Coupled fixed point theorems for mappings satisfies contractive condition in cone metric space

Introduction

Fixed point theory is a well-known and significant area in mathematics, with a wide range of applications. In 2007, Huang and Zhang [9] introduced the concept of cone metric spaces as a generalization of traditional metric spaces. They proved the existence of a unique fixed point for contractive mappings in complete cone metric spaces. Dajun Guo and V. Lakshmikantham [2] established existence theorems for coupled fixed points for both continuous and discontinuous operators, with applications to initial value problems of ordinary differential equations with discontinuous right-hand sides. Bhaskar and Lakshmikantham [19] further developed the theory by proving the existence of coupled fixed points for mixed monotone mappings in partially ordered metric spaces. In 2008, C. Di Bari [1] presented a common fixed point theorem in cone metric spaces. Later, in 2009 and 2010, I. Altun [6,7] established several common fixed point theorems in cone metric spaces and ordered cone metric spaces. Additionally, M. Arshad [13] and S. Radenović [16] , both in 2009, contributed further by proving common fixed point theorems in cone metric spaces.

Preliminaries

Definition 2.1: [8] Let E be a real Banach Spaces. A subset P of E is called a cone if and only if A). P is closed, non-empty and 𝑝 ≠ 0 𝐵) 𝑎, 𝑏 ∈ 𝑅, 𝑎, 𝑏 ≥ 0 and 𝑥, 𝑦 ∈ 𝑃 imply 𝑎𝑥 + 𝑏𝑦 ∈ 𝑃 𝐶) 𝑃 ∩ (-𝑃) = {0} Given a cone 𝑃 ⊂ 𝐸 we define the partial ordering ≤ with respect to P by 𝑥 ≤ 𝑦 if and only if 𝑦 -𝑥 ∈ 𝑃. We write 𝑥 < 𝑦 to denote that 𝑥 ≤ 𝑦 but ≠ 𝑦, while 𝑥 << 𝑦 will stand for 𝑦 -𝑥 ∈ 𝑖𝑛𝑡. 𝑃. Definition 2.2: [9] Let X be a nonempty set. Suppose the mapping 𝑑: 𝑋 × 𝑋 → 𝐸 satiafies the following condition:  0 < d(x, y) ∀ x, y ∈ X and d(x, y) = 0 ⟺ x = y  d(x, y) = d(y, x), ∀ x, y ∈ X  d(x, y) ≤ d(x, y) + d(x, y), ∀ x, y ∈ X Then d is called a cone metric on X and (𝑋, 𝑑) is called a cone metric space. Definition 2.3: [9] Let (𝑋, 𝑑) be a cone metric space, {𝑥 𝑛 } a sequence in X, {𝑥 𝑛 } is a convergent sequence if there is some 𝑘 ∈ ℕ such that, for all 𝑛 ≥ 𝑘, 𝑑(𝑥 𝑛 , 𝑥) ≪ 𝑐;

Then 𝑥 is called limit of the sequence {𝑥 𝑛 } Definition 2.4: [9] Let (𝑋, 𝑑) be a cone metric space, {𝑥 𝑛 } a sequence in X, {𝑥 𝑛 } is a Cauchy sequence if there is some 𝑘 ∈ ℕ such that, for all 𝑛, 𝑚 ≥ 𝑘, 𝑑(𝑥 𝑛 , 𝑥 𝑚 ) ≪ 𝑐;

Note that: (i) Every convergent sequence in a cone metric space 𝑋 is a Cauchy sequence. (ii) A cone metric space 𝑋 is said to be complete if every Cauchy sequence in 𝑋 is convergent in 𝑋 Bhashkar and Lakshmikantham in [19] introduced the concept of coupled fixed point of a mapping 𝐹: 𝑋 × 𝑋 → 𝑋 and investigated some coupled fixed point theorems in partially ordered sets. They also discussed an application of their result by investigating the existence and uniqueness of solution for a periodic boundary value problem. Sabetghadam et al. in [5] introduced this concept in cone metric spaces. Definition 2.5: [5] Let 𝐹: 𝑋 × 𝑋 → 𝑋 be mapping, an element (𝑥, 𝑦) ∈ 𝑋 × 𝑋 is called a coupled fixed point of mapping 𝐹 if 𝑥 = 𝐹(𝑥, 𝑦) and 𝑦 = 𝐹(𝑦, 𝑥)

