SIGN-CHANGING SOLUTIONS OF A DISCRETE FOURTH-ORDER LIDSTONE PROBLEM WITH THREE PARAMETERS

Yuhua Long; Qinqin Zhang

doi:10.11948/20220148

2022 Volume 12 Issue 3

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2022 > 12(3): 1118-1140

Next Article Previous Article

Yuhua Long, Qinqin Zhang. SIGN-CHANGING SOLUTIONS OF A DISCRETE FOURTH-ORDER LIDSTONE PROBLEM WITH THREE PARAMETERS[J]. Journal of Applied Analysis & Computation, 2022, 12(3): 1118-1140. doi: 10.11948/20220148

Citation:

Yuhua Long, Qinqin Zhang. SIGN-CHANGING SOLUTIONS OF A DISCRETE FOURTH-ORDER LIDSTONE PROBLEM WITH THREE PARAMETERS[J]. Journal of Applied Analysis & Computation, 2022, 12(3): 1118-1140. doi: 10.11948/20220148

SIGN-CHANGING SOLUTIONS OF A DISCRETE FOURTH-ORDER LIDSTONE PROBLEM WITH THREE PARAMETERS

Yuhua Long^1,2,,
Qinqin Zhang^3, ,

1.
School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, China
2.
Guangzhou Center for Applied Mathematics, Guangzhou University, Guangzhou, 510006, China
3.
Department of Foundational Courses, Software Engineering Institute of Guangzhou, Guangzhou, 510900, China

Dedicated to Professor Jibin Li on the occasion of his 80th birthday.
Corresponding author: Email address: qqzhang@gzhu.edu.cn(Q. Zhang)

Abstract

Abstract

By combining the method of the invariant sets of descending flow with variational technique, we give a series of criteria in terms of different values of $ \lambda $ to ensure that a discrete fourth-order Lidstone problem with three parameters possesses at least four solutions. It is further shown that these four solutions consist of one sign-changing solution, one positive solution, one negative solution and one trivial solution. Finally, three examples are also provided to illustrate our theoretical results.
- Sign-changing solutions /
- positive solutions /
- negative solutions /
- invariant sets of descending flow /
- discrete Lidstone boundary value problem
MSC: 39A10, 35B20

FullText(HTML)

