Abstract. We establish a **common** **coupled** **fixed** **point** theorem for hybrid pair of map- pings under weak ψ − ϕ contraction on a non-complete metric space, which is not partially ordered. It is to be noted that to find **coupled** coincidence **point**, we do not em- ploy the condition of continuity of any mapping involved therein. Moreover, an example and an application to integral equations are given here to illustrate the usability of the obtained results. We improve, extend, and generalize several known results.

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Theorem 2.9. Let (X, d) be a complex valued metric space, F : X × X → X and g : X → X be two mappings which satisfy all the conditions of Theorem 2.5. If F and g are w−compatible, then F and g have unique **common** **coupled** **fixed** **point**. Moreover, **common** **fixed** **point** of F and g has the form (u, u) for some u ∈ X.

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In this paper, we establish two **common** **coupled** **fixed** **point** theorems for two hybrid pairs of mappings satisfying an implicit relation under the conditions of weakly commuta- tivity and w−compatibility on a complete metric space, which is not partially ordered. To prove our theorems we do not use condition of continuity of any mapping. We improve, extend and generalize the result of Sedghi et al. [29].

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The notion of a **coupled** **fixed** **point** was introduced and studied by Guo and Lakshmikantham [18] and Bhaskar and Lakshmikantham [14]. In subsequent papers several authors proved various **coupled** and **common** **coupled** **fixed** **point** theorems in (partially ordered) metric spaces (see, e.g., [19, 24, 25, 37]). These results were applied for investigation of solutions of differential and integral equations. **Fixed** **point** and **coupled** **fixed** **point** results in partially ordered G-metric spaces were obtained in, e.g., [8, 9, 15, 16, 26, 34, 38, 39, 40].

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In this paper, we obtain a unique **common** **coupled** ﬁxed **point** theorem by using ( ψ , α , β )-contraction in ordered partial metric spaces. We give an application to integral equations as well as homotopy theory. Also we furnish an example which supports our theorem.

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Then there exist x ∈ X such that x Fx, x, that is, F admits a unique **fixed** **point** in X. Let φt kt, where 0 < k < 1, the following by Lemma 1, we get the following. Corollary 2 see 6. Let a ∗ b ≥ ab for all a, b ∈ 0, 1 and X, M, ∗ be a complete fuzzy metric space such that M has n-property. Let F : X × X → X and g : X → X be two functions such that

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Bhaskar and Lakshmikantham [], Lakshmikantham and Ćirić [] discussed the mixed monotone mappings and gave some **coupled** ﬁxed **point** theorems, which can be used to discuss the existence and uniqueness of solution for a periodic boundary value problem. Sedghiet al. [] gave a **coupled** ﬁxed **point** theorem for contractions in fuzzy metric spaces, and Jin-xuan Fang [] gave some **common** ﬁxed **point** theorems for compatible and weakly compatible φ-contractions mappings in Menger probabilistic metric spaces. Xin-Qi Hu [] proved a **common** ﬁxed **point** theorem for mappings under ϕ -contractive conditions in fuzzy metric spaces. Many authors [–] proved ﬁxed **point** theorems in (intuitionistic) fuzzy metric spaces or probabilistic metric spaces.

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In this manuscript, we generalize, improve, enrich and extend the above **coupled** ﬁxed **point** results. It is worth mentioning that our results do not rely on the continuity of map- pings involved therein. We also state some examples to illustrate our results. This paper can be considered as a continuation of the remarkable works of Abbas et al. [, ] and Sabetghadam et al. [].

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Abstract. In this paper, we prove some **coupled** **fixed** **point** theorems, which generalized the result of Hu [6], Hu et al.[7] from fuzzy metric spaces to intuitionistic fuzzy metric spaces for semi-compatible mappings, which is weaker form of compatible mappings. Keywords: **Coupled** **fixed** **point**, intuitionistic fuzzy metric space, **coupled** **common** **fixed** **point**, semi-compatible maps.

