Superstability and Solution of The Pexiderized Trigonometric Functional Equation

The present work continues the study for the superstability and solution of the Pexider type functional equation f(x + y) + g(x − y) = λ · h(x)k(y), which is the mixed functional equation represented by sum of the sine, cosine, tangent, hyperbolic trigonometric, and exponential functions. The stability of the cosine (d’Alembert) functional equation and the Wilson equation was researched by many authors: Baker [7], Badora [5], Kannappan [14], Kim ([16, 19]), and Fassi, etc [11]. The stability of the sine type equations was researched by Cholewa [10], Kim ([18], [20]). The stability of the difference type equation f(x + y) − f(x − y) = λ · h(x)k(y) for the above equation was studied by Kim ([21], [22]). In this paper, we investigate the superstability of the sine functional equation and the Wilson equation from the Pexider type difference functional equation f(x+y)−g(x−y) = λ ·h(x)k(y), which is the mixed equation represented by the sine, cosine, tangent, hyperbolic trigonometric functions, and exponential functions. Also, we obtain additionally that the Wilson equation and the cosine functional eqaution in the obtained results can be represented by the composition of a homomorphism. In here, the domain (G,+) of functions f, g, h, k : G → A is a noncommutative semigroup (or 2-divisible Abelian group), and A is an unital commutative normed algebra with unit 1A. The obtained results can be applied and expanded to the stability for the difference type’s functional equation which consists of the (hyperbolic) secant, cosecant, logarithmic functions. AMS Subject Classification 2010: 39B82, 39B52


