Fixed point theory approach to boundary value problems for second-order difference equations on non-uniform lattices
© Area et al.; licensee Springer. 2014
Received: 6 September 2013
Accepted: 10 December 2013
Published: 9 January 2014
In this paper, by means of the appropriate Green’s function, an integral representation for the solutions of certain boundary value problems for second-order difference equations on (quadratic and q-quadratic) non-uniform lattices is presented. As a consequence, using fixed point theory, new results for the existence and uniqueness of the solution are proved on non-uniform lattices.
MSC:34B27, 39A20, 34A05, 34B05.
In this paper we solve certain boundary value problems for second-order difference equations on the most general (quadratic and q-quadratic) non-uniform lattices . These problems appear in the discretization of differential operators when the convenient grid or mesh is not uniform. For our purposes, we shall use the Green’s function approach [2, 3].
As indicated by Roach , ‘Boundary value problems are an almost inevitable consequence of using mathematics to study problems in the real world’. In the monograph cited, the author examines in detail the one particular method which requires the construction of an auxiliary function known as a Green’s function. Some historical developments of Green’s functions, definitions, and applications to circuit theory, statics, wave equation, heat equation, quantum physics, finite elements, infinite products, Helmholtz equation or lattice Schrödinger operators can be found in [2–11].
A well-posed problem for an ordinary differential equation, partial differential equation or difference equation should have a unique solution that continuously depends on the sources. For linear equations, the operator that transforms the data into the solution is usually a linear integral operator . The kernel is the Green’s function. A list of formulas of such Green’s functions for different situations and problems can be found in . According to  the most appropriate way to solve a boundary value problem (BVP) is by calculating its Green’s function and by means of the integral expression, it is also possible obtain some additional qualitative information about the solutions of the considered problem, such as their sign, oscillation properties, a priori bounds or their stability.
Since the Green’s function does not depend on the data, it is clear that (2) expresses in a very simple manner the dependence of the solution on the data f, α, β.
the solution (2) provides a representation of the inverse operator as an integral operator with kernel the Green’s function .
One of the main advantages of the Green’s function is the fact that it is independent on the source and recently  an algorithm that calculates explicitly the Green’s function related to a linear ordinary differential equation, with constant coefficients, coupled with two-point linear boundary conditions has been developed.
For these reasons the theory of Green’s functions is a fundamental tool in the analysis of differential equations. It has been widely studied in the literature [2, 3, 6, 12] and it has a great importance for the use of monotone iterative techniques [14, 15], lower and upper solutions , fixed point theorems [17, 18], fractional calculus approach [19–23] or variational methods .
Although several results in the discrete case are similar to those already known in the continuous case, the adaptation from the continuous case to the discrete case is not direct but requires some special devices . Despite the fact that BVP of differential equations has been studied by many authors using various methods and techniques, there are scarce techniques for studying the BVP of difference equations. In the last decades, the fixed point theorems have been improved and generalized in different directions for solving boundary value problems (see e.g. [26–31]). These results were usually obtained by analytic techniques and various fixed point theorems. For example, the upper and lower solution method, the conical shell fixed point theorems, the Brouwer and Schauder fixed point theorems or topological degree theory (see  and references therein).
As indicated at the beginning of this section, we use the Green’s function to study some boundary value problems for second-order difference equations on non-uniform lattices. As in the classical case, separation of the variables and representation of solutions in terms of the basic Fourier series might also give explicit solutions of these boundary value problems . Finally, fixed point theorems are used to prove the existence and uniqueness of solutions for the boundary value problems analyzed in this paper.
2 Basic definitions and notations
with , and , , are constants.
Linear lattices if we choose in (6) and .
Quadratic lattices if we choose in (6) .
q-linear lattices if we choose in (5) and .
q-quadratic lattices if we choose in (5) .
Some properties of these classical lattices are presented in . Applications of the theory of basic Fourier series to q-analogues of several equations of mathematical physics such as the q-heat equation, q-wave equation and q-Laplace equation were studied in . Besides, linear second-order partial q-difference equations of the hypergeometric type in two variables on q-linear lattices have been recently analyzed in .
The Green’s function approach for difference equations on linear and q-linear lattices being well known (see e.g. [42–44]), we shall analyze certain boundary value problems for second-order difference equations on quadratic and q-quadratic lattices.
Remark 1 The q-quadratic lattice, in its general non-symmetrical form, is the most general case and the other lattices can be found from this by limiting processes. For and , we obtain the Askey-Wilson lattice . Moreover, if we choose , , and , we obtain the Racah lattice . Askey-Wilson and Racah polynomials  both satisfy a second-order difference equation on non-uniform lattices [1, 38, 39].
