# On Lipschitz Perturbations of a Self-Adjoint Strongly Positive Operator

Authors: Dinu Teodorescu; N. Hussain
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

In this paper we study semilinear equations of the formAu+λF(u)=f, where A is a linear self-adjoint operator, satisfying a strong positivity condition, and F 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.


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## Body

## 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.LetH be a real Hilbert space endowed with the inner product 〈·,·〉 and the norm ∥·∥.In [1, 2] semilinear equations of the form Au-F(u)=0 were studied, where A:D(A)⊂H→H is a self-adjoint linear operator with the resolvent set R(A) and F:H→H is a Gateaux differentiable gradient operator. If there exist real numbers a<b such that [a,b]⊂R(A) and
(1)a≤〈F(u)-F(v),u-v〉∥u-v∥2≤b,
for all u,v∈H, u≠v (i.e., F interacts suitably with the spectrum of A), then it is proved in [2] that the equation Au-F(u)=0 has exactly one solution.The author in [3] presented an existence and uniqueness result for the semilinear equation Au+F(u)=f, where A:D(A)⊂H→H is a linear maximal monotone operator, satisfying a strong positivity condition, and the nonlinearity F:H→H is a Lipschitz operator.LetX be a real Banach space, ordered by a cone K. A cone K is a closed convex subset of X with λK⊆K  (λ≥0), and K∩(-K)={0}. As usual x≤y⇔y-x∈K.Definition 1.
LetM be a nonempty subset of an ordered Banach space X with order ≤. Two mappings S,T:M→M are said to be weakly isotone increasing if Sx≤TSx and Tx≤STx hold for all x∈M. Similarly, we say that S and T are weakly isotone decreasing if Tx≥STx and Sx≥TSx hold for all x∈M. The mappings S and T 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. Letℬ(X) denote the collection of all nonempty bounded subsets of X and 𝒲(X) the subset of ℬ(X) consisting of all weakly compact subsets of X. Also, let Br denote the closed ball centered at 0 with radius r.Definition 2 (see [4]).
A functionψ:ℬ(X)→ℝ+ is said to be a measure of weak noncompactness if it satisfies the following conditions.(1)
The familyker(ψ)={M∈ℬ(X):ψ(M)=0} is nonempty, and ker(ψ) is contained in the set of relatively weakly compact sets of X.(2)
M1⊆M2⇒ψ(M1)≤ψ(M2).(3)
ψ(co¯(M))=ψ(M), where co¯(M) is the closed convex hull of M.(4)
ψ(λM1+(1-λ)M2)≤λψ(M1)+(1-λ)ψ(M2) for λ∈[0,1].(5)
If(Mn)n≥1 is a sequence of nonempty weakly closed subsets of X with M1 bounded and M1⊇M2⊇⋯⊇Mn⊇⋯ such that limn→∞ψ(Mn)=0, then M∞:=⋂n=1∞‍Mn is nonempty.The familykerψ described in (1) is said to be the kernel of the measure of weak noncompactness ψ. Note that the intersection set M∞ from (5) belongs to kerψ since ψ(M∞)≤ψ(Mn) for every n, and limn→∞ψ(Mn)=0. Also, it can be easily verified that the measure ψ satisfies
(2)ψ(Mw¯)=ψ(M),
where Mw¯ is the weak closure of M.A measure of weak noncompactnessψ is said to be regular if
(3)ψ(M)=0iffMisrelativelyweaklycompact,subadditive if
(4)ψ(M1+M2)≤ψ(M1)+ψ(M2),homogeneous if
(5)ψ(λM)=|λ|ψ(M),λ∈ℝ,
and set additive (or has the maximum property) if
(6)ψ(M1∪M2)=max(ψ(M1),ψ(M2)).The first important example of a measure of weak noncompactness has been defined by de Blasi [5] as follows:
(7)w(M)=inf{r>0:thereexistsW∈𝒲(X)withM⊆W+Br},
for each M∈ℬ(X).Notice thatw(·) is regular, homogeneous, subadditive, and set additive (see [5]).By a measure of noncompactness on a Banach spaceX, we mean a map ψ:ℬ(X)→ℝ+ which satisfies conditions (1)–(5) in Definition 2 relative to the strong topology instead of the weak topology.Definition 3.
LetX be a Banach space and ψ a measure of (weak) noncompactness on X. Let A:D(A)⊆X→X be a mapping. If A(D(A)) is bounded and for every nonempty bounded subset M of D(A) with ψ(M)>0, we have ψ(A(M))<ψ(M); then A is called ψ-condensing. If there exists k, 0≤k≤1, such that A(D(A)) is bounded and for each nonempty bounded subset M of D(A), we have ψ(A(M))≤kψ(M); then A is called k-ψ-contractive.Definition 4 (see [6]).
A mapA:D(A)→X is said to be ws-compact if it is continuous, and for any weakly convergent sequence (xn)n∈ℕ in D(A) the sequence (Axn)n∈ℕ has a strongly convergent subsequence in X.Definition 5.
A mapA:D(A)→X is said to be ww-compact if it is continuous, and for any weakly convergent sequence (xn)n∈ℕ in D(A) the sequence (Axn)n∈ℕ has a weakly convergent subsequence in X.Definition 6.
LetX be a Banach space. A mapping T:D(T)⊆X→X is called a nonlinear contraction if there exists a continuous and nondecreasing function φ:ℝ+→ℝ+ such that
