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• Propose an OD competitive network equilibrium model for ride-sourcing market. • Adopt finite-dimensional variational inequality based on non-cooperative game theory. • Incorporate heavy-asset and light-asset transportation network companies (TNCs). • Derive analytical monotonicity of equilibrium optima and sensitivity analysis. • Gain managerial insights for both the government and TNCs. The ride-sourcing market (RSM) consists of three parts: ride-sourcing platforms or transportation network companies (TNCs) that set the price for passengers and wage of drivers, passengers requesting on-demand ride services, and drivers offering the services. This paper proposes an origin–destination (OD) competitive network equilibrium model for RSM with heterogeneous TNCs. A finite-dimensional variational inequality is adopted to formulate the RSM competitive network equilibrium model based on the non-cooperative game theory. It assumes that each TNC and individual drivers aim to maximize profits or earnings, while passengers of demand regions seek utility maximization. The monotonicity of equilibrium optima is analytically derived, and the sensitivity analysis of parameters is theoretically conducted and numerically illustrated. The managerial insights are gained both for the government and TNCs. The analytical findings include: (a) market equilibrium price increases and market equilibrium travel quantity decreases in either the transaction cost, driver cost, or TNC cost; (b) market equilibrium price decreases in the number of light-asset TNCs, while the market equilibrium travel quantity increases in it; and (c) raising RSM entry requirements improves the market equilibrium price and reduces travel demand on RSM. The numerical results illustrate the plausibility and flexibility of the analytical model. The proposed model is general and capable of analyzing the network equilibrium with multi-type drivers, demand regions, and heterogeneous ride-sourcing platforms. [ABSTRACT FROM AUTHOR]