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Fixed points of multivalued contractions with nonclosed, nonconvex values

Salvatore A. Marano — 1994

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

For a class of multivalued contractions with nonclosed, nonconvex values, the set of all fixed points is proved to be nonempty and arcwise connected. Two applications are then developed. In particular, one of them is concerned with some properties of the set of all classical trajectories corresponding to continuous controls for a given nonlinear control system.

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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