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[tex]\varphi(x)=x+1-\int^{x}_0 \varphi(y)dy[/tex]

If I start from ##\varphi_0(x)=1## or ##\varphi_0(x)=x+1## I will get solution of this equation using Picard method in following way

[tex]\varphi_1(x)=x+1-\int^{x}_0 \varphi_0(y)dy[/tex]

[tex]\varphi_2(x)=x+1-\int^{x}_0 \varphi_1(y)dy[/tex]

[tex]\varphi_3(x)=x+1-\int^{x}_0 \varphi_2(y)dy[/tex]

...

Then solution is given by

[tex]\varphi(x)=\lim_{n \to \infty}\varphi_n(x)[/tex].

When I could say that this sequence will converge to solution of integral equation. How to see if there is some fixed point? I know how to use this method, but I am not sure from the form of equation, when I can use this method. Thanks for the answer.

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# A Integral equations -- Picard method of succesive approximation

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