When does the integral equation given by
i) a unique solution ???
ii) no solution???
Some head start with this question will be of great help
Have you tried differentiating the integral in order to get a differential equation to solve?
Differentiation under the integral sign - Wikipedia, the free encyclopedia
Also Lipschitz continuity is a good way to show existence of differential equations.
Lipschitz continuity - Wikipedia, the free encyclopedia
in the integral cos(x-t) = cos(x)cos(t)+sin(x)sin(t)
Leading to y = 1+lambda *(A*cos(x)+B*sin(x))
where A and B are constant (definite integrals where there is no x into them)
Bringing back y = 1+lambda *(A*cos(x)+B*sin(x)) into the equation, after integration we obtain two relationships so that the coefficient of sin(x) and cos(x) are = 0.
This leads to : A = B = -4 / (pi+2-(4/lambda))
Conclusion : There is one solution.