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Integral equations of the first kind of Sonine type. (English) Zbl 1034.45007
The authors consider the first kind Volterra integral equation $(K\varphi) (x):\equiv \int\limits_{-\infty}^x k(x-t)\varphi (t) dt =f(x), x\in R^1$ with the Sonine kernel. A kernel $$k(x)\in L_1^{loc} (R^1_+)$$ is called a Sonine kernel if there exists a kernel $$\ell (x)\in L_1^{loc} (R^1_+)$$ such that $$\int\limits_{0}^x k(x-t)\ell (t) dt \equiv 1$$. Under some assumptions on $$\ell (x)$$ and using the construction of left inverse operator in the Marshaud form as a limit (in $$L_p$$) of corresponding truncated operator, that is $(K^{-1}f)(x)=\ell (\infty)f(x)+\lim\limits_{\varepsilon\to 0 }\int_\varepsilon^\infty\ell^\prime (t) [f(x-t)-f(x)]dt,$ they give the description of the range $$K(L_p (R^1)), p\geq 1$$. It should be noted that such description is based on the language of Orlicz spaces.

##### MSC:
 45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) 45H05 Integral equations with miscellaneous special kernels 26A33 Fractional derivatives and integrals 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46N20 Applications of functional analysis to differential and integral equations
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