Fuzzy stability of quadratic-cubic functional equations

In this paper, the direct method and the fixed point alternative method are implemented to give Hyers-Ulam-Rassias stability of the functional equation

Functional equations on abelian groups with involution

Summary We produce complete solution formulas of selected functional equations of the formf(x +y) ±f(x + σ (ν)) = Σ _I ² =₁ g _l (x)h _l (y),x, y∈G, where the functionsf,g ₁,h ₁ to be determined are complex valued functions on an abelian groupG and where σ:G→G is an involution ofG. The special case of σ=−I encompasses classical functional equations like d’Alembert’s, Wilson’s first generalization of it, Jensen’s equation and the quadratic equation. We solve these equations, the equation for symmetric second differences in product form and similar functional equations for a general involution σ.     

A generalized mixed type of quartic–cubic–quadratic–additive functional equations

We determine the general solution of the functional equation f(x + ky) + f(x-ky) = g(x + y) + g(x-y) + h(x) + h(y) for fixed integers with k ≠ 0; ±1 without assuming any regularity conditions for the unknown functions f, g, h, and0020[(h)\tilde] \tilde{h} . The method used for solving these functional equations is elementary but it exploits an important result due to Hosszú. The solution of this functional equation can also be obtained in groups of certain type by using two important results due to Székelyhidi.     

A fixed point approach to stability of a quadratic equation

Using the fixed point alternative theorem we establish the orthogonal stability of the quadratic functional equation of Pexider type f (x+y)+g(x−y) = h(x)+k(y), where f, g, h, k are mappings from a symmetric orthogonality space to a Banach space, by orthogonal additive mappings under a necessary and sufficient condition on f.     

Stability of the Quadratic Equation of Pexider Type

We will investigate the stability problem of the quadratic equation (1) and extend the results of Borelli and Forti, Czerwik, and Rassias. By applying this result and an improved theorem of the author, we will also prove the stability of the quadratic functional equation of Pexider type,f ₁ (x +y) + f₂(x -y) =f ₃(x) +f ₄(y), for a large class of functions.     

Existence, uniqueness and stability of C m solutions of iterative functional equations

In this paper we discuss a relatively general kind of iterative functional equation G(x,f(x), ...,f ⁿ (x)) = 0 (for allx ∈J), whereJ is a connected closed subset of the real number axis ℝ,G∈C ^m (J ⁿ⁺¹, ℝ) andn ≥ 2. Using the method of approximating fixed points by small shift of maps, choosing suitable metrics on functional spaces and finding a relation between uniqueness and stability of fixed points of maps of general spaces, we prove the existence, uniqueness and stability ofCm solutions of the above equation for any integer m ≥ 0 under relatively weak conditions, and generalize related results in reference in different aspects.     

Decomposer and associative functional equations

In the previous researches [2,3] b-integer and b-decimal parts of real numbers were introduced and studied by M.H. Hooshmand. The b-parts real functions have many interesting number theoretic explanations, analytic and algebraic properties, and satisfy the functional equation f (f(x) + y - f(y)) = f(x). These functions have led him to a more general topic in semigroups and groups (even in an arbitrary set with a binary operation [4] and the following functional equations have been introduced: Associative equations:

f(xf(yz))=f(f(xy)z),f(xf(yz))=f(f(xy)z)=f(xyz)

On some characterization of probability distributions in Hilbert spaces

Summary The aim of this paper is to prove the following theorem about characterization of probability distributions in Hilbert spaces:Theorem. — Let x₁, x₂, …, x_n be n (n≥3) independent random variables in the Hilbert spaceH, having their characteristic functionals f_k(t) = E[e^i(t,x _k)], (k=1, 2, …, n): let y₁=x₁ + x_n, y₂=x₂ + x_n, …, y_n−1=x_n−1 + x_n. If the characteristic functional f(t₁, t₂, …, t_n−1) of the random variables (y₁, y₂, …, y_n−1) does not vanish, then the joint distribution of (y₁, y₂, …, y_n−1) determines all the distributions of x₁, x₂, …, x_n up to change of location.     

