On a Stability of Logarithmic-Type Functional Equation in Schwartz Distributions

and Applied Analysis 3 see 8–11 . However, when we try to consider the Hyers-Ulam stability problems of functional equations, the differentiation is not available for solving them in both the space of infinitely differentiable functions and the space of distributions. In the paper 12 , using convolutional approach we initiated the following distributional version of the well-known Hyers-Ulam stability problem for the Cauchy functional equation: ‖u ◦A − u ◦ P1 − u ◦ P2‖ ≤ , 1.9 where ◦ is the pullback. See Section 3 for the pullback and see below the definition of the norm ‖ · ‖ in 1.9 . Using the heat kernel Et x 4πt −n/2 exp ( −|x| 2 4t ) , x ∈ R, t > 0, 1.10 we proved the stability problems 1.9 in the space of tempered distributions 7 by converting the inequality 1.9 to the classical stability problems ∣ ∣U ( x y, t s ) −U x, t −U(y, s)∣∣ ≤ 1.11 for all x, y ∈ R, t, s > 0, where U is an infinitely differentiable function in R × 0,∞ given by U x, t u ∗ Et x . We also refer the reader to 13 for the stability of Pexider equations in the space of tempered distributions. In 14 we extend the stability problems in the space of tempered distributions to the space of distributions. Instead of the heat kernel, using the regularizing function δt x : t−nδ x/t , x ∈ R, t > 0, where

where L : R → R is a logarithmic function. In 1950, Schwartz introduced the theory of distributions in his monograph Théorie des distributions 7 . In this book Schwartz systematizes the theory of generalized functions, basing it on the theory of linear topological spaces, relating all the earlier approaches, and obtaining many important results. After his elegant theory appeared, many important concepts and results on the classical spaces of functions have been generalized to the space of distributions.
Making use of differentiation of distributions, several authors have dealt with functional equations in the spaces of Schwartz distributions, converting given functional equations to differential equations, and finding the solutions in the space of distributions see [8][9][10][11] . However, when we try to consider the Hyers-Ulam stability problems of functional equations, the differentiation is not available for solving them in both the space of infinitely differentiable functions and the space of distributions. In the paper 12 , using convolutional approach we initiated the following distributional version of the well-known Hyers-Ulam stability problem for the Cauchy functional equation: where • is the pullback. See Section 3 for the pullback and see below the definition of the norm · in 1.9 . Using the heat kernel we proved the stability problems 1.9 in the space of tempered distributions 7 by converting the inequality 1.9 to the classical stability problems We also refer the reader to 13 for the stability of Pexider equations in the space of tempered distributions. In 14 we extend the stability problems in the space of tempered distributions to the space of distributions. Instead of the heat kernel, using the regularizing function δ t x : we prove that the unknown distributions in the functional inequalities are tempered distributions and then use the same method as in 12, 13 . In this paper, developing the previous method in 12-14 , we consider a distributional version of the Hyers-Ulam stability of 1.7 in the space of distributions as • denotes the pullback of distributions and the inequality · ≤ in 1.14 means | ·, ϕ | ≤ ϕ L 1 for all test functions ϕ x, y defined on R 2 . Since the tempered distributions are defined in whole real line or whole space R n , the methods used in 14 are not available for the inequality 1.14 . For the proof of the above problem, we need some technical method than those employed in 12-14 . Indeed, we will show a method to control a functional inequality satisfied in a subset of R 2 . As a direct consequence of the result, we obtain the Hyers-Ulam stability of 1.7 in L ∞ -sense, that is, the Hyers-Ulam stability of the inequality will be obtained. Finally, we also find locally integrable solutions of 1.7 as a consequence of the stability of the inequality 1.15 .

