Global structure of positive solutions for third-order semipositone integral boundary value problems

Zhonghua Bi; Sanyang Liu; Zhonghua Bi; Sanyang Liu

doi:10.3934/math.2024353

AIMS Mathematics

2024, Volume 9, Issue 3: 7273-7292. doi: 10.3934/math.2024353

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Research article

Global structure of positive solutions for third-order semipositone integral boundary value problems

Zhonghua Bi ^,,
Sanyang Liu

School of Mathematics and Statistics, Xidian University, Xi'an, 710126, China

Received: 20 December 2023 Revised: 31 January 2024 Accepted: 01 February 2024 Published: 20 February 2024
MSC : 34B10, 34B18

In this paper, we were concerned with the global behavior of positive solutions for third-order semipositone problems with an integral boundary condition

$ \begin{equation*} \begin{split} &y'''+\beta y''+\alpha y'+\lambda f(t,y) = 0,\; \; \; t\in(0,1),\\ &y(0) = y'(0) = 0,\; \; \; y(1) = \chi\int^1_0y(s)ds, \end{split} \end{equation*} $

where $ \alpha\in(0, \infty) $ and $ \beta\in(-\infty, \infty) $ are two constants, $ \lambda, \chi $ are two positive parameters, and $ f\in C([0, 1]\times[0, \infty), \mathbb{R}) $ with $ f(t, 0) < 0 $. Our analysis mainly relied on the bifurcation theory.
- positive solutions,
- semipositone,
- third-order integral boundary value problems,
- bifurcation
Citation: Zhonghua Bi, Sanyang Liu. Global structure of positive solutions for third-order semipositone integral boundary value problems[J]. AIMS Mathematics, 2024, 9(3): 7273-7292. doi: 10.3934/math.2024353

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Abstract

In this paper, we were concerned with the global behavior of positive solutions for third-order semipositone problems with an integral boundary condition

$ \begin{equation*} \begin{split} &y'''+\beta y''+\alpha y'+\lambda f(t,y) = 0,\; \; \; t\in(0,1),\\ &y(0) = y'(0) = 0,\; \; \; y(1) = \chi\int^1_0y(s)ds, \end{split} \end{equation*} $

where $ \alpha\in(0, \infty) $ and $ \beta\in(-\infty, \infty) $ are two constants, $ \lambda, \chi $ are two positive parameters, and $ f\in C([0, 1]\times[0, \infty), \mathbb{R}) $ with $ f(t, 0) < 0 $. Our analysis mainly relied on the bifurcation theory.

