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March  2020, 40(3): 1665-1702. doi: 10.3934/dcds.2020089

On the fundamental solution and its application in a large class of differential systems determined by Volterra type operators with delay

Department of Mathematics University of Pannonia, 8200 Veszprém, Egyetem u. 10., Hungary

* Corresponding author

Received  May 2019 Published  December 2019

Fund Project: The research of the authors has been supported by Hungarian National Foundations for Scientific Research Grant No. K120186.

The variation-of-constants formula is one of the principal tools of the theory of differential equations. There are many papers dealing with different versions of the variation-of-constants formula and its applications. Our main purpose in this paper is to give a variation-of-constants formula for inhomogeneous linear functional differential systems determined by general Volterra type operators with delay. Our treatment of the delay in the considered systems is completely different from the usual methods. We deal with the representation of the studied Volterra type operators. Some existence and uniqueness theorems are obtained for the studied linear functional differential and integral systems. Finally, some applications are given.

Citation: István Győri, László Horváth. On the fundamental solution and its application in a large class of differential systems determined by Volterra type operators with delay. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1665-1702. doi: 10.3934/dcds.2020089
References:
[1] C. D. Aliprantis and O. Burkinshaw, Principles of Real Analysis, Academic Press, Inc., Boston, MA, 1990.  doi: 10.1016/C2009-0-22273-2.
[2]

O. Arino and E. Sánches, A variation of constants formula for an abstract functional differential equation of retarded type, Differential Integral Equations, 9 (1996), 1305-1320. 

[3]

J. BaštinecL. BerezanskyJ. Diblik and Z. Šmarda, On a delay population model with a quadratic nonlinearity without positive steady state, Appl. Math. Comput., 227 (2014), 622-629.  doi: 10.1016/j.amc.2013.11.061.

[4]

L. C. Becker, Principal matrix solutions and variation of parameters for a Volterra integro-differential equation and its adjoint, Electron. J. Qual. Theory Differ. Equ., 2006 (2006), 1-22.  doi: 10.14232/ejqtde.2006.1.14.

[5]

A. Carrasco and H. Leiva, Variation of constants formula for functional parabolic partial differential equations, Electron. J. Differential Equations, (2007), 20pp.

[6]

R. CollegariM. Federson and M. Frasson, Linear FDEs in the frame of generalized ODEs: Variation-of-constants formula, Czechoslovak Math. J., 68 (2018), 889-920.  doi: 10.21136/CMJ.2018.0023-17.

[7] C. Corduneanu, Integral Equations and Applications, Cambridge University Press, Cambridge, 1991.  doi: 10.1017/CBO9780511569395.
[8]

C. Corduneanu, Y. Li and M. Mahdavi, Functional Differential Equations: Advances and Applications, Pure and Applied Mathematics, John Wiley & Sons, Inc., NJ, 2016. doi: 10.1002/9781119189503.

[9]

R. D. Driver, Ordinary and Delay Differential Equations, Applied Mathematical Sciences, 20, Springer-Verlag, New York-Heidelberg, 1977. doi: 10.1007/978-1-4684-9467-9.

[10]

P. EloeM. Islam and B. Zhang, Uniform asymptotic stability in linear Volterra integrodifferential equations with application to delay systems, Dynam. Systems Appl., 9 (2000), 331-344. 

[11]

T. Faria and G. Röst, Persistence, permanence and global stability for an n-dimensional Nicholson system, J. Dynam. Differential Equations, 26 (2014), 723-744.  doi: 10.1007/s10884-014-9381-2.

[12]

M. FunakuboT. Hara and S. Sakata, On the uniform asymptotic stability for a linear integro-differential equation of Volterra type, J. Math. Anal. Appl., 324 (2006), 1036-1049.  doi: 10.1016/j.jmaa.2005.12.053.

[13]

T. Furumochi and S. Matsuoka, Stability and boundedness in Volterra integro-differential equations, Mem. Fac. Sci. Eng. Shimane Univ. Ser. B Math. Sci., 32 (1999), 25-40. 

[14]

R. Grimmer and G. Seifert, Stability properties of Volterra integrodifferential equations, J. Differential Equations, 19 (1975), 142-166.  doi: 10.1016/0022-0396(75)90025-X.

