zbMATH — the first resource for mathematics

On the existence of $$m$$-point boundary value problem at resonance for higher order differential equation. (English) Zbl 1046.34029
The paper considers an $$m$$-point boundary value problem for a higher-order differential equation of the form $x^{(k)}(t)=f(t,x(t),x'(t),\dots,x^{(k-1)}(t))+e(t), \quad t\in (0,1),$
$x'(0)=0,\;x''(0)=0, \dots, x^{(k-1)}(0)=0, \quad x(1)=\sum_{i=1}^{m-2} a_ix(\xi_i),$ where $$f:[0,1]\times \mathbb{R}^k \to \mathbb{R}$$ and $$e:[0,1]\to \mathbb{R}$$ are continuous functions. Further, $$m\geq 3$$, $$k\geq 2$$ are two integers, $$a_i\in \mathbb{R}$$, $$\xi_i \in (0,1)$$, $$i=1,2,\dots, m-2$$, $$\xi_1 <\xi_2<\dots <\xi_{m-2}$$. The authors study the problem at resonance because they assume that $\sum_{i=1}^{m-2}a_i=1.$ Moreover, they do not need that all $$a_i$$’s, $$1\leq i\leq m-2$$, have the same sign.
The authors prove a new existence result under certain sign and growth conditions imposed on $$f$$. The growth of $$f$$ in some variables can be superlinear. The proofs are based on Mawhin’s continuation theorem. Some examples illustrate the obtained result.

MSC:
 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations
Full Text:
References:
 [1] Il’in, V.A.; Moiseev, E.I., Nonlocal boundary value problem of the first kind for a sturm – liouville operator, J. differential equations, 23, 803-810, (1987) · Zbl 0668.34025 [2] Il’in, V.A.; Moiseev, E.I., Nonlocal boundary value problems of the second kind for a sturm – liouville operator, J. differential equations, 23, 979-987, (1987) · Zbl 0668.34024 [3] Gupta, C.P., A second order m-point boundary value problem at resonance, Nonlinear anal., 24, 1483-1489, (1995) · Zbl 0824.34023 [4] Gupta, C.P., Solvability of a multiple boundary value problem at resonance, Results math., 28, 270-276, (1995) · Zbl 0843.34023 [5] Gupta, C.P., Existence theorems for a second order m-point boundary value problem at resonance, Internat. J. math. sci., 18, 705-710, (1995) · Zbl 0839.34027 [6] O’Regan, D., Existence theory for nonlinear ordinary differential equations, (1997), Kluwer Academic Dordrecht · Zbl 1077.34505 [7] Ma, R.Y., Existence theorems for second order m-point boundary value problems, J. math. anal. appl., 211, 545-555, (1997) · Zbl 0884.34024 [8] Gupta, C.P., A Dirichlet type multi point boundary value problem for second order ordinary differential equations, Nonlinear anal., 26, 925-931, (1996) · Zbl 0847.34018 [9] Karakostas, G.L.; Tsamatos, P.Ch., On a nonlocal boundary value problem at resonance, J. math. anal. appl., 259, 209-218, (2001) · Zbl 1002.34057 [10] Przeradzki, B.; Stańczy, R., Solvability of m-point boundary value problems at resonance, J. math. anal. appl., 264, 253-261, (2001) · Zbl 1043.34016 [11] Feng, W.; Webb, J.R.L., Solvability of three-point boundary value problems at resonance, Nonlinear anal., 30, 3227-3238, (1997) · Zbl 0891.34019 [12] Feng, W.; Webb, J.R.L., Solvability of m-point boundary value problems with nonlinear growth, J. math. anal. appl., 212, 467-480, (1997) · Zbl 0883.34020 [13] Gupta, C.P.; Ntouyas, S.K.; Tsamatos, P.Ch., Solvability of an m-point boundary value problem for second order ordinary differential equations, J. math. anal. appl., 189, 575-584, (1995) · Zbl 0819.34012 [14] Gupta, C.P.; Ntouyas, S.K.; Tsamatos, P.Ch., On an m-point boundary value problem for second order ordinary differential equations, Nonlinear anal., 23, 1427-1436, (1994) · Zbl 0815.34012 [15] Liu, B.; Yu, J., Solvability of multiple point boundary value problem at resonance, Appl. math. comput., 136, 353-377, (2003) · Zbl 1053.34016 [16] Mawhin, J., Topological degree methods in nonlinear boundary value problems, NSF-CBMS regional conference series in math., 40, (1979), American Mathematical Society Providence, RI [17] Mawhin, J., Topological degree and boundary value problems for nonlinear differential equations, (), 74-142 · Zbl 0798.34025
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.