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Global structure of positive solutions for superlinear 2 m th-boundary value problems

Ruyun Ma, Yulian An (2010)

Czechoslovak Mathematical Journal

We consider boundary value problems for nonlinear 2 m th-order eigenvalue problem ( - 1 ) m u ( 2 m ) ( t ) = λ a ( t ) f ( u ( t ) ) , 0 < t < 1 , u ( 2 i ) ( 0 ) = u ( 2 i ) ( 1 ) = 0 , i = 0 , 1 , 2 , ⋯ , m - 1 . where a ∈ C ( [ 0 , 1 ] , [ 0 , ∞ ) ) and a ( t 0 ) > 0 for some t 0 ∈ [ 0 , 1 ] , f ∈ C ( [ 0 , ∞ ) , [ 0 , ∞ ) ) and f ( s ) > 0 for s > 0 , and f 0 = ∞ , where f 0 = lim s → 0 + f ( s ) / s . We investigate the global structure of positive solutions by using Rabinowitz’s global bifurcation theorem.

Infinitely many solutions of a second-order p -Laplacian problem with impulsive condition

Libo Wang, Weigao Ge, Minghe Pei (2010)

Applications of Mathematics

Using the critical point theory and the method of lower and upper solutions, we present a new approach to obtain the existence of solutions to a p -Laplacian impulsive problem. As applications, we get unbounded sequences of solutions and sequences of arbitrarily small positive solutions of the p -Laplacian impulsive problem.

Multiple positive solutions for a second order delay boundary value problem on the half-line

K. G. Mavridis, Ch. G. Philos, P. Ch. Tsamatos (2006)

Annales Polonici Mathematici

Second order nonlinear delay differential equations are considered, and Krasnosel'skiĭ's fixed point theorem is used to establish a result on the existence of positive solutions of a boundary value problem on the half-line. This result can be used to guarantee the existence of multiple positive solutions. A specification of the result obtained to the case of second order nonlinear ordinary differential equations as well as to a particular case of second order nonlinear delay differential equations...

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