## 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
Full Text:

### References:

 [1] Agarwal, Singular Differential and Integral Equations with Applications (2003) [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 [3] Du, On a third-order multi-point boundary value problem at resonance, J. Math. Anal. Appl. 302 pp 217– (2005) · Zbl 1072.34012 [4] Du, Some higher-order multi-point boundary value problem at resonance, J. Comput. Appl. Math. 177 pp 55– (2005) · Zbl 1059.34010 [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 [6] Feng, Solvability of m-point boundary value problems with nonlinear growth, J. Math. Anal. Appl. 212 pp 467– (1997) · Zbl 0883.34020 [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 [10] Liu, A note on multi-point boundary value problems, Nonlinear Anal. 67 pp 2680– (2007) · Zbl 1127.34006 [11] Ma, Multiplicity results for a third order boundary value problem at resonance, Nonlinear Anal. 32 pp 493– (1998) · Zbl 0932.34014 [12] Mawhin, Topological Degree Methods in Nonlinear Boundary Value Problems, NSFCBMS Regional Conference Series in Mathematics (1979) · Zbl 0414.34025 [13] Nagle, On a third-order nonlinear boundary value problems at resonance, J. Math. Anal. Appl. 195 pp 148– (1995) · Zbl 0847.34026 [14] Rachånkové, Topological degree method in functional boundary value problems at resonance, Nonlinear Anal. 27 pp 271– (1996) · Zbl 0853.34062
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.