Positivity

Strong convergence of viscosity approximation methods with strong pseudocontraction for Lipschitz pseudocontractive mappings Yisheng Song Abstract. In this paper, for a Lipschitz pseudocontractive mapping T , we study the strong convergence of iterative schemes generated by xn+1 = (1 − αn − βn )xn + αn f (xn ) + βn T xn , where f is a Lipschitz strong pseudocontractive mapping and {βn }, {αn } sat∞ β2 isfy (i) lim αn = 0; (ii) αn = ∞; (iii) lim αnn = 0. n→∞

n=1

n→∞

Mathematics Subject Classification (2000). 47H06, 47J05, 47J25, 47H10, 47H17. Keywords. Lipschitz Pseudocontractions; Viscosity approximations; Strong convergence; uniformly Gˆ ateaux differentiable norm.

1. Introduction Let K be a nonempty closed convex subset of a smooth Banach space E. Let T : K → K be a continuous pseudocontractive mapping and f : K → K be any fixed Lipschitzian strongly pseudocontractive mapping. Then the mapping Tft = tf + (1 − t)T obviously is a continuous strongly pseudocontractive mapping from K to K for each fixed t ∈ (0, 1). Therefore, Tft has a unique fixed point in K ([18, Corollary 2]), i.e., for any given t ∈ (0, 1), there exists xt ∈ K such that xt = tf (xt ) + (1 − t)T xt .

(1.1)

As t → 0, the strong convergence of the path {xt } have recently been studied by Song, et al. [16,36,39] and others. The special cases of the path (1.1), in which f is a constant mapping (f (x) = u for some u ∈ K) was introduced and studied by Browder [1] for a nonexpansive mapping T in Hilbert space, by Morales–Jung [28] and Udomene [45] for a continuous pseudocontractive mapping T . The cases in URL: http://songys.mysinamail.com or http://songys.16f.cn/.

644

Y. Song

Positivity

which f : K → K is a contraction have been introduced and studied by Moudafi [26] for a nonexpansive self-mapping T defined on a Hilbert space, by Song–Chen [38] for a nonexpansive nonself-mapping T defined on a reflexive Banach space which has a weakly continuous duality mapping J. Halpern [23] was the first who introduced the following iteration scheme for a nonexpansive mapping T which was referred to as Halpern iteration: for u, x0 ∈ K, αn ∈ [0, 1], xn+1 = αn u + (1 − αn )T xn , n ≥ 0.

(1.2)

Convergence of this schemes have been studied by many researchers with various types of additional conditions. For the studies of a nonexpansive mapping T , see Bruck [3,4], Reich [29,30], Song-Xu [40], Takahashi-Ueda [43], Suzuki [42], and others. On the other hand, Mann [27] introduced the following iteration for T in a Hilbert space: xn+1 = αn xn + (1 − αn )T xn , n ≥ 0,

(1.3)

where {αn } is a sequence in [0, 1]. Later, Reich [31] studied this iteration in a uniformly convex Banach space with a Fr´ echet differentiable norm and obtained its weak convergence. This Mann’s iteration process has extensively been studied over the last twenty years for constructions of fixed points of nonlinear mappings and of solutions of nonlinear operator equations involving monotone, accretive and pseudocontractive operators(see, e.g., [5,8–12,14,17,19,20,22,25,35] and others). In an infinite-dimensional Hilbert space, the normal Mann’s iteration algorithm (1.3) has only weak convergence, in general, even for nonexpansive mappings. In order to get a strong convergence result, one has to modify the normal Mann’s iteration algorithm. Some attempts have been made and several important results have been reported(see, e.g., [7,13,15,21,32,37,45,47] and others). Recently, Chidume–Ofoedu [14] and Song [34] tried to found the strong convergence of the following iteration {xn } which is independent of the path xt ; xn+1 = (1 − λn αn )xn + λn αn T xn − λn θn (xn − x1 ), n ≥ 1.

