Convergence theorems for ψ -expansive and accretive mappings

Nonlinear Analysis 66 (2007) 73–82 www.elsevier.com/locate/na

Convergence theorems for ψ-expansive and accretive mappings Habtu Zegeye a , Naseer Shahzad b,∗ a Bahir Dar University, P.O.Box. 859, Bahir Dar, Ethiopia b Department of Mathematics, King Abdul Aziz University, P. O. B. 80203, Jeddah 21589, Saudi Arabia

Received 17 August 2005; accepted 8 November 2005

Abstract Let E be a real Banach space, and let A : D(A) ⊆ E → E be a Lipschitz, ψ-expansive and accretive mapping such that co(D(A)) ⊆ ∩λ>0 R(I + λA). Suppose that there exists x0 ∈ D(A), where one of the following holds: (i) There exists R > 0 such that ψ(R) > 2A(x0 ); or (ii) There exists a bounded neighborhood U of x0 such that t (x − x0 ) ∈ Ax for x ∈ ∂U ∩ D(A) and t < 0. An iterative sequence {xn } is constructed to converge strongly to a zero of A. Related results deal with the strong convergence of this iteration process to fixed points of ψ-expansive and pseudocontractive mappings in real Banach spaces. The convergence results established in this paper are new for this more general class of ψ-expansive and accretive or pseudocontractive mappings. c 2005 Elsevier Ltd. All rights reserved. MSC: 47H04; 47H06; 47H30; 47J05; 47J25 Keywords: Accretive mapping; ψ-expansive mapping; Pseudocontractive mapping; Banach space

1. Introduction Let E be a real normed linear space with dual E ∗ . We denote by J the normalized duality ∗ mapping from E to 2 E defined by J x = { f ∗ ∈ E ∗ : x, f ∗  = x2 =  f ∗ 2 }, where ., . denotes the generalized duality pairing. It is well known that if E ∗ is strictly convex then J is single-valued. ∗ Corresponding author.

E-mail addresses: [email protected] (H. Zegeye), [email protected] (N. Shahzad). c 2005 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2005.11.011

74

H. Zegeye, N. Shahzad / Nonlinear Analysis 66 (2007) 73–82

A map A : D(A) ⊆ E → E is called strongly accretive if for each x, y ∈ D(A), there exist j (x − y) ∈ J (x − y) and a real number 0 < k < 1 such that

Ax − Ay, j (x − y) ≥ kx − y2 ,

(1.1)

and it is called β-strongly accretive if ∀x, y ∈ D(A), there exist j (x − y) ∈ J (x − y) and a continuous function β : [0, ∞) → [0, ∞) with β(0) = 0 and β(r ) > 0 for r > 0 such that

Ax − Ay, j (x − y) ≥ β(x − y)x − y.

(1.2)

The map A is called φ-uniformly accretive if ∀x, y ∈ D(A), there exist j (x − y) ∈ J (x − y) and a continuous function φ : [0, ∞) → [0, ∞) with φ(0) = 0 and φ(r ) > 0 for r > 0 such that

Ax − Ay, j (x − y) ≥ φ(x − y).

(1.3)

A map A is called accretive (see [1]) if ∀x, y ∈ D(A), there exists j (x − y) ∈ J (x − y) such that

Ax − Ay, j (x − y) ≥ 0.

(1.4)

Following Kato [11], we find an equivalent definition of (1.4) as follows: A is said to be accretive if for each x, y ∈ D(A) and λ ≥ 0 we have that x − y ≤ (x − y) + λ(Ax − Ay). The map A is called ψ-expansive if ∀x, y ∈ D(A), we have that Ax − Ay ≥ ψ(x − y),

(1.5)

where ψ : [0, ∞) → [0, ∞) is a continuous function such that ψ(0) = 0 and ψ(r ) > 0 for r > 0. If E is a Hilbert space, accretive operators are also called monotone. An operator A is called m-accretive if it is accretive and R(I + r A), the range of (I + r A), is E for all r > 0; and A is said to satisfy the range condition if cl(D(A)) ⊆ R(I + r A), ∀r > 0. Closely related to the class of accretive mappings is the class of pseudocontractive mappings. A map T : D(T ) ⊆ E → E is called strongly pseudocontractive if for each x, y ∈ D(T ), there exist j (x − y) ∈ J (x − y) and a real number 0 < s < 1 such that

