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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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