Abe, who in principle could remain in power until his third and final term as LDP party leader ends in September 2021, said he feels "great responsibility" for the rest of his tenure.
Modi said he was convinced of the success of the forthcoming annual Summit which will further deepen the India-Japan Special Strategic and Global Partnership.
The two leaders held a meeting on the sidelines of the East Asia Summit here during which China’s fast expanding military and economic expansionism in the Indo-Pacific region figured prominently among several other key issues, officials said
Modi, who arrived in Russia on a two-day visit, is the first Indian prime minister to visit to the Russian Far East Region.
A defence ministry official said on Friday that Singh is scheduled to meet Japanese Prime Minister Shinzo Abe and Lee Nak-Yon, Prime Minister of the Republic of Korea.
"It will not be easy to reconcile our differences, but it is becoming clearer what the issues we need to overcome are," Shinzo Abe said.
G20 Summit Live Updates: Ahead of the formal opening of the G-20 Summit, Prime Minister Narendra Modi held a trilateral meeting with US President Donald Trump and Japanese premier Shinzo Abe in Osaka, on Friday.
Shinzo Abe stressed the importance of global leaders overcoming their differences and joining hands in tackling difficult issues to send a strong message to the rest of the world.
Japanese Prime Minister Shinzo Abe during a two-day trip to Iran sought to mediate a de-escalation in tensions between Washington and Tehran and offer to broker dialogue between the two countries, but his bid was rejected by Iran, officials here said on Friday.
The trilateral, which would be an expansion of the bilateral between Trump and Abe, is part of the series of meetings Trump would have later this week on the sidelines of the G-20 Summit in Argentina.
In November 2008, 10 terrorists of Lashkar-e-Taiba (LeT) had sailed into Mumbai from Karachi and carried out coordinated attacks, killing 166 people and injuring over 300.
Progress made on the Mumbai-Ahmedabad High Speed Rail project was also reviewed during the meet. The two sides signed an agreement on yen loan for the project.
Inviting the Indian community in Japan to contribute actively in building a "new India", Modi said that India is continuously working with the spirit of Indian solutions-global applications."
It is the first time that Abe has invited a foreign political leader to his holiday home in the village of Narusawa in the prefecture.
Prime Minister Modi was an outstanding leader of his great country, the Japanese premier said.
Prime Minister Narendra Modi meets Japan Prime Minister Shinzo Abe in Yamanashi
Abe's gesture can be seen as the response to PM Modi's 'hometown diplomacy'. PM Modi had hosted Abe in his home state Gujarat during the latter's visit to India.
The meeting will focus on bilateral, regional and global issues of mutual interests, the External Affairs Ministry said on Friday.
"Securing clean water and improving sanitary conditions is a common challenge in the world. We hope for the further progress of each country's efforts to address the challenge through active discussions at this convention (MGISC)," Mr Abe said in the message.
Abe took 80 percent of parliamentarian votes, while support from regional rank-and-file party members was at 55 percent, though it fell within projections.
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