Comments on fixed point results in classes of function with values in a b–metric space

We prove and discuss several fixed point results for nonlinear operators, acting on some classes of functions with values in a b-metric space. Thus we generalize and extend a recent theorem of Dung and Hang (J Math Anal Appl 462:131–147, 2018), motivated by several outcomes in Ulam type stability. As a simple consequence we obtain, in particular, that approximate (in some sense) eigenvalues of some linear operators, acting in some function spaces, must be eigenvalues while approximate eigenvectors are close to eigenvectors with the same eigenvalue. Our results also provide some natural generalizations and extensions of the classical Banach Contraction Principle.


Introduction
The notion of Ulam stability (see [8,16,22] for details) has motivated several generalizations of Banach Contraction Principle for various function spaces (see, e.g., [3,4,6,7,9]). Let us recall for instance the main result in [9] (see also [4,Theorem 2]). To formulate it we need the following hypothesis for operators Λ : R + E → R + E (R + stands for the set of nonnegative reals, E is a nonempty set and A B denotes the family of all functions mapping a nonempty set B into a nonempty set A): The term b-metric has been used for the first time in [10] with η = 2 and next in [11] for any η ≥ 1.
Let us yet remind that, in the b-metric spaces, the notions of convergence, Cauchy sequence and a limit of a sequence are defined analogously as in the classical metric spaces. Namely, if (Y , d, η) is a b-metric space, then: -a sequence (x n ) n∈N ∈ Y N is convergent to x ∈ Y if lim n→∞ d(x, x n ) = 0 and then we say that x is a limit (which must be unique) of the sequence and write x = lim n→∞ x n ; -a sequence (x n ) n∈N ∈ Y N is Cauchy if lim n,m→∞ d(x m , x n ) = 0; -(Y , d) is complete if every Cauchy sequence in Y is convergent to some point x ∈ Y .
The result in [13] can be formulated in the following way.
where θ = log 2η 2 and Λ : R + E → R + E is given by Then limit (1) exists for every t ∈ U and the function ψ : U → Y so defined is a fixed point Moreover, if with some real number M > 0, then ψ is the unique fixed point of T satisfying (4).
The reasoning presented in Remark 4 also leads to the following two examples (cf. [2, Examples 1.1 and 1.2]).
Moreover, note that if n ∈ N, a 1 , . . . , a n ∈ (0, ∞) and d 1 , . . . , d n are b-metrics in a nonempty set X , then d and d 0 also are b-metrics in X , where Let us yet recall the following very useful result.
In particular, if d is a metric, then D d = d.

Remark 6
Assume that (X , · , η) is a quasi-normed space, i.e., η ≥ 1 is a fixed real number, X is a real or complex linear space and · : X → R + satisfies the following three conditions: (a) x = 0 if and only if x = 0; (b) αx = |α| x for x ∈ X and all scalars α; Let Y ⊂ X be nonempty and d : Then it is easily seen that (Y , d, η) is a b-metric space.
In view of Aoki-Rolewicz Theorem (see, e.g., [20,Theorem 1]), each quasi-norm is equivalent to some p-norm, but there exist p-Banach spaces which are not normable (see, e.g., [20,Examples 1 and 2]). Let us yet recall that, if X is a a real (complex, respectively) linear space and p > 0, then a mapping · : X → R + is a p-norm provided conditions (a) and (b) hold and (c1) x + y p ≤ x p + y p for x, y ∈ X .
In the next section we provide some generalizations of Theorem 2 by showing that similar fixed point results hold also for operators Λ of more general forms. In this way we extend Theorem 1 and several similar outcomes in [4][5][6]9] to the case of b-metric spaces.
Finally, let us mention that in [14,19,23,25,28,29] and references contained therein, the interested readers can find various further recent metric-type fixed point results. Maybe some of them could be extended and applied in a similar way as in this paper. It seems that the concept of ν-generalized metric is very promising in this respect (see, e.g., [28]).

