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Jeffers, Benjamin L.; Lyons, Jeffrey W.

Solutions of the variational equation for an \(n\)-th order boundary value problem with an integral boundary condition. (English) Zbl 1471.34047

Involve 14, No. 1, 155-166 (2021).
Summary: We discuss differentiation of solutions to the boundary value problem \[y^{(n)} = f(x, y, y', y'', \ldots, y^{(n-1)}), \quad a<x<b,\] \[y^{(i)}(x_j) = y_{ij}, \quad 0\leq i \leq m_j,\, 1 \leq j \leq k-1,\] \[y^{(i)}(x_k) + \text style\int_c^d p y(x)\,dx = y_{ik}, \quad 0 \leq i \leq m_k,\,\sum_{i=1}^km_i=n,\] with respect to the boundary data. We show that under certain conditions, partial derivatives of the solution \(y(x)\) of the boundary value problem with respect to the various boundary data exist and solve the associated variational equation along \(y(x)\).

MSC:

34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations

Keywords:

variational equation; integral condition; continuous dependence; smoothness; Peano theorem
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References:

[1] 10.1017/S0004972716000022 · Zbl 1345.34147 · doi:10.1017/S0004972716000022
[2] ; Benchohra, Dynam. Systems Appl., 23, 133 (2014)
[3] 10.1080/10236199808808163 · Zbl 0921.39003 · doi:10.1080/10236199808808163
[4] 10.1006/jdeq.1993.1008 · Zbl 0848.34014 · doi:10.1006/jdeq.1993.1008
[5] 10.1016/0362-546X(94)E0084-T · Zbl 0841.34067 · doi:10.1016/0362-546X(94)E0084-T
[6] ; Ehme, PanAmer. Math. J., 10, 13 (2000)
[7] ; Ehme, Differ. Equ. Dyn. Syst., 1, 59 (1993) · Zbl 0872.34043
[8] 10.1016/j.jmaa.2006.09.055 · Zbl 1148.34009 · doi:10.1016/j.jmaa.2006.09.055
[9] 10.1016/0022-247X(84)90255-5 · Zbl 0533.34015 · doi:10.1016/0022-247X(84)90255-5
[10] 10.1016/0022-247X(87)90233-2 · Zbl 0613.34014 · doi:10.1016/0022-247X(87)90233-2
[11] 10.2140/involve.2015.8.629 · Zbl 1328.39003 · doi:10.2140/involve.2015.8.629
[12] 10.1155/S0161171291001011 · Zbl 0762.39004 · doi:10.1155/S0161171291001011
[13] 10.4171/ZAA/1214 · Zbl 1120.34008 · doi:10.4171/ZAA/1214
[14] ; Henderson, Comm. Appl. Nonlinear Anal., 1, 47 (1994) · Zbl 0856.49007
[15] 10.1090/S0002-9939-04-07647-6 · Zbl 1061.34009 · doi:10.1090/S0002-9939-04-07647-6
[16] 10.2140/involve.2008.1.167 · Zbl 1151.34016 · doi:10.2140/involve.2008.1.167
[17] ; Hopkins, Comm. Appl. Nonlinear Anal., 16, 1 (2009) · Zbl 1178.49017
[18] ; Janson, Dynam. Systems Appl., 23, 493 (2014)
[19] 10.1016/S0377-0427(01)00449-6 · Zbl 1036.34015 · doi:10.1016/S0377-0427(01)00449-6
[20] 10.14232/ejqtde.2011.1.51 · Zbl 1340.34069 · doi:10.14232/ejqtde.2011.1.51
[21] 10.1080/10236198.2013.837461 · Zbl 1297.39010 · doi:10.1080/10236198.2013.837461
[22] ; Lyons, J. Math. Stat. Sci., 1, 43 (2015)
[23] 10.2140/involve.2018.11.95 · Zbl 1372.34046 · doi:10.2140/involve.2018.11.95
[24] 10.1016/0022-247X(76)90080-9 · Zbl 0342.34008 · doi:10.1016/0022-247X(76)90080-9
[25] 10.1016/0022-247X(78)90029-X · Zbl 0386.34022 · doi:10.1016/0022-247X(78)90029-X
[26] 10.4153/CMB-1975-051-8 · Zbl 0321.34014 · doi:10.4153/CMB-1975-051-8
[27] 10.1216/RMJ-1976-6-2-357 · Zbl 0326.34018 · doi:10.1216/RMJ-1976-6-2-357
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