Journal of Inequalities and Applications  Full text  Fixed point theorems for a generalized almost $(phi,varphi)$ contraction with respect to S in ordered metric spaces
Fixed point theorems for a generalized almost ( . ) contraction with respect to S in ordered metric spaces
* Corresponding author: Poom Kumam poom.kum@kmutt.ac.th
Author Affiliations
Abstract
In this paper, the existence theorems of fixed points and common fixed points for two weakly increasing mappings satisfying a new condition in ordered metric spaces are proved. Our results extend, generalize and unify most of the fundamental metrical fixed point theorems in the literature.
1 Introduction and preliminaries
The classical Banach contraction principle is one of the most useful results in nonlinear analysis. In a metric space, the full statement of the Banach contraction principle is given by the following theorem.
Theorem 1.1 Let ( X. d ) be a complete metric space and T. X → X . If T satisfies
d ( T x. T y ) ≤ k d ( x. y ) (1.1)
for all x. y ∈ X . where k ∈ [ 0. 1 ) . then T has a unique fixed point .
Due to its simplicity and usefulness, it has become a very popular tool in solving existence problems in many branches of mathematical analysis and its has many applications in solving nonlinear equations. Then, several authors studied and extended it in many direction; for example, see [1 19 ] and the references therein.
Despite these important features, Theorem 1.1 suffers from one drawback: the contractive condition (1.1) forces T to be continuous on X . It was then natural to ask if there exist weaker contractive conditions which do not imply the continuity of T . In 1968, this question was answered in confirmation by Kannan [20 ], who extended Theorem 1.1 to mappings that need not be continuous on X (but are continuous at their fixed point, see [21 ]).
On the other hand, Sessa [22 ] introduced the notion of weakly commuting mappings, which are a generalization of commuting mappings, while Jungck [23 ] generalized the notion of weak commutativity by introducing compatible mappings and then weakly compatible mappings [24 ].
In 2004, Berinde [25 ] defined the notion of a weak contraction mapping which is more general than a contraction mapping. However, in [26 ] Berinde renamed it as an almost contraction mapping, which is more appropriate. Berinde [25 ] proved some fixed point theorems for almost contractions in complete metric spaces. Afterward, many authors have studied this problem and obtained significant results (see [27 36 ]). Moreover, in [25 ] Berinde proved that any strict contraction, the Kannan [26 ] and Zamfirescu [37 ] mappings as well as a large class of quasicontractions are all almost contractions.
Let T and S be two self mappings in a metric space ( X. d ) . The mapping T is said to be a S contraction if there exists k ∈ [ 0. 1 ) such that d ( T x. T y ) ≤ k d ( S x. S y ) for all x. y ∈ X .
In 2006, AlThagafi and Shahzad [38 ] proved the following theorem which is a generalization of many known results.
Theorem 1.2 ([ [38 ], Theorem 2.1])
Let E be a subset of a metric space ( X. d ) and S . T be two selfmaps of E such that T ( E ) ⊆ S ( E ) . Suppose that S and T are weakly compatible . T is an S — contraction and S ( E ) is complete . Then S and T have a unique common fixed point in E .
Recently Babu et al. [39 ] defined the class of mappings satisfying condition (B) as follows.
Definition 1.3 Let ( X. d ) be a metric space. A mapping T. X → X is said to satisfy condition (B) if there exist a constant δ ∈ ( 0. 1 ) and some L ≥ 0 such that
for all x. y ∈ X .
They proved a fixed point theorem for such mappings in complete metric spaces. They also discussed quasicontraction, almost contraction and the class of mappings that satisfy condition (B) in detail.
In recent year, Ćirić et al. [40 ] defined the following class of mappings satisfying an almost generalized contractive condition.
Definition 1.4 Let ( X. d ) be a metric space, and let S. T. X → X . A mapping T is called an almost generalized contraction if there exist δ ∈ [ 0. 1 ) and L ≥ 0 such that
Definition 1.5 Let ( X. ≤ ) be a partial ordered set. We say that x. y ∈ X are comparable if x ≤ y or y ≤ x holds.
Definition 1.6 Let ( X. ≤ ) be a partial ordered set. A mapping T. X → X is said to be nondecreasing if T x ≤ T y . whenever x. y ∈ X and x ≤ y .
Definition 1.7 Let ( X. ≤ ) be a partial ordered set. Two mappings S. T. X → X are said to be strictly increasing if S x T S x and T x S T x for all x ∈ X .
In 2004, Ran and Reurings [41 ] proved the following result.
Theorem 1.8 Let ( X. ≤ ) be a partially ordered set such that every pair ⊂ X has a lower and an upper bound . Suppose that d is a complete metric on X . Let T. X → X be a continuous and monotone mapping . Suppose that there exists δ ∈ [ 0. 1 ) such that d ( T x. T y ) ≤ δ d ( x. y ) for all comparable x. y ∈ X . If there exists x 0 ∈ X such that x 0 ≤ T x 0 . then T has a unique fixed point p ∈ X .
Ćirić et al. in [40 ] established fixed point and common fixed point theorems which are more general than Theorem 1.8 and several comparable results in the existing literature regarding the existence of a fixed point in ordered spaces.
In this paper, we introduce a new class which extends and unifies mappings satisfying the almost generalized contractive condition and establish the result on the existence of fixed points and common fixed points in a complete ordered space. This result substantially generalizes, extends and unifies the main results of Ćirić et al. [ [40 ], Theorems 2.1, 2.2, 2.3, 2.6], Theorem 1.8, and several comparable results in the existing literature regarding the existence of a fixed and a common fixed point in ordered spaces.
2 Fixed point theorems for a generalized almost ( ϕ. φ ) contraction
First we introduce the notion of generalized almost ( ϕ. φ ) contraction mappings.
Definition 2.1 Let ( X. ≤ ) be a partially ordered set, and let a metric d exist on X . A mapping T. X → X is called a generalized almost ( ϕ. φ ) contraction if there exist two mappings ϕ. X → [ 0. 1 ) which ϕ ( T x ) ≤ ϕ ( x ) and φ. X → [ 0. ∞ ) such that
Theorem 2.2 Let ( X. ≤ ) be a partially ordered set . and let a complete metric d exist on X . Let T. X → X be a strictly increasing continuous mapping with respect to ≤ and a generalized almost ( ϕ. φ ) — contraction mapping . If there exists x 0 ∈ X such that x 0 ≤ T x 0 . then T has a unique fixed point in X .
Proof If T x 0 = x 0 . then x 0 is fixed of T and we finish the proof. Now, we may assume that T x 0 ≠ x 0 . that is, x 0 T x 0 . We construct the sequence in X by
x n + 1 = T n + 1 x 0 = T x n (2.2)
for all n ≥ 0 . Since T is strictly increasing, we have that T x 0 T 2 x 0 ⋯ T n x 0 T n + 1 x 0 ⋯ . Thus, x 1 x 2 ⋯ x n x n + 1 ⋯ , which implies that the sequence is strictly increasing. Note that
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