Displaying similar documents to “Approximations by the Cauchy-type integrals with piecewise linear densities”

A survey and some new results on the existence of solutions of IPBVPs for first order functional differential equations

Yuji Liu (2009)

Applications of Mathematics

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This paper deals with the periodic boundary value problem for nonlinear impulsive functional differential equation x ' ( t ) = f ( t , x ( t ) , x ( α 1 ( t ) ) , ⋯ , x ( α n ( t ) ) ) for a.e. t ∈ [ 0 , T ] , Δ x ( t k ) = I k ( x ( t k ) ) , k = 1 , ⋯ , m , x ( 0 ) = x ( T ) . We first present a survey and then obtain new sufficient conditions for the existence of at least one solution by using Mawhin’s continuation theorem. Examples are presented to illustrate the main results.

Boundary value problems for higher order ordinary differential equations

Armando Majorana, Salvatore A. Marano (1994)

Commentationes Mathematicae Universitatis Carolinae

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Let f : [ a , b ] × ℝ n + 1 → ℝ 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.

A note on the Cauchy problem for first order linear differential equations with a deviating argument

Robert Hakl, Alexander Lomtatidze (2002)

Archivum Mathematicum

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Conditions for the existence and uniqueness of a solution of the Cauchy problem u ' ( t ) = p ( t ) u ( τ ( t ) ) + q ( t ) , u ( a ) = c , established in [2], are formulated more precisely and refined for the special case, where the function τ maps the interval ] a , b [ into some subinterval [ τ 0 , τ 1 ] ⊆ [ a , b ] , which can be degenerated to a point.