A lower and upper solution method is introduced for control problems related to abstract operator equations. The method is illustrated on a control problem for the Lotka–Volterra model with seasonal harvesting and applied to a control problem of cell evolution after bone marrow transplantation.
Lorand Gabriel Parajdi
Department of Mathematics West Virginia University P.O. Box 6201, Morgantown, WV 26506, USA e-mail: firstname.lastname@example.org
Department of Mathematics Babes–Bolyai University M. Kogalniceanu, Cluj-Napoca, Romania
Department of Mathematics Babes-Bolyai University, Cluj-Napoca, Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy
Department of Internal Medicine Iuliu Hatieganu University of Medicine and Pharmacy Victor Babes Street, Cluj-Napoca, Romania
control problem, lower and upper solutions, fixed point, approximation algorithm, numerical solution, medical application.
L.G. Parajdi, R. Precup, I.-S. Haplea, A method of lowert and upper solutions for control problems and application to a model of bone Marrow transplantation, Int. J. Appl. Math. Comput. Sci., 33 (2023) no. 3, 409–418, http://doi.org/10.34768/amcs-2023-0029
Int. J. Appl. Math. Comput. Sci.
Sciendo (Walter de Gruyter)
google scholar link
 Barbu, V. (2016). Differential Equations, Springer, Cham.
 Coron, J.-M. (2007). Control and Nonlinearity, Mathematical Surveys and Monographs, Vol. 136, American Mathematical Society, Providence.
 DeConde, R., Kim, P.S., Levy, D. and Lee, P.P. (2005). Post-transplantation dynamics of the immune response to chronic myelogenous leukemia, Journal of Theoretical Biology 236(1): 39–59.
 Foley, C. and Mackey, M.C. (2009). Dynamic hematological disease: A review, Journal of Mathematical Biology 58(1): 285–322.
 Haplea, I. ¸S., Parajdi, L.G. and Precup, R. (2021). On the controllability of a system modeling cell dynamics related to leukemia, Symmetry 13(10): 1867.
 Kelley, C.T. (1995). Iterative Methods for Linear and Nonlinear Equations, SIAM, Philadelphia.
 Kim, P.S., Lee, P.P. and Levy, D. (2007). Mini-transplants for chronic myelogenous leukemia: A modeling perspective, in I. Queinnec (Ed.), Biology and Control Theory: Current Challenges, Lecture Notes in Control and Information Sciences, Vol. 357, Springer, Berlin, pp. 3–20.
 Langtangen, H.P. and Mardal, K.A. (2019). Introduction to Numerical Methods for Variational Problems, Springer, Cham.
 Parajdi, L.G. (2020). Stability of the equilibria of a dynamic system modeling stem cell transplantation, Ricerche di Matematica 69(2): 579–601.
 Parajdi, L.G., Patrulescu, F.-O., Precup, R. and Haplea, I. ¸S. (2023). Two numerical methods for solving a nonlinear system of integral equations of mixed Volterra-Fredholm type arising from a control problem related to leukemia, Journal of Applied Analysis & Computation, DOI:10.11948/20220197, (online first).
 Precup, R. (2002). Methods in Nonlinear Integral Equations, Kluwer Academic Publishers, Dordrecht.
 Precup, R. (2022). On some applications of the controllability principle for fixed point equations, Results in Applied Mathematics 13: 100236.
 Precup, R., Dima, D., Tomuleasa, C., ¸Serban, M.-A. and Parajdi, L.-G. (2018). Theoretical models of hematopoietic cell dynamics related to bone marrow transplantation, in Atta-ur-Rahman and S. Anjum (Eds.), Frontiers in Stem Cell and Regenerative Medicine Research, Vol. 8, Bentham Science Publishers, Sharjah, pp. 202–241.
 Precup, R., ¸Serban, M.-A. and Trif, D. (2013). Asymptotic stability for a model of cell dynamics after allogeneic bone marrow transplantation, Nonlinear Dynamics and Systems Theory 13(1): 79–92.
 Precup, R., ¸Serban, M.-A., Trif, D. and Cucuianu, A. (2012). A planning algorithm for correction therapies after allogeneic stem cell transplantation, Journal of Mathematical Modelling and Algorithms 11(3): 309–323.
 Precup, R., Trif, D., ¸Serban, M.-A. and Cucuianu, A. (2010). A mathematical approach to cell dynamics before and after allogeneic bone marrow transplantation, Annals of the Tiberiu Popoviciu Seminar of Functional Equations, Approximation and Convexity 8: 167–175.
 Rahmani Doust, M.H. (2015). The efficiency of harvested factor: Lotka–Volterra predator-prey model, Caspian Journal of Mathematical Sciences 4(1): 51–59.