Abstract
In this paper, the second-order differential equations and systems of Kolmogorov type are defined. With reference to population dynamics models, unlike the first-order equations which give the expression of the per capita rate, in the case of the second-order equations, the law of change of the per capita rate is given. Several control problems with fixed final time and fixed final state, with additive and multiplicative control, are studied. Their controllability is proved with fixed-point methods, the theorems of Banach, Schauder, Krasnoselskii, Avramescu and Perov.
Authors
Radu Precup
Faculty of Mathematics and Computer Science and Institute of Advanced Studies in Science and Technology Babes-Bolyai University, Cluj-Napoca Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania
Alexandru Hofman
Faculty of Mathematics and Computer Science Babes-Bolyai University, Cluj-Napoca Romania
Keywords
Kolmogorov system; Lotka–Volterra system; control problem; fixed point; matrix convergent to zero; Volterra–Fredholm integral equation.
Paper coordinates
Al. Hofman, R. Precup, A fixed-point approach to control problems for Kolmogorov type second-order equations and systems, 27 (2025), art. no. 7, https://doi.org/10.1007/s11784-024-01160-5
freely available at the publisher
About this paper
Journal
Journal of Fixed Point Theory and Applications
Publisher Name
Print ISSN
1661-7738
Online ISSN
1661-7746
google scholar link
[1] Allen, L.J.S.: An Introduction to Mathematical Biology. Pearson Education, London (2006)
[2] Avramescu, C.: On a fixed point theorem (in Romanian). Studii ¸si Cercet˘ari Matematice 22(2), 215–221 (1970)
[3] Brauer, F., Castillo-Chavez, C.: Mathematical Models in Population Biology and Epidemiology. Springer, Berlin (2012)
[4] Coron, J.M.: Control and Nonlinearity, Mathematical Surveys and Monographs, vol. 136. American Mathematical Society, Providence (2007)
[5] Granas, A., Dugundji, J.: Fixed Point Theory. Springer, New York (2003)
[6] Haplea, I.S¸, Parajdi, L.G., Precup, R.: On the controllability of a system modeling cell dynamics related to leukemia. Symmetry 13, 1867 (2021)
[7] He, X., Zhu, Z., Chen, J., Chen, F.: Dynamical analysis of a Lotka Volterra commensalism model with additive Allee effect. Open Math. 20, 646–665 (2022)
[8] Hofman, A.: An algorithm for solving a control problem for Kolmogorov systems, Studia. Universitatis Babes-Bolyai. Mathematica 68, 331–340 (2023)
[9] Hofman, A., Precup, R.: On some control problems for Kolmogorov type systems. Math. Model. Control 2, 90–99 (2022)
[10] Hofman, A., Precup, R.: Vector fixed point approach to control of Kolmogorov differential systems. Contemp. Math. 5(2), 1968–1981 (2024)
[11] Kolmogorov, A.N.: Sulla teoria di Volterra della lotta per l’esistenza. Giornale dell Istituto Italiano degli Attuari 7, 74–80 (1936)
[12] Krasnoselskii, M.A.: Some problems of nonlinear analysis. Am. Math. Soc. Transl. Ser. 2(10), 345–409 (1958)
[13] Li, X., Liu, Z.H., Mig´orski, S.: Approximate controllability for second-order nonlinear evolution hemivariational inequalities. Electron. J. Qual. Theory Differ. Equ. 2015, 100 (2015)
[14] Li, J.: Control Schemes to reduce risk of extinction in the Lotka-Volterra predator-prey model. J. Appl. Math. Phys. 2(7), 644–652 (2014)
[15] Li, J., Zhao, A., Yan, J.: The permanence and global attractivity of a Kolmogorov system with feedback controls. Nonlinear Anal. Real World Appl. 10, 506–518 (2009)
[16] Llibre, J., Salhi, T.: On the dynamics of a class of Kolmogorov systems. Appl. Math. Comput. 225, 242–245 (2013)
[17] Lois-Prados, C., Precup, R.: Positive periodic solutions for Lotka-Volterra systems with a general attack rate. Nonlinear Anal. Real World Appl. 52, 103024
(2020)
[18] Mahmudov, N.I., Udhayakumar, R., Vijayakumar, V.: On the approximate controllability of second-order evolution hemivariational inequalities. Results Math. 75, 160 (2020)
[19] Murray, J.D.: An Introduction to Mathematical Biology, vol. 1. Springer, New York (2011)
[20] Parajdi, L.G., P˘atrulescu, F., Precup, R., Haplea, I.S¸: Two numerical methods for solving a nonlinear system of integral equations of mixed Volterra-Fredholm type arising from a control problem related to leukemia. J. Appl. Anal. Comput. 13, 1797–1812 (2023)
[21] Perov, A.I.: On the Cauchy problem for a system of ordinary differential equations (Russian). Pviblizhen. Met. Reshen. Differ. Uvavn. 2, 115–134 (1964)
[22] Precup, R.: Methods in Nonlinear Integral Equations. Springer, Dordrecht (2002)
[23] Precup, R.: The role of matrices that are convergent to zero in the study of semilinear operator systems. Math. Comput. Model. 49, 703–708 (2009)
[24] Precup, R.: On some applications of the controllability principle for fixed point equations. Results Appl. Math. 13, 100236 (2022)
[25] Quinn, M.D., Carmichael, N.: An approach to non-linear control problems using fixed-point methods, degree theory and pseudo-inverses. Numer. Funct. Anal. Optim. 7, 197–219 (1985)
[26] Reich, S.: Fixed points of condensing functions. J. Math. Anal. Appl. 41, 460–467 (1973)
[27] Sigmund, K.: Kolmogorov and population dynamics. In: Charpentier, E., Lesne, A., Nikolski, N.K. (eds.) Kolmogorov’s Heritage in Mathematics. Springer, Berlin (2007)
[28] Tigan, G., Lazureanu, C., Munteanu, F., Sterbeti, C., Florea, A.: Analysis of a class of Kolmogorov systems. Nonlinear Anal. Real World Appl. 57, 103202 (2021)