(1.4)

Very recently, Zhou [47] obtained the strong convergence theorem of the iterative sequence (1.5) for λ−strict pseudocontraction T in 2-uniformly smooth Banach space: for u, x0 ∈ E, xn+1 = βn u + γn xn + (1 − βn − γn )[αn xn + (1 − αn )T xn ], n ≥ 0,

(1.5)

where {βn }, {γn } and {αn } in (0, 1) satisfy: (i) a ≤ αn ≤ Kλ2 for some a > 0 ∞ and for all n ≥ 0; (ii) lim βn = 0 and βn = ∞; (iii) lim |αn+1 − αn | = 0; n→∞

n=1

n→∞

(iv) 0 < lim inf γn ≤ lim sup γn < 1. Chidume–Souza [6] showed several strongly n→∞

n→∞

convergent theorems of the iteration (1.6) for Lipschitz pseudocontractive mappings in a real reflexive Banach space E with a uniformly Gˆ ateaux differentiable

Vol. 13 (2009)

Lipschitz pseudocontractive mappings

645

norm and with the fixed point property for nonexpansive self-mappings: for any u, x1 ∈ K, xn+1 = βn u + (1 − βn )[αn T xn + (1 − αn )xn ]

(1.6)

for all positive integers n, where {βn } and {αn } are real sequences in (0, 1) satis∞ βn = ∞; (iii) lim αβnn = 0. fying the conditions: (i) lim βn = 0; (ii) n→∞

n→∞

n=1

In this paper, we deal with iterative schemes generated by (1.7) for a Lipschitz pseudocontractive mapping T and a fixed Lipschitz strongly pseudocontractive mapping f : xn+1 = (1 − αn − βn )xn + αn f (xn ) + βn T xn ,

(1.7)

and obtain strong convergence of the iteration when {αn } and {βn } are real ∞ β2 αn = ∞; (iii) lim αnn = 0, sequences in (0, 1) satisfying: (i) lim αn = 0; (ii) n→∞

n=1

n→∞

which is one of the generalized results of main results of Chidume–Udomene [13] and Schu [32] and Chidume–Souza [6] and Zhou [47] and Chidume– Ofoedu [14] and Song [34] and others. In particular, the parameters of our iterative 1 . sequence are simpler, and the typical example is αn = βn = n+1

2. Preliminaries Throughout this paper, a Banach space E will always be over the real scalar field. We denote its norm by · and its dual space by E ∗ . The value of x∗ ∈ E ∗ at y ∈ E is denoted by y, x∗ , and the normalized duality mapping J from E into ∗ 2E is defined by J(x) = {f ∈ E ∗ : x, f = xf , x = f },

∀x ∈ E.

Let F (T ) = {x ∈ E : T x = x} denote the set of all fixed point for a mapping T . Let S(E) := {x ∈ E; x = 1} denote the unit sphere of a Banach space E. The space E is said to have (i) a Gˆ ateaux differentiable norm (we also say that E is smooth), if the limit x + ty − x (2.1) t exists for each x, y ∈ S(E); (ii) a uniformly Gˆ ateaux differentiable norm, if for each y in S(E), the limit (2.1) is uniformly attained for x ∈ S(E); (iii) a Fr´echet differentiable norm, if for each x ∈ S(E), the limit (2.1) is attained uniformly for y ∈ S(E); (iv) a uniformly Fr´echet differentiable norm (we also say that E is uniformly smooth), if the limit (2.1) is attained uniformly for (x, y) ∈ S(E) × S(E). (v) ﬁxed point property for non-expansive self-mappings, if each non-expansive selfmapping defined on any bounded closed convex subset K of E has at least a fixed point. A Banach space E is said to be (i) strictly convex if x = y = 1, x = y implies x+y 2 < 1; (ii) uniformly convex if for all ε ∈ [0, 2], ∃δε > 0 such that x = y = 1 with x − y ≥ ε implies x+y < 1 − δε . 2 lim

t→0

646

Y. Song

Positivity

Recall that a mapping T with domain D(T ) and range R(T ) in Banach space E is called to be strongly pseudocontractive if there exists k ∈ (0, 1) such that for all x, y ∈ D(T ), there is j(x − y) ∈ J(x − y) satisfying T x − T y, j(x − y) ≤ kx − y2

(2.2)

(x − T x) − (y − T y), j(x − y) ≥ (1 − k)x − y2

(2.3)

or, equivalently, while T is said to be pseudocontractive if (2.2) or (2.3) holds for k = 1. T is called λ-strictly pseudocontractive, if there exists λ ∈ (0, 1) such that for all x, y ∈ D(T ), there is j(x − y) ∈ J(x − y) satisfying T x − T y, j(x − y) ≤ ||x − y||2 − λ||x − y − (T x − T y)||2 .

