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Existence results for a class of high order differential equation associated with integral boundary conditions at resonance. (English) Zbl 1424.34081

The authors study the existence of solutions of the boundary value problem (BVP) \[\begin{gathered}u^{(n)}(t)=f(t,u(t),u'(t),\ldots,u^{(n-1)}(t)),\ t\in(0,1),\\ \alpha_{i}u^{(i-1)}(0)+\beta_{i}u^{(i-1)}(1)=\gamma_{i}\int_{0}^{1}u(s)\,ds,\ i=1,2,\ldots,n,\end{gathered}\tag{1}\] where \(\alpha_{i},\beta_{i},\gamma_{i}\) are real constants. The problem here is at resonance, in the sense that the associated linear operator is not invertible. The authors use a continuation principle for \(L\)-compact operators to prove that the BVP (1) has at least one solution. An illustrative example is provided in the last section of the paper.

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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