Abstract:
Assertion about existence and evaluation of solutions to equations $\Upsilon (x,x)=y,$ where the mapping $\Upsilon$ acting in partially ordered spaces is covering by the first argument and antitone by the second argument is derived. This result is used for the proof of the Chaplygin's type theorem on differential inequality with deviating argument.
Keywords:orderly covering mappings, differential equation with deviating argument, the Сauchy problem, Chaplygin’s type inequality.
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