(2.4)

T is said to be Lipschitzian if there exists L > 0 such that for all x, y ∈ D(T ), T x − T y ≤ Lx − y. The mapping T is called non-expansive if L = 1 and, further, T is said to be contractive if L < 1. It is obvious that λ-strictly pseudocontractive mapping is L-Lipschitzian with L = λ+1 λ , and nonexpansive mapping is a subclass of strictly pseudocontractive mapping in Hilbert space, (contractive) nonexpansive mapping is an important subclass of (Lipschitz strongly) pseudocontractive mapping, but the converse implication may be false. This can be seen from the existing examples (see, e.g., [12,14,47]). An important class of mappings closely related to the class of pseudocontractive mappings is that of accretive mappings. A is accretive if and only if (I − A) is pseudocontractive. The accretive mappings were independently introduced by Browder [2] and Kato [24] in 1967. It is now well known that if A is accretive then the solutions of the equation Ax = 0 correspond to the equilibrium points of some evolution systems. Lemma 2.1 (Song,et al. [36,39]). Let E be a reﬂexive Banach space which has both ﬁxed point property for non-expansive self-mappings and a uniformly Gˆ ateaux differentiable norm or be a reﬂexive and strictly convex Banach space with a uniformly Gˆ ateaux differentiable norm, and K be a nonempty closed convex subset of E. Suppose T is a continuous pseudocontractive mapping from K into itself with F (T ) = ∅ and f is a Lipschitzian strongly pseudocontractive mapping from K into itself. Then as t → 0, xt , deﬁned by xt = tf (xt ) + (1 − t)T xt converges strongly to a ﬁxed point p of T , where p is the unique solution in F (T ) to the following variational inequality: (f − I)p, J(u − p) ≤ 0

for all u ∈ F (T ).

(2.5)

Lemma 2.2 (Liu [25] and Xu [46]). Let {an } be a sequence of nonnegative real numbers satisfying the property: an+1 ≤ (1 − tn )an + bn + tn cn ,

∀n ≥ 0,

Vol. 13 (2009)

Lipschitz pseudocontractive mappings

647

where {tn }, {bn }, {cn } satisfy the restrictions: ∞ ∞ tn = ∞; (ii) bn < +∞; (iii) lim sup cn ≤ 0. (i) n=0

n=0

n→∞

Then {an } converges to zero as n → ∞.

3. Main results In this section, we shall study the strong convergence of the explicit viscosity approximation with a Lipschitzian strongly pseudocontractive mapping f for a Lipschitzian pseudocontractive mapping T : xn+1 = (1 − αn − βn )xn + αn f (xn ) + βn T xn , ∀n ≥ 0,

(3.1)

where {αn } and {βn } are real sequences in (0, 1) satisfying the conditions: +∞ β2 αn = +∞; (C2) αn + βn < 1, lim n = 0. (C1) lim αn = 0, n→∞ n→∞ αn n=1 Clearly, {αn } and {βn } are simpler than the results mentioned in ”Introduc1 . tion”. The typical example is αn = βn = n+1 Theorem 3.1. Let E be a reﬂexive Banach space which has both ﬁxed point property for non-expansive self-mappings and a uniformly Gˆ ateaux diﬀerentiable norm, and K be a nonempty closed convex subset of E. Suppose T : K → K is a Lipschitzian pseudo-contraction with a Lipschitz constant L1 > 0 and F (T ) = ∅, and f : K → K is a Lipschitzian strongly pseudocontractive mapping with a Lipschitz constant L2 > 0 and a pseudocontractive coeﬃcient k ∈ (0, 1). Assume that {xn } is a sequence given by (3.1) and {αn } and {βn } satisfy the conditions (C1) and (C2). Then, as n → ∞, {xn } converges strongly to to some ﬁxed point z of T , which is the unique solution in F (T ) to the variational inequality: (f − I)z, J(p − z) ≤ 0

for all p ∈ F (T ).

