Abstract
The existence and uniqueness of solutions and asymptotic estimate of solution formulas are studied for the following initial value problem: , , , where is a constant and . An approach which combines topological method of T. Wazewski and Schauder's fixed point theorem is used.
Highlights
Introduction and PreliminariesThe singular Cauchy problem for first-order differential and integro-differential equations resolved or unresolved with respect to the derivatives of unknowns is fairly well studied see, e.g., 1–16, but the asymptotic properties of the solutions of such equations are only partially understood
The singular Cauchy problems were widely considered by using various methods see, e.g., 1–13, 16–18, the method used here is based on a different approach
Our technique leads to the existence and uniqueness of solutions with asymptotic estimates in the right neighbourhood of a singular point
Summary
The singular Cauchy problem for first-order differential and integro-differential equations resolved or unresolved with respect to the derivatives of unknowns is fairly well studied see, e.g., 1–16 , but the asymptotic properties of the solutions of such equations are only partially understood. E.g., 19, 20 and Schauder’s fixed point theorem 21. An open subset Ω0 of the set Ω is called a u, v -subset of Ω with respect to system 1.2 if the following conditions are satisfied. Let S be a nonempty compact subset of Ω0 ∪ Ω0e such that the set S ∩ Ω0e is not a retract of S but is a retract Ω0e. There is at least one point t0, y0 ∈ S ∩ Ω0 such that the graph of a solution y t of the Cauchy problem y t0 y0 for 1.2 lies in Ω0 on its right-hand maximal interval of existence. If P is a continuous mapping of S into itself and PS is relatively compact the mapping P has at least one fixed point
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