Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2020 / Article
Special Issue

Advances in Delay Differential Equations and Applications

View this Special Issue

Research Article | Open Access

Volume 2020 |Article ID 8342735 | https://doi.org/10.1155/2020/8342735

Zhenguo Wang, Zhan Zhou, "Multiple Solutions for Boundary Value Problems of -Laplacian Difference Equations Containing Both Advance and Retardation", Mathematical Problems in Engineering, vol. 2020, Article ID 8342735, 8 pages, 2020. https://doi.org/10.1155/2020/8342735

Multiple Solutions for Boundary Value Problems of -Laplacian Difference Equations Containing Both Advance and Retardation

Guest Editor: Chuangxia Huang
Received05 May 2020
Accepted01 Jun 2020
Published10 Aug 2020

Abstract

This paper concerns the existence of solutions for the Dirichlet boundary value problems of -Laplacian difference equations containing both advance and retardation depending on a parameter . Under some suitable assumptions, infinitely many solutions are obtained when lies in a given open interval. The approach is based on the critical point theory.

1. Introduction

Let and be the sets of integers and real numbers, respectively. For , denotes the discrete interval if .

Assume that is a positive integer. In this paper, we consider the following boundary value problem of difference equation containing both advance and retardation:where is a positive real parameter, the forward difference operator is defined as . For every , , is a continuous function, is the -Laplacian operator, that is, , and .

As it is to know, difference equations have been widely used in various research fields such as computer science, economics, network, and control system. Relevant examples and mathematical models can be found in [13]. By means of critical point theory, many researchers devote themselves to the study of difference equations and achieve many excellent results, for example, the results on boundary value problems [416], periodic solutions [1722], and homoclinic solutions [2331] had been obtained.

Difference equations containing both advance and retardation have many applications in physical and biological phenomena [32]. Yu et al. [23] obtained the existence of nontrivial homoclinic orbits for the following second-order difference equation containing both advance and retardation:

The operator is a second-order difference operator given bywhere and are real valued for each and .

Mei and Zhou [21] considered the existence of the periodic and subharmonic solutions of a order -Laplacian difference equation containing both advances and retardations:

There are more about the results of the difference equations containing both advance and retardation which can be seen in [13, 19, 22, 24].

Here, we are interested in using critical point theory to investigate infinitely many solutions for (1) containing both advance and retardation.

In fact, there are some papers which studied the existence of infinitely many solutions for the boundary value problems of difference equations. Bonanno and Candito [33] in 2009 proved the existence of infinitely many solutions of the following discrete boundary value problem:

Recently, Zhou and Ling [10] studied the existence of infinitely many positive solutions for the following boundary value problem of the second order nonlinear difference equation with -Laplacian:

However, to the best of our knowledge, no similar results are obtained for problem (1) containing both advance and retardation. The main difficulty is caused by the advance and retardation. In this paper, we obtain some sufficient conditions to guarantee the existence of infinitely many solutions of (1). In fact, under some assumptions, we prove the existence of infinitely many solutions of equation (1) for each in Theorem 1. Moreover, Theorem 2 guarantees the existence of infinity positive solutions for (1) by applying a strong maximum principle. Finally, we show that (1) possesses a sequence of distinct solutions which converges to zero for each in Theorem 3.

This paper is organized as follows. In Section 2, some definitions and preliminaries on difference equations are collected. In Section 3, our main results are established. Finally, two examples are given to illustrate our main results.

2. Preliminaries

Let be a reflexive real Banach space and be a function satisfying the following structure hypothesis: Assume that is a real positive parameter. , , where , is coercive, that is, .

Provided that , write

Obviously, and . When or , we agree to read or as .

Lemma 1 (see [34]). Assume that condition holds, and one has(a)For every and every , the restriction of functional to admits a global minimum, which is a critical point (local minimum) of in .(b)If , then for each , the following alternative holds: possesses a global minimum There is a sequence of critical points (local minimum) of such that .(c)If , then for each , the following alternative holds: There is a global minimum of which is a local minimum of There is a sequence of pairwise distinct critical points (local minimum) of , with , which weakly converges to a global minimum of .

Consider the -dimensional Banach space:endowed with the norm

We define another two norms on as follows:

According to Lemma 2.2 in [12], we have the following inequality:for every .

Lemma 2. For every , one haswhere .

