On restricted functional inequalities associated with quadratic functional equations

Aequationes mathematicae - In this paper it is proved that, for a function $$f:mathcal {X}rightarrow E$$ mapping from a normed linear space $$mathcal {X}$$ into an inner product space E, the...     

On trigonometric functional equations of rectangular type

Summary Letf, G₁ × G₂ C, where G _i (i = 1, 2) denote arbitrary groups and C denotes the set of complex numbers. The general solutions of the following functional equationsf(x ₁ y ₁ ,x ₂ y ₂ ) +f(x ₁ y ₁ ,x ₂ y ₂ ^-1 ) +f(x ₁ y ₁ ^-1 ,x ₂ y ₂ ) +f(x ₁ y ₁ ^-1 ,x ₂ y ₂ ^-1 ) =f(x ₁ ,x ₂ )F(y ₁ ,y ₂ ) +F(x ₁ ,x ₂ )f(y ₁ ,y ₂ ) (1) andf(x ₁ y ₁ ,x ₂ y ₂ ) +f(x ₁ y ₁ ,x ₂ y ₂ ^-1 ) +f(x ₁ y ₁ ^-1 ,x ₂ y ₂ ) +f(x ₁ y ₁ ^-1 ,x ₂ y ₂ ^-1 ) =f(x ₁ ,x ₂ )f(y ₁ ,y ₂ ) +F(x ₁ ,x ₂ )F(y ₁ ,y ₂ ) (2) are determined assuming thatf satisfies the conditionf(x ₁y₁z₁, x₂) = f(x₁z₁y₁, x₂), f(x₁, x₂y₂z₂) = f(x₁, x₂z₂y₂) (C) for allx _i, y_i, x_i G_i (i = 1, 2). The functional equations (1) and (2) are generalizations of the well known rectangular type functional equationf(x ₁ + y₁, x₂ + y₂) + f(x₁ + y₁, x₂ – y₂) + f(x₁ – y₁, x₂ + y₂) + f(x₁ – y₁, x₂ – y₂) = 4f(x₁, x₂) studied by J. Aczel, H. Haruki, M. A. McKiernan and G. N. Sakovic in 1968.     

On the stability of a quadratic Jensen type functional equation

In this paper we obtain the general solution of the quadratic Jensen type functional equation

and prove the stability of this equation in the spirit of Hyers, Ulam, Rassias, and G vruta.     

关于Fermat 型函数方程的亚纯解

本文研究了三项Fermat 型函数方程亚纯解的存在性问题, 证明了: 如果(1/n) + (1/m) + (1/k) < (25/72) ; 则不存在非常数亚纯函数f, g, h 满足函数方程fⁿ + g^m + h^k ≡ 1:     

On some alternative quadratic equations

From the quadratic functional equation ?(x + y) + ?(x ? y) ? 2?(x) ? 2f(y) = 0 various alternative equations are derived here by grouping in different ways its terms and then equating norms. Some equivalence results are proved in the class of functionals ?: X → (?, ¦· ¦). Suitable examples concerning operators ?: X → (E,∥· ∥) with values in normed spaces show that in this more general setting such an equivalence can fail to be true.     

On the Hyers-Ulam stability of functional equations connected with additive and quadratic mappings

We investigate some inequalities connected with the Hyers-Ulam stability of three functional equations, which have a solution of the form φ=a+q, where a is an additive mapping and q is a quadratic one.     

On Urysohn-Volterra fractional quadratic integral equations

In this paper the authors study a fractional quadratic integral equation of Urysohn-Volterra type. They show that the integral equation has at least one monotonic solution in the Banach space of all real functions defined and continuous on the interval $[0,1]$. The main tools in the proof are a fixed point theorem due to Darbo and a monotonicity measure of noncompactness.     

Mean-value type functional equations

Summary In this work the following two conjectures concerning mean-value type functional equations are proved: then-dimensional octahedron and cube equations are equivalent (conjectured by D. Z. Djokovi and H. Haruki), and the continuous solutions of these equations on ⁿ are linear combinations of a given harmonic polynomial (conjectured by H. Haruki).     

On the alienation of the logarithmic and exponential Cauchy equations

In this paper, we give the solution of a problem formulated in Kominek and Sikorska (Aequationes Math 90:107–121, 2016) in connection with the functional equation

$$\begin{aligned} f(xy)-f(x)-f(y)=g(x+y)-g(x)g(y). \end{aligned}$$

Our result can also be interpreted in the way that, under some additional condition, the logarithmic and the exponential Cauchy equations are strongly alien.

     

Gelfand type problem for singular quadratic quasilinear equations

In this paper, we study the existence of positive solutions for the quasilinear elliptic singular problem

$$\left\{\begin{array}{ll}-\Delta u + c\,\frac{|\nabla u|^2}{u^\gamma} = \lambda\,f(u), \quad \quad \mbox{in $\Omega$},\\ u=0, \quad \qquad \qquad \qquad \quad \, \, \, \, \, \mbox{on $\partial$$\Omega$},\end{array}\right.$$

where \({c,\lambda >0}\), \({\gamma \in (0,1)}\), f is strictly increasing and derivable in \({[0,\infty)}\) with \({f(0)>0}\). We show that there exists \({\lambda^*>0}\) such that \({(0,\lambda^*]}\) is the maximal set of values such there exists solution. In addition, we prove that for \({\lambda<\lambda^*}\) there exists minimal and bounded solutions. Moreover, we give sufficient conditions for existence and regularity of solutions for \({\lambda=\lambda^*}\).

     

Local asymptotic attractivity for nonlinear quadratic functional integral equations

In this paper, an existence result for local asymptotic attractivity of the solutions is proved for a nonlinear quadratic functional integral equation under certain growth conditions which in turn gives the existence as well as asymptotic stability of solutions. A couple of examples are provided for indicating the natural realizations of abstract theory presented in the paper.     

On the Cauchy type problem for systems of functional differential equations

Using theorems on functional differential inequalities, we establish new efficient conditions for the solvability as well as unique solvability of the Cauchy type problem for systems of functional differential equations in both linear and nonlinear cases.     

On the complexity of quadratic programming with two quadratic constraints

Mathematical Programming - The complexity of quadratic programming problems with two quadratic constraints is an open problem. In this paper we show that when one constraint is a ball constraint...     

On viscosity solution of functional Hamilton-Jacobi type equations for hereditary systems

The paper is devoted to the development of the viscosity approach to the generalized solution of functional Hamilton-Jacobi type equations with coinvariant derivatives and a nonanticipatory Hamiltonian. These equations are naturally connected to problems of dynamical optimization of hereditary systems and, as compared with classical Hamilton-Jacobi equations, possess a number of additional peculiarities stipulated by the aftereffect. The definition of a viscosity solution that takes the above peculiarities into account is given. The consistency of this definition with the notion of a classical solution and with the minimax approach to the generalized solution is substantiated. The existence and uniqueness theorems are proved.