Introduction
In 1940, Ulam [24] conjectured the stability problem of the functional equation. Next year, Hyers [13] obtained partial answer for the case of additive mapping in this problem.
In 1983, Cholewa [10] investigated the superstability of the sine functional equation His result was improved by Kim ([18,20]) in the following generalized sine functional equation Kim [22] also proved the stability of functional equation Kim [17] has studied the superstability of the Pexider-type trigonometric functional equation Fassi, Kabbaj, and Kim in [11] also obtained the superstability of cosine functional equation from (C λ f ghk ).
(C λ f ghk ) A trigonometric functional equation: cos(x + y) − sin(x − y) = (cos x − sin x)(cos y + sin y) causes the functional The present work continues the study for the stability and solution of the sine function and Wilson equation from the following Pexider type functional equation The aim of this paper is to study for the solution and the stability of the sine function and Wilson equation from the Pexider type's trigonometric functional equation: under the conditions f (x + y) − g(x − y) − λh(x)k(y) ≤ ϕ(x) or ϕ(y) on an unital commutative normed algebra (A, · ). The obtained stability results for (T λ f ghk ) can be applied to the hyperbolic trigonometric functions, several exponential functions, and Jensen equation as follows: e x+y − e x−y = 2 e x 2 (e y − e −y ) = 2e x cosh(y) n(x + y) − n(x − y) = 2ny : for f (x) = nx.
When (G,+) relates to the sine functional equation (S), it is a 2-divisible Abelian group, but it is not related to (S), then it is a noncommutative semigroup.
Let C be a the set of complex numbers, and C * = C \ {0}, V is a vector space. A is an unital commutative normed algebra with unit 1 A , and a −1 is an invertible element of 0 = a ∈ A (i.e a −1 a = aa −1 = 1 A ). And also ε, λ > 0 be a constant, ϕ : G → [0, ∞) be a function.
Let us denote the cosine type equation, sine type equation, Wilson type equation, and trigonometric type equation as following: f 2 Superstability of the Wilson and the sine functional equation.
In this section, we will investigate the superstability of the Wilson and the sine functional equation from the mixed trigonometric functional equation (T λ f ghk ), Proof. (i) From assumption, we can choose {y n } such that k(y n ) −1 → 0 as n → ∞.
Putting y = y n (with n ∈ N) in (1), we have for all x, y n ∈ G. Then for all y ∈ G. As n → ∞ in (2), we get for all y ∈ G.
Replacing y by y + y n in (1), we obtain for all x, y, y n ∈ V. Substituting y by −y + y n in (1), we find for all x, y ∈ G. Then for all x, y ∈ G.
This implies that for all x, y ∈ G.
We notice that the right-hand side converges to zero as n → ∞. So we can define a limit function l k : G → F as follows for all y ∈ G, then l k (0) = 2λ −1 and l k is an even function. Letting n → ∞ in (7), we see from (3) that as desired.
Putting y = x in (9), we get the equation Keeping this odd in mind, by means of (9) and (10), we infer the equality This, in return, leads to the equation valid for all x, y ∈ G which, in the light of the unique 2-divisibility of G, states nothing else but (S).
(ii-2) For the case f (x) = g(−x), It is enough to show that h(0) = 0. Suppose that this is not the case. Putting x = 0 in (1), due to h(0) = 0 and f (y) = g(−y), we obtain the inequality This inequality means that k is globally bounded, which is a contradiction. Thus, the claimed h(0) = 0 holds. (iii) In the case k satisfies (C λ ), the limit l k states nothing else but k from (8), so h and k validate a required equation (W λ h k ).
Suppose that there exists a sequence {x n } in G such that lim n→∞ h(x n ) −1 = 0. Then, (i) k satisfies k(x + y) + k(x − y) = λk(x)l h (y), in which, l h is an even function such that l h (0) = 2λ −1 .
(ii) k satisfies (S) under one of the cases k(0) = 0 or f (x) = g(x), (iii) In particular, if h satisfies (C λ ), then k and h satisfies Proof. Let us choose {x n } in G such that lim n→∞ h(x n )) −1 = 0. Taking x = x n (with n ∈ N) in (12), dividing both sides by λ · h(x n ) , and passing to the limit as n → ∞, we obtain that for all x n , y ∈ G Replace (x, y) by (x n + y, x) and replace (x, y) by (x n − y, x) in (12). Thereafter we go through same procedure as the equations (4) ∼ (7) of Theorem 1, i.e., add the above two inequalities, and dividing by λ · h(x n ), then it gives us the existence of a limit function where the function l h : G → C satisfies the equation as desired.
The process of the remainders (ii) and (iii) are achieved by following the same steps as Theorem 1.
The following corollaries follow immediate from the Theorems 1, 2.
Corollary 3. Assume that f, g, h, k : G × G → A satisfy the inequality
In complex analysis, the trigonometric function can be represented by the exponential function. Hence, from the obtained results in Sections 2, the explicit solution of Wilson type equation (W f g ) is estimated and can be found by direct calculations and can also be obtained directly from Lemma.
Let us find the solution for the trigonometric equations contained cosine(d'Alembert), sine, Wilson type equations.(W f g ). And let us investigate a representation of a solution for them.
It is easy to verify shows that the following lemma holds.
Lemma 4. Let f, g : G × G → C * satisfy the Wilson type equation , Then, g satisfies (C λ ), and g, f are given by where c, d ∈ C, and E : G → C * is a homomorphism.
Letting λ = 2 in the case(ii) of Remark 1 in Section 2, then we obtain the stability of Wilson type equations.(wf g ) and (wf g ) from the stability results of (T λ f ghk ). It's solution's types be represented as following results.
Theorem 5. Assume that f, g, h, k : G × G → C * satisfy the inequality If k fails to be bounded, then (i) h and l k satisfy (W λ h l k ). In particular, if k satisfies (C), then h and k satisfies (W λ h k ),) and h, k are given by where c, d ∈ C, and E : G → C * is a homomorphism.
(ii) h satisfies (S) under one of the cases h(0) = 0 or f (−x) = −g(x), and h is of the form where A : G → C is an additive function, c ∈ C, E is as in (i).
Proof. The proof of the Corollary is enough from Theorem 1 except for the solution. However, they are immediate from the following: (i) Appealing to the solutions of the Wilson equation(wf g ) in paper [14] and [15](theorem 3.41, pp. 148) (see also [2,3]), then the given explicit solutions are taken from it.
(ii) Appealing to the solutions of (S) in paper ( [15], theorem 3.44, pp. 153) (see also [14]), the explicit shapes of h are as stated in the statement of the theorem. This completes the proof of the Corollary. Corollary 6. Assume that f, g, h, k : G × G → C * satisfy the inequality If h fails to be bounded, then (i) k and l h satisfy (W λ k l h ). In particular, if h satisfies (C), then k and h satisfies (W λ k h ), and k, h are given by where c, d ∈ C, and E : G → C * is a homomorphism.
(ii) k satisfies (S) under one of the cases h(0) = 0 or f (−x) = −g(x), and k is of the form where A : G → C is an additive function, c ∈ C, E is as in (i).
Corollary 7. Suppose that f, g, h, k : G → C * satisfy the inequality (a)If k fails to be bounded, then (i) h and l k satisfy (W λ h l k ). In particular, if k satisfies (C), then h and k satisfies (W λ h k ),) and h, k are given by where c, d ∈ C, and E : G → C * is a homomorphism.
(ii) h satisfies (S) under one of the cases h(0) = 0 or f (−x) = −g(x), and h is of the form where A : G → C is an additive function, c ∈ C, E is as in (i).
(b) If h fails to be bounded, then (i) k and l h satisfy (W λ k l h ). In particular, if h satisfies (C), then k and h satisfies (W λ k h ),) and k, h are given by where c, d ∈ C, and E : G → C * is a homomorphism.
(ii) k satisfies (S) under one of the cases h(0) = 0 or f (−x) = −g(x), and k is of the form where A : G → C is an additive function, c ∈ C, E is as in (i).

Application to the Banach space
In all results obtained in Section 2, the range of functions on the Abelian group can be extended to the semisimple commutative Banach space. We will represent only for the main equation (T λ f ghk ).
Theorem 8. Let (E, · ) be a semisimple commutative Banach space. Assume that f, g, h, k : G → E satisfy one of each inequalities For an arbitrary linear multiplicative functional x * ∈ E * , (a) Suppose that x * • k fails to be bounded, then (i) h satisfies (W λ h l k ) with an even function l k such that l k (0) = 2λ −1 . In particular, if k satisfies (C λ ), then h and k satisfies the (W λ h k ).
(ii) h satisfies (S) under one of the cases (x * • h)(0) = 0 or (x * • f )(x) = −(x * • g)(x). (b) Suppose that x * • h fails to be bounded, then (i) k satisfies (W λ k l h ) with an even function l h such that l h (0) = 2λ −1 . In particular, if h satisfies (C λ ), then k and h satisfies the (W λ k h ).
Proof. (a)(i). Assume that (16) holds and arbitrarily fixes a linear multiplicative functional x * ∈ E * . As is well known, we have x * = 1, hence, for every x, y ∈ G, we have