Stability of Unbounded Differential Equations in Menger k -Normed Spaces: A Fixed Point Technique

: We attempt to solve differential equations υ (cid:48) ( ν ) = Γ ( ν , υ ( ν )) and use the ﬁxed point technique to prove its Hyers–Ulam–Rassias stability in Menger k -normed spaces.


Hyers-Ulam-Rassias Stability in M-k-NLS
Recently, Cȃdariu and Radu [14] applied the fixed point method to the investigation of the Jensen's functional equation. Using such an idea, they could present a proof for the Hyers-Ulam stability of that equation (see [11,15,30]). In this section, by using the idea of Cȃdariu and Radu, we will prove the Hyers-Ulam-Rassias stability of the differential Equation (1). Hereinafter we suppose that * = * M = .
Proof. We show the set of all continuous map σ : J → R by and define the function δ on Σ, In [31], Miheţ and Radu proved that (Σ, δ) is a complete generalized metric space (see also [32]). Now, we consider the linear map Λ : Σ → Σ is defined by for all υ ∈ Σ.
In the last theorem, we have investigated the Hyers-Ulam-Rassias stability of the differential Equation (1) in M-k-NLS defined on a bounded and closed interval. We will now prove the theorem for the case of unbounded intervals. More precisely, Theorem 2 is also true if J is replaced by an unbounded interval such as (−∞, q], R, or [p, ∞) as we see in the following theorem. Theorem 3. Let J be (−∞, q] or R or [p, ∞) in which p, q ∈ R. Put m = p for I = [p, ∞) or m = q for J = (−∞, q], or if J = R, put m ∈ R being fixed. Consider the constant numbers ρ and β such that 0 < ρβ < 1 and continuous map Γ : J × R → R holds (2) for all ν j ∈ J and all υ j , ϑ j ∈ R. Let υ : J → R be continuous differentiable and satisfies (3) for all ν j ∈ J, in which ϕ : J k × ∞ → (0, 1] be a distribution function satisfying the condition (4) for any ν i ∈ J, (j = 1, 2, ..., k), so there is a unique continuous map υ 0 : J → R which satisfies (5) and (6) for all ν j ∈ J.

Hyers-Ulam Stability in M-k-NLS
In the following theorem, we prove the Hyers-Ulam stability of the differential Equation (1) defined on a finite and closed interval.

Examples
In this section, we show that there certainly exist functions υ(ν) which satisfy all the conditions given in Theorems 2, 3 and 4.
Author Contributions: All authors conceived the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript. All authors have read and agreed to the published version of the manuscript.

Funding:
The authors are grateful to the Basque Government by the support, of this work through Grant IT1207-19.