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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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