Existence of Solutions for the Discrete Dirichlet Problem Involving <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.52687pt" id="M1" height="13.4804pt" version="1.1" viewBox="-0.0657574 -8.95353 10.3455 13.4804" width="10.3455pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-113" d="M570 304C570 398 525 448 414 448C385 448 343 445 312 434L329 511L321 518C297 504 262 482 244 460L233 411C195 397 159 381 128 358L135 332C160 347 189 360 224 373L111 -147C97 -210 84 -218 17 -231L13 -257L254 -247L259 -218L233 -216C183 -212 177 -202 189 -142L218 -1C238 -10 266 -12 283 -12C351 3 429 48 483 105C543 168 570 242 570 304ZM482 289C482 161 380 33 304 33C278 33 248 51 233 69L303 396C326 400 352 403 369 403C428 403 482 380 482 289Z"/></g></svg>-</span>Mean Curvature Operator

Wang, Jianxia; Zhou, Zhan

doi:https://doi.org/10.1155/2020/6591523

Discrete Dynamics in Nature and Society

On this page

Abstract Introduction Examples Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Special Issue

Stability and Bifurcation Analysis of Discrete Dynamical Systems 2020

View this Special Issue

Research Article | Open Access

Volume 2020 | Article ID 6591523 | https://doi.org/10.1155/2020/6591523

Existence of Solutions for the Discrete Dirichlet Problem Involving -Mean Curvature Operator

Jianxia Wang^1,2and Zhan Zhou^1,2

Academic Editor: Abdul Qadeer Khan

Received04 Jul 2020

Accepted27 Jul 2020

Published29 Aug 2020

Abstract

This work is to discuss the Dirichlet boundary value problem of the difference equation with -mean curvature operator. Under some determinate growth conditions on the nonlinear term, the existence of one solution or two nontrivial solutions is obtained via variational methods and some analysis techniques. Examples are also given to illustrate our theorems in the last section.

1. Introduction

Let , and stand for the sets of natural numbers, integer numbers, and real numbers, respectively. For , define and when .

Consider the following Dirichlet boundary value problem:where is a given positive integer, for each , is the forward difference operator defined by , and . Here, is counted as a discretization of -mean curvature operator.

We may regard problem (1) as being a discrete analog of the following differential problem in one-dimensional space:where , named -mean curvature operator [1], is a generalization of mean curvature operator ([2], page 1212). When , it reduces to mean curvature operator [3–6]. When , it reduces to Laplacian operator.

Difference equations arise in different research fields, for instance, computer science, mechanical engineering, control systems, artificial or biological neural networks, economics, and so on, see [7–12].

In order to research discrete boundary problems, different methods have been used: fixed point theorems, upper and lower solutions techniques, see [3, 5] and the references given therein. Starting from Guo and Yu [13], various results have been achieved via variational methods, see [14–32].

In 2011, Candito and Bisci [33] studied the boundary value problem:where , is continuous with respect to for every . Through variational methods, they proved that problem (3) admits two nontrivial solutions.

In 2015, Zhou and Su [30] discussed the following problem:where is positive integer, , and . By critical point theory and some analytical skills, two nontrivial solutions were acquired.

Motivated by the works of [15, 30, 34–43], we deal with problem (1). Compared to problem (4), problem (1) is much more difficult to handle because of -mean curvature operator. It needs more analytical skills. For the convenience of analysis, we have no option but to split the problem into two types: and .

This paper is arranged as follows. In Section 2, we establish an appropriate Banach space and a suitable functional corresponding to problem (1). In order to gain solutions of problem (1), some preliminary knowledge is introduced. In Section 3, under certain growth conditions on the antiderivative of the nonlinearity , one solution or two nontrivial solutions for problem (1) are obtained. In Section 4, we give two concrete examples to demonstrate the applicability of our main results.

2. Mathematical Background

For problem (1), we naturally select the - dimensional Banach space:equipped with the normfor all .

Its infinity norm is

Apparently, one has

Furthermore, we need to use the following inequality. When , one hassee ([44], page 28).