Proof. Making use of (11), we obtainthen
For every , putwhere satisfies the following conditions: , , , for Direct computation ensures that is a functional of class on S withThen, is a critical point of on S if and only ifThat is, the function is just the variational framework of (1). For the reader’s convenience, we recall a consequence of strong maximum principle [8]. The strong maximum principle is used to obtain positive solutions to (1), that is, for every .

Lemma 3 (see [8]). Fix such thatThen, either or .

Remark 1. Assume is continuous for each and for all . Putthen for each .
Consider the following boundary value problem:From Lemma 3, all solutions of (19) are either zero or positive; hence, they are also solutions for (1). When (19) possesses nontrivial solutions, (1) possesses positive solutions, independently of the sign of .

3. Main Results

Put

Now, we consider the suitable oscillating behavior of when goes to . We have the following theorem.

Theorem 1. Assume that are satisfied and there exist two real sequences , with , such that

Then, problem (1) admits an unbounded sequence of solutions for each .

Proof. Our aim is to apply Lemma 1(b) to prove our conclusion. Fix , which clearly holds. Our conclusion needs to provide that . Put for all . Owing to (12), if , then for every , and we obtainLet for every , and . Clearly, and . From (21), we obtain , and one hasHence, .
Now, we verify that is unbounded from below. By (20), let be a positive real sequence with andFirst, we assume that , and then we can fix , and there exists such thatWe take a sequence in such that for every , , and one hasfor all . That is, .
Next, we assume that . Since , from (25), we can fix such that , arguing as before, there is a such thatBy choosing in as above, we obtain thatfor all . Hence, one has .
We have verified all assumptions of Lemma 1(b); then, there is a sequence of critical points (local minima) of such that .
According to Theorem 1, it is easy to obtain the following corollary.

Corollary 1. Let be satisfied, and assume that

Then, problem (1) admits an unbounded sequence of solutions for each .

Proof. Let be a real sequence with such thatWe take for all in (21); combining with , (30) and (31), we can apply Theorem 1 to reach the conclusion.
From the argument of Remark 1, we have the following theorem and corollary.

Theorem 2. If for every and the hypotheses of Theorem 1 hold. Then, problem (1) admits an unbounded sequence of positive solutions for each .

Corollary 2. If for every and the hypotheses of Corollary 1 are satisfied, then problem (1) admits an unbounded sequence of positive solutions for each .

Next, we consider the oscillating behavior of when goes to 0. We obtain the following theorem.

Theorem 3. Assume that are satisfied andwhere and . Then, problem (1) possesses a sequence of pairwise distinct solutions, which converges to zero for each .

Proof. We will check the part of Lemma 1. Fix , let be a positive real sequence such that andAs before, we let for each . In view of (12), note that implies for every , by the definition of , we havethen .
Clearly, 0 is a global minimum of in , and by .
Moreover, we can verify that 0 is not a local minimum of . Given a positive real sequence with such thatif , fix , and there exists such thatLet be a sequence satisfying for every , . Since as , we havefor all . Thus, 0 is not a local minimum of .
If , since , we can also find a positive real sequence with such that (35) holds. Fix , and there exists , such thatArguing as before, there exist a sequence , such thatfor all . Obviously, 0 is not a local minimum of , and of Lemma 1(c) is true.
A similar result to Theorem 2 is obtained as follows.

Theorem 4. If for every and the hypotheses of Theorem 3 hold, then problem (1) admits a sequence of pairwise distinct positive solutions, which converges to zero for each .

4. Examples

Finally, we give two examples to illustrate our results.

Example 1. Consider the boundary value problem (1) withfor . PutObviously, satisfies conditions and . It is easy to see thatClearly,andLet be sufficiently small, then . Hence, for each , (1) admits an unbounded sequence of solutions by Corollary 1. Moreover, we have for all ; according to Corollary 2, (1) admits an unbounded sequence of positive solutions.

Example 2. Consider the boundary value problem (1) withif and for . Letthen holds. Clearly, holds andBy computation, we obtainObviously, we have , when be sufficiently small. On the contrary, for all .
The computations ensure that all the assumptions of Theorem 4 are satisfied. Then, for each , (1) admits a sequence of pairwise distinct positive solutions, which converges to zero.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

All authors contributed equally to the manuscript and read and approved the final manuscript.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 11971126), Program for Changjiang Scholars and Innovative Research Team in University (Grant no. IRT_16R16), and Scientific and Technologial Innovation Programs of Higher Education Institutions in Shanxi (Grant no. 2019L0955).

