In this paper, we obtain the general solution and the generalized   Hyers-Ulam-Rassias stability in random normed spaces, in non-Archimedean spaces and also in $p$-Banach spaces and finally the stability via fixed point method for a functional equationbegin{align*}&D_f(x_{1},.., x_{m}):= sum^{m}_{k=2}(sum^{k}_{i_{1}=2}sum^{k+1}_{i_{2}=i_{1}+1}... sum^{m}_{i_{m-k+1}=i_{m-k}+1}) f(sum^{m}_{i=1, i...

Let G be a (p, q) graph. Let f : V (G) → {1, 2, . . . , k} be a map. For each edge uv, assign the label gcd (f(u), f(v)). f is called k-prime cordial labeling of G if |vf (i) − vf (j)| ≤ 1, i, j ∈ {1, 2, . . . , k} and |ef (0) − ef (1)| ≤ 1 where vf (x) denotes the number of vertices labeled with x, ef (1) and ef (0) respectively denote the number of edges labeled with 1 and not labeled with 1....

Let $G$ be a graph. Let $f:V(G)to{0,1,2, ldots, k-1}$ be a map where $k in mathbb{N}$ and $k>1$. For each edge $uv$, assign the label $left|f(u)-f(v)right|$. $f$ is called a $k$-total difference cordial labeling of $G$ if $left|t_{df}(i)-t_{df}(j)right|leq 1$, $i,j in {0,1,2, ldots, k-1}$ where $t_{df}(x)$ denotes the total number of vertices and the edges labeled with $x$.A graph with admits a...

Let $kge 1$ be an integer, and let $G$ be a finite and simple graph with vertex set $V(G)$.A weak signed Roman $k$-dominating function (WSRkDF) on a graph $G$ is a function$f:V(G)rightarrow{-1,1,2}$ satisfying the conditions that $sum_{xin N[v]}f(x)ge k$ for eachvertex $vin V(G)$, where $N[v]$ is the closed neighborhood of $v$. The weight of a WSRkDF $f$ is$w(f)=sum_{vin V(G)}f(v)$. The weak si...

The purpose of this paper is twofold. First we derive theoretically, using appropriate transformation on x(n), the closed-form solution of the nonlinear difference equation x(n+1) = 1/(±1 + x(n)), n ∈ N_0. The form of solution of this equation, however, was first obtained in [10] but through induction principle. Then, with the solution of the above equation at hand, we prove a case ...