Puttingthe function is defined byfor every , where

It is not difficult to verify that , and the critical points of on are exactly the solutions of problem (1).

One can see thatwhere is a -dimensional row vector, stands for the transpose of , and

Matrix is symmetric and positive definite. It is easy to see that has distinct positive eigenvalues which are given by

We have the following lemma, see ([36], page 918).

Lemma 1. For every , one has

Let E be a real Banach space, the functional is said to satisfy the Palais–Smale (P. S. for short) condition if any sequence in E such that is bounded and contains a convergent subsequence.

Let denote the open ball in E of radius about 0, and let denote its boundary.

Our main tool is Mountain pass lemma, see ([45], Theorem 2.2).

Lemma 2. Let E be a real Banach space and satisfying the P. S. condition. Suppose and(i)There are constants such that (ii)There exists an such that Then, admits a critical value . In addition, can be characterized aswhere

3. Main Results and Their Proofs

Theorem 1. Let , for every , be continuous. Suppose that .

Then, is coercive, and problem (1) admits at least one solution.

Proof. It is obvious that .
Owing to , we choose and such that, for every and , . Since is continuous on , there exists some constant such that , for all . Consequently, we follow thatfor every.
Using (9) and (20), Hölder inequality, and Lemma 1, we follow thatNote thatand we have . So, is coercive in S. Hence, is bounded from below. Set . According to the coercivity of the functional again, there is such that . Thus, . Taking into account that is a finite dimensional space and is continuous, possesses a critical point satisfying . The proof is completed.

Theorem 2. Let , for every , be continuous. Suppose that

Then, is anticoercive, and problem (1) has at least one solution.

Proof. To avoid repetition, we give only the key steps of the proof.
By , we chooseand such thatfor every .
By virtue of (9) and (25), Hölder inequality, and Lemma 1, we follow thatSincewe have .

Theorem 3. Let , for every , be continuous. Suppose that

Then, is coercive, and problem (1) possesses at least one solution.

Proof. Here, we give only the main steps of the proof.
By , we choose and such thatfor every .
Thanks to (9) and (30), Hölder inequality, and Lemma 1, and we follow thatSincewe have

Theorem 4. Let , for every , be continuous. Suppose that

Then, is anticoercive, and problem (1) admits at least one solution.

Proof. In the same way, we give only the crucial steps of the proof.
By , we chooseand such thatfor every .
Given any , applying (9) and (36), Hölder inequality, and Lemma 1, we follow thatNote thatwe have

Remark 1. If moreover, in Theorems 1–4, if for some , then the solution is nontrivial.

Corollary 1. Let , for every , be continuous, and satisfy one of the following conditions:

Then, problem (1) has at least one solution.

Proof. Here, we only prove that implies . The proof that implies refers to ([33], Corollary 2.1).
From , there exist and such thatfor each and .
Since is continuous with respect to , we also letThus, when , one followsWhen , we havePutTherefore,Hence, our argument holds, and the proof is complete.

Remark 2. We observe that there are functions that satisfy , but does not hold, see, such as example 1 in Section 4.
Similarly, we have the following result.

Corollary 2. Let , for every , be continuous, and satisfy one of the following conditions:

Then, problem (1) admits at least one solution.

The following results guarantee the existence of two nontrivial solutions to problem (1).

Theorem 5. Let , for every , be continuous, and holds. Additionally, assume thatThen, problem (1) has at least two nontrivial solutions.

Proof. Our aim is to apply Lemma 2. To this purpose, take and as in (11). Let . Using Theorem 1, it is evident that is anticoercive. Since the dimension of is finite, we can easily deduce that satisfies the P.S. condition. Next, we prove that verifies the mountain pass geometry.
From , there are and such thatfor every .
By (8), Lemma 1, and the inequality , for , we havefor with .
Writing and , then . Hence, satisfies condition (i) of Lemma 2. It is obvious that . Because , we may pick and with , which means satisfying condition (ii) of Lemma 2.
According to Lemma 2, has a critical value given by , whereSet be a critical point of and , then is a nonzero solution of problem (1). By Theorem 1, possesses a critical point with . Noting that , we have . If , then the proof is complete. If , we divide the following proof into two cases.

