Integrated Fractional Resolvent Operator Function and Fractional Abstract Cauchy Problem
We firstly prove that β-times integrated α-resolvent operator function ((α,β)-ROF) satisfies a functional equation which extends that of β-times integrated semigroup and α-resolvent operator function. Secondly, for the inhomogeneous α-Cauchy problem cDtαu(t)=Au(t)+f(t), t∈(0,T), u(0)=x0, u'(0)=...
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| Format: | Article |
| Language: | English |
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Wiley
2014-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2014/430418 |
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| author | Ya-Ning Li Hong-Rui Sun |
| author_facet | Ya-Ning Li Hong-Rui Sun |
| author_sort | Ya-Ning Li |
| collection | DOAJ |
| description | We firstly prove that β-times integrated α-resolvent operator function ((α,β)-ROF) satisfies a functional equation which extends that of β-times integrated semigroup and α-resolvent operator function. Secondly, for the inhomogeneous α-Cauchy problem cDtαu(t)=Au(t)+f(t), t∈(0,T), u(0)=x0, u'(0)=x1, if A is the generator of an (α,β)-ROF, we give the relation between the function v(t)=Sα,β(t)x0+(g1*Sα,β)(t)x1+(gα-1*Sα,β*f)(t) and mild solution and classical solution of it. Finally, for the problem cDtαv(t)=Av(t)+gβ+1(t)x, t>0, v(k)(0)=0, k=0,1,…,N-1, where A is a linear closed operator. We show that A generates an exponentially bounded (α,β)-ROF on a Banach space X if and only if the problem has a unique exponentially bounded classical solution vx and Avx∈L
loc
1(ℝ+,X). Our results extend and generalize some related results in the literature. |
| format | Article |
| id | doaj-art-df1c86b592cb4e859346ca5695a23f25 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-df1c86b592cb4e859346ca5695a23f252025-02-03T05:44:59ZengWileyAbstract and Applied Analysis1085-33751687-04092014-01-01201410.1155/2014/430418430418Integrated Fractional Resolvent Operator Function and Fractional Abstract Cauchy ProblemYa-Ning Li0Hong-Rui Sun1School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, ChinaSchool of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, ChinaWe firstly prove that β-times integrated α-resolvent operator function ((α,β)-ROF) satisfies a functional equation which extends that of β-times integrated semigroup and α-resolvent operator function. Secondly, for the inhomogeneous α-Cauchy problem cDtαu(t)=Au(t)+f(t), t∈(0,T), u(0)=x0, u'(0)=x1, if A is the generator of an (α,β)-ROF, we give the relation between the function v(t)=Sα,β(t)x0+(g1*Sα,β)(t)x1+(gα-1*Sα,β*f)(t) and mild solution and classical solution of it. Finally, for the problem cDtαv(t)=Av(t)+gβ+1(t)x, t>0, v(k)(0)=0, k=0,1,…,N-1, where A is a linear closed operator. We show that A generates an exponentially bounded (α,β)-ROF on a Banach space X if and only if the problem has a unique exponentially bounded classical solution vx and Avx∈L loc 1(ℝ+,X). Our results extend and generalize some related results in the literature.http://dx.doi.org/10.1155/2014/430418 |
| spellingShingle | Ya-Ning Li Hong-Rui Sun Integrated Fractional Resolvent Operator Function and Fractional Abstract Cauchy Problem Abstract and Applied Analysis |
| title | Integrated Fractional Resolvent Operator Function and Fractional Abstract Cauchy Problem |
| title_full | Integrated Fractional Resolvent Operator Function and Fractional Abstract Cauchy Problem |
| title_fullStr | Integrated Fractional Resolvent Operator Function and Fractional Abstract Cauchy Problem |
| title_full_unstemmed | Integrated Fractional Resolvent Operator Function and Fractional Abstract Cauchy Problem |
| title_short | Integrated Fractional Resolvent Operator Function and Fractional Abstract Cauchy Problem |
| title_sort | integrated fractional resolvent operator function and fractional abstract cauchy problem |
| url | http://dx.doi.org/10.1155/2014/430418 |
| work_keys_str_mv | AT yaningli integratedfractionalresolventoperatorfunctionandfractionalabstractcauchyproblem AT hongruisun integratedfractionalresolventoperatorfunctionandfractionalabstractcauchyproblem |