Recently there have been large advances in energy technologies for battery-operated systems, including green energy resources and high capacity batteries. The effective use of battery energy resources in wireless infrastructure networks to improve the versatility and reliability of wireless communications is an important issue. Emerging applications of smart cities, Internet of Things (IoT), and emergency responses highly rely on the basic communication network infrastructures that enable ubiquitous network connections. However, energy consumption by nodes in a wireless infrastructure network depends on the transmissions of other nodes in the network. Considering this inter-dependence is necessary to achieve efficient provision of energy in wireless networks. This paper studies the issue of energy provision for wireless relay nodes in Wireless Multihop Infrastructures (WMI) assuming constraints on the total energy provision. We introduce a scheme of Energy Provision Matching (Matching-EP) for WMI which optimizes energy provision based on matching of energy provision with estimates of differentiated position-dependent energy consumption by wireless nodes distributed in the network. The evaluation results show that Matching-EP with 4%⁻34% improvement in energy matching degree enables 10%⁻40% improvement of the network lifetime, and 5%⁻40% improvement of packet delivery compared with conventional WMI networks.