 Research
 Open access
 Published:
Total quasiϕasymptotically nonexpansive semigroups and strong convergence theorems in Banach spaces
Fixed Point Theory and Applications volume 2012, Article number: 153 (2012)
Abstract
The purpose of this article is to modify the HalpernManntype iteration algorithm for total quasiϕasymptotically nonexpansive semigroups to have the strong convergence under a limit condition only in the framework of Banach spaces. The results presented in the paper improve and extend the corresponding recent results announced by many authors.
MSC:47J05, 47H09, 49J25.
1 Introduction
Throughout this paper, we assume that E is a real Banach space with the dual {E}^{\ast}, C is a nonempty closed convex subset of E, and J:E\to {2}^{{E}^{\ast}} is the normalized duality mapping defined by
Let T:C\to E be a nonlinear mapping; we denote by F(T) the set of fixed points of T.
Recall that a mapping T:C\to C is said to be nonexpansive if
T:C\to C is said to be quasinonexpansive if F(T)\ne \mathrm{\varnothing} and
T:C\to C is said to be asymptotically nonexpansive if there exists a sequence \{{k}_{n}\}\subset [1,\mathrm{\infty}) with {k}_{n}\to 1 such that
T:C\to C is said to be quasiasymptotically nonexpansive if F(T)\ne \mathrm{\varnothing} and there exists a sequence \{{k}_{n}\}\subset [1,\mathrm{\infty}) with {k}_{n}\to 1 such that
A oneparameter family \mathcal{T}:=\{T(t):t\ge 0\} of mappings from C into C is said to be a nonexpansive semigroup if the following conditions are satisfied:

(i)
T(0)x=x for all x\in C;

(ii)
T(s+t)=T(s)T(t), \mathrm{\forall}s,t\ge 0;

(iii)
for each x\in C, the mapping t\mapsto T(t)x is continuous;

(iv)
\parallel T(t)xT(t)y\parallel \le \parallel xy\parallel, \mathrm{\forall}x,y\in C.
We use F(\mathcal{T}) to denote a common fixed point set of the nonexpansive semigroup \mathcal{T}, i.e., F(\mathcal{T}):={\bigcap}_{t\ge 0}F(T(t)).
A oneparameter family \mathcal{T}:=\{T(t):t\ge 0\} of mappings from C into C is said to be a quasinonexpansive semigroup if F(\mathcal{T})\ne \mathrm{\varnothing} and the above conditions (i)(iii) and the following condition (v) are satisfied:

(v)
\parallel T(t)xp\parallel \le \parallel xp\parallel, \mathrm{\forall}x\in C, p\in F(\mathcal{T}), t\ge 0.
A oneparameter family \mathcal{T}:=\{T(t):t\ge 0\} of mappings from C into C is said to be an asymptotically nonexpansive semigroup if there exists a sequence \{{k}_{n}\}\subset [1,\mathrm{\infty}) with {k}_{n}\to 1 such that the above conditions (i)(iii) and the following condition (vi) are satisfied:

(vi)
\parallel {T}^{n}(t)x{T}^{n}(t)y\parallel \le {k}_{n}\parallel xy\parallel, \mathrm{\forall}x,y\in C, n\ge 1, t\ge 0.
A oneparameter family \mathcal{T}:=\{T(t):t\ge 0\} of mappings from C into C is said to be a quasiasymptotically nonexpansive semigroup if F(\mathcal{T})\ne \mathrm{\varnothing} and there exists a sequence \{{k}_{n}\}\subset [1,\mathrm{\infty}) with {k}_{n}\to 1 such that the above conditions (i)(iii) and the following condition (vii) are satisfied:

