Journal of Generalized Lie Theory and Applications On The Stability of Conditional Homomorphisms in Lie C*-algebras

During the last years several papers studying conditional functional equations have appeared. They mostly deal with equations satisfied on some restricted domain and many among them concern equations postulated for orthogonal vectors. In this paper, we define the conditional homomorphisms with the predecessor defined by γ ( x )=γ ( y ) with an even mapping γ. Then, using a fixed point theorem, we investigate the stability of the conditional homomorphisms in Lie C * -algebras.


Introduction
The problem of solving equations on spheres is an important part of computational aspects, ordinary differential equations and partial differential equations [1][2][3]. The development of computational methods for solving partial differential equations on the sphere is complicated by problems that result from the spherical coordinate system itself. However, some of the numerical methods for solving vector differential equations are applicable to any vector differential equation on the sphere.
The functional equation (ξ) is stable if any function f satisfying the equation (ξ) approximately is near to the true solution of (ξ). An interested reader can find more information on such problems with the emphasis on functional equations in [4][5][6][7][8].
Let X and Y are two Banach spaces. Consider : → f X Y to be a mapping such that f (tx) is continuous in t ∈  for each fixed x ∈ X.
Assume that there exist constants ε ≥ 0 and p ∈ [0, 1) such that For all x, y ∈ X, Rassias [6] showed that there exists a unique  -linear mapping : → T X Y such that For all x ∈ X, A generalization of the theorem of Rassias was obtained by Gavruta [9] by replacing the unbounded Cauchy difference by a general control function : [0, ) ϕ × → ∞ G G in the spirit of Rassias approach.
In 1994, Alsina and Garcia-Roig [10] solved the conditional equation whenever ∥x∥=∥y∥ for continuous mapping f from a real inner product space (X, (.|.)) with dim X ≥ 2 into a real topological linear space Y. They recognized the connection between this equation and the orthogonally additive functional equation. They also obtained the linearity of such a function f in the case where Y=R n .
Sikorska [11] studied a generalized stability of Cauchy and Jensen functional equations, where the respective Cauchy or Jensen differences are approximated by arbitrary functions. Moreover, [12] Sikorska solved the conditional Pexider functional equation on prescribed sets being generalizations of spheres. We refer the reader to [13][14][15][16][17] for some interesting results on the stability of conditional functional equations.
In this paper, we apply a fixed point theorem to prove the stability of conditional homomorphisms using contractively subhomogeneous and expansively superhomogeneous functions.
The paper is organized as follows. In Section 2, some necessary preliminaries and summarization of some previous known results are presented. The concepts of conditional homomorphism and conditional Jordan homomorphism in Lie C*-algebras are introduced. In Section 3, we deal with the stability of conditional homomorphisms and conditional Jordan homomorphisms in Lie C*-algebras. Finally, a conclusion is given in Section 4.

If
: → f X Y satisfies the condition Then f is additive.
They dealt also with the Hyers-Ulam stability problem cf. [4,5] for such a more general version of (1.1). In order to make the above assumptions more readable, they gave the following example.
A C*-algebra A endowed with the Lie product [x, y]=xy −yx on A is called a Lie C*-algebra. The stability problems of functional equations between C*-algebras have been investigated by a number of authors [19][20][21][22].

Definition 2.4:
For explicitly later use, we state the following theorem.
Theorem 2.5: (Banach) Let (X, d) be a complete metric space and consider a mapping : Λ → X X as a strictly contractive mapping, that is For all x, y ∈ X and for some (Lipschitz constant) 0<L<1. Then there exists a unique a ∈ X such that Λa=a. Moreover, for each x ∈ X, Let A, B be real vector spaces. We recall that if there exists a constant L with 0<L<1 such that a n−times mapping : For all x, x j ∈ A (1 ≤ ≠ ≤ j i n ) and all positive integers λ, then we say that ρ is n-contractively sub homogeneous if ℓ=-1, and ρ is n-expansively superhomogeneous if l=−1. It follows by the above inequality that ρ satisfies the following properties: For all x, x j ∈ A ( 1 ≤ ≠ ≤ j i n ) and all positive integers λ.

Remark 2.6:
If ρ is n-contractively sub-additive then ρ is contractively sub homogeneous of degree n and if ρ is n-expansively super-additive, then ρ is and expansively super homogeneous of degree n.

The Main Results
Throughout this section, let A and B be Lie C*-algebras, dim A ≥ 2 and γ be an even mapping from A to a nonempty set Z, satisfying the conditions (c 1 ) − (c 3 ) and for all x, y ∈ A, γ (x)=γ (y) implies that be either 3-expansively super homogeneous mappings for ℓ=−1 or 3-contractively sub homogeneous mappings for be either 2expansively super homogeneous mappings for ℓ=−1 or 2 -contractively sub homogeneous mappings for ℓ=1, with constant 0<L=L (ℓ)<1.

Theorem 3.2: Suppose that
: For all 0 1 1/ ∈ n t  and x, y, z ∈ A with γ (x)=γ (y). If f 3 is an odd mapping and f i (0)=0 (i=1, 2, 3), then there exists a unique conditional homomorphism : → H  such that x y x x y x x y y y x y x y x y x y x y x y x y x y (3.4) For all x ∈ A and some y 0 ∈ A for which x and y 0 satisfy the condition (c 3 ) with λ=1.
It is easy to see that ( , ) d  is a complete metric space. Let us consider the linear mapping : For all x ∈ A. Let , f g ∈  and let C ∈ [0, ∞) be an arbitrary constant with d (f, g)<C. From the definition of d, we have For all x, y ∈ A with γ (x)=γ (y). By the assumption and the last inequality, we get For all x, y ∈ A with γ (x)=γ (y) and therefore So, Λ is a strictly contractive mapping with the Lipschitz constant L.
Also, by the evenness of γ we have Substituting both x, y in (3.1) by x + y 0 and x − y 0 , we have It follows from the inequalities (3.5) − (3.12) and the triangle inequality that If we replace x and y 0 in (3.13) with 2 x and 0 2 y , respectively, and divide by 2 the resulting inequality, then we have If we replace x and y 0 in (3.13) with 4 x and 0 4 y , respectively, and using 3-expansivity of φ, then We can reduce (3.14) and (3.15) to and then That is, Let x ∈ A and y 0 ∈ A be the element for which x and y 0 satisfy the condition (c 3 ) with λ=1. Utilizing the triangle inequality, the oddness of f 3 and the inequalities (3.5) and (3.6), we infer that Combining ( From the inequalities (3.17) and (3.18), we get For all ∈  n , whence And by 3-expansivity of φ we conclude that In addition, it is clear from (3.1) that the following inequality  For all x, y ∈ A with γ (x)=γ (y). Now, letting x, y ∈ A with γ (x)=γ (y) and n ∈  , by (3.23) and conditional additivity of H one obtains