Functional Equationf Research Articles

A FIXED POINT APPROACH TO STABILITY OF A CUBIC FUNCTIONAL EQUATION IN 2-BANACH SPACES

In this paper, we prove a new type of stability and hyperstability results forthe following cubic functional equationf (2x + y) + f (2x - y) = 2f (x + y) + 2f (x - y) + 12f(x)in 2-Banach spaces using fixed point approach.

Generalized Ulam-Hyers Stability Results of a Quadratic Functional Equation in Felbin’s Type Fuzzy Normed Linear Spaces

This paper presents the generalized Ulam-Hyers stability of the following quadratic functional equationF((x+y)/2 – z) + f(y+z)/2 – x) + f((z+x)/2 – y) = 3/4(f(z−x) + f(z−y) + f(x−y))in Felbin’s type fuzzy normed linear spaces (f-NLS) using direct and fixed point methods.

Fuzzy approximate m-mappings in quasi fuzzy normed spaces

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.

GENERALIZED HYERS-ULAM STABILITY OF MIXED TYPE ADDITIVE QUADRATIC FUNCTIONAL EQUATION OF RANDOM DERIVATION IN RANDOM NORMED ALGEBRAS.

By the fixed point method, I have proved the generalized stability of the random derivation of the following additive–quadratic functional equationf(x+2y)+f(x+y)+f(y-x)=f(2x+y)-f(x)+4f(y)+f(-y) in random normed algebras.

Investigations on the Hyers\u2013Ulam stability of generalized radical functional equations

In (Brzdęk and Schwaiger in Aeq Math 92: 975–991, 2018) solutions of far reaching generalizations of the so-called radical functional equationf(p(pi (x)+pi (y)))=f(x)+f(y) have been investigated. These investigations are continued here by analysing the corresponding stability results, which have been the main subject of several recent papers. We propose a very general and uniform approach.

A sine type functional equation on a topological group

In [13] H. Stetkaer obtained the complex valued solutions of the functional equation f(xyz0)f(xy−1z0) = f(x)2 − f(y)2, x, y ∈ G, where G is a topological group and z0 ∈ Z(G) (the center of G). Our main goal is first to remove this restriction and second, when G is 2-divisible and abelian, we will investigate the superstability of the above functional equation.

Solution of the Ulam stability problem for Euler–Lagrange k-quintic mappings

Abstract The “oldest quartic” functional equation f ⁢ ( x + 2 ⁢ y ) + f ⁢ ( x - 2 ⁢ y ) = 4 ⁢ [ f ⁢ ( x + y ) + f ⁢ ( x - y ) ] - 6 ⁢ f ⁢ ( x ) + 24 ⁢ f ⁢ ( y ) f(x+2y)+f(x-2y)=4[f(x+y)+f(x-y)]-6f(x)+24f(y) was introduced and solved by the second author of this paper (see J. M. Rassias, Solution of the Ulam stability problem for quartic mappings, Glas. Mat. Ser. III 34(54) 1999, 2, 243–252). Similarly, an interesting “quintic” functional equation was introduced and investigated by I. G. Cho, D. Kang and H. Koh, Stability problems of quintic mappings in quasi-β-normed spaces, J. Inequal. Appl. 2010 2010, Article ID 368981, in the following form: 2 ⁢ f ⁢ ( 2 ⁢ x + y ) + 2 ⁢ f ⁢ ( 2 ⁢ x - y ) + f ⁢ ( x + 2 ⁢ y ) + f ⁢ ( x - 2 ⁢ y ) = 20 ⁢ [ f ⁢ ( x + y ) + f ⁢ ( x - y ) ] + 90 ⁢ f ⁢ ( x ) . 2f(2x+y)+2f(2x-y)+f(x+2y)+f(x-2y)=20[f(x+y)+f(x-y)]+90f(x). In this paper, we generalize this “Cho–Kang–Koh equation” by introducing pertinent Euler–Lagrange k-quintic functional equations, and investigate the “Ulam stability” of these new k-quintic functional mappings.

On the general solution and hyperstability of the general radical quintic functional equation in quasi-β-Banach spaces

The goal of this paper is to study the general solution of the following general radical quintic functional equationf(ax5+by55)=rf(x)+sf(y) for f a mapping from the field of real numbers into a vector space, where a,b,r,s are fixed nonzero reals. Also, we prove the generalized hyperstability results for the general radical quintic functional equation by using the fixed point theorem (cf. Dung and Hang (2018) [15], Theorem 2.1) in quasi-β-Banach spaces. Namely, we show, under some weak natural assumptions, functions satisfying the above equation approximately (in some sense) must be actually solutions to it.

A variant of the quadratic functional equation on semigroups

Let S be a semigroup, let H be an abelian group which is uniquely 2-divisible, and let σ be an involutive automorphism of S. We express the solutions f : S → H of the following variant of the quadratic functional equation f(xy) + f(σ(y)x) = 2f(x) + 2f(y), x, y ∈ S, in terms of bi-additive maps and solutions of the symmetrized additive Cauchy equation.

Approximate solution of radical quartic functional equation related to additive mapping in 2-Banach spaces

The aim of this paper is first to reformulate the fixed point theorem [4, Theorem 1] in 2-Banach spaces, after it, we introduce and solve the radical quartic functional equationf(x4+y44)=f(x)+f(y). We also show that the fixed point methods allow to investigate Ulam's type stability of radical quartic functional equation in 2-Banach spaces.