(vii)
\parallel {T}^{n}(t)xp\parallel \le {k}_{n}\parallel xp\parallel, \mathrm{\forall}x\in C, p\in F(\mathcal{T}), t\ge 0, n\ge 1.
As is well known, the construction of fixed points of nonexpansive mappings (asymptotically nonexpansive mappings) and of common fixed points of nonexpansive semigroups (asymptotically nonexpansive semigroups) is an important problem in the theory of nonexpansive mappings and its applications; in particular, in image recovery, convex feasibility problem, and signal processing problem (see, for example, [1–3]).
Iterative approximation of a fixed point for nonexpansive mappings, asymptotically nonexpansive mappings, nonexpansive semigroups, and asymptotically nonexpansive semigroups in Hilbert or Banach spaces has been studied extensively by many authors (see, for example, [4–31] and the references therein).
The purpose of this article is to introduce the concept of total quasiϕasymptotically nonexpansive semigroups; to modify the Halpern and Manntype iteration algorithm [13, 14] for total quasiϕasymptotically nonexpansive semigroups; and to have the strong convergence under a limit condition only in the framework of Banach spaces. The results presented in the paper improve and extend the corresponding results of Kim [32], Suzuki [4], Xu [5], Chang et al. [6–8, 22, 23, 30, 33], Cho et al. [10], Thong [11], Buong [12], Mann [13], Halpern [14], Qin et al. [15], Nakajo et al. [18] and others.
2 Preliminaries
In the sequel, we assume that E is a smooth, strictly convex, and reflexive Banach space and C is a nonempty closed convex subset of E. In what follows, we always use \varphi :E\times E\to {\mathcal{R}}^{+} to denote the Lyapunov functional defined by
It is obvious from the definition of ϕ that
and
Following Alber [34], the generalized projection {\mathrm{\Pi}}_{C}:E\to C is defined by
Lemma 2.1 ([34])
Let E be a smooth, strictly convex, and reflexive Banach space and C be a nonempty closed convex subset of E. Then the following conclusions hold:

(a)
\varphi (x,{\mathrm{\Pi}}_{C}y)+\varphi ({\mathrm{\Pi}}_{C}y,y)\le \varphi (x,y) for all x\in C and y\in E;

(b)
If x\in E and z\in C, then z={\mathrm{\Pi}}_{C}x\iff \u3008zy,JxJz\u3009\ge 0, \mathrm{\forall}y\in C;

(c)
For x,y\in E, \varphi (x,y)=0 if and only if x=y.
Remark 2.2 If E is a real Hilbert space H, then \varphi (x,y)={\parallel xy\parallel}^{2} and {\mathrm{\Pi}}_{C}={P}_{C} (the metric projection of H onto C).
Definition 2.3 A mapping T:C\to C is said to be closed if, for any sequence \{{x}_{n}\}\subset C with {x}_{n}\to x and T{x}_{n}\to y, Tx=y.
Definition 2.4 (1) A mapping T:C\to C is said to be quasiϕnonexpansive, if F(T)\ne \mathrm{\varnothing} and

(2)
A mapping T:C\to C is said to be (\{{k}_{n}\})quasiϕasymptotically nonexpansive, if F(T)\ne \mathrm{\varnothing} and there exists a real sequence \{{k}_{n}\}\subset [1,\mathrm{\infty}), {k}_{n}\to 1 such that
\varphi (p,{T}^{n}x)\le {k}_{n}\varphi (p,x),\phantom{\rule{1em}{0ex}}\mathrm{\forall}n\ge 1,x\in C,p\in F(T). 
(3)
A mapping T:C\to C is said to be (\{{\nu}_{n}\},\{{\mu}_{n}\},\zeta )total quasiϕasymptotically nonexpansive if F(T)\ne \mathrm{\varnothing} and there exist nonnegative real sequences \{{\nu}_{n}\}, \{{\mu}_{n}\} with {\nu}_{n}\to 0, {\mu}_{n}\to 0 (as n\to \mathrm{\infty}) and a strictly increasing continuous function \zeta :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) such that
\varphi (p,{T}^{n}x)\le \varphi (p,x)+{\nu}_{n}\zeta (\varphi (p,x))+{\mu}_{n},\phantom{\rule{1em}{0ex}}\mathrm{\forall}n\ge 1,x\in C,p\in F(T).(2.5)
Remark 2.5 ([22])
From the definitions, it is obvious that a quasiϕnonexpansive mapping is a (\{{k}_{n}=1\})quasiϕasymptotically nonexpansive mapping and a (\{{k}_{n}\})quasiϕasymptotically nonexpansive mapping is a (\{{\nu}_{n}\},\{{\mu}_{n}\},\zeta )total quasiϕasymptotically nonexpansive mapping with {\nu}_{n}={k}_{n}1, {\mu}_{n}=0, \zeta (t)=t, \mathrm{\forall}t\ge 0. However, the converse is not true.
Example 2.6 ([23])
Let E be a uniformly smooth and strictly convex Banach space and A:E\to {E}^{\ast} be a maximal monotone mapping such that {A}^{1}0\ne \mathrm{\varnothing}, then {J}_{r}={(J+rA)}^{1}J is closed and quasiϕnonexpansive from E onto D(A).
Example 2.7 ([30])

(1)
Let C be a unit ball in a real Hilbert space {l}^{2} and let T:C\to C be a mapping defined by
T:({x}_{1},{x}_{2},\dots )\to (0,{x}_{1}^{2},{a}_{2}{x}_{2},{a}_{3}{x}_{3},\dots ),({x}_{1},{x}_{2},\dots )\in {l}^{2},(2.6)
where \{{a}_{i}\} is a sequence in (0,1) such that {\mathrm{\Pi}}_{i=2}^{\mathrm{\infty}}{a}_{i}=\frac{1}{2}. It is proved in [30] that T is (singlevalued) total quasiϕasymptotically nonexpansive.

