# 4. Let [a,b] C R be a compact interval and let k: [a,b] X [a,b] R...

## Transcribed Text

4. Let [a,b] C R be a compact interval and let k: [a,b] X [a,b] R be a continuous function. Define for u € C([a,6];R), b (i) Explain why we can deduce that Kop C((a,b);R) C([a,6];R) and also that Kop is a linear and bounded operator. (It is sufficient here to give the basic ideas without detailed arguments). (ii) Show that, / - max -b (iii) Suppose that HE R and let K := 05056 max !. b le(s, t) I dt. a Show that if 101 < :L then <1. What does this result imply for the solvability of the integral equa- tion, b u1s) -v r k(s,t)u(t)dt=g(s) for g E C([a,6];R)?

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