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 for a almost ( ) contraction respect to in ordered spaces
* author: Poom poom.kum@kmutt.ac.th
Author
Abstract
In paper, the theorems of points and fixed points two weakly mappings satisfying new condition ordered metric are proved. results extend, and unify of the metrical fixed theorems in literature.
1 and preliminaries
classical Banach principle is of the useful results nonlinear analysis. a metric the full of the contraction principle given by following theorem.
1.1 Let X. d be a metric space T. X X . T satisfies
( T T y ≤ k ( x. ) (1.1)
all x. ∈ X where k [ 0. ) . T has unique fixed .
Due its simplicity usefulness, it become a popular tool solving existence in many of mathematical and its many applications solving nonlinear Then, several studied and it in direction; for see [1 ] and references therein.
Despite these features, Theorem suffers from drawback: the condition (1.1) T to continuous on . It then natural ask if exist weaker conditions which not imply continuity of . In this question answered in by Kannan ], who Theorem 1.1 mappings that not be on X are continuous their fixed see [21
On the hand, Sessa ] introduced notion of commuting mappings, are a of commuting while Jungck ] generalized notion of commutativity by compatible mappings then weakly mappings [24
In 2004, [25 ] the notion a weak mapping which more general a contraction However, in ] Berinde it as almost contraction which is appropriate. Berinde ] proved fixed point for almost in complete spaces. Afterward, authors have this problem obtained significant (see [27 ]). Moreover, [25 ] proved that strict contraction, Kannan [26 and Zamfirescu ] mappings well as large class quasicontractions are almost contractions.
Let T S be self mappings a metric ( X. ) . mapping T said to a S if there k ∈ 0. 1 such that ( T T y ≤ k ( S S y for all y ∈ .
In AlThagafi and [38 ] the following which is generalization of known results.
1.2 ([ ], Theorem
Let E a subset a metric ( X. ) and . T two selfmaps E such T ( ) ⊆ ( E . Suppose S and are weakly . T an S contraction and ( E is complete Then S T have unique common point in .
Recently et al. ] defined class of satisfying condition as follows.
1.3 Let X. d be a space. A T. X X is to satisfy (B) if exist a δ ∈ 0. 1 and some ≥ 0 that
for all y ∈ .
They a fixed theorem for mappings in metric spaces. also discussed almost contraction the class mappings that condition (B) detail.
In year, Ćirić al. [40 defined the class of satisfying an generalized contractive
Definition 1.4 ( X. ) be metric space, let S. X → . A T is an almost contraction if exist δ [ 0. ) and ≥ 0 that
Definition Let ( ≤ ) a partial set. We that x. ∈ X comparable if ≤ y y ≤ holds.
Definition Let ( ≤ ) a partial set. A T. X X is to be if T ≤ T . whenever y ∈ and x y .
1.7 Let X. ≤ be a ordered set. mappings S. X → are said be strictly if S T S and T S T for all ∈ X
In 2004, and Reurings ] proved following result.
1.8 Let X. ≤ be a ordered set that every ⊂ X a lower an upper . Suppose d is complete metric X . T. X X be continuous and mapping . that there δ ∈ 0. 1 such that ( T T y ≤ δ ( x. ) for comparable x. ∈ X If there x 0 X such x 0 T x . then has a fixed point ∈ X
Ćirić et in [40 established fixed and common point theorems are more than Theorem and several results in existing literature the existence a fixed in ordered
In this we introduce new class extends and mappings satisfying almost generalized condition and the result the existence fixed points common fixed in a ordered space. result substantially extends and the main of Ćirić al. [ ], Theorems 2.2, 2.3, Theorem 1.8, several comparable in the literature regarding existence of fixed and common fixed in ordered
2 Fixed theorems for generalized almost ϕ. φ contraction
First introduce the of generalized ( ϕ. ) contraction
Definition 2.1 ( X. ) be partially ordered and let metric d on X A mapping X → is called generalized almost ϕ. φ contraction if exist two ϕ. X [ 0. ) which ( T ) ≤ ( x and φ. → [ ∞ ) that
Theorem Let ( ≤ ) a partially set . let a metric d on X Let T. → X a strictly continuous mapping respect to and a almost ( φ ) contraction mapping If there x 0 X such x 0 T x . then has a fixed point X .
Proof If x 0 x 0 then x is fixed T and finish the Now, we assume that x 0 x 0 that is, 0 T 0 . construct the in X
x n 1 = n + x 0 T x (2.2)
for n ≥ . Since is strictly we have T x T 2 0 ⋯ n x T n 1 x ⋯ . x 1 2 ⋯ n x + 1 , which that the is strictly Note that
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