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The Dirichlet Problem for the Fourth Order Nonlinear Ordinary Differential Equations at Resonance

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Abstract

Landesman-Lazer’s type efficient sufficient conditions are established for the solvability of the two-point boundary value problem \(u^{(4)}(t)=p(t)u(t)+f(t,u(t))+h(t)\) for \(a\leq t\leq b\), \(u^{(i)}(a)=0\), \(u^{(i)}(b)=0,\quad(i=0,1)\), where \(h,p\in L([a,b];R)\) and \(f\in K([a,b]\times R;R)\), in the case where the linear problem \(w^{(4)}(t)=p(t)w(t)\), \(w^{(i)}(a)=0\), \(w^{(i)}(b)=0\), \((i=0,1)\) has nontrivial solutions. The results obtained in the paper are optimal in the sense that if \(f\equiv 0\), i.e. when nonlinear equation turns to the linear equation, from our results follows the first part of Fredholm’s theorem.

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Funding

The research of S. Mukhigulashvil was supported by institutional grant RVO: 67985840.

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Mukhigulashvili, S., Manjikashvili, M. The Dirichlet Problem for the Fourth Order Nonlinear Ordinary Differential Equations at Resonance. J. Contemp. Mathemat. Anal. 55, 291–302 (2020). https://doi.org/10.3103/S1068362320050039

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