×

Some properties of nonlinear Volterra–Stieltjes integral operators. (English) Zbl 1093.47048

The authors consider the nonlinear Volterra–Stieltjes integral operator \[ (Ux)(t)=\int_0^tu(t,s,x(s))d_sg(t,s),\quad t\in I=[0,1], \] where \(d_s\) denotes integration with respect to \(s\). They find suitable conditions which guarantee that the function \(Ux\) is of bounded variation or is nondecreasing on \(I\), where \(x\in C(I)\), the space of real-valued continuous functions defined on \(I\) with the classical maximum norm. Moreover, they investigate the continuity as well as the compactness of the operator \(U\) acting in this space. These results generalize the earlier ones obtained by Banaś and O’Regan for operators having less general form. As applications, the authors formulate existence results concerning continuous, nondecreasing functions of bounded variation which are solutions to the equation \(x=p+Ux\), with \(p\) satisfying suitable assumptions and \(U\) being defined as above.

MSC:

47G10 Integral operators
45G10 Other nonlinear integral equations
26A42 Integrals of Riemann, Stieltjes and Lebesgue type
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Agarwal, R. P.; O’Regan, D.; Wong, P. J.Y., Positive Solutions of Differential, Difference and Integral Equations (1999), Kluwer Academic: Kluwer Academic New York · Zbl 0923.39002
[2] Argyros, I. K., Quadratic equations and applications to Chandrasekhar’s and related equations, Bull. Austral. Math. Soc., 32, 275-292 (1985) · Zbl 0607.47063
[3] Banaś, J.; Sadarangani, K., Solvability of Volterra-Stieltjes operator-integral equations and their applications, Computers Math. Applic., 41, 1535-1544 (2001) · Zbl 0986.45006
[4] Deimling, K., Nonlinear Functional Analysis (1985), Springer-Verlag: Springer-Verlag Dordrecht · Zbl 0559.47040
[5] Hu, S.; Khavanin, M.; Zhuang, W., Integral equations arising in the kinetic theory of gases, Appl. Analysis, 34, 261-266 (1989) · Zbl 0697.45004
[6] O’Regan, D.; Meehan, M., Existence Theory for Nonlinear Integral and Integrodifferential Equations (1998), Kluwer Academic · Zbl 0932.45010
[7] Banaś, J.; O’Regan, D., On Volterra-Stieltjes integral operators, Mathl. Comput. Modelling, 41, 2/3, 335-344 (2005) · Zbl 1083.45007
[8] J. Banaś and D. O’Regan, Integral operators of Urysohn-Stieltjes type (to appear).; J. Banaś and D. O’Regan, Integral operators of Urysohn-Stieltjes type (to appear).
[9] Dunford, N.; Schwartz, J. T., Linear Operators I (1963), Int. Publ.: Int. Publ. Dordrecht
[10] Natanson, I. P., Theory of Functions of a Real Variable (1960), Ungar: Ungar Leyden · Zbl 0091.05404
[11] Myškis, A. D., Linear Differential Equations With Retarded Argument (in Russian) (1972), Nauka: Nauka New York
[12] Appell, J.; Zabrejko, P. P., Nonlinear superposition operators, (Cambridge Tracts in Mathematics, Volume 95 (1990), Cambridge Univ. Press: Cambridge Univ. Press Moscow) · Zbl 0970.47049
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.