Nonlinear generalized cyclic contractions in complete G -metric spaces and applications to integral equations

. In this paper we introduce generalized cyclic contractions in G -metric spaces and establish some ﬁxed point theorems. The presented theorems extend and unify various known ﬁxed point results. Examples are given in the support of these results. Finally, an application to the study of existence and uniqueness of solutions for a class of nonlinear integral equations is given


Introduction
Fixed point theory is an important and actual topic of nonlinear analysis. Moreover, it is well known that the contraction mapping principle, formulated and proved in the PhD dissertation of Banach in 1920, which was published in 1922 is one of the most important theorems in classical functional analysis. During the last five decades, this theorem has undergone various generalizations either by relaxing the condition of contractivity or changing the underlying space or sometimes both. Due to the importance, generalizations of Banach fixed point theorem have been investigated heavily by many authors. This celebrated Banach contraction theorem can be stated as follow.
Theorem 1. (See [1].) Let (X, d) be a complete metric space and f be a mapping of X into itself satisfying: d(f x, f y) kd(x, y) ∀x, y ∈ X, where k is a constant in (0, 1). Then, f has a unique fixed point x * ∈ X.
The simplicity of its proof and the possibilities of attaining the fixed point by using successive approximations let this theorem become a very useful tool in analysis and its applications. There is a great number of generalizations of the Banach contraction principle in the literature (see, e.g., [2] and references cited therein).
It is important to note that the inequality (1) implies continuity of f . A natural question is that whether we can find contractive conditions which will imply existence of fixed point in a complete metric space but will not imply continuity.
The above posed question was answered by Kirk et al. [3] and turned the area of investigation of fixed point by introducing cyclic representations and cyclic contractions, which can be stated as: [3].) Let (X, d) be a metric space. Let p be a positive integer, A 1 , A 2 , . . . , A p be subsets of X, Y = p i=1 A i and f : Y → Y . Then Y is said to be a cyclic representation of Y with respect to f if (i) A i , i = 1, 2, . . . , p, are nonempty closed sets, and Notice that although a contraction is continuous, cyclic contraction need not to be. This is one of the important gains of this notion. Kirk et al. obtained, among others, cyclic versions of the Banach contraction principle [1], of the Boyd and Wong fixed point theorem [4] and of the Caristi fixed point theorem [5]. Following the paper [3], a number of fixed point theorems on cyclic representation of Y with respect to a self-mapping f have appeared (see, e.g., [6][7][8][9][10][11][12][13][14][15][16][17]).
An attempt to extend the investigation specified in [3] to G-metric spaces was done by Aydi in [31]. A new variant of cyclic contractive mapping, named as generalized cyclic contraction mapping satisfying generalized altering distance condition in G-metric spaces, is introduced in the present paper. It is followed by the proof of existence and uniqueness of fixed points for such mappings. The obtained result generalizes and improves many existing theorems in the literature. Some examples are given in the support of our results. In conclusion, we apply accomplished fixed point results for generalized cyclic contraction type mappings to the study of existence and uniqueness of solutions for a class of nonlinear integral equations.

