A New Common Fixed Point Theorem for Three Commuting Mappings

In the present paper, we propose a common fixed point theorem for three commuting mappings via a new contractive condition which generalizes fixed point theorems of Darbo, Hajji and Aghajani et al. An application is also given to illustrate our main result. Moreover, several consequences are derived, which are generalizations of Darbo’s fixed point theorem and a Hajji’s result.


Introduction and Preliminaries
Schauder's fixed point theorem [1] plays a crucial role in nonlinear analysis. Namely, Schauder [1] has proved that if a self-mapping T is continuous on compact and convex subset of Banach spaces, then T has at least one fixed point. In 1955, Darbo [2] has generalized the classical Schauder's fixed point theorem for α-set contraction that is, such that α(T(A)) ≤ kα(A), with k ∈ [0, 1), on a closed, bounded and convex subsets of Banach spaces. Since then, many interesting works have appeared. For example, in 1967, Sadovskii [3] proved the fixed point property for condensing functions on a closed, bounded and convex subset of Banach spaces, that is, those satisfying α(T(A)) < α(A), with α(A) = 0.
It should be noted that any α-set contraction is a condensing function, but the converse is not true in general (see Reference [4]). In 2007, Hajji and Hanebaly [5] have extended the above contractive conditions and show the existence of a common fixed point for commuting mappings satisfying α(T(A)) ≤ k sup i∈I (α(S i (A))), α(T(A)) < k sup i∈I (α(S i (A)), α(A)), on a closed, bounded and convex subset Ω of a locally convex space. Here, S i and T are continuous functions from Ω into itself, with S i are affine or linear. In 2013, Hajji [6] established a common fixed point theorems for commuting mappings verifying which generalize Darbo's and Sadovskii's fixed point theorems. Furthermore, as examples and applications, he studied the existence of common solutions of equations in Banach spaces using the measure of noncompactness. Recently, in Reference [7], we made use of some axioms of measure of noncompactness to establish the following contractive condition σ(H(A)) ≤ ϕ(S(A)) − ϕ(S(conv(T(A)))), giving rise to common fixed point theorem for three commuting and continuous mappings H, S and T on a closed, bounded and convex subset of Banach spaces, with H and S are affine. Here, σ satisfies some properties of the measure of noncompactness while the conditions on ϕ are not needed. For particular choices of ϕ, σ, H and S Darbo's fixed point theorem can be obtained. As illustration, we have provided a concrete example for which both the classical Darbo's theorem and its generalization due to Hajji [6] are not applicable.
The aim of this paper is to prove the existence of a common fixed point for three mappings H, S and T satisfying the following new contraction Our result generalizes the theorems of Darbo [2], Hajji [6], and Aghajani et al. [8].
As an application, we study the existence of common solutions of the following equations under appropriate assumptions on functions S, H, f and ξ. Motivated by contractive conditions investigated in b-metric spaces [9][10][11] and using a measure of noncompactness, we derive from our main theorem some consequences, which are generalizations of Darbo's fixed point theorem [2] and a Hajji's result [6].
The paper is outlined as follows. Section 2 presents the main result with its proof. An application is provided in Section 3. Finally, several consequences on fixed point results are given in Section 4.
We conclude this introductory section by fixing some notations and recalling basic definitions that will be needed in the sequel. Denote by N the set of nonnegative integers and put R + = [0, +∞). Let (X, . ) be a given Banach space. The symbols A and conv(A) stand for the closure and the convex hull of A, respectively. Moreover, we denote by M X the family of all nonempty and bounded subsets of X and by N X its subfamily consisting of all relatively compact sets. Definition 1 ([12]). A mapping µ : M X → R + is called a measure of noncompactness in X if it satisfies the following conditions: is a sequence of closed sets from M X such that A n+1 ⊆ A n for n = 1, 2, · · · , and if lim n→+∞ µ(A n ) = 0, then the set A ∞ = ∩ +∞ n=1 A n is nonempty. The family ker µ defined in axiom (i) is called the kernel of the measure of noncompactness.

