| Title:
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Summation equations with sign changing kernels and applications to discrete fractional boundary value problems (English) |
| Author:
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Goodrich, Christopher S. |
| Language:
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English |
| Journal:
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Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
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0010-2628 (print) |
| ISSN:
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1213-7243 (online) |
| Volume:
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57 |
| Issue:
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2 |
| Year:
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2016 |
| Pages:
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201-229 |
| Summary lang:
|
English |
| . |
| Category:
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math |
| . |
| Summary:
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We consider the summation equation, for $t\in[\mu-2,\mu+b]_{\mathbb{N}_{\mu-2}}$, \begin{align*} y(t)=\gamma_1(t)H_1\left(\sum_{i=1}^{n}a_iy\left(\xi_i\right)\right) & + \gamma_2(t)H_2\left(\sum_{i=1}^{m}b_iy\left(\zeta_i\right)\right) &+ \lambda\sum_{s=0}^{b}G(t,s)f(s+\mu-1,y(s+\mu-1)) \end{align*} in the case where the map $(t,s)\mapsto G(t,s)$ may change sign; here $\mu\in(1,2]$ is a parameter, which may be understood as the order of an associated discrete fractional boundary value problem. In spite of the fact that $G$ is allowed to change sign, by introducing a new cone we are able to establish the existence of at least one positive solution to this problem by imposing some growth conditions on the functions $H_1$ and $H_2$. Finally, as an application of the abstract existence result, we demonstrate that by choosing the maps $t\mapsto\gamma_1(t)$, $\gamma_2(t)$ in particular ways, we can recover the existence of at least one positive solution to various discrete fractional- or integer-order boundary value problems possessing Green's functions that change sign. (English) |
| Keyword:
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summation equation |
| Keyword:
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sign-changing kernel |
| Keyword:
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discrete fractional calculus |
| Keyword:
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positive solution |
| Keyword:
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nonlocal boundary condition |
| MSC:
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26A33 |
| MSC:
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39A05 |
| MSC:
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39A12 |
| MSC:
|
39A99 |
| MSC:
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47H07 |
| idZBL:
|
Zbl 1374.39001 |
| idMR:
|
MR3513445 |
| DOI:
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10.14712/1213-7243.2015.164 |
| . |
| Date available:
|
2016-07-05T15:09:48Z |
| Last updated:
|
2018-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/145754 |
| . |
| Reference:
|
[1] Anderson D.R.: Existence of three solutions for a first-order problem with nonlinear nonlocal boundary conditions.J. Math. Anal. Appl. 408 (2013), 318–323. Zbl 1314.34048, MR 3079969, 10.1016/j.jmaa.2013.06.025 |
| Reference:
|
[2] Atici F.M., Acar N.: Exponential functions of discrete fractional calculus.Appl. Anal. Discrete Math. 7 (2013), 343–353. Zbl 1299.39001, MR 3135934, 10.2298/AADM130828020A |
| Reference:
|
[3] Atici F.M., Eloe P.W.: A transform method in discrete fractional calculus.Int. J. Difference Equ. 2 (2007), 165–176. MR 2493595 |
| Reference:
|