References

[1]	D. R. Anderson and F. Minhós, A discrete fourth-order lidstone problem with parameters, Appl. Math. Comput., 2009, 214, 523–533. Google Scholar
[2]	G. Bonanno and D. O'Regan, A boundary value problem on the half-line via critical point methods, Dyn. Syst. Appl., 2006, 15, 395–408. Google Scholar
[3]	S. Du and Z. Zhou, On the existence of multiple solutions for a partial discrete Dirichlet boundary value problem with mean curvature operator, Adv. Nonlinear Anal., 2022, 11, 198–211. Google Scholar
[4]	L. Erbe, B. Jia and Q. Zhang, Homoclinic solutions of discrete nonlinear systems via viriational method, J. Appl. Anal. Compt., 2019, 9(1), 271–294. Google Scholar
[5]	J. R. Graef, S. Heidarkhani, L. Kong and M. Wang, Existence of solutions to a discrete fourth order boundary value problem, J. Difference Equ. Appl., 2018, 24, 849–858. doi: 10.1080/10236198.2018.1428963 CrossRef Google Scholar
[6]	J. R. Graef, L. Kong and M. Wang, Two nontrivial solutions for a discrete fourth order periodic boundary value problem, Commun. Appl. Anal., 2015, 19, 487–496. Google Scholar
[7]	S. Goldberg, Introduction to Difference Equations: With Illustrative Examples from Economics, Psychology, and Sociology, New York, John Wiley, 1958. Google Scholar
[8]	T. He and Y. Su, On discrete fourth-order boundary value problems with three parameters, J. Comput. Appl. Math., 2010, 233, 2506–2520. doi: 10.1016/j.cam.2009.10.032 CrossRef Google Scholar
[9]	S. Heidarkhani, G. A. Afrouzi, A. Salari and G. Caristi, Discrete fourth-order boundary value problems with four parameters, Appl. Math. Comput., 2019, 346, 167–182. Google Scholar
[10]	W. G. Kelly and A. C. Peterson, Difference Equations: An Introduction with Applications, Academic Press, New York, 1991. Google Scholar
[11]	G. Lin and J. Yu, Homoclinic solutions of periodic discrete Schrődinger equations with local superquadratic, SIAM J. MATH. ANAL., 2022, 54(2), 1966– 2005. doi: 10.1137/21M1413201 CrossRef Google Scholar
[12]	G. Lin, Z. Zhou and J. Yu, Ground state solutions of discrete asymptotically linear Schrödinge equations with bounded and non-periodic potentials, J. Dyn. Diff. Equat., 2020, 32(2), 527–555. doi: 10.1007/s10884-019-09743-4 CrossRef Google Scholar
[13]	X. Liu and J. Liu, On a boundary value problem in the half-space, J. Differential Equat., 2011, 250, 2099–2142. doi: 10.1016/j.jde.2010.11.001 CrossRef Google Scholar
[14]	Z. Liu and J. Sun, Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations, J. Differential Equat., 2001, 172(2), 257–299. doi: 10.1006/jdeq.2000.3867 CrossRef Google Scholar
[15]	Y. Long and L. Wang, Global dynamics of a delayed two-patch discrete SIR disease model, Commu. Nonlinear Sci. Numer. Simul., 2020, 83, 105117. doi: 10.1016/j.cnsns.2019.105117 CrossRef Google Scholar
[16]	Y. Long and X. Deng, Existence and multiplicity solutions for discrete Kirchhoff type problems, Appl. Math. Lett., 2022, 126, 107817. doi: 10.1016/j.aml.2021.107817 CrossRef Google Scholar
[17]	Y. Long and H. Zhang, Three nontrivial solutions for second order partial difference equation via Morse theory, J. Funct. Spaces, 2022. https://doi.org/10.1155/2022/1564961. doi: 10.1155/2022/1564961 CrossRef Google Scholar
[18]	Y. Long and J. Chen, Existence of multiple solutions to second-order discrete Neumann boundary value problems, Appl. Math. Lett., 2018, 83, 7–14. doi: 10.1016/j.aml.2018.03.006 CrossRef Google Scholar
[19]	Y. Long and B. Zeng, Multiple and sign-changing solutions for discrete Robin boundary value problem with parameter dependence, Open Math., 2017, 15, 1549–1557. doi: 10.1515/math-2017-0129 CrossRef Google Scholar
[20]	Y. Long, Existence of two homoclinic solutions for a nonperiodic difference equation with a perturbation, AIMS Math., 2021, 6(5), 4786–4802. doi: 10.3934/math.2021281 CrossRef Google Scholar
[21]	Y. Long, S. Wang and J. Chen, Multiple solutions of fourth-order difference equations with different boundary conditions, Bound. Value Probl., 2019, 2019, 152. doi: 10.1186/s13661-019-1265-2 CrossRef Google Scholar
[22]	Y. Long and S. Wang, Multiple solutions for nonlinear functional difference equations by the invariant sets of descending flow, J. Difference Equ. Appl., 2019, 25(12), 1768–1789. doi: 10.1080/10236198.2019.1694014 CrossRef Google Scholar
[23]	Y. Long, Multiple results on nontrivial solutions of discrete Kirchhoff type problems, J. Appl. Math. Comput., 2022, to appear. Google Scholar
[24]	Y. Long, Existence of multiple and sign-changing solutions for a second-order nonlinear functional difference equation with periodic coefficients, J. Difference Equ. Appl., 2020, 26, 966–986. doi: 10.1080/10236198.2020.1804557 CrossRef Google Scholar
[25]	A. Losota, A discrete boundary value problem, Ann. Polon. Math., 1968, 20, 183–190. doi: 10.4064/ap-20-2-183-190 CrossRef Google Scholar
[26]	J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer, Berlin, 1989. Google Scholar
[27]	P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, in: Proceedings of the CBMS Regional Conference Series in Mathematics, Vol. 65, American Mathematical Society. Providence, RI, 1986. Google Scholar
[28]	Y. Shi and J. Yu, Wolbachia infection enhancing and decaying domains in mosquito population based on discrete models, J. Biol. Dyn., 2020, 14(1), 679– 695. doi: 10.1080/17513758.2020.1805035 CrossRef Google Scholar
[29]	S. Stević, B. Iričanin, W. Kosmala et al., Note on a solution form to the cyclic bilinear system of difference equations, Appl. Math. Lett., 2020, 106690. Google Scholar
[30]	S. Wang and Y. Long, Multiple solutions of fourth-order functional difference equation with periodic boundary conditions, Appl. Math. Lett., 2020, 104, 106292. doi: 10.1016/j.aml.2020.106292 CrossRef Google Scholar
[31]	J. Yu and B. Zheng, Modeling Wolbachia infection in mosquito population via discrete dynamical model, J. Difference Equ. Appl., 2019, 25(11), 1549–1567. doi: 10.1080/10236198.2019.1669578 CrossRef Google Scholar
[32]	J. Yu and Z. Guo, On boundary value problems for a discrete generalized Emden-Fowler equation, J. Differential Equations, 2006, 231(1), 18–31. doi: 10.1016/j.jde.2006.08.011 CrossRef Google Scholar
[33]	J. Yu and J. Li, Discrete-time models for interactive wild and sterile mosquitoes with general time steps, Mathematical Biosciences, 2022. https://doi.org/10.1016/j.mbs.2022.108797. doi: 10.1016/j.mbs.2022.108797 CrossRef Google Scholar
[34]	B. Zhang, L. Kong, Y. Sun and X. Deng, Existence of positive solutions for BVPs of fourth-order difference equations, Appl. Math. Comput., 2002, 131(2– 3), 583–591. Google Scholar
[35]	Q. Zhang, Homoclinic orbits for discrete Hamiltonian systems with local superquadratic conditions, Commun. Pure Appl. Anal., 2019, 18, 425–434. doi: 10.3934/cpaa.2019021 CrossRef Google Scholar
[36]	B. Zheng and J. Yu, Existence and uniqueness of periodic orbits in a discrete model on W olbachia infection frequency, Adv. Nonlinear Anal., 2022, 11, 212– 224. Google Scholar
[37]	B. Zheng, J. Li and J. Yu, One discrete dynamical model on W olbachia infection frequency in mosquito populations, Sci. China Math., 2021. https://doi.org/10.1007/s11425-021-1891-7. doi: 10.1007/s11425-021-1891-7 CrossRef Google Scholar
[38]	Z. Zhou and J. Ling, Infinitely many positive solutions for a discrete two point nonlinear boundary value problem with ϕ_c-Laplacian, Appl. Math. lett., 2019, 91, 28–34. doi: 10.1016/j.aml.2018.11.016 CrossRef Google Scholar