# American Institute of Mathematical Sciences

October  2014, 19(8): 2401-2416. doi: 10.3934/dcdsb.2014.19.2401

## Systems described by Volterra type integral operators

 1 Faculty of Mathematics and Computer Science, University of Lodz, Banacha 22, 90-238 Lodz, Poland, Poland, Poland

Received  October 2013 Revised  March 2014 Published  August 2014

In the paper we consider a nonlinear Volterra integral operator defined on some subspace of absolutely continuous function. Some sufficient conditions for the operator considered to be a diffeomorphism are formulated. The proof of main result relies in essential way on variational method. Applications of results to control systems with feedback and a specific nonlinear Volterra equation are presented.
Citation: Dorota Bors, Andrzej Skowron, Stanisław Walczak. Systems described by Volterra type integral operators. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2401-2416. doi: 10.3934/dcdsb.2014.19.2401
##### References:
 [1] Z. Artstein, Continuous dependence of solutions of Volterra integral equations, SIAM J. Math. Anal., 6 (1975), 446-456. doi: 10.1137/0506039.  Google Scholar [2] T. M. Atanackovic and S. Pilipovic, On a class of equations arising in linear viscoelastic theory, ZAMM Z. Angew. Math. Mech., 85 (2005), 748-754. doi: 10.1002/zamm.200310209.  Google Scholar [3] D. Bors, Global solvability of Hammerstein equations with applications to BVP involving fractional Laplacian, Abstr. Appl. Anal., 2013, Art. ID 240863, 10 pp. doi: 10.1155/2013/240863.  Google Scholar [4] H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511543234.  Google Scholar [5] R. M. Christensen, Theory of Viscoelasticity, Academic Press, New York, 1982. doi: 10.1115/1.3408900.  Google Scholar [6] C. Corduneanu, Integral Equations and Applications, Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511569395.  Google Scholar [7] M. A. Darwish, A. A. El-Bary and W. G. El-Sayed, Solvability of Urysohn integral equation, Appl. Math. Comput., 145 (2003), 487-493. doi: 10.1016/S0096-3003(02)00504-0.  Google Scholar [8] A. Friedman, On integral equations of Volterra type, J. Analyse Math., 11 (1963), 381-413. doi: 10.1007/BF02789991.  Google Scholar [9] A. Friedman and M. Shinbrot, Volterra integral equations in Banach space, Trans. Amer. Math. Soc., 126 (1967), 131-179. doi: 10.1090/S0002-9947-1967-0206754-7.  Google Scholar [10] G. Gripenberg, An abstract nonlinear Volterra equation, Israel J. Math., 34 (1979), 198-212. doi: 10.1007/BF02760883.  Google Scholar [11] G. Gripenberg, S.-O. Londen and O. Staffans, Volterra Integral and Functional Equations, Cambridge University Press, Cambridge, 1990. doi: 10.1017/CBO9780511662805.  Google Scholar [12] D. Idczak, A. Skowron and S. Walczak, On the diffeomorphisms between Banach and Hilbert spaces, Adv. Nonlinear Stud., 12 (2012), 89-100.  Google Scholar [13] A. D. Ioffe and V. M. Tihomirov, Theory of Extremal Problems, Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam, 1979.  Google Scholar [14] M. Joshi, Existence theorems for Urysohn's integral equation, Proc. Amer. Math. Soc., 49 (1975), 387-392.  Google Scholar [15] T. Kiffe and M. Stecher, $L^{2}$ solutions of Volterra integral equations, SIAM J. Math. Anal., 10 (1979), 274-280. doi: 10.1137/0510026.  Google Scholar [16] V. Lakshmikantham and S. Leela, Differential and Integral Inequalities, I, Academic Press, New York, 1969.  Google Scholar [17] S. -O. Londen, Stability analysis on nonlinear point reactor kinetics, Adv. Sci. Tech., 6 (1972), 45-63. Google Scholar [18] A. G. J. MacFarlane, ed., Frequency-Response Methods in Control Systems, Selected Reprint Series, IEEE Press, New York, 1979. Google Scholar [19] J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences, 74, Springer, New York, 1989. doi: 10.1007/978-1-4757-2061-7.  Google Scholar [20] R. K. Miller, Nonlinear Volterra Integral Equations, Mathematics Lecture Note Series, W. A. Benjamin, Inc., Menlo Park, Calif., 1971.  Google Scholar [21] M. S. Mousa, R. K. Miller and A. N. Michel, Stability analysis of hybrid composite dynamical systems: descriptions involving operators and differential equations, IEEE Trans. Automat. Control, 31 (1986), 216-226. doi: 10.1109/TAC.1986.1104251.  Google Scholar [22] D. O'Regan, Volterra and Urysohn integral equations in Banach Spaces, J. Appl. Math. Stochastic Anal., 11 (1998), 449-464. doi: 10.1155/S1048953398000379.  Google Scholar [23] M. Z. Podowski, A study of nuclear reactor models with nonlinear reactivity feedbacks: Stability criteria and power overshot evaluation, IEEE Trans. Automat. Control, 31 (1986), 108-115. doi: 10.1109/TAC.1986.1104204.  Google Scholar [24] J. Prüss, Evolutionary Integral Equations and Applications, Modern Birkhäuser Classics, Springer, New York, 2012. doi: 10.1007/978-3-0348-8570-6.  Google Scholar [25] M. Renardy, W. J. Hrusa and J. A. Nohel, Mathematical Problems in Viscoelasticity, Pitman Monographs Pure Appl. Math.Longman Sci. Tech., Harlow, Essex, 1987.  Google Scholar [26] R. S. Sánchez-Peńa and M. Sznaier, Robust Systems Theory and Applications, Wiley-Interscience, New Jersey, 1998. Google Scholar

