December  2020, 13(12): 3285-3304. doi: 10.3934/dcdss.2020240

Fractional Cauchy problems for infinite interval case

1. 

Department of Mathematics, The University of Jordan, Amman, Jordan

2. 

Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato, 5 - 40126 Bologna, Italy

3. 

Takarazuka, Hirai Sanso 12-13,665-0817, Japan

* Corresponding author: Mohammed Al Horani

Received  December 2018 Revised  October 2019 Published  December 2020 Early access  January 2020

We are devoted with fractional abstract Cauchy problems. Required conditions on spaces and operators are given guaranteeing existence and uniqueness of solutions. An inverse problem is also studied. Applications from partial differential equations are given to illustrate the abstract fractional degenerate differential problems.

Citation: Mohammed Al Horani, Mauro Fabrizio, Angelo Favini, Hiroki Tanabe. Fractional Cauchy problems for infinite interval case. Discrete and Continuous Dynamical Systems - S, 2020, 13 (12) : 3285-3304. doi: 10.3934/dcdss.2020240
References:
[1]

M. Al HoraniA. Favini and H. Tanabe, Direct and inverse fractional abstract Cauchy problems, Mathematics, 7 (2019), 1-9. 

[2]

M. Al Horani, M. Fabrizio, A. Favini and H. Tanabe, Fractional Cauchy problems and applications, Discrete & Continuous Dynamical Systems-Series S, to appear.

[3]

M. Al HoraniM. FabrizioA. Favini and H. Tanabe, Direct and inverse problems for degenerate differential equations, Ann. Univ. Ferrara, 64 (2018), 227-241.  doi: 10.1007/s11565-018-0303-9.

[4]

E. G. Bazhlekova,, Fractional Evolution Equations in Banach Spaces, Eindhoven University of Technology, 2001.

[5]

A. Favaron, A. Favini and H. Tanabe, Perturbation methods for inverse problems on degenerate differential equations, preprint.

[6]

A. Favini, A. Lorenzi, G. Marinoschi and H. Tanabe, Perturbation methods and identification problems for degenerate evolution systems, Advances in Mathematics, Ed. Acad. Române, Bucharest, (2013), 145-156.

[7]

A. FaviniA. Lorenzi and H. Tanabe, Degenerate integro-differential equations of parabolic type with Robin boundary conditions, Journal of Mathematical Analysis and Applications, 447 (2017), 579-665.  doi: 10.1016/j.jmaa.2016.10.029.

[8]

A. FaviniA. Lorenzi and H. Tanabe, Direct and inverse degenerate parabolic differential equations with multi-valued operators, Electron. J. Diff. Equ., 2015 (2015), 1-22. 

[9]

A. FaviniA. Lorenzi and H. Tanabe, Singular integro-differential equations of parabolic type, Advances in Differential Equations, 7 (2002), 769-798. 

[10]

A. Favini and H. Tanabe, Degenerate differential equations of parabolic type and inverse problems, Proceeding, Seminar on Partial Differential Equations, Osaka University, Osaka (2015), 89-100.

[11]

A. Favini and A. Yagi, Multivalued linear operators and degenerate evolution equations, Annali Di Matematica Pura ed Applicata, 163 (1993), 353-384.  doi: 10.1007/BF01759029.

[12]

A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, Marcel Dekker. Inc. New York, 1999.

[13]

V. Fedorov and N. D. Ivanova, Identification problem for degenerate evolution equations of fractional order, Fractional Calculus and Applied Analysis, 20 (2017), 706-721. 

[14]

D. Guidetti, On maximal regularity for the Cauchy-Dirichlet mixed parabolic problem with fractional time derivative, Bruno Pini Mathematical Analysis Seminar, 9 (2018), 147-157. 

[15]

T. Kato, Fractional powers of dissipative operators, J. Math. Soc. Japan, 13 (1961), 246-274.  doi: 10.2969/jmsj/01330246.

[16]

J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems And Applications, Springer-Verlag, Berlin, 1972.

