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This article is devoted to study the class of backward stochasticdifferential equation with delayed generator. We suppose the terminal value andthe generator to be $L^{p}$-integrable with $p>1$. We derive a new type ofestimation related to this BSDE. Next, we establish the existence anduniqueness result in two ways. First, an approximation technics used by Briandet al. (Stochastic Process. Appl. 108 (2003) 109-129) and hence the well-knowPicard iterative procedure. Using Picard iterative procedure, we revisit theresult of Dos Reis et al. (Stochastic Process. Appl. 121 (9) (2011) 2114-2150),simplifying the proof and give an explicit existence and uniqueness conditionrelated to the Lipschitz constant $K$ and the terminal time $T$.

$L^{p}$-solutions of backward stochastic differential equations with time-delayed generators