T x − T y, j (x − y) ≤ sx − y2 , and it is called β-strongly pseudocontractive if ∀x, y ∈ D(T ), there exist j (x − y) ∈ J (x − y) and a continuous function β : [0, ∞) → [0, ∞) with β(0) = 0 and β(r ) > 0 for r > 0 such that

T x − T y, j (x − y) ≤ x − y2 − β(x − y)x − y. The map T is called φ-uniformly pseudocontractive if ∀x, y ∈ D(T ), there exist j (x − y) ∈ J (x − y) and a continuous function φ : [0, ∞) → [0, ∞) with φ(0) = 0 and φ(r ) > 0 for r > 0 such that

T x − T y, j (x − y) ≤ x − y2 − φ(x − y). The map T is called pseudocontractive if ∀x, y ∈ D(T ), there exists j (x − y) ∈ J (x − y) such that

T x − T y, j (x − y) ≤ x − y2 .

H. Zegeye, N. Shahzad / Nonlinear Analysis 66 (2007) 73–82

75

Observe that A is φ-uniformly accretive if and only if (I − A) is uniformly φpseudocontractive (where I denotes the identity operator). Moreover, a close look at the above definitions shows that the class of φ-uniformly accretive mappings or φ-uniformly pseudocontractive mappings includes several important classes of operators studied by various authors. It is known that (see [14]) if E is a Banach space for which each nonempty bounded closed convex subset has the fixed point property for nonexpansive self-mappings, and A : D(A) ⊆ E → 2 E is m-accretive (or satisfies the range condition), then the following are equivalent: (i) 0 ∈ R(A) (range of A); (ii) M := {x ∈ D(A) : t x ∈ Ax for some t < 0} is bounded; (iii) there exists x 0 ∈ D(A) and a bounded open neighborhood U of x 0 such that t (x − x 0 ) ∈ Ax for x ∈ ∂U ∩ D(T ) and t < 0. Using any one of the above conditions is common in the literature (see, e.g. [14,10,15]). The following example due to Garcia-Falset and Morales [9] shows that the class of ψ-expansive and accretive mappings is more general than the class of φ-uniformly accretive mappings. Let H be the real Hilbert space with the usual Euclidean inner product. Define A : R2 → R2 by A(x, y) = (y, −x). Then A is accretive and ψ-expansive with ψ(x) = x. However, A is not ψ-uniformly accretive. Notice that A satisfies (iii) with x 0 = (0, 0) and U = B1 (x 0 ). Thus, we observe that the class of ψ-expansive accretive maps includes properly the class of φ-uniformly accretive mappings. Let N(A) := {x ∈ D(A) : Ax = 0} and F(T ) := {x ∈ D(T ) : T x = x} denote the null space of A and the fixed-point set of T , respectively. Clearly a mapping T is pseudocontractive if and only if A := I − T is accretive so that solutions of the equation Ax = 0 for accretive operator A correspond to fixed points of T . In [17,18], Osilike observed that the class of mappings studied by Dunn [7] and Weng [20] are proper subclasses of the class of uniformly hemi-contractive mappings and proved that if K is a nonempty subset of a uniformly smooth Banach space and T : K → 2 K is a multi-valued uniformly hemi-contractive mapping, then a Mann-type iteration sequence (e.g., [13]) converges strongly to the fixed point of T . In [19], Zhang proved that if D is a nonempty subset of a real smooth Banach space E, and T : D → 2 E is a uniformly hemi-contractive mapping such that T (D) := ∪x∈D T x is a bounded subset of E then an Ishikawa-type sequence (see e.g., [8,12]) converges strongly to x ∗ ∈ F(T ), provided that the sequence is well defined ∀n ≥ 0, ∗ n+1 −x ) > 0, where ηn ∈ T yn , ξn+1 ∈ T x n+1 , ηn − ξn+1  → 0 and σ := infn∈N0 φ(x x n+1 −x ∗ 2 ∗ N0 := {n ∈ N : x n+1 = x } and φ : [0, ∞) → [0, ∞) is a strictly increasing function with φ(0) = 0. Recently, Chidume and Zegeye [2] proved that if E is a real normed linear space and A : E → E is a uniformly continuous and φ-uniformly quasi-accretive mapping then the sequence {x n } iteratively generated by x 1 ∈ E, x n+1 := x n − αn Ax n , n ≥ 1, where {αn } satisfies certain conditions, converges strongly to the unique solution of the equation Ax = 0. It is our purpose in this paper to construct an iteration process which converges strongly to the zero of ψ-expansive and accretive mappings in Banach spaces. Related results deal with the strong convergence of this iteration process to fixed points of ψ-expansive and pseudocontractive mappings in real Banach spaces. The convergence results established in this paper are new for this more general class of ψ-expansive and accretive or pseudocontractive mappings. 2. Preliminaries Let K ⊆ E be closed convex and Q a mapping of E onto K . Then Q is said to be sunny if Q(Qx + t (x − Qx)) = Qx for all x ∈ E and t ≥ 0. A mapping Q of E into E is said to be