- 1.In the q-quadratic lattice , with , and , the basis defined for as(7)satisfies, for ,(8)where the constants are explicitly given by(9)the q-Pochhammer symbol is given byand the q-number is defined by(10)
- 2.In the quadratic lattice , with , the basis is defined for as(11)
and denotes the Pochhammer symbol.
As a consequence is an exact lowering operator in these bases where , or , are constants with respect to s and they depend on the lattice type.
This definition reduces to the usual definition of the difference integral and the Thomae  and Jackson q-integrals [46, 51, 52] in the canonical forms of the linear and q-linear lattices, respectively.
- 1.An analogue of the fundamental theorem of calculus(16)
- 2.An analogue of integration by parts formula for two functions and(17)(18)
- 1.In the case of a q-quadratic lattice , with , and ,(19)
where and have been defined in (7) and (9).
- 2.In the case of a quadratic lattice , with , the following relation holds:(20)
where and have been defined in (11) and (13).
3 Boundary value problem on non-uniform lattices
and denotes the Green’s function which shall be computed explicitly (see (24) and (25) below).
- 1.In the q-quadratic lattice , with , and , we haveand with the boundary conditions (22) we obtain(24)
- 2.In the quadratic lattice , ,and therefore, with the boundary conditions (22) it yields(25)
As a consequence, (24) and (25) provide an explicit representation of the Green’s function for the BVP (21)-(22) for each type of non-uniform lattice considered in this paper (quadratic and q-quadratic).
where the constants are given for (q-quadratic lattice) in (9) and for (quadratic lattice) in (13).
In a similar way as in  we can prove the following result for the uniqueness of the solution of the boundary value problem, assuming that the function f satisfies a Lipchitz bound in the second variable.
where L is a Lipschitz constant. Then, the boundary value problem (21)-(22) has a unique solution provided , where is given by (26).
As , the conclusion follows from Banach’s contraction mapping principle . □
we have , which ensures the unicity of the solution of the BVP for .
Moreover, if f is a bounded function we have the following.
Theorem 2 If f is a bounded function, then the operator has a fixed point and, in consequence, the difference equation on a non-uniform lattice (21) with the boundary conditions (22) has a solution.
Proof The operator is compact since it is an integral operator  and the range of is contained in a closed ball since f is bounded.
The operator has a fixed point in view the classical Schaefer’s fixed point theorem [, Theorem 4.3.2]. □
We would like to mention that our goal here is not to exploit all possible boundary value problems covered by this approach, but to emphasize its systematic character and its simplicity, due to very recent results on bases associated with the difference operators considered in this article.
The authors would like to thank the referees for the many valuable comments and references. The work of I. Area and E. Godoy has been partially supported by the Ministerio de Economía y Competitividad of Spain under grant MTM2012-38794-C02-01, co-financed by the European Community fund FEDER. J.J. Nieto also acknowledges partial financial support by the Ministerio de Economía y Competitividad of Spain under grant MTM2010-15314, co-financed by the European Community fund FEDER.
- Nikiforov AF, Suslov SK, Uvarov VB: Classical Orthogonal Polynomials of a Discrete Variable. Springer, Berlin; 1991.MATHView ArticleGoogle Scholar
- Roach GF: Green’s Functions. 2nd edition. Cambridge University Press, Cambridge; 1982.MATHGoogle Scholar
- Stakgold I, Holst M: Green’s Functions and Boundary Value Problems. 3rd edition. Wiley, Hoboken; 2011.MATHView ArticleGoogle Scholar
- Beck JV, Cole KD, Haji-Sheikh A, Litkouhi B: Heat Conduction Using Green’s Functions. Hemisphere, London; 1992.Google Scholar
- Cole KD, Beck JV, Haji-Sheikh A, Litkouhi B: Heat Conduction Using Green’s Functions. 2nd edition. CRC Press, Boca Raton; 2011.MATHGoogle Scholar
- Duffy DG: Green’s Functions with Applications. Chapman & Hall/CRC, Boca Raton; 2001.MATHView ArticleGoogle Scholar
