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Existence results for a linear equation with reflection, non-constant coefficient and periodic boundary conditions. (English) Zbl 1317.34150

Summary: This work is devoted to the study of first order linear problems with involution and periodic boundary value conditions. We first prove a correspondence between a large set of such problems with different involutions to later focus our attention to the case of the reflection. We study then different cases for which a Green’s function can be obtained explicitly and derive several results in order to obtain information about its sign. Once the sign is known, maximum and anti-maximum principles follow. We end this work with more general existence and uniqueness of solution results.

MSC:

34K10 Boundary value problems for functional-differential equations
34K06 Linear functional-differential equations
34B27 Green’s functions for ordinary differential equations
34K12 Growth, boundedness, comparison of solutions to functional-differential equations
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