## On discrete sequential fractional boundary value problems.(English)Zbl 1236.39008

The author considers several different types of discrete sequential fractional boundary value problems. Our prototype equation is $-\Delta^{\mu_1}\Delta^{\mu_2} \Delta^{\mu_3} y(t)= f(t+ \mu_1+ \mu_2+ \mu_3-1,\,y(t+ \mu_1+ \mu_2+ \mu_3- 1))$ subject to the conjugate boundary conditions $$y(0)= 0= y(b+ 2)$$, where $f: [1,b+1]_{N_0}\times R\to[0,+\infty)$ is a continuous function and $$\mu_1,\mu_2,\mu_3\in (0,1)$$ satisfy $$1< \mu_1+ \mu_3< 2$$ and $$1< \mu_1+ \mu_2+ \mu_3< 2$$. He also obtains results for delta-nabla discrete fractional boundary value problems. As an application of our analysis, he gives conditions under which such problem will admit at least one positive solution.

### MSC:

 39A12 Discrete version of topics in analysis
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### References:

 [1] Atici, F. M.; Eloe, P. W., Two-point boundary value problems for finite fractional difference equations, J. Difference Equ. Appl., 17, 445-456 (2011) · Zbl 1215.39002 [2] Wei, Z.; Li, Q.; Che, J., Initial value problems for fractional differential equations involving Riemann-Liouville sequential fractional derivative, J. Math. Anal. Appl., 367, 260-272 (2010) · Zbl 1191.34008 [3] Agarwal, R. P.; Lakshmikantham, V.; Nieto, J. J., On the concept of solution for fractional differential equations with uncertainty, Nonlinear Anal., 72, 2859-2862 (2009) · Zbl 1188.34005 [4] 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 [5] Arara, A., Fractional order differential equations on an unbounded domain, Nonlinear Anal., 72, 580-586 (2010) · Zbl 1179.26015 [6] 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 [7] 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 [8] Diethelm, K.; Ford, N., Analysis of fractional differential equations, J. Math. Anal. Appl., 265, 229-248 (2002) · Zbl 1014.34003 [9] 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 [10] Kirane, M.; Malik, S. A., The profile of blowing-up solutions to a nonlinear system of fractional differential equations, Nonlinear Anal., 73, 3723-3736 (2010) · Zbl 1205.26013 [11] 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 [12] Nieto, J. J., Maximum principles for fractional differential equations derived from Mittag-Leffler functions, Appl. Math. Lett., 23, 1248-1251 (2010) · Zbl 1202.34019 [13] Xu, X.; Jiang, D.; Yuan, C., Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation, Nonlinear Anal., 71, 4676-4688 (2009) · Zbl 1178.34006 [14] Atici, F. M.; Eloe, P. W., Initial value problems in discrete fractional calculus, Proc. Amer. Math. Soc., 137, 981-989 (2009) · Zbl 1166.39005 [15] Goodrich, C. S., Solutions to a discrete right-focal fractional boundary value problem, Int. J. Difference Equ., 5, 195-216 (2010) [16] Goodrich, C. S., Continuity of solutions to discrete fractional initial value problems, Comput. Math. Appl., 59, 3489-3499 (2010) · Zbl 1197.39002 [17] Atici, F. M.; Şengül, S., Modeling with fractional difference equations, J. Math. Anal. Appl., 369, 1-9 (2010) · Zbl 1204.39004 [18] Bastos, N. R.O., Necessary optimality conditions for fractional difference problems of the calculus of variations, Discrete Contin. Dyn. Syst., 29, 417-437 (2011) · Zbl 1209.49020 [19] C.S. Goodrich, On a discrete fractional three-point boundary value problem, J. Difference Equ. Appl., doi:10.1080/10236198.2010.503240; C.S. Goodrich, On a discrete fractional three-point boundary value problem, J. Difference Equ. Appl., doi:10.1080/10236198.2010.503240 · Zbl 1253.26010 [20] Goodrich, C. S., Some new existence results for fractional difference equations, Int. J. Dyn. Syst. Differ. Equ., 3, 145-162 (2011) · Zbl 1215.39004 [21] Goodrich, C. S., Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions, Comput. Math. Appl., 61, 191-202 (2011) · Zbl 1211.39002 [22] Goodrich, C. S., Existence of a positive solution to a system of discrete fractional boundary value