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Remarks on existence of positive solutions of some integral equations

Jan Ligęza (2005)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

We study the existence of positive solutions of the integral equation x ( t ) = μ ∫ 0 1 k ( t , s ) f ( s , x ( s ) , x ' ( s ) , ... , x ( n - 1 ) ( s ) ) d s , n ≥ 2 in both C n - 1 [ 0 , 1 ] and W n - 1 , p [ 0 , 1 ] spaces, where p ≥ 1 and μ > 0 . Throughout this paper k is nonnegative but the nonlinearity f may take negative values. The Krasnosielski fixed point theorem on cone is used.

Resonance and multiplicity in periodic boundary value problems with singularity

Irena Rachůnková, Milan Tvrdý, Ivo Vrkoč (2003)

Mathematica Bohemica

The paper deals with the boundary value problem u ' ' + k u = g ( u ) + e ( t ) , u ( 0 ) = u ( 2 π ) , u ' ( 0 ) = u ' ( 2 π ) , where k ∈ ℝ , g I ↦ ℝ is continuous, e ∈ 𝕃 J and lim x → 0 + ∫ x 1 g ( s ) d s = ∞ . In particular, the existence and multiplicity results are obtained by using the method of lower and upper functions which are constructed as solutions of related auxiliary linear problems.

Results on Non-resonant Oscillations for some Nonlinear Vector Fourth Order Differential Systems

Awar Simon Ukpera, Olufemi Adeyinka Adesina (2012)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

This paper presents vector versions of some existence results recently published for certain fourth order differential systems based on generalisations drawn from possibilities arising from the underlying auxiliary equation. The results obtained also extend some known works involving third order differential systems to the corresponding fourth order.

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