(3.2)

Proof. The proof will be split into three steps. Step 1. {xn } is bounded. Take p ∈ F (T ) and L = max{L1 , L2 } and choose r > 0 sufficiently large such that 1−k r < r. x1 − p ≤ r, f (p) − p ≤ 2 We proceed by induction to show that xn − p ≤ r for all n ≥ 1. Assume that xn − p ≤ r for some n > 1. We show that xn+1 − p ≤ r. Suppose that xn+1 − p > r. Then, from the iterative sequence (3.1), we estimate as follows: xn+1 − xn ≤ αn f (xn ) − xn + βn T xn − xn ≤ αn (f (xn ) − f (p) + f (p) − p + p − xn ) +βn (T xn − p + p − xn ) ≤ αn [(L + 1)p − xn + f (p) − p] + βn (L + 1)p − xn ≤ (βn + αn )(L + 2)r.

(3.3)

648

Y. Song

Positivity

Since xn+1 − T xn+1 , J(xn+1 − p) = (xn+1 − p) + (T p − T xn+1 ), J(xn+1 − p) = (xn+1 − p2 − T xn+1 − T p, J(xn+1 − p)) ≥ 0, we also obtain xn+1 − p2 = xn − p + αn (f (xn ) − xn ) + βn (T xn − xn ), J(xn+1 − p) ≤ xn − pJ(xn+1 − p) − αn xn+1 − p, J(xn+1 − p) +αn f (xn ) − f (xn+1 ) + xn+1 − xn + f (p) − p, J(xn+1 − p) +αn f (xn+1 ) − f (p), J(xn+1 − p) − βn xn+1 − T xn+1 , J(xn+1 − p) +βn T xn − xn + xn+1 − T xn+1 , J(xn+1 − p) ≤ xn − pxn+1 − p − αn xn+1 − p2 + αn kxn+1 − p2 +αn (f (xn ) − f (xn+1 ) + xn+1 − xn + f (p) − p)xn+1 − p +βn (T xn − T xn+1 + xn+1 − xn )xn+1 − p ≤ xn − pxn+1 − p − αn (1 − k)xn+1 − p2 +[(αn + βn )(L + 1)xn+1 − xn + αn f (p) − p]xn+1 − p. So, substituting xn+1 − xn with the estimation of xn+1 − xn above, we have [1 + αn (1 − k)]xn+1 − p ≤ xn − p + (αn + βn )2 (L + 1)(L + 2)r +αn f (p) − p. 1−k r, then Since xn+1 − p > r and f (p) − p ≤ 2 1−k αn r + (αn + βn )2 (L + 1)(L + 2)r. [1 + αn (1 − k)]r < 1 + 2 Therefore, it follows that 1−k αn < (αn + βn )2 (L + 1)(L + 2), 2 that is, αn + 2βn +

βn2 1−k > 0, > αn 2(L + 1)(L + 2) 2 βn α n n→∞

which contradicts to the conditions lim αn = 0 and lim n→∞

= 0. Therefore,

xn+1 − p ≤ r. This proves the boundedness of the sequence {xn }, which implies that the sequences {f (xn )} and {T xn } are bounded. Step 2. For the unique solution z in F (T ) to the variational inequality (3.2), lim supf (z) − z, J(xn+1 − z) ≤ 0. n→∞

(3.4)

Let Sm : K → K be defined by Sm x := (1 − βm )x + βm T x for each x ∈ K and βm ∈ (0, 1). Then, we observe that for each m, Sm is a Lipschitz pseudocontractive

Vol. 13 (2009)

Lipschitz pseudocontractive mappings

649

mapping with Lipschitz constant 1+βm (L−1) > 0 and F (Sm ) = F (T ). Moreover, the definition of Sm reduces to lim Sm xn − xn = lim βm T xn − xn = 0.

m→∞

(3.5)

m→∞

Let ztm be defined by ztm = tf (ztm ) + (1 − t)Sm ztm for each t ∈ (0, 1) and each βm ∈ (0, 1). Then for each m, it follows from Lemma 2.1 that lim ztm = zm , t→0

where zm is the unique solution in F (Sm ) = F (T ) of the variational inequality (3.2). Then zm ≡ z by the uniqueness of solution, and hence lim ztm = z for each m.

