Stability of a Ramanujan Type Additive Functional Equation

In this paper, the authors achieve the generalized Ulam Hyers stability of a Ramanujan Type Additive Functional Equation in Paranormed Spaces and Modular spaces via classical Hyers Method.


Introduction
The stability of a functional equation initiated from a question raised by Ulam: when is it true that the solution of an equation differing slightly from a given one must of necessity be close to the solution of the given equation? (see [37] ). The first answer (in the case of Cauchys functional equation in Banach spaces) to Ulams question was given by Hyers in [11] .
Following his result, a abundant number of papers on the stability problems have been extensively available as generalizing Ulams problem and Hyers theorem in various directions; see for instance [3,10,28,29,31], and the references given there.

Ramanujan Numbers
The world of mathematics is renowned for a number of interesting and fascinating numbers. Now Ramanujan Number also makes such a place in the list. Ramanujan Numbers (preciously termed as Hardy-Ramanujan Numbers) are those numbers that are the smallest positive integers that can be represented or expressed as a sum of 2 positive integers in n ways. Now lets discuss the above ways in a mathematical way.
It is easy to verify that 2 = 1 3 + 1 3 9 = 2 3 + 1 3 16 = 2 3 + 2 3 · · · . Here all these numbers can be expressed as a sum of 2 cubes in a single way and so all these numbers from the above set can be expressed in this way.
The Ramanujans 1 -way solution can be converted to Ramanujan Additive Functional Equation of the form

Basic Concepts And Stability on Paranormed Spaces
Now, we give to adopt the usual terminologies, notations, definitions and properties of the theory of paranormed spaces given in [8,9,15,18,25,33,35].
The pair (X, P ) is called a paranormed space if P is a paranorm on X.
Definition 2.2 Let X be a paranormed space and let {x n } be a sequence in X then {x n } is called Cauchy if for any > 0 if P (x n −x m ) → 0 for sufficiently large m, n ∈ N.
Definition 3.2 A modular ρ defines the following vector space: and we say that X ρ is a modular space.
Definition 3.3 Let X ρ be a modular space and let {x n } be a sequence in X ρ then Definition 3.5 Let X ρ be a modular space and let {x n } be a sequence in X ρ . A subset K ⊆ X ρ is called ρ−complete if any ρ−Cauchy sequence is ρ−convergent to a point in K.
It is said that the modular ρ has the Fatou property if and only if ρ(x) ≤ lim inf Theorem 3.6 In modular spaces, x and a is a constant vector, then x n + a ρ → x + a, and (2) if x n ρ → x and y n ρ → y then αx n + βy n ρ → αx + βy, where α + β ≤ 1 and α, β ≥ 0.
, which implies ρ = 0. Therefore, we must have the ∆ 2 − constant k ≥ Γ if ρ is convex modular. On the other hand, many authors have investigated the stability using fixed point theorem of quasicontraction mappings in modular spaces without ∆ 2 − condition, which has been introduced by Khamsi [17]. Recently, the stability results of additive functional equations in modular spaces equipped with the Fatou property and ∆ 2 −condition were investigated in H. M. Kim, H. Y. Shin [16] and Sadeghi [34] who used Khamsis fixed point theorem. Also the stability of quadratic functional equations in modular spaces satisfying the Fatou property without using the ∆ 2 −condition was proved by Wongkum, Chaipunya and Kumam [38] .

