Positive solutions of $n$th-order $m$-point boundary value problems

Document Type: Research Paper

Authors

1 Department of Mathematics‎, ‎Ege University‎, ‎Bornova‎, ‎35100 Izmir‎, ‎Turkey.

2 Neuroscience Institute‎, ‎Georgia State University‎, ‎Atlanta‎, ‎Georgia‎, ‎30303 USA; Department of Mathematics‎, ‎Gazi University‎, ‎Teknikokullar‎, ‎06500 Ankara‎, ‎Turkey.

Abstract

‎In this paper‎, ‎by using four functionals fixed point theorem‎, ‎we obtain sufficient conditions for the existence of‎ ‎at least one positive solution of an $n$th-order $m$-point boundary value problem‎. ‎As an application‎, ‎we give an example to demonstrate our main result.

Keywords

Main Subjects


D. R. Anderson and R. Ma, Second-order n-point eigenvalue problems on time scales, Adv. Difference Equ. (2006), Article ID 59572, 17 pages.

R. Avery, J. Henderson and D. O'Regan, Four functionals fixed point theorem, Math. Comput. Model. 48 (2008), no. 7-8, 1081--1089.

J. R. Graef and B. Yang, Positive solutions to a multi-point higher order boundary value problem, J. Math. Anal. Appl. 316 (2006), no. 2, 409--421.

Y. Guo, Y. Ji and J. Zhang, Three positive solutions for a nonlinear nth-order m-point boundary value problem, Nonlinear Anal. 68 (2008) 3485--3492.

J. Henderson and R. Luca, Positive solutions for systems of nonlinear second-order multipoint boundary value problems, Math. Methods Appl. Sci. 37 (2014), no. 16, 2502--2516.

V. A. Il'in and E. I. Moiseev, A nonlocal boundary value problem of the second kind for the Sturm-Liouville operator, Differ. Uravn. 23 (1987), no. 8, 1422--1431.

V. A. Il'in and E. I. Moiseev, A nonlocal boundary value problem of the first kind for the Sturm-Liouville operator in differential and difference interpretations, Differ. Uravn. 23 (1987), no. 7, 1198--1207.

I. Y. Karaca and F. Tokmak, Existence of three positive solutions for m-point time scale boundary value problems on infinite intervals, Dynam. Systems Appl. 20 (2011), no. 2-3, 355--367.

I. Y. Karaca, Positive solutions of an nth order three-point boundary value problem, Rocky Mountain J. Math. 43 (2013) 205--224.

I. Y. Karaca and F. Tokmak, Eigenvalues for iterative systems of nonlinear m-point boundary value problems on time scales, Bound. Value Probl. 2014 (2014), no. 63, 17 pages.

L. Liu, B. Liu and Y. Wu, Nontrivial solutions for higher-order m-point boundary value problem with a sign-changing nonlinear term, Appl. Math. Comput. 217 (2010), no. 8, 3792--3800.

R. Luca, Positive solutions for a higher-order m-point boundary value problem, Mediterr. J. Math. 9 (2012), no. 2, 379--392.

R. Ma, Multiple positive solutions for nonlinear m-point boundary value problems, Appl. Math. Comput. 148 (2004), no. 1, 249--262.

Y. B. Sang, Z. Wei and W. Dong, Existence and uniqueness of positive solutions for second-order Sturm-Liouville and multi-point problems on time scales, Bull. Korean Math. Soc. 48 (2011), no. 5, 1047--1061.

H. Su and X. Wang, Positive solutions to singular semipositone m-point n-order boundary value problems, J. Appl. Math. Comput. 36 (2011), no. 1--2, 187--200.

F. Tokmak and I. Y. Karaca, Existence of symmetric positive solutions for a multipoint boundary value problem with sign-changing nonlinearity on time scales, Bound. Value Probl. 2013 (2013), no. 52, 12 pages.

Y. L. Zhao, H. B. Chen and X. G. Liu, Triple positive solutions of second-order m-point singular boundary value problems, Math. Pract. Theory 41 (2011), no. 1, 171--177.


Volume 42, Issue 6
November and December 2016
Pages 1429-1439
  • Receive Date: 11 December 2013
  • Revise Date: 18 August 2015
  • Accept Date: 15 September 2015