Stability in Classical Sense
In this section, we prove the Hyers-Ulam stability of the functional inequality where f, g, h : R → R and ≥ 0.
for all x > 0.
Proof. Let xy t, 1/x 1/y s. Then we have for all t > 0, s > 0 such that ts 2 ≥ 4. For given t, s > 0, choose a large u > 0 so that ts 2 u 2 ≥ 4, tsu 2 ≥ 4, su 2 ≥ 4, s 2 u 2 ≥ 2. Then in view of 2.3 , we have From 2.4 -2.7 , using the triangle inequality we have for all t, s > 0. Changing the roles of g and h in 2.3 , we can show that for all t, s > 0. Now we prove that for all t, s > 0. Replacing t by u and s by s/u in 2.3 , we have Similarly, we have 14 For given t, s > 0, let u 1/4 min{s 2 , t 2 s 2 , t 2 s, s}. Then, from 2.11 -2.14 , using the triangle inequality we get the inequality 2.10 . Now by Theorem 1.2, there exist functions L j : R → R, j 1, 2, 3, satisfying the logarithmic functional equation In view of 2.15 and 2.21 , we have for all t > 0, t / 1 and t n ≥ 2. Letting n → ∞ for t > 1 and letting n → −∞ for 0 < t < 1, Similarly we can show that L 1 L 3 . This completes the proof.
Letting g h f in Theorem 2.1, in view of the inequalities 2.4 , 2.5 , and 2.6 , using the triangle inequality we have for all t, s > 0. Thus by Theorem 1.2 we have the following.
for all x, y > 0. Then there exists a logarithmic function L : R → R such that for all x > 0.
c Ω such that for every compact set K ⊂ Ω there exist constants C > 0 and k ∈ N 0 for which Let Ω j be open subsets of R n j for j 1, 2, with n 1 ≥ n 2 .
Abstract and Applied Analysis 7 Definition 3.2. Let u j ∈ D Ω j and λ : Ω 1 → Ω 2 be a smooth function such that for each x ∈ Ω 1 the derivative λ x is surjective, that is, the Jacobian matrix ∇λ of λ has rank n 2 . Then there exists a unique continuous linear map λ * : D Ω 2 → D Ω 1 such that λ * u u • λ when u is a continuous function. We call λ * u the pullback of u by λ and is usually denoted by u • λ.
In particular, if S x, y x y, P 1 x, y x, P 2 x, y y, the pullbacks u • S, u • P 1 , u • P 2 can be written as for all test functions ϕ ∈ C ∞ c Ω . Also, if λ is a diffeomorphism a bijection with λ, λ −1 smooth functions , the pullback u • λ can be written as For more details of distributions we refer the reader to 7, 15 .

Stability in Schwartz Distributions
We employ a function δ on R n defined by

4.2
It is easy to see that δ x is an infinitely differentiable function with support {x : |x| ≤ 1}. Let δ t x : t −n δ x/t , t > 0 and u ∈ D R n . Then for each t > 0, u * δ t x u y , δ t x − y is a smooth function of x ∈ R n and u * δ t x → u as t → 0 in the sense that for all ϕ ∈ C ∞ c R n . Here after we denote by S, P, P 1 , P 2 : R 2 → R, R : R 2 → R, E : R → R by S x, y x y, P x, y xy, P 1 x, y x, P 2 x, y y,

4.4
Now we are in a position to prove the Hyers-Ulam stability of the inequality Recall that the inequality · ≤ in 4.5 means that | ·, ϕ | ≤ ϕ L 1 for all test functions ϕ x, y defined on R 2 .
Then J is a diffeomorphism with J −1 : V → U, J −1 x, y log 2 xy, log 2 x y /xy . Taking pullback by J in 4.5 and using 3.5 , we have Denoting by δ t ⊗ δ s x, y δ t x δ s y and convolving δ t ⊗ δ s in the left hand side of 4.9 we have, in view of 3.2 ,

