Let us recall the modular definitions of asymptotic pointwise nonexpansive mappings and associated notions, [18].

**Definition 3.1** Let

and let

be nonempty and

*ρ*-closed. A mapping

is called an asymptotic pointwise mapping if there exists a sequence of mappings

such that

- (i)

- (ii)
If
converges pointwise to
, then *T* is called asymptotic pointwise contraction.

- (iii)
If
for any
, then *T* is called asymptotic pointwise nonexpansive.

- (iv)

Denoting

, we note that without loss of generality we can assume that

*T* is asymptotically pointwise nonexpansive if

Define

. In view of (3.2), we have

The above notation will be consistently used throughout this paper.

By
we will denote the class of all asymptotic pointwise nonexpansive mappings
.

In this paper, we will impose some restrictions on the behavior of
and
. This type of assumptions is typical for controlling the convergence of iterative processes for asymptotically nonexpansive mappings, see, *e.g.*, [25].

**Definition 3.2** Define

as a class of all

such that

We recall the following concepts related to the modular uniform convexity introduced in [18]:

**Definition 3.3** Let

. We define the following uniform convexity type properties of the function modular

*ρ*: Let

,

,

. Define

and
if
. We will use the following notational convention:
.

**Definition 3.4** We say that

*ρ* satisfies

if for every

,

,

. Note that for every

,

, for

small enough. We say that

*ρ* satisfies

if for every

,

there exists

depending only on

*s* and

*ε* such that

We will need the following result whose proof is elementary. Note that for
, this result follows directly from Definition 3.4.

The notion of bounded away sequences of real numbers will be used extensively throughout this paper.

**Definition 3.5** A sequence
is called bounded away from 0 if there exists
such that
for every
. Similarly,
is called bounded away from 1 if there exists
such that
for every
.

We will need the following generalization of Lemma 4.1 from [18] and being a modular equivalent of a norm property in uniformly convex Banach spaces, see, *e.g.*, [36].

*Proof* Assume to the contrary that this is not the case and fix an arbitrary

. Passing to a subsequence if necessary, we may assume that there exists an

such that

Since

is bounded away from 0 and 1 there exist

such that

for all natural

*n*. Passing to a subsequence if necessary, we can assume that

. For every

and

, let us define

. Observe that the function

is a convex function. Hence that the function

is also convex on

, and consequently, it is a continuous function on

. Noting that

we conclude that

is a continuous function of

. Hence

By (3.12) the left-hand side of (3.13) tends to

as

while the right-hand side tends to

in view of (3.7). Hence

By

and by Lemma 3.1, there exists

satisfying

Combining (3.14) with (3.15) we get

Letting
we get a contradiction which completes the proof. □

Let us introduce a notion of a *ρ*-type, a powerful technical tool which will be used in the proofs of our fixed point results.

**Definition 3.6** Let

be convex and

*ρ*-bounded. A function

is called a

*ρ*-type (or shortly a type) if there exists a sequence

of elements of

*K* such that for any

there holds

Note that *τ* is convex provided *ρ* is convex. A typical method of proof for the fixed point theorems in Banach and metric spaces is to construct a fixed point by finding an element on which a specific type function attains its minimum. To be able to proceed with this method, one has to know that such an element indeed exists. This will be the subject of Lemma 3.3 below. First, let us recall the definition of the Opial property and the strong Opial property in modular function spaces, [15, 17].

**Definition 3.7** We say that

satisfies the

*ρ*-a.e. Opial property if for every

which is

*ρ*-a.e. convergent to 0 such that there exists a

for which

the following inequality holds for any

not equal to 0

**Definition 3.8** We say that

satisfies the

*ρ*-a.e. strong Opial property if for every

which is

*ρ*-a.e. convergent to 0 such that there exists a

for which

the following equality holds for any

**Remark 3.1** Note that the *ρ*-a.e. Strong Opial property implies *ρ*-a.e. Opial property [15].

**Remark 3.2** Also, note that, by virtue of Theorem 2.1 in [15], every convex, orthogonally additive function modular *ρ* has the *ρ*-a.e. strong Opial property. Let us recall that *ρ* is called orthogonally additive if
whenever
. Therefore, all Orlicz and Musielak-Orlicz spaces must have the strong Opial property.

Note that the Opial property in the norm sense does not necessarily hold for several classical Banach function spaces. For instance, the norm Opial property does not hold for
spaces for
while the modular strong Opial property holds in
for all
.

**Lemma 3.3**[27]

*Let*
. *Assume that*
*has the*
*ρ*-*a*.*e*. *strong Opial property*. *Let*
*be a nonempty*, *strongly*
*ρ*-*bounded and*
*ρ*-*a*.*e*. *compact convex set*. *Then any*
*ρ*-*type defined in C attains its minimum in*
*C*.

Let us finish this section with the fundamental fixed point existence theorem which will be used in many places in the current paper.

**Theorem 3.1**[18]

*Assume*
*is*
. *Let*
*C*
*be a*
*ρ*-*closed*
*ρ*-*bounded convex nonempty subset*. *Then any*
*asymptotically pointwise nonexpansive has a fixed point*. *Moreover*, *the set of all fixed points*
*is*
*ρ*-*closed*.