- Research Article
- Open Access
On a Nonlinear Integral Equation with Contractive Perturbation
© Huan Zhu. 2011
- Received: 19 December 2010
- Accepted: 19 February 2011
- Published: 13 March 2011
We get an existence result for solutions to a nonlinear integral equation with contractive perturbation by means of Krasnoselskii's fixed point theorem and especially the theory of measure of weak noncompactness.
- Banach Space
- Integral Equation
- Existence Result
- Fixed Point Theorem
- Nonlinear Integral Equation
The integral equations have many applications in mechanics, physics, engineering, biology, economics, and so on. It is worthwhile mentioning that some problems considered in the theory of abstract differential equations also lead us to integral equations in Banach space, and some foundational work has been done in [1–8].
in the Banach space .
Our paper is organized as follows.
In Section 2, some notations and auxiliary results will be given. We will introduce the main tools measure of weak noncompactness and Krasnoselskii's fixed point theorem in Section 3 and Section 4. The main theorem in our paper will be established in Section 5.
First of all, we give out some notations to appear in the following.
denotes the set of real numbers and . Suppose that is a separable locally compact Banach space with norm in the whole paper. (Remark: the locally compactness of will be used in Lemma 2.2). Let be a Lebesgue measurable subset of and denote the Lebesgue measure of .
is measurable in for any ;
is continuous in for almost all .
The following lemma which we will use in the proof of our main theorem explains the structure of functions satisfying Carathéodory conditions with the assumption that the space is separable and locally compact (see ).
Let be a bounded interval and be a functionsatisfying Carathéodory conditions. Then it possesses the Scorza-Dragoni property. That is each , there exists a closed subset of such that and is continuous.
for all and .
In this section we will recall the concept of measure of weak noncompactness which is the key point to complete our proof of main theorem in Section 5.
Let be a Banach space. and denote the collections of all nonempty bounded subsets and relatively weak compact subsets, respectively.
the family Ker is nonempty and Ker ;
if , we have ;
, where denotes the convex closed hull of ;
If is a decreasing sequence, that is, , every is weakly closed,
and , then is nonempty.
where denotes the closed ball in centered at 0 with radius .
for a nonempty and bounded subset of space .
for a nonempty and bounded subset of space .
It is easy to know that is a measure of weak noncompactness in space following the verification in .
The following is the Krasnoselskii's fixed point theorem which will be utilized to obtain the existence of solutions in the next section.
is a contraction mapping;
is relatively compact and is continuous.
Then there exists such that .
In , they proved the existence of solutions by means of Schauder fixed point theorem. With the presence of the Perturbation term in the integral equation, the Schauder fixed point theorem is invalid. To overcome this difficulty we will use the Kransnoselskii's fixed point theorem to obtain the existence of solutions.
We will see in the following section that the important step is the construction of by means of measure of weak noncompactness. This is the biggest difference between our paper from .
The Krasnoselskii's fixed point theorem was extended to general case in  (see also in ). In , they used the general Krasnoselskii's fixed point theorem to obtain the existence result. It can be seen in the next section of our paper that the classical Krasnoselskii's fixed point theorem is enough unless we need more general conditions on the perturbation term .
Our main theorem in this paper is stated as follows.
Suppose that the following assumptions are satisfied.
for and .
transforms the space into itself.
(H4) The function such that where is an arbitrary subset of , and is bounded by for all .
(H5) , where denotes the norm of the linear Volterra operator .
Then the integral equation (1.1) has at least one solution .
where is the linear Volterra integraloperator and is the superposition operator generated by the function .
The proof will be given in six steps.
There exists such that , where is a ball centered zero element with radius in .
where by assumption (H5).
for allbounded subset of .
Take a arbitrary numbers and such that .
It follows that by definition (3.2).
and then by definition (3.3).
From above, we then obtain for all bounded subset of .
We will construct a nonempty closed convex weakly compact set in on which we will apply fixed point theorem to prove the existence of solutions.
which yields that .
Denote , and then . By the definition of measure of weak noncompact we know that is nonempty. Moreover, .
is just nonempty closed convex weakly compact set which we need in the following steps.
is relatively compact in , where is just the set constructed in Step 3.
Considering the function on and on , we can find a closed subset of interval , such that , and such that and is continuous. Especially is uniformly continuous.
where denotes the modulusof continuity of the function on the set and . The last inequality of (5.12) is obtained since , where is just the one in the Step 1.
Taking into account the fact that the , we infer that the terms of the numerical sequence are arbitrarily small provided that the number is small enough.
Since is also arbitrarily small provided that the number is small enough, the right of (5.12) then tends to zero independent of as tends to zero. We then have is equicontinuous in the space .
From above, we then obtain that is equibounded in the space .
By assumption (H1),we have the operator is continuous. So forms a relatively compact set in the space .
Further observe that the above result does not depend on the choice of . Thus we can construct a sequence of closed subsets of the interval such that as and such that the sequence is relatively compact in every space . Passing to subsequence if necessary we can assume that is a cauchy sequence in each space .
for any , where .
for any .
which means that is a cauchy sequence in the space . Hence we conclude that is relatively compact in .
where we have made a transformation in the above process. Since by assumption (H6), we then get the fact that the operator is a contraction mapping.
