PDM KG-oscillators in cosmic string rainbow gravity spacetime in a non-uniform magnetic field

We consider position-dependent mass (PDM) Klein-Gordon (KG) particles in cosmic string rainbow gravity spacetime in a non-uniform magnetic field. The corresponding KG-equation is reduced into the one-dimensional form of the two-dimensional radial Schrödinger-oscillator like equation (hence the notion KG-oscillator). We first report on the effects of rainbow gravity on the energy levels of KG-oscillators with constant mass. Next, we include the PDM settings so that KG-oscillators like interaction are introduced. The effects of PDM on the spectra of KG-oscillators in cosmic string rainbow gravity spacetime are also reported. In both cases four pairs of rainbow functions are considered: (a) $$, g_{_{1}}\left( y\right) =\sqrt{1-εy^{2}% }, (b) $g_{{}_0}\left(y\right)=1$, g_{_{1}}\left( y\right) =\sqrt{% 1-εy}, © $g_{{}_0}\left(y\right)=g_{{}_1}\left(y\right)={(1-\epsilon y)}^{-1}$, and (d) $g_{{}_0}\left(y\right)=(e^{\epsilon y}-1)\mathrm{/}\epsilon y$, $g_{{}_1}\left(y\right)=1$, where $y=E\mathrm{/}{E}_p$ and $\epsilon$ is the rainbow parameter. We report that, as long as KG-particles are in point, the rainbow functions pair in (a) is the only pair to emphasis that the Planck energy is the maximum possible energy the KG-particles may have.