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On equation $$P(\text{D})u=f(u^{(m)})+g\bigl (t,(u^{(j)})\bigr )$$ on the line. (English) Zbl 1116.34023
Let $$D=-i (d/dt)$$, $$f\in C(\mathbb R,\mathbb R)$$ with $$| f(x)| \leq K| x|$$ near $$x=0$$. Moreover, let $$g \: \mathbb R\times \mathbb R^k\to \mathbb R$$ be Carathéodory continuous such that $$| g(t,y_1,\cdots ,y_k)| \leq h(t)$$ for $$h\in L_2(R)$$. Additional conditions are considered in order to show the existence of a real solution for a differential equation $P(D)u=f(u^{(m)})+g\bigl (t,(u^{(j)})_{j=j_1,\dots ,j_k}\bigr )$
in the Sobolev space $$H_n(\mathbb R)$$, where $$n$$ is the degree of the linear differential operator $$P(\text{D})$$.
##### MSC:
 34B40 Boundary value problems on infinite intervals for ordinary differential equations 34C11 Growth and boundedness of solutions to ordinary differential equations
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##### References:
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