4.10
Abstract and Applied Analysis 9 Similarly we have, in view of 3.3 and 3.4 , Thus the inequality 4.9 is converted to the classical stability problem for all x 2y ≥ 5 and 0 < t < 1, 0 < s < 1. From now on, we assume that 0 < t < 1, 0 < s < 1.
For given x, y ∈ R, choose z ≥ 1/2 5 |x| 2|y| . Then in view of 4.13 -4.16 , using triangle inequality, we have for all x, y ∈ R. Replacing x, t by y, s , y, s by x, t in 4.12 and changing the positions of v and w, we have for all x, y ∈ R. Now we prove that for all x, y ∈ R. From the inequality 4.12 , we have for all x, y, z such that 2x 2y − z ≥ 5, 2x y − z ≥ 5, 2y − z ≥ 5, and y − z ≥ 5. For given x, y ∈ R, choose z ≤ −5 − 2|x| − 2|y|. Then in view of 4.20 , using triangle inequality, we have

4.21
Letting s → 0 in 4.21 , we get the inequality 4.19 . Now in view of 4.17 , 4.18 , and 4.19 , it follows from Theorem 1.1 that for each 0 < t < 1, there exist functions A j ·, t , j 1, 2, 3, satisfying for all x ∈ R. Now we prove that A 1 A 2 A 3 . From 4.12 , using the triangle inequality, we have for all x 2y ≥ 5. Since u * δ t * δ s x → u * δ s x as t → 0 , in view of 4.26 it is easy to see that exists for all x ∈ R. Putting y 0 in 4.12 and letting s → 0 so that w * δ s 0 → h 0 , we have for all x ≥ 5. Similarly, we have for all x ≥ 5/2. Using 4.23 , 4.24 , 4.29 , and the triangle inequality, we have for all x ≥ 5. From 4.22 and 4.31 , we have for all x ∈ R, x / 0 and all integers k with kx ≥ 5. Letting k → ∞ if x > 0 and letting Similarly, using 4.23 , 4.25 , and 4.30 we can show that A 1 A 3 . Finally we prove that A 1 is independent of t. Fixing x ∈ R and letting t → 0 so that v * δ t x → g x in 4.12 , we have u * δ s x y − g x − w * δ s y ≤ 4.33 for all x 2y ≥ 5. The same substitution as the inequalities 4.13 -4.16 gives g x y − g x − g y g 0 ≤ 4 .

4.34
Using the stability Theorem 2 , we obtain that there exists a unique function A satisfying the Cauchy functional equation Now we show that A 1 x, t A x for all x ∈ R and 0 < t < 1. Putting y 0 in 4.33 , we have for all x ≥ 5. From 4.23 , 4.36 , and 4.37 , using the triangle inequality, we have for all x ≥ 5. From 4.38 , using the method of proving A 1 A 2 , we can show that A 1 x, t A x for all x ∈ R and t. Thus we have for some c 2 ∈ C. Similarly, letting t → 0 in 4.25 so that w * δ t 0 → h 0 , we have for some c 3 ∈ C. Now we prove the inequality for some c 1 ∈ C. Putting x −5, y 5 in 4.33 and using the triangle inequality, we have

4.43
Finally we show that the solution A of the Cauchy equation 4.35 has the form A x ax for some a ∈ C. Recall that g is the supremum limit of a collection of continuous functions v * δ t , 0 < t < 1. Thus, if we let g g 1 ig 2 , then both g 1 and g 2 are Lebesgue measurable functions. Now, as we see in the proof of Hyers-Ulam stability Theorem 2 , the function A is given by Then there exist a, c 1 , c 2 , c 3 ∈ C such that

4.46
Proof. Every locally integrable function f defines a distribution via the equation Viewing f, g, h as distributions, the inequality 4.45 implies for all ϕ ∈ C ∞ c R . Viewing C ∞ c as a subspace of L 1 dense subspace and using the Hahn-Banach theorem we obtain that the inequalities 4.49 hold for all ϕ ∈ L 1 . Now since L ∞ L 1 we get the inequalities 4.46 . This completes the proof.
As a direct consequence of the above result we solve the functional equation in L ∞ -sense, that is, we obtain the following.