t→0

Clearly, ztm − z2 = tf (ztm ) − z, J(ztm − z) + (1 − t)Sm ztm − z, J(ztm − z) ≤ tf (ztm ) − f (z) + f (z) − z, J(ztm − z) + (1 − t)ztm − z2 ≤ ktztm − z2 + tf (z) − zztm − z + (1 − t)ztm − z2 . Then (1 − k)tztm − z2 ≤ tf (z) − zztm − z. 1 Thus, ztm − z ≤ 1−k f (z) − z (also see Song,et al.[36,39]), and hence, there exists a positive constant M such that M ≥ xn − ztm . Since Sm is a pseudocontractive mapping for each m, then using the equality

(1 − t)(ztm − Sm ztm ) = t(f (ztm ) − ztm ), we have tf (ztm ) − ztm , J(xn − ztm ) = (1 − t)ztm − xn + xn − Sm xn + Sm xn − Sm ztm , J(xn − ztm ) = (1 − t)[−ztm − xn 2 + xn − Sm xn , J(xn − ztm ))

+Sm xn − Sm ztm , J(xn − ztm )] ≤ (1 − t)xn − Sm xn , J(xn − ztm )

≤ (1 − t)Sm xn − xn ztm − xn . Thus,

f (ztm ) − ztm , J(xn − ztm ) ≤

xn − Sm xn M. t

It follows from (3.5) that lim sup f (ztm ) − ztm , J(xn − ztm ) ≤ 0. m→∞

Therefore, for any ε > 0, there exists a positive integer N such that for all m ≥ N , ε (3.6) f (ztm ) − ztm , J(xn − ztm ) < . 2

650

Y. Song

Positivity

On the other hand, since J is norm topology to weak∗ topology uniformly continuous on bounded sets and lim ztN − z = 0, we have that t→0

|f (z) − z, J(xn − z) − f (ztN ) − ztN , J(xn − ztN )| = |f (z) − z, J(xn − z) − J(xn − ztN ) + f (z) − f (ztN ) + ztN − z, J(xn − ztN )| ≤ |f (z) − z, J(xn − z) − J(xn − ztN )| + (1 + L)M ztN − z → 0, as t → 0. Hence, for the above ε > 0, ∃δ > 0, such that ∀t ∈ (0, δ), for all n, we have ε f (z) − z, J(xn − z) < f (ztN ) − ztN , J(xn − ztN ) + . 2 By (3.6), we obtain that lim supf (z) − z, J(xn − z) ≤ n→∞

ε ε + = ε. 2 2

Since ε is arbitrary, (3.4) is proved. Step 3. lim xn − z = 0. n→∞

Using the pseudocontraction of T (noting xn+1 − T xn+1 , J(xn+1 − z) ≥ 0) and the strong pseudocontraction of f , we make the following estimates: xn+1 − z2 = (1 − αn )xn − z, J(xn+1 − z) − βn xn+1 − T xn+1 , J(xn+1 − z) + αn f (xn ) − f (xn+1 ) + f (xn+1 ) − f (z) + f (z) − z, J(xn+1 − z) + βn T xn − xn + xn+1 − T xn+1 , J(xn+1 − z) ≤ (1 − αn )xn − zxn+1 − z + kαn xn+1 − z2 + αn f (xn ) − f (xn+1 )xn+1 − z + αn f (z) − z, J(xn+1 − z) + βn (T xn − T xn+1 + xn+1 − xn )xn+1 − z xn − z2 + xn+1 − z2 + kαn xn+1 − z2 2 + αn Lxn − xn+1 xn+1 − z + αn f (z) − z, J(xn+1 − z)

≤ (1 − αn )

+ βn (L + 1)xn+1 − xn xn+1 − z. Thus, by the final inequality in (3.3), we have (1 + (1 − 2k)αn )xn+1 − z2 ≤ (1 − αn )xn − z2 + 2αn f (z) − z, J(xn+1 − z) + 2(L + 1)(L + 2)r2 (αn + βn )2 . Since limn→∞ αn = 0, there exists a positive integer N1 such that |1 − 2k|αn < 1,

∀n ≥ N1 .

Vol. 13 (2009)

Lipschitz pseudocontractive mappings

651

1 β2 [(L + 1)(L + 2)r2 (αn + 2βn + n ) + f (z) − z, J(xn+1 − z)] and 1−k αn 2(1 − k)αn γn = , then for all n ≥ N1 , we have 1 + (1 − 2k)αn

Set λn =

xn+1 − z2 ≤ (1 − γn )xn − z2 + γn λn .