Main Results

In this theorem, we extend and unify several well-known comparable results in the literature and results of M. Abbas et al. [12] . Theorem 3.1: Let X a non-empty set, (𝑋, 𝑑) be a Cone metric space with cone P having non empty interior, 𝐹, 𝐺: 𝑋 × 𝑋 → 𝑋 be mapping satisfying the following conditions: 𝑑(𝐺(𝑥, 𝑦), 𝐹(𝑢, 𝑣)) ≤ 𝛼{𝑑(𝑥, 𝑢) + 𝑑(𝑦, 𝑣)} + 𝛽 {𝑑(𝑢, 𝐹(𝑢, 𝑣)) + 𝑑(𝑢, 𝐺(𝑥, 𝑦))} + 𝛾{𝑑(𝑥, 𝐺(𝑥, 𝑦))} ∀ 𝑥, 𝑦, 𝑢, 𝑣 ∈ 𝑋 where 𝛼, 𝛽, 𝛾 𝜖(0,

References

1. Di Bari C, Vetro P. U-pairs and common fixed points in cone metric spaces. Rend Circ Mat Palermo. 2008;57:279-85.
2. Guo D, Lakshmikantham V. Coupled fixed points of nonlinear operation with applications. Nonlinear Anal Theory Methods Appl. 1987;11:623-32.
3. Ilic D, Rakocevic V. Common fixed points for maps on cone metric space. J Math Anal Appl. 2008;341:876-82.
4. Ilic D, Rakocevic V. Quasi-contraction on cone metric space. Appl Math Lett. 2009;22:728-31.
5. Sabetghadam F, Masiha HP, Sanatpour AH. Some coupled fixed point theorems in cone metric space. Fixed Point Theory Appl. 2009;2009:1254260.
6. Altun I, Damjanovic B, Djoric D. Fixed point and common fixed point theorems on ordered cone metric spaces. Appl Math Lett. 2010;23:310-6.
7. Altun I, Durmaz G. Some fixed point results in cone metric spaces. Rend Circ Mat Palermo. 2009;58:319-25.
8. Deimling K. Nonlinear Functional Analysis. Berlin: Springer-Verlag; 1985.
9. Huang LG, Zhang X. Cone metric spaces and fixed point theorems of contractive mappings. J Math Anal Appl. 2007;332:1467-75.
10. Abbas M, Rhoades BE. Fixed and periodic point results in cone metric spaces. Appl Math Lett. 2009;22:511-5.
11. Abbas M, Rhoades BE, Nazir T. Common fixed points for four maps in cone metric spaces. Appl Math Comput. 2010;216:80-6.
12. Abbas M, Khan MA, Radenovic S. Common coupled fixed point theorems in cone metric spaces for w-compatible mappings. Appl Math Comput. 2010;217:195-202.
13. Arshad M, Azam A, Vetro P. Some common fixed point results in cone metric spaces. Fixed Point Theory Appl. 2009;2009:493965.
14. Jankovic S, Kadelburg Z, Radenovic S, Rhoades BE. Assad-Kirk-type fixed point theorems for a pair of nonself mappings on cone metric spaces. Fixed Point Theory Appl. 2009;2009:761086.
15. Radenovic S, Rhoades BE. Fixed point theorem for two non-self mappings in cone metric spaces. Comput Math Appl. 2009;57:1701-7.
16. Radenovic S. Common fixed points under contractive conditions in cone metric spaces. Comput Math Appl. 2009;58(6):1273-8.
17. Rezapour S, Hamlbarani R. Some note on the paper "Cone metric spaces and fixed point theorems of contractive mappings." J Math Anal Appl. 2008;345:719-24.
18. Rezapour Sh, Haghi RH, Shahzad N. Some notes on fixed points of quasi-contraction maps. Appl Math Lett. 2010;23:498502.
19. Bhashkar TG, Lakshmikantham V. Fixed point theorems in partially ordered cone metric spaces and applications. Nonlinear Anal Theory Methods Appl. 2006;65(7):825-32.