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Báo cáo hóa học: " Infinitely many solutions to superlinear second order m-point boundary value problems"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Infinitely many solutions to superlinear second order m-point boundary value problems | Ma et al. Boundary Value Problems 2011 2011 14 http www.boundaryvalueproblems.eom content 2011 1 14 o Boundary Value Problems a SpringerOpen Journal RESEARCH Open Access Infinitely many solutions to superlinear second order m-point boundary value problems Ruyun Ma1 Chenghua Gao1 and Xiaoqiang Chen2 Correspondence mary@nwnu. edu.cn department of Mathematics Northwest Normal University Lanzhou 730070 PR China Full list of author information is available at the end of the article Abstract We consider the boundary value problem u x g u x p x u x u x 0 x e 0 1 m 2 u 0 0 u i 2 aiu m i 1 where 1 m 3 hi e 0 1 and a 0 with A Ym-2 ai 1 2 g R R is continuous and satisfies g s s 0 s 0 and lim 3 p 0 1 X R2 R is continuous and satisfies p x u v C p u x e 0 1 u v e R2 for some C 0 and b e 0 1 2 . We obtain infinitely many solutions having specified nodal properties by the bifurcation techniques. MSC 2000 . 34B15 58E05 47J10 Keywords Nodal solutions Second order equations Multi-point boundary value problems Bifurcation 1 Introduction We consider the nonlinear boundary value problem u x g u x p x u x u x 0 x e 0 1 1.1 Springer m-2 u 0 0 u 1 2 aiu ni i 1 where 1.2 2011 Ma et al licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License http creativecommons.org licenses by 2.0 which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. Ma et al. Boundary Value Problems 2011 2011 14 http www.boundaryvalueproblems.eom content 2011 1 14 Page 2 of 11 H1 m 3 hi e 0 1 and ai 0 with m 2 A ai 1 i 1 H2 g R R is continuous and satisfies g s s 0 s 0 1.3 and lim x 1.4 s x s H3 p 0 1 X R2 R is continuous and satisfies p x u v C p u x e 0 1 u v e R2 1.5 for some C 0 and b e 0 1 2 . In order to state our results we first recall some standard notations to describe the nodal properties of solutions. For any integer n 0 Cn 0 1 will denote the usual Banach space of n-times .