The main result
In what follows, we always assume that (Y , d, η) is a complete b-metric space. Moreover, E denotes a nonempty set and, given ν, μ ∈ Y E , we define a mapping d(ν, μ) : E → R + by the formula: Similarly, as in the usual metric spaces, if (χ n ) n∈N is a sequence in Y E , then we say that χ ∈ Y E is a pointwise limit of (χ n ) n∈N when We say that a subset F = ∅ of Y E is p-closed (u-closed, respectively) if each χ ∈ Y E that is a pointwise (uniform, resp.) limit of a sequence of elements of F , is an element of F .
First note that from Theorem 1 we can easily deduce the following partial generalization of Theorem 2 (i.e., of [13, Theorem 2.1]), in which we do not need to presume any particular form of Λ, but has to replace (4) with a somewhat different estimate (see the definition of ε * in (2)) and formulate the statement on uniqueness similarly as in Theorem 1.
where ξ := log 2η 2 and ε ξ : Then the limit exists for each t ∈ E and the function ψ ∈ Y E , defined in this way, is a unique fixed point of T with Proof According to Theorem 5, (7) and the Λ-contractivity of T , for every ν, μ ∈ Y E and t ∈ E. Moreover, note that the completeness of (Y , d, η) implies the completeness of the metric space (Y , D d ) (see (6)). So, by Theorem 5 and Theorem 1 (with ρ = D d and Λ and ε replaced by Λ ξ and ε ξ ), we obtain the statement.
The next fixed point theorem provides a further, a bit different, generalization of Theorem 2; it is the main result of this paper. To simplify some formulas, we denote by Λ 0 the identity and one of the following two hypotheses is valid.
Then, for each t ∈ E, the limit exists and the function ψ ∈ Y E , so defined, belongs to C and fulfils the inequalities Moreover, the following three statements are valid.
(a) If l ∈ N and lim inf then ψ is a fixed point of operator T l .
then ψ(t) = ω(t) for every fixed point ω ∈ C of T l , satisfying the inequality Proof Let D d be the metric defined as in Theorem 5. Since T n is Λ n -contractive, (8) implies that and consequently (8), which is (10). Since, in the case of (ii), we have it is easily seen that (10) implies that ψ ∈ C also in this case. Next, observe that (16) implies Hence, if (11) is valid for a fixed l ∈ N, then with n → ∞ we get and consequently T l ψ = ψ. Now, let l ∈ N, γ : E → R + and ω ∈ C be a fixed point of T l satisfying (13). Then, for each n ∈ N, we get Consequently, letting n → ∞, by (9), we can easily see that ω(t) = ψ(t) for each t ∈ E such that (12) holds. Finally, let t ∈ E and τ ∈ C be such that ε * n (t) = 0 for n ∈ N and (14) is valid. Write .
Then there is a strictly increasing sequence (t n ) n∈N in N such that and consequently, by (16), whence we get τ (t) = ψ(t) with n → ∞ (in view of (8)).

Remark 9
From statement (b) of Theorem 8 we can deduce, in particular, the following: If ψ is not a fixed point of T l for some l ∈ N, then the same is true for every function ω ∈ C, which satisfies (13) with some γ ∈ l , where l denotes the family of all functions γ : E → R + such that (12) holds for each t ∈ E.
But, if ψ is a fixed point of T l for some l ∈ N, then for every χ ∈ C, t ∈ E, and every sequence ( In fact, if γ : E → R + and χ ∈ C satisfies the inequality d(χ, ψ) ≤ γ , then for each t ∈ E and each sequence (k n ) n∈N in N such that (17) holds, we have which implies that To avoid any ambiguity in the sequel, we need one more definition.

Definition 10
Let (D, ρ, η) be a b-metric space, T be a nonempty subset of D, S : T → D and λ : R + → R + . We say that S is a λ-contraction provided for every t, s ∈ T and every δ ∈ R + such that ρ(t, s) ≤ δ.
Note yet that, if E has only one element, i.e., E = {t 0 }, then Y E can be identified with Y . Therefore, Theorem 8 yields at once the following generalization of the Banach Contraction Principle.

Corollary 11
Let T : Y → Y and λ n : R + → R + for n ∈ N. Suppose that T n is a λ ncontraction for n ∈ N and there exist e ∈ R + and φ ∈ Y such that Then there exists the limit where λ 0 (d) = d for d ∈ R + . Moreover, the following statements are valid.

Remark 12
Analogously as in Remark 9, from statement (β) of Corollary 11 we can deduce, in particular, the following: If ψ 0 is not a fixed point of T l for some l ∈ N, then the same is However, if ψ 0 is a fixed point of T l for some l ∈ N, then (again analogously as in Remark 9) it is easy to show that Theorem 8 also yields the subsequent.

Corollary 13
Let C ⊂ Y E be nonempty, Λ : R + E → R + E , T : C → C be Λ-contractive, and functions ε ∈ R + E and ϕ ∈ C be such that Next, assume one of the following two hypotheses.
(i) C is p-closed.
(ii) C is u-closed and the sequence n i=0 (Λ i ε) ξ n∈N tends uniformly to ε .
Then, for each t ∈ E, there exists the limit and the function ψ ∈ Y E , so defined, belongs to C and fulfils the inequalities Moreover, the following statements are valid.
then ψ is a fixed point of T .
Proof It is easy to see that T n is Λ n -contractive for each n ∈ N. So, it is enough to use Theorem 8 with l = 1 and Λ n := Λ n for n ∈ N.
Then (C 0 ) holds for such Λ (see Remark 3). This means that Theorem 1 is a simple consequence of Corollary 13. We show that this is also the case for Theorem 2. First note that (C 0 ) and (18) imply (21). Therefore the statement of Theorem 2 that ψ is a fixed point of T can be deduced from (A). We are yet to show that the statement on uniqueness of ψ can be derived from (B).
To this end fix γ ∈ R + E and M > 0 such that We prove that, for each n ∈ N 0 , Clearly, the case n = 0 is trivial in view of (23). Next, fix n ∈ N 0 and suppose that (24) is valid. Then Thus, by induction, we have proved (24). Therefore, in this case, (24) yields (22). Consequently, (22) results also from the condition This means that (5) (cf. [13, (2.6)]) implies that and therefore from (B) we can infer that ψ is the unique fixed point of T , satisfying the inequality d(ϕ, ψ) ≤ ε * 0 . Thus we have shown that Theorem 2 (i.e., [13,Theorem 2.1]) is a simple consequence of Corollary 13.
The next example shows that, in some natural situations, assumption (8) can be significantly weaker than (18) with Λ := Λ 1 . It is a bit trivial, but this allows to expose clearly the main differences between (8) and (18).

