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Zbl 0781.45007
Šmarda, Z.
On the uniqueness of solutions of the singular problem for certain class of integro-differential equations.
(English)
[J] Demonstr. Math. 25, No.4, 835-841 (1992). ISSN 0420-1213

The uniqueness and asymptotic behavior of a solution to the singular problem $$g(x)y'(x)=y(x)+\int\sp x\sb{0\sp +}\left[\sum\sp m\sb{\vert\alpha\vert=2}u\sb \alpha(x) v\sb \alpha(t)y\sp{i\sb 1}(x)y\sp{i\sb 2}(t)(y'(x))\sp{i\sb 3}(y'(t))\sp{i\sb 4}\right]dt, \tag 1$$ $y(0\sp +)=0$, $\vert\alpha\vert=i\sb 1+i\sb 2+i\sb 3+i\sb 4$, $\alpha=(i\sb 1,\ldots,i\sb 4)$, $i\sb k\in N\cup\{0\}$, $k=1,\ldots,4$, $m\in N$, is considered. The definition of a singular function $g(x)$ with respect to (1) is introduced. The considerations are an extension of the author's previous results.
[I.Foltyńska (Poznań)]
MSC 2000:
*45J05 Integro-ordinary differential equations
45M05 Asymptotic theory of integral equations

Keywords: integro-differential equations; asymptotic behavior; singular problem; singular function

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