Third kind Fredholm integral equation
$begingroup$
Let us consider the following integral equation
$$a(x)u(x) + intlimits_0^1 {K(s,x)u(s)ds} = f(x)$$
Let f in $L^p(0,1)$ for some $p in [1,infty] and let $ $K in L^q((0,1) times (0,1))$. Assume that $a$ doesn't vanishe in any point of (0,1). My question is: What are the optimal assumptions to ensure that this equation has a solution?
In my opinion, if $K$ and $f$ are continuous with $K$ is lipschitz kernel then we can apply Picard's iterations to prove the existence, what about $L^p$?
Thank you.
real-analysis functional-analysis functional-equations compact-operators integral-equations
asked Dec 20 '18 at 16:39
GustaveGustave
734211
$endgroup$
add a comment |
$begingroup$
Let us consider the following integral equation
$$a(x)u(x) + intlimits_0^1 {K(s,x)u(s)ds} = f(x)$$
Let f in $L^p(0,1)$ for some $p in [1,infty] and let $ $K in L^q((0,1) times (0,1))$. Assume that $a$ doesn't vanishe in any point of (0,1). My question is: What are the optimal assumptions to ensure that this equation has a solution?
In my opinion, if $K$ and $f$ are continuous with $K$ is lipschitz kernel then we can apply Picard's iterations to prove the existence, what about $L^p$?
Thank you.
real-analysis functional-analysis functional-equations compact-operators integral-equations
asked Dec 20 '18 at 16:39
GustaveGustave
734211
$endgroup$
add a comment |
$begingroup$
Let us consider the following integral equation
$$a(x)u(x) + intlimits_0^1 {K(s,x)u(s)ds} = f(x)$$
Let f in $L^p(0,1)$ for some $p in [1,infty] and let $ $K in L^q((0,1) times (0,1))$. Assume that $a$ doesn't vanishe in any point of (0,1). My question is: What are the optimal assumptions to ensure that this equation has a solution?
In my opinion, if $K$ and $f$ are continuous with $K$ is lipschitz kernel then we can apply Picard's iterations to prove the existence, what about $L^p$?
Thank you.
real-analysis functional-analysis functional-equations compact-operators integral-equations
asked Dec 20 '18 at 16:39
GustaveGustave
734211
$endgroup$
Let us consider the following integral equation
$$a(x)u(x) + intlimits_0^1 {K(s,x)u(s)ds} = f(x)$$
Let f in $L^p(0,1)$ for some $p in [1,infty] and let $ $K in L^q((0,1) times (0,1))$. Assume that $a$ doesn't vanishe in any point of (0,1). My question is: What are the optimal assumptions to ensure that this equation has a solution?
In my opinion, if $K$ and $f$ are continuous with $K$ is lipschitz kernel then we can apply Picard's iterations to prove the existence, what about $L^p$?
Thank you.
real-analysis functional-analysis functional-equations compact-operators integral-equations
real-analysis functional-analysis functional-equations compact-operators integral-equations
asked Dec 20 '18 at 16:39
GustaveGustave
734211
asked Dec 20 '18 at 16:39
GustaveGustave
734211
edited Dec 20 '18 at 21:37
Gustave
asked Dec 20 '18 at 16:39
GustaveGustave
734211
asked Dec 20 '18 at 16:39
GustaveGustave
734211
asked Dec 20 '18 at 16:39
GustaveGustave
734211
734211
add a comment |
add a comment |
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