There are solid reasons for China to claim ownership of the Diaoyu Islands.
The islands have been under the jurisdiction of China's navy as islands affiliated with Taiwan since the Ming Dynasty (1368-1644). The Ming and Qing dynasties of China exercised sovereignty over the Diaoyu Islands and their affiliated islands, and officially included them in maps.
Japan has its own claims. It believes that Japan and China signed the Treaty of Shimonoseki in 1895 and as a result, China ceded Japan the island of Taiwan together with all its affiliated islands. Japan put the Diaoyu Islands and the affiliated islands under the jurisdiction of Okinawa Prefecture. This was the origin of backroom deals between Japan and the US regarding the jurisdiction of the Diaoyu Islands thereafter.
However, the international community has the final say on the ownership of the islands.
On December 1, 1943, China, the US and the UK issued the Cairo Declaration, which stated in explicit terms that the Three Great Allies were fighting this war to "restrain and punish the aggression of Japan" and required that "all the territories Japan has stolen from the Chinese, such as Manchuria, Formosa, and Pescadores" shall be restored to the Chinese side.
According to international law, the Diaoyu Islands and their affiliated islands have belonged to China since then. On July 26, 1945, the eve of the victory of the World Anti-Fascist War, China, the US and the UK issued the Potsdam Proclamation to reaffirm that
"the terms of the Cairo Declaration shall be carried out."
The Cairo Declaration used "stolen," a word rarely seen in international legal instruments, and it was reaffirmed by the Potsdam Proclamation. This clearly defines the illegal nature of Japan's aggression and the stealing of Chinese territory, and makes it clear to the whole world that China is the owner and Japan is the thief.
Some political figures in Japan question the authority and legitimacy of the Cairo Declaration and the Potsdam Proclamation. It is arrogant, if not ignorant, and even hysterical to a certain extent.
The Cairo Declaration and the Potsdam Proclamation are important outcomes of the victory of the World Anti-Fascist War, which lay the legal foundation for the postwar international order. They are of universally undoubted authority and legitimacy. The Japanese side stirred up tension when "purchasing and nationalizing islands," and further denied the outcomes of the victory of the World Anti-Fascist War. It constitutes a threat to the peace and stability of the Asia-Pacific region.