Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences) ( IF 0.3 ) Pub Date : 2020-11-09 , DOI: 10.3103/s1068362320050039 S. Mukhigulashvili , M. Manjikashvili
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.
中文翻译:
四阶非线性常微分方程的Dirichlet问题
摘要
建立了两点边值问题\(u ^ {(4)}(t)= p(t)u(t)+ f(t,u(t)的可解性的Landesman-Lazer型有效充分条件)+ h(t)\)表示\(a \ leq t \ leq b \),\(u ^ {(i)}(a)= 0 \),\(u ^ {{i)}(b) = 0,\ quad(i = 0,1)\),其中\(h,p \ in L([a,b]; R)\)和\(f \ in K([a,b] \ times R; R)\),在线性问题\(w ^ {(4)}(t)= p(t)w(t)\)的情况下,\(w ^ {(i)}(a) = 0 \),\(w ^ {(i)}(b)= 0 \),\((i = 0,1)\)具有非平凡解。在以下情况下,本文获得的结果是最佳的:\(f \ equiv 0 \),即当非线性方程变成线性方程时,根据我们的结果,遵循弗雷德霍姆定理的第一部分。