An Analog of the Peano Theorem for Fractional-Order Quasilinear Equations in Compactly Embedded Scales of Banach Spaces

Abstract

In the present paper, we consider solvability conditions for the Cauchy problem for ordinary differential equations of fractional order α> 1 in Banach spaces with so-called “worsening” righthand sides. For integer-order (first- and higher-order) equations with “worsening” operators, there are classical results due to Nagumo and Ovsyannikov (e.g., see [1–6]), based on the analysis of the convergence of the successive approximation method in scales of continuously embedded Banach spaces. A number of further results in this direction was obtained in [7–9]. Similar results for fractionalorder equations with “worsening” operators were considered in [10]. Nazarov [11, 12] developed a new method for first-order differential equations in scales of compactly embedded Banach spaces on the basis of Tonelli approximations. In the present paper, we suggest a modification of his method, which results in a number of assertions on the (nonlocal!) solvability of the Cauchy problem for differential equations of integer or fractional order strictly greater than unity.