American Institute of Mathematical Sciences

September  2014, 19(7): 2159-2168. doi: 10.3934/dcdsb.2014.19.2159

Inverse problems for singular differential-operator equations with higher order polar singularities

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

In this paper we study an inverse problem for strongly degenerate differential equations in Banach spaces. Projection method on suitable subspaces will be used to solve the given problem. A number of concrete applications to ordinary and partial differential equations is described.
Citation: Mohammed Al Horani, Angelo Favini. Inverse problems for singular differential-operator equations with higher order polar singularities. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2159-2168. doi: 10.3934/dcdsb.2014.19.2159
References:
 [1] M. Al Horani and A. Favini, Perturbation method for first and complete second order differential equations,, Preprint., ().   Google Scholar [2] M. Al Horani and A. Favini, An identification problem for first-order degenerate differential equations,, Journal of Optimization Theory and Applications, 130 (2006), 41.  doi: 10.1007/s10957-006-9083-y.  Google Scholar [3] R. Cross, A. Favini and Y. Yakubov, Perturbation results for multivalued linear operators,, Progress in Nonlinear Differential Equations and Their Applications, 80 (2011), 111.  doi: 10.1007/978-3-0348-0075-4_7.  Google Scholar [4] G. V. Demidenko and S. V. Uspenskii, Partial Differential Equations and Systems not Solvable With Respect to the Highest-Order Derivative,, Marcel Dekker, (2003).  doi: 10.1201/9780203911433.  Google Scholar [5] A. Favaron and A. Favini, Fractional powers and interpolation theory for multivalued linear operators and applications to degenerate differential equations,, Tsukuba J. Math., 35 (2011), 259.   Google Scholar [6] A. Faviniand and G. Marinoschi, Identification for degenerate problems of hyperbolic type,, Applicable Analysis, 91 (2012), 1511.  doi: 10.1080/00036811.2011.630665.  Google Scholar [7] A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces,, Marcel Dekker. Inc. New York, (1999).   Google Scholar [8] F. Kappel and H. W. Knobloch, Gewöhnliche Differentialgleichungen,, B. G. Teubner, (1974).   Google Scholar [9] A. E. Taylor, Introduction to Functional Analysis,, John Wiley & Sons, (1958).   Google Scholar [10] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators,, North-Holland, (1978).   Google Scholar [11] L. A. Vlasenko, Evolutionary Models with Implicit and Degenerate Differential Equations,, (rus.)- Dnepropetrovsk: System Technology, (2006).   Google Scholar [12] K. Yosida, Functional Analysis,, $6^{th}$ ed, (1980).   Google Scholar [13] S. Yakubov and Y. Yakubov, Differential-operator Equations. Ordinary and Partial Differential Equations,, Chapman & Hall, (2000).   Google Scholar

