Application of Fixed Point Theorem of Convex-power Operators to Nonlinear Volterra Type Integral Equations

This paper firstly generalizes a kind of new operator, i.e. convex-power condensing operator, which was obtained by Sun Jingxian in paper [1], and defines a new class of operator, i.e. − P convex-power condensing operator in locally convex space. Also, a new fixed point theorem of this new operator is proved. Finally we apply the results obtained to investigate the existence of solutions for nonlinear Volterra type integral equations in locally convex spaces.


Introduction
The theory of differential equations, integral equations and integraldifferential equation in abstract space, developing in the first half of the twentieth Century, is a very active research area.Combining the theory of differential equations and functional analysis, using theory and methods of

Definition and Lemma
Before providing the main result, we need to introduce some basic facts about locally convex spaces.We give the definitions as following.
Paper [8] introduced Kuratowski's measure of non-compactness and basic properties of the bounded set in the Banach space.as the Kuratowski's measure of non-compactness, non-compactness measure for short, this indicates the diameter of i S for diam ) ( i S .Obviously, Paper [2] gives Kuratowski's measure of non-compactness, determined by the half of its range in locally convex space.
Definition 1.2 Let X be a locally convex space, whose topology generated by these mi-norm family Obviously, Paper [3] explains in detail about Kuratowski's measure of non-compactness in the locally convex space.
As we all know, the famous Schauder fixed point theorem is an important conclusion, extremely widely applied.This conclusion, however, requires operator to be completely continuous, which is a very hard condition.To weaken this condition, we have proposed condensing operator concept.We turn the condition from the completely continuous operator condensing operators to condensing operator, which defined by condensing operators with non-compact measure in the Schauder fixed point theorem. Definition Paper [1] proposed the new concept of convex-power condensing operator and basically generalizes it.We also apply the Sadovskii fixed point theorem to condensing operators further.First, we give you a mark.
Let E be real Banach space, . For any given , we calls A the convex-power condensing operator.If A is continuous and bounded, and it has D x ∈ 0 and positive integer 0 n , allowing for any non-relative compactness bounded sets , S is a relatively compact set in E .Clearly, the condensing operator must be cohesive by convex power.What's more, paper [1] establishes a new fixed point theorem about the newly defined convex-power condensing operator, namely the following lemma 2.6 and Lemma 2.7.
Let D be a nonempty bounded closed convex set of the Banach space, is convex-power condensing operator, there must be fixed points of A in D .
, as well as a positive integer 0 n , making , there must be fixed points of A in D .
Like the notation and definitions of convex-power condensing operator in Banach space of in the paper [1], we give the definition of − P convex-power condensing operator in locally convex space.

Let
) , ( P X be complete Hausdorff locally convex space,

Convex Spaces Nonlinear
This section examines the existence of Nonlinear Volterra integral equations in locally convex spaces.Among them, is a collection of continuous images of all slaves from J to X ., then

If
, we can conclude that Similar to the proof of Lemma in the paper [1], it is easy to prove the following lemma.

Lemma 2.2 Let
, so the same as

Lemma 2.4 For any given
With the above lemma and obtained fixed point theorem in − P convex-power condensing operator, Lemma 1.3, we can give the existence results of integral equation (2.1).Theorem 2.1 Let f meet: , and there is a continuous function 0 ) ( ≥ s a and real .
Then we prove that: is convex-power condensing operator.Take

Definition 1 . 1
Let E be a Banach space and S is a bounded set in E , Ω 's non-compactness measure about half of the range of α p diameter, determined by the half of the range of α p .
[1]h the convex Banach space power condensing operators fixed point theorem obtained by paper[1], namely Lemma 2.6, we can get − P convex-power condensing operator's fixed point theorem in locally convex spaces.Because they are proved similarly, the progress is omitted here.Lemma 1.4 Let D be a non-empty bounded closed convex set in a D .
mapped to F , and A is continuous bounded.Known by Lemma 2.4, F is