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Singular Cauchy initial value problem for certain classes of integro-differential equations. (English) Zbl 1192.45002

Summary: The existence and uniqueness of solutions and asymptotic estimate of solution formulas are studied for the following initial value problem: \[ g(t)y'(t)=ay(t)[1+f(t,y(t),\int^t_{0^+}K(t,s,y(t),y(s))\,ds)],\quad y(0^+)=0, \quad t\in (0,t_0], \] where \(a>0\) is a constant and \(t_0>0\). An approach which combines topological method of T. Ważewski [cf. R. Srzednicki, “Ważewski method and Conley index”, Handbook of Differential Equations, 591–684 (2004; Zbl 1091.37006)] and Schauder’s fixed point theorem is used.

MSC:

45J05 Integro-ordinary differential equations
45G05 Singular nonlinear integral equations

Citations:

Zbl 1091.37006
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References:

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