Let X be a non-empty set and F : X × X → X be a given mapping. An element ( x , y ) ∈ X × X is said to be a coupled fixed point of the mapping F if F ( x , y ) = x and F ( y , x ) = y . In this paper, we consider the case when X is a complete metric space endowed with a partial order. We define generalized Meir–Keeler type functions and we prove some coupled fixed point theorems under a generalized Meir–Keeler contractive condition. Some applications of our obtained results are given. The presented theorems extend and complement the recent fixed point theorems due to Bhaskar and Lakshmikantham [T. Gnana Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (2006) 1379–1393].

Coupled fixed point theorems for a generalized Meir–Keeler contraction in partially ordered metric spaces

We prove that the recent fixed point theorem for contractions of integral type due to Branciari is a corollary of the famous Meir-Keeler fixed point theorem. We also prove that Meir-Keeler contractions of integral type are still Meir-Keeler contractions.

/pdf/meir-keeler-contractions-of-integral-type-are-still-meir-39k1i1yd2z.pdf

Meir-keeler contractions of integral type are still meir-keeler contractions

In this paper, we prove a common fixed point theorem for weakly compatible mappings satisfying an implicit relation. Our theorem generalizes many fixed point theorems.

/pdf/some-fixed-point-theorems-for-weakly-compatible-mappings-4dznugdkls.pdf

Some fixed point theorems for weakly compatible mappings satisfying an implicit relation

Common fixed point theorems for some new generalized contractive type mappings

We prove a common fixed point theorem for two pairs of occasionally weakly compatible mappings satisfying an implicit relation of a new type that involves an altering distance, so generalizing a theorem of Aliouche and Djoudi (Common fixed point theorems for mappings satisfying an implicit relation without decreasing assumption, Hacettepe J. Math. Stat. 36(1), 11–18, 2007). As a consequence, we obtain a fixed point theorem for two pairs of mappings satisfying an implicit relation of integral type, providing a strong generalization to a known result from Kumar, Chugh and Kumar (Fixed point theorems for compatible mappings satisfying a contractive condition of integral type, Soochow J. Math. 33(2), 181–185, 2007).

/pdf/altering-distance-and-common-fixed-points-under-implicit-jtqgtkfuwk.pdf

Altering Distance and Common Fixed Points under Implicit Relations

We analyze the existence of fixed points for mappings defined on
complete metric spaces ( X , d ) satisfying a general contractive
inequality of integral type. This condition is analogous to
Banach-Caccioppoli's one; in short, we study mappings f : X → X for which there exists a real number
 c ∈ ] 0 , 1 [ , such that for each x , y ∈ X we have
 ∫ 0 d ( f x , f y ) φ ( t ) d t ≤ c ∫ 0 d ( x , y ) φ ( t ) d t , where φ : [ 0 , + ∞ [ → [ 0 , + ∞ ] is a Lebesgue-integrable mapping which is summable on each compact
subset of [ 0 , + ∞ [ , nonnegative and such that for each
 0$" id="E8" xmlns:mml="http://www.w3.org/1998/Math/MathML"> e > 0 , 0$" id="E9" xmlns:mml="http://www.w3.org/1998/Math/MathML"> ∫ 0 e φ ( t ) d t > 0 .

/pdf/a-fixed-point-theorem-for-mappings-satisfying-a-general-2hxxsx4vvs.pdf

A fixed point theorem for mappings satisfying a general contractive condition of integral type

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A fixed point theorem for mappings satisfying a general contractive condition of integral type