Formula for a solution ofu t +H(u,Du) =g

We study the continuous as well as the discontinuous solutions of Hamilton-Jacobi equationu _t +H(u,Du) =g in ℝ ⁿ x ℝ₊ withu(x, 0) =u ₀(x). The HamiltonianH(s,p) is assumed to be convex and positively homogeneous of degree one inp for eachs in ℝ. IfH is non increasing ins, in general, this problem need not admit a continuous viscosity solution. Even in this case we obtain a formula for discontinuous viscosity solutions.     

On dual integral equations in the semiconservative case

A dual integral equations on the whole real axis with an unknown function f is considered. It is supposed that the kernel functions of the equations are even and representable as superposition of exponents. The equation is transferred to a system of integral equations on the positive semi-axis with two unknown functions: f ₁(x) = f(x) and f ₂(x) = f(−x). Applying a factorization method and using the solution of Ambartsumian equation, a system of Laplace transforms α ₁, α ₂ of functions f ₁, f ₂ is obtained. Under some conditions on the free term, the existence and uniqueness of the solution of that system is proved in the semi-conservative case. A construction of the functions α ₁, α ₂ is given by means of successive approximations, and a construction method of the solution (f ₁, f ₂) by α ₁, α ₂ is described.     

The stability of d'Alembert and Jensen type functional equations

The aim of this paper is to study the stability problem of the d'Alembert type and Jensen type functional equations:

f(x+y)+f(x+σy)=2g(x)f(y),     

Jensen’s functional equation on the symmetric group S<Subscript>n</Subscript>

Two natural extensions of Jensen’s functional equation on the real line are the equations f(xy) + f(xy ⁻¹) =  2f(x) and f(xy) + f(y ⁻¹ x) =  2f(x), where f is a map from a multiplicative group G into an abelian additive group H. In a series of papers (see Ng in Aequationes Math 39:85–99, 1990; Ng in Aequationes Math 58:311–320, 1999; Ng in Aequationes Math 62:143–159, 2001), Ng solved these functional equations for the case where G is a free group and the linear group GL_n(R), R=\mathbbZ,\mathbbR{{GL_n(R), R=\mathbb{Z},\mathbb{R}}} , is a quadratically closed field or a finite field. He also mentioned, without a detailed proof, in the above papers and in (see Ng in Aequationes Math 70:131–153, 2005) that when G is the symmetric group S _n , the group of all solutions of these functional equations coincides with the group of all homomorphisms from (S _n , ·) to (H, + ). The aim of this paper is to give an elementary and direct proof of this fact.     

Superstability of the generalized orthogonality equation on restricted domains

Chmielinski has proved in the paper [4] the superstability of the generalized orthogonality equation |〈f(x), f(y)〉| = |〈x,y〉|. In this paper, we will extend the result of Chmielinski by proving a theorem: LetD _n be a suitable subset of ℝⁿ. If a function f:D _n → ℝⁿ satisfies the inequality ∥〈f(x), f(y)〉| |〈x,y〉∥ ≤ φ(x,y) for an appropriate control function φ(x, y) and for allx, y ∈ D _n, thenf satisfies the generalized orthogonality equation for anyx, y ∈ D _n.     

On an Asymptotic Behavior of Exponential Functional Equation

The stability problems of the exponential （functional） equation on a restricted domain will be investigated, and the results will be applied to the study of an asymptotic property of that equation. More precisely, the following asymptotic property is proved： Let X be a real （or complex） normed space. A mapping f ： X → C is exponential if and only if f（x ＋ y） - f（x）f（y） → 0 as ｜｜x｜｜ ＋ ｜｜y｜｜ → ∞ under some suitable conditions.     

