Extension of the Kantorovich Theorem to Equations in Vector Metric Spaces: Applications to Functional Differential Equations

The equation G ( x , x ) = y ˜ , where G : X × X → Y , and X , Y are vector metric spaces (meaning that the values of a distance between the points in these spaces belong to some cones E + , M + of a Banach space E and a linear space M , respectively), is considered. This operator equation is compared with a “model” equation, namely, g ( t , t ) = 0 , where a continuous map g : E + × E + → M + is orderly covering in the first argument and antitone in the second one. The idea to study equations comparing them with “simpler” ones goes back to the Kantorovich fixed-point theorem for an operator acting in a Banach space. In this paper, the conditions under which the solvability of the “model” equation guarantees the existence of solutions to the operator equation are obtained. The statement proved extends the recent results about fixed points and coincidence points to more general equations in more general vector metric spaces. The results obtained for the operator equation are then applied to the study of the solvability, as well as to finding solution estimates, of the Cauchy problem for a functional differential equation.