In this paper, we attempt to deal with two types of set-valued quadratic ρ-functional inequalities, which are related to quadratic type set-valued functional equation. We also discuss the Hyers–Ulam stability of such set-valued quadratic ρ-functional inequalities by applying the fixed point approach.

In this paper, we establish the general solution of the following functional inequality ll2f(x) +2 f(y)+2 f(z)- f(x+y)- f(y+z)ll S llf(x+z)ll, investigate the generalized Hyers-Ulam stability of this inequality in Banach spaces and non-Archimedean Banach spaces by using two different approaches.

The purpose of this paper is first to introduce the notation of matrix intuitionistic fuzzy normed spaces, and then by virtue of this notation to study the Hyers-Ulam stability results concerning the mixed type additive-quadratic functional equation \begin{document}$ 2k[f(x+ky)+f(kx+y)] = k(1-s+k+ks+2k^{2})f(x+y)\\+k(1-s-3k+ks+2k^{2})f(x-y) \\ +2kf(kx)+2k(s+k-ks-2k^{2})f(x)+2(1-k-s)f(ky)+2ksf(y)$\end{document} in the setting of matrix intuitionistic fuzzy normed spaces by applying two different methods, where $ s $ is a parameter, $ k > 1 $ and $ s\neq 1-2k $. Moreover, the interdisciplinary relation between the theory of matrix intuitionistic fuzzy normed spaces and the theory of functional equations are also presented in this paper.

In this paper, we investigate the generalized Hyers-Ulam stability of the following mixed type quadratic-cubic functional equation \begin{align*} 2f(2x+y)+2f(2x-y) = 4f(x+y)+4f(x-y)+4f(2x)+f(2y)-8f(x)-8f(y) \end{align*} in non-Archimedean $(n,\beta)$-normed spaces.

Using the direct method and fixed point method, we investigate the Hyers-Ulam stability of the following cubic ?-functional equation f(x+2y) + f(x-2y)- 2f(x+y)-2f(x-y)-12f(x) = ?(4f(x+y/2) + 4f(x-y/2)-f(x+y)-f(x-y)-6f(x)) in matrix non-Archimedean random normed spaces, where ? is a fixed real number with ? ? 2.

Let 1≤m≤4 be a fixed integer and let f:X→Y be a mapping with X,Y two real vector spaces. For any fixed integers a with a≠0,±1, the following functional equationf(ax+y)+f(ax−y)=am−2[f(x+y)+f(x−y)]+2(a2−1)[am−2f(x)+(m−2)(1−(m−2)2)6f(y)] is said to be additive when m=1, quadratic when m=2, cubic when m=3 and quartic when m=4, respectively. For convenience, a solution of the above functional equation will be called an m-mapping. In this paper, for each m=1,2,3,4, we apply the fixed point method to investigate Hyers-Ulam stability results concerning the above functional equation in quasi fuzzy p-normed spaces. We also discuss the fuzzy continuity behavior of fuzzy approximate m-mappings in quasi fuzzy p-normed spaces. As applications, we establish Hyers-Ulam stability results of approximate m-mappings from a linear space into a quasi p-normed space.

Let k >= 2 be an integer. The purpose of this paper is first to introduce the notation of Felbin's type fuzzy normed linear spaces, and then by virtue of this notation to study some stability results concerning the more general cubic functional equation of the form f (x + ky) + f (x - ky) + f (kx) = k(2) f (x + y) + k(2) f (x - y) + (k(3) - 2k(2) + 2) f (x) in the setting of Felbin's type fuzzy normed linear spaces by employing the direct and fixed point methods. Then some applications of our results for the stability of the cubic functional equation from a real normed space to a Banach space will be demonstrated. Furthermore, the interdisciplinary relation between the theory of Felbin's type fuzzy spaces and the theory of functional equations are also presented in this paper.

In this paper, we study two functional equations with two unknown functions from an Abelian group into a commutative ring without zero divisors. The two equations are generalizations of Swiatak's functional equations with an involution. We determine the general solutions of the two functional equations and the properties of the general solutions of the two functional equations under three different hypotheses, respectively. For one of the functional equations, we establish the Hyers-Ulam stability in the case that the unknown functions are complex valued.

In this paper, by using fixed point method, we approximate a stable map of higher *-derivation in NA C*-algebras and of Lie higher *-derivations in NA Lie C*-algebras associated with the following additive functional equation Sigma(m)(i=1) T(u(i) + 1/m Sigma(m)(j=1, j not equal i) u(j)) + T(1/m Sigma(m)(i=1) u(i)) = 2T(Sigma(m)(i=1) u(i)), where m >= 2.

In this paper, using the fixed point method, we prove some results related to the generalized Hyers-Ulam stability of homomorphisms and derivations in non-Archimedean random C*-algebras and non-Archimedean random Lie C*-algebras for the generalized additive functional equation ?1 ? i < j ?n f(xi+xj/2 + ?n-2 l=1,kl?i,j xkl) = (n-1)2/2 ?n,i=1 f(xi) where n ? N is a fixed integer with n ? 3.

