Fixed Point Problems on Generalized Metric Spaces in Perov’s Sense

: The aim of this paper is to give some ﬁxed point results in generalized metric spaces in Perov’s sense. The generalized metric considered here is the w -distance with a symmetry condition. The operators satisfy a contractive weakly condition of Hardy–Rogers type. The second part of the paper is devoted to the study of the data dependence, the well-posedness, and the Ulam–Hyers stability of the ﬁxed point problem. An example is also given to sustain the presented results. L.G., M.-F.B. and A.N.; formal analysis, M.-F.B.; investigation, L.G.; resources, A.N.; data curation, A.N.; L.G.; and M.-F.B., L.G.; visualization, A.N.; L.G.;


Introduction and Preliminaries
The well-known Banach contraction principle was extended by Perov in 1964 to the case of spaces endowed with vector-valued metrics. In [1], Perov introduced the concept of vector-valued metric as follows.
Let X be a nonempty set. A mapping d : · · · d m (x, y)    for every m ∈ N is called vector-valued metric on X if the following properties are satisfied.
In this case, the pair (X, d) is called a generalized metric space in Perov's sense. Some examples of fixed points on the sense of vector-valued metric are given in [2][3][4][5][6]. Throughout this paper M m,m (R + ) will denote the set of all m × m matrices with positive elements. We also denote by Θ the zero m × m Recall that a matrix A is said to be convergent to zero if and only if A n → Θ as n → ∞. Let us recall the following theorem, which is useful for the proof of the main result, see [7]. Definition 1. A mapping w : X × X → [0, ∞) is a w-distance on X if it satisfies the following conditions for any x, y, z ∈ X.
In [20], we find the definition of w 0 -distance as follows.
Definition 2. Let (X, d) be a metric space. A mapping w : X × X → [0, ∞) is called w 0 -distance if it is w-distance on X with w(x, x) = 0 for every x ∈ R. Remark 1. Each metric is a w 0 -distance, but the reverse is not true.
For the following notations see I.A. Rus [21,22], I.A. Rus, A. Petruşel, A. Sîntȃmȃrian [23], and A. Petruşel [24]. Definition 3. Let (X,d) be a metric space and f : X → X be a single-valued operator. f is a weakly Picard operator (briefly WPO) if the sequence of successive approximations for f starting from x ∈ X, ( f n (x)) n∈N , converges, for all x ∈ X and its limit is a fixed point for f .
If f is a WPO, then we consider the operator Definition 4. Let (X,d) be a metric space, f : X → X be a WPO and c > 0 be a real number. By definition, the single-valued operator f is c-weakly Picard operator (briefly c-WPO) if and only if the following inequality holds, For the theory of weakly Picard operators, for single-valued operators, see [21]. I.A. Rus gave in [22] the definition of Ulam-Hyers stability as follows.

Definition 5.
Let (X,d) be a metric space and f : X → X be a single-valued operator. By definition, the fixed point equation is Ulam-Hyers stable if there exists a real number c f > 0 such that for each ε > 0 and each solution y * of the inequation d(y, f (y)) ≤ ε (2) there exists a solution x * of Equation (1) such that Remark 2. If f is a c-weakly Picard operator, then the fixed point Equation (1) is Ulam-Hyers stable.
We present in the first part of this paper some fixed point results in generalized metric spaces in Perov's sense. The operator satisfies a contractive condition of Hardy-Rogers type. In the second part of the paper, we study the data dependence of the fixed point set. The well-posedness of the fixed point problem and the Ulam-Hyers stability are also studied.

