On solutions vanishing at infinity of infinite systems of quadratic Urysohn integral equations
DOI:
https://doi.org/10.12775/TMNA.2023.046Słowa kluczowe
Space of continuous and bounded functions, sequence space, measure of noncompactness, Schauder fixed point theorem, infinite system of integral equationsAbstrakt
The paper is devoted to present a result on the existence of solutions of an infinite system of quadratic integral equations of the Urysohn type considered on the real half-axis. Our investigations are conducted in the Banach space consisting of bounded and continuous functions defined on the real half-axis with values in the space of real sequences converging to zero. That space is equipped with the standard supremum norm. The main tools used in our study is the technique of measures of noncompactness and the Schauder fixed point principle. We illustrate our result by a suitable example.Bibliografia
E. Ablet, L. Cheng, Q. Cheng and W. Zhuang, Every Banach space admits a homogeneous measure of non-compactness not equivalent to the Hausdorff measure, Sci. China Math. 62 (2019), 147–156.
R.R. Akhmerov, M.I. Kamenskiı̆, A.S. Potapov, A.E. Rodkin a and B.N. Sadovskiı̆, Measures of Noncompactness and Condensing Operators, Oper. Theory Adv. Appl., vol. 55, Birkhäuser, Basel, 1992.
J.M. Ayerbe Toledano, T. Dominguez Benavides and G. Lopez Acedo, Measures of Noncompactness in Metric Fixed Point Theory, Oper. Theory Adv. Appl., vol. 99, Birkhäuser, Basel, 1997.
J. Banaś and A. Chlebowicz, On solutions of an infinite system of nonlinear integral equations on the real half-axis, Banach J. Math. Anal. 13 (2019), 944–968.
J. Banaś, A. Chlebowicz and W. Woś, On measures of noncompactness in the space of functions defined on the half-axis with values in a Banach space, J. Math. Anal. Appl. 489, (2020), 124187.
J. Banaś and K. Goebel, Measures of Noncompactness in Banach Spaces, Lect. Notes Pure Appl. Math., vol. 60, Marcel Dekker, New York, 1980.
J. Banaś and M. Mursaleen, Sequence Spaces and Measures of Noncompactness with Applications to Differential and Integral Equations, Springer, New Delhi, 2014.
J. Banaś, R. Nalepa and B. Rzepka, The study of the solvability of infinite systems of integral equations via measures of noncompactness, Numer. Funct. Anal. Optim. 43, (2022), 961–986.
J. Banaś and L. Olszowy, On solutions of a quadratic Urysohn integral equation on an unbounded interval, Dyn. Syst. Appl. 17 (2008), 255–270.
J. Banaś and B. Rzepka, The technique of Volterra–Stieltjes integral equations in the application to infinite systems of nonlinear integral equations of fractional orders, Comput. Math. Appl. 64 (2012), 3108–3116.
J. Banaś and B. Rzepka, On solutions of infinite systems of integral equations of Hammerstein type, J. Nonlinear Convex Anal. 18 (2017), 261–278.
J. Banaś and W. Woś, Solvability of an infinite system of integral equations on the real half-axis, Adv. Nonlinear Anal. 10 (2021), 202–216.
C. Corduneanu, Integral Equations and Applications, Cambridge University Press, Cambridge, 1991.
K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985.
G.M. Fichtenholz, Differential and Integral Calculus, vol. II, PWN, Warsaw, 1980 (in Polish).
J. Garcia-Falset and K. Latrach, Nonlinear Functional Analysis and Applications, Series in Nonlinear Analysis and Applications, vol. 41, De Gruyter, 2023.
I.T. Gohberg, L.S. Goldenštein and A.S. Markus, Investigations of some properties of bounded linear operators with their q-norms, Učen. Zap. Kishinevsk. Univ. 29 (1957), 29–36.
L.S. Goldenštein and A.S. Markus, On the measure of noncompactness of bounded sets and of linear operators, Studies in Algebra and Math. Anal., Izdat. “Karta Moldovenjaske”, Kishinev, 1965, pp. 45–54.
T. Jalal and A.H. Jan, Measures of noncompactness in the Banach space BC(R+ × R+ , E) and its application to infinite system of integral equations in two variables, Filomat (2022) (to appear).
M.A. Krasnosel’skiı̆, P.P. Zabreı̆ko, E.I. Pustylnik and P.E. Sobolevskiı̆, Integral Operators in Spaces of Summable Functions, Noordhoff, Leyden, 1976.
K. Kuratowski, Sur les espaces complets, Fundam. Math. 15 (1930), 301–309.
I.A. Malik and T. Jalal, Infinite system of integral equations in two variables of Hammerstein type in c0 and l1 spaces, Filomat 33 (2019), 3441–3455.
E. Malkowsky and V. Rakočević, Advanced Functional Analysis, CRC Press, Taylor and Francis Group, Boca Raton, 2019.
J. Mallet-Paret and R.D. Nussbaum, Inequivalent measures of noncompactness and the radius of the essential spectrum, Proc. Amer. Math. Soc. 139 (2011), 917–930.
H. Mehravaran and H. Amiri Kayvanloo, Solvability of infinite system of nonlinear convolution type integral equations in the tempered sequence space mβ (%, p), AsianEuropean J. of Mathematics 16 (2023), 2350004.
W. Pogorzelski, Integral Equations and Their Applications, Pergamon Press, Oxford, New York, Frankfurt, PWN Polish Scientific Publishers, Warsaw, 1966.
P.P. Zabreı̆ko, A.I. Koshelev, M.A. Krasnosel’skiı̆, S.G. Mikhlin, L.S. Rakovshchik and V.J. Stetsenko, Integral Equations, Noordhoff, Leyden, 1975.
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Prawa autorskie (c) 2024 Józef Banaś, Justyna Madej
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