ON THE STABILITY OF GENERALIZED GAMMA FUNCTIONAL EQUATION

We obtain the Hyers-Ulam stability and modified Hyers-Ulam stability for the equations of the formg(x+p)=φ(x)g(x) in the following settings: |g(x+p)−φ(x)g(x)| ≤ δ, |g(x+p)−φ(x)g(x)| ≤φ(x), |(g(x+p)/φ(x)g(x))−1| ≤ψ(x). As a consequence we obtain the stability theorems for the gamma functional equation.


Introduction. The functional equation
is said to have the Hyers-Ulam stability if for an approximate solution f , such that for some fixed constant δ ≥ 0, there exists a solution g of (1.1) such that for some positive constant ε depending only on δ.Sometimes we call f a δ-approximate solution of (1.1) and g ε-close to f .Such an idea of stability was given by Ulam [13] for Cauchy equation f (x + y) = f (x)+ f (y) and his problem was solved by Hyers [4].Later, the Hyers-Ulam stability was studied extensively (see, e.g., [6,8,10,11]).Moreover, such a concept is also generalized in [2,3,12].As in [5] we say (1.1) has the generalized Hyers-Ulam-Rassias stability if for an approximate solution f , such that for some fixed function ψ(x), there exists a solution g of (1.1) such that f (x)− g(x) ≤ Φ(x) (1.5) for some fixed function Φ(x) depending only on ψ(x).We say (1.1) has the stability in the sense of Ger if for an approximate solution f , such that for some fixed function ψ(x), there exists a solution g of (1.1) such that for some fixed functions α(x) and β(x) depending only on ψ(x).
The three senses of the Hyers-Ulam stability are discussed in [5] for the generalized gamma functional equation where p > 0 is a fixed real constant.It is proved that (1.8) has the Hyers-Ulam stability if for a nonnegative constant n 0 , has the generalized Hyers-Ulam-Rassias stability if the function ψ(x) in (1.4) satisfies for a nonnegative constant n 0 , and has the stability in the sense of Ger if the function for a nonnegative constant n 0 .In [5] conditions (1.9), (1.10), and (1.11) are checked with the concrete equation g(x + 1) = xg(x), which the well-known gamma function Γ (x) = ∞ 0 e −t t x−1 dt satisfies.
Proof.Consider the sequence {u j (x)} defined by by (2.1).By ratio test we see that the series (1.9) converges for all x > n 0 .By [5,Theorem 2.1] we obtain the Hyers-Ulam stability.
A similar idea to give conditions of stability by use of inferior limit was once taken in [7].
Example 2.2.It is easier to see that the gamma functional equation has the Hyers-Ulam stability because in this case a and condition (2.1) in Theorem 2.1 is satisfied.
Example 2.3.As in [9], the G-functional equation has the Hyers-Ulam stability because we consider a(x) = Γ (x), which obviously satisfies the same as in (2.5).
Similarly, (1.8) also has the Hyers-Ulam stability when a(x) = x r where the real r > 0 or a(x) = log x, sinh x, which are not power functions, because (2.5) holds in these cases.
Example 2.4.The functional equation has the Hyers-Ulam stability because in this case a(x) = arctan x satisfies and condition (2.1) in Theorem 2.1 is satisfied.
Example 2.5.With notations that where q ∈ (0, 1), the equation called q-Gamma functional equation, is considered in [1,14].On {x ∈ C : x > 0} it has solutions (2.11) In particular, the first one is called Jackson's q-Gamma function.Restricted to real line, namely to (0, +∞), this equation has the Hyers-Ulam stability because in this case a(x) = [x] and which implies that condition (2.1) in Theorem 2.1 is satisfied.
Theorem 2.1 also provides a method to discuss cases of divergent a(x).
Example 2.6.Consider the functional equation Different from Example 2.6, in some cases the fact lim inf x→+∞ a(x) > 1 does not hold, but we can still discuss the Hyers-Ulam stability with Theorem 2.1.
Remark that in the first three cases lim k→∞ ψ(x + p(k − 1))/ψ(x + pk) converges but in case (iv) this limit may not exist.
Example 3.3.Consider ψ(x) = sin x for (1.8) where a(x) = x and a(x) = Γ (x) separately.They are in case (iv) of the corollary although lim k→∞ ψ(x + p(k − 1))/ψ(x + pk) does not converge.Therefore both gamma functional equation and G-functional equation have the generalized Hyers-Ulam-Rassias stability with such an ψ(x).Besides, the q-Gamma functional equation (2.10) can be considered in cases (i), (ii), and (iii), so it has the generalized Hyers-Ulam-Rassias stability with ψ(x) in the forms of polynomial, logarithm, and exponential function r x where r < 1/(1 − q).
Remark that in Theorem 4.1 we do not require condition (1.9).This condition, required in [5,Theorem 3.2], is in fact unnecessary.In the proof of [5,Theorem 3.2] the convergence in (1.11) guarantees that {log P n (x)} is a Cauchy sequence.Thus L(x) := lim n→∞ log P n (x) exists and so does lim n→∞ P n (x).The restriction of a(x) is given by the convergence in (1.11) and the range of ψ in (0, 1) because it is required that |f Corollary 4.2.Suppose that the function ψ : (0, +∞) → (0, 1) is continuous and decreasing such that for some constant η > 1.Then (1.8) has the stability in the sense of Ger. Proof.Obviously, Taking summation, we obtain where s > 1, n 0 ≥ 0, and δ > 0. Clearly ψ(x) := δ/x s satisfies (4.3).Thus the gamma equation (2.4) has the stability in the sense of Ger with such a ψ(x).

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
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