On elliptic equations and systems with critical growth in dimension two

We consider nonlinear elliptic equations of the form −Δu = g(u) in Ω, u = 0 on ∂Ω, and Hamiltonian-type systems of the form −Δu = g(v) in Ω, −Δv = f(u) in Ω, u = 0 and v = 0 on ∂Ω, where Ω is a bounded domain in ℝ² and f, g ∈ C(ℝ) are superlinear nonlinearities. In two dimensions the maximal growth (= critical growth) of f and g (such that the problem can be treated variationally) is of exponential type, given by Pohozaev-Trudinger-type inequalities. We discuss existence and nonexistence results related to the critical growth for the equation and the system. A natural framework for such equations and systems is given by Sobolev spaces, which provide in most cases an adequate answer concerning the maximal growth involved. However, we will see that for the system in dimension 2, the Sobolev embeddings are not sufficiently fine to capture the true maximal growths. We will show that working in Lorentz spaces gives better results. Dedicated to Professor S. Nikol’skii on the occasion of his 100th birthday     

Functional equations for Hecke-Maaass series

The Dirichlet (Hecke-Maass) series associated with the eigenfuctionsf andg of the invariant differential operator Δ_k=−y²(∂²/∂x²)+iky∂/∂x of weightk are investigated. It is proved that any relation of the form (f/kM)=g for thek-action of the groupSL ₂ SL ₂(ℝ) is equivalent to a pair of functional equations relating the Hecke-Maass series forf andg and involving only traditional gamma factors. This work was supported by the Russian Foundation for Basic Research (grant No. 96-01-10439). Institute of Applied Mathematics, Far East Division of Russian Academy of Sciences. Translated from Funktional'nyi Analiz i Ego Prilozheniya, Vol. 34, No. 2, pp. 23–32, April–June, 2000. Translated by V. M. Volosov     

On the behavior of solutions of a semilinear parabolic or elliptic equation satisfying a nonlinear boundary condition in a cylindrical domain

One considers a semilinear parabolic equation u _t = Lu − a(x)f(u) or an elliptic equation u _tt + Lu − a(x)f(u) = 0 in a semi-infinite cylinder Ω × ℝ₊ with the nonlinear boundary condition , where L is a uniformly elliptic divergent operator in a bounded domain Ω ∈ ℝⁿ; a(x) and b(x) are nonnegative measurable functions in Ω. One studies the asymptotic behavior of solutions of such boundary-value problems for t → ∞. __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 26, pp. 368–389, 2007.     

Functions of ultraexponential and infralogarithm types and general solution of the Abel functional equation

We propose generalized forms of ultraexponential and infralogarithm functions introduced and studied earlier by the author and present two classes of special functions, namely, ultraexponential and infralogarithm f -type functions. As a result of this investigation, we obtain a general solution of the Abel equation α(f(x)) = α (x) + 1 under some conditions on a real function f and prove a new completely different uniqueness theorem for the Abel equation stating that an infralogarithm f -type function is its unique solution. We also show that an infralogarithm f -type function is an essentially unique solution of the Abel equation. Similar theorems are proved for ultraexponential f -type functions and their functional equation β(x) = f(β(x − 1)), which can be considered as dual to the Abel equation. We also solve a certain problem unsolved before and study some properties of two considered functional equations and some relations between them.     

The pexiderized Go?a?b-Schinzel functional equation

Let X be a linear space over a commutative field K. We characterize a general solution f,g,h,k:X→K of the pexiderized Go?a?b-Schinzel equation f(x+g(x)y)=h(x)k(y), as well as real continuous solutions of the equation.     

Two-Point Boundary Value Problems for Duffing Equations across Resonance

In this paper, we consider the equation y″+f(x,y)=0 with a nonresonance condition of the form A≤f _y (x,y)≤B, where (k−1)²<A≤k ²<⋅⋅⋅<m ²≤B<(m+1)², k,m∈ℤ⁺. With optimal control theory and the Schauder fixed-point theorem, by introducing a new cost functional, we obtain a new existence and uniqueness result for the above equation with two-point boundary-value conditions. This work was supported by NSFC Grant 10501017 and 985 Project of Jilin University.