(8)∥Tx-Ty∥≤φ(∥x-y∥),
for all x,y∈D(T), where φ(r)<r for r>0.In this paper we consider the semilinear equation(9)Au+λF(u)=f,
where A:H→H is a linear self-adjoint operator, satisfying a strong positivity condition, F:H→H is a nonlinear Lipschitz operator, and λ 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 A of the semilinear equation with the hypothesis that A 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.
LetA:H→H be a linear self-adjoint operator and F:H→H nonlinear, satisfying the following conditions:(i)
F is a Lipschitz operator, that is, there is a constant M>0 such that
(10)∥F(x)-F(y)∥≤M∥x-y∥,for allx,y∈H;(ii)
A is a strongly positive operator, that is, there is a constant c>0 such that
(11)〈Ax,x〉≥c∥x∥2,for allx∈H.
Then the equationAu+λF(u)=f  has a unique solution for all f∈H and λ∈(0;c/M).Proof.
Let us chooses in the spectrum of A. We have
(12)s≥inf{〈Ax,x〉∣x∈H,∥x∥=1}≥c,
and we obtain that every real number ω∈(-∞;c) is in the resolvent set of the operator A. Consequently, we have
(13)Rg(A-ωI)=H,
for all ω<c, where I is the identity of H.
Letδ<0. We write (9) in the equivalent form
(14)Aδu+Fδ(u)=f,
where Aδ=A-δI  and Fδ=λF+δI. We have Rg(Aδ)=H and
(15)∥Fδ(x)-Fδ(y)∥≤(λM+|δ|)∥x-y∥,
for all x, y∈H.
Also(16)〈Aδx,x〉=〈Ax,x〉-δ∥x∥2≥(c-δ)∥x∥2=(c+|δ|)∥x∥2∀x∈H.
From (16) we obtain
(17)∥Aδx∥≥(c+|δ|)∥x∥∀x∈H.
Consequently, there existsAδ-1:H→H which is linear and continuous, that is Aδ-1∈ℒ(H), the Banach space of all linear and bounded operators from H to H. Moreover, we have
(18)∥Aδ-1∥ℒ(H)≤1c+|δ|.
Now (14) can be equivalently written as
(19)u+Aδ-1Fδu=Aδ-1f.
We consider the operatorT:H→H defined by
(20)Tu=-Aδ-1Fδu+Aδ-1f.
Therefore (19) becomes
(21)u=Tu,
and so, the problem of the solvability of (9) is reduced to the study of fixed points of the operator T. We have
(22)∥Tx-Ty∥=∥Aδ-1Fδx-Aδ-1Fδy∥=∥Aδ-1(Fδx-Fδy)∥≤∥Aδ-1∥ℒ(H)∥Fδx-Fδy∥≤λM+|δ|c+|δ|∥x-y∥∀x,y∈H.
It results thatT is a strict contraction from  H  to  H  because λM<c. According to the Banach fixed point theorem, T 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  f.Theorem 8.
Under the assumptions from the hypothesis of Theorem7, let i∈{1,2}, and let ui  be the unique solution of the equation
(23)Au+λF(u)=fi,i∈{1,2};f1,f2∈H.
Then
(24)∥u1-u2∥≤1c-λM∥f1-f2∥.Proof.
According to the equivalent form (19) of (9), we have
(25)∥u1-u2∥=∥-Aδ-1Fδu1+Aδ-1f1+Aδ-1Fδu2-Aδ-1f2∥≤∥Aδ-1Fδu1-Aδ-1Fδu2∥+∥Aδ-1f1-Aδ-1f2∥≤λM+|δ|c+|δ|∥u1-u2∥+1c+|δ|∥f1-f2∥.
It results that(26)(c-λM)∥u1-u2∥≤∥f1-f2∥,
and thus our assertion is proved.In fact, Theorem8 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 operatorA+λF is invertible for all λ∈(0;c/M) and the inverse C=(A+λF)-1 is a Lipschitz operator. From (24) it results that the operator C satisfies
(27)∥C(f1)-C(f2)∥≤1c-λM∥f1-f2∥∀f1,f2∈H.Suppose now thatc>1 and λ∈(0;(c-1)/M)⊂(0;c/M). We obtain Mλ<c-1 or 1/(c-λM)<1, and, consequently, C is a strict contraction. It results that there exists an unique element u*∈H such that C(u*)=u* which is equivalent to (A+λF)(u*)=u*. So we obtained the following fixed point theorem.Theorem 9.
LetA:H→H be a linear self-adjoint operator and F:H→H nonlinear, satisfying the following conditions:(i)
F is a Lipschitz operator, that is, there is a constant M>0 such that
(28)∥F(x)-F(y)∥≤M∥x-y∥,for allx,y∈H;(ii)
A is a strongly positive operator, that is, there is a constant c>0 such that
(29)〈Ax,x〉≥c∥x∥2,for allx∈H.
Ifc>1 and λ∈(0;(c-1)/M), then the operator A+λF has a unique fixed point.The way used in obtaining Theorem9 can be applied in the study of the following problem: extracting operators which have the unique fixed point property from a family of operators {Vλ/λ∈R}.LetE be a Banach space, and ℱ={Vλ:E→E/λ∈R}, Vλ satisfying some constraints for anyλ∈R. Our intention is to extract a subfamily 𝒢⊂ℱ, so that Vλ can have a unique fixed point for all Vλ∈𝒢. It is easy to observe, using the same method as in obtaining Theorem 9,that the following result holds.Theorem 10.
If(i)
there existsJ⊂R, J≠ϕ, so that Vλ is invertible and
(30)∥Vλ-1x-Vλ-1y∥≤g(λ)∥x-y∥,x,y∈E;g(λ)>0,for allλ∈J;(ii)
there existsI⊂J, I≠ϕ, so that g(λ)<1 for all λ∈I, then Vλ has a unique fixed point for all λ∈I.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.
LetA, F, λ, M and c be as in Theorem 9, T=(A+λF)-1, and let H be an ordered Hilbert space. Let D be a nonempty closed convex subset of H and ψ a set additive measure of noncompactness on H. Let S:H→H be a mapping satisfying the following:(i)