Example 3
Assume that (X , · , η) is a real quasi-normed space. Let Y = X 2 and define d : Clearly, (Y , d, η) is a b-metric space. Fix τ 1 , τ 2 ∈ X E and define the operator T : for every φ = (φ 1 , φ 2 ), μ = (μ 1 , μ 2 ) ∈ Y E . Therefore, with any Λ 1 : ∈ Y E and we can take Λ n δ(t) = 0 for δ ∈ R + E , t ∈ E and n ∈ N, n ≥ 2. This means that the second inequality in (8) is fulfilled for every ε ∈ R + E . Fix δ ∈ R + E and u ∈ X with u = 1. Let functions φ, ψ ∈ Y E be defined by: Clearly, Assume now that T is Λ 1 -contractive with some Λ 1 : Then Thus we have shown that Consequently, by an easy induction, for each n ∈ N, we obtain

Ulam stability
Now, we present a simple Ulam stability result that can be derived from our considerations in the previous sections. To this end, we need the following hypothesis.
The following corollary can be easily deduced from Corollary 13.

Corollary 15
Let hypothesis (H1) be valid, where Λ : R + E → R + E is given by Then the limit exists for each t ∈ E, with T given by: and ψ ∈ Y E , defined by (28), is a solution of the functional equation such that d(ϕ(t), ψ(t)) ≤ (4ε (t)) 1/ξ =: ε(t), then ψ is the unique function in Y E that satisfies (29) and (30).
Stability of functional equations of form (29) (or related to it) has already been studied by several authors and for further information we refer to [1,5,8]. A very particular case of (29) is the linear functional equation of the form with j ∈ N, f 1 , . . . , f j ∈ E E , L 1 , . . . , L j : E → R + and for functions φ ∈ Y E (provided Y is a real linear space); some recent results concerning stability of less general cases of it can be found in [8,17,18,21]. A result similar to Corollary 15 can also be obtained for the functional equation with Y = E = R and d(x, y) = |x − y| p for x, y ∈ R, with some fixed p ∈ [1, ∞) (see Remark 4). Then we take T : Y E → Y E given by and it is easy to show that such T is Λ-contractive with Λ : R + E → R + E defined by the formula: To this end it is enough to prove by a simple induction that using the inequality which is very easy to check, because only the case max {x 1 , x 2 } = x 1 , max {y 1 , y 2 } = y 2 must be verified and then we have the following two situations:

Approximate eigenvalues
Assume additionally that (Y , · , η) is a real quasi-normed space and d(x, y) = x − y for x, y ∈ Y . Clearly, then Y E is also a real liner space with the operations defined pointwise in the usual way: The next corollary is an example of a result (that can be derived from the previous sections) concerning approximate eigenvalues of some linear operators on Y E . Actually the assumption of linearity of the operators is not necessary in the proof, but the notion of eigenvalue might be ambiguous without it (see, e.g., [26]) and therefore we confine only to the linear case.
Then γ is an eigenvalue of T 0 , the limit exists for every t ∈ E, and the function ψ 0 ∈ Y E , given by is an eigenvector of T 0 , with the eigenvalue γ , and Proof Let ϕ := γ h and T : Y X → Y X be given by: Then, in view of the Λ 0 -contractivity and linearity of T 0 , for every μ, ν ∈ Y E and δ ∈ R + E with d(μ, ν) ≤ δ we have which means that T is Λ -contractive. Next, we can write (32) in the form Hence, by Corollary 13 (with C = Y E ), the limit (19) exists for each t ∈ E and the function ψ : E → Y , so defined, is a fixed point of T with d(ψ(t), ϕ(t)) ≤ ε * (t), t ∈ E.
Clearly, under suitable additional assumptions in Corollary 16, we can deduce from Corollary 13 some statements on the uniqueness of ψ, and consequently of the uniqueness of ψ 0 .
Given ε ∈ R + E , let us introduce the following definition: γ ∈ R \ {0} is an ε-eigenvalue of a linear operator T 0 : Y E → Y E provided there exists h ∈ Y E such that T 0 h − γ h ≤ ε.
Corollary 16 yields the following simple result.
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