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Zbl 0653.45005
Ashirov, S.; Mamedov, Ya.D.
On an integral equation of Volterra type. (Russian)
[J] Ukr. Mat. Zh. 40, No.4, 510-515 (1988). ISSN 0041-6053

The following integral equation is investigated (1) $x(t)=x\sb 0+\int\sp{t}\sb{-t}K(t,s,x(s))ds,$ $t\in [-1,1]$. Assume that the function K is continuous on the set $[-1,1]\times [-1,1]\times [x\sb 0- r,x\sb 0+r]$ and satisfies the condition $\lim\sb{t\sb 2\to t\sb 1}\int\sp{t\sb 1}\sb{-t\sb 1}\vert K(t\sb 2,s,x)-K(t\sb 1,s,x)\vert ds=0,$ where $t\sb 1,t\sb 2\in [-1,1]$. Then the equation (1) possesses at least one solution on [-1,1] which may be obtained by Tonelli 's procedure or by the method of successive approximation.
[J.Banaś]

MSC 2000:: *45G10 Nonsingular nonlinear integral equations
45L05 Theoretical approximation of solutions of integral equations

Keywords: nonlinear Volterra integral equation; Tonelli method; method of successive approximation

Cited in: Zbl 0666.45009

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