Table 2 Problem formulation of the optimal bandwidth sharing among mobile nodes

$\begin{array}{cc}\underset{\mathbf{x}\ge 0}{\max }U\left(\mathbf{x}\right)=& \sum _{i\in \mathcal{\mathscr{H}},j\in \mathcal{R}}U\left(w_1{x}_{\mathit{\text{ij}}}-w_2{C}_{\mathit{\text{ij}}}\left(x_{\mathit{\text{ij}}}\right)\right)\end{array}$

(3)

$\begin{array}{lll}0\le {\sum }_{i\in \mathcal{\mathscr{H}}}{x}_{\mathit{\text{ij}}}\le x_{j,max}& \text{,}& \forall j\in \mathcal{R}\end{array}$

(4)

$\begin{array}{ccc}{\sum }_{j\in \mathcal{R}}{x}_{\mathit{\text{ij}}}\le c_i& \text{,}& \forall i\in \mathcal{\mathscr{H}}\end{array}$

(5)

$\begin{array}{lll}{\sum }_{i:j\in \mathcal{P}\left(l\right)}{x}_{\mathit{\text{ij}}}\le w& \text{,}& \forall l\in \mathcal{ℒ}\end{array}$

(6)

$\begin{array}{lll}{x}_{\mathit{\text{ij}}}\ge 0& ,& \forall (i,j)\in \mathcal{\mathscr{H}}\times \mathcal{R}\end{array}$

(7)