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...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2012-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2012/495487 |
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| Summary: | 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. |
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| ISSN: | 1085-3375 1687-0409 |