[4] Atici F.M., Eloe P.W.: Discrete fractional calculus with the nabla operator.Electron. J. Qual. Theory Differ. Equ. (2009), Special Edition I, 12 pp. Zbl 1189.39004, MR 2558828 |
| Reference:
|
[5] Atici F.M., Eloe P.W.: Initial value problems in discrete fractional calculus.Proc. Amer. Math. Soc. 137 (2009), 981–989. Zbl 1166.39005, MR 2457438, 10.1090/S0002-9939-08-09626-3 |
| Reference:
|
[6] Atici F.M., Eloe P.W.: Two-point boundary value problems for finite fractional difference equations.J. Difference Equ. Appl. 17 (2011), 445–456. Zbl 1215.39002, MR 2783359, 10.1080/10236190903029241 |
| Reference:
|
[7] Atici F.M., Eloe P.W.: Linear systems of fractional nabla difference equations.Rocky Mountain J. Math. 41 (2011), 353–370. Zbl 1218.39003, MR 2794443, 10.1216/RMJ-2011-41-2-353 |
| Reference:
|
[8] Atici F.M., Eloe P.W.: Gronwall's inequality on discrete fractional calculus.Comput. Math. Appl. 64 (2012), 3193–3200. Zbl 1268.26029, MR 2989347, 10.1016/j.camwa.2011.11.029 |
| Reference:
|
[9] Atici F.M., Şengül S.: Modeling with fractional difference equations.J. Math. Anal. Appl. 369 (2010), 1–9. Zbl 1204.39004, MR 2643839, 10.1016/j.jmaa.2010.02.009 |
| Reference:
|
[10] Atici F.M., Uyanik M.: Analysis of discrete fractional operators.Appl. Anal. Discrete Math. 9 (2015), 139–149. MR 3362702, 10.2298/AADM150218007A |
| Reference:
|
[11] Baoguo J., Erbe L., Goodrich C.S., Peterson A.: The relation between nabla fractional differences and nabla integer differences.Filmoat(to appear). |
| Reference:
|
[12] Baoguo J., Erbe L., Goodrich C.S., Peterson A.: Monotonicity results for delta fractional differences revisited.Math. Slovaca(to appear). |
| Reference:
|
[13] Bastos N.R.O., Mozyrska D., Torres D.F.M.: Fractional derivatives and integrals on time scales via the inverse generalized Laplace transform.Int. J. Math. Comput. 11 (2011), 1–9. MR 2800417 |
| Reference:
|
[14] Dahal R., Duncan D., Goodrich C.S.: Systems of semipositone discrete fractional boundary value problems.J. Difference Equ. Appl. 20 (2014), 473–491. Zbl 1319.39002, MR 3173559, 10.1080/10236198.2013.856073 |
| Reference:
|
[15] Dahal R., Goodrich C.S.: A monotonicity result for discrete fractional difference operators.Arch. Math. (Basel) 102 (2014), 293–299. Zbl 1330.39022, MR 3181719, 10.1007/s00013-014-0620-x |
| Reference:
|
[16] Dahal R., Goodrich C.S.: Erratum to “R. Dahal, C.S. Goodrich, A monotonicity result for discrete fractional difference operators, Arch. Math. (Basel) 102.(2014), 293–299”, Arch. Math. (Basel) 104 (2015), 599–600. MR 3181719, 10.1007/s00013-014-0620-x |
| Reference:
|
[17] Erbe L., Peterson A.: Positive solutions for a nonlinear differential equation on a measure chain.Math. Comput. Modelling 32 (2000), 571–585. Zbl 0963.34020, MR 1791165, 10.1016/S0895-7177(00)00154-0 |
| Reference:
|
[18] Erbe L., Peterson A.: Eigenvalue conditions and positive solutions.J. Difference Equ. Appl. 6 (2000), 165–191. Zbl 0949.34015, MR 1760156, 10.1080/10236190008808220 |
| Reference:
|
[19] Ferreira R.A.C.: Nontrivial solutions for fractional $q$-difference boundary value problems.Electron. J. Qual. Theory Differ. Equ. (2010), 10 pp. Zbl 1207.39010, MR 2740675 |
| Reference:
|
[20] Ferreira R.A.C.: Positive solutions for a class of boundary value problems with fractional $q$-differences.Comput. Math. Appl. 61 (2011), 367–373. Zbl 1216.39013, MR 2754144, 10.1016/j.camwa.2010.11.012 |
| Reference:
|
[21] Ferreira R.A.C.: A discrete fractional Gronwall inequality.Proc. Amer. Math. Soc. 140 (2012), 1605–1612. Zbl 1243.26012, MR 2869144, 10.1090/S0002-9939-2012-11533-3 |
| Reference:
|
[22] Ferreira R.A.C.: Existence and uniqueness of solution to some discrete fractional boundary value problems of order less than one.J. Difference Equ. Appl. 19 (2013), 712–718. Zbl 1276.26013, MR 3049050, 10.1080/10236198.2012.682577 |
| Reference:
