Let

*C* be a nonempty subset of a Banach space

*X* and

a semigroup of mappings from

*C* into itself. A sequence

in

*C* is said to an

*approximating fixed point sequence of* ℱ if

for all

. The family

is demiclosed at zero if

is a sequence in

*C* weakly converging to

and

for all

imply

for all

. Following [

18], we say that ℱ has property (

) if for every bounded set

in

*C*, we have

In [30], Schu introduced the concept of asymptotically pseudo-contractive mapping as follows:

Let

*H* be a real Hilbert space whose inner product and norm are denoted by

and

respectively. Let

*C* be a nonempty subset of

*H* and

a mapping. Then

*T* is called an asymptotically pseudo-contractive mapping if there exists a sequence

in

with

such that

The class of asymptotically pseudo-contractive mappings contain properly the class of asymptotically nonexpansive mappings. The following example shows that a continuous asymptotically pseudo-contractive mapping is not necessarily asymptotically nonexpansive.

**Example 5.1** Let

and

. Define

by

Note that *T* is a pseudo-contractive mapping which is not Lipschitzian (see [31]). Since *T* is not Lipschitzian, it is not asymptotically nonexpansive. It is shown in [30] that *T* is an asymptotically pseudo-contractive mapping with sequence
.

Let

be a semigroup of mappings from

*C* into itself. Then ℱ is said to be pseudo-contractive if

**Remark 5.2**

- (i)
The semigroup ℱ is pseudo-contractive if and only if the following holds:

- (ii)
Every nonexpansive semigroup must be a continuously pseudo-contractive semigroup.

We say ℱ is asymptotically pseudo-contractive if there exists a function

with

such that

**Example 5.3** Let

,

,

and

. For

, define

by

Set

and

. Note that

For

and

, we have

, and hence

Therefore,
is an asymptotically pseudo-contractive semigroup with function
. Moreover, for each
,
is discontinuous at
and hence ℱ is not a Lipschitzian semigroup.

We begin with the following:

**Theorem 5.4** (Demiclosedness Principle)

*Let C be a nonempty closed convex bounded subset of a real Hilbert space*
*H*. *Let*
*be a strongly continuous semigroup of uniformly continuous nearly uniformly*
*L*-*Lipschitzian asymptotically pseudo*-*contractive mappings from*
*C*
*into itself*. *Then the family*
*is demiclosed at zero*.

*Proof* Assume that

is a sequence in

*C* weakly converging to

*z* and

for all

. Let

with

and let

be a sequence in

*G* defined by

for all

. Notice that

. Fix

and define

Since *T* is uniformly continuous, we have
as
for fixed
.

Indeed, for fixed

, we have

for all

. Since ℱ is a uniformly continuous semigroup, it follows that

for each fixed

. Noticing that ℱ is an asymptotically pseudo-contractive semigroup, for fixed

, we have

Since

and

, it follows from (5.1) that

Letting

in (5.2), we obtain that

. It follows from the continuity of

that

Therefore,
for all
. □

The following result extends the celebrated convergence theorem of Browder [32] and many results concerning Browder’s convergence theorem to a semigroup of uniformly continuous nearly uniformly *L*-Lipschitzian asymptotically pseudo-contractive mappings.

**Theorem 5.5**
*Let*
*C*
*be a nonempty closed convex bounded subset of a real Hilbert space*
*H*
*and*
*a strongly continuous semigroup of uniformly continuous nearly uniformly*
*L*-

*Lipschitzian asymptotically pseudo*-

*contractive mappings from*
*C*
*into itself*.

*Let*
*be a sequence in*
*and*
*a sequence in*
*such that*
*for all*
,

*and*
.

*Then*:

- (a)
*There exists a sequence*
*in*
*C*
*defined by*

- (b)
*If* ℱ

*has property* (

),

*then*
*and*
*converges strongly to*
*such that*

*Proof* (a) Let

be a strongly continuous semigroup of asymptotically pseudo-contractive mappings with a net

. Set

. Note

for all

, it follows that

and hence

for all

. Then, for each

, the mapping

defined by

is continuous and strongly pseudo-contractive. Indeed, for

*x*,

*y* in

*C*, we have

Therefore, by Lemma 2.1, there exists a sequence
in *C* described by (5.3).

(b) Assume that ℱ has property (
). From (5.3), we have
as
. The property (
) of ℱ gives that
as
for all
. Since
is bounded, we can assume that a subsequence
of
such that
for some
. By Theorem 5.4, we have
.

For

, we have

Since

and

*C* is bounded, it follows from (5.5) that

We claim that the set

is sequentially compact. For

, we have

By the weak compactness of

*C*, there exists a weakly convergent subsequence

. Suppose that

as

. Since

is an approximating fixed point sequence of ℱ, we infer from Theorem 5.4 that

. In (5.7), interchange

*v* and

to obtain that

Since
, we get that
. Hence the set
is sequentially compact.

Next, we show that

. Suppose, for contradiction, that

is another subsequence of

such that

. It is easy to see that

. Observe that

Since

, we get

Adding inequalities (5.8) and (5.9) yields

a contradiction. In a similar way it can be shown that each cluster point of the sequence
is equal to
. Therefore, the entire sequence
converges strongly to
. It is easy to see, from (5.6), that the inequality (5.4) holds. □

**Theorem 5.6**
*Let*
*C*
*be a nonempty closed convex bounded subset of a real Hilbert space*
*H*
*and*
*a strongly continuous semigroup of uniformly continuous nearly uniformly*
*L*-*Lipschitzian asymptotically pseudo*-*contractive mappings from*
*C*
*into itself*. *Suppose that* ℱ *has property* (
). *Then*
*and*
*is a sunny nonexpansive retract of*
*C*.

*Proof* Assume that
is a semigroup of asymptotically pseudo-contractive mappings from *C* into itself with a function
with
. Without loss of generality, we may assume that
in
and
in
such that
for all
,
and
. Then, for an arbitrarily fixed element
, there exists a sequence
in *C* defined by (5.3). By Theorem 5.5(b),
.

By Theorem 5.5(b),

converges strongly to an element

such that the inequality (5.4) holds. Define a mapping

by

In view of (5.4), we have

Therefore, by Lemma 2.2, we conclude that *Q* is sunny nonexpansive. □