<abstract xmlns="http://www.w3.org/1999/xhtml">We consider a scalar integral equation <inline-formula><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="../graphic/v10127-011-0003-7#0007_graphic1.png"></inline-graphic></inline-formula>where <italic>|G(t,z)| ≤ ϕ(t)|z|, C </italic> is convex, and <inline-formula><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="../graphic/v10127-011-0003-7#0007_graphic2.png"></inline-graphic></inline-formula>. Related to this is the linear equation <inline-formula><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="../graphic/v10127-011-0003-7#0007_graphic3.png"></inline-graphic></inline-formula> and the resolvent equation <inline-formula><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="../graphic/v10127-011-0003-7#0007_graphic4.png"></inline-graphic></inline-formula>. A Liapunov functional is constructed which gives qualitative results about all three equations. We have two goals. First, we are interested in conditions under which properties of <italic>C</italic> are transferred into properties of the resolvent <italic>R</italic> which is used in the variation-of-parameters formula. We establish conditions on <italic>C</italic> and functions <italic>b</italic> so that <inline-formula><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="../graphic/v10127-011-0003-7#0007_graphic5.png"></inline-graphic></inline-formula> as t→∞ and is in <italic>L</italic>2[0, ∞] implies that <inline-formula><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="../graphic/v10127-011-0003-7#0007_graphic6.png"></inline-graphic></inline-formula> as t→∞ and is in <italic>L</italic>2[0, ∞]. Such results are fundamental in proving that the solution <italic>z</italic> satisfies <italic>z(t) →a(t)</italic> as t→∞ and that <inline-formula><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="../graphic/v10127-011-0003-7#0007_graphic7.png"></inline-graphic></inline-formula>. This is in final form and no other version will be submitted. </abstract>

We consider a scalar integral equation where |G(t,z)| ≤ ϕ(t)|z|, C  is convex, and . Related to this is the linear equation  and the resolvent equation . A Liapunov functional is constructed which gives qualitative results about all three equations. We have two goals. First, we are interested in conditions under which properties of C are transferred into properties of the resolvent R which is used in the variation-of-parameters formula. We establish conditions on C and functions b so that  as t→∞ and is in L2[0, ∞] implies that  as t→∞ and is in L2[0, ∞]. Such results are fundamental in proving that the solution z satisfies z(t) →a(t) as t→∞ and that . This is in final form and no other version will be submitted.

Kernel-resolvent relations for an integral equation

Northwest Research Institute 732 Caroline St. Port Angeles Washington 98362 U.S.A.

Tatra Mountains Mathematical Publications

{"article-title":"Kernel-resolvent relations for an integral equation"}

We consider a scalar integral equation where |G(t,z)| ≤ ϕ(t)|z|, C  is convex, and . Related to this is the linear equation  and the resolvent equation . A...

Kernel-resolvent relations for an integral equation

Published Online: Nov 13, 2012

Page range: 25 - 40

DOI: https://doi.org/10.2478/v10127-011-0003-7

Keywords
integral equations, Liapunov functionals.

This content is open access.

Kernel-resolvent relations for an integral equation

Published Online: Nov 13, 2012

Page range: 25 - 40

DOI: https://doi.org/10.2478/v10127-011-0003-7

Keywordsintegral equations, Liapunov functionals.

This content is open access.

Keywords
integral equations, Liapunov functionals.