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

Nhan, Le Cong; Hoang, Do Huy; Truong, Le Xuan

Arch. Math., Brno 53, No. 2, 111-130 (2017).

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.

Reviewer: Gennaro Infante (Arcavata di Rende)

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

Keywords:

coincidence degree; higher-order differential equation; resonance