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On the cardinality of solutions of multilinear differential equations and applications. (English) Zbl 0663.34008
The author considers differential equations of the form $$Mu=(L_ 1u)(L_ 2u)...(L_ mu)=0,$$ where $$L_ i=\sum^{n(i)}_{j=0}c_{ij}D^ j,$$ n(1)$$\geq...\geq n(m)$$, $$i=1,...,m$$, in the space $$C^{n(1)}[a,b]$$, $$c_{ij}\in C^{n(1)}[a,b]$$. He obtains necessary and sufficient conditions in order that the given equation have branching solutions of a given form, and studies their cardinality. He considers the initial problem in a neighborhood of a branching point. He applies the results obtained to specific equations, constructs solutions, and gives their quantitative estimate in various cases.
MSC:
 34A30 Linear ordinary differential equations and systems 34A99 General theory for ordinary differential equations
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