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One of the main pillar in the study of **fixed** **point** theory is Banach Contraction priciple which was done by Banach in 1922. **Fixed** In 2011 Akbar Azam et al., 2011 introduced the concept of complex valued metric space. The concept of **coupled** **fixed** **point** was first introduced by Bhaskar and Laxikantham in 2006. Recently some researchers prove some **coupled** **fixed** **point** theorems in complex valued metric space in (Kang et al., 2013;

In this work, we show that the mixed g-monotone property in **common** **coupled** ﬁxed **point** theorems in ordered cone metric spaces can be replaced by another property due to Ðorić et al. []. This property is automatically satisﬁed in the case of a totally ordered space. Therefore, these results can be applied in a much wider class of problems. Our results generalize and extend many well-known comparable results in the literature. An illustrative example is presented in this work when our results can be used in proving the existence of a **common** **coupled** ﬁxed **point**, while the results of Nashine et al. [] cannot.

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As a generalization of b-metric spaces and cone metric spaces, Hussain and Shah [] announced cone b-metric spaces, which was in . They built up some topological prop- erties in such spaces and upgraded some latest results about KKM mappings in the setting of a cone b-metric space. Hussain and Shah [] have done initial work that stimulated many authors to prove ﬁxed **point** theorems, as well as **common** ﬁxed **point** theorems for two or more mappings on cone b-metric spaces (see [–] and the references therein).

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Corollary 4. Let F : X × X ® X and g : X ® X be two mappings which satisfy all the conditions of Corollary 2. If F and g are w-compatible, then F and g have a unique **common** **coupled** **fixed** **point**. Moreover, the **common** **fixed** **point** of F and g is of the form (u, u) for some u Î X.

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Bhaskar and Lakshimkantham proved the existence of **coupled** **fixed** **point** for a single valued mapping under weak contractive conditions and as an application they proved the existence of a unique solution of a boundary value problem associated with a first order ordinary differential equation. Recently, Lakshmikantham and Ćirić obtained a **coupled** coincidence and **coupled** **common** **fixed** **point** of two single valued maps. In this article, we extend these concepts to multi-valued mappings and obtain **coupled** coincidence points and **common** **coupled** **fixed** **point** theorems involving hybrid pair of single valued and multi-valued maps satisfying generalized contractive conditions in the frame work of a complete metric space. Two examples are presented to support our results.

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Bhaskar and Lakshmikantham [12] established some **coupled** **fixed** **point** theorems and applied these to study the existence and uniqueness of solution for periodic boundary value problems. Lakshmikantham and Ciric [24] proved **coupled** coincidence and **common** **coupled** **fixed** **point** theorems for nonlinear contractive mappings in partially ordered complete metric spaces and extended the results of Bhaskar and Lakshmikantham [12].

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Consequently, (u, u) is a **coupled** coincidence **point** of F and g, and therefore (gu, gu) is a **coupled** **point** of coincidence of F and g, and by its uniqueness, we get gu = gx. Thus, we obtain F(u, u) = gu = u. Therefore, (u, u) is the unique **common** **coupled** **fixed** **point** of F and g. This completes the proof.

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Using the concept of a mixed g-monotone mapping, we prove some **coupled** coincidence and **coupled** **common** **fixed** **point** theorems for nonlinear contractive mappings in partially ordered complete quasi-metric spaces with a Q-function q. The presented theorems are generalizations of the recent **coupled** **fixed** **point** theorems due to Bhaskar and Lakshmikantham 2006, Lakshmikantham and ´ Ciri´c 2009 and many others.

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Theorem 3.3 . In addition to the hypotheses of Theorem 3.1., suppose that, for any (x, y), (x*, y*) Î X × X, there exists (u, v) Î X × X such that (F(u, v), F(v, u)) is comparable with (F(x, y), F(y, x)) and (F(x*, y*), F(y*, x*)). Then F and g have a unique **coupled** **common** **fixed** **point**, that is, there exists a unique (x, y) Î X × X such that x = gx = F (x, y) and y = gy = F(y, x).

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Samet and Vetro [26] introduced the concept of **coupled** …xed **point** for mul- tivalued mapping and later several authors proved existence of **coupled** …xed points for multivalued mappings under di¤erent conditions. Subsequently, many results in this direction were given (see, e.g., [14; 15; 16; 18; 21; 26])

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Later on, Lakshmikantham and Ciríc [8] proved coupled coincidence and coupled common fixed point theorems for nonlinear contractive mappings in partially ordered complete metric spaces..[r]