(3.7)

By the condition (C1) and (C2) and (3.4), we obtain lim sup λn ≤ 0 and n→∞

+∞

γn = +∞.

n=1

Hence, Applying Lemma 2.2 to the inequality (3.7), we conclude that lim

n→∞

xn − z = 0. This completes the proof.

We remark that if E is strictly convex then the property that E has the fixed point property for nonexpansive self-mappings may be dropped. In fact, we have the following theorem. Theorem 3.2. Let E be a reﬂexive and strictly convex Banach space with a uniformly Gˆ ateaux differentiable norm. suppose K, T, f, {xn }, {αn }, {βn } are as Theorem 3.1. Then, as n → ∞, {xn } converges strongly to some ﬁxed point z of T , which is the unique solution in F (T ) to the variational inequality (3.2). Proof. This follows from Lemma 2.1 and the proof of Theorem 3.1

Theorem 3.3. Let E be a reﬂexive Banach space which has both ﬁxed point property for non-expansive self-mappings and a uniformly Gˆ ateaux differentiable norm or be a reﬂexive and strictly convex Banach space with a uniformly Gˆ ateaux differentiable norm. Assumed that A : E → E is a Lipschitzian accretive mapping with A−1 (0) = {y ∈ E; Ay = 0} = ∅, and S : E → E is a Lipschitzian strongly accretive mapping. Let {xn } be be a sequence given by xn+1 = xn − αn S(xn ) − βn Axn ,

∀n ≥ 0.

(3.8)

If {αn } and {βn } are sequences in (0, 1) satisfying the conditions (C1) and (C2). Then, as n → ∞, {xn } converges strongly to a zero x of A, which is the unique solution in A−1 (0) to the variational inequality: Sx, J(p − x) ≥ 0

for all p ∈ A−1 (0).

(3.9)

Proof. Since A is accretive and S is strongly accretive, T = I − A is Lipschitz pseudocontractive with Lipschitz constant L = (L + 1) and the fixed point of T is the solution of the equation Ax = 0, and f = I − S is strongly pseudocontractive ([44]), respectively. Thus (3.8) can be rewritten as follows: xn+1 = (1 − αn − βn )xn + αn f (xn ) + βn T xn ,

∀n ≥ 0.

The conclusion follows from Theorem 3.1 and 3.2. This completes the proof.

As a direct consequence of Theorem 3.1 and 3.2, we obtain the following corollaries.

652

Y. Song

Positivity

Corollary 3.4. Suppose E, K, f, {xn }, {αn }, {βn } are as Theorem 3.1 or 3.2, and T : K → K is a nonexpansive mapping. Then {xn } converges strongly to some ﬁxed point z of T . Corollary 3.5. Suppose E, K, T, {αn }, {βn } are as Theorem 3.1 or 3.2. Let {xn } be be a sequence given by xn+1 = αn u + βn T xn + (1 − αn − βn )xn ,

∀n ≥ 0.

(3.10)

Then, as n → ∞, {xn } converges strongly to some ﬁxed point z of T . Remark 1. (i) Corollary 3.5 include Chidume–Chidumeb–Zegeye [6, Theorem 3.1, 3.6] as a special case. (1.6) can be rewritten as follows: xn+1 = βn u + (1 − βn )αn T xn + (1 − βn − (1 − βn )αn )xn . αn α2 = 0 implies the condition lim n = 0, but the converse n→∞ βn n→∞ βn 1 implication may be false. For example, αn = βn = n+1 . (ii) There are plenty of Banach spaces with the fixed point property for nonexpansive mappings. For example, finite-dimensional Banach spaces, uniformly convex Banach space, uniformly smooth Banach space, reflexive Banach space with the Opial’s property, reflexive Banach space with normal structure and so on. In Theorem 3.1 and 3.2, if in addition, K is bounded, then we have that {xn } is bounded and hence the requirements that F (T ) = ∅ is not needed. In fact, we have the following corollary. the condition lim

Corollary 3.6. Let E, f, {αn }, {βn } be as Theorem 3.1 or 3.2, and K be a nonempty bounded closed convex subset of E. Assume that T : K → K is a Lipschitzian pseudocontractive mapping and {xn } is a sequence given by (3.1) or (3.10). Then, as n → ∞, {xn } converges strongly to some ﬁxed point z of T . Remark 2. All the theorems of this paper remain true if f is a contractive selfmapping in the recursion formula (3.1). Using argumentation technique of Suzuki [41] or Song [33], for a nonexpansive mapping T , we can obtain the equivalence of the convergence between Halpern type iterations and viscosity approximations methods with a contractive mapping. Remark 3. For a Lipschitzian pseudocontractive mapping T (even a nonexpansive mapping T ), it is not known whether the convergence of Halpern type iterations implies the convergence of viscosity approximations methods with a strong pseudocontractive mapping.