Case I. . In such a case, we may see that . Using again, there exists satisfying . Since and , there exists such that . This shows that is a critical point of . Hence, problem (1) admits two nontrivial solutions.

Case II. . Through the definition of , we show that for every . Considering that , there exists which is linearly independent with . For every , definethen . Thus, we have satisfying . Obviously, when , and is a sequence of pairwise distinct critical points of in . Consequently, problem (1) has infinitely many nontrivial solutions. The proof is finished.

Theorem 6. Let , for every , be continuous, and holds. Moreover, suppose that

Then, problem (1) admits at least two nontrivial solutions.

Proof. Let and as in (11). Owing to Theorem 2, is anticoercive. Noting that is a finite dimensional space, it is obvious that satisfies the P.S. condition. Now, we show that satisfies the mountain pass geometry.
By , we pick such thatfor each .
Hence, there exists , and we follow thatwhen
Thanks to (8), for with , we getFor with exploiting Lemma 1, we getBy Lemma 1 and the inequality , we follow thatfor with .
Let , then combining (56) and (58), we havewhen with .
Picking and , then for every , . The rest of the proof runs as the proof of Theorem 5.

Theorem 7. Let , for every , be continuous. Suppose that and are satisfied. Then, problem (1) has at least two nontrivial solutions.

Proof. Take and as in (11) and denote . Applying Theorem 3, we can derive that is anticoercive. Here, we only verify that satisfies condition (i) of Lemma 2.
Since , there exists such that .
Using Taylor’s formula, we have the following inequality:From (9) and the above inequality, we follow thatOwing to , there exist and such that (49) is verified.
Because of the fact thatfor the above , there is such thatwhenever .
Picking , we have for all with .
Through (49), (61), and (63), we obtainfor with .
Taking and , we follow that if . The proof is complete.
Applying the method analogous to that used above, we can draw the following conclusion.

Theorem 8. Let , for every , be continuous. Suppose that and are fulfilled. Then, problem (1) has at least two nontrivial solutions.

4. Examples

4.1. Example 1

Let , and be a function which is defined as follows:

Sincethrough direct computations, we follow thatfor every . Thus, the conditions and are satisfied. Theorem 5 ensures that problem (1) admits at least two nontrivial solutions.

Moreover, we show that condition does not hold taking into account thatfor every .

4.2. Example 2

When , consider the following problem:

Noting thatwe havefor every . According to Theorem 8, problem (69) has at least two nontrivial solutions.

Data Availability

Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.

Conflicts of Interest

The authors declare that they have no conflicts ofinterest.

Authors’ Contributions

Both the authors contributed equally to the writing of this paper, and they read and approved the final manuscript.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 11971126) and the Program for Changjiang Scholars and Innovative Research Team in University (Grant no. IRT_16R16).

