Solvability of boundary value problems for a class of partial difference equations on the combinatorial domain
- Stevo Stević^{1, 2}Email author
Received: 17 June 2016
Accepted: 5 October 2016
Published: 18 October 2016
Abstract
Keywords
MSC
1 Introduction
Let \(\mathbb{N}\) denote the set of natural numbers and \(\mathbb{N} _{0}=\mathbb{N}\cup \{0\}\). Let \(k,l\in \mathbb{N}_{0}\) be such that \(k< l\), then the notation \(j=\overline{k,l}\) means \(k\le j\le l\). In the rest of this section we give some motivation for the study, as well as notions that will be used in the rest of the paper.
Motivated by [29] and numerous recent applications of equation (8) (for example, those in [9, 10, 13–17, 19, 20]), here we show that there is a closed form formula for solutions to an extension of equation (9) on domain \(A\setminus \{(0,0)\}\) in terms of given boundary values \(d_{n,0}\) and \(d_{n,n}\), \(n\in \mathbb{N}\).
2 Main results
We show how general solution to (9) on set \(A\setminus \{(0,0) \}\) can be found by using a method in [29], which we call the method of half-lines. Namely, the domain is divided into some half-lines, and (9) is regarded on each line as an equation of type (8). It is solved, and based on the obtained formulas one gets the general solution. However, the method cannot be applied directly, so it needs some modifications.
Now we are in a position to state and prove our first result based on the above consideration.
Theorem 1
Proof
By choosing \(l=n-k\) in (17), using the conditions in (18), and some calculations, formula (19) is obtained. □
Remark 1
Corollary 1
Remark 2
Note that under the conditions of Corollary 1 equation (22) gives a closed form formula for such solutions due to equation (6).
Theorem 2
Now we prove the following lemma.
Lemma 1
Proof
Hence, the following result holds.
Theorem 3
Declarations
Acknowledgements
The author would like to express his sincere thanks to the referee of the paper for her/his comments and for suggesting some references on the solvability of partial difference equations.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Jordan, C: Calculus of Finite Differences. Chelsea Publishing Company, New York (1956) MATHGoogle Scholar
- Mitrinović, DS, Kečkić, JD: Methods for Calculating Finite Sums. Naučna Knjiga, Beograd (1984) (in Serbian) Google Scholar
- Yablonskiy, SV: Introduction to Discrete Mathematics. Mir Publishers, Moscow (1989) Google Scholar
- Riordan, J: Combinatorial Identities. Wiley, New York (1968) MATHGoogle Scholar
- Brand, L: A sequence defined by a difference equation. Am. Math. Mon. 62(7), 489-492 (1955) MathSciNetView ArticleGoogle Scholar
- Brand, L: Differential and Difference Equations. Wiley, New York (1966) MATHGoogle Scholar
- Krechmar, VA: A Problem Book in Algebra. Mir Publishers, Moscow (1974) Google Scholar
- Levy, H, Lessman, F: Finite Difference Equations. Dover, New York (1992) MATHGoogle Scholar
- Stević, S: More on a rational recurrence relation. Appl. Math. E-Notes 4, 80-85 (2004) MathSciNetMATHGoogle Scholar
- Stević, S, Diblik, J, Iričanin, B, Šmarda, Z: On the difference equation \(x_{n}=a_{n}x_{n-k}/(b_{n}+c_{n}x_{n-1}\cdots x_{n-k})\). Abstr. Appl. Anal. 2012, Article ID 409237 (2012) MATHGoogle Scholar
- Stević, S: On a solvable system of difference equations of kth order. Appl. Math. Comput. 219, 7765-7771 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Stević, S, Diblik, J, Iričanin, B, Šmarda, Z: On a third-order system of difference equations with variable coefficients. Abstr. Appl. Anal. 2012, Article ID 508523 (2012) MathSciNetMATHGoogle Scholar
- Aloqeili, M: Dynamics of a kth-order rational difference equation. Appl. Math. Comput. 181, 1328-1335 (2006) MathSciNetView ArticleMATHGoogle Scholar
- Berg, L, Stević, S: On some systems of difference equations. Appl. Math. Comput. 218, 1713-1718 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Papaschinopoulos, G, Stefanidou, G: Asymptotic behavior of the solutions of a class of rational difference equations. Int. J. Difference Equ. 5(2), 233-249 (2010) MathSciNetGoogle Scholar
- Stević, S: On the system of difference equations \(x_{n}=c_{n}y _{n-3}/(a_{n}+b_{n}y_{n-1}x_{n-2}y_{n-3})\), \(y_{n}=\gamma_{n} x_{n-3}/( \alpha_{n}+\beta_{n} x_{n-1}y_{n-2}x_{n-3})\). Appl. Math. Comput. 219, 4755-4764 (2013) MathSciNetView ArticleGoogle Scholar
- Stević, S: Solvable subclasses of a class of nonlinear second-order difference equations. Adv. Nonlinear Anal. 5(2), 147-165 (2016) MathSciNetMATHGoogle Scholar
- Stević, S, Diblik, J, Iričanin, B, Šmarda, Z: On some solvable difference equations and systems of difference equations. Abstr. Appl. Anal. 2012, Article ID 541761 (2012) MathSciNetMATHGoogle Scholar
- Stević, S, Diblik, J, Iričanin, B, Šmarda, Z: On a solvable system of rational difference equations. J. Differ. Equ. Appl. 20(5-6), 811-825 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Stević, S, Diblik, J, Iričanin, B, Šmarda, Z: Solvability of nonlinear difference equations of fourth order. Electron. J. Differ. Equ. 2014, Article ID 264 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Stević, S: First-order product-type systems of difference equations solvable in closed form. Electron. J. Differ. Equ. 2015, Article ID 308 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Stević, S, Iričanin, B, Šmarda, Z: On a product-type system of difference equations of second order solvable in closed form. J. Inequal. Appl. 2015, Article ID 327 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Cheng, SS: Partial Difference Equations. Taylor & Francis, London (2003) View ArticleMATHGoogle Scholar
- Cheng, SS, Hsieh, LY, Chao, ZT: Discrete Lyapunov inequality conditions for partial difference equations. Hokkaido Math. J. 19, 229-239 (1990) MathSciNetView ArticleMATHGoogle Scholar
- Cheng, SS, Lin, JY: Green’s function and stability of a linear partial difference scheme. Comput. Math. Appl. 35(5), 27-41 (1998) MathSciNetView ArticleMATHGoogle Scholar
- Cheng, SS, Lu, YF: General solutions of a three-level partial difference equation. Comput. Math. Appl. 38(7-8), 65-79 (1999) MathSciNetView ArticleMATHGoogle Scholar
- Lin, YZ, Cheng, SS: Stability criteria for two partial difference equations. Comput. Math. Appl. 32(7), 87-103 (1996) MathSciNetView ArticleMATHGoogle Scholar
- Slavik, A, Stehlik, P: Explicit solutions to dynamic diffusion-type equations and their time integrals. Appl. Math. Comput. 234, 486-505 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Stević, S: Note on the binomial partial difference equation. Electron. J. Qual. Theory Differ. Equ. 2015, Article ID 96 (2015) MathSciNetView ArticleMATHGoogle Scholar