Multiple Monotone Positive Solutions to 3rd-Order Boundary Value Problems Involving Riemann-Stieltjes Integral Conditions

ZHANG Hai-e

doi:10.3879/j.issn.1000-0887.2015.07.010

Volume 36 Issue 7

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2015 > 36(7): 779-786

ZHANG Hai-e. Multiple Monotone Positive Solutions to 3rd-Order Boundary Value Problems Involving Riemann-Stieltjes Integral Conditions[J]. Applied Mathematics and Mechanics, 2015, 36(7): 779-786. doi: 10.3879/j.issn.1000-0887.2015.07.010

Citation:

PDF( 331 KB)

Multiple Monotone Positive Solutions to 3rd-Order Boundary Value Problems Involving Riemann-Stieltjes Integral Conditions

doi: 10.3879/j.issn.1000-0887.2015.07.010

ZHANG Hai-e

Department of Basic Science, Tangshan College, Tangshan, Hebei 063000, P.R.China

Received Date: 2014-10-27
Rev Recd Date: 2015-05-18
Publish Date: 2015-07-15

Abstract

Abstract

A class of 3rd-order nonlocal boundary value problems (BVPs) with Riemann-Stieltjes integral conditions were studied. The existence of positive solutions to BVPs was explored via perturbed Hammerstein integral equations. Through the construction of the Green functions and discussion on their properties, the existence criterion for at least 3 or 2n-1 positive solutions was obtained by means of the generalization of the Leggett-Williams fixed point theorem. The results generalize and improve some known results of the latest literatures, and fully reflect the influence of nonlinear terms involving derivatives on the existence of positive solutions. An example was also included to illustrate the main results.
- Leggett-Williams fixed point theorem,
- positive solution,
- nonlocal,
- integral condition

FullText(HTML)

References(15)

References

[1]	Gregus M. Third Order Linear Differential Equations[M]. Mathematics and Its Applications22. Dordrecht: D Reidel Publishing Company, 1987.
[2]	Aftabizadeh A R, Gupta C P, XU Jian-ming. Existence and uniqueness theorems for three-point boundary value problems[J]. SIAM Journal on Mathematical Analysis,1989,20(3): 716-726.
[3]	Aftabizadeh A R, Deimling K. A three-point boundary value problem[J]. Differential and Integral Equations,1991,4(1): 189-194.
[4]	Bernis F, Peletier L A. Two problems from draining flows involving third-order ordinary differential equations[J]. SIAM Journal on Mathematical Analysis,1996,27(2): 515-527.
[5]	Graef J R, Yang B. Positive solutions of a third order nonlocal boundary value problem[J]. Discrete and Continuous Dynamical Systems—Series S,2008,1(1): 89-97.
[6]	YAO Qing-liu. Positive solutions of singular third-order three-point boundary value problems[J]. Journal of Mathematical Analysis and Applications,2009,354(1): 207-212.
[7]	WANG You-yu, GE Wei-gao. Existence of solutions for a third order differential equation with integral boundary conditions[J]. Computers & Mathematics With Applications,2007,53(1): 144-154.
[8]	ZHAO Jun-fang, WANG Pei-guang, GE Wei-gao. Existence and nonexistence of positive solutions for a class of third order BVP with integral boundary conditions in Banach spaces[J]. Communications in Nonlinear Science and Numerical Simulation,2011,16(1): 402-413.
[9]	ZHANG Hai-e. Multiple positive solutions of nonlinear BVPs for differential systems involving integral conditions[J]. Boundary Value Problems,2014,2014: 61. doi: 10.1186/1687-2770-2014-61.
[10]	张建元, 赵书芬, 韩艳. K-解析函数的Riemann边值问题[J]. 应用数学和力学, 2014,35(7): 805-814.(ZHANG Jian-yuan, ZHAO Shu-fen, HAN Yan. Riemann’s boundary value problem of K-analytic functions[J]. Applied Mathematics and Mechanics,2014,35(7): 805-814.(in Chinese))
[11]	罗李平, 俞元洪, 曾云辉. 三阶非线性时滞微分方程振动性的新准则[J]. 应用数学和力学, 2013,34(9): 941-947.(LUO Li-ping, YU Yuan-hong, ZENG Yun-hui. New criteria for oscillation of third order nonlinear delay differential equations[J]. Applied Mathematics and Mechanics,2013,34(9): 941-947.(in Chinese))
[12]	Webb J R L, Infante G. Positive solutions of nonlocal boundary value problems: a unified approach[J]. Journal of the London Mathematical Society,2006,74(3): 673-693.
[13]	Webb J R L, Infante G. Non-local boundary value problems of arbitrary order[J]. Journal of the London Mathematical Society,2009,79(1): 238-258.
[14]	ZHANG Hai-e, SUN Jian-ping. Existence and iteration of monotone positive solutions for third-order nonlocal BVPs involving integral conditions[J]. Electronic Journal of Qualitative Theory of Differential Equations,2012(18): 1-9.
[15]	BAI Zhan-bing, GE Wei-gao. Existence of three positive solutions for some second-order boundary value problems[J]. Computers & Mathematics With Applications,2004,48(5/6): 699-707.