In this section, we introduce two new notions of a cyclic contraction and establish new results for such mappings.

In the sequel, we fixed the set of functions by
such that

(i) ℱ is nondecreasing, continuous, and
for every
;

(ii) *ψ* is nondecreasing, right continuous, and
for every
.

Define
and
.

We state the notion of a cyclic generalized
-contraction as follows.

**Definition 2.1** Let
be a metric space. Let *p* be a positive integer,
be nonempty subsets of *X* and
. An operator
is said to be a cyclic generalized
-contraction for some
,
, and
if

(a)
is a cyclic representation of *Y* with respect to *T*;

Our first main result is the following.

**Theorem 2.1**
*Let*
*be a complete metric space*,
,
*be nonempty closed subsets of*
*X*, *and*
. *Suppose*
*is a cyclic generalized*
-*contraction mapping for some*
*and*
. *Then*
*T*
*has a unique fixed point*. *Moreover*, *the fixed point of*
*T*
*belongs to*
.

*Proof* Let

(such a point exists since

). Define the sequence

in

*X* by

If, for some

*k*, we have

, then (2) follows immediately. So, we can suppose that

for all

*n*. From the condition (a), we observe that for all

*n*, there exists

such that

. Then, from the condition (b), we have

On the other hand, we have

Suppose that

for some

. Then

, so

Thus, from (4) and (5), we get

for all

. Now, from

and the property of *ψ*, we obtain
, and consequently (2) holds.

Now, we shall prove that

is a Cauchy sequence in

. Suppose, on the contrary, that

is not a Cauchy sequence. Then there exists

for which we can find two sequences of positive integers

and

such that for all positive integers

*k*,

Further, corresponding to

, we can choose

in such a way that it is the smallest integer with

satisfying (6). Then we have

Using (6), (7), and the triangular inequality, we get

Passing to the limit as

in the above inequality and using (2), we obtain

On the other hand, for all

*k*, there exists

such that

. Then

(for

*k* large enough,

) and

lie in different adjacently labeled sets

and

for certain

. Using (b), we obtain

for all

*k*. Using the triangular inequality, we get

which implies from (8) that

Again, using the triangular inequality, we get

Passing to the limit as

in the above inequality, using (14) and (12), we get

Passing to the limit as

, using (2) and (12), we obtain

Now, it follows from (12)-(16) and the continuity of

*φ* that

Passing to the limit as

in (9), using (17), (18), (19), and the condition (ii), we obtain

which is a contradiction. Thus, we proved that
is a Cauchy sequence in
.

Since

is complete, there exists

such that

From the condition (a), and since
, we have
. Since
is closed, from (20), we get that
. Again, from the condition (a), we have
. Since
is closed, from (20), we get that
. Continuing this process, we obtain (21).

Now, we shall prove that

is a fixed point of

*T*. Indeed, from (21), since for all

*n* there exists

such that

, applying (b) with

and

, we obtain

for all

*n*. On the other hand, we have

Passing to the limit as

in the above inequality and using (20), we obtain that

Passing to the limit as

in (22), using (23) and (20), we get

Suppose that

. In this case, we have

a contradiction. Then we have
, that is,
is a fixed point of *T*.

Finally, we prove that

is the unique fixed point of

*T*. Assume that

is another fixed point of

*T*, that is,

. From the condition (a), this implies that

. Then we can apply (b) for

and

. We obtain

Since

and

are fixed points of

*T*, we can show easily that

and

. If

, we get

a contradiction. Then we have
, that is,
. Thus, we proved the uniqueness of the fixed point. □

In the following, we deduce some fixed point theorems from our main result given by Theorem 2.1.

If we take
and
in Theorem 2.1, then we get immediately the following fixed point theorem.

*for all*
. *Then*
*T*
*has a unique fixed point*.

**Remark 2.1** Corollary 2.1 extends and generalizes many existing fixed point theorems in the literature [1, 16–21].

**Corollary 2.2**
*Let*
*be a complete metric space*,
,
*be nonempty closed subsets of*
*X*,
, *and*
. *Suppose that there exist*
*and*
*such that*

(a′)
*is a cyclic representation of*
*Y*
*with respect to*
*T*;

*Then*
*T*
*has a unique fixed point*. *Moreover*, *the fixed point of*
*T*
*belongs to*
.

**Remark 2.2** Corollary 2.2 is similar to Theorem 2.1 in [7].

**Remark 2.3** Taking in Corollary 2.2
with
, we obtain a generalized version of Theorem 1.3 in [6].

**Corollary 2.3**
*Let*
*be a complete metric space*,
,
*be nonempty closed subsets of*
*X*,
, *and*
. *Suppose that there exist*
*and*
*such that*

(a′)
*is a cyclic representation of*
*Y*
*with respect to*
*T*;

*Then*
*T*
*has a unique fixed point*. *Moreover*, *the fixed point of*
*T*
*belongs to*
.

**Remark 2.4** Taking in Corollary 2.3
with
, we obtain a generalized version of Theorem 3 in [13].

**Corollary 2.4**
*Let*
*be a complete metric space*,
,
*be nonempty closed subsets of*
*X*,
, *and*
. *Suppose that there exist*
*and*
*such that*

(a′)
*is a cyclic representation of*
*Y*
*with respect to*
*T*;

*Then*
*T*
*has a unique fixed point*. *Moreover*, *the fixed point of*
*T*
*belongs to*
.

**Remark 2.5** Taking in Corollary 2.4
with
, we obtain a generalized version of Theorem 5 in [13].

**Corollary 2.5**
*Let*
*be a complete metric space*,
,
*be nonempty closed subsets of*
*X*,
, *and*
. *Suppose that there exist*
*and*
*such that*

(a)
*is a cyclic representation of*
*Y*
*with respect to*
*T*;

*Then*
*T*
*has a unique fixed point*. *Moreover*, *the fixed point of*
*T*
*belongs to*
.

We provide some examples to illustrate our obtained Theorem 2.1.

**Example 2.1** Let

with the usual metric. Suppose

and

and

. Define

such that

for all

. It is clear that

is a cyclic representation of

*Y* with respect to

*T*. Let

be defined by

and

of the form

,

. For all

and

, we have

So, *T* is a cyclic generalized
-contraction for any
. Therefore, all conditions of Theorem 2.1 are satisfied (
), and so *T* has a unique fixed point (which is
).

**Example 2.2** Let

with the usual metric. Suppose

and

and

. Define the mapping

by

Clearly, we have
and
. Moreover,
and
are nonempty closed subsets of *X*. Therefore,
is a cyclic representation of *Y* with respect to *T*.

Now, let

with

and

, we have

On the other hand, we have

Define the function

by

and

of the form

,

and

. Then we have

Moreover, we can show that (24) holds if
or
. Similarly, we also get (24) holds for
.

Now, all the conditions of Theorem 2.1 are satisfied (with
), we deduce that *T* has a unique fixed point
.