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We consider the functional equation f(ax+y)+f(ax-y)-a(a+1)f(x)-a(a-1)f(-x)-[f(y)+f(-y)]+k[f(x+y)+f(x-y)-2f(x)-f(y)-f(-y)]=0 for a fixed rational number a with a≠1,-1,0 and a fixed real number k. We study the solution of the equation between linear spaces and prove the generalized Hyers-Ulam stability for it when the target space is a fuzzy normed space.

We investigate the following typical form of a certain class of quadratic functional equations: . Furthermore, we provide a systematic program to prove the generalized Hyers-Ulam stability for the class of functional equations via the stability for the typical form.

We would like to study the relationship between the integral of the derivative and the derivative of the integral. This is the study of whether these two are the same or not. Also, this study began with the question of Kreyszig’s assumptions presented in [8]. The research method used two tools: the Riemann integral and its generalized concept, the Lebesgue integral. The obtained research results give some answer to this question. [ABSTRACT FROM AUTHOR]

In this paper, we consider the generalized Hyers-Ulam stability for the following cubic functional equation f (x + 2y) - 3f (x + y) + 3f (x) - f (x - y) - 6f (y) + Gf (x, y) = 0. with an extra term Gf which is a functional operator of f. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the orthogonally Euler-Lagrange type cubic functional equation f(ax + by) + f(ax - by) - ab²[f(x + y) + f(x - y)] - 2a(a² - b²)f(x) + c[f(x + 2y) - 3f(x + y) + 3f(x) - f(x - y) - 6f(y)] = 0; x?y for fixed non-zero rational numbers a; b and a fixed non-zero real number c with a² ≠ b² and a ≠ ±1 and prove the generalized Hyers-Ulam stability for it by using the fixed point method, [ABSTRACT FROM AUTHOR]

In this paper, we prove the generalized Hyers-Ulam stability for the following functional inequality p(f(x+y)+f(x-y)-2f(x)-2f(y))≥ p(k[f(ax+by)+f(ax-by)-2a²f(x)-2b²f(y)]) in modular spaces without Δ-conditions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the functional inequality N(f(2x + y) + f(2x - y) - 6f(x) - 2f(-x) - f(y) - f(-y); t) ≥ N(f(x + y) + f(x - y) - 2f(x) - f(y) - f(-y); kt) for some fixed real number k and prove the generalized Hyers-Ulam stability for it in fuzzy Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper, we prove the generalized Hyers-Ulam stability for the following cubic functional equation with involution f(2x + y) + f(2x + σ(y)) - 2f(x + y) - 2f(x + σ(y)) - 12f(x) = 0 in modular spaces by using a fixed point theorem. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the functional inequality N(f(x + 2y) - 3f(x + y) + 3f(x) - f(x - y) - 6f(y), t) ∁ N(f(2x + y) + f(2x - y) - 2f(x + y) - 2f(x - y) - 12f(x); kt) for some fixed nonzero real number k and prove the generalized Hyers-Ulam stability for it in fuzzy Banach spaces by fixed point methods. [ABSTRACT FROM AUTHOR]

In this paper, we prove the generalized Hyers-Ulam stability of the following additive-quadratic p-functional inequalities N(f(x+y) + f(x-y) - 2f(x) - f(y) - f(-y), -p[4f(x+y/2) + 4f(x-y/2) - 3f(x) + f(-x) - f(-y)],t) ≥ t/t+ɸ(x,y), (p≠1, 1/2) and n(4f(x+y/2)+4f(x-y) -2f(x)-f(y)-f(y) - f(-y)], t) ≥ t/t+ɸ(x,y), (p ≠ 0,1,2) in fuzzy Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the following typical form of a class of cubic functional equations: f (x + 2y) - 3f (x + y) + 3f (x) - f (x - y) - 6f (y) + k[f (mx + y) + f (mx - y)- m[f (x + y) + f (x - y)] - 2(m3 -m)f (x)] = 0 for some rational number m and some real number k. Furthermore, we provide a systematic program to prove the generalized Hyers-Ulam stability for the class of functional equations via the stability for the typical form in fuzzy normed spaces. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following functional equation af(x + y) + bf(x - y) + cf(y - x) = (a + b)f(x) + cf(-x) + (a + c)f(y) + bf(-y) for a xed real numbers a; b; c with a = b + c and a ≠0. We study the fuzzy version of the generalized Hyers-Ulam stability for it in the sense of Mirmostafaee and Moslehian. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a fixed point theorem for a mapping without the condition of bounded orbit in a modular space, whose induced modular is lower semi-continunous. Using this fixed point theorem, we prove the generalized Hyers-Ulam stability for an additive-cubic functional equation in modular spaces without Δ2-conditions and the convexity. [ABSTRACT FROM AUTHOR]

In this paper, we consider the generalized Hyers-Ulam stability for the following quadratic functional equation.... Here Gf is a functional operator of f . We consider some sufficient conditions on Gf which can be applied easily for the generalized Hyers-Ulam stability, and illustrate some new functional equations by using them. [ABSTRACT FROM AUTHOR]

We introduce a viewpoint to analyze functional inequalities for the generalized version of Hyers-Ulam stability problems. Especially, this article focuses the following generalized version of Cauchy functional inequality containing an general extra term Gf. ||f(x + y) - f(x) - f(y) - Gf(x,y)|| ≤ φ (x,y) We investigate the functional operator properties of Gf. And by virtue of this observation, we provide a method to produce lots of interesting additive functional equations as applications. [ABSTRACT FROM AUTHOR]