Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 147409, 9 pages http://dx.doi.org/10.1155/2013/147409 Research Article The Existence and Attractivity of Solutions of an Urysohn Integral Equation on an Unbounded Interval Mohamed Abdalla Darwish, 1 Józef BanaV, 2 and Ebraheem O. Alzahrani 3 1 Mathematics Department, Science Faculty for Girls, King Abdulaziz University, Jeddah, Saudi Arabia 2 Department of Mathematics, Rzesz´ ow University of Technology, al. Powsta´ nc´ ow Warszawy 8, 35-959 Rzesz´ ow, Poland 3 Mathematics Depertment, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia Correspondence should be addressed to J´ ozef Bana´ s; jbanas@prz.edu.pl Received 21 August 2013; Accepted 4 September 2013 Academic Editor: Mohammad Mursaleen Copyright © 2013 Mohamed Abdalla Darwish et al. his is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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. he main tool used in investigations is the technique associated with the measures of noncompactness and a ixed point theorem of Darbo type. An example showing the utility of the obtained results is also included. 1. Introduction he theory of nonlinear functional integral equations creates an important branch of the modern nonlinear analysis. he large part of that theory describes a lot of classical nonlinear integral equations such as nonlinear Volterra integral equa- tions, Hammerstein integral equations, and Urysohn integral equations with solutions deined on a bounded interval (cf. [1–4]). Nevertheless, more important and simultaneously, a more diicult part of that theory is connected with the study of solutions of the mentioned integral equations deined on an unbounded domain. Obviously, there are some known results concerning the existence of solutions of those integral equations in such a setting but, in general, they are mostly obtained under rather restrictive assumptions [2, 4–8]. On the other hand, the use of some tools of nonlinear analysis enables us to obtain several valuable results under less restrictive assumptions (cf. [1, 9–15]). It turns out that the technique of measures of noncompactness creates a very convenient tool for the study of the solvability of nonlinear functional integral equations of various types. It is caused by the fact that the approach to the study of solutions of those equations with the use of the technique of measures of noncompactness gives not only the possibility to obtain existence results, but also allows us to look for solutions of mentioned equations having some desired properties such as monotonicity, attractivity, and asymptotic stability (cf. [16– 19], for instance). In the paper, we will use the above described approach associated with the technique of measures of noncompact- ness in order to obtain a result on the existence of solu- tions of a quadratic Urysohn integral equation. Applying the mentioned technique in conjunction with a ixed point theorem of Darbo type, we show that the equation in question has solutions deined, continuous, and bounded on the nonnegative real half axis R + which are uniformly attractive (asymptotic stable) on R + . he results obtained in this paper generalize several results obtained earlier in numerous papers treating non- linear functional integral equations, which were quoted above. Particularly, we generalize the results concerning the Urysohn or Hammerstein integral equations obtained in the papers [9, 10, 20]. 2. Notation, Definitions, and Auxiliary Facts In this section, we establish some notations, and we collect auxiliary facts which will be used in the sequel.