# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Existence of a positive solution to a system of discrete fractional boundary value problems. (English) Zbl 1215.39003

This paper is concerned with a pair of discrete fractional difference equations

$\begin{array}{cc}& -{{\Delta }}^{{\nu }_{1}}{y}_{1}\left(t\right)={\lambda }_{1}{a}_{1}\left(t+{\nu }_{1}-1\right){f}_{1}\left({y}_{1}\left(t+{\nu }_{1}-1\right),{y}_{2}\left(t+{\nu }_{2}-1\right)\right),\hfill \\ & -{{\Delta }}^{{\nu }_{2}}{y}_{2}\left(t\right)={\lambda }_{2}{a}_{2}\left(t+{\nu }_{2}-1\right){f}_{2}\left({y}_{1}\left(t+{\nu }_{1}-1\right),{y}_{2}\left(t+{\nu }_{2}-1\right)\right),\hfill \end{array}$

for $t\in {\left[0,b\right]}_{{ℕ}_{0}},$ subject to boundary conditions

${y}_{1}\left({\nu }_{1}-2\right)={\psi }_{1}\left({y}_{1}\right),\phantom{\rule{4pt}{0ex}}{y}_{2}\left({\nu }_{2}-2\right)={\psi }_{2}\left({y}_{2}\right),\phantom{\rule{4pt}{0ex}}{y}_{1}\left({\nu }_{1}+b\right)={\varphi }_{1}\left({y}_{1}\right),\phantom{\rule{4pt}{0ex}}{y}_{2}\left({\nu }_{2}+b\right)={\varphi }_{2}\left({y}_{2}\right),$

where ${\lambda }_{1},{\lambda }_{2}>0;$ ${a}_{1},{a}_{2}:ℝ\to \left[0,+\infty \right)$ and ${\nu }_{1},{\nu }_{2}\in \left(1,2\right];$ and ${\psi }_{1},{\psi }_{2},{\varphi }_{1},{\varphi }_{2}:{ℝ}^{b+3}\to ℝ$ are functionals. A standard transformation of the problem is achieved by means of the Green’s function method, and then using the Krasnoselski fixed point theorem for mappings on Banach spaces with cones, two existence theorems are derived for positive solutions of the above boundary value problem based on further restrictions on ${\lambda }_{1},{\lambda }_{2},{a}_{1},{a}_{2},{f}_{1},{f}_{2}$ and the functionals. Since the system is more general than some existing ones, the existence results are extensions of some, if not all, of the existing results for second order difference boundary value problems.