From Step 3, we know that , where is the set constructed in Step 3.
From Step 5, we know that is a contraction mapping.
From the Step 4 and assumptions (H1), (H2), is relatively compact and is continuous.
We apply Theorem 4.1, and then obtain that (1.1) has at least one solution in .
When , in  they said is weakly sequence compact in their Step 1 of main proof. From our proof, we know that their proof is not precise, since in Step 4, one of the crucial conditions to prove the relatively compactness of is that is weakly compact. We can only obtain that is weakly sequence compact as a mapping from to which is the weakly compact set defined in Step 3. The construction of set overcomes the fault in , and we obtain the existence result finally.
- Liang J, Liu JH, Xiao T-J: Nonlocal problems for integrodifferential equations. Dynamics of Continuous, Discrete & Impulsive Systems. Series A 2008,15(6):815-824.MathSciNetMATHGoogle Scholar
- Liang J, Xiao T-J: Semilinear integrodifferential equations with nonlocal initial conditions. Computers & Mathematics with Applications 2004,47(6-7):863-875. 10.1016/S0898-1221(04)90071-5MathSciNetView ArticleMATHGoogle Scholar
- Liang J, Zhang J, Xiao T-J: Composition of pseudo almost automorphic and asymptotically almost automorphic functions. Journal of Mathematical Analysis and Applications 2008,340(2):1493-1499. 10.1016/j.jmaa.2007.09.065MathSciNetView ArticleMATHGoogle Scholar
- Xiao T-J, Liang J: Approximations of Laplace transforms and integrated semigroups. Journal of Functional Analysis 2000,172(1):202-220. 10.1006/jfan.1999.3545MathSciNetView ArticleMATHGoogle Scholar
- Xiao T-J, Liang J: Existence of classical solutions to nonautonomous nonlocal parabolic problems. Nonlinear Analysis, Theory, Methods and Applications 2005,63(5–7):e225-e232.View ArticleMATHGoogle Scholar
- Xiao T-J, Liang J: Blow-up and global existence of solutions to integral equations with infinite delay in Banach spaces. Nonlinear Analysis. Theory, Methods & Applications 2009,71(12):e1442-e1447. 10.1016/j.na.2009.01.204MathSciNetView ArticleMATHGoogle Scholar
- Xiao T-J, Liang J, van Casteren J: Time dependent Desch-Schappacher type perturbations of Volterra integral equations. Integral Equations and Operator Theory 2002,44(4):494-506. 10.1007/BF01193674MathSciNetView ArticleMATHGoogle Scholar
- Xiao T-J, Liang J, Zhang J: Pseudo almost automorphic solutions to semilinear differential equations in Banach spaces. Semigroup Forum 2008,76(3):518-524. 10.1007/s00233-007-9011-yMathSciNetView ArticleMATHGoogle Scholar
- Banaś J, Chlebowicz A: On existence of integrable solutions of a functional integral equation under Carathéodory conditions. Nonlinear Analysis. Theory, Methods & Applications 2009,70(9):3172-3179. 10.1016/j.na.2008.04.020MathSciNetView ArticleMATHGoogle Scholar
- Taoudi MA: Integrable solutions of a nonlinear functional integral equation on an unbounded interval. Nonlinear Analysis. Theory, Methods & Applications 2009,71(9):4131-4136. 10.1016/j.na.2009.02.072MathSciNetView ArticleMATHGoogle Scholar
- Ricceri B, Villani A: Separability and Scorza-Dragoni's property. Le Matematiche 1982,37(1):156-161.MathSciNetMATHGoogle Scholar
- Lucchetti R, Patrone F: On Nemytskii's operator and its application to the lower semicontinuity of integral functionals. Indiana University Mathematics Journal 1980,29(5):703-713. 10.1512/iumj.1980.29.29051MathSciNetView ArticleMATHGoogle Scholar
- Djebali S, Sahnoun Z:Nonlinear alternatives of Schauder and Krasnosel'skij types with applications to Hammerstein integral equations in spaces. Journal of Differential Equations 2010,249(9):2061-2075. 10.1016/j.jde.2010.07.013MathSciNetView ArticleMATHGoogle Scholar
- De Blasi FS: On a property of the unit sphere in Banach spaces. Bulletin Mathématique de la Société des Sciences Mathématiques de Roumanie 1997, 21: 259-262.MathSciNetMATHGoogle Scholar
- Appel J, De Pascale E: Su alcuni parametri connessi con la misura di non compatteza di Hausdorff in spazi di funzioni misurabilli. Bollettino della Unione Matematica Italiana. B 1984,3(6):497-515.Google Scholar
- Banaś J, Knap Z: Measures of weak noncompactness and nonlinear integral equations of convolution type. Journal of Mathematical Analysis and Applications 1990,146(2):353-362. 10.1016/0022-247X(90)90307-2MathSciNetView ArticleMATHGoogle Scholar
- Latrach K, Taoudi MA:Existence results for a generalized nonlinear Hammerstein equation on spaces. Nonlinear Analysis. Theory, Methods & Applications 2007,66(10):2325-2333. 10.1016/j.na.2006.03.022MathSciNetView ArticleMATHGoogle Scholar
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