Bosonic string theories in non-commutative space-time with deformed dispersion relations

In this paper, we study string theories with deformed commutation relations and ordinary constraints to derive the Magueijo-Smolin form of deformed relativistic dispersion relations. A closed energy-dependent constraints algebra is obtained. We quantize these theories, and we find that the characteristics of the spectrum change with respect to the total energy functions. This deformation is equivalent to work in an energy- and mass-dependent potential. In a particular choice of the energy-dependent functions, the tachyonic state can be eliminated and the next excited states accumulate below the Planck scale.