On the existence of square integrable solutions and their derivatives to fourth and fifth order differential equations. (English) Zbl 0711.34012

Consider the fourth-order nonlinear differential equation \[ (*)\quad x^{iv}(t)+ax'''(t)+bx''(t)+cx'(t)+h(x(t))=p(t), \] where \(a,b,c\in {\mathbb{R}}^+\) are constants with \(ab>c\); \(h(x)\in C^ 1({\mathbb{R}})\), \(h'(x)<0\), \(h(0)=0\), \(| h(x)| \leq H\), \(| p(t)| \leq P\), \(\liminf_{| x| \to \infty}| h(x)| >| p(0)|\) for all t,x\(\in {\mathbb{R}}\). Assuming, in addition, that \(h'(x)\) is bounded on (0,\(\infty)\) and \(\limsup_{t\to \infty}| \int^{t}_{0}p(\tau)d\tau | <\infty,\) it is shown that \(x^{(j)}(t)\in L_ 2(0,\infty)\) for \(j=0,1,2,3\). In the second section, for the fifth-order equation analogous to (*), with corresponding assumptions on the constants and the functions h(x), p(t) it is shown that the validity of \(x^{(j)}(t)\in L_ 2(0,\infty)\), \(j=0,...,4\), is attainable with a certain restriction of \(h'(x)\) on the interval (0,\(\infty)\), only.
Reviewer: P.Talpalaru


34A30 Linear ordinary differential equations and systems
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