| Title:
|
Existence of a positive solution to a nonlocal semipositone boundary value problem on a time scale (English) |
| Author:
|
Goodrich, Christopher S. |
| Language:
|
English |
| Journal:
|
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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54 |
| Issue:
|
4 |
| Year:
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2013 |
| Pages:
|
509-525 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We consider the existence of at least one positive solution to the dynamic boundary value problem \begin{align*} -y^{\Delta\Delta}(t) & = \lambda f(t,y(t))\text{, }t\in [0,T]_{\mathbb{T}} y(0) & = \int_{\tau_1}^{\tau_2}F_1(s,y(s)) \Delta s y\left(\sigma^2(T)\right) & = \int_{\tau_3}^{\tau_4}F_2(s,y(s)) \Delta s, \end{align*} where $\mathbb{T}$ is an arbitrary time scale with $0<\tau_1<\tau_2<\sigma^2(T)$ and $0<\tau_3<\tau_4<\sigma^2(T)$ satisfying $\tau_1$, $\tau_2$, $\tau_3$, $\tau_4\in \mathbb{T}$, and where the boundary conditions at $t=0$ and $t=\sigma^2(T)$ can be both nonlinear and nonlocal. This extends some recent results on second-order semipositone dynamic boundary value problems, and we illustrate these extensions with some examples. (English) |
| Keyword:
|
time scales |
| Keyword:
|
integral boundary condition |
| Keyword:
|
second-order boundary value problem |
| Keyword:
|
cone |
| Keyword:
|
positive solution |
| MSC:
|
26E70 |
| MSC:
|
34B10 |
| MSC:
|
34B15 |
| MSC:
|
34B18 |
| MSC:
|
34N05 |
| MSC:
|
39A10 |
| MSC:
|
47H07 |
| idZBL:
|
Zbl 06373981 |
| idMR:
|
MR3125073 |
| . |
| Date available:
|
2013-10-01T21:15:01Z |
| Last updated:
|
2016-01-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143473 |
| . |
| Reference:
|
[1] Agarwal R., Meehan M., O'Regan D.: Fixed Point Theory and Applications.Cambridge University Press, Cambridge, 2001. Zbl 1159.54001, MR 1825411, 10.1017/CBO9780511543005.008 |
| Reference:
|
[2] Anderson D.R.: Second-order $n$-point problems on time scales with changing-sign nonlinearity.Adv. Dynamical Sys. Appl. 1 (2006), 17–27. Zbl 1119.34310, MR 2287632 |
| Reference:
|
[3] Anderson D.R.: Existence of solutions for first-order multi-point problems with changing-sign nonlinearity.J. Difference Equ. Appl. 14 (2008), 657–666. Zbl 1158.34006, MR 2417015, 10.1080/10236190701736682 |
| Reference:
|
[4] Anderson D.R., Zhai C.: Positive solutions to semi-positone second-order three-point problems on time scales.Appl. Math. Comput. 215 (2010), 3713–3720. Zbl 1188.34119, MR 2578954, 10.1016/j.amc.2009.11.010 |
| Reference:
|
[5] Anuradha V., Hai D.D., Shivaji R.: Existence results for superlinear semipositone BVPs.Proc. Amer. Math. Soc. 124 (1996), 757–763. MR 1317029, 10.1090/S0002-9939-96-03256-X |
| Reference:
|
[6] Boucherif A.: Second-order boundary value problems with integral boundary conditions.Nonlinear Anal. 70 (2009), 364–371. Zbl 1169.34310, MR 2468243, 10.1016/j.na.2007.12.007 |
| Reference:
|
[7] Bohner M., Peterson A.C.: Dynamic Equations on Time Scales: An Introduction with Applications.Birkhäuser, Boston, 2001. Zbl 0978.39001, MR 1843232 |
| Reference:
|
[8] Dahal R.: Positive solutions of semipositone singular Dirichlet dynamic boundary value problems.Nonlinear Dyn. Syst. Theory 9 (2009), 361–374. Zbl 1205.34128, MR 2590764 |
| Reference:
|
[9] Dahal R.: Positive solutions for a second-order, singular semipositone dynamic boundary value problem.Int. J. Dyn. Syst. Differ. Equ. 3 (2011), 178–188. Zbl 1214.34090, MR 2797048 |
| Reference:
|
[10] Erbe L.H., Peterson A.C.: 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:
|
[11] Feng M.: Existence of symmetric positive solutions for a boundary value problem with integral boundary conditions.Appl. Math. Lett. 24 (2011), 1419–1427. Zbl 1221.34062, MR 2793645, 10.1016/j.aml.2011.03.023 |
| Reference:
|
[12] Goodrich C.S.: Existence of a positive solution to a system of discrete fractional boundary value problems.Appl. Math. Comput. 217 (2011), 4740–4753. Zbl 1215.39003, MR 2745153, 10.1016/j.amc.2010.11.029 |
| Reference:
|