76

H. Zegeye, N. Shahzad / Nonlinear Analysis 66 (2007) 73–82

a retraction if Q 2 = Q. If a mapping Q is a retraction, then Qz = z for every z ∈ R(Q), the range of Q. A subset K of E is said to be a sunny nonexpansive retract of E if there exists a sunny nonexpansive retraction of E onto K and it is said to be a nonexpansive retract of E if there exists a nonexpansive retraction of E onto K . If E = H , the metric projection PK is a sunny nonexpansive retraction from H to any closed convex subset of H . In what follows, we shall make use of the following lemmas. Lemma 2.1 (See e.g., [2]). Let E be a real normed linear space and J the normalized duality map on E. Then for any given x, y ∈ E, the following inequality holds: x + y2 ≤ x2 + 2 y, j (x + y),

∀ j (x + y) ∈ J (x + y).

}, {αn } and {γn } be sequences of nonnegative numbers satisfying Lemma 2.2 (See [5]). Let {λn  γn the conditions: lim αn = 0, ∞ 1 αn = ∞, and αn → 0, as n → ∞. Let the recursive inequality λn+1 ≤ λn − αn ψ(λn+1 ) + γn ,

n = 1, 2, . . . ,

(2.1)

be given where ψ : [0, ∞) → [0, ∞) is a strictly increasing function such that it is positive on (0, ∞) and ψ(0) = 0. Then λn → 0 as n → ∞. Theorem M1 ([9]). Let E be a Banach space, and let A : D(A) ⊆ E → E be an accretive and ψ-expansive mapping on D(A) such that co(D(A)) ⊆ ∩λ>0 R(I + λA). Suppose that there exists x 0 ∈ D(A), where one of the following holds: (i) There exists R > 0 such that ψ(R) > 2A(x 0 ); or (ii) There exists a bounded neighborhood U of x 0 such that t (x − x 0 ) = Ax for x ∈ ∂U ∩ D(A) and t < 0. Then A has a unique zero z ∈ D(A) where Jλ x → z as λ → ∞ for an arbitrary x ∈ co(D(A)), where Jλ := (I + λA)−1 . Theorem M2 ([9]). Let E be a Banach space, and let A : D(A) ⊆ E → E be an accretive and ψ-expansive mapping on D(A) such that co(D(A)) ⊆ ∩λ>0 R(I + λA). Suppose U is a bounded neighborhood of x 0 ∈ D(A) such that Ax 0  < Ax for all x ∈ ∂U ∩ D(A). Then there exists a unique z ∈ D(A) such that Az = 0. In addition limλ→∞ Jλ x = z for each x ∈ ∩λ>0 R(I + λA). Theorem I ([6], p. 221). Let A be a continuous and accretive operator on the real Banach space E with D(A) = E. Then A is m-accretive. For the rest of this paper, {λn } and {θn } are real sequences in (0, 1] satisfying the following  θn−1  θn −1 λn conditions: (i) lim θn = 0; (ii) λn θn = ∞, limn→∞ θn = 0; (iii) lim = 0. λ n θn n→∞