References

Z. H. Chen and Y. T. Shen, “Infinitely many solutions of Dirichlet problem for p-mean curvature operator,” Applied Mathematics-A Journal of Chinese Universities Series B, vol. 18, no. 2, pp. 161–172, 2003.
View at: Publisher Site | Google Scholar
P. De Nápoli and M. C. Mariani, “Mountain pass solutions to equations of p-Laplacian type,” Nonlinear Analysis: Theory, Methods and Applications, vol. 54, no. 7, pp. 1205–1219, 2003.
View at: Publisher Site | Google Scholar
C. Bereanu and J. Mawhin, “Boundary value problems for second-order nonlinear difference equations with discrete ϕ-Laplacian and singular ϕ,” Journal of Difference Equations and Applications, vol. 14, no. 10-11, pp. 1099–1118, 2008.
View at: Publisher Site | Google Scholar
D. Bonheure, P. Habets, F. Obersnel, and P. Omari, “Classical and non-classical solutions of a prescribed curvature equation,” Journal of Differential Equations, vol. 243, no. 2, pp. 208–237, 2007.
View at: Publisher Site | Google Scholar
A. Cabada and V. Otero-Espinar, “Existence and comparison results for difference ϕ-Laplacian boundary value problems with lower and upper solutions in reverse order,” Journal of Mathematical Analysis and Applications, vol. 267, no. 2, pp. 501–521, 2002.
View at: Publisher Site | Google Scholar
J. Mawhin, “Periodic solutions of second order nonlinear difference systems with ϕ-Laplacian: a variational approach,” Nonlinear Analysis, vol. 75, pp. 4672–4687, 2012.
View at: Publisher Site | Google Scholar
R. P. Agarwal, Difference Equations and Inequalities, Theory, Methods, and Applications, Marcel Dekker Incorporated, New York, NY, USA, 2000.
R. P. Agarwal and P. J. Y. Wong, Advanced Topics in Difference Equations, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997.
S. Elaydi, An Introduction to Difference Equations, Springer-Verlag, New York, NY, USA, Third edition, 2011.
W. G. Kelly and A. C. Peterson, Difference Equations: An Introduction with Applications, Academic Press, San Diego, CA, USA, 1991.
Y. H. Long and L. Wang, “Global dynamics of a delayed two-patch discrete SIR disease model,” Communications in Nonlinear Science and Numerical Simulation, vol. 83, Article ID 105117, 2020.
View at: Publisher Site | Google Scholar
J. S. Yu and B. Zheng, “Modeling Wolbachia infection in mosquito population via discrete dynamical model,” Journal of Difference Equations and Applications, vol. 25, no. 11, pp. 1549–1567, 2019.
View at: Publisher Site | Google Scholar
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
P. Candito and G. D’Aguì, “Three solutions to a perturbed nonlinear discrete dirichlet problem,” Journal of Mathematical Analysis and Applications, vol. 375, no. 2, pp. 594–601, 2011.
View at: Publisher Site | Google Scholar
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, pp. 383–397, 2017.
View at: Google Scholar
L. Erbe, B. G. Jia, and Q. Q. Zhang, “Homoclinic solutions of discrete nonlinear systems via variational method,” Journal of Applied Analysis & Computation, vol. 9, no. 1, pp. 271–294, 2019.
View at: Publisher Site | Google Scholar
L. Q. Jiang and Z. Zhou, “Three solutions to Dirichlet boundary value problems for advances in difference equations,” Advances in Difference Equations, vol. 2008, no. 1, Article ID 345916, 2008.
View at: Publisher Site | Google Scholar
J. H. Kuang and Z. M. Guo, “Heteroclinic solutions for a class of p-Laplacian difference equations with a parameter,” Applied Mathematics Letters, vol. 100, Article ID 106034, 2020.
View at: Publisher Site | Google Scholar
G. H. Lin and Z. Zhou, “Homoclinic solutions of discrete ϕ-Laplacian equations with mixed nonlinearities,” Communications on Pure and Applied Analysis, vol. 17, pp. 1723–1747, 2018.
View at: Publisher Site | Google Scholar
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
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
Y. H. Long and S. H. Wang, “Multiple solutions for nonlinear functional difference equations by the invariants sets of descending flow,” Journal of Difference Equations and Applications, vol. 25, no. 12, pp. 1768–1789, 2019.
View at: Publisher Site | Google Scholar
H. P. Shi, “Periodic and subharmonic solutions for second-order nonlinear difference equations,” Journal of Applied Mathematics and Computing, vol. 48, pp. 157–171, 2015.
View at: Publisher Site | Google Scholar
X. H. Tang, “Non-Nehari manifold method for periodic discrete super linear Schrodinger equation,” Acta Mathematica Sinica, English Series, vol. 32, no. 4, pp. 463–473, 2016.
