On the Stability of the Generalized Psi Functional Equation

: In this paper, we investigate the generalized Hyers–Ulam stability for the generalized psi functional equation f ( x + p ) = f ( x ) + ϕ ( x ) by the direct method in the sense of P. Gˇavruta and the Hyers–Ulam–Rassias stability.

Equations with functional perturbations are interesting from many points of view [17,18] and enjoy various applications especially in the theory of integral [19] and functional-differential equations [18].
The psi (digamma) function is defined by where Γ(x) stands for the Gamma function. The Gamma functional equation is the following: The stability for this functional equation is proved in Jung [13] and Kim [15]. Since the Gamma functional equation implies that it follows that the psi function (2) constitutes the solution of the equation: which is the so-called psi functional equation. Due to (3), we can consider the functional equation in which f , ϕ are unknown functions, and x, p are positive real numbers. Let us recall that, in the Peano axioms, n = n + 1 is called the successor of n. Therefore, the functional equation with the unit step is implied, which can be called the unit successor functional equation with unit step. More generally, the functional equation can be considered the α-successor functional equation with p-step, where the constant α p = α depends on a fixed positive real number p. The aim of the present paper is to investigate the generalized Hyers-Ulam stability for the functional Equation (4), in the sense of P. Gǎvruta [21] and the Hyers-Ulam-Rassias stability [22].
As a corollary, we obtain stability results of the successor functional Equations (5) and (6) and the psi functional Equation (3).
Throughout this paper, let R and R + be the set of real numbers and the set of all positive real numbers, respectively. Set R * := R + ∪ {0}. Let p, δ > 0 be fixed real numbers, and n be a non-negative integer.

Stability of the Functional Equation (4)
In this section, we will investigate the Hyers-Ulam-Rassias stability as well as the stability in the sense of P. Gǎvruta, for the functional Equation (4) Theorem 1. Let a mapping θ : R + → R * satisfy the inequality Assume that f : R + −→ R * satisfies the inequality Then, there exists a unique solution F : Proof. For any x > 0 and for every positive integer n, we define By (8), we have The right-hand-side of (12) converges to zero as m −→ ∞, by (7). In view of (12), the sequence {P n (x)} is a Cauchy sequence for all x ∈ R + .
Hence, we can define a function F : R + → R * by By induction on n, we show that for all n. For n = 1, the inequality (13) follows immediately from (8). Assume that (13) holds true for some n. Then, from (11) and (13), it follows that Therefore, (13) holds true for all positive integers n.
From the definition of P n , it follows that F satisfies the functional Equation (4) F If G : R + −→ R * is another function which satisfies (9) and (4), then it follows from (10) and (9) that for all n, it holds Thus, the uniqueness of the solution of Equation (4) is established, and this completes the proof of Theorem 1.
For the stability in the sense of Gǎvruta [21] to be valuable, there must exist a convergent sequence which satisfies the assumption (7) of the Theorem.
We can show that the infinite series of the undefined function θ of the condition (7) converges, by the improper integral test, the p-series test, or the ratio test for the infinite series.
By replacing the function θ in the stability inequality (8) by an arbitrary exponential function, the assumption (7) of Theorem 1 can be omitted.
Then, there exists a unique solution F : Proof. The limit of the ratio test implies that respectively.
Then, there exists a unique solution F : R + −→ R * of the Equation (4) with Proof. Set θ(x) = δ x r in Theorem 1. Since the convergence condition of Ψ is satisfied by the p-series test in the case when r > 1, Corollary 2 follows.
The result (14) of Corollary 2 is the following: where · stands for the Gaussian notation.
The results below concern the Hyers-Ulam-Rassias stability of the successor functional Equations (5) and (6), and the psi functional Equation (3).
for fixed r > 1, and constant α p , which depends on p.
Then, there exists a unique solution F : R + −→ R * of the equation Proof. Let ϕ(x) := ϕ(p) = α p that is a constant. Namely, we define The following process is similar to that of Theorem 1.
The next result constitutes the Hyers-Ulam-Rassias stability for the psi functional Equation (3).
for a fixed real number r > 1.
Then, there exists a unique solution F : R + −→ R * of Equation (3) with in Theorem 1. By applying the p-series test, the result follows.
Then, there exists a unique solution F : R + −→ R * of the equation

Proof. Setting
in Theorem 1, and applying the p-series test, the result follows.

Remark 1.
By setting this result can be immediately extended to the more general form f (φ(x)) = f (x) + ϕ(x).

Conclusions
In this paper, we proved the generalized Hyers-Ulam stability for the generalized psi functional equation by the direct method in the sense of P. Gǎvruta and the Hyers-Ulam-Rassias stability. As corollaries, we obtain the generalized Hyers-Ulam stability of the unit successor functional Equation (5) with unit step and the α p -successor functional Equation (6) with p-step.
Author Contributions: The authors contributed equally for the preparation of this paper. All authors have read and agree to the published version of the manuscript.
Funding: The first author of this work was supported by Kangnam University Research Grant in 2018.