Abstract

Suppose (i) f ( t , u ) : ( 0 , 1 ) × ( 0 , + ∞ ) → [ 0 , + ∞ ) is continuous and is increasing on u; (ii) there exists a function g : [ 1 , ∞ ) → ( 0 , + ∞ ) , g ( b ) < b and g ( b ) / b 2 is integrable on ( 1 , + ∞ ) such that f ( t , bu ) ⩽ g ( b ) f ( t , u ) , ∀ ( t , u ) ∈ ( 0 , 1 ) × ( 0 , + ∞ ) . Consider the singular boundary value problem (★) u ″ ( t ) + f ( t , u ( t ) ) = 0 , 0 < t < 1 , u ( 0 ) = au ( η ) , u ( 1 ) = 0 . Then a necessary and sufficient condition for Eq. (★) to have C [ 0 , 1 ] positive solutions is that 0 < ∫ 0 1 s ( 1 - s ) f ( s , 1 ) d s < ∞ , a necessary and sufficient condition for Eq. (★) to have C 1 [ 0 , 1 ] positive solutions is that 0 < ∫ 0 1 f ( s , 1 - s ) d s < ∞ . Also, the uniqueness, iterative methods and computational methods of C 1 [ 0 , 1 ] positive solutions have been studied. Our nonlinearity may be singular at t = 0 and/or t = 1 .

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.