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On solvability of nonlinear boundary value problems for the equation ( x ' + g ( t , x , x ' ) ) ' = f ( t , x , x ' ) with one-sided growth restrictions on f

Staněk, Svatoslav (2002)

Archivum Mathematicum

We consider boundary value problems for second order differential equations of the form ( x ' + g ( t , x , x ' ) ) ' = f ( t , x , x ' ) with the boundary conditions r ( x ( 0 ) , x ' ( 0 ) , x ( T ) ) + ϕ ( x ) = 0 , w ( x ( 0 ) , x ( T ) , x ' ( T ) ) + ψ ( x ) = 0 , where g , r , w are continuous functions, f satisfies the local Carathéodory conditions and ϕ , ψ are continuous and nondecreasing functionals. Existence results are proved by the method of lower and upper functions and applying the degree theory for α -condensing operators.

On some three-point problems for third-order differential equations

Irena Rachůnková (1992)

Mathematica Bohemica

This paper is concerned with existence and uniqueness of solutions of the three-point problem u ' ' ' = f ( t , u , u ' , u ' ' ) , u ( c ) = 0 , u ' ( a ) = u ' ( b ) . u ' ' ( a ) = u ' ' ( b ) , a ≤ c ≤ b . The problem is at resonance, in the sense that the associated linear problem has non-trivial solutions. We use the method of lower and upper solutions.

On the existence and the stability of solutions for higher-order semilinear Dirichlet problems

Marek Galewski, M. Płócienniczak (2007)

Czechoslovak Mathematical Journal

We investigate the existence and stability of solutions for higher-order two-point boundary value problems in case the differential operator is not necessarily positive definite, i.e. with superlinear nonlinearities. We write an abstract realization of the Dirichlet problem and provide abstract existence and stability results which are further applied to concrete problems.

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