Second-order functional problems with a resonance of dimension one. (English) Zbl 1347.34034

From the author’s abstract: We obtain solvability conditions, for all possible resonance scenarios, for a differential equation of the form \[ u''=f(t,u,u') \] with linear functional conditions \(B_1(u)=0\) and \(B_2(u)=0\) such that the kernel of the linear map \[ L:\{ u\in C^1[0,1]\, :\, B_1(u)=B_2(u)=0\}\to L_1[0,1],\quad Lu=u'' \] has dimension 1. Our work generalizes and improves the results of Z. Zhao and J. Liang [J. Math. Anal. Appl. 373, No. 2, 614–634 (2011; Zbl 1208.34020)] and Y. Cui [Electron. J. Differ. Equ. 2012, Paper No. 45, 9 p. (2012; Zbl 1244.34020)]. We also construct an example of a nonlinear functional problem for a pendulum equation which not only satisfies the assumptions of an existence theorem but also has a closed form solution.


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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