
Previous Article
Periodic stability of ellipticparabolic variational inequalities with timedependent boundary double obstacles
 PROC Home
 This Issue

Next Article
Nodal properties of radial solutions for a class of polyharmonic equations
Optimal constants for two point boundary value problems
1.  Department of Mathematics, Ryerson University, Toronto, Ontario, M5B 2K3, Canada 
2.  Department of Computation Science, Chengdu University of Information Technology, Chengdu, Sichuan 610225, China 
$u''(t) + \mug(t)u(t)$ = 0 a.e. on [0, 1],
subject to the general separated boundary conditions (BCs) are estimated. It is shown that $m$ < $\mu_1$ < $M(a, b)$, where $m$ and $M(a, b)$ are computable definite integrals related to the kernels arising from the above boundary value problems. The mimimum values for $M(a, b)$ are discussed when $g \stackrel{}{=}$ 1 and $g(s) = 1/s^\alpha (\alpha > 0)$ for some of these BCs. All of these values obtained here are useful in studying the existence of nonzero positive solutions for the nonlinear differential equations of the form
$u''(t) + g(t)f(t, u(t)) = 0$ a.e. on [0, 1],
subject to the above BCs.
[1] 
Abdelkader Boucherif. Positive Solutions of second order differential equations with integral boundary conditions. Conference Publications, 2007, 2007 (Special) : 155159. doi: 10.3934/proc.2007.2007.155 
[2] 
John R. Graef, Bo Yang. Multiple positive solutions to a three point third order boundary value problem. Conference Publications, 2005, 2005 (Special) : 337344. doi: 10.3934/proc.2005.2005.337 
[3] 
Qiong Meng, X. H. Tang. Multiple solutions of secondorder ordinary differential equation via Morse theory. Communications on Pure & Applied Analysis, 2012, 11 (3) : 945958. doi: 10.3934/cpaa.2012.11.945 
[4] 
Johnny Henderson, Rodica Luca. Existence of positive solutions for a system of nonlinear secondorder integral boundary value problems. Conference Publications, 2015, 2015 (special) : 596604. doi: 10.3934/proc.2015.0596 
[5] 
K. Q. Lan. Positive solutions of semiPositone Hammerstein integral equations and applications. Communications on Pure & Applied Analysis, 2007, 6 (2) : 441451. doi: 10.3934/cpaa.2007.6.441 
[6] 
Chunhua Jin, Jingxue Yin, Zejia Wang. Positive periodic solutions to a nonlinear fourthorder differential equation. Communications on Pure & Applied Analysis, 2008, 7 (5) : 12251235. doi: 10.3934/cpaa.2008.7.1225 
[7] 
Yu Tian, John R. Graef, Lingju Kong, Min Wang. Existence of solutions to a multipoint boundary value problem for a second order differential system via the dual least action principle. Conference Publications, 2013, 2013 (special) : 759769. doi: 10.3934/proc.2013.2013.759 
[8] 
Wenming Zou. Multiple solutions results for twopoint boundary value problem with resonance. Discrete & Continuous Dynamical Systems  A, 1998, 4 (3) : 485496. doi: 10.3934/dcds.1998.4.485 
[9] 
TsungFang Wu. Multiplicity of positive solutions for a semilinear elliptic equation in $R_+^N$ with nonlinear boundary condition. Communications on Pure & Applied Analysis, 2010, 9 (6) : 16751696. doi: 10.3934/cpaa.2010.9.1675 
[10] 
John R. Graef, Johnny Henderson, Bo Yang. Positive solutions to a fourth order three point boundary value problem. Conference Publications, 2009, 2009 (Special) : 269275. doi: 10.3934/proc.2009.2009.269 
[11] 
Zhiming Guo, ZhiChun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a nonlocal differential equation with homogeneous Dirichlet boundary conditionA nonmonotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 18251838. doi: 10.3934/cpaa.2012.11.1825 
[12] 
Galina V. Grishina. On positive solution to a second order elliptic equation with a singular nonlinearity. Communications on Pure & Applied Analysis, 2010, 9 (5) : 13351343. doi: 10.3934/cpaa.2010.9.1335 
[13] 
Yuri V. Rogovchenko, Fatoş Tuncay. Interval oscillation of a second order nonlinear differential equation with a damping term. Conference Publications, 2007, 2007 (Special) : 883891. doi: 10.3934/proc.2007.2007.883 
[14] 
Alexander Arguchintsev, Vasilisa Poplevko. An optimal control problem by parabolic equation with boundary smooth control and an integral constraint. Numerical Algebra, Control & Optimization, 2018, 8 (2) : 193202. doi: 10.3934/naco.2018011 
[15] 
Wenying Feng, Guang Zhang, Yikang Chai. Existence of positive solutions for second order differential equations arising from chemical reactor theory. Conference Publications, 2007, 2007 (Special) : 373381. doi: 10.3934/proc.2007.2007.373 
[16] 
Robert Stegliński. On homoclinic solutions for a second order difference equation with pLaplacian. Discrete & Continuous Dynamical Systems  B, 2018, 23 (1) : 487492. doi: 10.3934/dcdsb.2018033 
[17] 
MarieFrançoise BidautVéron, Marta GarcíaHuidobro, Cecilia Yarur. Large solutions of elliptic systems of second order and applications to the biharmonic equation. Discrete & Continuous Dynamical Systems  A, 2012, 32 (2) : 411432. doi: 10.3934/dcds.2012.32.411 
[18] 
Jiyu Zhong, Shengfu Deng. Two codimensiontwo bifurcations of a secondorder difference equation from macroeconomics. Discrete & Continuous Dynamical Systems  B, 2018, 23 (4) : 15811600. doi: 10.3934/dcdsb.2018062 
[19] 
To Fu Ma. Positive solutions for a nonlocal fourth order equation of Kirchhoff type. Conference Publications, 2007, 2007 (Special) : 694703. doi: 10.3934/proc.2007.2007.694 
[20] 
John V. Baxley, Philip T. Carroll. Nonlinear boundary value problems with multiple positive solutions. Conference Publications, 2003, 2003 (Special) : 8390. doi: 10.3934/proc.2003.2003.83 
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]