∈ X. Assume that {σ n } n≥0 , {δ n } n≥0 , {a n } n≥0 , {b n } n≥0 , {x n } n≥0 , {y n } n≥0 , and {u n } n≥0 are as in Theorem 2.1 satisfying (2.1)- (2.4).Suppose that either the sequences {x n + Tx n } n≥0 and {y n + T y n } n≥0 or the sequences {Tx n } n≥0 and {T y n } n≥0 are bounded.Then, for u 0 = x 0 ∈ X, the following assertions are equivalent: (iii) the Mann iteration sequence with errors {u n } n≥0 converges strongly to the unique solution of the equation x + Tx = f ; (iv) the Ishikawa iteration sequence with errors {x n } n≥0 converges strongly to the unique solution of the equation Then, for u 0 = x 0 , the following assertions are equivalent: (v) the Mann iteration sequence {u n } n≥0 converges strongly to the fixed point of T; (vi) the Ishikawa iteration sequence {x n } n≥0 converges strongly to the fixed point of T.
Proof.Since T is φ-strongly pseudocontractive and F(T) = ∅, it follows that T has a unique fixed point q ∈ K and  Thus T is φ-strongly pseudocontractive in K. Consequently, Theorem 2.6 ensures the equivalence of the Mann iteration method and the Ishikawa iteration method for φstrongly pseudocontractive operator T in K.But the results in [1,[3][4][5][6][7]18] are not applicable since the subset K is unbounded and T is not strongly pseudocontractive.In fact, for any t > 1 there exist (x t , y t ) = ((t − 1)/2,0) ∈ K × K such that that is, T is not strongly pseudocontractive in K.
Tx t − T y t ,x t − y t −