American Institute of Mathematical Sciences

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September  2014, 3(3): 355-361. doi: 10.3934/eect.2014.3.355

First-order inverse evolution equations

Mohammed Al Horani 1, and  Angelo Favini 2,

1. 

Department of Mathematics, The University of Jordan, Amman
2. 

Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta S. Donato, 5, 40126 Bologna

Received  May 2013 Revised  April 2014 Published  August 2014

We are concerned with an inverse problem for first order linear evolution equations. We indicate sufficient conditions for existence and uniqueness of a solution to these problems. All the results apply well to inverse problems for equations from mathematical physics. Indeed, as a possible application of the abstract theorems, some examples of partial differential equations are given.
Keywords: Inverse problem, analytic semigroup., $c_{0}-$semigroup, first-order problem.
Mathematics Subject Classification: Primary: 34G10; Secondary: 34A5.
Citation: Mohammed Al Horani, Angelo Favini. First-order inverse evolution equations. Evolution Equations & Control Theory, 2014, 3 (3) : 355-361. doi: 10.3934/eect.2014.3.355
References:
[1]

M. Al Horani, An identification problem for some degenerate differential equations, Matematiche (Catania), 57 (2002), 217-227.  Google Scholar

[2]

M. Al Horani, Projection method for solving degenerate first-order identification problem, J. Math. Anal. Appl., 364 (2010), 204-208. doi: 10.1016/j.jmaa.2009.10.033.  Google Scholar

[3]

M. Al Horani and A. Favini, An identification problem for first-order degenerate differential equations, Journal of Optimization Theory and Applications, 130 (2006), 41-60. doi: 10.1007/s10957-006-9083-y.  Google Scholar

[4]

M. Al Horani and A. Favini, Degenerate first-order identification problems in Banach spaces,, Differential Equations, (): 1.  doi: 10.1201/9781420011135.ch1.  Google Scholar

[5]

M. Al Horani and A. Favini, Perturbation method for first and complete second order differential equations,, Preprint., ().   Google Scholar

[6]

W. Desch and W. Schappacher, On relatively bounded perturbations of linear $C_0$-semigroups, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 11 (1984), 327-341.  Google Scholar

[7]

N. Dunford and T. Schwarz, Linear Operators, I, Wiley (Interscience), New York, 1958. Google Scholar

[8]

K.-J. Engel and R. Nagel, One Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Math., Springer-Verlag, Berlin-Heidelgerg-New York, 2000.  Google Scholar

[9]

A. Favini and A. Lorenzi, Identification problems for singular integro-differential equations of parabolic type I, Dynamics of Continuous, Discrete and Impulsive Systems, Series A; Mathematical Analysis, 12 (2005), 303-328.  Google Scholar

[10]

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

[11]

A. Favini and S. Romanelli, Analicity semigroups on $C[0,1]$ generated by some classes of second order differential equations, Semigroup Forum, 56 (1998), 362-372.  Google Scholar

[12]

G. Greiner, Spectral properties and asymptotic behavior of the linear transport equation, Math. Zeit., 185 (1984), 167-177. doi: 10.1007/BF01181687.  Google Scholar

[13]

A. Lorenzi, An Introduction to Identification Problems Via Functional Analysis, VSP, Utrecht, The Netherland, 2001. doi: 10.1515/9783110940923.  Google Scholar

[14]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, $1^{ist}$ ed, Birkhäuser, Basel, 1995.  Google Scholar

[15]

I. Prilepko, G. Orlovsky and A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, Marcel Dekker. Inc., New York, 2000.  Google Scholar

[16]

E. Sinestrari, Wave equation with memory, Discrete Contin. Dynam. Systems, 5 (1999), 881-896. doi: 10.3934/dcds.1999.5.881.  Google Scholar

[17]

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

show all references

References:
[1]

M. Al Horani, An identification problem for some degenerate differential equations, Matematiche (Catania), 57 (2002), 217-227.  Google Scholar

[2]

M. Al Horani, Projection method for solving degenerate first-order identification problem, J. Math. Anal. Appl., 364 (2010), 204-208. doi: 10.1016/j.jmaa.2009.10.033.  Google Scholar

[3]

M. Al Horani and A. Favini, An identification problem for first-order degenerate differential equations, Journal of Optimization Theory and Applications, 130 (2006), 41-60. doi: 10.1007/s10957-006-9083-y.  Google Scholar

[4]

M. Al Horani and A. Favini, Degenerate first-order identification problems in Banach spaces,, Differential Equations, (): 1.  doi: 10.1201/9781420011135.ch1.  Google Scholar

[5]

M. Al Horani and A. Favini, Perturbation method for first and complete second order differential equations,, Preprint., ().   Google Scholar

[6]

W. Desch and W. Schappacher, On relatively bounded perturbations of linear $C_0$-semigroups, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 11 (1984), 327-341.  Google Scholar

[7]

N. Dunford and T. Schwarz, Linear Operators, I, Wiley (Interscience), New York, 1958. Google Scholar

[8]

K.-J. Engel and R. Nagel, One Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Math., Springer-Verlag, Berlin-Heidelgerg-New York, 2000.  Google Scholar

[9]

A. Favini and A. Lorenzi, Identification problems for singular integro-differential equations of parabolic type I, Dynamics of Continuous, Discrete and Impulsive Systems, Series A; Mathematical Analysis, 12 (2005), 303-328.  Google Scholar

[10]

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

[11]

A. Favini and S. Romanelli, Analicity semigroups on $C[0,1]$ generated by some classes of second order differential equations, Semigroup Forum, 56 (1998), 362-372.  Google Scholar

[12]

G. Greiner, Spectral properties and asymptotic behavior of the linear transport equation, Math. Zeit., 185 (1984), 167-177. doi: 10.1007/BF01181687.  Google Scholar

[13]

A. Lorenzi, An Introduction to Identification Problems Via Functional Analysis, VSP, Utrecht, The Netherland, 2001. doi: 10.1515/9783110940923.  Google Scholar

[14]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, $1^{ist}$ ed, Birkhäuser, Basel, 1995.  Google Scholar

[15]

I. Prilepko, G. Orlovsky and A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, Marcel Dekker. Inc., New York, 2000.  Google Scholar

[16]

E. Sinestrari, Wave equation with memory, Discrete Contin. Dynam. Systems, 5 (1999), 881-896. doi: 10.3934/dcds.1999.5.881.  Google Scholar

[17]

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

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