(2)
Let I=[0,1], X=C(I) (the Banach space of continuous functions defined on I with the uniform convergence norm {\parallel f\parallel}_{C}={sup}_{t\in I}f(t)), D=\{f\in X:f(x)\ge 0,\mathrm{\forall}x\in I\} and a, b be two constants in (0,1) with a<b. Let T:D\to {2}^{D} be a multivalued mapping defined by
T(f)=\{\begin{array}{cc}\{g\in D:a\le f(x)g(x)\le b,\mathrm{\forall}x\in I\},\hfill & \text{if}f(x)1,\mathrm{\forall}x\in I;\hfill \\ \{0\},\hfill & \text{otherwise}.\hfill \end{array}(2.7)
It is proved that T:C\to {2}^{C} is a multivalued total quasiϕasymptotically nonexpansive mapping.
Example 2.8 Let {\mathrm{\Pi}}_{C} be the generalized projection from a smooth, reflexive, and strictly convex Banach space E onto a nonempty closed convex subset C of E, then {\mathrm{\Pi}}_{C} is a closed and quasiϕnonexpansive from E onto C.
Lemma 2.9 Let E be a smooth, reflexive, and strictly convex real Banach space such that both E and {E}^{\ast} have the KadecKlee property, and let C be a nonempty closed and convex subset of E. Let T:C\to C be a closed and (\{{\nu}_{n}\},\{{\mu}_{n}\},\zeta )total quasiϕasymptotically nonexpansive mapping, then F(T) is a closed convex subset of C.
Proof Let \{{x}_{n}\} be any sequence in F(T) such that {x}_{n}\to {x}^{\ast}. Now, we prove that {x}^{\ast}\in F(T). In fact, since T{x}_{n}={x}_{n}\to {x}^{\ast} and T is closed, we have {x}^{\ast}=T{x}^{\ast}, i.e., F(T) is a closed subset in C.
Next, we prove that F(T) is convex. In fact, let x,y\in F(T) and p=tx+(1t)y, where t\in (0,1). Now, we prove that p=Tp. Indeed, since T is (\{{\nu}_{n}\},\{{\mu}_{n}\},\zeta )total quasiϕasymptotically nonexpansive, for any n\ge 1, we have
and
On the other hand, it follows from (2.4) that
and
It follows from (2.8)(2.11) that
and
Multiplying t and (1t) on both sides of (2.12) and (2.13), respectively and then adding up these two inequalities, we have that
Letting n\to \mathrm{\infty}, we have that \varphi (p,{T}^{n}p)\to 0. Hence, it follows from (2.2) that
and so
E is reflexive and so is {E}^{\ast}. Without loss of generality, we can assume that J({T}^{n}p)\rightharpoonup {x}^{\ast} (some point in {E}^{\ast}). In view of the reflexivity of E, we have J(E)={E}^{\ast}. This shows that there exists an element x\in E such that Jx={x}^{\ast}. Hence, we have
Taking {lim}_{n\to \mathrm{\infty}} on both sides of the equality above, we obtain that
This implies that JpJx=0. Therefore, we have J({T}^{n}p)\rightharpoonup Jp. Since {E}^{\ast} has the KadecKlee property, this together with (2.15) shows that J({T}^{n}p)\to Jp. Since E is reflexive and strictly convex, {J}^{1} is normweakcontinuous, {T}^{n}p\rightharpoonup p. Again, since E has the KadecKlee property, this together with (2.14) shows that {T}^{n}p\to p (as n\to \mathrm{\infty}). Therefore, T{T}^{n}p={T}^{n+1}p\to p. By virtue of the closeness of T, it follows that p=Tp, i.e., p\in F(T). The convexity of F(T) is proved.
This completes the proof of Lemma 2.9. □
Definition 2.10 (I) Let E be a real Banach space, C be a nonempty closed convex subset of E. \mathcal{T}:=\{T(t):t\ge 0\} be a oneparameter family of mappings from C into C. \mathcal{T} is said to be

(1)
a quasiϕnonexpansive semigroup if \mathcal{F}={\bigcap}_{t\ge 0}F(T(t))\ne \mathrm{\varnothing} and the following conditions are satisfied:

(i)
T(0)x=x for all x\in C;

(ii)
T(s+t)=T(s)T(t) for all s,t\ge 0;

(iii)
for each x\in C, the mapping t\mapsto T(t)x is continuous;

(iv)
\varphi (p,T(t)x)\le \varphi (p,x), \mathrm{\forall}t\ge 0, p\in \mathcal{F}, x\in C.

