GLOBAL STRUCTURE OF POSITIVE SOLUTIONS FOR FIRST-ORDER DISCRETE PERIODIC BOUNDARY VALUE PROBLEMS WITH INDEFINITE WEIGHT

Xiaoxiao Su; Meng Yan; Ruyun Ma

doi:10.11948/20230082

2024 Volume 14 Issue 1

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Xiaoxiao Su, Meng Yan, Ruyun Ma. GLOBAL STRUCTURE OF POSITIVE SOLUTIONS FOR FIRST-ORDER DISCRETE PERIODIC BOUNDARY VALUE PROBLEMS WITH INDEFINITE WEIGHT[J]. Journal of Applied Analysis & Computation, 2024, 14(1): 119-132. doi: 10.11948/20230082

Citation:

Xiaoxiao Su, Meng Yan, Ruyun Ma. GLOBAL STRUCTURE OF POSITIVE SOLUTIONS FOR FIRST-ORDER DISCRETE PERIODIC BOUNDARY VALUE PROBLEMS WITH INDEFINITE WEIGHT[J]. Journal of Applied Analysis & Computation, 2024, 14(1): 119-132. doi: 10.11948/20230082

GLOBAL STRUCTURE OF POSITIVE SOLUTIONS FOR FIRST-ORDER DISCRETE PERIODIC BOUNDARY VALUE PROBLEMS WITH INDEFINITE WEIGHT

1.
School of Mathematics and Statistics, Xidian University, Xi'an, Shannxi 710071, China

Author Bio: Email: yanmeng161999@163.com(M. Yan); Email: ryma@xidian.edu.cn(R. Ma)
Corresponding author: Email: suxiaoxiao2856@163.com(X. Su)
Fund Project: The authors were supported by National Natural Science Foundation of China (12061064) and Shaanxi Fundamental Science Research Project for Mathematics and Physics (No. 22JSY018)

Abstract

Abstract

We are concerned with the first-order discrete periodic boundary value problem
$ \begin{align} \left\{\begin{array}{ll} -D u(t)= \lambda a(t) f(u(t)),\; \; \; t\in\{1,2,\cdots,T\},\\ u(0)=u(T), \end{array} \right.\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(P) \end{align} $
where $ \lambda>0 $ is a parameter, $ T>2 $ is an integer, $ D u(t)=u(t+1)-u(t) $, $ a:\{1,2,\cdots,T\}\to\mathbb{R} $, $ f:\mathbb{R}\to\mathbb{R} $ is continuous and $ f(0)=0 $. Depending on the behavior of $ f $ near 0 and $ \infty $, we obtain that there exist $ 0<\lambda_*\leq\lambda^* $ such that for any $ \lambda>\lambda^* $, problem $ (P) $ possesses at least two positive solutions, while it has no solution for $ \lambda\in(0,\lambda_*) $. The proof of our main results are based upon bifurcation technique.
- Periodic boundary value problem /
- difference equation /
- positive solution /
- indefinite weight /
- bifurcation
MSC: 39A27, 39A28, 39A70

