Streaming graph processing is increasingly critical across various domains, requiring continuous edge updates and real-time analysis of evolving graph structures. With its massive parallelism and high-bandwidth memory access, the GPU is a promising platform for accelerating streaming graph processing. However, streaming graph processing on GPUs faces inefficiencies in graph updates and computations. To address these issues, we design and implement GASgraph, a GPU-Accelerated Streaming graph processing system. Specifically, to mitigate the high overhead of expansion and global rebalancing, we propose a novel data structure subHPMAs, which integrate an adaptive hybrid update strategy with a subPMAs-based graph representation. We implement a GPU-optimized incremental computation engine based on subHPMAs and provide a user-friendly programming interface. Extensive experiments show that GASgraph achieves significant performance improvements, with average speedups of \(66.79\times \) and \(1.25\times \) over GPMA and LPMA during graph updates, and \(9.06\times \) and \(3.46\times \) over Tigr and KickStarter in graph analytics.