Spectral Criteria for Solvability of Boundary Value Problems and Positivity of Solutions of Time-Fractional Differential Equations
We investigate mild solutions of the fractional order nonhomogeneous Cauchy problem Dtαu(t)=Au(t)+f(t), t>0, where 0<α<1. When A is the generator of a C0-semigroup (T(t))t≥0 on a Banach space X, we obtain an explicit representation of mild solutions of the above problem in terms of the sem...
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Wiley
2013-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2013/614328 |
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| author | Valentin Keyantuo Carlos Lizama Mahamadi Warma |
| author_facet | Valentin Keyantuo Carlos Lizama Mahamadi Warma |
| author_sort | Valentin Keyantuo |
| collection | DOAJ |
| description | We investigate mild solutions of the fractional order nonhomogeneous Cauchy problem Dtαu(t)=Au(t)+f(t), t>0, where 0<α<1. When A is the generator of a C0-semigroup (T(t))t≥0 on a Banach space X, we obtain an explicit representation of mild solutions of the above problem in terms of the semigroup. We then prove that this problem under the boundary condition u(0)=u(1) admits a unique mild solution for each f∈C([0,1];X) if and only if the operator I-Sα(1) is invertible. Here, we use the representation Sα(t)x=∫0∞Φα(s)T(stα)x ds, t>0 in which Φα is a Wright type function. For the first order case, that is, α=1, the corresponding result was proved by Prüss in 1984. In case X is a Banach lattice and the semigroup (T(t))t≥0 is positive, we obtain existence of solutions of the semilinear problem Dtαu(t)=Au(t)+f(t,u(t)), t>0, 0<α<1. |
| format | Article |
| id | doaj-art-54aad13bc93b4f9c8233b464164b996c |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
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| series | Abstract and Applied Analysis |
| spelling | doaj-art-54aad13bc93b4f9c8233b464164b996c2025-02-03T01:01:07ZengWileyAbstract and Applied Analysis1085-33751687-04092013-01-01201310.1155/2013/614328614328Spectral Criteria for Solvability of Boundary Value Problems and Positivity of Solutions of Time-Fractional Differential EquationsValentin Keyantuo0Carlos Lizama1Mahamadi Warma2Department of Mathematics, Faculty of Natural Sciences, University of Puerto Rico, Rio Piedras Campus, P.O. Box 70377, San Juan, PR 00936-8377, USADepartamento de Matemática, Facultad de Ciencias, Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, ChileDepartment of Mathematics, Faculty of Natural Sciences, University of Puerto Rico, Rio Piedras Campus, P.O. Box 70377, San Juan, PR 00936-8377, USAWe investigate mild solutions of the fractional order nonhomogeneous Cauchy problem Dtαu(t)=Au(t)+f(t), t>0, where 0<α<1. When A is the generator of a C0-semigroup (T(t))t≥0 on a Banach space X, we obtain an explicit representation of mild solutions of the above problem in terms of the semigroup. We then prove that this problem under the boundary condition u(0)=u(1) admits a unique mild solution for each f∈C([0,1];X) if and only if the operator I-Sα(1) is invertible. Here, we use the representation Sα(t)x=∫0∞Φα(s)T(stα)x ds, t>0 in which Φα is a Wright type function. For the first order case, that is, α=1, the corresponding result was proved by Prüss in 1984. In case X is a Banach lattice and the semigroup (T(t))t≥0 is positive, we obtain existence of solutions of the semilinear problem Dtαu(t)=Au(t)+f(t,u(t)), t>0, 0<α<1.http://dx.doi.org/10.1155/2013/614328 |
| spellingShingle | Valentin Keyantuo Carlos Lizama Mahamadi Warma Spectral Criteria for Solvability of Boundary Value Problems and Positivity of Solutions of Time-Fractional Differential Equations Abstract and Applied Analysis |
| title | Spectral Criteria for Solvability of Boundary Value Problems and Positivity of Solutions of Time-Fractional Differential Equations |
| title_full | Spectral Criteria for Solvability of Boundary Value Problems and Positivity of Solutions of Time-Fractional Differential Equations |
| title_fullStr | Spectral Criteria for Solvability of Boundary Value Problems and Positivity of Solutions of Time-Fractional Differential Equations |
| title_full_unstemmed | Spectral Criteria for Solvability of Boundary Value Problems and Positivity of Solutions of Time-Fractional Differential Equations |
| title_short | Spectral Criteria for Solvability of Boundary Value Problems and Positivity of Solutions of Time-Fractional Differential Equations |
| title_sort | spectral criteria for solvability of boundary value problems and positivity of solutions of time fractional differential equations |
| url | http://dx.doi.org/10.1155/2013/614328 |
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