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

MSC:

 34A30 Linear ordinary differential equations and systems
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References:

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