A New Fixed Point Result of Perov Type and Its Application to a Semilinear Operator System

: In this paper, we present a new generalization of the Perov ﬁxed point theorem on vector-valued metric space. Moreover, to show the signiﬁcance of our result, we present both a nontrivial comparative example and an application to a kind of semilinear operator system about the existence of its solution.


Introduction and Preliminaries
The well-known Banach contraction mapping principle plays a crucial role in the functional analysis and ensures the existence and uniqueness of a fixed point on a complete metric space. Many generalizations of this principle have been given either by taking into account more general contractive inequality or by changing the structure of space. In this context Perov [1] has presented this principle in vector-valued metric spaces. Many contributions in this aspect have been obtained (see, for example, Abbas et al. [2], Altun and Olgun [3], Cvetković and Rakočević [4,5], Flip and Petruşel [6], Ilić et al. [7] and Vetro and Radenović [8]). As we can see in [1,9], the results in this aspect can be used to guarantee the existence of solutions of some Cauchy problems. In order to talk about the contribution of Perov, we need to remember the following notations: Let R m be the set of m × 1 real matrices, X be a nonempty set and d : X × X → R m be a function. Then d is said to be a vector-valued metric and in this case (X, d) is said to be vector-valued metric space, if the following properties are satisfied: For all x, y, z ∈ X where 0 is the zero m × 1 matrix and is the coordinate-wise ordering on R m , that is, for α = For the rest of this paper α β and β α will be the same and α ≺ β will be α i < β i for each i ∈ {1, 2, . . . , m}. Moreover, we denote by R + the set of non-negative real numbers, by R m + the set of m × 1 real matrices with non-negative elements, by M m m (R + ) the set of all m × m matrices with non-negative elements, by θ the zero m × m matrix, by I the identity m × m matrix. If M ∈ M m m (R + ), then the symbol M T stands for the transpose matrix of M. For the sake of simplicity, we will make an identification between row and column vectors in R m .
Notice that the convergence and Cauchyness of a sequence and completeness of the space in a vector-valued metric space are defined in a similar manner as in the usual metric space. Let M ∈ M m m (R + ), then M is said to be convergent to zero if and only if M n → θ as n → ∞ (See [10]).
Theorem 1 ([10]). Let M ∈ M m m (R + ). Then the following conditions are equivalent: 1. M is convergent to zero, 2. the eigenvalues of M are in the open unit disc, that is, |λ| < 1 for every λ ∈ C with det(M − λI) = 0, 3. the matrix I − M is nonsingular and We can find some examples of matrices convergent to zero in the literature.
with a + b < 1, converges to zero.    for all x 0 ∈ X, the sequence of successive approximations {x n } defined by x n = T n x 0 is convergent to z, 3. one has the following estimation:
Define a mapping T : X → X by Now we claim that T is not a Perov contraction. Assume the contrary. Then there exists a matrix M ∈ M 2 2 (R + ) such that M is convergent to zero and holds for all x, y ∈ X. Let M = a b c d , then from Equation (1), for x = x n and y = 0, we get Therefore, since lim n→∞ x n+1 x n = 1, this last inequality implies that a + b ≥ 1 and c + d ≥ 1. On the other hand one of the eigenvalues of M is and the by routine calculation we can see that λ ≥ 1. Therefore, since one of the eigenvalues of M does not lie in the open unit disc, then from Theorem 1, M does not converge to zero. This is a contradiction. Hence T is not a Perov contraction.
In this paper, by considering the recent technique of Jleli and Samet [11], we present a new generalization of the Perov fixed point theorem. This technique is known as θ-contraction in the literature and there are many studies using this technique (See for example [12][13][14]). Let Θ : R m +0 → R m +1 be a function, where R m +j is the set of m × 1 real matrices with every element being greater than j. For the sake of completeness, we will consider the following conditions: (Θ3) There exist r ∈ (0, 1) and l ∈ (0, ∞] such that lim α i →0 + We denote by Ξ m the set of all functions Θ satisfying (Θ1)-(Θ3).
By considering the class Ξ m , we introduce the concept of Perov type Θ-contraction as follows: Here we use the notation for all x, y ∈ X with d(Tx, Ty) 0, then T is called a Perov type Θ-contraction.
then Equation (2) By considering some different function Θ belonging to Ξ m in Equation (2), we can obtain new type contractions on vector-valued metric spaces.

