Strong convergence theorems for uniformly L-Lipschitzian asymptotically pseudocontractive mappings in Banach spaces

In the sequel, we denote the single-valued normalized duality mapping by j. In a Hilbert space H, j is the identity mapping.

for all $n\ge 1$.

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 $T:K\to K$ be a completely continuous, uniformly L-Lipschitzian and asymptotically pseudocontractive mapping with a sequence $\{k_n\}\subset [1,+\mathrm{\infty })$ satisfying the following conditions:

(a-1) $k_n\to 1$ as $n\to \mathrm{\infty }$;

(a-2) ${\sum }_{n=1}^{\mathrm{\infty }}(q_n^2-1)<\mathrm{\infty }$, where $q_n=2k_n-1$.

for all $n\ge 1$. Then $\{x_n\}$ 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 $T:K\to K$ be an asymptotically pseudocontractive mapping with a sequence $\{k_n\}\subset [1,+\mathrm{\infty })$, ${lim}_{n\to \mathrm{\infty }}{k}_n=1$ and $F(T)\ne \mathrm{\varnothing }$, where $F(T)$ is the set of fixed points of T in K. Let $\{{\alpha }_n\}$ be a sequence in $[0,1]$ satisfying the following conditions:

(a-1) ${\alpha }_n\to 0$ as $n\to \mathrm{\infty }$;

(a-2) ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$.

for all $x\in K$ and $n\ge 0$, where $x^{\ast }\in F(T)$, then $x_n\to x^{\ast }$ as $n\to \mathrm{\infty }$.

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 $T:K\to K$ be a uniformly L-Lipschitzian asymptotically pseudocontractive mapping with a sequence $\{k_n\}\subset [1,+\mathrm{\infty })$, ${lim}_{n\to \mathrm{\infty }}{k}_n=1$ and $x^{\ast }\in F(T)$. Let $\{{\alpha }_n\}$ be a sequence in $[0,1]$ satisfying the following conditions:

(a-1) ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$;

(a-2) ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_n^2<\mathrm{\infty }$;

(a-3) ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_n(k_n-1)<\mathrm{\infty }$.

Theorem 1.4 [13]

Let E be a real Banach space. Let K be a nonempty closed and convex subset of E, $T:K\to K$ be a uniformly L-Lipschitzian asymptotically pseudocontractive mapping with a sequence $\{k_n\}\subset [1,+\mathrm{\infty })$ with ${lim}_{n\to \mathrm{\infty }}{k}_n=1$ and $x^{\ast }\in F(T)$. Let $\{a_n\}$, $\{b_n\}$ and $\{c_n\}$ be real sequences in $[0,1]$ satisfying the following conditions:

(a-1) $a_n+b_n+c_n=1$;

(a-2) ${\sum }_{n\ge 0}(b_n+c_n)=\mathrm{\infty }$;

(a-3) ${\sum }_{n\ge 0}{(b_n+c_n)}^2<\mathrm{\infty }$;

(a-4) ${\sum }_{n\ge 0}(b_n+c_n)(k_n-1)<\mathrm{\infty }$;

(a-5) ${\sum }_{n\ge 0}{c}_n<\mathrm{\infty }$.

for all $x\in K$. Then ${\{x_n\}}_{n\ge 0}$ converges strongly to $x^{\ast }\in F(T)$.

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, $T_i:K\to K$ be two uniformly $L_i$-Lipschitzian mappings with $F(T_1)\cap F(T_2)\ne \mathrm{\varnothing }$ and $x^{\ast }\in F(T_1)\cap F(T_2)$. Let $\{k_n\}\subset [1,+\mathrm{\infty })$ be a sequence with ${lim}_{n\to \mathrm{\infty }}{k}_n=1$. Let $\{{\alpha }_n\}$ and $\{{\beta }_n\}$ be two sequences in $[0,1]$ satisfying the following conditions:

(a-1) ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$;

(a-2) ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_n^2<\mathrm{\infty }$;

(a-3) ${\sum }_{n=0}^{\mathrm{\infty }}{\beta }_n<\mathrm{\infty }$;

(a-4) ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_n(k_n-1)<\mathrm{\infty }$.

for all $j(x-x^{\ast })\in J(x-x^{\ast })$ and $x\in K$, $i=1,2$, then $\{x_n\}$ converges strongly to $x^{\ast }$.

