The spread of fake news on online social networks (OSNs) has become a matter of concern. These platforms are also used for propagating important authentic information. Thus, there is a need for mitigating fake news without significantly influencing the spread of real news. We leverage users' inherent capabilities of identifying fake news and propose a warning-based control mechanism to curb this spread. Warnings are based on previous users' responses that indicate the authenticity of the news. We use population-size dependent continuous-time multi-type branching processes to describe the spreading under the warning mechanism. We also have new results towards these branching processes. The (time) asymptotic proportions of the individual populations are derived using stochastic approximation tools. Using these, relevant type 1, type 2 performances are derived and an appropriate optimization problem is solved. The proposed mechanism effectively controls fake news, with negligible influence on the propagation of authentic news. We validate performance measures using Monte Carlo simulations on network connections provided by Twitter data.