Main Result
Here we present our main result. Theorem 3. Let (X, d) be a complete vector-valued metric space and T : X → X be a Perov type Θ-contraction, then T has a unique fixed point.
Proof. Let x 0 ∈ X be an arbitrary point and define a sequence {x n } in X by x n = Tx n−1 for n ∈ {1, 2, . . . }. If x n 0 +1 = x n 0 for some n 0 ∈ {0, 1, . . . }, then Tx n 0 = x n 0 , and so T has a fixed point.
Now let x n+1 = x n for every n ∈ {0, 1, . . . } and let d(x n+1 , x n ) = (λ (1) n , λ Suppose that l < ∞. In this case, let B = l 2 > 0. From the definition of the limit, there exists n 0 ∈ N such that, for all n ≥ n 0 Λ for all i ∈ {1, 2, . . . , m}. This implies that, for all n ≥ n 0 , Suppose now that l = ∞. Let B > 0 is an arbitrary positive number. From the definition of the limit, there exists n 0 ∈ N such that, for all n ≥ n 0 , or all i ∈ {1, 2, . . . , m}. This implies that, for all n ≥ n 0 and for all i ∈ {1, 2, . . . , m} Bn λ

Considering these two cases and Equation (3) we have
Bn λ for all i ∈ {1, 2, . . . , m} and for some B > 0. Letting n → ∞ in Equation (4), we obtain that In order to show that {x n } is a Cauchy sequence consider k, l ∈ N such that k > l ≥ n 0 . Using the triangular inequality for the vector-valued metric and from Equation (6), we have By the convergence of the series ∞ ∑ j=1 1 j 1/r , passing to limit l → ∞, we get d(x l , x k ) → 0. This yields that {x n } is a Cauchy sequence in (X, d). Since (X, d) is a complete vector-valued metric space, the sequence {x n } converges to some point z ∈ X, that is, lim n→∞ x n = z.
On the other hand from (Θ1) and Equation (2), we can get d(Tx, Ty) d(x, y) for all x, y ∈ X. Therefore, we have d(Tx n , Tz) d(x n , z) as n → ∞. So we have lim n→∞ x n = Tz and hence Tz = z. The uniqueness of the fixed point can be easily seen by Equation (2). Here, since max{k i : i ∈ {1, 2, . . . , m}} < 1, the matrix M is convergent to zero.

Remark 1. By taking
Now we present an illustrative and at the same time comparative example.
Therefore by considering the unboundedness of y, we have a + b ≥ 1 and c + d ≥ 1. This shows that M does not converges to zero, which shows T is not a Perov contraction. Now, we claim that T is a Perov type Θ-contraction with and k = (exp(− 1 2 ), exp(− 1 2 )). To see this we have to show that Θ(d(Tx, Ty)) [Θ(d(x, y))] [k] for all x, y ∈ X with d(Tx, Ty) 0. For this, it is sufficient to show Θ(d(Tx, Ty)) [Θ(d(x, y))] [k] or equivalently Tx + Ty for all x, y ∈ X with d(Tx, Ty) 0. First, observe that d(Tx, Ty) 0 ⇔ the set {x, y} ∩ {0, 1} is singleton or empty.
Since Equation (7) is symmetric with respect to x and y, we may assume x > y in the following cases. Case 1. Let {x, y} ∩ {0, 1} be singleton. Then Tx + Ty = x − 1 and x + y ≤ x + 1, and so we have Case 2. Let {x, y} ∩ {0, 1} be empty. Then Tx + Ty = x + y − 2 and so we have Therefore by Theorem 3, T has a unique fixed point.

Semilinear Operator System
Let (B, · ) be a Banach space and N, M : B 2 → B be two nonlinear operators. In this section we will give an existence result for a semilinear operator system of the form Since initial or boundary value problems for nonlinear differential systems can be written in the operator form of Equation (8), such systems appear in various applications of mathematics. We can see that various fixed point theorems such as Schauder, Leray-Schauder, Krasnoselskii and Perov fixed point theorems were applied in the existence of solutions of such systems in [9].
Let X = B 2 and define d : . Then it can be seen that (X, d) is a complete vector-valued metric space. If we define a mapping T : X → X by Tu = (Nu, Mu), then Equation (8) can be written as a fixed point problem in the space X. Therefore, we will use the Theorem 3 to investigate the sufficient conditions that guarantee the existence of a solution of the fixed point problem Equation (9).

Conclusions
In this paper, by using the recent technique named as Θ-contraction we give a new generalization of the Perov fixed point theorem on vector-valued metric space. Then we present an existence result of solution of a kind of semilinear operator system.