[15] G. GripenbergS.-O. Londen and O. Staffans, Volterra Integral and Functional Equations, Encyclopedia of Mathematics and its Applications, 34, Cambridge University Press, Cambridge, 1990.  doi: 10.1017/CBO9780511662805.
[16]

S. I. Grossman and R. K. Miller, Perturbation theory for Volterra integrodifferential systems, J. Differential Equations, 8 (1970), 457-474.  doi: 10.1016/0022-0396(70)90018-5.

[17]

I. Győri and L. Horváth, Sharp estimation for the solutions of delay differential and Halanay type inequalities, Discrete Contin. Dyn. Syst., 37 (2017), 3211-3242.  doi: 10.3934/dcds.2017137.

[18]

I. Győri and L. Horváth, Sharp estimation for the solutions of inhomogeneous delay differential and Halanay-type inequalities, Electron. J. Qual. Theory Differ. Equ., (2018), 1–18. doi: 10.14232/ejqtde.2018.1.54.

[19]

Y. HinoS. MurakamiT. Naito and N. V. Minh, A variation-of-constants formula for abstract functional differential equations in the phase space, J. Differential Equations, 179 (2002), 336-355.  doi: 10.1006/jdeq.2001.4020.

[20]

J. Horváth, Topological Vector Spaces and Distributions, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1966.

[21]

L. Horváth, An integral inequality, Math. Inequal. Appl., 4 (2001), 507-513.  doi: 10.7153/mia-04-45.

[22]

L. V. Kantorovich and G. P. Akilov, Functional Analysis in Normed Spaces, International Series of Monographs in Pure and Applied Mathematics, 46, The Macmillan Co., New York, 1964.

[23] J. L. Massera and J. J. Schäffer, Linear Differential Equations and Function Spaces, Pure and Applied Mathematics, 21, Academic Press, New York-London, 1966. 
[24]

S. Murakami, Linear periodic functional differential equations with infinite delay, Funkcial. Ekvac., 29 (1986), 335-361. 

[25]

M. Rama Mohana Rao and P. Srinivas, Asymptotic behavior of solutions of Volterra integro-differential equations, Proc. Amer. Math. Soc., 94 (1985), 55-60.  doi: 10.1090/S0002-9939-1985-0781056-5.

[26]

J. Šremr, On differentiation of a Lebesgue integral with respect to a parameter, Math. Appl. (Brno), 1 (2012), 91-116.  doi: 10.13164/ma.2012.06.

[27]

C. Tunç and O. Tunç, New qualitative criteria for solutions of Volterra integro-differential equations, Arab J. Basic Appl. Sciences, 25 (2018), 158-165.  doi: 10.1080/25765299.2018.1509554.

[28]

K. Yosida and E. Hewitt, Finitely additive measures, Trans. Amer. Math. Soc., 72 (1952), 46-66.  doi: 10.1090/S0002-9947-1952-0045194-X.

show all references

References:
[1] C. D. Aliprantis and O. Burkinshaw, Principles of Real Analysis, Academic Press, Inc., Boston, MA, 1990.  doi: 10.1016/C2009-0-22273-2.
[2]

O. Arino and E. Sánches, A variation of constants formula for an abstract functional differential equation of retarded type, Differential Integral Equations, 9 (1996), 1305-1320. 

[3]

J. BaštinecL. BerezanskyJ. Diblik and Z. Šmarda, On a delay population model with a quadratic nonlinearity without positive steady state, Appl. Math. Comput., 227 (2014), 622-629.  doi: 10.1016/j.amc.2013.11.061.

[4]

L. C. Becker, Principal matrix solutions and variation of parameters for a Volterra integro-differential equation and its adjoint, Electron. J. Qual. Theory Differ. Equ., 2006 (2006), 1-22.  doi: 10.14232/ejqtde.2006.1.14.

[5]

A. Carrasco and H. Leiva, Variation of constants formula for functional parabolic partial differential equations, Electron. J. Differential Equations, (2007), 20pp.

[6]

R. CollegariM. Federson and M. Frasson, Linear FDEs in the frame of generalized ODEs: Variation-of-constants formula, Czechoslovak Math. J., 68 (2018), 889-920.  doi: 10.21136/CMJ.2018.0023-17.

[7] C. Corduneanu, Integral Equations and Applications, Cambridge University Press, Cambridge, 1991.  doi: 10.1017/CBO9780511569395.
[8]

C. Corduneanu, Y. Li and M. Mahdavi, Functional Differential Equations: Advances and Applications, Pure and Applied Mathematics, John Wiley & Sons, Inc., NJ, 2016. doi: 10.1002/9781119189503.