Acknowledgments The authors would like to thank Editors and the anonymous referee for their valuable suggestions which helps to improve this manuscript.

Vol. 13 (2009)

Lipschitz pseudocontractive mappings

653

References [1] F.E. Browder, Fixed-point theorems for noncompact mappings in Hilbert space, Proc. Natl. Acad. Sci. USA 53 (1965) 1272–1276. [2] F.E. Browder, Nonlinear equations of evolution and nonlinear accretive operators in Banach spaces, Bull. Amer. Math. Soc. 73 (1967), 470–475, Part 2 (1976). [3] R.E. Bruck, A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces, Israel J. Math. 32 (1979), 107–116. [4] R.E. Bruck, On the convex approximation property and the asymptotic behavior of nonlinear contractions in Banach spaces, Israel J. Math. 38 (1981) 304–314. [5] Y.J. Cho, X. Qin, Viscosity approximation methods for a family of m -accretive mappings in reflexive Banach spaces, Positivity 12 (2008), 483–494. [6] C.O. Chidume, G. De Souza, Convergence of a Halpern-type iteration algorithm for a class of pseudo-contractive mappings, Nonlin. Anal. 69 (2008), 2286–2292. [7] C.E. Chidume, C.O. Chidume, Iterative approximation of fixed points of nonexpansive mappings, J. Math. Anal Appl. 318 (2006) 288–295. [8] C.E. Chidume, C. Moore, Fixed point iteration for pseudocontractive maps, Proc. Amer. Math. Soc. 127 (4)(1999), 1163–1170. [9] C.E. Chidume, Iterative approximation of Lipschitz strictly pseudocontractive mappings, Proc. Amer. Math. Soc., 99 (2) (1987), 283–288. [10] C.E. Chidume, Approximation of fixed points of strongly pseudocontractive mappings, Proc. Amer. Math. Soc. 120 (2) (1994), 545–551. [11] C.E. Chidume, Global iteration schemes for strongly pseudocontractive maps, Proc. Amer. Math. Soc. 126 (9) (1998), 2641–2649. [12] C.E. Chidume, S.A. Mutangadura, An example on the Mann iteration method for Lipschitz pseudocontractions, Proc. Amer. Math. Soc. 129 (8) (2001), 2359–2363. [13] C.E. Chidume, A. Udomene, Strong convergence theorems for uniformly continuous pseudocontractive maps, J. Math. Anal. Appl. 323 (2006), 88–99. [14] C.E. Chidume, E.U. Ofoedu, A new iteration process for generalized Lipschitz pseudocontractive and generalized Lipschitz accretive mappings, Nonlinear Anal. 67 (2007), 307–315. [15] C.E. Chidume, H. Zegeye, Approximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive maps, Proc. Amer. Math. Soc. 132 (2004), 831–840. [16] R. Chen, Y. Song, H. Zhou, Viscosity Approximation Methods for Continuous Pseudocontractive Mappings, Acta Mathematica Sinica, Chinese Series, 49 (2006), 1275–1278. [17] R. Chen, Y. Song, H. Zhou, Convergence Theorems for Implicit Iteration Process for a Finite Family of Continuous Pseudocontractive Mappings, J. Math. Anal. Appl. 314 (2006), 701–709. [18] K. Deimling, Zero of accretive operators, Manuscripta Math. 13 (1974), 365–374. [19] L. Deng, On Chidumes open problems, J. Math. Anal. Appl. 174 (2) (1993), 441–449. [20] L. Deng, X.P. Ding, Iterative approximation of Lipschitz strictly pseudocontractive mappings in uniformly smooth Banach spaces, Nonlinear Anal. 24 (7) (1995), 981– 987.