Using the fixed point method, we prove the Hyers-Ulam stability of the following additive functional inequality and quadratic functional inequalitybroken vertical bar broken vertical bar f (x+y) - f(x) - f(y) broken vertical bar broken vertical bar <= broken vertical bar broken vertical bar f(x+y/2) -1/2f(x) -1/2f(y) broken vertical bar broken vertical bar,broken vertical bar broken vertical bar f(x+y) + f(x-y) - 2f(x) - 2f(y) broken vertical bar broken vertical bar<= broken vertical bar broken vertical bar f(x+y)/2 + f(x-y)/2 -1/2 f(x) - 1/2 f(y) broken vertical bar broken vertical barin matrix Banach spaces, respectively.

In this paper, we prove some theorems about the Hyers-Ulam stability of the functional equation f(2x+ y) + f(2x− y) = 2f(x+ y) + 2f(x− y) + 2[f(2x) − 2f(x)] in paranormed spaces. From these theorems, as corollaries, we obtain the stability of the above functional equation with weaker conditions controlled by product of powers of norms and mixed-type product-sum of powers of norms. c ©2017 All rights reserved.

In this paper, we solve the following quadratic rho-functional inequalitiesparallel to f(x + y) + f(x - y) - 2f(x) - 2f(y)parallel to <= parallel to rho(2f(x + y)/2) + 2f(x - y/2) - f(x) - f(y))parallel to,where rho is a fixed complex number with |rho| < 1, andparallel to 2f(x + y/2)+2f(x - y/2) - f(x) - f(y)parallel to <= parallel to rho(f(x + y) + f(x - y) - 2f(x) - 2f(y))parallel to,where rho is a fixed complex number with |rho| < 1/2. By using the direct method, we prove the Hyers-Ulam stability of these inequalities in complex matrix normed spaces, and prove the Hyers-Ulam stability of quadratic rho-functional equations associated with these inequalities in complex matrix normed spaces. (C) 2016 All rights reserved.

Using fixed point method, we prove some new stability results for Lie (alpha, beta, gamma)-derivations and Lie C*-algebra homomorphisms on Lie C*-algebras associated with the Euler-Lagrange type additive functional equationSigma(n)(j=1) f(-r(j)x(j) + Sigma(1 <= ii <= n,i not equal j) r(i)x(i) ) +2 Sigma(n)(i=1) r(i)f(x(i)) = nf (Sigma(n)(i=1) r(i)x(i))where r(1),..., r(n), is an element of R are given and r(i), r(j) not equal 0 for some 1 <= i < j <= n.

Using the fixed point method, we investigate the generalized Hyers–Ulam stability of the ternary homomorphisms and ternary derivations between fuzzy ternary Banach algebras for the additive functional equation of n -Apollonius type, namely ∑_i=1^n f(z-x_i) = -1/n∑_1 ≤ i < j ≤ n f(x_i+x_j) + n f (z-1/n^2∑_i=1^nx_i), where n ≥ 2 is a fixed positive integer.

In this paper, we prove some stability results concerning the generalized quadratic and quartic type functional equation in the context of nonArchimedean fuzzy normed spaces in the spirit of Hyers-Ulam-Rassias. As applications, we establish some results of approximately generalized quadratic and quartic type mapping in non-Archimedean normed spaces. Also, we show that the assumption of the non-Archimedean absolute value of 2 is less than 1 cannot be omitted in our corollaries. The results improve and extend some recent results.

The aim of the present paper is to investigate the Hyers-Ulam stability of the Pexiderized quadratic functional equation, namely of f (x + y) + f (x - y) = 2 1 (x) + 2h(y) in paranormed spaces. More precisely, first we examine the stability for odd and even functions and then we apply our results to prove the Hyers-Ulam stability of the quadratic functional equation f (x + y) + f (x y) = 2 f(x) + 2 f(y) in paranormed spaces for a general function.

This paper considers the Jensen type cubic fuzzy set-valued functional equation and the n -dimensional cubic fuzzy set-valued functional equation. We establish the Hyers–Ulam stability of these two types of cubic fuzzy set-valued functional equations by using the fixed point method. Our results can be regarded as two extensions of stability results corresponding to single-valued functional equations and set-valued functional equations, respectively.

Using the fixed point method, we prove some results concerning the stability of the functional equation 2n?i=1 f(xi-1/2n 2n?j=1 xj)=2n?i=1 f (xi)-2nf(1/2n 2n?i=1 xi) where f is defined on a vector space and taking values in a fuzzy Banach space, which is said to be a functional equation related to a characterization of inner product spaces.

In this paper, we prove the Hyers-Ulam-Rassias stability of the following quadratic functional equations in. Serstnev probabilistic normed space endowed with Pi(M) triangle function: f(x + y) + f(x - y) = 2f(x) + 2f(y), f(ax + by) + f(ax - by) = 2a(2) f(x) + 2b(2) f(y) for nonzero real numbers a; b with a not equal +/- 1. More precisely, we show under some suitable conditions that an approximately quadratic function can be approximated by a quadratic mapping in above mentioned spaces.