n→∞

1 1 Examples of real sequences which satisfy these conditions are λn = (n+1) , a and θn = (n+1)b where 0 < b < a and a + b < 1. Under the conditions of Theorem M1, let {yn } denote tn 1 the sequence defined by yn := ytn = (I + 1−t A)−1 x 1 , where tn = 1+θ , ∀n ≥ 1 and n n x 1 ∈ D(A), then, by the same theorem, we have that yn → z ∈ N(A). Moreover, we note that θn (x 1 − yn ) − Ayn = 0 for each n ≥ 1.

H. Zegeye, N. Shahzad / Nonlinear Analysis 66 (2007) 73–82

77

3. Main results We now prove the following theorems. Theorem 3.1. Let E be a real Banach space, and let A : D(A) ⊆ E → E be a Lipschitz, ψ-expansive accretive mapping such that co(D(A)) ⊆ ∩λ>0 R(I + λA). Assume that D(A) is a nonexpansive retract of E with Q as the nonexpansive retraction. Suppose that there exists x 0 ∈ D(A) where one of the following holds: (i) There exists R > 0 such that ψ(R) > 2A(x 0); or (ii) There exists a bounded neighborhood U of x 0 such that t (x − x 0 ) = Ax for x ∈ ∂U ∩ D(A) and t < 0. For arbitrary x 1 ∈ D(A), the projection perturbed Mann iteration sequence {x n } is defined by x n+1 := Q(x n − λn Ax n − λn θn (x n − x 1 )),

(3.1)

for all positive integers n. Then {x n } converges strongly to a zero of A. 1 Proof. Since λθnn → 0 there exists N0 > 0 such that λθnn ≤ d := 2(1+L)(2+L) , ∀n ≥ N0 , where L is Lipschitz constant for A. Furthermore, by Theorem M1, N(A) = ∅. Let x ∗ ∈ N(A) and r > 0 be sufficiently large such that x N0 ∈ Br (x ∗ ) and x 1 ∈ B 2r (x ∗ ). Now we prove that {x n } is bounded. It suffices to show by induction that {x n } belongs to B := Br (x ∗ ) for all integers n ≥ N0 . Now, x N0 ∈ B by construction. Hence we may assume x n ∈ B for any n > N0 and claim that x n+1 ∈ B. Suppose not. Then x n+1 − x ∗  > r and thus from the recursion formula (3.1) and Lemma 2.1, we get that

x n+1 − x ∗ 2 = Q(x n − λn Ax n − λn θn (x n − x 1 )) − x ∗ 2 ≤ x n − x ∗ − λn (Ax n + θn (x n − x 1 ))2 ≤ x n − x ∗ 2 − 2λn Ax n + θn (x n − x 1 ), j (x n+1 − x ∗ ) = x n − x ∗ 2 − 2λn θn x n+1 − x ∗ 2 + 2λn θn (x n+1 − x n ) − Ax n + θn (x 1 − x ∗ ) + Ax n+1 − Ax n+1 , j (x n+1 − x ∗ ).

(3.2)

But, since A is accretive, we have Ax n+1 − Ax ∗ , j (x n+1 − x ∗ ) ≥ 0. Thus, (3.2) and the Lipschitz property of A give x n+1 − x ∗ 2 ≤ x n − x ∗ 2 − 2λn θn x n+1 − x ∗ 2 + 2λn θn (x n+1 − x n ) + θn (x 1 − x ∗ ) + (Ax n+1 − Ax n ), j (x n+1 − x ∗ ) ≤ x n − x ∗ 2 − 2λn θn x n+1 − x ∗ 2   + 2λn (1 + L)x n+1 − x n  + θn x 1 − x ∗  x n+1 − x ∗  = x n − x ∗ 2 − 2λn θn x n+1 − x ∗ 2  + 2λn (1 + L)λn (Ax n − Ax ∗ ) + θn (x n − x ∗ + x ∗ − x 1 )  + θn x 1 − x ∗  x n+1 − x ∗  ≤ x n − x ∗ 2 − 2λn θn x n+1 − x ∗ 2

