On a Functional Equation Associated with (a,k)-Regularized Resolvent Families
Let a∈Lloc1(ℝ+) and k∈C(ℝ+) be given. In this paper, we study the functional equation R(s)(a*R)(t)-(a*R)(s)R(t)=k(s)(a*R)(t)-k(t)(a*R)(s), for bounded operator valued functions R(t) defined on the positive real line ℝ+. We show that, under some natural assumptions on a(·) and k(·), every solution of...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
2012-01-01
|
| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2012/495487 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1832546239337463808 |
|---|---|
| author | Carlos Lizama Felipe Poblete |
| author_facet | Carlos Lizama Felipe Poblete |
| author_sort | Carlos Lizama |
| collection | DOAJ |
| description | Let a∈Lloc1(ℝ+) and k∈C(ℝ+) be given. In this paper, we study the functional equation R(s)(a*R)(t)-(a*R)(s)R(t)=k(s)(a*R)(t)-k(t)(a*R)(s), for bounded operator valued functions R(t) defined on the positive real line ℝ+. We show that, under some natural assumptions on a(·) and k(·), every solution of the above mentioned functional equation gives rise to a commutative (a,k)-resolvent family R(t) generated by Ax=lim t→0+(R(t)x-k(t)x/(a*k)(t)) defined on the domain D(A):={x∈X:lim t→0+(R(t)x-k(t)x/(a*k)(t)) exists in X} and, conversely, that each (a,k)-resolvent family R(t) satisfy the above mentioned functional equation. In particular, our study produces new functional equations that characterize semigroups, cosine operator families, and a class of operator families in between them that, in turn, are in one to one correspondence with the well-posedness of abstract fractional Cauchy problems. |
| format | Article |
| id | doaj-art-f7db4c5c83fa4b6c9aa7114dcb356517 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-f7db4c5c83fa4b6c9aa7114dcb3565172025-02-03T07:23:32ZengWileyAbstract and Applied Analysis1085-33751687-04092012-01-01201210.1155/2012/495487495487On a Functional Equation Associated with (a,k)-Regularized Resolvent FamiliesCarlos Lizama0Felipe Poblete1Departamento de Matemática y Ciencia de la Computación, Facultad de Ciencias, Universidad de Santiago de Chile, Casilla 307, Correo 2, 9160000 Santiago, ChileFacultad de Ciencias, Universidad de Chile, Las Palmeras 3425, 7810000 Santiago, ChileLet a∈Lloc1(ℝ+) and k∈C(ℝ+) be given. In this paper, we study the functional equation R(s)(a*R)(t)-(a*R)(s)R(t)=k(s)(a*R)(t)-k(t)(a*R)(s), for bounded operator valued functions R(t) defined on the positive real line ℝ+. We show that, under some natural assumptions on a(·) and k(·), every solution of the above mentioned functional equation gives rise to a commutative (a,k)-resolvent family R(t) generated by Ax=lim t→0+(R(t)x-k(t)x/(a*k)(t)) defined on the domain D(A):={x∈X:lim t→0+(R(t)x-k(t)x/(a*k)(t)) exists in X} and, conversely, that each (a,k)-resolvent family R(t) satisfy the above mentioned functional equation. In particular, our study produces new functional equations that characterize semigroups, cosine operator families, and a class of operator families in between them that, in turn, are in one to one correspondence with the well-posedness of abstract fractional Cauchy problems.http://dx.doi.org/10.1155/2012/495487 |
| spellingShingle | Carlos Lizama Felipe Poblete On a Functional Equation Associated with (a,k)-Regularized Resolvent Families Abstract and Applied Analysis |
| title | On a Functional Equation Associated with (a,k)-Regularized Resolvent Families |
| title_full | On a Functional Equation Associated with (a,k)-Regularized Resolvent Families |
| title_fullStr | On a Functional Equation Associated with (a,k)-Regularized Resolvent Families |
| title_full_unstemmed | On a Functional Equation Associated with (a,k)-Regularized Resolvent Families |
| title_short | On a Functional Equation Associated with (a,k)-Regularized Resolvent Families |
| title_sort | on a functional equation associated with a k regularized resolvent families |
| url | http://dx.doi.org/10.1155/2012/495487 |
| work_keys_str_mv | AT carloslizama onafunctionalequationassociatedwithakregularizedresolventfamilies AT felipepoblete onafunctionalequationassociatedwithakregularizedresolventfamilies |