(2)
\mathcal{T} is said to be a (\{{k}_{n}\})quasiϕasymptotically nonexpansive semigroup if the set \mathcal{F}={\bigcap}_{t\ge 0}F(T(t)) is nonempty, and there exists a sequence \{{k}_{n}\}\subset [1,\mathrm{\infty}) with {k}_{n}\to 1 such that the conditions (i)(iii) and the following condition (v) are satisfied:

(v)
\varphi (p,{T}^{n}(t)x)\le {k}_{n}\varphi (p,x), \mathrm{\forall}t\ge 0, p\in \mathcal{F}, n\ge 1, x\in C.

(3)
\mathcal{T} is said to be a (\{{\nu}_{n}\},\{{\mu}_{n}\},\zeta )total quasiϕasymptotically nonexpansive semigroup if the set \mathcal{F}={\bigcap}_{t\ge 0}F(T(t)) is nonempty, and there exists nonnegative real sequences \{{\nu}_{n}\}, \{{\mu}_{n}\} with {\nu}_{n}\to 0, {\mu}_{n}\to 0 (as n\to \mathrm{\infty}) and a strictly increasing continuous function \zeta :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) with \zeta (0)=0 such that the conditions (i)(iii) and the following condition (vi) are satisfied:

(vi)
\varphi (p,{T}^{n}(t)x)\le \varphi (p,x)+{\nu}_{n}\zeta (\varphi (p,x))+{\mu}_{n}, \mathrm{\forall}n\ge 1, x\in C, p\in F(T).

(II)
A total quasiϕasymptotically nonexpansive semigroup \mathcal{T} is said to be uniformly Lipschitzian if there exists a bounded measurable function L:[0,\mathrm{\infty})\to (0,\mathrm{\infty}) such that
\parallel {T}^{n}(t)x{T}^{n}(t)y\parallel \le L(t)\parallel xy\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in C,\mathrm{\forall}n\ge 1,t\ge 0.
3 Main results
Theorem 3.1 Let E be a reflexive, strictly convex, and smooth Banach space such that both E and {E}^{\ast} have the KadecKlee property, and let C be a nonempty closed convex subset of E. Let \mathcal{T}:=\{T(t):t\ge 0\} be a closed, uniformly LLipschitz and (\{{\nu}_{n}\},\{{\mu}_{n}\},\zeta )total quasiϕasymptotically nonexpansive semigroup. Let \{{\alpha}_{n}\} be a sequence in [0,1] and \{{\beta}_{n}\} be a sequence in (0,1) satisfying the following conditions:

(i)
{lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0;

(ii)
0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\beta}_{n}\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\beta}_{n}<1.
Let \{{x}_{n}\} be a sequence generated by
where \mathcal{F}:={\bigcap}_{t\ge 0}^{\mathrm{\infty}}F(T(t)), {\xi}_{n}={\nu}_{n}{sup}_{p\in \mathcal{F}}\zeta (\varphi (p,{x}_{n}))+{\mu}_{n}, {\mathrm{\Pi}}_{{C}_{n+1}} is the generalized projection of E onto {C}_{n+1}. If \mathcal{F} is bounded in C, then \{{x}_{n}\} converges strongly to {\mathrm{\Pi}}_{\mathcal{F}}{x}_{1}.
Proof (I) First, we prove that \mathcal{F} and {C}_{n}, n\ge 1 all are closed and convex subsets in C.
In fact, it follows from Lemma 2.9 that F(T(t)), t\ge 0 is a closed and convex subset of C. Therefore, \mathcal{F} is closed and convex in C.
Again, by the assumption that {C}_{1}=C is closed and convex, suppose that {C}_{n} is closed and convex for some n\ge 2. In view of the definition of ϕ, we have that
This shows that {C}_{n+1} is closed and convex. The conclusion is proved.

(II)
Now, we prove that \mathcal{F}\subset {C}_{n}, \mathrm{\forall}n\ge 1.
In fact, it is obvious that \mathcal{F}\subset {C}_{1}=C. Suppose that \mathcal{F}\subset {C}_{n} for some n\ge 2. Letting
it follows from (2.3) that for any u\in \mathcal{F}\subset {C}_{n}, we have
and
Therefore, we have
where {\xi}_{n}={\nu}_{n}{sup}_{p\in \mathcal{F}}\zeta (\varphi (p,{x}_{n}))+{\mu}_{n}. This shows that u\in {C}_{n+1}, and so \mathcal{F}\subset {C}_{n+1}. The conclusion is proved.

