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EXISTENCE OF POSITIVE SOLUTIONS FOR BOUNDARY-VALUE PROBLEMS WITH SINGULARITIES IN PHASE VARIABLES

Published online by Cambridge University Press:  27 May 2004

Ravi P. Agarwal ,
Donal O’Regan  and
Svatoslav Staněk
Ravi P. Agarwal
Affiliation:
Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, Florida 32901-6975, USA (agarwal@fit.edu)
Donal O’Regan
Affiliation:
Department of Mathematics, National University of Ireland, Galway, Ireland (donal.oregan@nuigalway.ie)
Svatoslav Staněk
Affiliation:
Department of Mathematical Analysis, Faculty of Science, Palacký University, Tomkova 40, 779 00 Olomouc, Czech Republic (stanek@risc.upol.cz)
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Abstract

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The singular boundary-value problem $(g(x'))'=\mu f(t,x,x')$, $x'(0)=0$, $x(T)=b>0$ is considered. Here $\mu$ is the parameter and $f(t,x,y)$, which satisfies local Carathéodory conditions on $[0,T]\times(\mathbb{R}\setminus\{b\})\times(\mathbb{R}\setminus\{0\})$, may be singular at the values $x=b$ and $y=0$ of the phase variables $x$ and $y$, respectively. Conditions guaranteeing the existence of a positive solution to the above problem for suitable positive values of $\mu$ are given. The proofs are based on regularization and sequential techniques and use the topological transversality theorem.

AMS 2000 Mathematics subject classification: Primary 34B16; 34B18

Keywords

singular boundary-value problemmixed conditionpositive solutiontopological transversality theorem
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 47 , Issue 1 , February 2004 , pp. 1 - 13
Copyright
Copyright © Edinburgh Mathematical Society 2004