Fixed Point Results
First, let us we recall the notion of generalized w-distance defined in [36] by L. Guran. Definition 6. Let (X, d) be a generalized metric space. The mapping w : X × X → R m + is called generalized w-distance on X if it satisfies the following conditions.
(1) w(x, y) ≤ w(x, z) + w(z, y), for every x, y, z ∈ X; (2) w is lower semicontinuous with respect to the second variable.; (3) For any ε := Examples of generalized w-distance and some of its useful properties are also given in [36] and [37]. In the same framework, let us give the definition of generalized w 0 -distance. Definition 7. Let (X, d) be a generalized metric space. A mapping w : Let us recall the following useful result. Lemma 1. Let (X, d) be a generalized metric space, and let w : X × X → R m + be a generalized w-distance on X. Let (x n ) and (y n ) be two sequences in X, let α n := be two sequences such that α n(i) and β n (i) converge to zero for each i ∈ {1, 2, . . . , m}. Let x, y, z ∈ X. Then, the following assertions hold, for every x, y, z ∈ X.
(1) If w(x n , y) ≤ α n and w(x n , z) ≤ β n for any n ∈ N, then y = z.
(2) If w(x n , y n ) ≤ α n and w(x n , z) ≤ β n for any n ∈ N, then (y n ) converges to z.
(3) If w(x n , x m ) ≤ α n for any n, m ∈ N with m > n, then (x n ) is a Cauchy sequence.
(4) If w(y, x n ) ≤ α n for any n ∈ N, then (x n ) is a Cauchy sequence.
Next, let us give the definition of single-valued weakly Hardy-Rogers type operator on generalized metric space in Perov's sense. Definition 8. Let (X, d) be a generalized metric space in Perov's sense, w : X × X → R m + be a generalized w-distance, and f : X → X be a single-valued operator. We say that f is a weakly Hardy-Rogers type operator if the following inequality is satisfied, for all x, y ∈ R and A, B, C ∈ M m,m (R + ).
The first fixed point result of this paper is the following. Theorem 2. Let (X, d) be a complete generalized metric space in Perov's sense, w : X × X → R m + be a generalized w 0 -distance. Let f : X → X be a single-valued weakly Hardy-Rogers type operator such that We get the inequality For the next step, we have .
Using (3) we obtain the inequality with M ∈ M m,m (R + ) and n ∈ N. We will prove next that (x n ) n∈N is a Cauchy sequence, by estimating w(x n , x m ), for every m, n ∈ N with m > n. w By Lemma 1 (3) the sequence (x n ) n∈N is a Cauchy sequence.
as n → ∞. By the uniqueness of the limit, we get This We can replace the continuity condition on the operator f and we obtain the following fixed point theorem. Theorem 3. Let (X, d) be a complete generalized metric space in Perov's sense and w : X × X → R m + be a generalized w 0 -distance. Let f : X → X be a single-valued weakly Hardy-Rogers type operator such that the following conditions are satisfied, Proof. Following the same steps as in the previous theorem, Theorem 2, we have the estimation with M ∈ M m,m (R + ) and n ∈ N.
Assume that x * = f (x * ). Then, for every x ∈ X, by hypothesis (a) we have This is a contradiction. Therefore For the proof of the last part of this theorem we use the same steps as is the previous theorem, Theorem 2.
Further we give a more general fixed point result concerning this new type of operators. Then Proof. Following the same steps as in Theorem 2, we get the estimation with M ∈ M m,m (R + ) and n ∈ N. By Lemma 1 (3) the sequence (x n ) n∈N is a Cauchy sequence; since (X, d) is complete there exists x * ∈ X such that x n d → x * . Let n ∈ N be fixed. Then, as (x m ) m∈N d → x * , w(x n , ·) is lower semicontinuous and letting n → ∞ we have Let f (x * ) ∈ X. By triangle inequality and using (6) we obtain Using Lemma 1(1), by Equations (10) and (11), we get For the last part of the proof we use the same steps as in Theorem 2.
Another fixed point result concerning the single-valued weakly Hardy-Rogers operators in generalized metric space is the following.
Theorem 5. Let (X, d) be a complete generalized metric space in Perov' sense, w : X × X → R m + be a generalized w 0 -distance and f : X → X be a single-valued Hardy-Rogers type operator. Suppose that all the hypothesis of Theorem 2 hold. Then, we have (2) There exists a sequence (x n ) n∈N ∈ X such that x n+1 = f (x n ), for all n ∈ N and converge to a fixed point of f .
and f 2 (x, y) = 6y 5 − 1, for (x, y) ∈ R 2 , with x ≤ 5; Next, let us give some common fixed point results. Then, f and g have a common fixed point x * ∈ X.
Then, we have w(x 2n , x 2n+1 ) ≤ (I − (B + C)) −1 (A + B + C) w(x 2n−1 , x 2n ) = M w(x 2n−1 , x 2n ). By the same argument as above, we get Further, we obtain w(x n , x n+1 ) ≤ M n w(x 0 , x 1 ) for each n ∈ N. Following the same steps as in the proof of Theorem 2 we estimate w(x n , x m ), for every m, n ∈ N with m > n.
Using the lower semicontinuity of the generalized w-distance, by relation (8) we have w(x n , x * ) d → 0 1×m as n → ∞. Then, we have w(x 2n , x * ) d → 0 1×m as n → ∞. By the continuity of f it follows By the uniqueness of the limit we get x * = f (x * ).
By the uniqueness of the limit we get x * = g(x * ). Then, x * is a common fixed point for f and g.
By replacing the continuity condition for the mappings f and g, we can state the following result. Then f and g have a common fixed point x * ∈ X.
Proof. (1) As in the proof of the previous theorem, Theorem 6, for x 0 ∈ X we consider (x n ) n∈N the sequence of successive approximations for f and g, defined by x 2n+1 = f (x 2n ), n = 0, 1, ...
x 2n+2 = g(x 2n+1 ), n = 0, 1, ... We define the sequence (x n ) nN ∈ X such that Further, we obtain w(x n , x n+1 ) ≤ M n d(x 0 , x 1 ) for each n ∈ N. Following the same steps as in the proof of Theorem 6 we estimate w(x n , x m ), for every m, n ∈ N with m > n and we get w(x n , x m ) ≤ M n (I − M) −1 w(x 0 , x 1 )).