(III)
Next, we prove that \{{x}_{n}\} is bounded and \{\varphi ({x}_{n},{x}_{1})\} is convergent.
In fact, since {x}_{n}={\mathrm{\Pi}}_{{C}_{n}}{x}_{1}, from Lemma 2.1(b), we have
Again, since \mathcal{F}\subset {C}_{n}, \mathrm{\forall}n\ge 1, we have
It follows from Lemma 2.1(a) that for each u\in \mathcal{F} and for each n\ge 1,
Therefore, \{\varphi ({x}_{n},{x}_{1})\} is bounded. By virtue of (2.2), \{{x}_{n}\} is also bounded.
Again, since {x}_{n}={\mathrm{\Pi}}_{{C}_{n}}{x}_{1}, {x}_{n+1}={\mathrm{\Pi}}_{{C}_{n+1}}{x}_{1}, and {x}_{n+1}\in {C}_{n+1}\subset {C}_{n}, \mathrm{\forall}n\ge 1, we have
This implies that \{\varphi ({x}_{n},{x}_{1})\} is nondecreasing and bounded. Hence, {lim}_{n\to \mathrm{\infty}}\varphi ({x}_{n},{x}_{1}) exists. The conclusions are proved.

(IV)
Next, we prove that {x}_{n}\to {p}^{\ast} (some point in C).
In fact, since \{{x}_{n}\} is bounded and the space E is reflexive, we may assume that there exists a subsequence of \{{x}_{{n}_{i}}\} such that {x}_{{n}_{i}}\rightharpoonup {p}^{\ast}. Since {C}_{n}, \mathrm{\forall}n\ge 1 is closed and convex, we see that {p}^{\ast}\in {C}_{n}, \mathrm{\forall}n\ge 1. This implies that \varphi ({x}_{{n}_{i}},{x}_{1})\le \varphi ({p}^{\ast},{x}_{1}), \mathrm{\forall}{n}_{i}. On the other hand, it follows from the weakly lower semicontinuity of the norm that
which implies that \varphi ({x}_{{n}_{i}},{x}_{1})\to \varphi ({p}^{\ast},{x}_{1}) (as {n}_{i}\to \mathrm{\infty}). Hence, \parallel {x}_{{n}_{i}}\parallel \to \parallel {p}^{\ast}\parallel (as {n}_{i}\to \mathrm{\infty}). In view of the KadecKlee property of E, we see that {x}_{{n}_{i}}\to {p}^{\ast} (as {n}_{i}\to \mathrm{\infty}).
If there exists another subsequence \{{x}_{{n}_{j}}\}\subset \{{x}_{n}\} such that {x}_{{n}_{j}}\to {q}^{\ast}\in C, we have
i.e., {p}^{\ast}={q}^{\ast}. This shows that {x}_{n}\to {p}^{\ast}. Therefore, we have

