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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.

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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