##### MSC:
 39A12 Discrete version of topics in analysis 39A22 Growth, boundedness, comparison of solutions (difference equations) 39A70 Difference operators 47B39 Difference operators (operator theory)
##### References:
 [1] Infante, G.; Webb, J. R. L.: Positive solutions of nonlocal boundary value problems: a unified approach, J. London math. Soc. 74, No. 2, 673-693 (2006) · Zbl 1115.34028 · doi:10.1112/S0024610706023179 [2] Henderson, J.; Ntouyas, S. K.; Purnaras, I. K.: Positive solutions for systems of nonlinear discrete boundary value problems, J. differ. Equat. appl. 15, 895-912 (2009) · Zbl 1185.39003 · doi:10.1080/10236190802350649 [3] Su, X.: Boundary value problem for a coupled system of nonlinear fractional differential equations, Appl. math. Lett. 22, 64-69 (2009) · Zbl 1163.34321 · doi:10.1016/j.aml.2008.03.001 [4] Anderson, D. R.: Solutions to second-order three-point problems on time scales, J. differ. Equat. appl. 8, 673-688 (2002) · Zbl 1021.34011 · doi:10.1080/10236919021000000726 [5] Kaufmann, E. R.; Raffoul, Y. N.: Positive solutions for a nonlinear functional dynamic equation on a time scale, Nonlinear anal. TMA 62, 1267-1276 (2005) · Zbl 1090.34054 · doi:10.1016/j.na.2005.04.031 [6] Cheung, W.; Ren, J.; Wong, P. J. Y.; Zhao, D.: Multiple positive solutions for discrete nonlocal boundary value problems, J. math. Anal. appl. 330, 900-915 (2007) · Zbl 1120.39016 · doi:10.1016/j.jmaa.2006.08.034 [7] Agarwal, R.; O’regan, D.: A coupled system of difference equations, Appl. math. Comput. 114, 39-49 (2000) · Zbl 1023.39001 · doi:10.1016/S0096-3003(99)00073-9 [8] Anderson, D. R.: Existence of solutions for first-order multi-point problems with changing-sign nonlinearity, J. differ. Equat. appl. 14, 657-666 (2008) · Zbl 1158.34006 · doi:10.1080/10236190701736682 [9] Avery, R. I.; Henderson, J.: Twin solutions of boundary value problems for ordinary differential equations and finite difference equations, Comput. math. Appl. 42, 695-704 (2001) · Zbl 1006.34022 · doi:10.1016/S0898-1221(01)00188-2 [10] Henderson, J.; Wong, P. J. Y.: Double symmetric solutions for discrete lidstone boundary value problems, J. differ. Equat. appl. 7, 811-828 (2001) · Zbl 1001.39025 · doi:10.1080/10236190108808304 [11] Ma, R.; Raffoul, Y. N.: Positive solutions of three-point nonlinear discrete second order boundary value problem, J. differ. Equat. appl. 10, 129-138 (2004) · Zbl 1056.39024 · doi:10.1080/1023619031000114323 [12] Wang, C.; Han, X.; Li, C.: Positive solutions to nonlinear second-order three-point boundary-value problems for difference equation with change of sign, Electon. J. Differ. eqs., No. 87, 10 (2008) · Zbl 1165.39015 · doi:emis:journals/EJDE/Volumes/2008/87/abstr.html [13] Wang, P.; Wang, Y.: Existence of positive solutions for second-order m-point boundary value problems on time scales, Acta mathmeticae applicatae sinica, English series 22, 457-468 (2006) · Zbl 1108.34014 · doi:10.1007/s10255-006-0322-7 [14] Almeida, R.; Torres, D. F. M.: Calculus of variations with fractional derivatives and fractional integrals, Appl. math. Lett. 22, 1816-1820 (2009) · Zbl 1183.26005 · doi:10.1016/j.aml.2009.07.002 [15] Arara, A.: Fractional order differential equations on an unbounded domain, Nonlinear anal. TMA 72, 580-586 (2010) [16] Babakhani, A.; Daftardar-Gejji, V.: Existence of positive solutions of nonlinear fractional differential equations, J. math. Anal. appl. 278, 434-442 (2003) · Zbl 1027.34003 · doi:10.1016/S0022-247X(02)00716-3 [17] Bai, Z.; Lü, H.: Positive solutions for boundary value problem of nonlinear fractional differential equation, J. math. Anal. appl. 311, 495-505 (2005) · Zbl 1079.34048 · doi:10.1016/j.jmaa.2005.02.052 [18] Devi, J. V.: Generalized monotone method for periodic boundary value problems of Caputo fractional differential equation, Commun. appl. Anal. 12, 399-406 (2008) · Zbl 1184.34014 [19] Diethelm, K.; Ford, N.: Analysis of fractional differential equations, J. math. Anal. appl. 265, 229-248 (2002) · Zbl 1014.34003 · doi:10.1006/jmaa.2001.7194 [20] Goodrich, C. S.: Existence of a positive solution to a class of fractional differential equations, Appl. math. Lett. 23, 1050-1055 (2010) · Zbl 1204.34007 · doi:10.1016/j.aml.2010.04.035 [21] Malinowska, A.; Torres, D. F. M.: Generalized natural boundary conditions for fractional variational problems in terms of the Caputo derivative, Comput. math. Appl. 59, 3110-3116 (2010) · Zbl 1193.49023 · doi:10.1016/j.camwa.2010.02.032 [22] Xu, X.; Jiang, D.; Yuan, C.: Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation, Nonlinear anal. TMA 71, 4676-4688 (2009) · Zbl 1178.34006 · doi:10.1016/j.na.2009.03.030 [23] Atici, F. M.; Eloe, P. W.: Fractional q-calculus on a time scale, J. nonlinear math. Phys. 14, No. 3, 333-344 (2007) [24] Atici, F. M.; Eloe, P. W.: A transform method in discrete fractional calculus, Int. J. Differ. equat. 2, No. 2, 165-176 (2007) [25] Atici, F. M.; Eloe, P. W.: Initial value problems in discrete fractional calculus, Proc. amer. Math. soc. 137, No. 3, 981-989 (2009) · Zbl 1166.39005 · doi:10.1090/S0002-9939-08-09626-3 [26] F.M. Atici, P.W. Eloe, Two-point boundary value problems for finite fractional difference equations, J. Differ. Equat. Appl., in press, doi:10.1080/10236190903029241. [27] F.M. Atici, S. Scedil;engül, Modeling with fractional difference equations, J. Math. Anal. Appl., 2010, doi:10.1016/j.jmaa.2010.02.009. [28] Goodrich, C. S.: Continuity of solutions to discrete fractional initial value problems, Comput. math. Appl. 59, 3489-3499 (2010) · Zbl 1197.39002 · doi:10.1016/j.camwa.2010.03.040 [29] C.S. Goodrich, Solutions to a discrete right-focal fractional boundary value problem, Int. J. Differ. Equat. 5 (2010), in press. [30] C.S. Goodrich, On a discrete fractional three-point boundary value problem, J. Differ. Equat. Appl., doi:10.1080/10236198.2010.503240. [31] C.S. Goodrich, Some new existence results for fractional difference equations, Int. J. Dyn. Syst. Differ. Equat., in press. · Zbl 1215.39004 · doi:10.1504/IJDSDE.2011.038499 [32] C.S. Goodrich, Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions, Comput. Math. Appl., doi:10.1016/j.camwa.2010.10.041. [33] C.S. Goodrich, On positive solutions to nonlocal fractional and integer-order difference equations, submitted for publication. [34] C.S. Goodrich, On the operational properties of the fractional difference applied to a fractional delta-nabla difference equation with a nonlocal condition, submitted for publication. [35] N.R.O. Bastos, et al., Discrete-time fractional variational problems, Signal Process., 2010, doi:10.1016/j.sigpro.2010.05.001. [36] Anastassiou, G. A.: Principles of delta fractional calculus on time scales and inequalities, Math. comput. Model. 52, 556-566 (2010) · Zbl 1201.26001 · doi:10.1016/j.mcm.2010.03.055 [37] Atici, F. M.; Eloe, P. W.: Discrete fractional calculus with the nabla operator, Electron. J. Qualitative theor. Differ. equat. (spec. Ed. I) 3, 1-12 (2009) · Zbl 1189.39004 · doi:emis:journals/EJQTDE/sped1/103.pdf [38] Anastassiou, G. A.: Foundations of nabla fractional calculus on time scales and inequalities, Comput. math. Appl. 59, 3750-3762 (2010) · Zbl 1198.26033 · doi:10.1016/j.camwa.2010.03.072 [39] Dunninger, D.; Wang, H.: Existence and multiplicity of positive solutions for elliptic systems, Nonlinear anal. TMA 29, 1051-1060 (1997) · Zbl 0885.35028 · doi:10.1016/S0362-546X(96)00092-2 [40] Agarwal, R.; Meehan, M.; O’regan, D.: Fixed point theory and applications, (2001) [41] Henderson, J.; Thompson, H. B.: Existence of multiple solutions for second-order discrete boundary value problems, Comput. math. Appl. 43, 1239-1248 (2002) · Zbl 1005.39014 · doi:10.1016/S0898-1221(02)00095-0