654

Y. Song

Positivity

[21] T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal. 61 (2005) 51–60. [22] T.L. Hicks, J.R. Kubicek, On the Mann iteration process in Hilbert space, J. Math. Anal. Appl. 59 (1977), 498–504. [23] B. Halpern, Fixed points of nonexpansive maps, Bull. Amer. Math. Soc. 73 (1967) 957–961. [24] T. Kato, Nonlinear semigroups and evolution equations, J. Math. Soc. Japan 19 (1967), 508–520. [25] L.S. Liu, Ishikawa and Mann iteration process with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl. 194 (1995), 114–125. [26] A. Moudafi, Viscosity Approximation Methods for Fixed-Points Problems, J. Math. Anal. Appl. 241 (2000), 46–55. [27] W.R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953) 506–510. [28] C.H. Morales, J.S. Jung, Convergence of paths for pseudocontractive mappings in Banach spaces, Proc. Amer. Math. Soc. 128 (2000), 3411–3419. [29] S. Reich, Approximating zeros of accretive operators, Proc. Amer. Math. Soc. 51 (1975) 381–384. [30] S. Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces J. Math. Anal Appl. 75 (1980) 287–292. [31] S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces J. Math. Anal Appl. 67 (1979) 274–276. [32] J. Schu, Approximating fixed points of Lipschitzian pseudocontractive mappings, Houston J. Math. 19 (1993), 107–115. [33] Y. Song, Iterative approximation to common fixed points of a countable family of nonexpansive mappings, Applicable Analysis, 86 (11) (2007), 1329–1337. [34] Y. Song, A note on the paper “A new iteration process for generalized Lipschitz pseudocontractive and generalized Lipschitz accretive mappings”, Nonlinear Anal., 68 (2008), 3047–3049. [35] Y. Song, On a Mann type implicit iteration process for continuous pseudocontractive mappings, Nonlinear Anal. 67 (2007), 3058–3063. [36] Y. Song, R. Chen, Convergence theorems of iterative algorithms for continuous pseudocontractive mappings, Nonlinear Anal. 67 (2007), 486–497. [37] Y. Song, R. Chen, Strong convergence of an iterative method for non-expansive mappings, Mathematische Nachrichten, 281 (8) (2008), 1196–1204. [38] Y. Song, R. Chen, Viscosity approximation methods for nonexpansive nonself-mappings, J. Math. Anal. Appl. 321 (2006), 316–326. [39] Y. Song, R. Chen, An approximation method for continuous pseudocontractive mappings, J. Inequal. Appl., Volume 2006, Article ID 28950, Pages 1–9, DOI 10.1155/ JIA/2006/28950. [40] Y. Song, S. Xu, Strong convergence theorems for nonexpansive semigroup in Banach spaces, J. Math. Anal. Appl., 338 (2008) 152–161. [41] T. Suzuki, Moudafi’s viscosity approximations with Meir-Keeler contractions, J. Math. Anal. Appl. 325 (2007) 342–352.

Vol. 13 (2009)

Lipschitz pseudocontractive mappings

655

[42] T. Suzuki, Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces, Fixed Point Theory and Applications, 2005 (1) (2005) 103–123. doi:10.1155/FPTA.2005.103. [43] W. Takahashi, Y. Ueda, On Reich’s strong convergence for resolvents of accretive operators, J. Math. Anal. Appl. 104 (1984), 546–553. [44] W. Takahashi Nonlinear Functional Analysis– Fixed Point Theory and its Applications, Yokohama Publishers inc, Yokohama, 2000. [45] A. Udomene, Path convergence, approximation of fixed points and variational solutions of Lipschitz pseudocontractions in Banach spaces, Nonlinear Anal. 67 (2007), 2403–2414. [46] H.K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. 66 (2002), 240–256. [47] H. Zhou, Convergence theorems for λ-strict pseudo-contractions in 2-uniformly smooth Banach spaces, Nonlinear Analysis, 69 (9) (2008), 3160–3173. Yisheng Song College of Mathematics and Information Science Henan Normal University XinXiang 453007 P.R. China e-mail: [email protected] URL: http://songys.mysinamail.com or http://songys.16f.cn/ Received 22 March 2008; accepted 17 September 2008

To access this journal online: www.birkhauser.ch/pos