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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 902563, 6 pages
http://dx.doi.org/10.1155/2013/902563
Research Article

On Lipschitz Perturbations of a Self-Adjoint Strongly Positive Operator

1Department of Mathematics, Valahia University of Targoviste, Boulevard Unirii 18, 130024 Targoviste, Romania
2Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 5 May 2013; Accepted 13 September 2013

Academic Editor: Satit Saejung

Copyright © 2013 Dinu Teodorescu and N. Hussain. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper we study semilinear equations of the form , where is a linear self-adjoint operator, satisfying a strong positivity condition, and is a nonlinear Lipschitz operator. As applications we develop Krasnoselskii and Ky Fan type approximation results for certain pair of maps and to illustrate the usability of the obtained results, the existence of solution of an integral equation is provided.

1. Introduction and Preliminaries

The study of abstract operator equations involving linear or nonlinear operators has generated over time useful instruments in the approach of some concrete equations. Therefore, we consider as interesting to present some aspects regarding the semilinear abstract operator equations in Hilbert spaces.

Let be a real Hilbert space endowed with the inner product and the norm .

In [1, 2] semilinear equations of the form were studied, where is a self-adjoint linear operator with the resolvent set and is a Gateaux differentiable gradient operator. If there exist real numbers such that and for all , (i.e., interacts suitably with the spectrum of ), then it is proved in [2] that the equation has exactly one solution.

The author in [3] presented an existence and uniqueness result for the semilinear equation , where is a linear maximal monotone operator, satisfying a strong positivity condition, and the nonlinearity is a Lipschitz operator.

Let be a real Banach space, ordered by a cone . A cone is a closed convex subset of with   , and . As usual .

Definition 1. Let be a nonempty subset of an ordered Banach space with order . Two mappings are said to be weakly isotone increasing if and hold for all . Similarly, we say that and are weakly isotone decreasing if and hold for all . The mappings and are said to be weakly isotone if they are either weakly isotone increasing or weakly isotone decreasing.

In our considerations, the following definition will play an important role. Let denote the collection of all nonempty bounded subsets of and the subset of consisting of all weakly compact subsets of . Also, let denote the closed ball centered at with radius .

Definition 2 (see [4]). A function is said to be a measure of weak noncompactness if it satisfies the following conditions.(1) The family is nonempty, and is contained in the set of relatively weakly compact sets of .(2). (3), where is the closed convex hull of .(4) for .(5) If is a sequence of nonempty weakly closed subsets of with bounded and such that , then is nonempty.

The family described in (1) is said to be the kernel of the measure of weak noncompactness . Note that the intersection set from (5) belongs to since for every , and . Also, it can be easily verified that the measure satisfies where is the weak closure of .

A measure of weak noncompactness is said to be regular if subadditive if homogeneous if and set additive (or has the maximum property) if

The first important example of a measure of weak noncompactness has been defined by de Blasi [5] as follows: for each .

Notice that is regular, homogeneous, subadditive, and set additive (see [5]).

By a measure of noncompactness on a Banach space , we mean a map which satisfies conditions (1)–(5) in Definition 2 relative to the strong topology instead of the weak topology.

Definition 3. Let be a Banach space and a measure of (weak) noncompactness on . Let be a mapping. If is bounded and for every nonempty bounded subset of with , we have ; then is called -condensing. If there exists , , such that is bounded and for each nonempty bounded subset of , we have ; then is called --contractive.

Definition 4 (see [6]). A map is said to be ws-compact if it is continuous, and for any weakly convergent sequence in the sequence has a strongly convergent subsequence in .

Definition 5. A map is said to be ww-compact if it is continuous, and for any weakly convergent sequence in the sequence has a weakly convergent subsequence in .

Definition 6. Let be a Banach space. A mapping is called a nonlinear contraction if there exists a continuous and nondecreasing function such that for all , where for .

