Assignment:
Consider the nonlinear Fredholm equation x(t) = ba f( t, s, x(s)) ds + g(t) where g:ℜ→ℜn is continuous on [a,b] and f:ℜn+2→ℜn is continuous and satisfies a Lipschitz condition: | f(t, s, v1) - (t, s, v2)) ≤ L |v1 -v2|on the set S = {(t,s,v ):t, s ∈ [a, b] , v ∈ Rn} . Show that the integral equation has a unique solution on [a,b] if |λ|< 1/ (L( b- a)) .
Provide complete and step by step solution for the question and show calculations and use formulas.