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##### References:
 [1] Z. Artstein, Continuous dependence of solutions of Volterra integral equations, SIAM J. Math. Anal., 6 (1975), 446-456. doi: 10.1137/0506039.  Google Scholar [2] T. M. Atanackovic and S. Pilipovic, On a class of equations arising in linear viscoelastic theory, ZAMM Z. Angew. Math. Mech., 85 (2005), 748-754. doi: 10.1002/zamm.200310209.  Google Scholar [3] D. Bors, Global solvability of Hammerstein equations with applications to BVP involving fractional Laplacian, Abstr. Appl. Anal., 2013, Art. ID 240863, 10 pp. doi: 10.1155/2013/240863.  Google Scholar [4] H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511543234.  Google Scholar [5] R. M. Christensen, Theory of Viscoelasticity, Academic Press, New York, 1982. doi: 10.1115/1.3408900.  Google Scholar [6] C. Corduneanu, Integral Equations and Applications, Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511569395.  Google Scholar [7] M. A. Darwish, A. A. El-Bary and W. G. El-Sayed, Solvability of Urysohn integral equation, Appl. Math. Comput., 145 (2003), 487-493. doi: 10.1016/S0096-3003(02)00504-0.  Google Scholar [8] A. Friedman, On integral equations of Volterra type, J. Analyse Math., 11 (1963), 381-413. doi: 10.1007/BF02789991.  Google Scholar [9] A. Friedman and M. Shinbrot, Volterra integral equations in Banach space, Trans. Amer. Math. Soc., 126 (1967), 131-179. doi: 10.1090/S0002-9947-1967-0206754-7.  Google Scholar [10] G. Gripenberg, An abstract nonlinear Volterra equation, Israel J. Math., 34 (1979), 198-212. doi: 10.1007/BF02760883.  Google Scholar [11] G. Gripenberg, S.-O. Londen and O. Staffans, Volterra Integral and Functional Equations, Cambridge University Press, Cambridge, 1990. doi: 10.1017/CBO9780511662805.  Google Scholar [12] D. Idczak, A. Skowron and S. Walczak, On the diffeomorphisms between Banach and Hilbert spaces, Adv. Nonlinear Stud., 12 (2012), 89-100.  Google Scholar [13] A. D. Ioffe and V. M. Tihomirov, Theory of Extremal Problems, Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam, 1979.  Google Scholar [14] M. Joshi, Existence theorems for Urysohn's integral equation, Proc. Amer. Math. Soc., 49 (1975), 387-392.  Google Scholar [15] T. Kiffe and M. Stecher, $L^{2}$ solutions of Volterra integral equations, SIAM J. Math. Anal., 10 (1979), 274-280. doi: 10.1137/0510026.  Google Scholar [16] V. Lakshmikantham and S. Leela, Differential and Integral Inequalities, I, Academic Press, New York, 1969.  Google Scholar [17] S. -O. Londen, Stability analysis on nonlinear point reactor kinetics, Adv. Sci. Tech., 6 (1972), 45-63. Google Scholar [18] A. G. J. MacFarlane, ed., Frequency-Response Methods in Control Systems, Selected Reprint Series, IEEE Press, New York, 1979. Google Scholar [19] J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences, 74, Springer, New York, 1989. doi: 10.1007/978-1-4757-2061-7.  Google Scholar [20] R. K. Miller, Nonlinear Volterra Integral Equations, Mathematics Lecture Note Series, W. A. Benjamin, Inc., Menlo Park, Calif., 1971.  Google Scholar [21] M. S. Mousa, R. K. Miller and A. N. Michel, Stability analysis of hybrid composite dynamical systems: descriptions involving operators and differential equations, IEEE Trans. Automat. Control, 31 (1986), 216-226. doi: 10.1109/TAC.1986.1104251.  Google Scholar [22] D. O'Regan, Volterra and Urysohn integral equations in Banach Spaces, J. Appl. Math. Stochastic Anal., 11 (1998), 449-464. doi: 10.1155/S1048953398000379.  Google Scholar [23] M. Z. Podowski, A study of nuclear reactor models with nonlinear reactivity feedbacks: Stability criteria and power overshot evaluation, IEEE Trans. Automat. Control, 31 (1986), 108-115. doi: 10.1109/TAC.1986.1104204.  Google Scholar [24] J. Prüss, Evolutionary Integral Equations and Applications, Modern Birkhäuser Classics, Springer, New York, 2012. doi: 10.1007/978-3-0348-8570-6.  Google Scholar [25] M. Renardy, W. J. Hrusa and J. A. Nohel, Mathematical Problems in Viscoelasticity, Pitman Monographs Pure Appl. Math.Longman Sci. Tech., Harlow, Essex, 1987.  Google Scholar [26] R. S. Sánchez-Peńa and M. Sznaier, Robust Systems Theory and Applications, Wiley-Interscience, New Jersey, 1998. Google Scholar
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