In the sequel, we let the function *ϕ* : [0, ∞) → [0, ∞) satisfies the following conditions:

(*ϕ*
_{1}) *ϕ* : [0, ∞) → [0, ∞) is a weaker Meir-Keeler function;

(*ϕ*
_{2}) *ϕ*(*t*) *>* 0 for *t >* 0 and *ϕ*(0) = 0;

(*ϕ*
_{3}) for all *t* ∈ (0, ∞),
is decreasing;

(*ϕ*
_{4}) for *t*
_{
n
} ∈ [0, ∞), we have that

(a) if lim_{
n→∞}
*t*
_{
n
}= *γ >* 0, then lim_{
n→∞}
*ϕ*(*t*
_{
n
}) *< γ*, and

(b) if lim_{
n→∞ }
*t*
_{
n
}= 0, then lim_{
n→∞ }
*ϕ*(*t*
_{
n
}) = 0.

Let the function *ψ* : [0, ∞) → [0, 1) satisfies the following conditions:

(*ψ*
_{1}) *ψ* : [0, ∞) → [0, 1) is a stronger Meir-Keeler function;

(*ψ*
_{2}) *ψ*(*t*) *>* 0 for *t >* 0 and *ϕ*(0) = 0.

And, we let the function *φ* : [0, ∞) → [0, ∞) satisfies the following conditions:

(*φ*
_{1}) for all *t* ∈ (0, ∞), lim_{
n→∞ }
*t*
_{
n
}= 0 if and only if lim_{
n→∞ }
*φ*(*t*
_{
n
}) = 0;

(*φ*
_{2}) *φ*(*t*) *>* 0 for *t >* 0 and *φ*(0) = 0;

(*φ*
_{3}) *φ* is subadditive, that is, for every *μ*
_{1}, *μ*
_{2} ∈ [0, ∞), *φ*(*μ*
_{1} + *μ*
_{2}) ≤ *φ*(*μ*
_{1}) + *φ*(*μ*
_{2}).

Using the functions *ϕ* and *φ*, we first introduce the notion of the (*ϕ-φ*)-weak contraction mapping and prove a theorem which assures the existence of a periodic point for the (*ϕ-φ*)-weak contraction mapping.

**Definition 7**
*Let* (

*X*,

*d*)

*be a g.m.s, and let f* :

*X* →

*X be a function satisfying*
*for all x*, *y* ∈ *X. Then f is said to be a* (*ϕ - φ*)-*weak contraction mapping*.

**Theorem 1**
*Let* (*X*, *d*) *be a Hausdorff and complete g.m.s, and let f be a* (*ϕ - φ*)-*weak contraction mapping. Then f has a periodic point μ in X, that is, there exists μ* ∈ *X such that*
for some
.

*Proof*. Given

*x*
_{0} and define a sequence {

*x*
_{
n
}} in

*X* by

**Step 1**. We shall prove that

Using the inequality (1), we have that for each

Since

is decreasing, it must converge to some

*η* ≥ 0. We claim that

*η* = 0. On the contrary, assume that

*η >* 0. Then by the definition of weaker Meir-Keeler function

*ϕ*, corresponding to

*η* use, there exists

*δ >* 0 such that for

*x*
_{0},

*x*
_{1} ∈

*X* with

*η* ≤

*φ*(

*d*(

*x*
_{0},

*x*
_{1}))

*< δ* +

*η*, there exists

such that

. Since lim

_{
n→∞ }
*ϕ*
^{
n
}(

*φ*(

*d*(

*x*
_{0},

*x*
_{1}))) =

*η*, there exists

such that

*η* ≤

*ϕ*
^{
p
}(

*φ*(

*d*(

*x*
_{0},

*x*
_{1})))

*< δ* +

*η*, for all

*p* ≥

*p*
_{0}. Thus, we conclude that

. So we get a contradiction. Therefore lim

_{
n→∞ }ϕ

^{
n
}(

*φ*(

*d*(

*x*
_{0},

*x*
_{1}))) = 0, that is,

Using the inequality (1), we also have that for each

Since

is decreasing, by the same proof process, we also conclude

Next, we claim that {*x*
_{
n
}} is *g.m.s* Cauchy. We claim that the following result holds:

**Step 2**. Claim that
, that is, for every *ε >* 0, there exists
such that if *p*, *q* ≥ *n* then *φ*(*d*(*x*
_{
p
}, *x*
_{
q
})) *< ε*.

Suppose the above statement is false. Then there exists

*ε >* 0 such that for any

, there are

with

*p*
_{
n
}
*> q*
_{
n
} ≥

*n* satisfying

Further, corresponding to

*q*
_{
n
} ≥

*n*, we can choose

*p*
_{
n
} in such a way that it the smallest integer with

*p*
_{
n
}
*> q*
_{
n
} ≥

*n* and

. Therefore

. By the rectangular inequality and (2), (3), we have

Letting

*n* → ∞. Then we get

On the other hand, we have

Letting

*n* → ∞. Then we get

Using the inequality (1), we have

Letting

*n* → ∞, by the definitions of the functions

*ϕ* and

*φ*, we have

So we get a contradiction. Therefore
, by the condition (*φ*
_{1}), we have
. Therefore {*x*
_{
n
}} is *g.m.s* Cauchy.

