We proposed to give some new ψ − coupled fixed point theorems using simulation function coupled with other control functions in a complete partially ordered metric space which includes many related results. Further we prove the existence of solution of a fractional integral equation by using this fixed point theorem and explain it with the help of an example.

We will investigate the superstability of the trigonometric functional equation from the following Pexider type functional equation: f ( x + y ) − f ( x − y ) = λ · g ( x ) h ( y ) , λ is constant, which is a trigonometric functional equation mixed by the sine and cosine function. More- over, the equation can be considered by the mixed functional equation of the hyperbolic trigonometric functions, several exponential type functions, and Jensen type equation.

In this paper, we estabilish the general solution of sexvigintic functional equation f ( x + 13 y ) − 26 f ( x + 12 y ) + 325 f ( x + 11 y ) − 2600 f ( x + 10 y ) + 14950 f ( x + 9 y ) −65780f (x + 8y) + 230230f (x + 7y) − 657800f (x + 6y) + 1562275f (x + 5y)−3124550f (x + 4y) + 5311735f (x + 3y) − 7726160f (x + 2y) + 9657700f (x + y)−10400600f (x) + 9657700f (x − y) − 7726160f (x − 2y) + 5311735f (x − 3y)−3124550f (x − 4y) + 1562275f (x − 5y) − 657800f (x − 6y) + 230230f (x − 7y)−65780f (x − 8y) + 14950f (x − 9y) − 2600f (x − 10y) + 325f (x − 11y)−26f(x − 12y) + f (x − 13y) = 26!f(y) and investigate the Hyers-Ulam stability of this functional equation in Banach spaces using two different methods.