# American Institute of Mathematical Sciences

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 & 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.  Google Scholar [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.  Google Scholar [3] J. Banaś and K. Goebel, Measures of Noncompactness in Banach Spaces, Marcel Dekker, New York, 1980.  Google Scholar [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.   Google Scholar [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.  Google Scholar [6] L. Cheng, Q. Cheng, Q. Shen, K. 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.  Google Scholar [7] K. Deimling, Ordinary Differential Equations in Banach Spaces, Springer, Berlin, 1977.  Google Scholar [8] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [3] J. Banaś and K. Goebel, Measures of Noncompactness in Banach Spaces, Marcel Dekker, New York, 1980.  Google Scholar [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.   Google Scholar [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.  Google Scholar [6] L. Cheng, Q. Cheng, Q. Shen, K. 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.  Google Scholar [7] K. Deimling, Ordinary Differential Equations in Banach Spaces, Springer, Berlin, 1977.  Google Scholar [8] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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