Research Article | Open Access
Safeer Hussain Khan, Mujahid Abbas, Talat Nazir, "Fixed Point Results for Generalized Chatterjea Type Contractive Conditions in Partially Ordered -Metric Spaces", The Scientific World Journal, vol. 2014, Article ID 341751, 7 pages, 2014. https://doi.org/10.1155/2014/341751
Fixed Point Results for Generalized Chatterjea Type Contractive Conditions in Partially Ordered -Metric Spaces
In the framework of ordered -metric spaces, fixed points of maps that satisfy the generalized -Chatterjea type contractive conditions are obtained. The results presented in the paper generalize and extend several well known comparable results in the literature.
1. Introduction and Preliminaries
The study of fixed points of mappings satisfying certain contractive conditions has been at the center of rigorous research activity. Mustafa and Sims  generalized the concept of a metric space. Based on the notion of generalized metric spaces, Mustafa et al. [1–5] obtained some fixed point theorems for mappings satisfying different contractive conditions. Abbas and Rhoades  initiated the study of a common fixed point theory in generalized metric spaces. Abbas et al.  and Chugh et al.  obtained some fixed point results for maps satisfying property in -metric spaces. Recently, Shatanawi  proved some fixed point results for self mappings in a complete-metric space under some contractive conditions related to a nondecreasing map with for all . Recently, Saadati et al.  proved some fixed point results for contractive mappings in partially ordered -metric spaces.
Ran and Reurings  extended Banach contraction principle in partially ordered metric spaces with some applications to linear and nonlinear matrix equations, while Nieto and Rodríguez-López  extended the result of Ran and Reurings and applied their main result to obtain a unique solution for a first order ordinary differential equation with periodic boundary conditions. Bhaskar and Lakshmikantham  introduced the concept of mixed monotone mappings and obtained some coupled fixed point results. Also, they applied their results to a first order differential equation with periodic boundary conditions.
Alber and Guerre-Delabriere  introduced the concept of weakly contractive mappings and proved the existence of fixed points of such mappings in Hilbert spaces. Thereafter, in 2001, Rhoades  proved the fixed point theorem which is one of the generalizations of Banach’s contraction mapping principle. Weakly contractive mappings are closely related to the mappings of Boyd and Wong  and of Reich types . Recently, Dorić  proved a common fixed point theorem for generalized -weakly contractive mappings. Fixed point problems involving weak contractions and mappings satisfying weak contractive type inequalities have been studied by many authors (see [8, 14, 15, 18–21] and references cited therein).
In this paper, we generalize the Chatterjea type contraction mappings to generalized -Chatterjea type contraction mappings and derive some fixed point results for single-valued mappings in ordered generalized metric spaces.
Consistent with Mustafa and Sims , the following definitions and results will be needed in the sequel.
Definition 1. Letbe a nonempty set. Suppose that a mappingsatisfies(G1) if ;(G2) for all , with ;(G3) for all , with ;(G4) (symmetry in all three variables);(G5) for all .Then, is called a -metric on and is called a -metric space.
Definition 2. A sequencein a-metric spaceis(i)a-Cauchy sequence if, for every , there is a natural number such that for all ,(ii)a-Convergent sequence if, for any , there is an and an , such that for all .A -metric space onis said to be-complete if every-Cauchy sequence inis-convergent in. It is known that -converges to if and only if as .
Proposition 3 (see ). Let be a -metric space. Then, the following are equivalent.(1)The sequence is -convergent to .(2) as . (3) as . (4) as
Proposition 4 (see ). Let be a -metric space. Then, the following are equivalent.(1)The sequence is -Cauchy.(2)For every , there exists , such that for all , ; that is, if as .
Definition 5. A -metric onis said to be symmetric if for all .
Proposition 6. Every -metric on defines a metriconby For a symmetric -metric space, one obtains However, ifis not symmetric, then the following inequality holds: First, we recall some basic definitions and notations.
Definition 7. We define two classes of mappings as follows: is continuous and nondecreasing with if and only if and is lower semi-continuous with if and only if .
Definition 8. An ordered partial -metric space is said to have a sequential limit comparison property if for every nondecreasing sequence (nonincreasing sequence) insuch that as implies that , respectively.
2. Fixed Point Results
In this section, we obtain fixed point results for mappings satisfying generalized-Chatterjea type contractive conditions on partially ordered complete generalized metric space. We start with the following result.
Theorem 9. Let be a partially ordered set and be a nondecreasing self mapping on a complete -metric space satisfying where , with for all with . Suppose that there exists with . If is continuous or a sequential limit comparison property, then has a fixed point in .
Proof. If , there is nothing to prove. Suppose that . Since and is nondecreasing, we have Define a sequenceby so that . We may assume that for every. If not, then for some and becomes a fixed point of . Using (4), we obtain where If we take for some , it follows that , a contradiction. Therefore, for all , so that . Now is a decreasing sequence, so there exists such that . This gives . By lower semicontinuity of , We claim that . Taking the upper limits as on both sides of we have This implies and we conclude that
It follows that is a -Cauchy sequence and by -completeness of , there exists such that -converges to as . If is continuous, then it is clear that . Next, if has a sequential limit comparison property, then we have for all . From (4), we have where This implies that . Thus, from (32), we obtain This givesso thatand, hence, .
Corollary 10. Letbe a partially ordered set andbe a nondecreasing self mapping on a complete -metric spacesatisfying where , with for all with with for all with . Suppose that there exists with . If is continuous or has a sequential limit comparison property, then has a fixed point in .
Now we give an example to illustrate above result.
Example 11. Let and be a -metric on . Define by We take and for all , .
Corollary 12. Let be a partially ordered set and be a nondecreasing self mapping on a complete -metric space satisfying where, for all with . Suppose that there exists with . If is continuous or has a sequential limit comparison property, then has a fixed point in .
Corollary 13. Let be a partially ordered set and be a nondecreasing self mapping on a complete -metric space satisfying where , for all with , where for with . Suppose that there existswith. Ifis continuous orhas a sequential limit comparison property, then has a fixed point in.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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