Volterra and Functional Differential Equations

SOME SYNTHESIS PROBLEMS FOR INTEGRAL EQUATIONS

J. J. Levin

Department of Mathenaties University of Nisconsin Madison, Wisconsin

DOI: 10.1201/9781003420026-6

We consider the equations

$$x^{\prime }\left(t\right)+{{\displaystyle \int }}_{-\infty }^{\infty }8\left(x\left(t-\xi \right)\right)dM\left(\xi \right)=f\left(t\right),-\infty <t<\infty$$ (1f)

$$x\left(t\right)+{{\displaystyle \int }}_{-\infty }^{-\infty }8\left(x\left(t-\xi \right)\right)d\lambda \left(\xi \right)=f\left(t\right),-\infty <t<\infty ,$$ (2f)

where ‘$=\frac{d}{dt}$,

$$\begin{array}{ll}\hfill & g=C\left({\varphi }^N,{\varphi }^N\right),A=\left(A_{ij}\right),A_{1j}\in B\vee \left({ℝ}^1,{\varphi }^1\right)\left(1,j=1,\dots ,N\right)\hfill \\ \hfill & f=L^{″}\left({ℝ}^1,{\varphi }^N\right),\underset{t\to \infty }{\text{lim}}f\left(t\right)=f\left(m\right)\text{ exists and, in }\left(2f\right),f=B\left({ℝ}^1,{\varphi }^N\right).\hfill \end{array}$$

B denotes Borel neasurable. The limit equations associated ...