Preliminaries
For more details on the following definitions and results, we refer the reader to [19].
Definition 2. Let X be a nonempty set and let G : X × X × X → R + be a function satisfying the following properties: for all x, y ∈ X with x = y; (G3) G(x, x, y) G(x, y, z) for all x, y, z ∈ X with z = y; (G4) G(x, y, z) = G(x, z, y) = G(y, z, x) = · · · (symmetry in all three variables); (G5) G(x, y, z) G(x, a, a) + G(a, y, z) for all x, y, z, a ∈ X (rectangle inequality).
Then the function G is called a G-metric on X and the pair (X, G) is called a G-metric space.
Note that it can be easily deduced from (G4) and (G5) that holds for all x, y, z ∈ X.
Definition 3. Let (X, G) be a G-metric space and let (x n ) be a sequence of points in X.
(i) A point x ∈ X is said to be the limit of sequence (x n ) if lim n,m→∞ G(x,x n ,x m ) = 0, and one says that the sequence (x n ) is G-convergent to x.
(ii) The sequence (x n ) is said to be a G-Cauchy sequence if, for every ε > 0, there is a positive integer N such that G(x n , x m , x l ) < ε for all n, m, l N ; that is, if G(x n , x m , x l ) → 0 as n, m, l → ∞.
Thus, if x n → x in a G-metric space (X, G), then for any ε > 0, there exists a positive integer N such that G(x, x n , x m ) < ε for all n, m N . It was shown in [19] that the G-metric induces a Hausdorff topology and that the convergence, as described in the above definition, is relative to this topology. The topology being Hausdorff, a sequence can converge to at most one point. Lemma 1. Let (X, G) be a G-metric space, (x n ) a sequence in X and x ∈ X. Then the following are equivalent: (ii) G(x n , x n , x) → 0 as n → ∞; Lemma 2. If (X, G) is a G-metric space, then the following are equivalent: (ii) for every ε > 0, there exists a positive integer N such that G(x n , x m , x m ) < ε for all n, m N .
holds for arbitrary x, y ∈ X. If this is not the case, the space is called non-symmetric.
To every G-metric on the set X a standard metric can be associated by If G is symmetric, then obviously d G (x, y) = 2G(x, y, y), but in the case of a nonsymmetric G-metric, only holds for all x, y ∈ X.
The following are some easy examples of G-metric spaces.
Examples. (a) Let (X, d) be an ordinary metric space. Define G s by and extend G to X × X × X by using the symmetry in the variables. Then it is clear that (X, G) is a non-symmetric G-metric space. Khan et al. [32] introduced the concept of altering distance function: is called an altering distance function if the following properties are satisfied: (i) ψ is continuous and non-decreasing; (ii) ψ(t) = 0 if and only if t = 0.

Main results
First we define the notion of generalized cyclic contraction under an altering distance function in a G-metric space.
The following is the main result of this paper.
Without loss of the generality, we can assume that We shall prove that lim By the assumption, G(x n , x n+1 , x n+1 ) > 0 for all n. From the condition (I), we observe that for all n, there exists Therefore, we have and By (4), we get that (G(x n , x n+1 , x n+1 )) n∈N is a non-increasing sequence. Hence there is r 0 such that lim n→∞ G(x n , x n+1 , x n+1 ) = r.
Therefore ψ(r) = 0 and hence r = 0. Thus, (3) is proved. Next, we show that, (x n ) is a G-Cauchy sequence in X. Suppose the contrary, that is, (x n ) is not G-Cauchy. Then there exists > 0 for which we can find two subsequences (x m(k) ) and (x n(k) ) of (x n ) such that n(k) is the smallest index for which This means that From (6), (7) and (G5), we get < + G(x n(k)−1 , x n(k) , x n(k) ).
Passing to the limit as k → ∞ in the above inequality, and using (11) and (9), we get Similarly, we have Passing to the limit as k → ∞, and using (3) and (9), we obtain Similarly, we have Using (II), we obtain Passing to the limit as k → ∞, using (8)- (14) and the continuity of ψ, we get ψ( ) βψ( ).
Since (X, G) is G-complete, there exists x * ∈ X such that We shall prove that From condition (I), and since x 0 ∈ A 1 , we have (x np ) n 0 ⊆ A 1 . Since A 1 is closed, from (15), we get that x * ∈ A 1 . Again, from condition (I), we have (x np+1 ) n 0 ⊆ A 2 . Since A 2 is closed, from (15), we get that x * ∈ A 2 . Continuing this process, we obtain (16). Now, we shall prove that x * is a fixed point of f . Indeed, from (16), since for all n, there exists i(n) ∈ {1, 2, . . . , p} such that x n ∈ A i(n) , applying (II) with x = y = x * and z = x n , we obtain Passing to the limit as n → ∞ in the above inequality and using (15), we obtain that hence f x * = x * , that is, x * is a fixed point of f . Finally, we prove that x * is the unique fixed point of f . Assume that y * is another fixed point of f , that is, f y * = y * . By the condition (I), this implies that y * ∈ p i=1 A i . Then we can apply (II) for x = x * and y = z = y * . We obtain ψ G(f x * , f y * , f y * ) βψ max G(x * , y * , y * ), G(x * , f x * , f x * ), G(y * , f y * , f y * ), Since x * and y * are fixed points of f , we get that βψ G(x * , y * , y * ) .