Remark 3.
In the case of common fixed points, the generalized w-distance must not necessarily be a generalized w 0 -distance.

Ulam-Hyers Stability, Well-Posedness, and Data Dependence of Fixed Point Problem
We begin this section with the extension of Ulam-Hyers stability for fixed point equation for the case of single-valued operators on generalized metric space in Perov's sense. Then, let us recall the definition of weakly Ulam-Hyers stability. Definition 9. Let (X, d) be a metric space, w : X × X → R m + be a generalized w-distance, and f : X → X be an operator. By definition, the fixed point equation is weakly Ulam-Hyers stable if there exists a real positive matrix N ∈ M m,m (R+) such that, for each ε > 0 and each solution y * of the inequation w(y, f (y)) ≤ εI 1×m (15) there exists a solution x * of the Equation (14) such that d(y * , x * ) ≤ NεI 1×m .
Theorem 8. Let (X, d) be a generalized metric space in Perov's sense, w : X × X → R m + be a generalized w 0 -distance and f : X → X be a single-valued Hardy-Rogers type operator defined in (8). There exist matrices A, B, C ∈ M m,m (R + ) such that Then, the fixed point Equation (14) is weakly Ulam-Hyers stable.
The following result assures the well-posedness of the fixed point problem with respect to the generalized w 0 -distance w. Theorem 9. Let (X, d) be a generalized metric space in Perov's sense, w : X × X → R m + be a generalized w 0 -distance, and f : X → X be a single-valued Hardy-Rogers type operator defined in Equation (8). If all the hypothesis of Theorem 2 (respectively, 3 and 4) are satisfied, the fixed point Equation (14) is well-posed with respect to the generalized w 0 -distance w, i.e., if Fix( f ) = {x * } and x n ∈ N, with n ∈ N, such that w(x n , f (x n )) → 0 1×m as n → ∞, then x n → x * as n → ∞.
Proof. Let x * ∈ Fix( f ) and let (x) n∈N ∈ X such that w(x n , f (x n )) d → 0 1×m as n → ∞. That means w(x n−1 , x n ) d → 0 1×m as n → ∞. By the lower semicontinuity of the generalized w-distance, using (8) we have Then, using Lemma 1 (3) we get x n d → x * as n → ∞.
The next theorem presents a data dependence result. Theorem 10. Let (X, d) be a generalized metric space in Perov's sense, w : X × X → R m + be a generalized w 0 -distance, and f 1 , f 2 : X → X be single-valued operators, which satisfy the following conditions, (i) for A, B, C, M ∈ M m,m (R + ) with M = [I − (B + C)] −1 (A + B + C) a matrix convergent to Θ such that, for every x, y ∈ X and i ∈ {1, 2}, we have: w( f i (x), f i (y)) ≤ A w(x, y) + B[ w(x, f i (x)) + w(y, f i (y))] + C[ w(x, f i (y)) + w(y, f i (x))]; (ii) there exists η > 0 such that w( f 1 (x), f 2 (x)) ≤ η I, for all x ∈ X.
If we consider the sequence (x n ) n∈N ∈ X converges to x * 2 , we have x * 2 = f (x * 2 ). Moreover, for each n, p ∈ N we have w(x n , x n+p ) ≤ M n (I − M) −1 w(x 0 , x 1 ).

Conclusions
The purpose of this paper is to establish some fixed point results in generalized metric spaces in Perov's sense. The generalized metric considered here is the w-distance, for which the symmetry condition is not satisfied. The operators satisfy a contractive weakly condition of Hardy-Rogers type. The second part of the paper is devoted to the study of the data dependence, as well as the well-posedness and the Ulam-Hyers stability of the fixed point problem. In order to prove our main results we had to impose a symmetry condition for the w-distance. The results presented in this paper generalize some recent ones. Funding: The second author wish to thanks Babes , -Bolyai University, Cluj-Napoca, Romania, for the financial support.

Conflicts of Interest:
The authors declare no conflicts of interest.