[13] Goodrich C.S.: Existence of a positive solution to a first-order $p$-Laplacian BVP on a time scale.Nonlinear Anal. 74 (2011), 1926–1936. Zbl 1236.34112, MR 2764390 |
| Reference:
|
[14] Goodrich C.S.: Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions.Comput. Math. Appl. 61 (2011), 191–202. Zbl 1211.39002, MR 2754129, 10.1016/j.camwa.2010.10.041 |
| Reference:
|
[15] 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:
|
[16] Goodrich C.S.: The existence of a positive solution to a second-order $p$-Laplacian BVP on a time scale.Appl. Math. Lett. 25 (2012), 157–162. MR 2843745, 10.1016/j.aml.2011.08.005 |
| Reference:
|
[17] Goodrich C.S.: Positive solutions to boundary value problems with nonlinear boundary conditions.Nonlinear Anal. 75 (2012), 417–432. MR 2846811, 10.1016/j.na.2011.08.044 |
| Reference:
|
[18] Goodrich C.S.: Nonlocal systems of BVPs with asymptotically superlinear boundary conditions.Comment. Math. Univ. Carolin. 53 (2012), 79–97. Zbl 1249.34054, MR 2880912 |
| Reference:
|
[19] 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:
|
[20] Goodrich C.S.: On nonlocal BVPs with boundary conditions with asymptotically sublinear or superlinear growth.Math. Nachr. 285 (2012), 1404–1421. MR 2959967 |
| Reference:
|
[21] Goodrich C.S.: On discrete fractional boundary value problems with nonlocal, nonlinear boundary conditions.Commun. Appl. Anal. 16 (2012), 433–446. MR 3051308 |
| Reference:
|
[22] Goodrich C.S.: Nonlocal systems of BVPs with asymptotically sublinear boundary conditions.Appl. Anal. Discrete Math. 6 (2012), 174–193. MR 3012670, 10.2298/AADM120329010G |
| Reference:
|
[23] Goodrich C.S.: On nonlinear boundary conditions satisfying certain asymptotic behavior.Nonlinear Anal. 76 (2013), 58–67. Zbl 1264.34030, MR 2974249, 10.1016/j.na.2012.07.023 |
| Reference:
|
[24] 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:
|
[25] Goodrich C.S.: On semipositone discrete fractional boundary value problems with nonlocal boundary conditions.J. Difference Equ. Appl., doi: 10.1080/10236198.2013.775259. |
| Reference:
|
[26] J. Graef, L. Kong: Positive solutions for third order semipositone boundary value problems.Appl. Math. Lett. 22 (2009), 1154–1160. Zbl 1173.34313, MR 2532528, 10.1016/j.aml.2008.11.008 |
| Reference:
|
[27] Hilger S.: Analysis on measure chains – a unified approach to continuous and discrete calculus.Results Math. 18 (1990), 18–56. Zbl 0722.39001, MR 1066641, 10.1007/BF03323153 |
| Reference:
|
[28] Jia M., Liu X.: Three nonnegative solutions for fractional differential equations with integral boundary conditions.Comput. Math. Appl. 62 (2011), 1405–1412. Zbl 1235.34016, MR 2824728, 10.1016/j.camwa.2011.03.026 |
| Reference:
|
[29] Infante G.: Nonlocal boundary value problems with two nonlinear boundary conditions.Commun. Appl. Anal. 12 (2008), 279–288. Zbl 1198.34025, MR 2499284 |
| Reference:
|
[30] Infante G., Pietramala P.: Existence and multiplicity of non-negative solutions for systems of perturbed Hammerstein integral equations.Nonlinear Anal. 71 (2009), 1301–1310. Zbl 1169.45001, MR 2527550, 10.1016/j.na.2008.11.095 |
| Reference:
|
[31] Infante G., Pietramala P.: Eigenvalues and non-negative solutions of a system with nonlocal BCs.Nonlinear Stud. 16 (2009), 187–196. Zbl 1184.34027, MR 2527180 |
| Reference:
|
[32] Infante G., Pietramala P.: A third order boundary value problem subject to nonlinear boundary conditions.Math. Bohem. 135 (2010), 113–121. Zbl 1224.34036, MR 2723078 |
| Reference:
|
[33] G. Infante, F. Minhós, P. Pietramala: Non-negative solutions of systems of ODEs with coupled boundary conditions.Commun. Nonlinear Sci. Numer. Simul. 17 (2012), 4952–4960. MR 2960289, 10.1016/j.cnsns.2012.05.025 |
| Reference:
|
[34] Sun J.P., Li W.T.: Existence of positive solutions to semipositone Dirichlet BVPs on time scales.Dynam. Systems Appl. 16 (2007), 571–578. MR 2356340 |
| Reference:
|
[35] Sun J.P., Li W.T.: Solutions and positive solutions to semipositone Dirichlet BVPs on time scales.Dynam. Systems Appl. 17 (2008), 303–312. MR 2436566 |
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