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Existence and multiplicity of solutions to \(2m\)th-order ordinary differential equations. (English) Zbl 1119.34014

Summary: The existence and multiplicity of solutions are obtained for the \(2m\)th-order ordinary differential equation two-point boundary value problems \[ (-1)^mu^{(2 m)}(t)+\sum^m_{i=1}(-1)^{m-i}a_i u^{(2(m-i))}(t)=f(t,u(t))\text{ for all }t\in [0,1] \] subject to Dirichlet, Neumann, mixed and periodic boundary value conditions, respectively, where \(f\) is continuous, \(a_i\in\mathbb{R}\) for all \(i=1,2,\dots,m\). Since these four boundary value problems have some common properties and they can be transformed into the integral equation of form \[ u+ \sum^m_{i=1}a_iT^iu=T^m{\mathbf f}u, \] we firstly deal with this nonlinear integral equation. By using the strongly monotone operator principle and the critical point theory, we establish some conditions on \(f\) which are able to guarantee that the integral equation has a unique solution, at least one nonzero solution, and infinitely many solutions. Furthermore, we apply the abstract results on the integral equation to the above four \(2m\)th-order two-point boundary problems and successfully resolve the existence and multiplicity of their solutions.

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
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