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Additive-quadratic ๐žบ-functional inequalities in fuzzy Banach spaces

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ํ•œ๊ตญ์ˆ˜ํ•™๊ต์œกํ•™ํšŒ|2016.05|์ด๋ก ์ˆ˜ํ•™๊ณผ ๊ต์ง์ˆ˜ํ•™ (Theoretical Mathematics and Pedagogical Mathematics)|์ œ23๊ถŒ ์ œ2ํ˜ธ|pp. 163-179

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fuzzy Banach spaceadditiveโ€“quadratic ๐žบ โ€“functional inequalityHyers Ulam stability
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Let \begin{eqnarray*} M_1f(x,y) : & = & \frac{3}{4} f(x+y) - \frac{1}{4}f(-x-y) + \frac{1}{4} f(x-y) + \frac{1}{4} f(y-x) - f(x) - f(y),\\ M_2 f(x,y): &= &2 f\left( \frac{x+y}{2} \right) + f\left( \frac{x-y}{2}\right) + f\left( \frac{y-x}{2}\right) - f(x) - f(y). \end{eqnarray*} Using the direct method, we prove the Hyers-Ulam stability of the additive-quadratic ๐žบ-functional inequalities \begin{eqnarray} N\left(M_1f(x,y)-\๐žบ M_2f(x,y), t \right)\ge \frac{t}{t+\varphi(x, y)} \end{eqnarray} and \begin{eqnarray} N\left(M_2f(x,y) -\๐žบM_1f(x,y) , t\right) \ge \frac{t}{t+\varphi(x, y)} \end{eqnarray} in fuzzy Banach spaces, where ๐žบ is a fixed real number with ๐žบ \ne 1.

Abstract

Let \begin{eqnarray*} M_1f(x,y) : & = & \frac{3}{4} f(x+y) - \frac{1}{4}f(-x-y) + \frac{1}{4} f(x-y) + \frac{1}{4} f(y-x) - f(x) - f(y),\\ M_2 f(x,y): &= &2 f\left( \frac{x+y}{2} \right) + f\left( \frac{x-y}{2}\right) + f\left( \frac{y-x}{2}\right) - f(x) - f(y). \end{eqnarray*} Using the direct method, we prove the Hyers-Ulam stability of the additive-quadratic ๐žบ-functional inequalities \begin{eqnarray} N\left(M_1f(x,y)-\๐žบ M_2f(x,y), t \right)\ge \frac{t}{t+\varphi(x, y)} \end{eqnarray} and \begin{eqnarray} N\left(M_2f(x,y) -\๐žบM_1f(x,y) , t\right) \ge \frac{t}{t+\varphi(x, y)} \end{eqnarray} in fuzzy Banach spaces, where ๐žบ is a fixed real number with ๐žบ \ne 1.

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