Solvability of an integral equation of Volterra-Wiener-Hopf type. (English) Zbl 1474.45015

Summary: The paper presents results concerning the solvability of a nonlinear integral equation of Volterra-Stieltjes type. We show that under some assumptions that equation has a continuous and bounded solution defined on the interval \(\left[0, \left.\infty\right)\right.\) and having a finite limit at infinity. As a special case of the mentioned integral equation we obtain an integral equation of Volterra-Wiener-Hopf type. That fact enables us to formulate convenient and handy conditions ensuring the solvability of the equation in question in the class of functions defined and continuous on the interval \(\left[0, \left.\infty\right)\right.\) and having finite limits at infinity.


45D05 Volterra integral equations
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[1] Chandrasekhar, S., Radiative Transfer (1950), London, UK: Oxford University Press, London, UK · Zbl 0037.43201
[2] Deimling, K., Nonlinear Functional Analysis (1985), Berlin, Germany: Springer, Berlin, Germany · Zbl 0559.47040
[3] Dunford, N.; Schwartz, J. T., Linear Operators (1963), Leiden, The Netherlands: International Publishing, Leiden, The Netherlands · Zbl 0128.34803
[4] Podlubny, I., Fractional Differential Equations. Fractional Differential Equations, Mathematics in Science and Engineering, 198 (1999), San Diego, Calif, USA: Academic Press, San Diego, Calif, USA · Zbl 0924.34008
[5] Zabrejko, P. P.; Koshelev, A. I.; Krasnosel’skii, M. A.; Mikhlin, S. G.; Rakovschik, L. S.; Stetsenko, J., Integral Equations (1975), Leyden, Mass, USA: Nordhoff, Leyden, Mass, USA · Zbl 0293.45001
[6] Hopf, E., Mathematical Problems of Radiative Equilibria, Cambridge Tract 31 (1934), Cambridge, UK: Cambridge University Press, Cambridge, UK
[7] Copson, E. T., On an integral equation arising in the theory of diffraction, The Quarterly Journal of Mathematics: Oxford: Second Series, 17, 19-34 (1946) · Zbl 0063.00965
[8] Carlson, J. F.; Heins, A. E., The reflection of an electromagnetic plane wave by an infinite set of plates. I, Quarterly of Applied Mathematics, 4, 313-329 (1947) · Zbl 0031.13804
[9] Abrahams, I. D., On the application of the Wiener-Hopf technique to problems in dynamic elasticity, Wave Motion, 36, 4, 311-333 (2002) · Zbl 1163.74304 · doi:10.1016/S0165-2125(02)00027-6
[10] Antipov, Y. A., Diffraction of a plane wave by a circular cone with an impedance boundary condition, SIAM Journal on Applied Mathematics, 62, 4, 1122-1152 (2002) · Zbl 1004.65121 · doi:10.1137/S0036139900363324
[11] Banaś, J., Measures of noncompactness in the study of solutions of nonlinear differential and integral equations, Central European Journal of Mathematics, 10, 6, 2003-2011 (2012) · Zbl 1277.47067 · doi:10.2478/s11533-012-0120-9
[12] Banaś, J.; Goebel, K., Measures of Noncompactness in Banach Spaces. Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics, 60 (1980), New York, NY, USA: Marcel Dekker, New York, NY, USA · Zbl 0441.47056
[13] Appell, J.; Banaś, J.; Merentes, N., Bounded Variation and Around. Bounded Variation and Around, De Gruyter Series in Nonlinear Analysis and Applications, 17 (2014), Berlin, Germany: Walter de Gruyter, Berlin, Germany · Zbl 1282.26001
[14] Natanson, I. P., Theory of Functions of Real Variable (1960), New York, NY, USA: Ungar, New York, NY, USA · Zbl 0091.05404
[15] Banaś, J.; Zając, T., A new approach to the theory of functional integral equations of fractional order, Journal of Mathematical Analysis and Applications, 375, 2, 375-387 (2011) · Zbl 1210.45005 · doi:10.1016/j.jmaa.2010.09.004
[16] Zając, T., Solvability of fractional integral equations on an unbounded interval through the theory of Volterra-Stieltjes integral equations, Zeitschrift für Analysis und ihre Anwendungen, 33, 1, 65-85 (2014) · Zbl 1292.45006 · doi:10.4171/ZAA/1499
[17] Rudin, W., Real and Complex Analysis (1987), New York, NY, USA: McGraw-Hill Book, New York, NY, USA · Zbl 0925.00005
[18] Sikorski, R., Real Functions, Warsaw, Poland: PWN, Warsaw, Poland · Zbl 0064.05502
[19] Corduneanu, C., Integral Equations and Applications (1991), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 0714.45002 · doi:10.1017/CBO9780511569395
[20] Stańczy, R., Hammerstein equations with an integral over a noncompact domain, Annales Polonici Mathematici, 69, 1, 49-60 (1998) · Zbl 0919.45004
[21] Agarwal, R. P.; Banaś, J.; Banaś, K.; O’Regan, D., Solvability of a quadratic Hammerstein integral equation in the class of functions having limits at infinity, Journal of Integral Equations and Applications, 23, 2, 157-181 (2011) · Zbl 1223.45006 · doi:10.1216/JIE-2011-23-2-157
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