In this paper we consider the semilinear equation where is a linear self-adjoint operator, satisfying a strong positivity condition, is a nonlinear Lipschitz operator, and is a positive parameter. Using the Banach fixed point theorem, we prove an existence and uniqueness result about the considered equation. Thus, we obtain here the same type of result as in [2], replacing the maximal monotonicity of linear part of the semilinear equation with the hypothesis that is self-adjoint. So, the principal result of this paper can be applied in the study of nonlinear Lipschitz perturbations of a linear integral operator with symmetric kernel. Further a result of continuous dependence on the free term and a fixed point theorem are presented. As applications we present some common fixed point theorems and approximation results for a pair of nonlinear mappings. Finally, the existence of solution of an integral equation is provided to illustrate the usability of the obtained results.

2. Results

Theorem 7. Let be a linear self-adjoint operator and nonlinear, satisfying the following conditions:(i) is a Lipschitz operator, that is, there is a constant such that for all ;(ii) is a strongly positive operator, that is, there is a constant such that for all .
Then the equation has a unique solution for all and .

Proof. Let us choose in the spectrum of . We have and we obtain that every real number is in the resolvent set of the operator . Consequently, we have for all , where is the identity of .
Let . We write (9) in the equivalent form where and . We have and for all , .
Also
From (16) we obtain
Consequently, there exists which is linear and continuous, that is , the Banach space of all linear and bounded operators from to . Moreover, we have
Now (14) can be equivalently written as
We consider the operator defined by
Therefore (19) becomes and so, the problem of the solvability of (9) is reduced to the study of fixed points of the operator . We have
It results that is a strict contraction from    to    because . According to the Banach fixed point theorem, has a unique fixed point, and thus the proof of Theorem 7 is complete.

Let us consider now the dependence of solution of (9) on the data .

Theorem 8. Under the assumptions from the hypothesis of Theorem 7, let , and let be the unique solution of the equation Then

Proof. According to the equivalent form (19) of (9), we have
It results that and thus our assertion is proved.

In fact, Theorem 8 establishes the continuous dependence of the solution of (9) on the free term and signifies the stability of the solution.

3. Consequences of Principal Result

The previous results prove that, under the considered assumptions, the operator is invertible for all and the inverse is a Lipschitz operator. From (24) it results that the operator satisfies

Suppose now that and . We obtain or , and, consequently, is a strict contraction. It results that there exists an unique element such that which is equivalent to . So we obtained the following fixed point theorem.

Theorem 9. Let be a linear self-adjoint operator and nonlinear, satisfying the following conditions:(i) is a Lipschitz operator, that is, there is a constant such that for all ;(ii) is a strongly positive operator, that is, there is a constant such that for all .
If and , then the operator has a unique fixed point.

The way used in obtaining Theorem 9 can be applied in the study of the following problem: extracting operators which have the unique fixed point property from a family of operators .

Let be a Banach space, and , satisfying some constraints for any. Our intention is to extract a subfamily , so that can have a unique fixed point for all . It is easy to observe, using the same method as in obtaining Theorem 9, that the following result holds.

Theorem 10. If(i) there exists , , so that is invertible and for all ;(ii) there exists , , so that for all , then has a unique fixed point for all .

As an application of the observations established above, we develop here the Krasnoselskii and Ky Fan type approximation results for certain pair of maps.

Theorem 11. Let , , , and be as in Theorem 9, , and let be an ordered Hilbert space. Let be a nonempty closed convex subset of and a set additive measure of noncompactness on . Let be a mapping satisfying the following:(i) and ,(ii) is continuous and -condensing,(iii) and are weakly isotone.
Then there exists a unique such that .

Proof. By Theorem 9, is a contraction with contractive constant . Thus is a shrinking mapping. Now all of the conditions of Corollary 1.18 [7] are satisfied so there exists an such that which implies that , and hence .

