We obtain the Hyers‐Ulam stability and modified Hyers‐Ulam stability for the equations of the form g(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.

The aim of this paper is to study the superstability related to the d'Alembert, the Wilson, the sine functional equations for the trigonometric functional equations as follows: .

A new coupled fixed point theorem related to the Pata contraction for mappings having the mixed monotone property in partially ordered complete metric spaces is established. It is shown that the coupled fixed point can be unique under some extra suitable conditions involving mid point lower or upper bound properties. Also the corresponding convergence rate is estimated when the iterates of our function converge to its coupled fixed point. MSC:47H10, 34B15, 54H25.

The purpose of this article is to present some fixed point theorems for (ψ, φ)-weak contractive mappings in a complete metric space endowed with a partial order. As an application of the main result, we give an existence theorem for the solution of a periodic boundary value problem.

In this paper, we investigate the stability of an additive-quadratic-quartic functional equation f ( x + 2 y ) + f ( x − 2 y ) − 2 f ( x + y ) − 2 f ( − x − y ) − 2 f ( x − y ) − 2 f ( y − x ) + 4 f ( − x ) + 2 f ( x ) − f ( 2 y ) − f ( − 2 y ) + 4 f ( y ) + 4 f ( − y ) = 0 $$\begin{align*}f(x+2y)& +f(x-2y)-2f(x+y)-2f(-x- y)-2f(x-y)-2f(y-x)\nonumber \\ &+4f(-x)+ 2f(x)-f(2y)-f(-2y)+4f(y)+4f(-y)=0 \end{align*}$$by the direct method in the sense of Găvruta.