(V)
Now, we prove that {p}^{\ast}\in \mathcal{F}.
In fact, since {x}_{n+1}\in {C}_{n+1}, {x}_{n}\to {p}^{\ast} and {\alpha}_{n}\to 0, it follows from (3.1) and (3.5) that
This implies that for each t\ge 0,
Therefore,
and so
This shows that \{J({y}_{n,t})\} is uniformly bounded. E is reflexive and so is {E}^{\ast}. Without loss of generality, we can assume that J({y}_{n,t})\rightharpoonup {y}^{\ast} (some point in {E}^{\ast}). Since E is reflexive, J(E)={E}^{\ast}. Hence, there exists y\in E such that Jy={y}^{\ast}. This implies that J({y}_{n,t})\rightharpoonup Jy. Since
Letting n\to \mathrm{\infty}, from (3.6), we have
which shows that J{p}^{\ast}=Jy, and so
This together with (3.9) and the KadecKlee property of {E}^{\ast} shows that J({y}_{n,t})\to J{p}^{\ast}. Since {J}^{1} is normweakcontinuous, we have
It follows from (3.8), (3.11) and the KadecKlee property of E, we have
On the other hand, since \{{x}_{n}\} is bounded and \mathcal{T}:=\{T(t):t\ge 0\} is a (\{{\nu}_{n}\},\{{\mu}_{n}\},\zeta )total quasiϕasymptotically nonexpansive semigroup, for any given p\in \mathcal{F}, we have
This implies that {\{{T}^{n}(t){x}_{n}\}}_{t\ge 0} is uniformly bounded. Again, since
it implies that {\{{w}_{n,t}\}}_{t\ge 0} is also uniformly bounded.
Since {\alpha}_{n}\to 0, from (3.1), we have
It follows from (3.12) that J{w}_{n,t}\to J{p}^{\ast} (as n\to \mathrm{\infty}), uniformly in t\ge 0. Therefore, we have
Since
This together with (3.14) shows that
Since {x}_{n}\to {p}^{\ast}, we have J{x}_{n}\to J{p}^{\ast}, and so for each t\ge 0,
By condition (ii), we have that
Since {J}^{1} is normweaklycontinuous, this implies that
It follows from (3.16) that for each t\ge 0,
This together with (3.17) and the KadecKlee property of E shows that
Again, by the assumptions that the semigroup \mathcal{T}:=\{T(t):t\ge 0\} is closed and uniformly LLipschitzian, we have
Since {lim}_{n\to \mathrm{\infty}}{T}^{n}(t){x}_{n}={p}^{\ast} uniformly in t\ge 0, {x}_{n}\to {p}^{\ast} and L(t):[0,\mathrm{\infty})\to [0,\mathrm{\infty}) is a bounded and measurable function, these together with (3.9) imply that
and so
i.e.,
In view of the closeness of the semigroup \mathcal{T}, it yields that T(t){p}^{\ast}={p}^{\ast}, i.e., {p}^{\ast}\in F(T(t)). By the arbitrariness of t\ge 0, we have {p}^{\ast}\in \mathcal{F}:={\bigcap}_{t\ge 0}F(T(t)).

