- Research Article
- Open Access
Fixed Point Theorems for Generalized Weakly Contractive Condition in Ordered Metric Spaces
© H. K. Nashine and I. Altun. 2011
- Received: 30 September 2010
- Accepted: 11 February 2011
- Published: 10 March 2011
Fixed point results with the concept of generalized weakly contractive conditions in complete ordered metric spaces are derived. These results generalize the existing fixed point results in the literature.
- Fixed Point Theorem
- Contractive Condition
- Lower Semicontinuity
- Cauchy Sequence
- Common Fixed Point
There are a lot of generalizations of the Banach contraction mapping principle in the literature. One of the most interesting of them is the result of Khan et al. . They addressed a new category of fixed point problems for a single self-map with the help of a control function which they called an altering distance function.
Khan et al.  given the following result.
In fact, Khan et al.  proved a more general theorem of which the above result is a corollary. Another generalization of the contraction principle was suggested by Alber and Guerre-Delabriere  in Hilbert Spaces by introducing the concept of weakly contractive mappings.
Rhoades  showed that most results of  are still valid for any Banach space. Also, Rhoades  proved the following very interesting fixed point theorem which contains contractions as special case .
In fact, Alber and Guerre-Delabriere  assumed an additional condition on which is . But Rhoades  obtained the result noted in Theorem 1.2 without using this particular assumption. Also, the weak contractions are closely related to maps of Boyd and Wong  and Reich type . Namely, if is a lower semicontinuous function from the right, then is an upper semicontinuous function from the right, and moreover, (1.2) turns into . Therefore, the weak contraction is of Boyd and Wong type. And if we define for and , then (1.2) is replaced by . Therefore, the weak contraction becomes a Reich-type one.
Recently, the following generalized result was given by Dutta and Choudhury  combining Theorem 1.1 and Theorem 1.2.
Also, Zhang and Song  given the following generalized version of Theorem 1.2.
In recent years, many results appeared related to fixed point theorem in complete metric spaces endowed with a partial ordering in the literature [10–25]. Most of them are a hybrid of two fundamental principle: Banach contraction theorem and the weakly contractive condition. Indeed, they deal with a monotone (either order-preserving or order-reversing) mapping satisfying, with some restriction, a classical contractive condition, and such that for some , either or , where is a self-map on metric space. The first result in this direction was given by Ran and Reurings [22, Theorem 2.1] who presented its applications to matrix equation. Subsequently, Nieto and Rodŕiguez-López  extended the result of Ran and Reurings  for nondecreasing mappings and applied to obtain a unique solution for a first-order ordinary differential equation with periodic boundary conditions.
Further, Harjani and Sadarangani  proved the ordered version of Theorem 1.2, Amini-Harandi and Emami  proved the ordered version of Rich type fixed point theorem, and Harjani and Sadarangani  proved ordered version of Theorem 1.3.
We will begin with a single map. The following theorem is a generalized version of Theorems 2.1 and 2.2 of Harjani and Sadarangani .
The following corollary is a generalized version of Theorems 1.2 and 1.3 of Harjani and Sadarangani .
and hence, has a unique fixed point. If condition (2.32) fails, it is possible to find examples of functions with more than one fixed point. There exist some examples to illustrate this fact in .
Also, it is easy to see that the other conditions of Theorem 2.1 are satisfied, and so has a fixed point in . Also, note that the weak contractive condition of Theorem 1.3 of this paper and Corollary 2.2 of  is not satisfied.
Now, we will give a common fixed point theorem for two maps. For this, we need the following definition, which is given in .
Note that two weakly increasing mappings need not be nondecreasing. There exist some examples to illustrate this fact in .
Proof of Theorem 2.6.
In this section, we present some applications of previous sections, first we obtain some fixed point theorems for single mapping and pair of mappings satisfying a general contractive condition of integral type in complete partially ordered metric spaces. Second, we give an existence theorem for common solution of two integral equations.
Consider the integral equations (3.14).
Then, the integral equations (3.14) have a unique common solution in .
Then, is a partially ordered set. Also, is a complete metric space. Moreover, for any increasing sequence in converging to , we have for any . Also, for every , there exists which is comparable to and .
The authors thank the referees for their appreciation, valuable comments, and suggestions.
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