References

  1. W. G. Kelly and A. C. Peterson, Difference Equations: an Introduction with Applications, Academic Press, Cambridge, MA, USA, 1991.
  2. R. P. Agarwal, Difference Equations and Inequalities: Theory, Methods, and Applications, Marcel Dekker, Inc., New York City, NY, USA, 2000.
  3. J. S. Yu and B. Zheng, “Modeling wolbachia infection in mosquito population via discrete dynamical models,” Journal of Difference Equations and Applications, vol. 25, no. 11, pp. 1549–1567, 2019. View at: Publisher Site | Google Scholar
  4. G. Bonanno and P. Candito, “Nonlinear difference equations investigated via critical point methods,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 9, pp. 3180–3186, 2009. View at: Publisher Site | Google Scholar
  5. G. Bonanno, P. Candito, and G. D’Aguì, “Variational methods on finite dimensional banach space and discrete problems,” Advanced Nonlinear Studies, vol. 14, no. 4, pp. 915–939, 2014. View at: Publisher Site | Google Scholar
  6. G. Bonanno, P. Candito, and G. D’Aguì, “Positive solutions for a nonlinear parameter-depending algebraic system,” Electronic Journal of Differential Equations, vol. 2015, no. 17, pp. 1–14, 2015. View at: Google Scholar
  7. G. Bonanno, P. Jebelean, and C. Şerban, “Superlinear discrete problems,” Applied Mathematics Letters, vol. 52, pp. 162–168, 2016. View at: Publisher Site | Google Scholar
  8. G. D’Aguì, J. Mawhin, and A. Sciammetta, “Positive solutions for a discrete two point nonlinear boundary value problem with p-laplacian,” Journal of Mathematical Analysis and Applications, vol. 447, no. 1, pp. 383–397, 2017. View at: Publisher Site | Google Scholar
  9. R. P. Agarwal and R. Luca, “Positive solutions for a system of second order discrete boundary value problem,” Advances in Difference Equations, vol. 470, p. 17, 2018. View at: Google Scholar
  10. Z. Zhou and J. X. Ling, “Infinitely many positive solutions for a discrete two point nonlinear boundary value problem with ϕc-laplacian,” Applied Mathematics Letters, vol. 91, pp. 28–34, 2019. View at: Publisher Site | Google Scholar
  11. J. X. Ling and Z. Zhou, “Positive solutions of the discrete dirichlet problem involving the mean curvature operator,” Open Mathematics, vol. 17, no. 1, pp. 1055–1064, 2019. View at: Publisher Site | Google Scholar
  12. L. Q. Jiang and Z. Zhou, “Three solutions to dirichlet boundary value problems for p-laplacian difference equations,” Advances in Difference Equations, vol. 2008, Article ID 345916, 10 pages, 2007. View at: Google Scholar
  13. Z. Zhou and M. T. Su, “Boundary value problems for 2n-order ϕc-laplacian difference equations containing both advance and retardation,” Applied Mathematics Letters, vol. 41, pp. 7–11, 2015. View at: Publisher Site | Google Scholar
  14. Y. H. Long and J. L. Chen, “Existence of multiple solutions to second-order discrete neumann boundary value problems,” Applied Mathematics Letters, vol. 83, pp. 7–14, 2018. View at: Publisher Site | Google Scholar
  15. Y. H. Long and S. H. Wang, “Multiple solutions for nonlinear functional difference equations by the invariant sets of descending flow,” Journal of Difference Equations and Applications, vol. 25, no. 12, pp. 1768–1789, 2019. View at: Publisher Site | Google Scholar
  16. S. H. Wang and Y. H. Long, “Multiple solutions of fourth-order functional difference equation with periodic boundary conditions,” Applied Mathematics Letters, vol. 104, Article ID 106292, 2020. View at: Publisher Site | Google Scholar
  17. Z. M. Guo and J. S. Yu, “Existence of periodic and subharmonic solutions for second-order superlinear difference equations,” Science in China Series A: Mathematics, vol. 46, no. 4, pp. 506–515, 2003. View at: Publisher Site | Google Scholar
  18. Z. Zhou, J. S. Yu, and Z. M. Guo, “Periodic solutions of higher-dimensional discrete systems,” Proceedings of the Royal Society of Edinburgh: Section A Mathematics, vol. 134, no. 5, pp. 1013–1022, 2004. View at: Publisher Site | Google Scholar
