Fixed Point Results for Generalized Chatterjea Type Contractive Conditions in Partially Ordered 𝐺-Metric Spaces

Abstract

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 [1] 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 [6] initiated the study of a common fixed point theory in generalized metric spaces. Abbas et al. [7] and Chugh et al. [8] obtained some fixed point results for maps satisfying property in -metric spaces. Recently, Shatanawi [9] 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. [10] proved some fixed point results for contractive mappings in partially ordered -metric spaces.

Ran and Reurings [11] extended Banach contraction principle in partially ordered metric spaces with some applications to linear and nonlinear matrix equations, while Nieto and Rodríguez-López [12] 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 [13] 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 [14] introduced the concept of weakly contractive mappings and proved the existence of fixed points of such mappings in Hilbert spaces. Thereafter, in 2001, Rhoades [15] 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 [16] and of Reich types [17]. Recently, Dorić [18] 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 [1], 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 [1]). 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 [1]). 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.

Let be a metric space. A map is said to be(a)Kannan type (see [22]) if there exists a such thatfor all;(b)Chatterjea type [20] if there exists a such that for all .

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

Next, we show that is a -Cauchy sequence in. If not, then there exist and integers and with such that A joint effect of (13) and (14) on yields Also, implies that .

On the other hand, combined with (13) and (16) results in so that Now, gives that , and implies by (13) and (19) that Hence, Also, from (16) and we obtain .

But from together with (13) and (16), we get . Thus, This gives and so Also, From (4), we obtain which on taking the upper limit as implies that a contradiction as.

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 , .

Now, for all with , we have So that Thus, (35) is satisfied with , where . Hence, the conditions of Corollary 10 are satisfied and is the fixed point of .

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.