Theorem 12. Let , , , , and be as in Theorem 9, , and let be an ordered Hilbert space. Let be a nonempty closed convex subset of and a set additive measure of noncompactness on . Let be a mapping satisfying the following:(i) and ,(ii) is a nonlinear contraction,(iii) and are weakly isotone.
Then there exists a unique such that .

Proof. By Theorem 9, is a contraction with contractive constant . Thus is a shrinking mapping. Now all of the conditions of Corollary 1.19 [7] are satisfied so there exists an such that which implies .

As an application of Corollaries 1.24 or 1.25 and 1.30 or 1.31 [7], we obtain the following results, respectively

Theorem 13. Let , , , , and be as in Theorem 9, , and let be an ordered Hilbert space. Let be a nonempty closed convex subset of and a set additive measure of weak noncompactness on . Assume that are sequentially weakly continuous mappings satisfying the following:(i) and ,(ii) is -condensing or is a nonlinear contraction,(iii) and are weakly isotone.
Then there exists a unique point such that .

Theorem 14. Let , , , , and be as in Theorem 9, , and let be an ordered Hilbert space. Let be a nonempty closed convex subset of and a set additive measure of weak noncompactness on . Assume that , satisfy the following:(i) is a ww-compact mapping,(ii) is continuous ws-compact and -condensing or is continuous ws-compact and nonlinear contraction,(iii) and are weakly isotone,(iv) and .
Then there exists a unique such that .

Theorem 15. Let , , , , , and be as in Theorem 9 and sequentially weakly continuous. Assume that is nonempty closed bounded convex subset of and is sequentially weakly continuous mapping satisfying the following: (i) is relatively weakly compact,(ii).
Then there exists a unique point such that .

Proof. By Theorem 9, is a strict contraction with contractive constant . Now Theorem 2.1 [6] implies that there exists an such that which implies that .

Theorem 16. Let , , , , and be as in Theorem 9, , and let be an ordered Hilbert space. Let be a nonempty closed convex subset of and a set additive measure of noncompactness on . Let be a mapping satisfying the following:(i), (ii) is continuous and -condensing or ( is a nonlinear contraction),(iii) and are weakly isotone,
where is the proximity map on . Then there exists such that

More precisely, either(1) and have a common fixed point , or(2)there exists with

Proof. Let be the proximity map on ; that is, for each , we have . It is well known that is nonexpansive in . As is shrinking map, so is also shrinking mapping. By Theorem 4.1 (or 4.2), there exists such that . Thus we obtain, as in Theorem 4.1 [8] the desired conclusion.

Following the proof of Corollary 4.5 [8] and using Theorem 16, we obtain the following common fixed point theorem.

Theorem 17. Let , , , , and be as in Theorem 9, , and let be an ordered Hilbert space. Let be a nonempty closed convex subset of and a set additive measure of noncompactness on . Assume that is a mapping satisfying the following:(i), (ii) is continuous and -condensing or ( is a nonlinear contraction),(iii) and are weakly isotone,
where is the proximity map on . Suppose that satisfies one of the following conditions for each , with :(i) for some in ;(ii)there is a such that and ;(iii); (iv)for each , ;(v)there exists an such that, ;(vi)there exists such that, .
Then .

4. An Application

Fixed point theorems for certain operators have found various applications in differential and integral equations (see [710] and references therein). In this section, we present an application of our Theorem 7 to establish a solution of a nonlinear integral equation.

Let be a continuous function, and suppose that is symmetric (i.e., for all ). We consider the linear operator defined by . It is easy to observe that is a self-adjoint operator. Now let defined by ; that is, where is the identity of . is a self-adjoint strongly positive operator satisfying ( and signify the inner product and the norm in ).

Define , , having partial derivative of first order in the second variable and for all . defined by is a Lipschitz operator of constant from into .

Define nonlinear integral equation where . All of the conditions of Theorem 7 are satisfied, so the above-mentioned integral equation has a unique solution for all . Moreover, by Theorem 8 the solution depends continuously on the free term .

Acknowledgment

The work of first author was supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0087.

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