Abstract
In this manuscript, we examine an inverse problem for abstract semilinear differential inclusions. We identify the unknown parameter using the given overdetermined condition. We establish the existence of mild solution to the considered inverse problem. Further, the continuous dependence of the solution to the initial data is discussed.The main tools to obtain the results are Banach fixed point theorem, Grönwall’s inequality, set-valued theory and semigroup theory. Moreover, we provide several examples and some graphical illustrations to validate our findings.
Funding source: National Board of Higher Mathematics
Award Identifier / Grant number: 02011/23/2021 NBHM (R.P)/RDII/8776
Funding statement: The first author’s Ph.D. research is funded by the National Board of Higher Mathematics (NBHM) through project grant number 02011/23/2021 NBHM (R.P)/RDII/8776.
References
[1] M. Al Horani and A. Favini, First-order inverse evolution equations, Evol. Equ. Control Theory 3 (2014), no. 3, 355–361. 10.3934/eect.2014.3.355Search in Google Scholar
[2] J. P. Aubin and H. Frankowska, Differential Inclusions, Birkhäuser, Boston, 2009. 10.1007/978-0-8176-4848-0_10Search in Google Scholar
[3] F. Awawdeh, Perturbation method for abstract second-order inverse problems, Nonlinear Anal. 72 (2010), no. 3–4, 1379–1386. 10.1016/j.na.2009.08.021Search in Google Scholar
[4] V. Barbu, Differential Equations, Springer Undergrad. Math. Ser., Springer, Cham, 2016. 10.1007/978-3-319-45261-6Search in Google Scholar
[5] V. Barbu and G. Marinoschi, An identification problem for a linear evolution equation in a Banach space, Discrete Contin. Dyn. Syst. Ser. S 13 (2020), no. 5, 1429–1440. 10.3934/dcdss.2020081Search in Google Scholar
[6] T. Cardinali and P. Rubbioni, On the existence of mild solutions of semilinear evolution differential inclusions, J. Math. Anal. Appl. 308 (2005), no. 2, 620–635. 10.1016/j.jmaa.2004.11.049Search in Google Scholar
[7] M. Cichoń, Differential inclusions and abstract control problems, Bull. Aust. Math. Soc. 53 (1996), no. 1, 109–122. 10.1017/S0004972700016774Search in Google Scholar
[8] G. Di Blasio and A. Lorenzi, Identification problems for parabolic delay differential equations with measurement on the boundary, J. Inverse Ill-Posed Probl. 15 (2007), no. 7, 709–734. 10.1515/jiip.2007.039Search in Google Scholar
[9] V. E. Fedorov and N. D. Ivanova, Identification problem for degenerate evolution equations of fractional order, Fract. Calc. Appl. Anal. 20 (2017), no. 3, 706–721. 10.1515/fca-2017-0037Search in Google Scholar
[10] V. E. Fedorov, A. V. Nagumanova and M. Kostić, A class of inverse problems for fractional order degenerate evolution equations, J. Inverse Ill-Posed Probl. 29 (2021), no. 2, 173–184. 10.1515/jiip-2017-0099Search in Google Scholar
[11] A.-G. M. Ibrahim, On the differentiability of set-valued functions defined on a Banach space and mean value theorem, Appl. Math. Comput. 74 (1996), no. 1, 79–94. 10.1016/0096-3003(95)00095-XSearch in Google Scholar
[12] V. K. Ivanov, V. V. Vasin and V. P. Tanana, Theory of Linear Ill-Posed Problems and its Applications. Vol. 36, Walter de Gruyter Berlin, 2013. Search in Google Scholar
[13] M. Kamenskii, V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, De Gruyter Ser. Nonlinear Anal. Appl. 7, Walter de Gruyter, Berlin, 2001. 10.1515/9783110870893Search in Google Scholar
[14] S. Kubo, Inverse Problems, Atlanta Technology, Dordrecht, 1992. Search in Google Scholar
[15] R. I. Leine and H. Nijmeijer, Dynamics and Bifurcations of Non-Smooth Mechanical Systems, Lect. Notes Appl. Comput. Mech. 18, Springer, Berlin, 2013. Search in Google Scholar
[16] E. N. Mahmudov, Approximation and Optimization of Discrete and Differential Inclusions, Elsevier, Amsterdam, 2011. 10.1016/B978-0-12-388428-2.00006-0Search in Google Scholar
[17] M. Malik and S. Ruhil, Inverse problem for an abstract neutral differential equation of Sobolev-type, Results Control Optim. 11 (2023), Article ID 100235. 10.1016/j.rico.2023.100235Search in Google Scholar
[18] M. Malik, S. Ruhil and R. Dhayal, An investigation of an inverse problem for second-order abstract differential equation, Indian J. Pure Appl. Math. 56 (2025), no. 2, 512–525. 10.1007/s13226-023-00498-9Search in Google Scholar
[19] A. Mohamad-Djafari, Inverse Problems in Vision and 3D Tomography, John Wiley & Sons, New York, 2013. 10.1002/9781118603864Search in Google Scholar
[20] M. Muslim, Approximation of solutions to history-valued neutral functional differential equations, Comput. Math. Appl. 51 (2006), no. 3–4, 537–550. 10.1016/j.camwa.2005.07.013Search in Google Scholar
[21] D. O’Regan, Existence Theory for Nonlinear Ordinary Differential Equations, Math. Appl. 398, Kluwer Academic, Dordrecht, 1997. 10.1007/978-94-017-1517-1Search in Google Scholar
[22] Z. Pizlo, Perception viewed as an inverse problem, Vision Res. 41 (2001), no. 24, 3145–3161. 10.1016/S0042-6989(01)00173-0Search in Google Scholar
[23] A. I. Prilepko, D. G. Orlovsky and I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, Monog. Textb. Pure Appl. Math. 231, Marcel Dekker, New York, 2000. Search in Google Scholar
[24] S. Raczynski, Dynamics of economic growth: Uncertainty treatment using differential inclusions, MethodsX 6 (2019), 615–632. 10.1016/j.mex.2019.02.029Search in Google Scholar PubMed PubMed Central
[25] S. Ruhil and M. Malik, Inverse problem for the Atangana–Baleanu fractional differential equation, J. Inverse Ill-Posed Probl. 31 (2023), no. 5, 763–779. 10.1515/jiip-2022-0025Search in Google Scholar
[26] S. Ruhil and M. Malik, Inverse problem for abstract delay differential equation with impulsive effects, Evol. Equ. Control Theory 13 (2024), no. 3, 751–766. 10.3934/eect.2024004Search in Google Scholar
[27] G. V. Smirnov, Introduction to the Theory of Differential Inclusions, Grad. Stud. Math. 41, American Mathematical Society, Providence, 2022. Search in Google Scholar
[28] A. Tolstonogov, Differential Inclusions in a Banach Space, Math. Appl. 524, Kluwer Academic, Dordrecht, 2012. Search in Google Scholar
[29]
I. I. Vrabie,
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