Let E be a real Banach space and let J denote the normalized duality mapping from E into defined by
for all , where denotes the dual space of E and denotes the generalized duality pairing, respectively. The normalized duality mapping J has the following properties:
-
(1)
J is an odd mapping, i.e., .
-
(2)
J is positive homogeneous, i.e., for any , .
-
(3)
J is bounded, i.e., for any bounded subset A of E, is a bounded subset of .
-
(4)
If E is smooth (or is strictly convex), then J is single-valued.
In the sequel, we denote the single-valued normalized duality mapping by j. In a Hilbert space H, j is the identity mapping.
Let D be a nonempty closed convex subset of E. A mapping is said to be asymptotically nonexpansive with a sequence and if, for all ,
(1.1)
for all . The mapping T is said to be asymptotically pseudocontractive with a sequence and if, for any , there exists such that
(1.2)
for all . Furthermore, the mapping T is said to be uniformly L-Lipschitzian if, for any , there exists a constant such that
(1.3)
for all .
It is easy to see that if T is an asymptotically nonexpansive mapping, then it is both asymptotically pseudocontractive and uniformly L-Lipschitzian. The converse is not true in general. Therefore, it is interesting to study these mappings in fixed point theory and its applications. In fact, the asymptotically nonexpansive and asymptotically pseudocontractive mappings were first introduced by Goebel-Kirk [1] and Schu [2], respectively. Since then, some authors have studied several iterative sequences for asymptotically nonexpansive and asymptotically pseudocontractive mappings in Hilbert spaces and Banach spaces (see [3–11]).
In [2], Schu proved the following theorem.
Theorem 1.1 [2]
Let H be a Hilbert space, K be a nonempty bounded closed convex subset of H and be a completely continuous, uniformly L-Lipschitzian and asymptotically pseudocontractive mapping with a sequence satisfying the following conditions:
(a-1) as ;
(a-2) , where .
Suppose further that and are two sequences in such that for all , where and . For any , let be an iterative sequence defined by
for all . Then converges strongly to a fixed point of T in K.
In [12], Chang extended above Theorem 1.1 to the setting of real uniformly smooth Banach spaces and proved the following.
Theorem 1.2 [12]
Let E be a uniformly smooth Banach space, K be a nonempty bounded closed convex subset of E and be an asymptotically pseudocontractive mapping with a sequence , and , where is the set of fixed points of T in K. Let be a sequence in satisfying the following conditions:
(a-1) as ;
(a-2) .
For any , let be an iterative sequence defined by
for all . If there exists a strictly increasing function with such that
for all and , where , then as .
In [13], Ofoedu extended Theorem 1.2 in a uniformly smooth Banach space to the setting of arbitrary real Banach spaces and dropped the boundedness assumption in Theorem 1.2.
Theorem 1.3 [13]
Let E be a real Banach space, K be a nonempty closed convex subset of E and be a uniformly L-Lipschitzian asymptotically pseudocontractive mapping with a sequence , and . Let be a sequence in satisfying the following conditions:
(a-1) ;
(a-2) ;
(a-3) .
For any , let be an iterative sequence defined by
for all . If there exists a strictly increasing function with such that
for all and . Then
-
(1)
is bounded;
-
(2)
converges strongly to .
Theorem 1.4 [13]
Let E be a real Banach space. Let K be a nonempty closed and convex subset of E, be a uniformly L-Lipschitzian asymptotically pseudocontractive mapping with a sequence with and . Let , and be real sequences in satisfying the following conditions:
(a-1) ;
(a-2) ;
(a-3) ;
(a-4) ;
(a-5) .
For arbitrary , let be a sequence in K iteratively defined by
for all , where is a bounded sequence in K. Suppose that there exists a strictly increasing continuous function with such that
for all . Then converges strongly to .
Very recently, in [14], Chang et al. proved the following theorem.
Theorem 1.5 [14]
Let E be a real Banach space. Let K be a nonempty closed convex subset of E, be two uniformly -Lipschitzian mappings with and . Let be a sequence with . Let and be two sequences in satisfying the following conditions:
(a-1) ;
(a-2) ;
(a-3) ;
(a-4) .
For any , let be an iterative sequence in K defined by
for all . If there exists a strictly increasing function with such that
for all and , , then converges strongly to .
Also, some authors have studied the modified Halpern, Mann and Ishikawa iterative sequences for nonlinear mappings in Hilbert spaces and Banach spaces (see [15, 16]).
The aim of this paper is to give some strong convergence theorems for uniformly L-Lipschitzian and asymptotically pseudo contractive mappings in Banach spaces. Our results not only include the past ones known in [3–11], but also provide quite a different proof method.
For our main purpose, we recall some concepts and lemmas.
Definition 1.6 [17]
For arbitrary , the sequence in D defined by
(1.4)
for all is called the modified Ishikawa iteration with errors, where , , , are four real sequences in satisfying , and , are any bounded sequences in D.
In particular, if in (1.4), then the sequence defined by
(1.5)
for all is called the modified Mann iteration with errors.
If in (1.4) and (1.5), then the sequence defined by
(1.6)
and
(1.7)
for all is called the modified Ishikawa iteration and the modified Mann iteration, respectively.
Lemma 1.7 [18]
Let E be a real Banach space and be a normalized duality mapping. Then
for all and .
Lemma 1.8 [19]
Let , and be three nonnegative real sequences and be a strictly increasing continuous function with satisfying the following inequality:
for all , where with and . Then as .