In this paper, we consider the power splitting relaying (PSR) and time switching relaying (TSR) protocols for decode-and-forward (DF)-based relay networks consisting of a source, a relay, and a destination. For the networks, the relay is assumed to have a rechargeable battery with a certain amount of remaining energy for energy harvesting through the received signal transmitted from the source. Specifically, for the PSR protocol with and without the direct-path, we present the outage probability expression with a given power splitting coefficient, and we obtain the optimum power splitting coefficient to maximize the transmission rate and derive its transmission rate and outage probability, where the optimum coefficient depends on channel conditions and remaining energy. For the TSR protocol with and without the direct-path, we present the outage probability expression with a given time switching coefficient, and we obtain the optimum time switching coefficient to maximize the transmission rate and derive its transmission rate, where the optimum coefficient also depends on channel conditions and remaining energy.