Examples
Now we present some examples showing how our Theorem 2 can be used. Example 1. Let X = R and G : X × X × X → X be defined by G(x, y, z) = max |x − y|, |x − z|, |y − z| for all x, y, z ∈ X. Clearly, (X, G) is a G-complete G-metric space. Consider the closed subsets A 1 and A 2 of X defined by Define the selfmap f on X by Clearly, we have f (A 1 ) ⊆ A 2 and f (A 2 ) ⊆ A 1 .
Next we show that f satisfies condition (II). Let (x, y, z) ∈ A 1 × A 2 × A 2 with x, y, z = 0. We have for β = 1/4 and ψ(t) = t. Similarly, one can show that the previous inequality holds for (x, y, z) ∈ A 2 × A 1 × A 1 , as well as when some of x, y, z is equal to 0. Hence, all the conditions of Theorem 2 are satisfied (with p = 2), and we deduce that f has a unique fixed point Example 2. Let X = R 3 and let G : X 3 → R + be given as where (x 4 , y 4 , z 4 ) = (x 1 , y 1 , z 1 ). It is easy to see that (X, G) is a G-complete G-metric space. Consider the following closed subsets of X: and the mapping f : Y → Y given by It is clear that Y = A 1 ∪ A 2 ∪ A 3 is a cyclic representation of Y with respect to f . We will check that f satisfies the contraction condition (II). Take ψ(t) = t 2 (which is an altering distance function) and β = 1/4. Let, e.g., (x, y, z) ∈ A 1 × A 2 × A 2 (the other two cases are treated analogously), and let x = (a, 0, 0), y = (0, b, 0), z = (0, c, 0). Without loss of generality, we can assume that b c.
Then f x = (0, a/2, 0), f y = (0, 0, b/4), f z = (0, 0, c/4) and with symmetry in all variables (see [25]). Note that G is non-symmetric since G(a, a, b) = G (a, b, b). Let A 1 = {a, b} and A 2 = {a, c} and consider the mapping f : X → X given by f a = f c = a and f b = c. Obviously, A 1 ∪ A 2 = X is a cyclic representation with respect to f .
We have to check the inequality L(x, y, z) R(x, y, z) in the following cases: 1.
x ∈ {a, b} and y, z ∈ {a, c}. All the conditions of Corollary 1 are fulfilled and f has a unique fixed point a ∈ A 1 ∩ A 2 .

An application to integral equations
In this section we apply Theorem 2 to study the existence and uniqueness of solutions for a class of nonlinear integral equations. We consider the nonlinear integral equation for u, v, w, ∈ X. Then (X, G) is a G-complete metric space.
Consider the self-map f : X → X defined by www.mii.lt/NA Clearly, u is a solution of (17) if and only if u is a fixed point of f . We will prove the existence and uniqueness of the fixed point of f under the following conditions: (II) K(s, ·) is a non-increasing function for any fixed s ∈ [0, 1], that is, x, y ∈ R + , x y =⇒ K(s, x) K(s, y).
Proof. In order to prove the existence of a (unique) fixed point of f we construct the closed subsets A 1 and A 2 of X as follows: and A 2 = u ∈ X: u(t) α(t), t ∈ [a, b] .
We shall prove that f (A 1 ) ⊆ A 2 and f (A 2 ) ⊆ A 1 .
Let u ∈ A 1 , that is, u(s) γ(s) for all s ∈ [a, b].
Since G(t, s) 0 for all t, s ∈ [a, b], we deduce from (II) and (IV) that b a G(t, s)K s, u(s) ds b a G(t, s)K s, γ(s) ds α(t) for all t ∈ [a, b]. Then we have f u ∈ A 2 . Similarly, the other inclusion is proved. Hence, Y = A 1 ∪ A 2 is a cyclic representation of Y with respect to f . Finally we will show that, for each u ∈ A 1 and v, w ∈ A 2 one has G(f u, f v, f w) βG(u, v, w).
To this end, let u ∈ A 1 and (v, w) ∈ A 2 × A 2 . Therefore by (III), we deduce that, for each t ∈ Using the same technique, we can show that the above inequality also holds if we take (u, v, w) ∈ A 2 × A 1 × A 1 . Consequently, by Corollary 1, f has a unique fixed point u * ∈ A 1 ∩ A 2 , that is, u * ∈ C is the unique solution to (17).