Jensen’s and the quadratic functional equations with an endomorphism

We determine the solutions f : S → H of the generalized Jensen’s functional equationf (x + y) + f (x + φ(y)) = 2f (x), x,y ∈ S,and the solutions f : S → H of the generalized quadratic functional equationf (x + y) + f (x + φ(y)) = 2f (x) + 2f (y), x,y ∈ S,where S is a commutative semigroup, H is an abelian group (2-torsion free in the first equation and uniquely 2-divisible in the second) and φ is an endomorphism of S.

On the hyperstability of a quartic functional equation in Banach spaces

In this paper, we establish some hyperstability results of the following functional equationf (2x + y) + f (2x - y) = 4(f (x + y) + f (x - y)) + 24f (x) - 6f (y)in Banach spaces.

A Variant of Jensen’s Functional Equation on Semigroups

Abstract We determine the solutions f : S → H of the following functional equation f(xy) + f(σ(y)x) = 2f(x); x; y ∈ S; and the solutions f1; f2; f3 : M → H of the functional equation f1(xy) + f2(σ(y)x) = 2f31(x); x; y ∈ M; where S is a semigroup, M is a monoid, H is an abelian group 2-torsion free, and σ is an involutive automorphism.

Approximate Drygas mappings on a set of measure zero

Let R be the set of real numbers, Y be a Banach space and f : R →Y. We prove the Hyers-Ulam stability for the Drygas functional equationf (x + y) + f (x - y) = 2f (x) + f (y) + f (-y) for all (x, y) ∈ Ω, where Ω⊂ R2 is of Lebesgue measure 0.

The stability of a generalized affine functional equation in fuzzy normed spaces

We obtain the general solution of the following functional equation f(kx1+x2+???+xk)+f(x1+kx2+???+xk)+???+f(x1+x2+???+kxk)+f(x1)+ f(x2)+???+ f(xk)= 2kf(x1+ x2+???+xk), k ? 2. We establish the Hyers-Ulam-Rassias stability of the above functional equation in the fuzzy normed spaces. More precisely, we show under suitable conditions that a fuzzy q-almost affine mapping can be approximated by an affine mapping. Further, we determine the stability of same functional equation by using fixed point alternative method in fuzzy normed spaces.

Stability of a Quartic Functional Equation in Restricted Domains

LetXbe a real normed space andYa Banach space andf:X→Y. We prove the Ulam-Hyers stability theorem for the quartic functional equationf(2x+y)+f(2x-y)-4f(x+y)-4f(x-y)-24f(x)+6f(y)=0in restricted domains. As a consequence we consider a measure zero stability problem of the above inequality whenf:R→Y.

On the Stability of Quadratic Functional Equations inF-Spaces

The Hyers-Ulam-Rassias stability of quadratic functional equationf(2x+y)+f(2x-y)=f(x+y)+f(x-y)+6f(x)and orthogonal stability of the Pexiderized quadratic functional equationf(x+y)+f(x-y)=2g(x)+2h(y)inF-spaces are proved.

APPROXIMATE QUADRATIC MAPPINGS IN QUASI-β-NORMED SPACES

Abstract. In this article, we consider a modi ed quadratic func-tional equation and then investigate its generalized Hyers{Ulamstability theorem in quasi- -normed spaces. 1. IntroductionIn 1940, S.M. Ulam [17] raised the question concerning the stabilityof group homomorphisms: Let Gbe a group and let G 0 be a metricgroup with the metric d(; ). Given >0, does there exist a >0 suchthat if a mapping f: G!G 0 satis es the inequalityd(f(xy);f(x)f(y)) <for all x;y2G, then there exists a homomorphism F : G!G 0 withd(f(x);F(x)) <for all x2G? The case of approximately additivemappings was solved by D.H. Hyers [8] under the assumption that XandY are Banach spaces. A generalization of Hyers’ theorem was providedby Th.M. Rassias [12] in 1978 and by P. Gavruta [6] in 1994.We recall that the following functional equationf(x+ y) + f(x y) = 2[f(x) + f(y)]is called a quadratic functional equation which may be originated fromthe important parallelogram equality kx+yk 2 +kx yk

Ulam-Hyers Stability of Trigonometric Functional Equation with Involution

LetSandGbe a commutative semigroup and a commutative group, respectively,CandR+the sets of complex numbers and nonnegative real numbers, respectively, andσ:S→Sorσ:G→Gan involution. In this paper, we first investigate general solutions of the functional equationf(x+σy)=f(x)g(y)-g(x)f(y)for allx,y∈S, wheref,g:S→C. We then prove the Hyers-Ulam stability of the functional equation; that is, we study the functional inequality|f(x+σy)-f(x)g(y)+g(x)f(y)|≤ψ(y)for allx,y∈G, wheref,g:G→Candψ:G→R+.

On a functional equation originating from a mixed additive and cubic equation and its stability

In this paper, we study solutions of the 2-variable mixed additive andcubic functional equationf (2x + y, 2z + t) + f (2x − y, 2z − t) = 2f (x + y, z + t)+ 2f (x − y, z − t) + 2f (2x, 2z) − 4f (x, z),which has the cubic form f (x, y) = ax+ bx y + cxy+ dy as a solution. Also the Hyers–Ulam–Rassias stability of this equation in thenon-Archimedean Banach spaces is investigated.