General and alien solutions of a functional equation and of a functional inequality

2013/07/02 by Włodzimierz Fechner, Fechner, Włodzimierz, Eszter Gselmann +1
Mathematics · #39B52 #39B62 #39B72 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.CA #msc:39B52 #msc:39B62 #msc:39B72

paper · pdf · doi:10.48550/arxiv.1307.0653

9 pages, published in Publicationes Mathematicae (Debrecen) in 2012

arxiv created 2013/07/02 · arxiv updated 2013/07/03

Abstract

The purpose of the present paper is to solve (under some assumption on the domain) the equation g(x+y)-g(x)-g(y)=xf(y)+yf(x). After determining the general solutions, we will investigate the so--called alien solutions. %More precisely, we will examine the cases when the above equation implies that %g(x+y)=g(x)+g(y) and xf(y)+yf(x)=0. %Concerning this, necessary and sufficient conditions will be provided. Finally, we will discuss the real solutions of the following related functional inequality: g(x+y)-g(x)-g(y)≥ xf(y)+yf(x).

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