Second order boundary value problems with sign-changing nonlinearities and nonhomogeneous boundary conditions

John R. Graef; Lingju Kong; Qingkai Kong; Bo Yang

Mathematica Bohemica (2011)

  • Volume: 136, Issue: 4, page 337-356
  • ISSN: 0862-7959

Abstract

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The authors consider the boundary value problem with a two-parameter nonhomogeneous multi-point boundary condition u ' ' + g ( t ) f ( t , u ) = 0 , t ∈ ( 0 , 1 ) , u ( 0 ) = α u ( ξ ) + λ , u ( 1 ) = β u ( η ) + μ . C r i t e r i a f o r t h e e x i s t e n c e o f n o n t r i v i a l s o l u t i o n s o f t h e p r o b l e m a r e e s t a b l i s h e d . T h e n o n l i n e a r t e r m f ( t , x ) m a y t a k e n e g a t i v e v a l u e s a n d m a y b e u n b o u n d e d f r o m b e l o w . C o n d i t i o n s a r e d e t e r m i n e d b y t h e r e l a t i o n s h i p b e t w e e n t h e b e h a v i o r o f f ( t , x ) / x f o r x n e a r 0 a n d ± ∞ , a n d t h e s m a l l e s t p o s i t i v e c h a r a c t e r i s t i c v a l u e o f a n a s s o c i a t e d l i n e a r i n t e g r a l o p e r a t o r . T h e a n a l y s i s m a i n l y r e l i e s o n t o p o l o g i c a l d e g r e e t h e o r y . T h i s w o r k c o m p l e m e n t s s o m e r e c e n t r e s u l t s i n t h e l i t e r a t u r e . T h e r e s u l t s a r e i l l u s t r a t e d w i t h e x a m p l e s .

How to cite

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Graef, John R., et al. "Second order boundary value problems with sign-changing nonlinearities and nonhomogeneous boundary conditions." Mathematica Bohemica 136.4 (2011): 337-356. <http://eudml.org/doc/196426>.

@article{Graef2011,
abstract = {The authors consider the boundary value problem with a two-parameter nonhomogeneous multi-point boundary condition \begin\{gather\}u^\{\prime \prime \}+g(t)f(t,u)=0, \quad t\in (0,1),\nonumber \\ u(0)=\alpha u(\xi )+\lambda ,\quad u(1)=\beta u(\eta )+\mu .\nonumber \unknown. Criteria for the existence of nontrivial solutions of the problem are established. The nonlinear term f(t,x) may take negative values and may be unbounded from below. Conditions are determined by the relationship between the behavior of f(t, x)/x for x near 0 and \pm \infty , and the smallest positive characteristic value of an associated linear integral operator. The analysis mainly relies on topological degree theory. This work complements some recent results in the literature. The results are illustrated with examples.\end\{gather\}},
author = {Graef, John R., Kong, Lingju, Kong, Qingkai, Yang, Bo},
journal = {Mathematica Bohemica},
keywords = {nontrivial solutions; nonhomogeneous boundary conditions; cone; Krein-Rutman theorem; Leray-Schauder degree; nontrivial solution; multi-point nonhomogeneous conditions; existence; Leray-Schauder degree; cone},
language = {eng},
number = {4},
pages = {337-356},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Second order boundary value problems with sign-changing nonlinearities and nonhomogeneous boundary conditions},
url = {http://eudml.org/doc/196426},
volume = {136},
year = {2011},
}

TY - JOUR
AU - Graef, John R.
AU - Kong, Lingju
AU - Kong, Qingkai
AU - Yang, Bo
TI - Second order boundary value problems with sign-changing nonlinearities and nonhomogeneous boundary conditions
JO - Mathematica Bohemica
PY - 2011
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 136
IS - 4
SP - 337
EP - 356
AB - The authors consider the boundary value problem with a two-parameter nonhomogeneous multi-point boundary condition \begin{gather}u^{\prime \prime }+g(t)f(t,u)=0, \quad t\in (0,1),\nonumber \\ u(0)=\alpha u(\xi )+\lambda ,\quad u(1)=\beta u(\eta )+\mu .\nonumber \unknown. Criteria for the existence of nontrivial solutions of the problem are established. The nonlinear term f(t,x) may take negative values and may be unbounded from below. Conditions are determined by the relationship between the behavior of f(t, x)/x for x near 0 and \pm \infty , and the smallest positive characteristic value of an associated linear integral operator. The analysis mainly relies on topological degree theory. This work complements some recent results in the literature. The results are illustrated with examples.\end{gather}
LA - eng
KW - nontrivial solutions; nonhomogeneous boundary conditions; cone; Krein-Rutman theorem; Leray-Schauder degree; nontrivial solution; multi-point nonhomogeneous conditions; existence; Leray-Schauder degree; cone
UR - http://eudml.org/doc/196426
ER -

References

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  3. Graef, J. R., Kong, L., 10.1017/S0013091507000788, Proc. Edinb. Math. Soc., II. Ser. 52 (2009), 79-95. (2009) Zbl1178.34024MR2475882DOI10.1017/S0013091507000788
  4. Graef, J. R., Kong, L., 10.1017/S0308210509000523, Proc. R. Soc. Edinb., Sect. A, Math. 140 (2010), 597-616. (2010) Zbl1200.34077MR2651375DOI10.1017/S0308210509000523
  5. Guo, D., Lakshmikantham, V., Nonlinear Problems in Abstract Cones, Academic Press Orlando (1988). (1988) Zbl0661.47045MR0959889
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  10. Kong, L., Kong, Q., 10.1016/j.aml.2009.05.009, Appl. Math. Lett. 22 (2009), 1633-1638. (2009) Zbl1181.34021MR2569054DOI10.1016/j.aml.2009.05.009
  11. Krasnosel'skii, M. A., Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon Press New York (1964). (1964) MR0159197
  12. Liu, L., Liu, B., Wu, Y., 10.1016/j.cam.2008.05.007, J. Comput. Appl. Math. 224 (2009), 373-382. (2009) MR2474239DOI10.1016/j.cam.2008.05.007
  13. Ma, R., 10.1016/S0893-9659(00)00102-6, Appl. Math. Lett. 14 (2001), 1-5. (2001) Zbl0989.34009MR1758592DOI10.1016/S0893-9659(00)00102-6
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