show all references

References:
 [1] M. Al Horani and A. Favini, Perturbation method for first and complete second order differential equations,, Preprint., ().   Google Scholar [2] M. Al Horani and A. Favini, An identification problem for first-order degenerate differential equations,, Journal of Optimization Theory and Applications, 130 (2006), 41.  doi: 10.1007/s10957-006-9083-y.  Google Scholar [3] R. Cross, A. Favini and Y. Yakubov, Perturbation results for multivalued linear operators,, Progress in Nonlinear Differential Equations and Their Applications, 80 (2011), 111.  doi: 10.1007/978-3-0348-0075-4_7.  Google Scholar [4] G. V. Demidenko and S. V. Uspenskii, Partial Differential Equations and Systems not Solvable With Respect to the Highest-Order Derivative,, Marcel Dekker, (2003).  doi: 10.1201/9780203911433.  Google Scholar [5] A. Favaron and A. Favini, Fractional powers and interpolation theory for multivalued linear operators and applications to degenerate differential equations,, Tsukuba J. Math., 35 (2011), 259.   Google Scholar [6] A. Faviniand and G. Marinoschi, Identification for degenerate problems of hyperbolic type,, Applicable Analysis, 91 (2012), 1511.  doi: 10.1080/00036811.2011.630665.  Google Scholar [7] A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces,, Marcel Dekker. Inc. New York, (1999).   Google Scholar [8] F. Kappel and H. W. Knobloch, Gewöhnliche Differentialgleichungen,, B. G. Teubner, (1974).   Google Scholar [9] A. E. Taylor, Introduction to Functional Analysis,, John Wiley & Sons, (1958).   Google Scholar [10] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators,, North-Holland, (1978).   Google Scholar [11] L. A. Vlasenko, Evolutionary Models with Implicit and Degenerate Differential Equations,, (rus.)- Dnepropetrovsk: System Technology, (2006).   Google Scholar [12] K. Yosida, Functional Analysis,, $6^{th}$ ed, (1980).   Google Scholar [13] S. Yakubov and Y. Yakubov, Differential-operator Equations. Ordinary and Partial Differential Equations,, Chapman & Hall, (2000).   Google Scholar
 [1] Viorel Barbu, Gabriela Marinoschi. An identification problem for a linear evolution equation in a banach space. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-12. doi: 10.3934/dcdss.2020081 [2] Alfredo Lorenzi, Ioan I. Vrabie. An identification problem for a linear evolution equation in a Banach space and applications. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 671-691. doi: 10.3934/dcdss.2011.4.671 [3] Atsushi Kawamoto. Hölder stability estimate in an inverse source problem for a first and half order time fractional diffusion equation. Inverse Problems & Imaging, 2018, 12 (2) : 315-330. doi: 10.3934/ipi.2018014 [4] Vladimir E. Fedorov, Natalia D. Ivanova. Identification problem for a degenerate evolution equation with overdetermination on the solution semigroup kernel. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 687-696. doi: 10.3934/dcdss.2016022 [5] 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 [6] Rehana Naz, Fazal M Mahomed, Azam Chaudhry. First integrals of Hamiltonian systems: The inverse problem. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020121 [7] M. Nakamura, Tohru Ozawa. The Cauchy problem for nonlinear wave equations in the Sobolev space of critical order. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 215-231. doi: 10.3934/dcds.1999.5.215 [8] Nguyen Huy Tuan, Mokhtar Kirane, Long Dinh Le, Van Thinh Nguyen. On an inverse problem for fractional evolution equation. Evolution Equations & Control Theory, 2017, 6 (1) : 111-134. doi: 10.3934/eect.2017007 [9] Fabio Camilli, Francisco Silva. A semi-discrete approximation for a first order mean field game problem. Networks & Heterogeneous Media, 2012, 7 (2) : 263-277. doi: 10.3934/nhm.2012.7.263 [10] Monika Laskawy. Optimality conditions of the first eigenvalue of a fourth order Steklov problem. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1843-1859. doi: 10.3934/cpaa.2017089 [11] Alfredo Lorenzi, Eugenio Sinestrari. An identification problem for a nonlinear one-dimensional wave equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5253-5271. doi: 10.3934/dcds.2013.33.5253 [12] Luciano Pandolfi. Riesz systems, spectral controllability and a source identification problem for heat equations with memory. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 745-759. doi: 10.3934/dcdss.2011.4.745 [13] Valter Pohjola. An inverse problem for the magnetic Schrödinger operator on a half space with partial data. Inverse Problems & Imaging, 2014, 8 (4) : 1169-1189. doi: 10.3934/ipi.2014.8.1169 [14] Gen Nakamura, Michiyuki Watanabe. An inverse boundary value problem for a nonlinear wave equation. Inverse Problems & Imaging, 2008, 2 (1) : 121-131. doi: 10.3934/ipi.2008.2.121 [15] Gisella Croce. An elliptic problem with degenerate coercivity and a singular quadratic gradient lower order term. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 507-530. doi: 10.3934/dcdss.2012.5.507 [16] Ciro D'Apice, Olha P. Kupenko, Rosanna Manzo. On boundary optimal control problem for an arterial system: First-order optimality conditions. Networks & Heterogeneous Media, 2018, 13 (4) : 585-607. doi: 10.3934/nhm.2018027 [17] Martino Bardi, Paola Mannucci. On the Dirichlet problem for non-totally degenerate fully nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2006, 5 (4) : 709-731. doi: 10.3934/cpaa.2006.5.709 [18] Yanbo Hu, Tong Li. The regularity of a degenerate Goursat problem for the 2-D isothermal Euler equations. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3317-3336. doi: 10.3934/cpaa.2019149 [19] Michael V. Klibanov. A phaseless inverse scattering problem for the 3-D Helmholtz equation. Inverse Problems & Imaging, 2017, 11 (2) : 263-276. doi: 10.3934/ipi.2017013 [20] Kenichi Sakamoto, Masahiro Yamamoto. Inverse source problem with a final overdetermination for a fractional diffusion equation. Mathematical Control & Related Fields, 2011, 1 (4) : 509-518. doi: 10.3934/mcrf.2011.1.509

2018 Impact Factor: 1.008