- Economou EN: Green’s Functions in Quantum Physics. 3rd edition. Springer, Berlin; 2006.View ArticleGoogle Scholar
- Hartmann F: Green’s Functions and Finite Elements. Heidelberg, Springer; 2013.MATHView ArticleGoogle Scholar
- Cabada A: Green’s Functions in the Theory of Ordinary Differential Equations. Springer, Berlin; 2014.MATHView ArticleGoogle Scholar
- Melnikov YA: Green’s Functions and Infinite Products. Birkhäuser/Springer, New York; 2011.MATHView ArticleGoogle Scholar
- Bourgain J Annals of Mathematics Studies 158. In Green’s Function Estimates for Lattice Schrödinger Operators and Applications. Princeton University Press, Princeton; 2005.View ArticleGoogle Scholar
- Kythe PK Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series. In Green’s Functions and Linear Differential Equations. CRC Press, Boca Raton; 2011.Google Scholar
- Cabada A, Cid JÁ, Máquez-Villamarín B: Computation of Green’s functions for boundary value problems with Mathematica . Appl. Math. Comput. 2012, 219(4):1919–1936. 10.1016/j.amc.2012.08.035MATHMathSciNetView ArticleGoogle Scholar
- Ladde GS, Lakshmikantham V, Vatsala AS: Monotone Iterative Techniques for Nonlinear Differential Equations. Pitman, Boston; 1985.MATHGoogle Scholar
- Nieto JJ: An abstract monotone iterative technique. Nonlinear Anal. 1997, 28(12):1923–1933. 10.1016/S0362-546X(97)89710-6MATHMathSciNetView ArticleGoogle Scholar
- De Coster C, Habets P: Two-Point Boundary Value Problems: Lower and Upper Solutions. Elsevier, Amsterdam; 2006.Google Scholar
- Webb JRL: Solutions of nonlinear equations in cones and positive linear operators. J. Lond. Math. Soc. (2) 2010, 82(2):420–436. 10.1112/jlms/jdq037MATHMathSciNetView ArticleGoogle Scholar
- Nieto JJ: Generalized quasilinearization method for a second order ordinary differential equation with Dirichlet boundary conditions. Proc. Am. Math. Soc. 1997, 125(9):2599–2604. 10.1090/S0002-9939-97-03976-2MATHView ArticleGoogle Scholar
- Baleanu D, Agarwal RP, Mohammadi H, Rezapour S: Some existence results for a nonlinear fractional differential equation on partially ordered Banach spaces. Bound. Value Probl. 2013., 2013: Article ID 112Google Scholar
- Abdeljawad T, Baleanu D, Jarad F, Agarwal RP: Fractional sums and differences with binomial coefficients. Discrete Dyn. Nat. Soc. 2013., 2013: Article ID 104173Google Scholar
- Nyamoradi N, Baleanu D, Agarwal RP: On a multipoint boundary value problem for a fractional order differential inclusion on an infinite interval. Adv. Math. Phys. 2013., 2013: Article ID 823961Google Scholar
- Babakhani A, Baleanu D, Agarwal RP: The existence and uniqueness of solutions for a class of nonlinear fractional differential equations with infinite delay. Abstr. Appl. Anal. 2013., 2013: Article ID 592964Google Scholar
- Debbouche A, Baleanu D, Agarwal RP: Nonlocal nonlinear integrodifferential equations of fractional orders. Bound. Value Probl. 2012., 2012: Article ID 78Google Scholar
- Struwe M: Variational Methods. 4th edition. Springer, Berlin; 2008.MATHGoogle Scholar
- Agarwal RP, Lalli BS: Discrete polynomial interpolation, Green’s functions, maximum principles, error bounds and boundary value problems. Comput. Math. Appl. 1993, 25(8):3–39. 10.1016/0898-1221(93)90169-VMATHMathSciNetView ArticleGoogle Scholar
- Zhang P: Existence of positive solutions for nonlocal second-order boundary value problem with variable parameter in Banach spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 43Google Scholar
- Du X, Zhao Z: On fixed point theorems of mixed monotone operators. Fixed Point Theory Appl. 2011., 2011: Article ID 563136Google Scholar
- Wang G, Zhang L, Song G: Boundary value problem of a nonlinear Langevin equation with two different fractional orders and impulses. Fixed Point Theory Appl. 2012., 2012: Article ID 200Google Scholar
- Wu J, Liu Y: Fixed point theorems for monotone operators and applications to nonlinear elliptic problems. Fixed Point Theory Appl. 2013., 2013: Article ID 134Google Scholar
- Ahmad B, Nieto JJ: Solvability of nonlinear Langevin equation involving two fractional orders with Dirichlet boundary conditions. Int. J. Differ. Equ. 2010., 2010: Article ID 649486Google Scholar
- Wardowski D: Fixed points of a new type of contractive mappings in complete metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 94Google Scholar
