The Study Of Solutions For A Class Of Nonlinear Integral Equations Of Convolution Type

The research about the solutions of integral equations, we often studied the existence and uniqueness of the solution, and a big part studies also focus on the asymptotic behavior of the solution. The existence and uniqueness of the solution to the equation is the basic theorem in the theory of ordinary differential equation. While studying the asymptotic behavior of the unique solution of this equation, more precisely, is studying the existence of monotone solutions of the equation. In recent years, a number of domestic and foreign mathematicians have dedicated to research in this area. In 2008 O.Lipovan studied the existence and uniqueness of solutions of a class of nonlinear integral equations, and discussed the asymptotic behavior of the solution in 2014. This paper extends the results of O.Lipovan, by using the similar methods, it is discussed that the existence and uniqueness of the solution about the equation as well as its asymptotic behavior, which are a more general class of convolution nonlinear integral equations when p≥1, it obtains the following results:In the first part, we show that there exists an unique nonnegative solution for the equation. Let us consider the integral equation:three functions are involved in the equa-tion including　Φ、L(t) and P(t), where needs to meet three conditions we set, and these conditions ensure that　Φ is strictly increasing, with the inverse　Φ-1:[0, ∞)�' [0, ∞); respectively, L (t) and P (t) are continuous positive functions on the interval [0,∞), and P (t) is not identically zero, it is easy to know that L (t) and P (t) are continuously differentiable, let p≥ 1 and u(t) is the unknown function. Under these assumptions, we study the existence and uniqueness of solutions of the promoted equation, at the same time, we prove that the equation exists a nonnegative solution when p≥1 by using the Schauder fixed point theorem and related lemma.In the second part, we mainly study the asymptotic behavior of the unique solution of this equation. For this purpose, we give the same conditions ensuring the function　Φ and inverse Φ-1 are monotonous; respectively, L (t) and P (t) are continuous positive functions on the interval [0, ∞), and P (t) is not identically zero, as well as u(t) is the unknown function. Under these conditions, by use of the related knowledge of mathematical anal-ysis and apagoge, we prove that if P is a non-decreasing function and p≥1, the equa-tion has the following asymptotic behavior of solution, namely u(t) is strictly decreasing on [0, ∞) when L’(t) and U0P(t) are less than zero; on the other hand, if P is a non-increasing function and p≥ 1, u(t) is strictly increasing on [0, ∞) when L’(t) and U0P(t) are greater than zero.