  19. P. Chen and H. Fang, “Existence of periodic and subharmonic solutions for second-order p-laplacian difference equations,” Advances in Difference Equations, vol. 2007, Article ID 42530, 10 pages, 2007. View at: Google Scholar
  20. Z. Zhou, J. S. Yu, and Y. M. Chen, “Periodic solutions of a 2nth-order nonlinear difference equation,” Science in China Series A: Mathematics, vol. 53, no. 1, pp. 41–50, 2010. View at: Publisher Site | Google Scholar
  21. P. Mei and Z. Zhou, “Periodic and subharmonic solutions for a 2nth-order p-Laplacian difference equation containing both advances and retardations,” Open Mathematics, vol. 16, no. 1, pp. 1435–1444, 2018. View at: Publisher Site | Google Scholar
  22. P. Mei, Z. Zhou, and G. H. Lin, “Periodic and subharmonic solutions for a 2n th-order φc -laplacian difference equation containing both advances and retardations,” Open Mathematics, vol. 12, no. 7, pp. 2085–2095, 2019. View at: Google Scholar
  23. J. S. Yu, H. P. Shi, and Z. M. Guo, “Homoclinic orbits for nonlinear difference equations containing both advance and retardation,” Journal of Mathematical Analysis and Applications, vol. 352, no. 2, pp. 799–806, 2009. View at: Publisher Site | Google Scholar
  24. P. Chen and X. H. Tang, “Existence of homoclinic orbits for 2nth-order nonlinear difference equations containing both many advances and retardations,” Journal of Mathematical Analysis and Applications, vol. 381, no. 2, pp. 485–505, 2011. View at: Publisher Site | Google Scholar
  25. Z. Zhou and J. S. Yu, “Homoclinic solutions in periodic nonlinear difference equations with superlinear nonlinearity,” Acta Mathematica Sinica, vol. 29, no. 9, pp. 1809–1822, 2013. View at: Publisher Site | Google Scholar
  26. G. H. Lin and Z. Zhou, “Homoclinic solutions in periodic difference equations with mixed nonlinearities,” Mathematical Methods in the Applied Sciences, vol. 39, no. 2, pp. 245–260, 2016. View at: Publisher Site | Google Scholar
  27. Q. Q. Zhang, “Homoclinic orbits for discrete hamiltonian Systems with indefnite linear part,” Communications on Pure and Applied Analysis, vol. 14, no. 5, pp. 1929–1940, 2017. View at: Publisher Site | Google Scholar
  28. G. H. Lin and Z. Zhou, “Homoclinic solutions in non-periodic discrete ϕ-laplacian equations with mixed nonlinearities,” Applied Mathematics Letters, vol. 64, pp. 15–20, 2017. View at: Google Scholar
  29. G. H. Lin and Z. Zhou, “Homoclinic solutions of discrete ϕ-laplacian equations with mixed nonlinearities,” Communications on Pure and Applied Analysis, vol. 17, no. 5, pp. 1723–1747, 2018. View at: Publisher Site | Google Scholar
  30. Q. Q. Zhang, “Homoclinic orbits for discrete hamiltonian systems with local super-quadratic conditions,” Communications on Pure and Applied Analysis, vol. 18, no. 1, pp. 425–434, 2019. View at: Publisher Site | Google Scholar
  31. G. H. Lin, Z. Zhou, and J. S. Yu, “Ground state solutions of discrete asymptotically linear schrodinger equations with bounded and non-periodic potentials,” Journal of Dynamics and Differential Equations, vol. 32, no. 2, pp. 527–555, 2020. View at: Publisher Site | Google Scholar
  32. D. Smets and M. Willem, “Solitary waves with prescribed speed on infinite lattices,” Journal of Functional Analysis, vol. 149, no. 1, pp. 266–275, 1997. View at: Publisher Site | Google Scholar
  33. G. Bonanno and P. Candito, “Infinitely many solutions for a class of discrete nonlinear boundary value problem,” Applicable Analysis, vol. 88, no. 4, pp. 605–616, 2009. View at: Publisher Site | Google Scholar
  34. G. Bonanno and B. G. Molica, “Infinitely many solutions for a boundary value problem with discontinuous nonlinearities,” Boundary Value Problems, vol. 2009, Article ID 670675, 20 pages, 2009. View at: Publisher Site | Google Scholar

Copyright © 2020 Zhenguo Wang and Zhan Zhou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


More related articles

 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder
Views115
Downloads240
Citations

Related articles

Article of the Year Award: Outstanding research contributions of 2020, as selected by our Chief Editors. Read the winning articles.