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A priori bounds and existence results for singular boundary value problems. (English) Zbl 1363.34067

Summary: This article derives qualitative properties for solutions to two-point boundary value problems (BVPs) which are systems of singular, second-order, nonlinear ordinary differential equations. The right-hand side of the differential equation is allowed to be unrestricted in the growth of its variables and may depend on the derivative of the solution. A new approach is introduced by using a singular differential inequality that ensures that all possible solutions satisfy certain a priori bounds, including their “derivatives”. Topological methods, in particular Schauder’s fixed point theorem are then applied to generate new existence results for solutions to the singular boundary value problems. Many of the results are novel for both the singular and the nonsingular cases.

MSC:

34B16 Singular nonlinear boundary value problems for ordinary differential equations
34C11 Growth and boundedness of solutions to ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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