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The existence theorem for solutions of differential inclusion with pseudo-Lipschitz right-hand side


E.S. Polovinkin

We study the solvability of the differential inclusion . The solution is sought in the space of absolutely integrable functions defined on a finite time interval , with values in a separable Banach space . These functions have the integrable derivative belonging to the space of Lebesgue - Bochner. Two theorems on the existence of solutions and some consequences of them are proved. In Theorem 1 we obtain the local existence of solutions for sufficiently small time interval for a given initial condition. In this theorem, we assume that the multivalued map takes convex values, is locally compact limited, measurable to the first argument and continuous to the second argument in the Hausdorff metric. The differential inclusion reduces to the integral inclusion with the Aumann integral. We prove that the operator generated by the integral inclusion has a fixed point, which is a solution to the original problem. Then we introduce the concept of measurable-pseudo-Lipschitz map in the vicinity of an absolutely continuous function and an integrable function . It means that: 1) has a measurable branch for any continuous second argument, 2) the map obtained by the restriction of to the neighborhood of , when intersecting with some neighborhood of defines a continuous on Lipschitz function with Lipschitz constant integrable on . In particular, is automatically the measurable-pseudo-Lipschitz map if it takes closed values, measurable in the first argument and Lipschitz continuous in the second argument with the Lipschitz constant integrable on . In Theorem 2 we obtain the existence of solutions on the whole interval to a differential inclusion with a measurable-pseudo-Lipschitz right-hand side for a given initial condition. An iterative process for finding a pair of functions - the solution and its derivative is considered. We prove the possibility of constructing of a convergent iterative process on the basis of the theorem on a measurable selection by C. Castaing. The result of Theorem 2 is a generalization of the well-known theorem of A.F. Filippov on the existence of solutions of differential inclusions with Lipschitz right-hand side in the case where the right-hand side of the inclusion is a measurable-pseudo-Lipschitz and the inclusion is considered in a separable Banach space.

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