References

[1]	Z. Bi, S. Liu, Positive solutions for third-order boundary value problems with indefinite weight, Mediterr. J. Math., 20 (2023), 306. https://doi.org/10.1007/s00009-023-02507-x doi: 10.1007/s00009-023-02507-x
[2]	R. Graef, L. Webb, Third order boundary value problems with nonlocal boundary conditions, Nonlinear Anal., 71 (2009), 1542–1551. https://doi.org/10.1016/j.na.2008.12.047 doi: 10.1016/j.na.2008.12.047
[3]	J. Henderson, N. Kosmatov, Three-point third-order problems with a sign-changing nonlinear term, Electron J. Differ. Equations, 175 (2014), 1–10.
[4]	T. Jankowski, Existence of positive solutions to third order differential equations with advanced arguments and nonlocal boundary conditions, Nonlinear Anal., 75 (2012), 913–923. https://doi.org/10.1016/j.na.2011.09.025 doi: 10.1016/j.na.2011.09.025
[5]	R. Ma, Y. Lu, Disconjugacy and extremal solutions of nonlinear third-order equations, Commun. Pure Appl. Anal., 13 (2014), 1223–1236. https://doi.org/10.3934/cpaa.2014.13.1223 doi: 10.3934/cpaa.2014.13.1223
[6]	J. Ren, Z. Cheng, Y. Chen, Existence results of periodic solutions for third-order nonlinear singular differential equation, Math. Nachr., 286 (2013), 1022–1042. https://doi.org/10.1002/mana.200910173 doi: 10.1002/mana.200910173
[7]	H. Yu, H. Lv, Y. Liu, Multiple positive solutions to third-order three-point singular semipositone boundary value problem, Proc. Math. Sci., 114 (2004), 409–422. https://doi.org/10.1007/BF02829445 doi: 10.1007/BF02829445
[8]	W. A. Coppel, Disconjugacy, Springer-Verlag, 1971.
[9]	A. Ambrosetti, D. Arcoya, B. Buffoni, Positive solutions for some semipositone problems via bifurcation theory, Differ. Integral Equations, 7 (1994), 655–663. https://doi.org/10.57262/die/1370267698 doi: 10.57262/die/1370267698
[10]	V. Anuradha, D. D. Hai, R. Shivaji, Existence results for superlinear semipositone BVP's, Proc. Amer. Math. Soc., 124 (1996), 757–763. https://doi.org/10.1090/S0002-9939-96-03256-X doi: 10.1090/S0002-9939-96-03256-X
[11]	A. Castro, L. Sankar, R. Shivaji, Uniqueness of nonnegative solutions for semipositone problems on exterior domains, J. Math. Anal. Appl., 394 (2012), 432–437. https://doi.org/10.1016/j.jmaa.2012.04.005 doi: 10.1016/j.jmaa.2012.04.005
[12]	R. Dhanya, Q. Morris, R. Shivaji, Existence of positive radial solutions for superlinear, semipositone problems on the exterior of a ball, J. Math. Anal. Appl., 434 (2016), 1533–1548. https://doi.org/10.1016/j.jmaa.2015.07.016 doi: 10.1016/j.jmaa.2015.07.016
[13]	D. Hai, R. Shivaji, Positive radial solutions for a class of singular superlinear problems on the exterior of a ball with nonlinear boundary conditions, J. Math. Anal. Appl., 456 (2017), 872–881. https://doi.org/10.1016/j.jmaa.2017.06.088 doi: 10.1016/j.jmaa.2017.06.088
[14]	R. Ma, Connected component of positive solutions for singular superlinear semi-positone problems, Topol. Methods Nonlinear Anal., 55 (2020), 51–62. https://doi.org/10.12775/TMNA.2019.084 doi: 10.12775/TMNA.2019.084
[15]	R. Ma, S. Wang, Positive solutions for some semi-positone problems with nonlinear boundary conditions via bifurcation theory, Mediterr. J. Math., 17 (2020), 12. https://doi.org/10.1007/s00009-019-1443-6 doi: 10.1007/s00009-019-1443-6
[16]	A. Cabada, R. Rochdi, Multiplicity results for fourth order problems related to the theory of deformations beams, Discrete Contin. Dyn. Syst., 25 (2020), 489–505. https://doi.org/10.3934/dcdsb.2019250 doi: 10.3934/dcdsb.2019250
[17]	A. Cabada, R. Rochdi, Existence results for a clamped beam equation with integral boundary conditions, Electron. J. Qual. Theory Differ. Equations, 70 (2020), 17. https://doi.org/10.14232/ejqtde.2020.1.70 doi: 10.14232/ejqtde.2020.1.70
[18]	A. Cabada, Green's functions in the theory of ordinary differential equations, Springer-Verlag, 2014. https://doi.org/10.1007/978-1-4614-9506-2
[19]	K. Deimling, Nonlinear functional analysis, Springer-Verlag, 1985. https://doi.org/10.1007/978-3-662-00547-7
[20]	U. Elias, Eigenvalue problems for the equations $Ly+\lambda p(x)y = 0$, J. Differ. Equations, 29 (1978), 28–57. https://doi.org/10.1016/0022-0396(78)90039-6 doi: 10.1016/0022-0396(78)90039-6
[21]	Z. Ma, S. Yuan, K. Meng, S. Mei, Mean-square stability of uncertain delayed stochastic systems driven by G-Brownian motion, Mathematics, 11 (2023), 2405. https://doi.org/10.3390/math11102405 doi: 10.3390/math11102405
[22]	Q. Yang, X. Wang, X. Cheng, B. Du, Y. Zhao, Positive periodic solution for neutral-type integral differential equation arising in epidemic model, Mathematics, 11 (2023), 2701. https://doi.org/10.3390/math11122701 doi: 10.3390/math11122701
[23]	Y. Zhao, Q. Zhu, Stabilization of stochastic highly nonlinear delay systems with neutral term, IEEE Trans. Autom. Control, 68 (2023), 2544–2551. https://doi.org/10.1109/TAC.2022.3186827 doi: 10.1109/TAC.2022.3186827
[24]	G. Li, Y. Zhang, Y. Guan, W. Li, Stability analysis of multi-point boundary conditions for fractional differential equation with non-instantaneous integral impulse, Math. Biosci. Eng., 20 (2023), 7020–7041. https://doi.org/10.3934/mbe.2023303 doi: 10.3934/mbe.2023303
[25]	M. Xia, L. Liu, J. Fang, Y. Zhang, Stability analysis for a class of stochastic differential equations with impulses, Mathematics, 11 (2023), 1541. https://doi.org/10.3390/math11061541 doi: 10.3390/math11061541

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