(VI)
Finally, we prove that {x}_{n}\to {p}^{\ast}={\mathrm{\Pi}}_{\mathcal{F}}{x}_{1}.
Let w={\mathrm{\Pi}}_{\mathcal{F}}{x}_{1}. Since w\in \mathcal{F}\subset {C}_{n} and {x}_{n}={\mathrm{\Pi}}_{{C}_{n}}{x}_{1}, we have \varphi ({x}_{n},{x}_{1})\le \varphi (w,{x}_{1}), \mathrm{\forall}n\ge 1. This implies that
In view of the definition of {\mathrm{\Pi}}_{\mathcal{F}}{x}_{1}, from (3.10) we have {p}^{\ast}=w. Therefore, {x}_{n}\to {p}^{\ast}={\mathrm{\Pi}}_{\mathcal{F}}{x}_{1}. This completes the proof of Theorem 3.1. □
From Theorem 3.1, we can obtain the following.
Theorem 3.2 Let E, C, \{{\alpha}_{n}\}, \{{\beta}_{n}\} be the same as in Theorem 3.1. Let \mathcal{T}:=\{T(t):t\ge 0\} be a closed, uniformly LLipschitz and (\{{k}_{n}\})quasiϕasymptotically nonexpansive semigroup with \{{k}_{n}\}\subset [1,\mathrm{\infty}), {k}_{n}\to 1. Let \{{x}_{n}\} be a sequence generated by
where \mathcal{F}:={\bigcap}_{t\ge 0}^{\mathrm{\infty}}F(T(t)), {\xi}_{n}=({k}_{n}1){sup}_{p\in \mathcal{F}}\varphi (p,{x}_{n}), {\mathrm{\Pi}}_{{C}_{n+1}} is the generalized projection of E onto {C}_{n+1}. If \mathcal{F} is bounded in C, then \{{x}_{n}\} converges strongly to {\mathrm{\Pi}}_{\mathcal{F}}{x}_{1}.
Proof It follows from Definition 2.10 that if \mathcal{T}:=\{T(t):t\ge 0\} is a closed, uniformly LLipschitz and (\{{k}_{n}\})quasiϕasymptotically nonexpansive semigroup, then it must be a closed, uniformly LLipschitz (\{{\nu}_{n}\},\{{\mu}_{n}\},\zeta )total quasiϕasymptotically nonexpansive semigroup with {\nu}_{n}={k}_{n}1, {\mu}_{n}=0, \mathrm{\forall}n\ge 1 and \zeta (t)=t, t\ge 0. Therefore, all the conditions in Theorem 3.1 are satisfied. The conclusion of Theorem 3.2 can be obtained from Theorem 3.1 immediately. □
Theorem 3.3 Let E, C, \{{\alpha}_{n}\}, \{{\beta}_{n}\} be the same as in Theorem 3.1. Let \mathcal{T}:=\{T(t):t\ge 0\} be a closed, quasiϕnonexpansive semigroup such that the set \mathcal{F}:={\bigcap}_{t\ge 0}F(T(t)) is nonempty. Let \{{x}_{n}\} be a sequence generated by
Then the sequence \{{x}_{n}\} converges strongly to {\mathrm{\Pi}}_{\mathcal{F}}{x}_{1}.
Proof Since \mathcal{T}:=\{T(t):t\ge 0\} is a closed, quasiϕnonexpansive semigroup, by Remark 2.5, it is a closed, uniformly Lipschitzian and quasiϕasymptotically nonexpansive semigroup with the sequence \{{k}_{n}=1\}. Hence, {\xi}_{n}=({k}_{n}1){sup}_{u\in \mathcal{F}}\varphi (u,{x}_{n})=0. Therefore, the conditions appearing in Theorem 3.1: ‘\mathcal{F} is a bounded subset in C’ and ‘\mathcal{T}:=\{T(t):t\ge 0\} is uniformly Lipschitzian’ are of no use here. Therefore, all conditions in Theorem 3.1 are satisfied. The conclusion of Theorem 3.3 can be obtained from Theorem 3.2 immediately. □
Remark 3.4 Theorems 3.1, 3.2 and 3.3 improve and extend the corresponding results of Suzuki [4], Xu [5], Chang et al. [6–8, 22, 23, 30], Cho et al. [10], Thong [11], Buong [12], Mann [13], Halpern [14], Qin et al. [15], Nakajo et al. [18] and others.
References
Byrne C: A unified treatment of some iterative algorithms in signal processing and image construction. Inverse Probl. 2004, 20: 103–120. 10.1088/02665611/20/1/006
Combettes PL: The convex feasibility problem in image recovery. 95. In Advances in Imaging and Electron Physics. Academic Press, New York; 1996:155–270.
Kitahara S, Takahashi W: Image recovery by convex combinations of sunny nonexpansive retractions. Topol. Methods Nonlinear Anal. 1993, 2: 333–342.
Suzuki T: On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces. Proc. Am. Math. Soc. 2003, 131: 2133–2136. 10.1090/S0002993902068442
Xu HK: A strong convergence theorem for contraction semigroups in Banach spaces. Bull. Aust. Math. Soc. 2005, 72: 371–379. 10.1017/S000497270003519X
Chang SS, Yang L, Liu JA: Strong convergence theorem for nonexpansive semigroups in Banach spaces. Appl. Math. Mech. 2007, 28: 1287–1297. 10.1007/s104830071002x
Zhang S: Convergence theorem of common fixed points for Lipschitzian pseudocontraction semigroups in Banach spaces. Appl. Math. Mech. 2009, 30(2):145–152. 10.1007/s104830090202y
Chang SS, Lee HWJ, Chan CK: Convergence theorem of common fixed point for asymptotically nonexpansive semigroups in Banach spaces. Appl. Math. Comput. 2009, 212: 60–65. 10.1016/j.amc.2009.01.086
Cho SY, Kang SM: Approximation of fixed points of pseudocontraction semigroups based on a viscosity iterative process. Appl. Math. Lett. 2011, 24: 224–228. 10.1016/j.aml.2010.09.008
Cho YJ, Ciric L, Wang S:Convergence theorems for nonexpansive semigroups in CAT(0) spaces. Nonlinear Anal. 2011. doi:10.1016/j.na.2011.05.082
Thong DV: An implicit iteration process for nonexpansive semigroups. Nonlinear Anal. 2011. doi:10.1016/j.na.2011.05.090
Buong N: Hybrid Ishikawa iterative methods for a nonexpansive semigroup in Hilbert space. Comput. Math. Appl. 2011, 61: 2546–2554. 10.1016/j.camwa.2011.02.047
