Fixed-point theorem in classes of function with values in a dq-metric space

We prove a fixed point result for nonlinear operators, acting on some classes of functions with values in a dq-metric space, and show some applications of it. The result has been motivated by some issues arising in Ulam stability. We use a restricted form of a contraction condition.


Introduction
The name of Ulam has been somehow connected with various definitions of stability (see, e.g., [1,12,16,24]), but roughly speaking, the following one describes our considerations in this paper (A B denotes the family of all functions mapping a nonempty set B into a nonempty set A, R stands for the set of all real numbers and R + :=[0, ∞)). Definition 1. Let (Y, d) be a metric space, E be a nonempty set, D 0 ⊂ D ⊂ Y E and E ⊂ R + E be nonempty, T : D → Y E and S : E → R + E . We say that the equation is S-stable in D 0 provided, for any ψ ∈ D 0 and δ ∈ E with there is a solution φ ∈ D of the equation, such that d(φ(t), ψ(t)) ≤ Sδ(t), t∈ E.
Let d be a dq-metric in a nonempty set Y . We say that x ∈ Y is a limit of a sequence ( then we write x n → x or x = lim n→∞ x n ; in view of (A2), it is easy to note that such a limit must be unique. Next, we say that a sequence ( d is complete if every Cauchy sequence in Y has a limit in Y .

Remark 2.
Usually, in a dq-metric space, the Cauchy sequence is defined in a somewhat different way; e.g., in a metric-like space (Y, d), a sequence (x n ) ∞ n=1 is said to be Cauchy if the limit lim N →∞ sup m,n N d(x n , x m ) exists and is finite (see [3]). However, such definitions are too weak and would exclude from our considerations the metric and quasi-metric spaces. The same concerns the notion of completeness.
Our definition of a limit of a sequence is stronger than the usual, but this seems to be necessary in the proof of the main result; moreover, it actually corresponds to our definition of the Cauchy sequence and makes such limit unique (which is not the case in general) and, therefore, more useful.

The main result
In what follows, we always assume that (Y, d) is a complete dq-metric space, i.e., d is a complete dq-metric in a nonempty set Y . Moreover, E denotes a nonempty set and d : Analogously, as in the classical metric spaces, if (χ n ) n∈N is a sequence of elements of Y E , then a function χ ∈ Y E is a pointwise limit of (χ n ) n∈N provided A nonempty subset F of Y E is called p-closed (u-closed, respectively) if every χ ∈ Y E , which is a pointwise (uniform, resp.) limit of a sequence (χ n ) n∈N of elements of F, belongs to F.
Furthermore, given f, Finally, to simplify some formulas, we denote by Λ 0 the identity operator on R + E , i.e., Λ 0 δ = δ for each δ ∈ R + E . Now, we are in a position to present the fixed-point theorem, which is the main result of this paper. Theorem 3. Let C ⊂ Y E be nonempty, Λ n : R + E → R + E for n ∈ N, and T : C → C. Assume that there exist functions ε 1 , ε 2 ∈ R + E and ϕ ∈ C, such that and write ε * (t):= max{ε 1 (t), ε 2 (t)} for t ∈ E. Let T n be (ε * , Λ n )-contractive for n ∈ N and one of the following two hypotheses be valid.
(i) C is p-closed.
(ii) C is u-closed and the sequence n i=0 Λ i ε j n∈N tends uniformly to ε * j on E for j = 1, 2.
Then, for each t ∈ E, there exists the limit Vol. 20 (2018) Fixed-point theorem in a dq-metric space Page 5 of 16 143 and the function ψ ∈ C, defined in this way, is a fixed point of T with Moreover, the following two statements are valid: then ψ is the unique fixed point of T with Proof. Clearly, (3) implies that, for any k, l ∈ N and n ∈ N 0 whence Therefore, by (2), (T n ϕ(t)) n∈N is a Cauchy sequence in Y for each t ∈ E. Since Y is complete, this sequence is convergent. Consequently, (5) defines a function ψ ∈ C. Letting k → ∞ in (9) and l → ∞ in (10), on account of (5), we get which is (6). Next, using (11), we get whence letting n → ∞, we get ξ = ψ. It remains to prove statement (b). Therefore, assume that (7) holds and Now, let j, l ∈ {1, 2} and (k n ) n∈N be a sequence of positive integers with Letting n → ∞, we get (8).
Theorem 3 implies at once the following.
(i) C is p-closed.
(ii) C is u-closed and the sequence n i=0 Λ i ε j n∈N tends uniformly to ε * j on E for j = 1, 2.
Then, for each t ∈ E, there exists the limit and the function ψ ∈ C, defined in this way, is a fixed point of T with Moreover, the following two statements are valid: (a) For every sequence (k n ) n∈N of positive integers with lim n→∞ k n = ∞, ψ is the unique fixed point of T with then ψ is the unique fixed point of T with for every sequence (k n ) n∈N of positive integers with lim n→∞ Λ kn ε * m (t) = 0 for t ∈ E and m ∈ {j, l}.
Proof. It is enough to notice that T n is (ε * , Λ n )-contractive for each n ∈ N and use Theorem 3.

