April  2019, 12(2): 189-202. doi: 10.3934/dcdss.2019013

On solutions of semilinear upper diagonal infinite systems of differential equations

1. 

Department of Nonlinear Analysis, Rzeszów University of Technology, Al. Powstańców Warszawy 8, 35-959 Rzeszów, Poland

2. 

Institute of Economic and Management, State Higher School of Technology and Economics in Jarosław, ul. Czarnieckiego 16, 37-500 Jarosław, Poland

* Corresponding author: Józef Banaś

Dedicated to Professor Vicentiu Radulescu on the occasion of his 60th anniversary

Received  August 2017 Revised  December 2017 Published  August 2018

The goal of the paper is to investigate the existence of solutions for semilinear upper diagonal infinite systems of differential equations. We will look for solutions of the mentioned infinite systems in a Banach tempered sequence space. In our considerations we utilize the technique associated with the Hausdorff measure of noncompactness and some existence results from the theory of ordinary differential equations in abstract Banach spaces.

Citation: Józef Banaś, Monika Krajewska. On solutions of semilinear upper diagonal infinite systems of differential equations. Discrete and Continuous Dynamical Systems - S, 2019, 12 (2) : 189-202. doi: 10.3934/dcdss.2019013
References:
[1]

R. R. Akhmerov, M. I. Kamenskii, A. S. Potapov, A. E. Rodkina and B. N. Sadovskii, Measures of Noncompactness and Condensing Operators, Birkhäuser, Basel, 1992. doi: 10.1007/978-3-0348-5727-7.

[2]

J. M. Ayerbe Toledano, T. Dominguez Benavides and G. Lopez Acedo, Measures of Noncompactness in Metric Fixed Point Theory, Birkhäuser, Basel, 1997. doi: 10.1007/978-3-0348-8920-9.

[3]

J. Banaś and K. Goebel, Measures of Noncompactness in Banach Spaces, Marcel Dekker, New York, 1980.

[4]

J. Banaś and M. Krajewska, Existence of solutions for infinite systems of differential equations in spaces of tempered sequences, Electronic J. Differential Equations, 2017 (2017), 1-28. 

[5]

J. Banaś and M. Mursaleen, Sequence Spaces and Measures of Noncompactness with Applications to Differential and Integral Equations, Springer, New Delhi, 2014. doi: 10.1007/978-81-322-1886-9.

[6]

L. ChengQ. ChengQ. ShenK. Tu and W. Zhang, A new approach to measures of noncompactness of Banach spaces, Studia Math., 240 (2018), 21-45.  doi: 10.4064/sm8448-2-2017.

[7]

K. Deimling, Ordinary Differential Equations in Banach Spaces, Springer, Berlin, 1977.

[8]

K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.

[9]

J. Mallet-Paret and R. D. Nussbaum, Inequivalent measures of noncompactness, Ann. Mat. Pura Appl., 190 (2011), 453-488.  doi: 10.1007/s10231-010-0158-x.

[10]

H. Mönch and G. H. von Harten, On the Cauchy problem for ordinary differential equations in Banach spaces, Arch. Math., 39 (1982), 153-160.  doi: 10.1007/BF01899196.

show all references

References:
[1]

R. R. Akhmerov, M. I. Kamenskii, A. S. Potapov, A. E. Rodkina and B. N. Sadovskii, Measures of Noncompactness and Condensing Operators, Birkhäuser, Basel, 1992. doi: 10.1007/978-3-0348-5727-7.

[2]

J. M. Ayerbe Toledano, T. Dominguez Benavides and G. Lopez Acedo, Measures of Noncompactness in Metric Fixed Point Theory, Birkhäuser, Basel, 1997. doi: 10.1007/978-3-0348-8920-9.

[3]

J. Banaś and K. Goebel, Measures of Noncompactness in Banach Spaces, Marcel Dekker, New York, 1980.

[4]

J. Banaś and M. Krajewska, Existence of solutions for infinite systems of differential equations in spaces of tempered sequences, Electronic J. Differential Equations, 2017 (2017), 1-28. 

[5]

J. Banaś and M. Mursaleen, Sequence Spaces and Measures of Noncompactness with Applications to Differential and Integral Equations, Springer, New Delhi, 2014. doi: 10.1007/978-81-322-1886-9.

[6]

L. ChengQ. ChengQ. ShenK. Tu and W. Zhang, A new approach to measures of noncompactness of Banach spaces, Studia Math., 240 (2018), 21-45.  doi: 10.4064/sm8448-2-2017.

[7]

K. Deimling, Ordinary Differential Equations in Banach Spaces, Springer, Berlin, 1977.

[8]

K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.

[9]

J. Mallet-Paret and R. D. Nussbaum, Inequivalent measures of noncompactness, Ann. Mat. Pura Appl., 190 (2011), 453-488.  doi: 10.1007/s10231-010-0158-x.

[10]

H. Mönch and G. H. von Harten, On the Cauchy problem for ordinary differential equations in Banach spaces, Arch. Math., 39 (1982), 153-160.  doi: 10.1007/BF01899196.

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