78

H. Zegeye, N. Shahzad / Nonlinear Analysis 66 (2007) 73–82

   + 2λn (1 + L)λn (1 + L)x n − x ∗  + θn x 1 − x ∗   + θn x 1 − x ∗  x n+1 − x ∗  ≤ x n − x ∗ 2 − 2λn θn x n+1 − x ∗ 2

θn + 2λn λn (1 + L)(L + 2)r + r x n+1 − x ∗ , 2

(3.3)

since x n ∈ B and x 1 ∈ B 2r (x ∗ ). But x n+1 − x ∗  > x n − x ∗ , so we have from (3.3)  that θn x n+1 − x ∗  ≤ λn (1 + L)(2 + L)r + θ2n r , and hence x n+1 − x ∗  ≤ r , since λn θn

1 ≤ 2(2+L)(1+L) , ∀n ≥ N0 . Thus we get a contradiction. As a result, x n ∈ B for all positive integers n ≥ N0 and hence the sequence {x n } is bounded. Next, we prove that x n − yn  → 0 as n → ∞. From the recursion formula (3.1) and Lemma 2.1 we have that

x n+1 − yn 2 ≤ x n − yn 2 − 2λn θn (x n+1 − yn ), j (x n+1 − yn ) + 2λn θn (x n+1 − yn ) − Ax n − θn (x n − x 1 ), j (x n+1 − yn ) = x n − yn 2 − 2λn θn x n+1 − yn 2 + 2λn θn (x n+1 − x n ) + [θn (x 1 − yn ) − Ayn ] − [ Ax n+1 − Ayn ] + [ Ax n+1 − Ax n ], j (x n+1 − yn ).

(3.4)

Observe that by the property of yn and accretivity of A we have θn (x 1 − yn ) − Ayn = 0 and

Ax n+1 − Ayn , j (x n+1 − yn ) ≥ 0 for all n ≥ 1. Thus, we have from (3.4) that x n+1 − yn 2 ≤ x n − yn 2 − 2λn θn x n+1 − yn 2 + 2λn θn (x n+1 − x n ) +(Ax n+1 − Ax n ), j (x n+1 − yn ) ≤ x n − yn 2 − 2λn θn x n+1 − yn 2 + 2λn (1 + L)x n+1 − x n .x n+1 − yn  ≤ x n − yn 2 − 2λn θn x n+1 − yn 2 + 2λ2n (1 + L)Ax n + θn (x n − x 1 ).x n+1 − yn .

(3.5)

But, since yn is convergent and x n is bounded, there exists M1 > 0 such that max{x n+1 − yn , Ax n + θn (x n − x 1 )} ≤ M1 . Thus from (3.5) we get that x n+1 − yn 2 ≤ x n − yn 2 − 2λn θn x n+1 − yn 2 + 2λ2n (2 + L)M1 . Moreover, by the accretivity of A, we have that



1

yn−1 − yn  ≤ yn−1 − yn + (Ayn−1 − Ayn )

θn   θn−1 − θn θn−1 ≤ (yn−1  + x 1 ) = − 1 (yn−1  + x 1 ). θn θn

(3.6)

(3.7)

Thus, from (3.6) and (3.7), we get that x n+1 − yn 2 ≤ x n − yn−1 2 − 2λn θn x n+1 − yn 2   θn−1 +M − 1 + 2λ2n (2 + L)M, θn

(3.8)

