Existence and Non-Existence of Positive Solutions of BVPs for Singular ODEs on Whole Lines

- Journal title : Kyungpook mathematical journal
- Volume 55, Issue 4, 2015, pp.997-1030
- Publisher : Department of Mathematics, Kyungpook National University
- DOI : 10.5666/KMJ.2015.55.4.997

Title & Authors

Existence and Non-Existence of Positive Solutions of BVPs for Singular ODEs on Whole Lines

LIU, YUJI; YANG, PINGHUA;

LIU, YUJI; YANG, PINGHUA;

Abstract

This paper is concerned with integral type boundary value problems of second order singular differential equations with quasi-Laplacian on whole lines. Sufficient conditions to guarantee the existence and non-existence of positive solutions are established. The emphasis is put on the non-linear term involved with the nonnegative singular function and the singular nonlinearity term f in differential equations. Two examples are given to illustrate the main results.

Keywords

Second order singular differential equation;integral type boundary value problem;positive solution;fixed point theorem;

Language

English

References

1.

R. P. Agarwal, Boundary value problems for higher order differential equations, World Scientific, Singapore, 1986.

2.

C. Avramescu and C. Vladimirescu, Existence of Homoclinic solutions to a nonlinear second order ODE, Dynamics of continuous, discrete and impulsive systems, Ser. A, Math Anal., 15(2008), 481-491.

3.

C. Avramescu and C. Vladimirescu, Existence of solutions to second order ordinary differential equations having finite limits at ${\pm}{\infty}$ , Electronic Journal of differential equations, 18(2004), 1-12.

4.

C. Avramescu and C. Vladimirescu, Limits of solutions of a perturbed linear differential equation, E. J. Qualitative Theory of Diff. Equ., 3(2002), 1-11.

5.

A. A. Boichuk and O. A. Pokutnyi, Dichotomy and boundary value problems on the whole line, Chaotic Modeling and Simulation (CMSIM), 2(2013), 247-255.

6.

B. Bianconi and F. Papalini, Non-autonomous boundary value problems on the real line, Discrete and continuous dynamical systems, 15(2006), 759-776.

7.

A. Calamai, Heteroclinic solutions of boundary value problems on the real line involving singular $\Phi$ -Laplacian operators, Journal of Mathematical Analysis and Applications, 378(2011), 667-679.

8.

A. Cabada and J. A. Cid, Heteroclinic solutions for non-autonomous boundary value problems with singular $\Phi$ -Laplacian operators, Discrete Contin. Dyn. Syst. 2009, Dynamical Systems, Differential Equations and Applications. 7th AIMS Conference, suppl., 118-122.

9.

G. Cupini, C. Marcelli and F. Papalini, Heteroclinic solutions of boundary value problems on the real line involving general nonlinear differential operators, Differ. Integral Equ., 24(7-8)(2011), 619-644.

10.

G. Cupini, C. Marcelli and F. Papalini, On the solvability of a boundary value problem on the real line, Bound. Value Probl., 26(2011), 2011.

11.

K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985.

12.

F. Y. Deren and N. A. Hamal, Second-order boundary-value problems with integral boundary conditions on the real line, Electronic Journal of Differential Equations, 19(2014), 1-13.

13.

W. Ge, Boundary Value Problems for Ordinary Differential Equations (in Chinese), Science Press, Beijing, 2007.

14.

V. A. Il'in and E. I. Moiseev, Nonlocal boundary-value problem of the second kind for a Sturm-Liouville operator, Differential Equations, 23(1987), 979-987.

15.

Y. Liu, Multiple positive solutions to mixed boundary value problems for singular ordinary differential equations on the whole line, Nonlinear Analysis: Modelling and Control, 17(4)(2012), 460-480.

16.

Y. Liu, Existence of multiple positive solutions of p-Laplacian boundary value problems, Math. Slovak., 57(2007), 225-242.

17.

Y. Liu, Solutions to second order non-homogeneous multi-point BVPs using a fixedpoint theorem, Electronic Journal of Differential Equations, 96(2008), 1-52.

18.

C. Marcelli, Existence of solutions to some boundary-value problems governed by general nonlinear non-autonomous differential operators, Electron. J. Differ. Equ., 171(2012), (2012).

19.

C. Marcelli, The role of boundary data on the solvability of some equations involving non-autonomous nonlinear differential operators, Boundary Value Problems, 2013, 2013: 252.

20.

C. Marcelli and F. Papalini, Heteroclinic connections for fully non-linear nonautonomous second-order differential equations, Journal of Differential Equations, 241(2007), 160-183.

21.

K. R. Prasad, A. K. Rao and S. N. Rao, Existence of positive solutions for the system of higher order two-point boundary value problems, Proceedings of the Indian Academy of Sciences Mathematical Sciences, 122(2012), 139-152.

22.

C. G. Philos and I. K. Purnaras, A boundary value problem on the whole line to second order nonlinear differential equations, Georgian Mathematical Journal, 17(2010), 241-252.