View at: Publisher Site | Google Scholar
Z. G. Wang and Z. Zhou, “Boundary value problem for a second-order difference equation with resonance,” Complexity, vol. 2020, Article ID 7527030, 10 pages, 2020.
View at: Publisher Site | Google Scholar
Q. Q. Zhang, “Homoclinic orbits for a class of discrete periodic Hamiltonian systems,” Proceedings of the American Mathematical Society, vol. 143, pp. 3155–3163, 2015.
View at: Publisher Site | Google Scholar
Q. Q. Zhang, “Homoclinic orbits for discrete Hamiltonian systems with indefinite linear part,” Communications on Pure and Applied Analysis, vol. 14, no. 5, pp. 1929–1940, 2017.
View at: Publisher Site | Google Scholar
Q. Q. Zhang, “Homoclinic orbits for discrete Hamiltonian systems with local super-quadratic conditions,” Communications on Pure and Applied Analysis, vol. 18, pp. 425–434, 2019.
View at: Publisher Site | Google Scholar
Z. Zhou and D. F. Ma, “Multiplicity results of breathers for the discrete nonlinear Schrodinger equations with unbounded potentials,” Science China Mathematics, vol. 58, no. 4, pp. 781–790, 2015.
View at: Publisher Site | Google Scholar
Z. Zhou and M. T. Su, “Boundary value problems for Laplacian difference equations containing both advance and retardation,” Applied Mathematics Letters, vol. 41, pp. 7–11, 2015.
View at: Google Scholar
Z. Zhou and J. S. Yu, ““Homoclinic solutions in periodic nonlinear difference equations with superlinear nonlinearity,” Acta Mathematica Sinica, English Series, vol. 29, no. 9, pp. 1809–1822, 2013.
View at: Publisher Site | Google Scholar
Z. Zhou, J. S. Yu, and Y. M. Chen, “Homoclinic solutions in periodic difference equations with saturable nonlinearity,” Science China Mathematics, vol. 54, no. 1, pp. 83–93, 2011.
View at: Publisher Site | Google Scholar
P. Candito and G. M. Bisci, “Existence of two solutions for a second-order discrete boundary value problem,” Advanced Nonlinear Studides, vol. 11, pp. 443–453, 2011.
View at: Publisher Site | Google Scholar
G. Bonanno, “A critical point theorem via the Ekeland variational principle,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 5, pp. 2992–3007, 2012.
View at: Publisher Site | Google Scholar
G. Bonanno and P. Candito, “Infinitely many solutions for a class of discrete non-linear boundary value problems,” Applicable Analysis, vol. 88, no. 4, pp. 605–616, 2009.
View at: Publisher Site | Google Scholar
G. Bonanno, P. Candito, and G. D’Aguì, “Variational methods on finite dimensional Banach spaces and discrete problems,” Advanced Nonlinear Studies, vol. 14, pp. 915–939, 2014.
View at: Publisher Site | Google Scholar
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
P. Candito, L. Li, and R. Livrea, “Infinitely many solutions for a perturbed nonlinear Navier boundary value problem involving the -biharmonic,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 17, pp. 6360–6369, 2012.
View at: Publisher Site | Google Scholar
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
M. K. Moghadsm, S. Heidarkhani, and J. Henderson, “Infinitely many solutions for perturbed difference equations,” Journal of Difference Equations and Applications, vol. 20, no. 7, pp. 1055–1068, 2014.
View at: Publisher Site | Google Scholar
S. A. Marano and D. Motreanu, “Infinitely many critical points of non-differentiable functions and applications to a Neumann type problem involving the p-Laplacian,” Journal of Differential Equations, vol. 182, pp. 108–120, 2002.
View at: Google Scholar
J. X. Wang and Z. Zhou, “Large constant-sign solutions of discrete Dirichlet boundary value problems with p-mean curvature operator,” Mathematics, vol. 8, no. 3, p. 12, 2020.
View at: Publisher Site | Google Scholar
Z. Zhou and J. X. Ling, “Infinitely many positive solutions for a discrete two point nonlinear boundary value problem with ϕ_c-Laplacian,” Applied Mathematics Letters, vol. 91, pp. 28–34, 2019.
View at: Publisher Site | Google Scholar
G. H. Hardy, J. E. Littewood, and G. Pólya, Inequalities, Cambridge University Press, Cambridge, UK, Second edition, 1988.
P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, vol. 65, American Mathematical Society, Providence, RI, USA, 1986.

Copyright

Copyright © 2020 Jianxia 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.

PDF Download Citation

Download other formats

Order printed copies

Views

496

Downloads

535

Citations