**Step 3**. We claim that *f* has a periodic point in *X*.

Suppose, on contrary,

*f* has no periodic point. Then {

*x*
_{
n
}} is a sequence of distinct points, that is,

*x*
_{
p
} ≠

*x*
_{
q
} for all

with

*p* ≠

*q*. By step 2, since

*X* is complete

*g.m.s*, there exists

*ν* ∈

*X* such that

*x*
_{
n
} →

*ν*. Using the inequality (1), we have

by the condition (

*φ*
_{1}), we get

As (*X*, *d*) is Hausdorff, we have *ν* = *fν*, a contradiction with our assumption that *f* has no periodic point. Therefore, there exists *ν* ∈ *X* such that
for some
. So *f* has a periodic point in *X*. □

Using the functions *ψ* and *φ*, we next introduce the notion of the (*ψ*-*φ*)-weak contraction mapping and prove a theorem which assures the existence of a periodic point for the (*ψ*-*φ*)-weak contraction mapping.

**Definition 8**
*Let* (

*X*,

*d*)

*be a g.m.s, and let f* :

*X* →

*X be a function satisfying*
*for all x*, *y* ∈ *X. Then f is said to be a* (*ψ* - *φ*)-*weak contraction mapping*.

**Theorem 2**
*Let* (*X*, *d*) *be a Hausdorff and complete g.m.s, and let f be a* (*ψ* - *φ*)-*weak contraction mapping. Then f has a periodic point μ in X*.

*Proof*. Given

*x*
_{0} and define a sequence {

*x*
_{
n
}} in

*X* by

**Step 1**. We shall prove that

Taking into account (4) and the definition of stronger Meir-Keeler function

*ψ*, we have that for each

Thus the sequence {

*φ*(

*d*(

*x*
_{
n
},

*x*
_{
n+1}))} is descreasing and bounded below and hence it is con-vergent. Let lim

_{
n → ∞ }
*φ*(

*d*(

*x*
_{
n
},

*x*
_{
n+1})) =

*η* ≥ 0. Then there exists

and

*δ >* 0 such that for all

with

*n* ≥

*n*
_{0}
Taking into account (7) and the definition of stronger Meir-Keeler function

*ψ*, corresponding to

*η* use, there exists

*γ*
_{
η
} ∈ [0, 1) such that

Thus, we can deduce that for each

with

*n* ≥

*n*
_{0} + 1

Since

*γ*
_{
η
} ∈ [0, 1), we get

Taking into account (4) and the definition of stronger Meir-Keeler function

*ψ*, we have that for each

Thus the sequence {

*φ*(

*d*(

*x*
_{
n
},

*x*
_{
n+2}))} is descreasing and bounded below and hence it is convergent. By the same proof process, we also conclude

Next, we claim that {*x*
_{
n
}} is *g.m.s* Cauchy.

**Step 2**. Claim that
, that is, for every *ε >* 0, corresponding to above *n*
_{0} use, there exists
with *n* ≥ *n*
_{0} +1 such that if *p*, *q* ≥ *n* then *φ*(*d*(*x*
_{
p
}, *x*
_{
q
})) *< ε*.

Suppose the above statement is false. Then there exists

*ε >* 0 such that for any

, there are

with

*p*
_{
n
}
*> q*
_{
n
} ≥

*n* ≥

*n*
_{0} + 1 satisfying

Following from Theorem 1, we have that

Using the inequality (4), we have

Letting

*n* → ∞, by the definitions of the functions

*ψ* and

*φ*, we have

So we get a contradiction. Therefore
, by the condition (*φ*
_{1}), we have
. Therefore {*x*
_{
n
}} is *g.m.s* Cauchy.

**Step 3**. We claim that *f* has a periodic point in *X*.

Suppose, on contrary,

*f* has no periodic point. Then {

*x*
_{
n
}} is a sequence of distinct points, that is,

*x*
_{
p
} ≠

*x*
_{
q
} for all

with

*p* ≠

*q*. By step 2, since

*X* is complete

*g.m.s*, there exists

*ν* ∈

*X* such that

*x*
_{
n
} →

*ν*. Using the inequality (4), we have

by the condition (

*φ*
_{1}), we get

As (*X*, *d*) is Hausdorff, we have *ν* = *fν*, a contradiction with our assumption that *f* has no periodic point. Therefore, there exists *ν* ∈ *X* such that
for some
. So *f* has a periodic point in *X*. □

In conclusion, by using the new concepts of (*ϕ-φ*)-weak contraction mappings and (*ψ - φ*)-weak contraction mappings, we obtain two theorems (Theorems 1 and 2) which assure the existence of a periodic point for these two types of weak contraction in complete generalized metric spaces. Our results generalize or improve many recent fixed point theorems in the literature.