In this paper we study the generalized version of weighted matching in bipartite networks. Consider a weighted matching in a bipartite network in which the nodes derive value from the split of the matching edge assigned to them if they are matched. The value a node derives from the split depends both on the split as well as the partner the node is matched to. We assume that the value of a split to the node is continuous and strictly increasing in the part of the split assigned to the node. A stable weighted matching is a matching and splits on the edges in the matching such that no two adjacent nodes in the network can split the edge between them so that both of them can derive a higher value than in the matching. We extend the weighted matching problem to this general case and study the existence of a stable weighted matching. We also present an algorithm that converges to a stable weighted matching. The algorithm generalizes the Hungarian algorithm for bipartite matching. Faster algorithms can be made when there is more structure on the value functions.