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 all $n\ge 1$ is called the modified Ishikawa iteration with errors, where $\{a_n\}$, $\{b_n\}$, $\{c_n\}$, $\{d_n\}$ are four real sequences in $[0,1]$ satisfying $a_n+c_n\le 1$, $b_n+d_n\le 1$ and $\{u_n\}$, $\{v_n\}$ are any bounded sequences in D.

for all $n\ge 1$ is called the modified Mann iteration with errors.

for all $n\ge 1$ is called the modified Ishikawa iteration and the modified Mann iteration, respectively.

Lemma 1.7 [18]

for all $x,y\in E$ and $j(x+y)\in J(x+y)$.

Lemma 1.8 [19]

for all $n\ge 0$, where ${\lambda }_n\in [0,1]$ with ${\sum }_{n=0}^{\mathrm{\infty }}{\lambda }_n=\mathrm{\infty }$ and ${\gamma }_n=o({\lambda }_n)$. Then ${\delta }_n\to 0$ as $n\to \mathrm{\infty }$.

Now, we give our main results in this paper.

Theorem 2.1 Let E be a real Banach space, D be a nonempty closed convex subset of E and $T:D\to D$ be a uniformly L-Lipschitzian asymptotically pseudocontractive mapping with a sequence $\{k_n\}\subset [1,+\mathrm{\infty })$, ${lim}_{n\to \mathrm{\infty }}{k}_n=1$ and $q\in F(T)=\{x\in D:Tx=x\}\ne \mathrm{\varnothing }$. Let $\{a_n\}$, $\{b_n\}$, $\{c_n\}$ and $\{d_n\}$ be four real sequences in $[0,1]$ satisfying the following conditions:

(A-1) $a_n,b_n,d_n\to 0$ as $n\to \mathrm{\infty }$;

(A-2) ${\sum }_{n=1}^{\mathrm{\infty }}{a}_n=\mathrm{\infty }$;

(A-3) $c_n=o(a_n)$.

for all $n\ge 1$, where $j(x-q)\in J(x-q)$. Then $\{x_n\}$ converges strongly to the fixed point q of T.

Proof Step 1. For any $n\ge 1$, $\{x_n\}$ is a bounded sequence.

Set $sup\{k_n:n\ge 1\}=k$. Then there exists $x_1\in D$ with $x_1\ne T{x}_1$ such that $r_0=(k+L){\parallel x_1-q\parallel }^2\in R(\mathrm{\Phi })$. Indeed, for any taking $x_1\in D$ and $x_1\ne T{x}_1$, we denote $r_0=(k+L){\parallel x_1-q\parallel }^2$. If $\mathrm{\Phi }(r)\to +\mathrm{\infty }$ as $r\to +\mathrm{\infty }$, then $r_0\in R(\mathrm{\Phi })$. If $sup\{\mathrm{\Phi }(r):r\in [0,+\mathrm{\infty })\}=r_1<+\mathrm{\infty }$ with $r_1<r_0$, then there exists a sequence $\{{\xi }_n\}\subset D$ such that ${\xi }_n\to q$ as $n\to \mathrm{\infty }$ with ${\xi }_n\ne q$, thus there exists a positive integer $n_0$ such that $(k+L){\parallel {\xi }_n-q\parallel }^2<\frac{r_1}{2}$ for all $n\ge n_0$. We redefine $x_1={\xi }_{n_0}$ and $(k+L){\parallel x_1-q\parallel }^2\in R(\mathrm{\Phi })$.

