Global attractivity results for nonlinear functional integral equations via a Krasnoselskii type fixed point theorem

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Abstract

In this paper, two results concerning the global attractivity and global asymptotic attractivity of the solutions for a nonlinear functional integral equation are proved via a variant of the Krasnoselskii fixed point theorem due to Dhage [B.C. Dhage, A fixed point theorem in Banach algebras with applications to functional integral equations, Kyungpook Math. J. 44 (2004) 145–155]. The investigations are placed in the Banach space of real functions defined, continuous and bounded on an unbounded interval. A couple of examples are indicated for demonstrating the natural realizations of the abstract results presented in the paper. Our results generalize the attractivity results of Banas and Rzepka [J. Banas, B. Rzepka, An application of measures of noncompactness in the study of asymptotic stability, Appl. Math. Lett. 16 (2003) 1–6] and Banas and Dhage [J. Banas, B.C. Dhage, Global asymptotic stability of solutions of a functional integral equations, Nonlinear Anal. (2007), doi:10.1016/j.na.2007.07.038], under weaker conditions with a different method.

Introduction

Let R be the real line and let R+ 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 x(t)=q(t)+f(t,x(t),x(α(t)))+0β(t)g(t,s,x(s),x(γ(s)))ds for all tR+, where q:R+R,f:R+×R×RR,g:R+×R+×R×RR and α,β,γ:R+R+.

By a solution of the FIE (1.1) we mean a function xC(R+,R) that satisfies the Eq. (1.1), where C(R+,R) is the space of continuous real-valued functions defined on R+.

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 x(t)=f(t,x(α(t))) has been studied in Kuczma [9], whereas the integral equation in the form x(t)=q(t)+0tg(s,x(γ(s)))ds 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 g(t,s,x)=k(t,s)h(s,x), f(t,x,y)=0, and β(t)=t=γ(t) in (1.1), it reduces to the Hammerstein integral equation, namely x(t)=0tk(t,s)h(s,x(s))ds. The integral equation in the form x(t)=f(t,x(t))+0tg(t,s,x(s))ds 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 x(t)=f(t,x(α(t)))+0β(t)g(t,s,x(γ(s)))ds 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 X be the infinite dimensional Banach space with the norm . A mapping Q:XX is called D-Lipschitz if there is a continuous and nondecreasing function ϕ:R+R+ satisfying QxQyϕ(xy). for all x,yX, where ϕ(0)=0. If ϕ(r)=kr, k>0, then Q is called Lipschitz with the Lipschitz constant k. In particular, if k<1, then Q is called a contraction on X with the contraction constant k. Further, if ϕ(r)<r for r>0, then Q is called a nonlinear D-contraction and the function ϕ is called a D

Global attractivity results

The Eq. (3.1) will be considered under the following assumptions:

  • (H0)

    The functions α,β,γ:R+R+ are continuous and α(t) as t.

  • (H1)

    The function q:R+R is continuous and bounded with K=supt0|q(t)|.

  • (H2)

    The function f:R+×R×RR is continuous and there exists a bounded function :R+R+ with bound L and a positive constant M such that |f(t,x1,x2)f(t,y1,y2)|(t)max{|x1y1|,|x2y2|}M+max{|x1y1|,|x2y2|} for tR+ and for x,yR. Moreover, we assume that LM.

  • (H3)

    The function tf(t,0,0) is bounded on R+ with F0=sup{|

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 f satisfying the hypothesis (H2).

Define a class Φ of functions ϕ:R+R+ satisfying the following properties:

  • (i)

    ϕ is continuous,

  • (ii)

    ϕ is nondecreasing, and

  • (iii)

    ϕ is subadditive, i.e., ϕ(x+y)ϕ(x)+ϕ(y) for all x,yR+.

Notice that if ϕΦ, then after simple computation it can be shown that |ϕ(x)ϕ(y)|ϕ(|xy|) for all x,yR+.

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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