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References

[1]	R. P. Agarwal, On multipoint boundary value problems for discrete equations, J. Math. Anal. Appl., 1983, 96(2), 520-534. doi: 10.1016/0022-247X(83)90058-6 CrossRef Google Scholar
[2]	A. Boscaggin and G. Feltrin, Positive periodic solutions to an indefinite Minkowski-curvature equation, J. Differential Equations, 2020, 269(7), 5595-5645. doi: 10.1016/j.jde.2020.04.009 CrossRef Google Scholar
[3]	A. Boscaggin, G. Feltrin and F. Zanolin, Pairs of positive periodic solutions of nonlinear ODEs with indefinite weight: a topological degree approach for the super-sublinear case, Proc. Roy. Soc. Edinburgh Sect. A, 2016, 146(3), 449-474. doi: 10.1017/S0308210515000621 CrossRef Google Scholar
[4]	A. Boscaggin and F. Zanolin, Pairs of positive periodic solutions of second order nonlinear equations with indefinite weight, J. Differential Equations, 2012, 252(3), 2900-2921. doi: 10.1016/j.jde.2011.09.011 CrossRef Google Scholar
[5]	A. Cabada, V. Otero-Espinar and R. L. Pouso, Existence and approximation of solutions for first-order discontinuous difference equations with nonlinear global conditions in the presence of lower and upper solutions, Comput. Math. Appl., 2000, 39(1-2), 21-33. doi: 10.1016/S0898-1221(99)00310-7 CrossRef Google Scholar
[6]	E. Dancer, On the structure of solutions of non-linear eigenvalue problems, Indiana Univ. Math. J., 1974, 23, 1069-1076. doi: 10.1512/iumj.1974.23.23087 CrossRef Google Scholar
[7]	D. Djurčić and A. Torgašev, Strong asymptotic equivalence and inversion of functions in the class $K_c$, J. Math. Analysis Applic., 2001, 255(2), 383-390. doi: 10.1006/jmaa.2000.7083 CrossRef $K_c$" target="_blank">Google Scholar
[8]	W. Guan, S. Ma and D. Wang, Periodic boundary value problems for first order difference equations, Electron. J. Qual. Theory Differ. Equ., 2012, 52, 1-9. Google Scholar
[9]	P. Korman and D. S. Schmidt, Global solution curves for first order periodic problems with applications, Nonlinear Anal. Real World Appl., 2020, 56, 1-18. Google Scholar
[10]	Y. Li, L. Zhu and P. Liu, Positive periodic solutions of nonlinear functional difference equations depending on a parameter, Comput. Math. Appl., 2004, 48(10-11), 1453-1459. doi: 10.1016/j.camwa.2004.08.006 CrossRef Google Scholar
[11]	Y. Lu, Positive periodic solutions of nonlinear first-order functional difference equations with a parameter, Electron. J. Differential Equations, 2011, 96, 1-9. Google Scholar
[12]	Y. Lu and J. Wang, Continuum branch of one-signed periodic solutions of first-order functional equations involving the nonlinearity with zeros, Monatsh. Math., 2019, 190(4), 769-788. doi: 10.1007/s00605-019-01286-6 CrossRef Google Scholar
[13]	M. Ma and J. Yu, Existence of multiple positive periodic solutions for nonlinear functional difference equations, J Math. Anal. Appl., 2005, 305(2), 483-490. doi: 10.1016/j.jmaa.2004.11.010 CrossRef Google Scholar
[14]	R. Ma and Y. An, Global structure of positive solutions for nonlocal boundary value problems involving integral conditions, Nonlinear Anal., 2009, 71(10), 4364-4376. doi: 10.1016/j.na.2009.02.113 CrossRef Google Scholar
[15]	R. Ma, T. Chen and Y. Lu, Positive periodic solutions of nonlinear first-order functional difference equations, Discrete Dyn. Nat. Soc., 2010. DOI: 10.1155/2010/419536. CrossRef Google Scholar
[16]	R. Ma, X. Su and Z. Zhao, Global structure of positive solutions for a Neumann problem with indefinite weight in Minkowski space, J. Fixed Point Theory Appl., 2023, 25(2), 1-13. Google Scholar
[17]	L. Maia, N. EI Khattabi and M. Frigon, Existence and multiplicity results for first-order Stieltjes differential equations, Adv. Nonlinear Stud., 2022, 22(1), 684-710. doi: 10.1515/ans-2022-0038 CrossRef Google Scholar
[18]	S. Peng, Positive solutions for first order periodic boundary value problem, Appl. Math. Comput., 2004, 158(2), 345-351. Google Scholar
[19]	P. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 1971, 7, 487-513. doi: 10.1016/0022-1236(71)90030-9 CrossRef Google Scholar
[20]	Y. N. Raffoul, Positive periodic solutions of nonlinear functional difference equations, Electron. J. Differential Equations, 2002, 55, 1-8. Google Scholar
[21]	R. Schmidt, Über divergente Folgen und lineare Mittelbildungen, Math. Z., 1925, 22(1), 89-152. doi: 10.1007/BF01479600 CrossRef Google Scholar
[22]	J. Sugie, Number of positive periodic solutions for first-order nonlinear difference equations with feedback, Appl. Math. Comput., 2021, 391, 1-13. Google Scholar
[23]	J. Sun, Positive solution for first-order discrete periodic boundary value problem, Appl. Math. Lett. 2006, 19(11), 1244-1248. doi: 10.1016/j.aml.2006.01.007 CrossRef Google Scholar
[24]	D. Wang, Multiple solutions of periodic boundary value problems for first-order difference equations, Bull. Aust. Math. Soc., 2008, 78(1), 1-11. doi: 10.1017/S0004972708000208 CrossRef Google Scholar