Remark 3.
There arises a natural question whether, in some situations, assumption (2) can be weaker than (12) with Λ:=Λ 1 . Below, we provide a somewhat trivial example that this is the case.

Remark 5.
Let j ∈ N and K be either the set of reals R or the set of complex numbers C. Fix f i : E → E and L i : E → K for i = 1, . . . , j. Then, the operator T : is (ω, Λ) contractive, with any ω ∈ R E + and Λ : Moreover, (C) holds. Next, for any function ε 0 : E → R + with ε * 0 given by [see (12)] and analogously, by induction, we get This means that (12) yields (14). Therefore, [9, Theorem 1] can be derived from Theorem 4.

Remark 6.
Let F : E × R + → R + be subadditive and non-decreasing with respect to the second variable (i. e., F (x, a + b) ≤ F (x, a) + F (x, b) and F (x, a) ≤ F (x, c) for a, b, c ∈ R + with a ≤ c and x ∈ E). Let f : E → E be given and Λ : R + E → R + E be defined by We show that for such Λ, condition (12) yields (13) and (14). Therefore, assume that (12) holds for some suitable ε j with j = 1, 2. Fix x ∈ E and define a function F 0 : F (x, a), a∈ R + .
Since F 0 is non-decreasing and Λ n ε 1 (f (x)) ≥ 0 for each n ∈ N 0 , we have Hence, by (12), we get F 0 (0) = 0. Fix j ∈ {1, 2}. Next, we prove that F 0 is continuous at 0 or there exists To this end suppose that F 0 is not continuous at 0 and there is a strictly increasing sequence k n n∈N of positive integers, such that Λ kn ε j (f (x)) = 0 for n ∈ N. Since F 0 is non-decreasing and F (0) = 0, there exists d > 0 with F 0 (c) > d for every c > 0, whence which contradicts to (12). Thus, we have proved that Furthermore, by subadditivity of F 0 , for every k, l ∈ N 0 , l > k, we get and consequently, by induction (with k = 0) Clearly, using those inequalities, we can easily deduce (13) and (14) from (12). Now, consider a very special situation when the set E has only one element, E = {s}. Then, actually, each C ⊂ Y E can be considered as a subset of Y of the form C:={φ(s) : φ ∈ C}.

Theorem 5.
Let T : Y → Y , λ n : R + → R + for n ∈ N, and λ 1 satisfy hypothesis (C 0 ). Suppose that there exist y 0 ∈ Y and ε 1 , ε 2 ∈ R + , such that and T n is (ε * , λ n )-contractive for n ∈ N with ε * := max {ε * 1 , ε * 2 }. Then, the limit exists and z 0 is a unique fixed point of T with Moreover, the following two statements are valid: (a) for every sequence (k n ) n∈N of positive integers with lim n→∞ k n = ∞, z 0 is the unique fixed point of T with then z 0 is the unique fixed point of T , such that Clearly, if there is λ ∈ R + , such that λ n (a) = λ n a for a ∈ R + and n ∈ N, then Theorem 5 becomes a natural modification of the Banach Contraction Principle (with a local contraction condition) and (16) means that λ < 1.
The following corollary also can be easily deduced from Theorem 2.
and Λ : R E + → R E + is given by Then, the limit exists for each t ∈ E, with T given by T ϕ(t):=Φ(t, ϕ(f 1 (t)), ..., ϕ(f j (t))), ϕ∈ Y E , t ∈ E, and the function ψ : E → Y , defined by (19), is the unique solution of the functional equation: such that Proof. Let us note that inequalities (17) and (18)  Stability of functional equations of form (20) (or related to it) has been already studied by several authors, and for further information, we refer to the survey papers [1,8]. A particular case of (20) is the linear functional equation of the form under the assumptions as in Remark 5; some recent results concerning stability of less general cases of it can be found in [10,18,19,23].
As an example of applications of Corollary 6 consider stability of the difference equation: where Φ : N × Y → Y is given and ψ : N → Y is unknown. Clearly, (22) is a very simple particular case of (20), with E = N, j = 1 and f 1 (i) = i + 1 for i ∈ X. Let (a n ) n∈N be a sequence of positive reals, such that For instance, we can take ρ ∈ (0, 1) and write a 2n = 1 ρ , a 2n−1 = ρ 2 , n∈ N.