Set $R={\mathrm{\Phi }}^{-1}(r_0)$. Then we obtain $\parallel x_1-q\parallel \le R$. Denote

Since $a_n,b_n,c_n,\frac{c_n}{a_n},d_n,k_n-1\to 0$ as $n\to \mathrm{\infty }$, without loss of generality, let $0\le a_n,b_n,c_n,\frac{c_n}{a_n},d_n,k_n-1\le {\tau }_0$ for any $n\ge 1$. Thus, we have

(2.3)

(2.4)

and

(2.5)

Since $a_n\to 0$ and $k_n\to 1$ as $n\to \mathrm{\infty }$, we have $2k_n{a}_n\to 0$ as $n\to \mathrm{\infty }$. Thus, without loss of generality, let $1-2k_n{a}_n>0$ for any $n\ge 1$. Then (2.6) implies that

(2.7)

which is a contradiction. Hence, $x_{n+1}\in B_1$, i.e., $\{x_n\}$ is a bounded sequence.

Step 2. We prove that $\parallel x_n-q\parallel \to 0$ as $n\to \mathrm{\infty }$.

Using Lemma 1.7, (2.6) and (2.8), we have

(2.9)

Therefore, by Lemma 1.8, we obtain ${lim}_{n\to \mathrm{\infty }}{\delta }_n=0$, i.e., $x_n\to q$ as $n\to \mathrm{\infty }$. This completes the proof. □

From Theorem 2.1, we have the following corollary.

Corollary 2.2 Let E be a real Banach space. Let D be a nonempty closed convex subset of E, $T:D\to D$ be a uniformly L-Lipschitzian asymptotically pseudocontractive mapping with a sequence $\{k_n\}\subset [1,+\mathrm{\infty })$, ${lim}_{n\to \mathrm{\infty }}{k}_n=1$ and $q\in F(T)$. Let $\{a_n\}$ and $\{c_n\}$ be two real sequences in $[0,1]$ satisfying the following conditions:

(A-1) $a_n\to 0$ as $n\to \mathrm{\infty }$;

(A-2) ${\sum }_{n=1}^{\mathrm{\infty }}{a}_n=\mathrm{\infty }$;

(A-3) $c_n=o(a_n)$.

for all $n\ge 1$, where $j(x-q)\in J(x-q)$. Then $\{x_n\}$ converges strongly to the fixed point q of T.

Proof In Theorem 2.1, letting $b_n=0$, $d_n=0$, we can get the convergence of the modified Mann iteration (1.5). □

Theorem 2.3 Let E be a real Banach space. Let D be a nonempty closed convex subset of E and $T_i:K\to K$ be two uniformly $L_i$-Lipschitzian mappings with $q\in F(T_1)\cap F(T_2)$. Let $\{k_n\}\subset [1,+\mathrm{\infty })$ be a sequence with $k_n\to 1$ as $n\to \mathrm{\infty }$. Let $\{a_n\}$, $\{b_n\}$, $\{c_n\}$ and $\{d_n\}$ be four real sequences in $[0,1]$ satisfying the following conditions:

(A-1) $a_n,b_n\to 0$ as $n\to \mathrm{\infty }$;

(A-2) ${\sum }_{n=1}^{\mathrm{\infty }}{a}_n=\mathrm{\infty }$;

(A-3) $c_n=o(a_n)$, $d_n\to 0$ as $n\to \mathrm{\infty }$.

for all $n\ge 1$ and $i=1,2$, where $j(x-q)\in J(x-q)$. Then $\{x_n\}$ converges strongly to the fixed point q of $T_1\cap T_2$.

Proof Similarly, we can obtain the result of Theorem 2.3 by using the proof method of Theorem 2.1. □

for all $n\ge 1$, $x\in D$ and $y\in F(T)$. Then, using the same methods, we can also prove that Theorem 2.1 holds for the more general class of weak uniformly Lipschitzian asymptotically pseudocontractive mappings. In practical application, it can be seen from the following example.

Therefore, T is weakly uniform Lipschitzian and satisfies (2.1) of Theorem 2.1.