## Abstract and Applied Analysis

### The Existence and Attractivity of Solutions of an Urysohn Integral Equation on an Unbounded Interval

#### Abstract

We prove a result on the existence and uniform attractivity of solutions of an Urysohn integral equation. Our considerations are conducted in the Banach space consisting of real functions which are bounded and continuous on the nonnegative real half axis. The main tool used in investigations is the technique associated with the measures of noncompactness and a fixed point theorem of Darbo type. An example showing the utility of the obtained results is also included.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 147409, 9 pages.

Dates
First available in Project Euclid: 26 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393443697

Digital Object Identifier
doi:10.1155/2013/147409

Mathematical Reviews number (MathSciNet)
MR3124027

Zentralblatt MATH identifier
1297.45008

#### Citation

Darwish, Mohamed Abdalla; Banaś, Józef; Alzahrani, Ebraheem O. The Existence and Attractivity of Solutions of an Urysohn Integral Equation on an Unbounded Interval. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 147409, 9 pages. doi:10.1155/2013/147409. https://projecteuclid.org/euclid.aaa/1393443697

#### References

• J. Banaś and K. Sadarangani, “Compactness conditions in the study of functional, differential, and integral equations,” Abstract and Applied Analysis, vol. 2013, Article ID 819315, 14 pages, 2013.
• K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985.
• M. A. Krasnosel'skii, P. P. Zabrejko, J. I. Pustylnik, and P. I. Sobolevskii, Integral Operators in Spaces of Summable Functions, Nordhoff, Leyden, Mass, USA, 1976.
• P. P. Zabrejko, A. I. Koshelev, M. A. Krasnosel'skii, S. G. Mikhlin, L. S. Rakovschik, and V. J. Stetsenko, Integral Equations, Nordhoff, Leyden, Mass, USA, 1975.
• I. J. Cabrera and K. B. Sadarangani, “Existence of solutions of a nonlinear integral equation on an unbounded interval,” Dynamic Systems and Applications, vol. 18, no. 3-4, pp. 551–570, 2009.
• C. Corduneanu, Intergral Equations and Applications, Cambridge University Press, Cambridge, UK, 1991.
• N. Dunford and J. T. Schwartz, Linear Operators I, International Publishing, Leyden, The Netherlands, 1963.
• D. O'Regan and M. Meehan, Existence Theory for Nonlinear Integral and Integrodifferential Equations, Kluwer Academic, Dordrecht, The Netherlands, 1998.
• R. P. Agarwal, J. Banaś, K. Banaś, and D. O'Regan, “Solvability of a quadratic Hammerstein integral equation in the class of functions having limits at infinity,” Journal of Integral Equations and Applications, vol. 23, no. 2, pp. 157–181, 2011.
• J. Banaś and L. Olszowy, “On solutions of a quadratic Urysohn integral equation on an unbounded interval,” Dynamic Systems and Applications, vol. 17, no. 2, pp. 255–270, 2008.
• M. A. Darwish and J. Henderson, “Nondecreasing solutions of a quadratic integral equation of Urysohn-Stieltjes type,” The Rocky Mountain Journal of Mathematics, vol. 42, no. 2, pp. 545–566, 2012.
• M. A. Darwish and K. Sadarangani, “Nondecreasing solutions of a quadratic Abel equation with supremum in the kernel,” Applied Mathematics and Computation, vol. 219, no. 14, pp. 7830–7836, 2013.
• M. Gil and S. Wedrychowicz, “Schauder-Tychonoff fixed-point theorem in the theory of superconductivity,” Journal of Function Spaces and Applications, vol. 2013, Article ID 692879, 12 pages, 2013.
• L. Olszowy, “Fixed point theorems in the Fréchet space \emphC(${\mathbb{R}}_{+}$) and functional integral equations on an unbounded interval,” Applied Mathematics and Computation, vol. 218, no. 18, pp. 9066–9074, 2012.
• L. Olszowy, “Nondecreasing solutions of a quadratic integral equation of Urysohn type on unbounded interval,” Journal of Convex Analysis, vol. 18, no. 2, pp. 455–464, 2011.
• B. C. Dhage and V. Lakshmikantham, “On global existence and attractivity results for nonlinear functional integral equations,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 72, no. 5, pp. 2219–2227, 2010.
• A. Aghajani and N. Sabzali, “Existence and local attractivity of solutions of a nonlinear quadratic functional integral equation,” Iranian Journal of Science and Technology, Transaction A, vol. 36, no. 4, pp. 453–460, 2012.
• M. A. Darwish, “Monotonic solutions of a convolution functional-integral equation,” Applied Mathematics and Computation, vol. 219, no. 22, pp. 10777–10782, 2013.
• X. Hu and J. Yan, “The global attractivity and asymptotic stability of solution of a nonlinear integral equation,” Journal of Mathematical Analysis and Applications, vol. 321, no. 1, pp. 147–156, 2006.
• R. Stańczy, “Hammerstein equations with an integral over a noncompact domain,” Annales Polonici Mathematici, vol. 69, no. 1, pp. 49–60, 1998.
• J. Banaś and K. Goebel, Measures of Noncompactness in Banach Spaces, vol. 60 of Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1980.
• G. M. Fichtenholz, Differential and Integral Calculus, vol. 2, Wydawnictwo Naukowe PWN, Warsaw, Poland, 2007, (Polish).