problems, Appl. Math. Comput., 217, 4740-4753 (2011) · Zbl 1215.39003 [23] C.S. Goodrich, A comparison result for the fractional difference operator, Int. J. Difference Equ. 6 (2011), in press.; C.S. Goodrich, A comparison result for the fractional difference operator, Int. J. Difference Equ. 6 (2011), in press. [24] Goodrich, C. S., On positive solutions to nonlocal fractional and integer-order difference equations, Appl. Anal. Discrete Math., 5, 122-132 (2011) · Zbl 1289.39008 [25] Bastos, N. R.O., Discrete-time fractional variational problems, Signal Process., 91, 513-524 (2011) · Zbl 1203.94022 [26] Bastos, N. R.O., Fractional derivatives and integrals on time scales via the inverse generalized Laplace transform, Int. J. Math. Comput., 11, 1-9 (2011) [27] Ferreira, R. A.C., Positive solutions for a class of boundary value problems with fractional $$q$$-differences, Comput. Math. Appl., 61, 367-373 (2011) · Zbl 1216.39013 [28] Anderson, D. R., Solutions to second-order three-point problems on time scales, J. Difference Equ. Appl., 8, 673-688 (2002) · Zbl 1021.34011 [29] Kaufmann, E. R.; Raffoul, Y. N., Positive solutions for a nonlinear functional dynamic equation on a time scale, Nonlinear Anal., 62, 1267-1276 (2005) · Zbl 1090.34054 [30] 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 [31] Atici, F. M.; Guseinov, G., On Greenʼs functions and positive solutions for boundary value problems on time scales, J. Comput. Appl. Math., 141, 75-99 (2002) · Zbl 1007.34025 [32] Anderson, D. R.; Hoffacker, J., A stacked delta-nabla self-adjoint problem of even order, Math. Comput. Modelling, 38, 5-6, 481-494 (2003) · Zbl 1106.39308 [33] DaCunha, J. J.; Davis, J. M.; Singh, P. K., Existence results for singular three point boundary value problems on time scales, J. Math. Anal. Appl., 295, 378-391 (2004) · Zbl 1069.34012 [34] Anderson, D. R.; Avery, R. I., An even-order three-point boundary value problem on time scales, J. Math. Anal. Appl., 291, 514-525 (2004) · Zbl 1056.34013 [35] Anderson, D. R.; Hoffacker, J., Even order self adjoint time scale problems, Electron. J. Differential Equations, 24 (2005), 9 pp · Zbl 1075.34010 [36] Guseinov, G., Self-adjoint boundary value problems on time scales and symmetric Greenʼs functions, Turkish J. Math., 29, 4, 365-380 (2005) · Zbl 1098.34009 [37] Anderson, D. R.; Hoffacker, J., Existence of solutions for a cantilever beam problem, J. Math. Anal. Appl., 323, 958-973 (2006) · Zbl 1115.34019 [38] Wang, P.; Wang, Y., Existence of positive solutions for second-order m-point boundary value problems on time scales, Acta Math. Appl. Sin. Engl. Ser., 22, 457-468 (2006) · Zbl 1108.34014 [39] Došlý, O., Reciprocity principle for even-order dynamic equations with mixed derivatives, Dynam. Systems Appl., 16, 4, 697-708 (2007) · Zbl 1148.34027 [40] M. Holm, Sum and difference compositions and applications in discrete fractional calculus, Cubo 3 (13) (2011), in press.; M. Holm, Sum and difference compositions and applications in discrete fractional calculus, Cubo 3 (13) (2011), in press. · Zbl 1248.39003 [41] Atici, F. M.; Eloe, P. W., A transform method in discrete fractional calculus, Int. J. Difference Equ., 2, 165-176 (2007) [42] Atici, F. M.; Eloe, P. W., Discrete fractional calculus with the nabla operator, Electron. J. Qual. Theory Differ. Equ., 3, 1-12 (2009), (Spec. Ed. I) · Zbl 1189.39004 [43] Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press New York · Zbl 0918.34010 [44] Agarwal, R.; Meehan, M.; OʼRegan, D., Fixed Point Theory and Applications (2001), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0960.54027 [45] Erbe, L. H.; Wang, H., On the existence of positive solutions of ordinary differential equations, Proc. Amer. Math. Soc., 120, 743-748 (1994) · Zbl 0802.34018 [46] Infante, G.; Webb, J. R.L., Positive solutions of nonlocal boundary value problems: a unified approach, J. Lond. Math. Soc., 74, 673-693 (2006) · Zbl 1115.34028
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