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This paper aims at solving one-dimensional backward stochastic differentialequations (BSDEs) under weaker assumptions. We establish general existence,uniqueness, and comparison results for bounded solutions, $L^p (p>1)$ solutionsand $L^1$ solutions of the BSDEs. The time horizon is allowed to be finite orinfinite, and the generator $g$ is allowed to have a general growth in $y$ anda quadratic growth in $z$. As compensation, the generator $g$ needs to satisfya kind of one-sided linear or super-linear growth condition in $y$, instead ofthe monotonicity condition in $y$ as is usually done. Many of our resultsimprove virtually some known results, even though for the case of the finitetime horizon and the case of the $L^2$ solution.

Bounded solutions, $L^p (p>1)$ solutions and $L^1$ solutions for one-dimensional BSDEs under general assumptions