|
[23] Ferreira R.A.C., Goodrich C.S.: Positive solution for a discrete fractional periodic boundary value problem.Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 19 (2012), 545–557. Zbl 1268.26010, MR 3058228 |
| Reference:
|
[24] Ferreira R.A.C., Torres D.F.M.: Fractional $h$-difference equations arising from the calculus of variations.Appl. Anal. Discrete Math. 5 (2011), 110–121. Zbl 1289.39007, MR 2809039, 10.2298/AADM110131002F |
| Reference:
|
[25] Gao L., Sun J.P.: Positive solutions of a third-order three-point BVP with sign-changing Green's function.Math. Probl. Eng. (2014), Article ID 406815, 6 pages. MR 3268274 |
| Reference:
|
[26] Goodrich C.S.: Solutions to a discrete right-focal boundary value problem.Int. J. Difference Equ. 5 (2010), 195–216. MR 2771325 |
| Reference:
|
[27] Goodrich C.S.: On discrete sequential fractional boundary value problems.J. Math. Anal. Appl. 385 (2012), 111–124. Zbl 1236.39008, MR 2832079, 10.1016/j.jmaa.2011.06.022 |
| Reference:
|
[28] Goodrich C.S.: On a discrete fractional three-point boundary value problem.J. Difference Equ. Appl. 18 (2012), 397–415. Zbl 1253.26010, MR 2901829, 10.1080/10236198.2010.503240 |
| Reference:
|
[29] Goodrich C.S.: On a first-order semipositone discrete fractional boundary value problem.Arch. Math. (Basel) 99 (2012), 509–518. Zbl 1263.26016, MR 3001554, 10.1007/s00013-012-0463-2 |
| Reference:
|
[30] Goodrich C.S.: On semipositone discrete fractional boundary value problems with nonlocal boundary conditions.J. Difference Equ. Appl. 19 (2013), 1758–1780. MR 3173516, 10.1080/10236198.2013.775259 |
| Reference:
|
[31] Goodrich C.S.: A convexity result for fractional differences.Appl. Math. Lett. 35 (2014), 58–62. Zbl 1314.26010, MR 3212846, 10.1016/j.aml.2014.04.013 |
| Reference:
|
[32] Goodrich C.S.: An existence result for systems of second-order boundary value problems with nonlinear boundary conditions.Dynam. Systems Appl. 23 (2014), 601–618. Zbl 1310.34035, MR 3241607 |
| Reference:
|
[33] Goodrich C.S.: Semipositone boundary value problems with nonlocal, nonlinear boundary conditions.Adv. Differential Equations 20 (2015), 117–142. Zbl 1318.34034, MR 3297781 |
| Reference:
|
[34] Goodrich C.S.: Coupled systems of boundary value problems with nonlocal boundary conditions.Appl. Math. Lett. 41 (2015), 17–22. Zbl 1312.34050, MR 3282393, 10.1016/j.aml.2014.10.010 |
| Reference:
|
[35] Goodrich C.S.: Systems of discrete fractional boundary value problems with nonlinearities satisfying no growth conditions.J. Difference Equ. Appl. 21 (2015), 437–453. Zbl 1320.39001, MR 3334521, 10.1080/10236198.2015.1013537 |
| Reference:
|
[36] Goodrich C.S.: On nonlinear boundary conditions involving decomposable linear functionals.Proc. Edinb. Math. Soc. (2) 58 (2015), 421–439. Zbl 1322.34038, MR 3341447, 10.1017/S0013091514000108 |
| Reference:
|
[37] Goodrich C.S.: Coercivity of linear functionals on finite dimensional spaces and its application to discrete boundary value problem.J. Difference Equ. Appl., doi: 10.1080/10236198.2015.1125896. MR 3516118, 10.1080/10236198.2015.1125896 |
| Reference:
|
[38] Goodrich C.S., Peterson A.C.: Discrete Fractional Calculus.Springer, Cham, 2015, doi: 10.1007/978-3-319-25562-0. MR 3445243, 10.1007/978-3-319-25562-0 |
| Reference:
|
[39] Graef J., Kong L., Wang H.: A periodic boundary value problem with vanishing Green's function.Appl. Math. Lett. 21 (2008), 176–180. Zbl 1135.34307, MR 2426975, 10.1016/j.aml.2007.02.019 |
| Reference:
|
[40] Graef J., Kong L.: Positive solutions for a class of higher order boundary value problems with fractional $q$-derivatives.Appl. Math. Comput. 218 (2012), 9682–9689. Zbl 1254.34010, MR 2916148, 10.1016/j.amc.2012.03.006 |
| Reference:
|
[41] Holm M.: Sum and difference compositions and applications in discrete fractional calculus.Cubo 13 (2011), 153–184. MR 2895482, 10.4067/S0719-06462011000300009 |