H. Zegeye, N. Shahzad / Nonlinear Analysis 66 (2007) 73–82

79

for some constant M > 0. By Lemma 2.2 and the conditions on {λn } and {θn }, we get x n+1 − yn → 0. Consequently, since yn → z ∈ N(A) we get that x n → z. The proof of the theorem is complete.  Theorem 3.2. Let E be a real Banach space, and let A : D(A) ⊆ E → E be a Lipschitz, ψ-expansive and accretive mapping such that co(D(A)) ⊆ ∩λ>0 R(I + λA). Assume that D(A) is a nonexpansive retract of E with Q as the nonexpansive retraction. Suppose that U is a bounded neighborhood of x 0 ∈ D(A) such that Ax 0  < Ax for all x ∈ ∂U ∩ D(A). For arbitrary x 1 ∈ D(A), the projection perturbed Mann iteration sequence {x n } is defined by x n+1 := Q(x n − λn Ax n − λn θn (x n − x 1 )), for all positive integers n. Then {x n } converges strongly to a zero of A. Proof. The proof follows from the proof of Theorem 3.1 with the help of Theorem M2 instead of Theorem M1.  Remark 3.3. If in Theorem 3.1 we have that D(A) is closed and convex and A is m-accretive then we get the following corollary. Theorem 3.4. Let E be a real Banach space, and let A : D(A) ⊆ E → E be a Lipschitz, ψ-expansive and m-accretive mapping with D(A) closed and convex. Assume that D(A) is a nonexpansive retract of E with Q as the nonexpansive retraction. Suppose that there exists x 0 ∈ D(A) where one of the following holds: (i) There exists R > 0 such that ψ(R) > 2A(x 0); or (ii) There exists a bounded neighborhood U of x 0 such that t (x − x 0 ) = Ax for x ∈ ∂U ∩ D(A) and t < 0. For arbitrary x 1 ∈ D(A), the projection perturbed Mann iteration sequence {x n } is defined by x n+1 := Q(x n − λn Ax n − λn θn (x n − x 1 )), for all positive integers n. Then {x n } converges strongly to a zero of A. Proof. Since A is m-accretive we have that co(D(A)) = D(A) ⊆ ∩λ>0 R(I + λA) = E and hence the conclusion follows from Theorem 3.1.  Remark 3.5. If in Theorem 3.4 we have that D(A) := E then the assumptions that A is m-accretive and D(A) is a nonexpansive retract of E may not be needed. Corollary 3.6. Let E be a real Banach space, and let A : E → E be a Lipschitz, ψ-expansive and accretive mapping. Suppose that there exists x 0 ∈ D(A) where one of the following holds: (i) There exists R > 0 such that ψ(R) > 2A(x 0); or (ii) There exists a bounded neighborhood U of x 0 such that t (x − x 0 ) = Ax for x ∈ ∂U ∩ D(A) and t < 0. For arbitrary x 1 ∈ E, the perturbed Mann iteration sequence {x n } is defined by x n+1 := x n − λn Ax n − λn θn (x n − x 1 ), for all positive integers n. Then {x n } converges strongly to a zero of A.

80

H. Zegeye, N. Shahzad / Nonlinear Analysis 66 (2007) 73–82

Proof. Since A is continuous and accretive with D(A) = E, by Theorem I, we have that A is m-accretive and hence the conclusion follows from Theorem 3.4.  The following corollary follows from Theorem 3.2. Corollary 3.7. Let E be a real Banach space, and let A : E → E be a Lipschitz, ψ-expansive and accretive mapping. Suppose that U is a bounded neighborhood of x 0 ∈ D(A) such that Ax 0  < Ax for all x ∈ ∂U ∩ D(A). For arbitrary x 1 ∈ D(A), the perturbed Mann iteration sequence {x n } is defined by x n+1 := x n − λn Ax n − λn θn (x n − x 1 ), for all positive integers n. Then {x n } converges strongly to a zero of A. Theorem 3.8. Let K be a closed convex subset of a Banach space E, and let T : K → K be a Lipschitz and pseudocontractive mapping such that I − T is ψ-expansive. Suppose that there exists x 0 ∈ K where one of the following holds: (i) There exists R > 0 such that ψ(R) > 2(I − T )(x 0 ); or (ii) There exists a bounded neighborhood U of x 0 such that t (x −x 0) = (I −T )x for x ∈ ∂U ∩ K and t < 0. Let {λn } and {θn } be real sequences satisfying conditions in Theorem 3.1 and λn (1 + θn ) ≤ 1 for all n ≥ 1. For arbitrary x 1 ∈ E, the perturbed Mann iteration sequence {x n } is defined by x n+1 := (1 − λn )x n + λn T x n − λn θn (x n − x 1 ), for all positive integers n. Then {x n } converges strongly to a fixed point of T . Proof. Since K is convex we get that x n is well defined. Let A := (I − T ); then we have that A is ψ-expansive accretive and Lipschitzian with constant L  = (1 + L), where L is Lipschitz constant for T . Also, K ⊆ ∩λ>0 R(I + λ(I − T )) by Lemma 4.1 of [4]. Thus, the conclusion follows from Theorem 3.1.  Corollary 3.9. Let E be a real Banach space, let T : E → E be a Lipschitz and pseudocontractive mapping such that I − T is ψ-expansive. Suppose that there exists x 0 ∈ D(T ) where one of the following holds: (i) There exists R > 0 such that ψ(R) > 2(I − T )(x 0 ); or (ii) There exists a bounded neighborhood U of x 0 such that t (x −x 0) = (I −T )x for x ∈ ∂U ∩ K and t < 0. For arbitrary x 1 ∈ E, the perturbed Mann iteration sequence {x n } is defined by x n+1 := (1 − λn )x n + λn T x n − λn θn (x n − x 1 ), for all positive integers n. Then {x n } converges strongly to a fixed point of T . If in Theorem 3.8 we have that K is a closed convex subset of a Banach space E for which each nonempty closed convex and bounded subset of K has the fixed point property for nonexpansive self-mappings and F(T ) = ∅, then by Theorem 1 of [14] we get that (ii) is satisfied since by Lemma 4.1 of [4] I − T satisfies the range condition. Thus, we may have the following corollary.

