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We prove that the class of positive operators from $L_\infty (\mu)$ to $Y$has the Bishop-Phelps-Bollob\'as property for any positive measure $\mu$,whenever $Y$ is a uniformly monotone Banach lattice with a weak unit. The sameresult also holds for the pair $(c_0, Y)$ for any uniformly monotone Banachlattice $Y.$ Further we show that these results are optimal in case that $Y$ isstrictly monotone.

Bishop-Phelps-Bollob\'as property for positive operators when the domain is $L_\infty $