inthis paper, the main aim is to introduce the class $mathcal{u}_p(lambda,alpha,beta,k_0)$ of $p$-harmonic mappings togetherwith its subclasses $mathcal{u}_p(lambda,alpha,beta,k_0)capmathcal {t}_p$ and $mathcal{u}_p(lambda,alpha,beta,k_0)capmathcal {t}_p^0$, andinvestigate the properties of the mappings in these classes. first,we give a sufficient condition for mappings to be in $mathcal{u}_p(l...

For some $$\beta \ge 1/2$$ , a $$\varDelta _{\beta }$$ -metric graph $$G=(V,E,w)$$ is complete edge-weighted such that $$w(v,v)=0$$ $$w(u,v)=w(v,u)$$ and $$w(u,v) \le \beta \cdot (w(u,x)+w(x,v))$$ for all vertices $$u,v,x\in V$$ . A $$H=(V', E')$$ called spanning subgraph of $$G=(V, E)$$ if $$V'=V$$ $$E'\subseteq E$$ Given positive integer p, let H be G satisfying the three conditions: (i) ther...

Consider $\beta > 1$ and $\lfloor \beta \rfloor$ its integer part. It is widely known that any real number $\alpha \in \Bigl[0, \frac{\lfloor \rfloor}{\beta - 1}\Bigr]$ can be represented in base $\beta$ using a development series of the form = \sum_{n 1}^\infty x_n\beta^{-n}$, where $x (x_n)_{n \geq 1}$ sequence taking values into alphabet $\{0,\; ...\; ,\; \lfloor \rfloor\}$. The so called $\...