T(D)⊆D and S(D)⊆D,(ii)
S is continuous and ψ-condensing,(iii)
T and S are weakly isotone.
Then there exists a uniquex*∈D such that ASx*+λFSx*=x*.Proof.
By Theorem9, T=(A+λF)-1:H→H is a contraction with contractive constant α=1/(c-λM)<1. Thus T is a shrinking mapping. Now all of the conditions of Corollary 1.18 [7] are satisfied so there exists an x*∈D such that Tx*=Sx*=x* which implies that (A+λF)oSx*=x*, and hence ASx*+λFSx*=x*.Theorem 12.
LetA, F, λ, M, and c be as in Theorem 9, T=(A+λF)-1, and let H be an ordered Hilbert space. Let D be a nonempty closed convex subset of H and ψ a set additive measure of noncompactness on H. Let S:H→H be a mapping satisfying the following:(i)
T(D)⊆D and S(D)⊆D,(ii)
S is a nonlinear contraction,(iii)
T and S are weakly isotone.
Then there exists a uniquex*∈D such that ASx*+λFSx*=x*.Proof.
By Theorem9, T=(A+λF)-1:H→H is a contraction with contractive constant α=1/(c-λM)<1. Thus T is a shrinking mapping. Now all of the conditions of Corollary 1.19 [7] are satisfied so there exists an x*∈D such that Tx*=Sx*=x* which implies ASx*+λFSx*=x*.As an application of Corollaries 1.24 or 1.25 and 1.30 or 1.31 [7], we obtain the following results, respectivelyTheorem 13.
LetA, F, λ, M, and c be as in Theorem 9, T=(A+λF)-1, and let H be an ordered Hilbert space. Let D be a nonempty closed convex subset of H and ψ a set additive measure of weak noncompactness on H. Assume that S,T:H→H are sequentially weakly continuous mappings satisfying the following:(i)
T(D)⊆D and S(D)⊆D,(ii)
S is ψ-condensing or S is a nonlinear contraction,(iii)
T and S are weakly isotone.
Then there exists a unique pointx*∈D such that ASx*+λFSx*=x*.Theorem 14.
LetA, F, λ, M, and c be as in Theorem 9, T=(A+λF)-1, and let H be an ordered Hilbert space. Let D be a nonempty closed convex subset of H and ψ a set additive measure of weak noncompactness on H. Assume that S, T satisfy the following:(i)
T is a ww-compact mapping,(ii)
S is continuous ws-compact and ψ-condensing or S is continuous ws-compact and nonlinear contraction,(iii)
T and S are weakly isotone,(iv)
T(D)⊆D and S(D)⊆D.
Then there exists a uniquex*∈D such that ASx*+λFSx*=x*.Theorem 15.
LetH, A, F, λ, M, and c be as in Theorem 9 and T=(A+λF)-1 sequentially weakly continuous. Assume that D is nonempty closed bounded convex subset of H and S:D→H is sequentially weakly continuous mapping satisfying the following: (i)
S(D) is relatively weakly compact,(ii)
(x=Tx+Sy;y∈D)⇒x∈D.
Then there exists a unique pointx*∈D such that (A+λF)o(I-S)x*=x*.Proof.
By Theorem9, T=(A+λF)-1:H→H is a strict contraction with contractive constant α=1/(c-λM)<1. Now Theorem 2.1 [6] implies that there exists an x*∈D such that Sx*+Tx*=x* which implies that (A+λF)o(I-S)x*=x*.Theorem 16.
LetA, F, λ, M, and c be as in Theorem 9, T=(A+λF)-1, and let H be an ordered Hilbert space. Let D be a nonempty closed convex subset of H and ψ a set additive measure of noncompactness on H. Let S:H→H be a mapping satisfying the following:(i)
S(D)⊆D,(ii)
S is continuous and ψ-condensing or (S is a nonlinear contraction),(iii)
PT and S are weakly isotone,
whereP is the proximity map on D. Then there exists x0∈D such that
(31)∥x0-Tx0∥=∥Sx0-Tx0∥=d(Tx0,D)=d(Tx0,ID(x0)¯).More precisely, either(1)
S and T have a common fixed point x0∈D, or(2)
there existsx0∈∂D with
(32)0<∥Tx0-Sx0∥=∥Tx0-x0∥=d(Tx0,D)=d(Tx0,ID(x0)¯).Proof.
LetP be the proximity map on D; that is, for each x∈H, we have ∥Px-x∥=d(x,D). It is well known that P is nonexpansive in H. As T is shrinking map, so PT:D→D is also shrinking mapping. By Theorem 4.1 (or 4.2), there exists x0∈M such that x0=Sx0=PTx0. 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.
LetA, F, λ, M, and c be as in Theorem 9, T=(A+λF)-1, and let H be an ordered Hilbert space. Let D be a nonempty closed convex subset of H and ψ a set additive measure of noncompactness on H. Assume that S:H→H is a mapping satisfying the following:(i)
S(D)⊆D,(ii)
S is continuous and ψ-condensing or (S is a nonlinear contraction),(iii)
PT and S are weakly isotone,
whereP is the proximity map on D. Suppose that T satisfies one of the following conditions for each x∈∂D, with x≠Tx:(i)
∥Tx-y∥<∥Tx-x∥ for some y in ID(x)¯;(ii)
there is aγ such that |γ|<1 and γx+(1-γ)Tx∈ID(x)¯;(iii)
Tx∈ID(x)¯;(iv)
for eachγ∈(0,1), x≠γTx;(v)
there exists anα∈(1,∞) such that, ∥Tx-x∥α≥∥Tx∥α-rα;(vi)
there existsβ∈(0,1) such that, ∥Tx-x∥β≤∥Tx∥β-rβ.
ThenF(S)∩F(T)≠∅.