H. Zegeye, N. Shahzad / Nonlinear Analysis 66 (2007) 73–82

81

Corollary 3.10. Let K be a closed convex subset of a Banach space E such that each nonempty bounded closed convex subset of K has the fixed point property for nonexpansive self-mappings, and let T : K → K be a Lipschitz and pseudocontractive mapping such that T is ψ-expansive with F(T ) = ∅. Let {λn } and {θn } be real sequences satisfying conditions in Theorem 3.1 and λn (1 + θn ) ≤ 1 for all n ≥ 1. For arbitrary x 1 ∈ E, the perturbed Mann iteration sequence {x n } is defined by x n+1 := (1 − λn )x n + λn T x n − λn θn (x n − x 1 ), for all positive integers n. Then {x n } converges strongly to a fixed point of T . Remark 3.11. We note that Corollary 3.10 improves Theorem 3.3 of [3], which assumes that E has uniformly Gˆateaux differentiable norm, where the assumption that T is ψ-expansive is actually made. Many convergence theorems exist in the literature for ψ-uniformly accretive and β-strongly accretive mappings (e.g., see [2,16]) but, as far we know, these convergence results seem new for the class of ψ-expansive mappings. Theorem 3.12. Let E be a real Banach space, and let A : D(A) ⊆ E → E be a Lipschitz and φ-uniformly accretive mapping with co(D(A)) ⊆ ∩λ>0 R(I + λA). Assume that D(A) is a nonexpansive retract of E with Q as the nonexpansive retraction. Suppose that there exists x 0 ∈ D(A) where one of the following holds: (i) There exists R > 0 such that ψ(R) > 2A(x 0); or (ii) There exists a bounded neighborhood U of x 0 such that t (x − x 0 ) = Ax for x ∈ ∂U ∩ D(A) and t < 0. For arbitrary x 1 ∈ D(A), the projection perturbed Mann iteration sequence {x n } is defined by x n+1 := Q(x n − λn Ax n − λn θn (x n − x 1 )), for all positive integers n. Then {x n } converges strongly to a zero of A. Proof. Since every φ-uniformly accretive mapping is ψ-expansive, the conclusion follows from Theorem 3.1.  The following corollaries are consequences of Theorem 3.12. Corollary 3.13. Let E be a real Banach space, and let A : E → E be a Lipschitz and φ-uniformly accretive mapping. Suppose that there exists x 0 ∈ D(A) where one of the following holds: (i) There exists R > 0 such that φ(R) > 2A(x 0 ); or (ii) There exists a bounded neighborhood U of x 0 such that t (x − x 0 ) = Ax for x ∈ ∂U ∩ D(A) and t < 0. For arbitrary x 1 ∈ D(A), the perturbed Mann iteration sequence {x n } is defined by x n+1 := x n − λn Ax n − λn θn (x n − x 1 ), for all positive integers n. Then {x n } converges strongly to a zero of A.

