Approximation of Mixed‐Type Functional Equations in Menger PN‐Spaces

Abstract

LetXandYbe vector spaces. We show that a functionf:X→Ywithf(0)=0satisfiesΔf(x1,…,xn)=0for allx1,…,xn∈X, if and only if there exist functionsC:X×X×X→Y,B:X×X→YandA:X→Ysuch thatf(x)=C(x,x,x)+B(x,x)+A(x)for allx∈X, where the functionCis symmetric for each fixed one variable and is additive for fixed two variables,Bis symmetric bi-additive,Ais additive andΔf(x1,…,xn)=∑k=2n(∑i1=2k∑i2=i1+1k+1⋯∑in-k+1=in-k+1n)f(∑i=1,i≠i1,…,in-k+1nxi-∑r=1n-k+1xir)+f(∑i=1nxi)-2n-2∑i=2n(f(x1+xi)+f(x1-xi))+2n-1(n-2)f(x1)(n∈N,n≥3) for allx1,…,xn∈X. Furthermore, we solve the stability problem for a given functionfsatisfyingΔf(x1,…,xn)=0, in the Menger probabilistic normed spaces.