Common Fixed Points for Two Contractive Mappings of Integral Type in Metric Spaces

In this paper, we obtain unique common fixed point theorems for two mappings satisfying the variable coefficient linear contraction of integral type and the implicit contraction of integral type respectively in metric spaces.


Introduction and Preliminaries
Throughout this paper, we assume that It is known that the Banach contraction principle has a lot of generalizations and various applications in many directions; see, for examples, [2]- [15] and the references cited therein.In 1962, Rakotch [11] extended the Banach contraction principle with replacing the contraction constant c in (1.1) by a contraction function γ and obtained the next theorem.

Theorem 1.3 ([12]
).Let f be a self-mapping on a complete metric space ( ) In 2011, Liu and Li [13] modified the method of Rakotch to generalize the Branciari's fixed point theorem with replacing the contraction constant c in (1.3) by contraction functions α and β and established the fol- lowing fixed point theorem:

Theorem 1.4 ([13]
).Let f be a self-mapping on a complete metric space ( ) ( ) ( ) Here, we will use the methods in [3] [9] [13] to discuss the unique existence problems of common fixed points for two self-mappings satisfying two different contractive conditions of integral type in a complete metric space.Theorem 2.1.Let ( ) , X d be a complete metric space, , :

Common Fixed Point Theorems
Then f and g have a unique common fixed point u, and the sequence { } x gx , Combining (2.4) and (2.5), we have Now, we prove that Otherwise, there exists 0 n N ∈ such that 0 0 1 .
We claim that { } x ∈ is a Cauchy sequence.Otherwise, there 0 ε > such that for k N ∈ , there exist ( ) m k denotes the least integer exceeding ( ) n k and satisfying the then .  , which is a contradiction, hence fu u = .Similarly, gu u = .So u is a common fixed point of f and g.The uni- queness is obvious.
From now on, we will discuss the second common fixed point problem for two mappings with implicit contraction of integral type.
Theorem 2.3.Let ( ) , X d be a complete metric space, , : where φ ∈ Φ is sub-additive and ψ ∈ Ψ .Then f and g have a unique common fixed point.

Proof.
We take any element 0 x X ∈ and consider the sequence { } This is a contradiction.Hence * x is the unique common fixed point of f and g.Using Theorem 2.3 and the Example 2.2, we have the next result.

≤ ∈
where ψ ∈ Ψ .Then f and g have a unique common fixed point u.

,
φ is Lebesgue integral, summable on each compact subset of R + and 's contraction principle is as follows:Theorem 1.1 ([1]).Let f be a self mapping on a complete metric space () .Then f has a unique fixed point x X ∈ 

,
and φ ∈ Φ .Then f has a unique fixed point x X ∈ 

Lemma 2 . 1 (
[13]).Let φ ∈ Φ and { } n n N r ∈ be a nonnegative sequence with lim n will give the first main result in this paper.

Theorem 2 . 4 .
Let ( ) , X d be a complete metric space, , : , , From Theorem 2.1, we obtain the next more general common fixed point theorem.
is a contradiction.Similarly, we obtain the same contradiction for the case that ( ) m k is odd and ( )* x y = , i.e., *x is the unique common fixed point of f and g.
. Then ψ ∈ Ψ .The function φ ∈ Φ is called to be sub-additive if and only if for all , This is a contradiction.Similarly, we obtain the same contradiction for the case that *y is another common fixed point of f and g, then ( ) Then f and g have a unique common.