Main Result
In this section, we present and prove our main result on a common fixed point for three commuting operators. We also deduce from the obtained result a corollary which belongs to the classical metric fixed point theory.

Theorem 1.
Let Ω be a nonempty, bounded, closed and convex subset of a Banach space X and let T, S and H be three continuous and commuting mappings from Ω into itself. Assume that the following conditions are satisfied (a) H is affine. (b) For any nonempty subset A of Ω, we have where σ, ϕ : P(Ω) → R + are mappings such that σ satisfies properties (2) If S is affine, then T, S and H have a common fixed point in Ω.

Proof.
(1) Consider the sequence {Ω n } defined as Define ω n = ϕ(Ω n ). From inequality (1), we have It implies that Hence, {ω n } is a non-increasing sequence of positive real numbers, so it converges to some ω ≥ 0 as n tends to +infinity. Using inequality (1) again, we get The rest of the proof needs to show that the sequence {Ω n } is nested. Indeed, for n = 1, we have Ω 1 ⊆ Ω 0 . Suppose that Ω n ⊆ Ω n−1 is true for some n ≥ 1. Then, By induction, we get Ω n ⊆ Ω n−1 for every n ≥ 1. It follows that H(Ω n ) ⊆ H(Ω n−1 ) for every n ≥ 1.
Passing to the limit, we get σ(Ω ∞ ) = 0, which together with property (i) of Definition 1 imply that Ω ∞ = Ω ∞ is compact and convex since H is affine. Note also that ST(Ω n ) ⊆ Ω n . Indeed, For n = 1, we have Assuming now that S(Ω n ) ⊆ Ω n is true for some n ≥ 1. Then By induction, we obtain S(Ω n ) ⊆ Ω n . Similarly as for S, we can prove H(Ω n ) ⊆ Ω n . So we get

ξ(σ(H(A))) ≤ ϕ(A) − ϕ(conv(ST(A))), for every A ⊆ E.
Then by part (1), the mapping S has a fixed point in E and therefore S and H have a common fixed point. In a similar way, we can show that T has a fixed point in F = {x ∈ Ω : S(x) = H(x) = x}. Thus, S, H and T have a common fixed point.

Remark 1.
By letting H and ξ be the identity mappings, and taking σ = µ and ϕ = ( 1 1−k )µ, where µ is a measure of nonocompactness and k ∈ [0, 1), one can deduce Hajji's fixed point theorem [6] and when we take furthermore S the identity mapping, we obtain Darbo's fixed point theorem [2].
Taking H and S the identity mappings, σ = µ and ϕ = ψ•µ, in Theorem 1, we obtain the following result due to Aghajani et al. [8].

Theorem 2.
Let Ω be a nonempty, bounded, closed and convex subset of a Banach space X and let T : Ω → Ω be a continuous mapping such that

Corollary 1.
Let Ω be a nonempty, bounded, closed and convex subset of a Banach space X and let T, S and H be commuting and continuous mappings from Ω into itself such that (a) H and S are affine.

Consequences
In this section, we establish several consequences of our main result.

Proof. Taking H and ξ as the identity functions and ϕ(A) = µ(A)
1−
The above result gives rise to two corollaries, which are also generalizations of the both theorems due to Darbo [2] and Hajji [6].

Corollary 3.
Let Ω be a nonempty, bounded, closed and convex subset of a Banach space X and let T, S : Ω → Ω are continuous mappings such that where µ is a measure of noncompactness defined in X and θ : (0, +∞) → (0, +∞) is function such that θ(t) t is non-increasing. Then the set {u ∈ Ω : T(u) = S(u) = u} is nonempty.
Proof. Let η(t) = t − θ(t), for each t > 0. Then η(t) < t, for each t > 0 and η(t) t is non-decreasing. Thus, the result is obtained by making use of Corollary 2.