Existence And Properties Of Solutions For Some Classes Of Nonlinear Functional Integral Equations

Along with the science and technology development, various nonlinear problems have aroused people's widespread attention day by day. So the nonlinear analysis and its applications has become one of important research directions in modern mathemat-ics. Many problems arising in physical sciences, mechanics, biology, vehicular traffic theory, economics, geology, engineering and applied mathematics can be attributed to mathematical models solved by nonlinear integral equations. The theory of nonlinear integral equations is an important branch of nonlinear analysis and its applications, at the same time, a fast growing field with important applications to a number of areas of analysis as well as other branches of science. Therefore, it becomes of great scientific and real significance to study the existence and properties of solutions for nonlinear integral equations.The present paper employs the theories, concepts, methods such as measure of noncompactness theory, measure of weak noncompactness theory, Schauder fixed point theorem, Krasnosel'skii fixed point theorem, weakly sequentially continuous and so on, to investigate the existence and properties of solutions for some classes of nonlinear integral equations. The obtained results are either new or intrinsically generalize and improve the previous relevant ones.The thesis is divided into four sections according to contents.Chapter 1 Preference, we introduce the main contents of this paper.Chapter 2 We consider the following nonlinear functional integral equation where, the functions g:R+×R→R, f:R+×R×R→R, u:R+×R+×R→R, andα,β,γ,η:R+→R+ are continuous. We work in the Banach space BC(R+) consisting of real functions which are continuous and bounded on unbounded interval R+. By using measures of noncompactness and Schauder fixed point theorem to study the existence and local attractivity of solutions for equation (2.1.1) in BC(R+). The result obtained in this paper generalizes some previous related ones. Chapter 3 we investigate the following nonlinear perturbed integral equation of fractional orderα∈(0,1) is a fixed number andГ(·) denotes the Gamma function. Here, the given functions g:R+×R→R and u:R+×R+×R→R are continuous. A:BC(R+)→BC(R+)(BC(R+):the Banach space consisting of real functions which are continuous and bounded on an unbounded interval R+.) is an operator. By using measures of noncompactness and Schauder fixed point theorem, we prove the existence of solutions for equation (3.1.1) in the space BC(R+). Moreover, we show that those solutions are uniformly locally attractive and uniformly locally asymptotically attractive under appropriate assumptions. The results obtained in this paper generalize and improve some previous relevant ones.Chapter 4 We discuss the following nonlinear functional integral equation where, the functions g, f:[0,1]×R→R satisfy the Caratheodory conditions, g, f and u are given Lebesgue integrable functions,φ1,φ2:[0,1]→* [0,1] are increasing, absolutely continuous functions, x∈L1[0,1] is an unknown function(L1 [0,1] the Banach space of lebesgue integrable functions on the interval [0,1]). By using the measures of weak noncompactness, weakly sequentially continuous and a Krasnosel'skii type fixed point theorem to prove the existence of monotonic solutions for equation (4.1.1) in L1[0,1]. The results presented in this paper extend the some previous relevant ones.