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doi: 10.3934/eect.2021051
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## Singular integro-differential equations with applications

 1 Department of Mathematics, The University of Jordan, Amman, 11942, 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 (horani@ju.edu.jo)

Received  January 2021 Revised  August 2021 Early access September 2021

We are devoted with singular integro-differential abstract Cauchyproblems. Required conditions on spaces and operators are givenguaranteeing existence and uniqueness of solutions. Applications from partial differential equations are given to illustrate the abstract singular integro-differential problem.

Citation: Mohammed Al Horani, Angelo Favini, Hiroki Tanabe. Singular integro-differential equations with applications. Evolution Equations and Control Theory, doi: 10.3934/eect.2021051
##### References:
 [1] M. Al Horani, M. Fabrizio, A. 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. [2] M. Al Horani, M. Fabrizio, A. Favini and H. Tanabe, Identification problems for degenerate integro-differential equations, In Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs, (eds. P. Colli, A. Favini, E. Rocca, G. Schimperna and J. Sprekels), Springer INDAM Series, 22 (2017), 55–75. [3] K. Balachandran and S. Kiruthika, Existence of solutions of abstract fractional integrodifferential equations of Sobolev type, Comput. Math. Appl., 64 (2012), 3406-3413.  doi: 10.1016/j.camwa.2011.12.051. [4] T. Binz and K. J. Engel, Operators with Wentzell boundary conditions and the Dirichlet-to-Neumann operator, Math. Nachr., 292 (2018), 733-746.  doi: 10.1002/mana.201800064. [5] A. Favaron and A. Favini, On the behaviour of singular semigroups in intermediate and interpolation spaces and its applications to maximal regularity for degenerate integro-differential evolution equations, Abstr. Appl. Anal., 2013 (2013), 1-37.  doi: 10.1155/2013/275494. [6] 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), 259-323.  doi: 10.21099/tkbjm/1331658708. [7] A. Favaron, Perturbation methods for inverse problems on degenerate differential equations, preprint, (2012), 83–103 [8] A. Favini, A. Lorenzi and H. Tanabe, Degenerate integrodifferential equations of parabolic type with Robin boundary conditions: $L^{p}-$theory, J. Math. Anal. Appl., 447 (2017), 579-665.  doi: 10.1016/j.jmaa.2016.10.029. [9] A. Favini, A. Lorenzi and H. Tanabe, Direct and inverse degenerate parabolic differential equations with multi-valued operators, Electronic J. Differential Equations, (2015), 22pp. [10] A. Favini, A. Lorenzi and H. Tanabe, Singular evolution integro-differential equations with kernels defined on bounded intervals, Appl. Anal., 84 (2005), 463-497.  doi: 10.1080/00036810410001724418. [11] A. Favini, A. Lorenzi and H. Tanabe, Singular integro-differential equations of parabolic type, Adv. Differential Equations, 7 (2002), 769-798. [12] 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. [13] A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, Marcel Dekker. Inc., New York, 1999. [14] O. A. Oleinik and E. V. Radkevich, The method of introducing a parameter in the study of evolutionary equations, Russian Mathematical Surveys, 33 (1978), 7-84. [15] O. A. Oleinik and E. V. Radkevich, Second order equations with nonnegative characteristic form, Mat. Anal., (1969), 7–252.

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##### References:
 [1] M. Al Horani, M. Fabrizio, A. 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. [2] M. Al Horani, M. Fabrizio, A. Favini and H. Tanabe, Identification problems for degenerate integro-differential equations, In Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs, (eds. P. Colli, A. Favini, E. Rocca, G. Schimperna and J. Sprekels), Springer INDAM Series, 22 (2017), 55–75. [3] K. Balachandran and S. Kiruthika, Existence of solutions of abstract fractional integrodifferential equations of Sobolev type, Comput. Math. Appl., 64 (2012), 3406-3413.  doi: 10.1016/j.camwa.2011.12.051. [4] T. Binz and K. J. Engel, Operators with Wentzell boundary conditions and the Dirichlet-to-Neumann operator, Math. Nachr., 292 (2018), 733-746.  doi: 10.1002/mana.201800064. [5] A. Favaron and A. Favini, On the behaviour of singular semigroups in intermediate and interpolation spaces and its applications to maximal regularity for degenerate integro-differential evolution equations, Abstr. Appl. Anal., 2013 (2013), 1-37.  doi: 10.1155/2013/275494. [6] 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), 259-323.  doi: 10.21099/tkbjm/1331658708. [7] A. Favaron, Perturbation methods for inverse problems on degenerate differential equations, preprint, (2012), 83–103 [8] A. Favini, A. Lorenzi and H. Tanabe, Degenerate integrodifferential equations of parabolic type with Robin boundary conditions: $L^{p}-$theory, J. Math. Anal. Appl., 447 (2017), 579-665.  doi: 10.1016/j.jmaa.2016.10.029. [9] A. Favini, A. Lorenzi and H. Tanabe, Direct and inverse degenerate parabolic differential equations with multi-valued operators, Electronic J. Differential Equations, (2015), 22pp. [10] A. Favini, A. Lorenzi and H. Tanabe, Singular evolution integro-differential equations with kernels defined on bounded intervals, Appl. Anal., 84 (2005), 463-497.  doi: 10.1080/00036810410001724418. [11] A. Favini, A. Lorenzi and H. Tanabe, Singular integro-differential equations of parabolic type, Adv. Differential Equations, 7 (2002), 769-798. [12] 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. [13] A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, Marcel Dekker. Inc., New York, 1999. [14] O. A. Oleinik and E. V. Radkevich, The method of introducing a parameter in the study of evolutionary equations, Russian Mathematical Surveys, 33 (1978), 7-84. [15] O. A. Oleinik and E. V. Radkevich, Second order equations with nonnegative characteristic form, Mat. Anal., (1969), 7–252.
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