- Gao L: Existence of multiple solutions for a second-order difference equation with a parameter. Appl. Math. Comput. 2010, 216(5):1592–1598. 10.1016/j.amc.2010.03.012MATHMathSciNetView ArticleGoogle Scholar
- Ey K, Ruffing A, Suslov S: Method of separation of the variables for basic analogs of equations of mathematical physics. Ramanujan J. 2007, 13(1–3):407–447. 10.1007/s11139-006-0260-2MATHMathSciNetView ArticleGoogle Scholar
- Bangerezako G: Variational calculus on q -nonuniform lattices. J. Math. Anal. Appl. 2005, 306: 161–179. 10.1016/j.jmaa.2004.12.029MATHMathSciNetView ArticleGoogle Scholar
- Magnus AP: Associated Askey-Wilson polynomials as Laguerre-Hahn orthogonal polynomials. Lecture Notes in Math. 1329. In Orthogonal Polynomials and Their Applications. Springer, Berlin; 1988:261–278.View ArticleGoogle Scholar
- Magnus AP: Special nonuniform lattice (snul) orthogonal polynomials on discrete dense sets of points. 65. Proceedings of the International Conference on Orthogonality, Moment Problems and Continued Fractions 1995, 253–265.Google Scholar
- 37. Witte, NS: Semi-classical Orthogonal Polynomial Systems on Non-uniform Lattices, Deformations of the Askey Table and Analogs of Isomonodromy. arXiv:1204.2328 (2011)Google Scholar
- Atakishiyev NM, Rahman M, Suslov SK: On classical orthogonal polynomials. Constr. Approx. 1995, 11(2):181–226. 10.1007/BF01203415MATHMathSciNetView ArticleGoogle Scholar
- Suslov SK: On the theory of difference analogues of special functions of hypergeometric type. Usp. Mat. Nauk 1989, 44(2(266)):185–226.MATHMathSciNetGoogle Scholar
- Suslov SK: An Introduction to Basic Fourier Series. Kluwer Academic, Dordrecht; 2003.MATHView ArticleGoogle Scholar
- Area I, Atakishiyev N, Godoy E, Rodal J: Linear partial q -difference equations on q -linear lattices and their bivariate q -orthogonal polynomial solutions. Appl. Math. Comput. 2013, 223: 520–536.MathSciNetView ArticleGoogle Scholar
- Hartman P: Difference equations: disconjugacy, principal solutions, Green’s functions, complete monotonicity. Trans. Am. Math. Soc. 1978, 246: 1–30.MATHGoogle Scholar
- Ahmad B, Alsaedi A, Ntouyas SK: A study of second-order q -difference equations with boundary conditions. Adv. Differ. Equ. 2012., 2012: Article ID 35Google Scholar
- Ahmad B, Nieto JJ: Basic theory of nonlinear third-order q -difference equations and inclusions. Math. Model. Anal. 2013, 18: 122–135. 10.3846/13926292.2013.760012MATHMathSciNetView ArticleGoogle Scholar
- Askey R, Wilson J: Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Am. Math. Soc. 1985., 54: Article ID 319Google Scholar
- Koekoek R, Lesky PA, Swarttouw RF: Hypergeometric Orthogonal Polynomials and Their q-Analogues. Springer, Berlin; 2010.MATHView ArticleGoogle Scholar
- Foupouagnigni M, Koepf K, Kenfack-Nangho K, Mboutngam S: On solutions of holonomic divided-difference equations on nonuniform lattices. Axioms 2013, 2(3):404–434. 10.3390/axioms2030404MATHView ArticleGoogle Scholar
- Suslov SK: Letter to the editors: ‘On the theory of difference analogues of special functions of hypergeometric type’. Usp. Mat. Nauk 1990, 45(3(273)):219. (Usp. Mat. Nauk 44(2(266)), 185–226 (1989))MathSciNetGoogle Scholar
- Thomae J: Beitrage zur Theorie der durch die Heinesche Reihe. J. Reine Angew. Math. 1869, 70: 258–281.MATHMathSciNetView ArticleGoogle Scholar
- Jackson F: On q -definite integrals. Q. J. Pure Appl. Math. 1910, 41: 193–203.MATHGoogle Scholar
- Gasper G, Rahman M: Basic Hypergeometric Series. 2nd edition. Cambridge University Press, Cambridge; 2004.MATHView ArticleGoogle Scholar
- Ismail MEH: Classical and Quantum Orthogonal Polynomials in One Variable. Cambridge University Press, Cambridge; 2005.MATHView ArticleGoogle Scholar
- Bustoz J, Suslov SK: Basic analog of Fourier series on a q -quadratic grid. Methods Appl. Anal. 1998, 5: 1–38.MATHMathSciNetGoogle Scholar
- Banach S: Sur les opérations dans les ensembles abstraits et leur applications aux équations intégrales. Fundam. Math. 1922, 3: 133–181.MATHGoogle Scholar
- Smart DR: Fixed Point Theorems. Cambridge University Press, London; 1974.MATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.