Mann WR: Mean value methods in iteration. Proc. Am. Math. Soc. 1953, 4: 506–510. 10.1090/S00029939195300548463
Halpern B: Fixed points of nonexpanding maps. Bull. Am. Math. Soc. 1967, 73: 957–961. 10.1090/S000299041967118640
Qin XL, Cho YJ, Kang SM, Zhou HY: Convergence of a modified Halperntype iterative algorithm for quasi ϕ nonexpansive mappings. Appl. Math. Lett. 2009, 22: 1051–1055. 10.1016/j.aml.2009.01.015
Wang ZM, Su YF, Wang DX, Dong YC: A modified Halperntype iteration algorithm for a family of hemirelative nonexpansive mappings and systems of equilibrium problems in Banach spaces. J. Comput. Appl. Math. 2011, 235: 2364–2371. 10.1016/j.cam.2010.10.036
Matsushita S, Takahashi W: A strong convergence theorem for relatively nonexpansive mappings in Banach spaces. J. Approx. Theory 2005, 134: 257–266. 10.1016/j.jat.2005.02.007
Nakajo K, Takahashi W: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl. 2003, 279: 372–379. 10.1016/S0022247X(02)004584
Nilsrakoo W, Sajung S: Strong convergence theorems by HalpernMann iterations for relatively nonexpansive mappings in Banach spaces. Appl. Math. Comput. 2011, 217(14):6577–6586. 10.1016/j.amc.2011.01.040
Su YF, Xu HK, Zhang X: Strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications. Nonlinear Anal. 2010, 73: 3890–3906. 10.1016/j.na.2010.08.021
Kang J, Su Y, Zhang X: Hybrid algorithm for fixed points of weak relatively nonexpansive mappings and applications. Nonlinear Anal. Hybrid Syst. 2010, 4(4):755–765. 10.1016/j.nahs.2010.05.002
Chang SS, Chan CK, Lee HWJ: Modified block iterative algorithm for quasi ϕ asymptotically nonexpansive mappings and equilibrium problem in Banach spaces. Appl. Math. Comput. 2011, 217: 7520–7530. 10.1016/j.amc.2011.02.060
Chang SS, Lee HWJ, Chan CK: A new hybrid method for solving a generalized equilibrium problem solving a variational inequality problem and obtaining common fixed points in Banach spaces with applications. Nonlinear Anal. TMA 2010, 73: 2260–2270. 10.1016/j.na.2010.06.006
Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 1994, 63(1/4):123–145.
Yao Y, Cho YJ, Liou YC: Hierarchical convergence of an implicit doublenet algorithm for nonexpansive semigroups and variational inequalities. Fixed Point Theory Appl. 2011., 2011: Article ID 101. doi:10.1186/1687–1812–2011–101
Yao Y, Shahzad N: New methods with perturbations for nonexpansive mappings in Hilbert spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 79. doi:10.1186/1687–1812–2011–79
Yao Y, Liou YC, Wong MM, Yao JC: Strong convergence of a hybrid method for monotone variational inequalities and fixed point problems. Fixed Point Theory Appl. 2011., 2011: Article ID 53. doi:10.1186/1687–1812–2011–53
Yao Y, Chen R: Regularized algorithms for hierarchical fixedpoint problems. Nonlinear Anal. 2011, 74: 6826–6834. 10.1016/j.na.2011.07.007
Yao Y, Shahzad N: Strong convergence of a proximal point algorithm with general errors. Optim. Lett. 2012. doi:10.1007/s11590–011–0286–2
Chang S, Wang L, Tang YK, Zao YH, Ma ZL: Strong convergence theorems of nonlinear operator equations for countable family of multivalued total quasi ϕ asymptotically nonexpansive mappings with applications. Fixed Point Theory Appl. 2012., 2012: Article ID 69. doi:10.1186/1687–1812–2012–69
Qin X, Agarwal RP, Cho SY, Kang SM: Convergence of algorithms for fixed points of generalized asymptotically quasiphinonexpansive mappings with applications. Fixed Point Theory Appl. 2012., 2012: Article ID 58. doi:10.1186/1687–1812–2012–58
Kim JK: Strong convergence theorems by hybrid projection methods for equilibrium problems and fixed point problems of asymptotically quasi ϕ nonexpansive mappings. Fixed Point Theory Appl. 2012., 2011: Article ID 10. doi:10.1186/1687–1812–10
Chang S, Kim JK, Lee HWJ, Chan CK: A generalization and improvement of Chidume theorems for total asymptotically nonexpansive mappings in Banach spaces. J. Inequal. Appl. 2012., 2012: Article ID 37. doi:10.1186/1029–242X2012–37
Alber YI: Metric and generalized projection operators in Banach spaces: properties and applications. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Edited by: Kartosator AG. Dekker, New York; 1996:15–50.
Acknowledgements
This work was supported by the Kyungnam University Research Fund, 2012.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
SsC and JKK conceived the study and participated in its design and coordination. JKK and LW suggested many good ideas that are useful for achievement this paper and made the revision. JKK and SsC prepared the manuscript initially and performed all the steps of proof in this research. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Chang, Ss., Kim, J.K. & Wang, L. Total quasiϕasymptotically nonexpansive semigroups and strong convergence theorems in Banach spaces. Fixed Point Theory Appl 2012, 153 (2012). https://doi.org/10.1186/168718122012153
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/168718122012153