"ceased their efforts" is a correct and usable phrase in written English.
You can use it when you want to indicate that somebody has stopped trying to do something. For example: After months of trying to find a solution, they eventually ceased their efforts and accepted defeat.
By 1543 the French had ceased their efforts to colonize the northeast portion of the New World.
They ceased their efforts when, during tunnelling, they heard a noise from above.
According to KPTV, a neighbor warned the cops "that goat only respects one man," after which they ceased their efforts.
If the story is true, the plotters ceased their efforts when the tenancy to the undercroft beneath the House of Lords became available.
Reforms introduced by Jaruzelski and Mieczysław Rakowski came too little and too late, especially as changes in the Soviet Union had bolstered the public's expectation that change must come, and the Soviets ceased their efforts to prop up Poland's failing regime.
Several of the scientists said they preferred not to speculate on when they might cease their efforts to find Dr. Gray.
They sent messages to Russian operatives saying they had been identified and warned them to cease their efforts.
I urge Republicans in Congress & @RealDonaldTrump to cease their efforts to undermine health care in America & work with Democrats to safeguard Medicare & Social Security for the next generation.
Because many aspects of the world defy easy explanation, however, most people are likely to cease their efforts at some point and to content themselves with whatever degree of understanding they have managed to achieve.
The mainstream media shills never cease their efforts to bully all seriousness -- all reality -- out of the voting process.
The Chinese government and the people will not cease their efforts until the battle to protect the ecological system and the environment is won.