| Reference:
|
[42] Infante G.: Nonlocal boundary value problems with two nonlinear boundary conditions.Commun. Appl. Anal. 12 (2008), 279–288. Zbl 1198.34025, MR 2499284 |
| Reference:
|
[43] Infante G., Pietramala P., Tenuta M.: Existence and localization of positive solutions for a nonlocal BVP arising in chemical reactor theory.Commun. Nonlinear Sci. Numer. Simul. 19 (2014), 2245–2251. MR 3157933, 10.1016/j.cnsns.2013.11.009 |
| Reference:
|
[44] Infante G., Pietramala P.: Multiple nonnegative solutions of systems with coupled nonlinear boundary conditions.Math. Methods Appl. Sci. 37 (2014), 2080–2090. Zbl 1312.34060, MR 3248749, 10.1002/mma.2957 |
| Reference:
|
[45] Jankowski T.: Positive solutions to fractional differential equations involving Stieltjes integral conditions.Appl. Math. Comput. 241 (2014), 200–213. Zbl 1334.34058, MR 3223422, 10.1016/j.amc.2014.04.080 |
| Reference:
|
[46] Jia B., Erbe L., Peterson A.: Two monotonicity results for nabla and delta fractional differences.Arch. Math. (Basel) 104 (2015), 589–597. Zbl 1327.39011, MR 3350348, 10.1007/s00013-015-0765-2 |
| Reference:
|
[47] Jia B., Erbe L., Peterson A.: Convexity for nabla and delta fractional differences.J. Difference Equ. Appl. 21 (2015), 360–373. Zbl 1320.39003, MR 3326277, 10.1080/10236198.2015.1011630 |
| Reference:
|
[48] Jia B., Erbe L., Peterson A.: Some relations between the Caputo fractional difference operators and integer order differences.Electron. J. Differential Equations (2015), No. 163, pp. 1–7. Zbl 1321.39024, MR 3375994 |
| Reference:
|
[49] Karakostas G.L.: Existence of solutions for an $n$-dimensional operator equation and applications to BVPs.Electron. J. Differential Equations (2014), No. 71, 17 pp. Zbl 1298.34118, MR 3193977 |
| Reference:
|
[50] Ma R.: Nonlinear periodic boundary value problems with sign-changing Green's function.Nonlinear Anal. 74 (2011), 1714–1720. MR 2764373, 10.1016/j.na.2010.10.043 |
| Reference:
|
[51] Picone M.: Su un problema al contorno nelle equazioni differenziali lineari ordinarie del secondo ordine.Ann. Scuola Norm. Sup. Pisa Cl. Sci. 10 (1908), 1–95. MR 1556636 |
| Reference:
|
[52] Sun J.P., Zhao J.: Multiple positive solutions for a third-order three-point BVP with sign-changing Green's function.Electron. J. Differential Equations (2012), No. 118, pp. 1–7. Zbl 1260.34049, MR 2967183 |
| Reference:
|
[53] Wang J., Gao C.: Positive solutions of discrete third-order boundary value problems with sign-changing Green's function.Adv. Difference Equ. (2015), 10 pp. MR 3315295 |
| Reference:
|
[54] Whyburn W.M.: Differential equations with general boundary conditions.Bull. Amer. Math. Soc. 48 (1942), 692–704. Zbl 0061.17904, MR 0007192, 10.1090/S0002-9904-1942-07760-3 |
| Reference:
|
[55] Wu G., Baleanu D.: Discrete fractional logistic map and its chaos.Nonlinear Dyn. 75 (2014), 283–287. MR 3144852, 10.1007/s11071-013-1065-7 |
| Reference:
|
[56] Yang Z.: Positive solutions to a system of second-order nonlocal boundary value problems.Nonlinear Anal. 62 (2005), 1251–1265. Zbl 1089.34022, MR 2154107, 10.1016/j.na.2005.04.030 |
| Reference:
|
[57] Yang Z.: Positive solutions of a second-order integral boundary value problem.J. Math. Anal. Appl. 321 (2006), 751–765. Zbl 1106.34014, MR 2241153, 10.1016/j.jmaa.2005.09.002 |
| Reference:
|
[58] Zeidler E.: Nonlinear Functional Analysis and Its Applications, I: Fixed-Point Theorems.Springer, New York, 1986. Zbl 0583.47050, MR 0816732 |
| Reference:
|
[59] Zhang P., Liu L., Wu Y.: Existence and uniqueness of solution to nonlinear boundary value problems with sign-changing Green's function.Abstr. Appl. Anal. (2013), Article ID 640183, 7 pp. MR 3121401 |
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