A Tauberian theorem for the logarithmic summability in ordered spaces

The present manuscript aims to extend a Tauberian theorem previously established for the Cesàro and weighted mean summability methods of single sequences in ordered spaces to the logarithmic summability method, also known as the (ℓ,1,1) method, for double sequences. In order to achieve this, we present several Tauberian conditions which address the O_L-oscillatory behavior of a double sequence (s_mn ) with respect to logarithmic summability in certain senses. These conditions facilitate the transition from (ℓ,1,1), (ℓ,1,0), and (ℓ,0,1) summability to P-convergence in ordered spaces.