Non trivial solution of Fredholm integral equation of second kind with constant kernel
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Let us consider the following integral equation $$f(x) + lambda int_0^1 {K(s,x)f(s)ds = 0,{text{ x}} in {text{(0}}{text{,1)}}{text{.}}} $$ I'm looking of the values of $lambda$ so that the above equation has only $f=0$ as solution with a constant kernel. Suppose that $K(s,x)=K$ , we obtain $$f(x) + lambda Kint_0^1 {f(s)ds = 0,{text{ x}} in {text{(0}}{text{,1)}}{text{.}}} $$ By taking the integral over $(0,1)$ , we get $$(1 + lambda K)int_0^1 {f(s)ds = 0} $$ . for all $f$ . Now, if $lambda$ is different of $-1/K$ , then $$int_0^1 {f(s)ds = 0} $$ . I don't see how this can be helpful. Any suggestions?. Thank you.
real-analysis integration functional-analysis differential-equations integral-equations
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