Contractive mappings and iteration procedures are some of the main tools in the study of fixed point theory. There are many contractive mappings and iteration schemes that have been introduced and developed by several authors to serve various purposes in the literature of this highly active research area, *viz.*, [1–12] among others.

Whether an iteration method used in any investigation converges to a fixed point of a contractive type mapping corresponding to a particular iteration process is of utmost importance. Therefore it is natural to see many works related to the convergence of iteration methods such as [13–22].

Fixed point theory is concerned with investigating a wide variety of issues such as the existence (and uniqueness) of fixed points, the construction of fixed points, *etc.* One of these themes is data dependency of fixed points. Data dependency of fixed points has been the subject of research in fixed point theory for some time now, and data dependence research is an important theme in its own right.

Several authors who have made contributions to the study of data dependence of fixed points are Rus and Muresan [23], Rus *et al.*[24, 25], Berinde [26], Espínola and Petruşel [27], Markin [28], Chifu and Petruşel [29], Olantiwo [30, 31], Şoltuz [32, 33], Şoltuz and Grosan [34], Chugh and Kumar [35] and the references therein.

This paper is organized as follows. In Section 1 we present a brief survey of some known contractive mappings and iterative schemes and collect some preliminaries that will be used in the proofs of our main results. In Section 2 we show that the convergence of a new multi-step iteration, which is a special case of the Jungck multistep-SP iterative process defined in [36], and S-iteration (due to Agarwal *et al.*) can be used to approximate the fixed points of contractive-like operators. Motivated by the works of Şoltuz [32, 33], Şoltuz and Grosan [34], and Chugh and Kumar [35], we prove two data dependence results for the new multi-step iteration and S-iteration schemes by employing contractive-like operators.

As a background of our exposition, we now mention some contractive mappings and iteration schemes.

In [

37] Zamfirescu established an important generalization of the Banach fixed point theorem using the following contractive condition. For a mapping

, there exist real numbers

*a*,

*b*,

*c* satisfying

,

such that, for each pair

, at least one of the following is true:

A mapping

*T* satisfying the contractive conditions (z

_{1}), (z

_{2}) and (z

_{3}) in (1.1) is called a Zamfirescu operator. An operator satisfying condition (z

_{2}) is called a

*Kannan operator*, while the mapping satisfying condition (z

_{3}) is called a

*Chatterjea operator*. As shown in [

13], the contractive condition (1.1) leads to

for all
, where
,
, and it was shown that this class of operators is wider than the class of Zamfirescu operators. Any mapping satisfying condition (b_{1}) or (b_{2}) is called a quasi-contractive operator.

Extending the above definition, Osilike and Udomene [

20] considered operators

*T* for which there exist real numbers

and

such that for all

,

Imoru and Olantiwo [

38] gave a more general definition: An operator

*T* is called a contractive-like operator if there exists a constant

and a strictly increasing and continuous function

, with

, such that for each

,

A map satisfying (1.4) need not have a fixed point, even if

*E* is complete. For example, let

and define

*T* by

WLOG, assume that
. Then, for
or
,
, and (1.4) is automatically satisfied.

If
, then
.

Define *φ* by
for any
. Then *φ* is increasing, continuous, and
. Also,
so that
.

for any
, and (1.4) is satisfied for
. But *T* has no fixed point.

However, using (1.4) it is obvious that if *T* has a fixed point, then it is unique.

From now on, we demand that ℕ denotes the set of all nonnegative integers. Let *X* be a Banach space, let
be a nonempty closed, convex subset of *X*, and let *T* be a self-map on *E*. Define
to be the set of fixed points of *T*. Let
,
,
and
,
,
be real sequences in
satisfying certain conditions.

In [

5] Rhoades and Şoltuz introduced a multi-step iterative procedure given by

The sequence

defined by

is known as the S-iteration process (see [12, 17, 39]).

Thianwan [

6] defined a two-step iteration

by

Recently Phuengrattana and Suantai [

7] introduced an SP iteration method defined by

We shall employ the following iterative process. For an arbitrary fixed order

,

**Remark 1** If each
, then SP iteration (1.8) reduces to two-step iteration (1.7). By taking
and
in (1.10), we obtain iterations (1.8) and (1.7), respectively.

We shall need the following definition and lemma in the sequel.

**Definition 1**[40]

Let

be two operators. We say that

is an approximate operator for

*T* if, for some

, we have

for all
.

**Lemma 1**[34]

*is satisfied*,

*where*
*for all*
,

*and*
,

.

*Then the following holds*: