An algorithm is proposed to seek a local optimal solution of the network utility maximization problem in a wireless mesh network, where the architecture being considered is an infrastructure/backbone wireless mesh network. The objective is to achieve proportional fairness amongst the end-to-end flows in wireless mesh networks. In order to establish the communication constraints of the flow rates in the network utility maximization problem, we have presented necessary and sufficient conditions for the achievability of the flow rates. Since wireless mesh networks are generally considered as a type of ad hoc networks, similarly as in wireless multi-hop network, the network utility maximization problem in wireless mesh network is a nonlinear nonconvex programming problem. Besides, the gateway/bridge functionalities in mesh routers enable the integration of wireless mesh networks with various existing wireless networks. Thus, the rate optimization problem in wireless mesh networks is more complex than in wireless multi-hop networks.