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Existence of \(AP_{r}\)-almost periodic solutions for some classes of functional differential equations. (English) Zbl 1281.34118

Summary: This paper presents a couple of existence results, related to the classes of functional equations of the form \[ x+k*x=f, \] or \[ \frac{d}{dt}[\dot{x}+k*x]=f \] with \(f, x\in AP_r(R,C)=\) the space of almost periodic functions defined by \[ AP_r(R,C)=\left\{f : f\simeq\sum_{j=1}^{\infty} f_j e^{i\lambda_j t},\;f_j\in C,\,\lambda_j\in R,\;\sum_{j=1}^{\infty}|f_j|^r <\infty\right\}, \] the norm being given by \(|f|_r= \left(\sum_{j=1}^{\infty}|f_j|^r\right)^{\frac{1}{r}}\), for each \(r\in [1, 2]\). The convolution product \(k*x\), \(k\in L^1(R,C)\), \(x\in AP_r(R,C)\), is defined by \[ (k*x)(t)= \sum_{j=1}^{\infty} x_j\left( \int_R k(s)e^{-\lambda_j s}ds\right) e^{i\lambda_j t}, \] where \(x(t)\simeq \sum_{j=1}^{\infty} x_j e^{i\lambda_j t}\).

MSC:

34K14 Almost and pseudo-almost periodic solutions to functional-differential equations
34K05 General theory of functional-differential equations
34K40 Neutral functional-differential equations
45B05 Fredholm integral equations
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References:

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