## 4. An Application

Fixed point theorems for certain operators have found various applications in differential and integral equations (see [7–10] and references therein). In this section, we present an application of our Theorem 7 to establish a solution of a nonlinear integral equation.LetK:[0,1]×[0,1]→R be a continuous function, and suppose that K is symmetric (i.e., K(x,y)=K(y,x) for all x,y∈[0,1]). We consider the linear operator B:LR2[0,1]→LR2[0,1] defined by Bu(t)=∫01K(t,s)u(s)ds. It is easy to observe that B is a self-adjoint operator. Now let A:LR2[0,1]→LR2[0,1] defined by Au(t)=u(t)+∬01K(t,s)K(s,z)u(z)dzds; that is, A=I+B2 where I is the identity of LR2[0,1]. A is a self-adjoint strongly positive operator satisfying 〈Au,u〉2=〈u,u〉2+〈B2u,u〉2≥∥u∥22(〈·,·〉2 and ∥·∥2 signify the inner product and the norm in LR2[0,1]).Definef:[0,1]×R→R, (t,u)↦f(t,u), having partial derivative of first order in the second variable u and |(∂f/∂u)(t,u)|≤M for all (t,u)∈[0,1]×R(M>0). F:LR2[0,1]→LR2[0,1] defined by Fu(t)=f(t,u(t)) is a Lipschitz operator of constant M from LR2[0,1] into LR2[0,1].Define nonlinear integral equation(33)u(t)+∫01∫01K(t,s)K(s,z)u(z)dzds+λf(t,u(t))=g(t),g∈LR2[0,1],
where λ∈(0;1/M). All of the conditions of Theorem 7 are satisfied, so the above-mentioned integral equation has a unique solution for all λ∈(0;1/M). Moreover, by Theorem 8 the solution depends continuously on the free term g.



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