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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.

Linear and nonlinear abstract differential equations of high order

Veli B. Shakhmurov (2015)

Open Mathematics

The nonlocal boundary value problems for linear and nonlinear degenerate abstract differential equations of arbitrary order are studied. The equations have the variable coefficients and small parameters in principal part. The separability properties for linear problem, sharp coercive estimates for resolvent, discreetness of spectrum and completeness of root elements of the corresponding differential operator are obtained. Moreover, optimal regularity properties for nonlinear problem is established....

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