## Banach Journal of Mathematical Analysis

### On solutions of an infinite system of nonlinear integral equations on the real half-axis

#### Abstract

We investigate the existence of solutions of an infinite system of integral equations of Volterra–Hammerstein type on the real half-axis. The method applied in our study is connected with the construction of a suitable measure of noncompactness in the space of continuous and bounded functions defined on the real half-axis with values in the space $c_{0}$ consisting of real sequences converging to zero and equipped with the classical supremum norm. We use the mentioned measure of noncompactness together with a fixed point theorem of Darbo type. Our investigations are illustrated by an example.

#### Article information

Source
Banach J. Math. Anal., Volume 13, Number 4 (2019), 944-968.

Dates
Accepted: 12 February 2019
First available in Project Euclid: 9 October 2019

https://projecteuclid.org/euclid.bjma/1570608169

Digital Object Identifier
doi:10.1215/17358787-2019-0008

Mathematical Reviews number (MathSciNet)
MR4016904

Zentralblatt MATH identifier
07118769

#### Citation

Banaś, Józef; Chlebowicz, Agnieszka. On solutions of an infinite system of nonlinear integral equations on the real half-axis. Banach J. Math. Anal. 13 (2019), no. 4, 944--968. doi:10.1215/17358787-2019-0008. https://projecteuclid.org/euclid.bjma/1570608169

#### References

• [1] E. Ablet, L. Cheng, Q. Cheng, and W. Zhang, Every Banach space admits a homogenous measure of noncompactness not equivalent to the Hausdorff measure, Sci. China Math. 62 (2019), no. 1, 147–156.
• [2] R. R. Akhmerov, M. I. Kamenskii, A. S. Potapov, A. E. Rodkina, and B. N. Sadovskii, Measures of Noncompactness and Condensing Operators, Oper. Theory Adv. Appl. 55, Birkhäuser, Basel, 1992.
• [3] J. M. Ayerbe Toledano, T. Domínguez Benavides, and G. López Acedo, Measures of Noncompactness in Metric Fixed Point Theory, Oper. Theory Adv. Appl. 99, Birkhäuser, Basel, 1997.
• [4] J. Banaś and K. Goebel, Measures of Noncompactness in Banach Spaces, Lect. Notes Pure Appl. Math. 60, Marcel Dekker, New York, 1980.
• [5] J. Banaś and A. Martinón, Measures of noncompactness in Banach sequence spaces, Math. Slovaca 42 (1992), no. 4, 497–503.
• [6] J. Banaś, N. Merentes, and B. Rzepka, “Measures of noncompactness in the space of continuous and bounded functions defined on the real half-axis” in Advances in Nonlinear Analysis via the Concept of Measure of Noncompactness, Springer, Singapore, 2017, 1–58.
• [7] J. Banaś and M. Mursaleen, Sequence Spaces and Measures of Noncompactness with Applications to Differential and Integral Equations, Springer, New Delhi, 2014.
• [8] J. Banaś, M. Mursaleen, and S. M. H. Rizvi, Existence of solutions to a boundary-value problem for an infinite system of differential equations, Electron. J. Differential Equations 2017, no. 262.
• [9] S. Chandrasekhar, Radiative Transfer, Dover, New York, 1960.
• [10] 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), no. 1, 21–45.
• [11] G. Darbo, Punti uniti in trasformazioni a condominio non compatto, Rend. Semin. Mat. Univ. Padova 24 (1955), 84–92.
• [12] K. Deimling, Ordinary Differential Equations in Banach Spaces, Lect. Notes Math 596, Springer, Berlin, 1977.
• [13] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985.
• [14] N. Dunford and J. T. Schwartz, Linear Operators, II: Spectral Theory, Wiley, New York, 1963.
• [15] I. T. Gohberg, L. S. Goldenštein, and A. S. Markus, Investigations of some properties of bounded linear operators with their $q$-norms, Učen Kishinersk. Univ. 29 (1957), 29–36.
• [16] L. S. Goldenštein and A. S. Markus, “On the measure of non-compactness of bounded sets and of linear operators” in Studies in Algebra and Mathematical Analysis, Izdat. “Karta Moldovenjaske,” Kishinev, 1965, 45–54.
• [17] K. Kuratowski, Sur les espaces complets, Fund. Math. 15 (1930), 301–309.
• [18] S. Łojasiewicz, An Introduction to the Theory of Real Functions, 3rd ed., Wiley, Chichester, 1988.
• [19] J. Mallet-Paret and R. D. Nussbaum, Inequivalent measures of noncompactness, Ann. Mat. Pura Appl. (4) 190 (2011), no. 3, 453–488.
• [20] J. Mallet-Paret and R. D. Nussbaum, Inequivalent measures of noncompactness and the radius of the essential spectrum, Proc. Amer. Math. Soc. 139 (2011), no. 3, 917–930.
• [21] M. Mursaleen and S. M. H. Rizvi, Solvability of infinite systems of second order differential equations in $c_{0}$ and $l_{1}$ by Meir-Keeler condensing operators, Proc. Amer. Math. Soc. 144 (2016), no. 10, 4279–4289.
• [22] W. Mydlarczyk, Coupled Volterra integral equations with blowing up solutions, J. Integral Equations Appl. 30 (2018), no. 1, 147–166.
• [23] R. D. Nussbaum, A generalization of the Ascoli theorem and an application to functional differential equations, J. Math. Anal. Appl. 35 (1971), 600–610.
• [24] W. Okrasiński and Ł. Płociniczak, Solution estimates for a system of nonlinear integral equations arising in optometry, J. Integral Equations Appl. 30 (2018), no. 1, 167–179.
• [25] W. Pogorzelski, Integral Equations and Their Applications, I, Int. Ser. Monogr. Pure Appl. Math. 88, Pergamon, Oxford, 1966.
• [26] P. P. Zabrejko, A. I. Koshelev, M. A. Krasnosel’skii, S. G. Mikhlin, L. S. Rakovschik, and J. Stetsenko, Integral Equations, Nordhoff, Leyden, 1975.