[9]

R. D. Driver, Ordinary and Delay Differential Equations, Applied Mathematical Sciences, 20, Springer-Verlag, New York-Heidelberg, 1977. doi: 10.1007/978-1-4684-9467-9.

[10]

P. EloeM. Islam and B. Zhang, Uniform asymptotic stability in linear Volterra integrodifferential equations with application to delay systems, Dynam. Systems Appl., 9 (2000), 331-344. 

[11]

T. Faria and G. Röst, Persistence, permanence and global stability for an n-dimensional Nicholson system, J. Dynam. Differential Equations, 26 (2014), 723-744.  doi: 10.1007/s10884-014-9381-2.

[12]

M. FunakuboT. Hara and S. Sakata, On the uniform asymptotic stability for a linear integro-differential equation of Volterra type, J. Math. Anal. Appl., 324 (2006), 1036-1049.  doi: 10.1016/j.jmaa.2005.12.053.

[13]

T. Furumochi and S. Matsuoka, Stability and boundedness in Volterra integro-differential equations, Mem. Fac. Sci. Eng. Shimane Univ. Ser. B Math. Sci., 32 (1999), 25-40. 

[14]

R. Grimmer and G. Seifert, Stability properties of Volterra integrodifferential equations, J. Differential Equations, 19 (1975), 142-166.  doi: 10.1016/0022-0396(75)90025-X.

[15] G. GripenbergS.-O. Londen and O. Staffans, Volterra Integral and Functional Equations, Encyclopedia of Mathematics and its Applications, 34, Cambridge University Press, Cambridge, 1990.  doi: 10.1017/CBO9780511662805.
[16]

S. I. Grossman and R. K. Miller, Perturbation theory for Volterra integrodifferential systems, J. Differential Equations, 8 (1970), 457-474.  doi: 10.1016/0022-0396(70)90018-5.

[17]

I. Győri and L. Horváth, Sharp estimation for the solutions of delay differential and Halanay type inequalities, Discrete Contin. Dyn. Syst., 37 (2017), 3211-3242.  doi: 10.3934/dcds.2017137.

[18]

I. Győri and L. Horváth, Sharp estimation for the solutions of inhomogeneous delay differential and Halanay-type inequalities, Electron. J. Qual. Theory Differ. Equ., (2018), 1–18. doi: 10.14232/ejqtde.2018.1.54.

[19]

Y. HinoS. MurakamiT. Naito and N. V. Minh, A variation-of-constants formula for abstract functional differential equations in the phase space, J. Differential Equations, 179 (2002), 336-355.  doi: 10.1006/jdeq.2001.4020.

[20]

J. Horváth, Topological Vector Spaces and Distributions, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1966.

[21]

L. Horváth, An integral inequality, Math. Inequal. Appl., 4 (2001), 507-513.  doi: 10.7153/mia-04-45.

[22]

L. V. Kantorovich and G. P. Akilov, Functional Analysis in Normed Spaces, International Series of Monographs in Pure and Applied Mathematics, 46, The Macmillan Co., New York, 1964.

[23] J. L. Massera and J. J. Schäffer, Linear Differential Equations and Function Spaces, Pure and Applied Mathematics, 21, Academic Press, New York-London, 1966. 
[24]

S. Murakami, Linear periodic functional differential equations with infinite delay, Funkcial. Ekvac., 29 (1986), 335-361. 

[25]

M. Rama Mohana Rao and P. Srinivas, Asymptotic behavior of solutions of Volterra integro-differential equations, Proc. Amer. Math. Soc., 94 (1985), 55-60.  doi: 10.1090/S0002-9939-1985-0781056-5.

[26]

J. Šremr, On differentiation of a Lebesgue integral with respect to a parameter, Math. Appl. (Brno), 1 (2012), 91-116.  doi: 10.13164/ma.2012.06.

[27]

C. Tunç and O. Tunç, New qualitative criteria for solutions of Volterra integro-differential equations, Arab J. Basic Appl. Sciences, 25 (2018), 158-165.  doi: 10.1080/25765299.2018.1509554.

[28]

K. Yosida and E. Hewitt, Finitely additive measures, Trans. Amer. Math. Soc., 72 (1952), 46-66.  doi: 10.1090/S0002-9947-1952-0045194-X.

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