82

H. Zegeye, N. Shahzad / Nonlinear Analysis 66 (2007) 73–82

Corollary 3.14. Let E be a real Banach space, and let T : E → E be a Lipschitz and φ-uniformly pseudocontractive mapping. Suppose that U is a bounded neighborhood of x 0 ∈ D(T ) such that (I − T )x 0  < (I − T )x for all x ∈ ∂U ∩ D(T ). For arbitrary x 1 ∈ D(T ), the perturbed Mann iteration sequence {x n } is defined by x n+1 := (1 − λn )x n + λn T x n − λn θn (x n − x 1 ), for all positive integers n. Then {x n } converges strongly to a zero of A. Acknowledgements The authors are indebted to the referee for valuable comments. The research of the first author is supported by a grant from TWAS (03-027 RG/MATHS/AF/Ac), Trieste, Italy. He is grateful to The Third World Academy of Sciences (TWAS) for a generous contribution towards his research. References [1] F.E. Browder, Nonlinear mappings of nonexpansive and accretive type in Banach spaces, Bull. Amer. Math. Soc. 73 (1967) 875–882. [2] C.E. Chidume, H. Zegeye, Approximation methods for nonlinear operator equations, Proc. Amer. Math. Soc. 131 (2003) 2467–2478. [3] C.E. Chidume, H. Zegeye, Approximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive maps, Proc. Amer. Math. Soc. 132 (2004) 831–840. [4] C.E. Chidume, H. Zegeye, Iterative solutions of nonlinear equations of accretive and pseudocontractive types, J. Math. Anal. Appl. 282 (2003) 756–765. [5] C.E. Chidume, H. Zegeye, Approximation of solutions of Nonlinear equations of monotone and Hammerstein type, Appl. Anal. 82 (8) (2003) 747–758. [6] I. Cioranescu, Geometry of Banach spaces, Duality mapping and Nonlinear Problems, Klumer Academic publishers, Amsterdam, 1990. [7] J.C. Dunn, Iterative construction of fixed points for multivalued operators of the monotone type, J. Funct. Anal. 27 (1978) 38–50. [8] S. Ishikawa, Fixed points by a new iteration Method, Proc. Amer. Math. Soc. 44 (1974) 147–150. [9] J. Garcia-Falset, C. Morales, Existence theorems for m-accretive operators in Banach spaces, J. Math. Anal. Appl. 309 (2005) 453–461. [10] A.G. Kartsatos, Some mappings for accretive operators in Banach spaces, J. Math. Anal. Appl. 82 (1981) 505–506. [11] T. Kato, Nonlinear semi-groups and evolution equations, J. Math. Soc. Japan 19 (1967) 508–520. [12] L. Liu, Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl. 194 (1995) 114–125. [13] W.R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953) 506–510. [14] C. Morales, Nonlinear equations involving m-accretive operators, J. Math. Anal. Appl. 97 (1983) 329–336. [15] C. Morales, Existence theorems for demicontinuous accretive operators in Banach spaces, Houston J. Math. 10 (1984) 535–543. [16] M.O. Osilike, Iterative solution of nonlinear equations of the ψ-strongly accretive type, J. Math. Anal. Appl. 200 (1996) 259–271. [17] M.O. Osilike, Iterative construction of fixed points of multi-valued operators of the accretive type, Soochow J. Math. 22 (1996) 485–494. [18] M.O. Osilike, Iterative construction of fixed points of multi-valued operators of the accretive type II, Soochow J. Math. 24 (1998) 141–146. [19] S. Zhang, On the convergence problems of Ishikawa and Mann iteration process with errors for ψ-pseudo contractive type mappings, Appl. Math. Mech. 21 (2000) 1–10. [20] X.L. Weng, Iterative construction of fixed points of a dissipative type operator, Tamkang J. Math. 23 (1992) 205–215.