In this section, we will introduce the class of generalized cyclic orbital stronger Meir-Keeler -contraction and we study the existence and uniqueness of fixed points for such mappings. Our results in this section extend and generalize several existing fixed-point theorems in the literature, including Theorem 2 and Theorem 3.
In the sequel, we denote by Θ the class of functions satisfying the following conditions:
() φ is a strictly increasing, continuous function in each coordinate;
() for all , , , , , and .
Example 1 Let denote
Then φ satisfies the above conditions and .
We now denote the below notion of generalized cyclic orbital stronger Meir-Keeler -contraction.
Definition 7 Let A and B be nonempty subsets of a metric space . Suppose is a cyclic map such that for some , there exist a stronger Meir-Keeler type mapping in X and such that
where
for all and . Then f is called a generalized cyclic orbital stronger Meir-Keeler -contraction.
Our main result is the following.
Theorem 5 Let A and B be two nonempty closed subsets of a complete metric space , and let be a stronger Meir-Keeler type mapping in X and . Suppose is a generalized cyclic orbital stronger Meir-Keeler -contraction. Then is nonempty and f has a unique fixed point in .
Proof Since is a generalized cyclic orbital stronger Meir-Keeler -contraction and for , we have . Put , for . Then we have that for each
and by the conditions of the function φ, we get
and
(2.1)
Similarly, we put and for each
and by the conditions of the function φ, we get
and
(2.2)
Using inequalities (2.1) and (2.2), we deduce that is a decreasing sequence and hence it is convergent. Let . Then there exists and such that for all ,
Taking into account the above inequality and the definition of stronger Meir-Keeler type mapping in X, corresponding to η use, there exists such that
(2.3)
Put , where is the integer part of . It follows from (2.1), (2.2) and (2.3) that we deduce that for all ,
(2.4)
and
(2.5)
It follows from (2.4) and (2.5) that for each
(2.6)
Since , we get
For with , we have
and hence , since . So, is a Cauchy sequence. Since is a complete metric space, A and B are closed, , there exists such that . Now is a sequence in A and is a sequence in B, and also both converge to ν. Since A and B are closed, , and so is nonempty. Next, we want to show that ν is a fixed point of f. Suppose that ν is not a fixed point of f. Then . Since and
where
we obtain that
This leads to a contradiction. So, , that is, ν is a fixed point of f.
Finally, we want to show the uniqueness of the fixed point. Let μ be another fixed point of f. By the cyclic character of f, we have . Since f is a generalized cyclic orbital stronger Meir-Keeler -contraction, we have
(2.7)
and
(2.8)
where
It follows from (2.7), (2.8) and the condition of the mapping φ that
This leads to a contradiction. Therefore, , and so ν is the unique fixed point of f. □
We give the following example to illustrate Theorem 5.
Example 2 Let and we define by
and let denote
We next define by
and let denote
Then f is a generalized cyclic orbital stronger Meir-Keeler -contraction and 0 is the unique fixed point.