[17]

J. L. Lions and J. Peetre, Sur une classe d'espaces d'interpolation, Publications Mathmatiques de l'IHES, 19 (1964), 5-68. 

[18]

A. Lorenzi, An Introduction to Identification Problems Via Functional Analysis, VSP, Utrecht, The Netherland, 2001.

[19]

G. A. Sviridyuk and V. E. Fedorov, Linear Sobolev Type Equations and Degenerate Semigroups of Operators, VSP, Utrecht, Boston, 2003.

[20]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amesterdam, 1978.

show all references

References:
[1]

M. Al HoraniA. Favini and H. Tanabe, Direct and inverse fractional abstract Cauchy problems, Mathematics, 7 (2019), 1-9. 

[2]

M. Al Horani, M. Fabrizio, A. Favini and H. Tanabe, Fractional Cauchy problems and applications, Discrete & Continuous Dynamical Systems-Series S, to appear.

[3]

M. Al HoraniM. FabrizioA. Favini and H. Tanabe, Direct and inverse problems for degenerate differential equations, Ann. Univ. Ferrara, 64 (2018), 227-241.  doi: 10.1007/s11565-018-0303-9.

[4]

E. G. Bazhlekova,, Fractional Evolution Equations in Banach Spaces, Eindhoven University of Technology, 2001.

[5]

A. Favaron, A. Favini and H. Tanabe, Perturbation methods for inverse problems on degenerate differential equations, preprint.

[6]

A. Favini, A. Lorenzi, G. Marinoschi and H. Tanabe, Perturbation methods and identification problems for degenerate evolution systems, Advances in Mathematics, Ed. Acad. Române, Bucharest, (2013), 145-156.

[7]

A. FaviniA. Lorenzi and H. Tanabe, Degenerate integro-differential equations of parabolic type with Robin boundary conditions, Journal of Mathematical Analysis and Applications, 447 (2017), 579-665.  doi: 10.1016/j.jmaa.2016.10.029.

[8]

A. FaviniA. Lorenzi and H. Tanabe, Direct and inverse degenerate parabolic differential equations with multi-valued operators, Electron. J. Diff. Equ., 2015 (2015), 1-22. 

[9]

A. FaviniA. Lorenzi and H. Tanabe, Singular integro-differential equations of parabolic type, Advances in Differential Equations, 7 (2002), 769-798. 

[10]

A. Favini and H. Tanabe, Degenerate differential equations of parabolic type and inverse problems, Proceeding, Seminar on Partial Differential Equations, Osaka University, Osaka (2015), 89-100.

[11]

A. Favini and A. Yagi, Multivalued linear operators and degenerate evolution equations, Annali Di Matematica Pura ed Applicata, 163 (1993), 353-384.  doi: 10.1007/BF01759029.

[12]

A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, Marcel Dekker. Inc. New York, 1999.

[13]

V. Fedorov and N. D. Ivanova, Identification problem for degenerate evolution equations of fractional order, Fractional Calculus and Applied Analysis, 20 (2017), 706-721. 

[14]

D. Guidetti, On maximal regularity for the Cauchy-Dirichlet mixed parabolic problem with fractional time derivative, Bruno Pini Mathematical Analysis Seminar, 9 (2018), 147-157. 

[15]

T. Kato, Fractional powers of dissipative operators, J. Math. Soc. Japan, 13 (1961), 246-274.  doi: 10.2969/jmsj/01330246.

[16]

J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems And Applications, Springer-Verlag, Berlin, 1972.

[17]

J. L. Lions and J. Peetre, Sur une classe d'espaces d'interpolation, Publications Mathmatiques de l'IHES, 19 (1964), 5-68. 

[18]

A. Lorenzi, An Introduction to Identification Problems Via Functional Analysis, VSP, Utrecht, The Netherland, 2001.

[19]

G. A. Sviridyuk and V. E. Fedorov, Linear Sobolev Type Equations and Degenerate Semigroups of Operators, VSP, Utrecht, Boston, 2003.

[20]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amesterdam, 1978.

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