Global attractivity results for nonlinear functional integral equations via a Krasnoselskii type fixed point theorem
Introduction
Let be the real line and let be the right half of the real axis, that is, the set of nonnegative real numbers. In this paper, we investigate the nonlinear functional integral equation (for short FIE) of the form for all , where and .
By a solution of the FIE (1.1) we mean a function that satisfies the Eq. (1.1), where is the space of continuous real-valued functions defined on .
The FIE (1.1) is of general interest since it includes several functional and functional integral equations studied earlier as special cases. For example, the functional equation in the form has been studied in Kuczma [9], whereas the integral equation in the form has been studied continuously in the literature for a long period of time as regards various aspects of the solutions, since its discovery by Volterra (see Burton [3], O’Regan and Meehan [11], Väth [13], Zeidler [14] and the references therein). In the special case when , , and in (1.1), it reduces to the Hammerstein integral equation, namely The integral equation in the form has been discussed in Banas and Rzepka [2] as regards the existence and asymptotic stability of the solution via the measure of noncompactness. Very recently, the functional integral equation in the form has been discussed in Banas and Dhage [1] as regards the existence and global attractivity results via an approach similar to that of Banas and Rzepka [2] under certain mixed Lipschitz and growth conditions. Thus, our FIE (1.1) is more general than all the above mentioned integral equations and the existence results of this paper include the existence results of all above mentioned integral equations as special cases.
It is well known that the integral equations are studied in the literature as regards various qualitative properties such as existence, uniqueness, boundedness, stability, monotonicity, extremality, attractivity and asymptotic behavior etc., of the solutions under suitable conditions on the functions involved in the equations. In this paper, we establish the global attractivity as well as global asymptotic attractivity results for the FIE (1.1) via a variant of the fixed point theorems of Krasnoselskii [10] and Burton [3] proved in Dhage [5] which generalize the earlier results of Banas and Rzepka [2] and Banas and Dhage [1] under a weaker condition with a different method. In the following section we give some preliminaries and auxiliary results which will be used in the sequel.
Section snippets
Preliminaries and auxiliary results
Let be the infinite dimensional Banach space with the norm . A mapping is called -Lipschitz if there is a continuous and nondecreasing function satisfying for all , where . If , , then is called Lipschitz with the Lipschitz constant . In particular, if , then is called a contraction on with the contraction constant . Further, if for , then is called a nonlinear -contraction and the function is called a
Global attractivity results
The Eq. (3.1) will be considered under the following assumptions:
- (H0)
The functions are continuous and as .
- (H1)
The function is continuous and bounded with .
- (H2)
The function is continuous and there exists a bounded function with bound and a positive constant such that for and for . Moreover, we assume that .
- (H3)
The function is bounded on with
Examples
In what follows, we show that the assumptions imposed in Theorem 3.1, Theorem 3.2 admit some natural realizations. First, we indicate some possible forms for expressing the function satisfying the hypothesis (H2).
Define a class of functions satisfying the following properties:
- (i)
is continuous,
- (ii)
is nondecreasing, and
- (iii)
is subadditive, i.e., for all .
Notice that if , then after simple computation it can be shown that for all .
Now
Remarks and conclusion
Although it has been mentioned in Dhage [6] that the classical fixed point theoretic approach is better than the measure theoretic approach for proving the attractivity of solutions for nonlinear integral equations, there is a disadvantage that these fixed point theorems do not yield automatically the characterizations of the solutions of such equations. The characterizations of the measures of noncompactness themselves automatically yield the various qualitative properties of the solutions of
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