Stability Theorem: Paranormed Spaces
To prove stability results, in this section let us take (N , P ) be a Fréchet space and (M, || · ||) be a Banach space.
Then there exists a unique additive mapping R bn : N → M which satisfying the functional equation (2) and the inequality where Γ = (α 3 1 + β 3 1 ) ; b = ±1 and R bn (y) is given by The mapping B : Proof: Fixing It follows from (7) It is ensure that the sequence is a Cauchy sequence in M. Since M is complete, there exists a limit function Also, by continuity of multiplication, we have Indeed, suppose if we replace y as Γ d y and divided by Γ d in (8), we get Letting p → ∞ in (8) and using (9), we arrive (4). If we replacing (x 1 , y 1 ) as (Γ c x 1 , Γ c y 1 , ) and divided by Γ c in (3) , we arrive for all x 1 , y 1 ∈ N . Letting c → ∞ in the above inequality and using (9), (P 5) and we see that R n (y) satisfies the Ramanujan Type additive functional equation (2) . It is clear that the existence of R n (y) is unique. Indeed, suppose that assume R 1 (y) be another additive mapping satisfying (2) and (9). So, one can easy to verify that for a positive integer d, we observe and for all y ∈ N . Now, for all y ∈ N . Again, taking y as y Γ in (7), we have In widely for a positive integer p, we arrive The rest of proof is similar to that of previous one. This completes the proof of the theorem.
for all x 1 , y 1 ∈ N and T > 0. Then there exists a unique additive mapping R bn : N → M which satisfying the functional equation (2) and the functional inequality for all y ∈ N .

Stability Theorem: Without Using The ∆ 2 − Condition
To prove stability results, in Sections 5 and 6, let us take N be an linear space and M ρ be an ρ−complete convex modular space.
Theorem 5.1 Assume M ρ satisfy the Fatou property. If R : N → M ρ is a function satisfying the inequality Then there exists a unique additive mapping R n : N → M ρ which satisfying the functional equation (2) and the functional inequality where Γ = (α 3 1 + β 3 1 ) and R n (y) is given by The mapping B : Proof: Fixing x 1 = y 1 = y in (18), we obtain Without using the ∆ 2 −condition it follows from (22) and (M S3)x for a positive integer c, we arrive It is ensure that the sequence is a ρ−Cauchy sequence in M ρ . Since M ρ is ρ−complete, there exists a ρ− limit function R 1 : N → M ρ defined by Indeed, suppose if we replace y as Γ d y and divided by Γ d in (23), we get It follows from the Fatou property that the inequality Thus, we arrive (19). If we replacing (x 1 , y 1 ) as (Γ c x 1 , Γ c y 1 , ) and divided by Γ c in (18) , we arrive for all x 1 , y 1 ∈ N . By convexity of ρ that for all x 1 , y 1 ∈ N . Taking p tends to infinity in the above inequality, we arrive for all x 1 , y 1 ∈ N . Thus, R n (y) satisfies the Ramanujan Type additive functional equation (2) . It is clear that the existence of R n (y) is unique.
Indeed, suppose that assume R 1 (y) be another additive mapping satisfying (2) and (24). So, one can easy to verify that for a positive integer d, we observe and for all y ∈ N . Now, for all y ∈ N .
Corollary 5.2 Let N be a normed space with norm || · || and M ρ satisfy the Fatou property. If R : N → M ρ is a function satisfying the inequality for all x 1 , y 1 ∈ N and T > 0. Then there exists a unique additive mapping R n : N → M ρ which satisfying the functional equation (2) and the functional inequality for all y ∈ N .
6. Stability Theorem: Using The ∆ 2 − Condition Theorem 6.1 Assume M ρ satisfy the Fatou property. If R : N → M ρ is a function satisfying the inequality Then there exists a unique additive mapping R n : N → M ρ which satisfying the functional equation (2) and the functional inequality where Γ = (a 3 1 + b 3 1 ) and R n (y) is given by The mapping B : Proof: Fixing x 1 = y 1 = y and again replace y = y Γ in (31), we reach Using the ∆ 2 −condition it follows from (35) and the convexity of the modular ρ that, Generalizing for a positive integer c > 0, we obtain So, for all c, d ≥ 0 with c ≥ d, we have Thus the sequence Γ c R y Γ c is a ρ−Cauchy sequence in M ρ . Since M ρ is ρ−complete, there exists a ρ− limit function R n : N → M ρ defined by Letting d = 0 and c → ∞ in (38) and using (39), we arrive (32). The rest of proof is similar to that of Theorem 5.1. This completes the proof of the theorem.

Conclusion
In this paper with the help of classical Hyers Method, we analyze the generalized Ulam -Hyers stability of a Ramanujan Type Additive Functional Equation in Paranormed Spaces and Modular spaces. The stability results in these two spaces are varying due to their respective definitions.