Homogeneous Fredholm integral equation of the first kind with positive symmetric kernel
up vote
0
down vote
favorite
Given the equation
$$int^{1}_{-1}K(|x-t|)varphi(t)dt=0,$$
where the kernel is positive: $K(x)>0$; equation is satisfied for $xin [-1,1]$. $K(x)$ and $varphi(x)$ are real and continuous functions. I'm looking for nontrivial solutions $varphi(x)$.
Unfortunately, the explicit form of kernel $K(x)$ is unknown, but approximately K(x) is close to $exp(-lambda x)$.
I suppose, that if kernel $K(x)$ is not constant, then only trivial solutions exist. (If kernel is constant, then any odd function satisfies the equation.) Is this so? And how to prove it?
integral-equations
asked Nov 20 at 20:06
And111
1
New contributor
And111 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
up vote
0
down vote
favorite
Given the equation
$$int^{1}_{-1}K(|x-t|)varphi(t)dt=0,$$
where the kernel is positive: $K(x)>0$; equation is satisfied for $xin [-1,1]$. $K(x)$ and $varphi(x)$ are real and continuous functions. I'm looking for nontrivial solutions $varphi(x)$.
Unfortunately, the explicit form of kernel $K(x)$ is unknown, but approximately K(x) is close to $exp(-lambda x)$.
I suppose, that if kernel $K(x)$ is not constant, then only trivial solutions exist. (If kernel is constant, then any odd function satisfies the equation.) Is this so? And how to prove it?
integral-equations
asked Nov 20 at 20:06
And111
1
New contributor
And111 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Given the equation
$$int^{1}_{-1}K(|x-t|)varphi(t)dt=0,$$
where the kernel is positive: $K(x)>0$; equation is satisfied for $xin [-1,1]$. $K(x)$ and $varphi(x)$ are real and continuous functions. I'm looking for nontrivial solutions $varphi(x)$.
Unfortunately, the explicit form of kernel $K(x)$ is unknown, but approximately K(x) is close to $exp(-lambda x)$.
I suppose, that if kernel $K(x)$ is not constant, then only trivial solutions exist. (If kernel is constant, then any odd function satisfies the equation.) Is this so? And how to prove it?
integral-equations
asked Nov 20 at 20:06
And111
1
New contributor
And111 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Given the equation
$$int^{1}_{-1}K(|x-t|)varphi(t)dt=0,$$
where the kernel is positive: $K(x)>0$; equation is satisfied for $xin [-1,1]$. $K(x)$ and $varphi(x)$ are real and continuous functions. I'm looking for nontrivial solutions $varphi(x)$.
Unfortunately, the explicit form of kernel $K(x)$ is unknown, but approximately K(x) is close to $exp(-lambda x)$.
I suppose, that if kernel $K(x)$ is not constant, then only trivial solutions exist. (If kernel is constant, then any odd function satisfies the equation.) Is this so? And how to prove it?
integral-equations
integral-equations
asked Nov 20 at 20:06
And111
1
New contributor
And111 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Nov 20 at 20:06
And111
1
New contributor
And111 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Nov 20 at 20:06
And111
1
New contributor
And111 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Nov 20 at 20:06
And111
1
asked Nov 20 at 20:06
And111
1
1
New contributor
And111 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
And111 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
And111 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
add a comment |
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
And111 is a new contributor. Be nice, and check out our Code of Conduct.
And111 is a new contributor. Be nice, and check out our Code of Conduct.
And111 is a new contributor. Be nice, and check out our Code of Conduct.
And111 is a new contributor. Be nice, and check out our Code of Conduct.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3006817%2fhomogeneous-fredholm-integral-equation-of-the-first-kind-with-positive-symmetric%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
