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Boundary value problems for higher order ordinary differential equations

Armando MajoranaSalvatore A. Marano — 1994

Commentationes Mathematicae Universitatis Carolinae

Let f : [ a , b ] × n + 1 be a Carath’eodory’s function. Let { t h } , with t h [ a , b ] , and { x h } be two real sequences. In this paper, the family of boundary value problems x ( k ) = f t , x , x ' , ... , x ( n ) x ( i ) ( t i ) = x i , i = 0 , 1 , ... , k - 1 ( k = n + 1 , n + 2 , n + 3 , ... ) is considered. It is proved that these boundary value problems admit at least a solution for each k ν , where ν n + 1 is a suitable integer. Some particular cases, obtained by specializing the sequence { t h } , are pointed out. Similar results are also proved for the Picard problem.

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