On a general bilinear functional equation

Let X, Y be linear spaces over a field $${\mathbb {K}}$$ K . Assume that $$f :X^2\rightarrow Y$$ f : X 2 → Y satisfies the general linear equation with respect to the first and with respect to the second variables, that is, for all $$x,x_i,y,y_i \in X$$ x , x i , y , y i ∈ X and with $$a_i,\,b_i \in {\mathbb {K}}{\setminus } \{0\}$$ a i , b i ∈ K \ { 0 } , $$A_i,\,B_i \in {\mathbb {K}}$$ A i , B i ∈ K ($$i \in \{1,2\}$$ i ∈ { 1 , 2 } ). It is easy to see that such a function satisfies the functional equation for all $$x_i,y_i \in X$$ x i , y i ∈ X ($$i \in \{1,2\}$$ i ∈ { 1 , 2 } ), where $$C_1:=A_1B_1$$ C 1 : = A 1 B 1 , $$C_2:=A_1B_2$$ C 2 : = A 1 B 2 , $$C_3:=A_2B_1$$ C 3 : = A 2 B 1 , $$C_4:=A_2B_2$$ C 4 : = A 2 B 2 . We describe the form of solutions and study relations between $$(*)$$ ( ∗ ) and $$(**)$$ ( ∗ ∗ ) .

Minimally deformed wormholes

This work is devoted to the study of wormhole solutions in the framework of gravitational decoupling by means of the minimal geometric deformation scheme. As an example, to analyze how this methodology works in this scenario, we have minimally deformed the well-known Morris–Thorne model. The decoupler function f(r) and the $$\theta $$ θ -sector are determined considering the following approaches: (i) the most general linear equation of state relating the $$\theta _{\mu \nu }$$ θ μ ν components is imposed and (ii) the generalized pseudo-isothermal dark matter density profile is mimicked by the temporal component of the $$\theta $$ θ -sector. It is found that the first approach leads to a non-asymptotically flat space-time with an unbounded mass function. To address this issue we have matched both the wormhole and the Schwarzschild vacuum solutions, via a thin-shell at the junction surface. Using the second approach, it can be seen that, on one hand, the solution for $$\gamma =1$$ γ = 1 does not give place to a bounded mass and it presents a topological defect at large distances; on the other hand, the wormhole manifold is asymptotically flat in the $$\gamma =2$$ γ = 2 case. In order to satisfy the flare-out condition, we have found restrictions on the value of the $$\alpha $$ α parameter, which is related with the amount of exotic matter distribution. Finally, the averaged weak energy condition has been analyzed by using the volume integral quantifier.

On a new class of functional equations satisfied by polynomial functions

The classical result of L. Székelyhidi states that (under some assumptions) every solution of a general linear equation must be a polynomial function. It is known that Székelyhidi’s result may be generalized to equations where some occurrences of the unknown functions are multiplied by a linear combination of the variables. In this paper we study the equations where two such combinations appear. The simplest nontrivial example of such a case is given by the equation $$\begin{aligned} F(x + y) - F(x) - F(y) = yf(x) + xf(y) \end{aligned}$$ F ( x + y ) - F ( x ) - F ( y ) = y f ( x ) + x f ( y ) considered by Fechner and Gselmann (Publ Math Debrecen 80(1–2):143–154, 2012). In the present paper we prove several results concerning the systematic approach to the generalizations of this equation.

A fixed point approach to the hyperstability of the general linear equation in β-Banach spaces

The purpose of this paper is first to reformulate the fixed point theorem (see Theorem 1 of [J. Brzdȩk, J. Chudziak and Z. Páles, A fixed point approach to stability of functional equations, Nonlinear Anal. 74 2011, 17, 6728–6732]) in β-Banach spaces. We also show that this theorem is a very efficient and convenient tool for proving the hyperstability results of the general linear equation in β-Banach spaces. Our main results state that, under some weak natural assumptions, functions satisfying the equation approximately (in some sense) must be actually solutions to it.

A NOTE ON THE GENERALISED HYPERSTABILITY OF THE GENERAL LINEAR EQUATION

Let $X$ and $Y$ be two normed spaces over fields $\mathbb{F}$ and $\mathbb{K}$, respectively. We prove new generalised hyperstability results for the general linear equation of the form $g(ax+by)=Ag(x)+Bg(y)$, where $g:X\rightarrow Y$ is a mapping and $a,b\in \mathbb{F}$, $A,B\in \mathbb{K}\backslash \{0\}$, using a modification of the method of Brzdęk [‘Stability of additivity and fixed point methods’, Fixed Point Theory Appl.2013 (2013), Art. ID 285, 9 pages]. The hyperstability results of Piszczek [‘Hyperstability of the general linear functional equation’, Bull. Korean Math. Soc.52 (2015), 1827–1838] can be derived from our main result.