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On the uniqueness of solutions of the singular problem for certain class of integro-differential equations. (English) Zbl 0781.45007

The uniqueness and asymptotic behavior of a solution to the singular problem

g(x)y ' (x)=y(x)+ 0 + x |α|=2 m u α (x)v α (t)y i 1 (x)y i 2 (t)(y ' (x)) i 3 (y ' (t)) i 4 dt,(1)

y(0 + )=0, |α|=i 1 +i 2 +i 3 +i 4 , α=(i 1 ,...,i 4 ), i k N{0}, k=1,...,4, mN, 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.

45J05Integro-ordinary differential equations
45M05Asymptotic theory of integral equations