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A note on a third-order multi-point boundary value problem at resonance. (English) Zbl 1229.34028

Summary: Based on the coincidence degree theory of Mawhin, we prove some existence results for the following third-order multi-point boundary value problem at resonance
\[ x'''(t)=f(t,x(t),x'(t),x''(t)),\quad t\in (0,1), \]
\[ x''(0)=\sum^m_{i=1}\alpha_ix''(\xi_i),\quad x'(0)=0,\;x(1)=\sum^n_{j=1}\beta_jx(\eta_j), \]
where \(f:[0,1]\times \mathbb R^3\to\mathbb R\) is a continuous function, \(0<\xi_1<\cdots<\xi_m < 1\), \(\alpha_i\in\mathbb R\), \(i=1,\dots,m\), \(m\geq 1\) and \(0<\eta_1<\eta_2<\cdots<\eta_n<1\), \(\beta_j\in\mathbb R\), \(j=1,2,\dots,n\), \(n\geq 2\). In this paper, the dimension of the linear space \(\text{Ker}\,L\) (the linear operator \(L\) is defined by \(Lx=x'''\)) is equal to 2. Since all the existence results for third-order differential equations obtained in previous papers are for the case \(\dim\text{Ker}\,L=1\), our work is new.

MSC:

34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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References:

[1] Agarwal, Singular Differential and Integral Equations with Applications (2003) · doi:10.1007/978-94-017-3004-4
[2] Du, Solvability of functional differential equations with multi-point boundary value problem at resonance, Comput. Math. Appl. 55 pp 2653– (2008) · Zbl 1142.34357 · doi:10.1016/j.camwa.2007.10.015
[3] Du, On a third-order multi-point boundary value problem at resonance, J. Math. Anal. Appl. 302 pp 217– (2005) · Zbl 1072.34012 · doi:10.1016/j.jmaa.2004.08.012
[4] Du, Some higher-order multi-point boundary value problem at resonance, J. Comput. Appl. Math. 177 pp 55– (2005) · Zbl 1059.34010 · doi:10.1016/j.cam.2004.08.003
[5] Du, Multiple solutions to a three-point boundary value problem for higher-order ordinary differential equations, J. Math. Anal. Appl. 335 pp 1207– (2007) · Zbl 1133.34011 · doi:10.1016/j.jmaa.2007.02.014
[6] Feng, Solvability of m-point boundary value problems with nonlinear growth, J. Math. Anal. Appl. 212 pp 467– (1997) · Zbl 0883.34020 · doi:10.1006/jmaa.1997.5520
[7] Ge, Boundary Value Problems for Ordinary Nonlinear Differential Equations (2007)
[8] Gupta, On a third-order boundary value problem at resonance, Differ. Integral Equ. 2 pp 1– (1989)
[9] Kosmatov, A multi-point boundary value problem with two critical conditions, Nonlinear Anal. 65 pp 622– (2006) · Zbl 1121.34023 · doi:10.1016/j.na.2005.09.042
[10] Liu, A note on multi-point boundary value problems, Nonlinear Anal. 67 pp 2680– (2007) · Zbl 1127.34006 · doi:10.1016/j.na.2006.09.032
[11] Ma, Multiplicity results for a third order boundary value problem at resonance, Nonlinear Anal. 32 pp 493– (1998) · Zbl 0932.34014 · doi:10.1016/S0362-546X(97)00494-X
[12] Mawhin, Topological Degree Methods in Nonlinear Boundary Value Problems, NSFCBMS Regional Conference Series in Mathematics (1979) · Zbl 0414.34025 · doi:10.1090/cbms/040
[13] Nagle, On a third-order nonlinear boundary value problems at resonance, J. Math. Anal. Appl. 195 pp 148– (1995) · Zbl 0847.34026 · doi:10.1006/jmaa.1995.1348
[14] Rachånkové, Topological degree method in functional boundary value problems at resonance, Nonlinear Anal. 27 pp 271– (1996) · Zbl 0853.34062 · doi:10.1016/0362-546X(95)00060-9
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