Volume 2009, Article ID 707406,6pages doi:10.1155/2009/707406
Research Article
A Note on Implicit Functions in
Locally Convex Spaces
Marianna Tavernise and Alessandro Trombetta
Dipartimento di Matematica, Universit`a degli Studi della Calabria, 87036 Arcavacata di Rende (CS), Italy
Correspondence should be addressed to Marianna Tavernise,[email protected]
Received 27 February 2009; Accepted 19 October 2009
Recommended by Fabio Zanolin
An implicit function theorem in locally convex spaces is proved. As an application we study the stability, with respect to a parameterλ, of the solutions of the Hammerstein equationxλKFxin a locally convex space.
Copyrightq2009 M. Tavernise and A. Trombetta. 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.
1. Introduction
Implicit function theorems are an important tool in nonlinear analysis. They have significant applications in the theory of nonlinear integral equations. One of the most important results is the classic Hildebrandt-Graves theorem. The main assumption in all its formulations is some differentiability requirement. Applying this theorem to various types of Hammerstein integral equations in Banach spaces, it turned out that the hypothesis of existence and continuity of the derivative of the operators related to the studied equation is too restrictive. In1it is introduced an interesting linearization property for parameter dependent operators in Banach spaces. Moreover, it is proved a generalization of the Hildebrandt-Graves theorem which implies easily the second averaging theorem of Bogoljubov for ordinary differential equations on the real line.
LetX X, · XandY Y, · Ybe Banach spaces,Λan open subset of the real lineRor of the complex planeC,Aan open subset of the product spaceΛ×XandLX, Y the space of all continuous linear operators fromX intoY. An operator Φ : A → Y and an operator functionL : Λ → LX, Yare calledosculatingatλ0, x0 ∈ Aif there exists a functionσ:R2 → 0,∞such that lim
ρ,r→0,0σρ, r 0 and
Φλ, x1−Φλ, x2−Lλx1−x2Y ≤σρ, rx1−x2X, 1.1
The notion of osculating operators has been considered from different points of view
see2,3. In this note we reformulate the definition of osculating operators. Our setting is a locally convex topological vector space. Moreover, we present a new implicit function theorem and, as an example of application, we study the solutions of an Hammerstein equation containing a parameter.
2. Preliminaries
Before providing the main results, we need to introduce some basic facts about locally convex topological vector spaces. We give these definitions following4–6. LetX be a Hausdorff locally convex topological vector space over the fieldK, whereKRorKC. A family of continuous seminormsPwhich induces the topology ofXis called acalibrationforX. Denote byPXthe set of all calibrations forX. Abasic calibrationforX isP ∈ PXsuch that the collection of all
Uε, px∈X:px≤ε, ε >0, p∈P 2.1
is a neighborhood base at 0. Observe thatP ∈ PXis a basic calibration forX if and only if for eachp1, p2 ∈ P there isp0 ∈ P such thatpix ≤ p0xfori 1,2 andx ∈ X. Given P ∈ PX, the family of all maxima of finite subfamily ofPis a basic calibration.
A linear operatorLonXis calledP-boundedif there exists a constantC >0 such that
pLx≤Cpx, x∈X, p∈P. 2.2
Denote byLXthe space of all continuous linear operators onXand byBPXthe space of
allP-bounded linear operatorsLonX. We haveBPX⊂ LX. Moreover, the spaceBPX is a unital normed algebra with respect to the norm
LP suppLx:x∈X, p∈P, px 1. 2.3
We say that a family{Lα : α ∈ I} ⊂ BPXisuniformlyP-boundedif there exists a constant C >0 such that
pLαx≤Cpx, x∈X, p∈P 2.4
for anyα∈I.
3. Main Result
LetΛbe an open subset of the real lineRor of the complex planeC. Consider the product spaceΛ×XofΛandXprovided with the product topology. LetAbe an open subset ofΛ×X andλ0, x0∈A. Consider a nonlinear operatorΦ:A → Xand the related equation
Φλ, x 0. 3.1
Assume thatλ0, x0is a solution of the above equation. A fundamental problem in nonlinear analysis is to study solutionsλ, xof3.1forλclose toλ0.
We say that an operatorΦ : A → X and an operator L : Λ → LX are called
P-osculating at λ0, x0 if there exist a function σ : R2 → 0,∞ and q ∈ P such that
limρ,r→0,0σρ, r 0 and for anyp∈P
pΦλ, x1−Φλ, x2−Lλx1−x2≤σρ, rpx1−x2, 3.2
when|λ−λ0| ≤ρandx1, x2∈x0Ur, q. Now we prove our main result.
Theorem 3.1. Suppose thatΦ:A → Xandλ0, x0satisfy the following conditions:
a λ0, x0is a solution of 3.1and the operatorΦ·, x0is continuous atλ0;
bthere exists an operator functionL : Λ → LXsuch thatΦandLareP-osculating at
λ0, x0;
cthe linear operatorLλis invertible andLλ−1 ∈ BPXfor eachλ ∈ Λ. Moreover the family{Lλ−1:λ∈Λ}is uniformlyP-bounded.
Then there areε >0,q∈P andδ >0such that, for eachλ∈Λwith|λ−λ0| ≤δ,3.1has a unique solutionxλ∈x0Uε, q.
Proof. LetΦandL:Λ → LXbeP-osculating atλ0, x0. Consider the operatorT :A → X defined by
Tλ, x x−Lλ−1Φλ, x. 3.3
Letp∈P. By the assumptioncthere existsC >0 such that
pTλ, x1−Tλ, x2≤CpΦλ, x1−Φλ, x2−Lλx1−x2 3.4
for anyλ, x1,λ, x2 ∈A. Moreover, sinceΦandLareP-osculating atλ0, x0, there are a functionσ:R2 → 0,∞andq∈Psuch that
for|λ−λ0| ≤ρandx1, x2∈x0Ur, q. Hence
pTλ, x1−Tλ, x2≤Cσρ, rpx1−x2 3.6
for|λ−λ0| ≤ρandx1, x2∈x0Ur, q. Chooseε >0 such that
pTλ, x1−Tλ, x2≤ 1
2px1−x2 3.7
for|λ−λ0| ≤εandx1, x2 ∈x0Uε, q. Therefore, for eachλ∈Λsuch that|λ−λ0| ≤ε, the operatorTλ,·fromx0Uε, qintoXis a contraction in the sense of7.
SinceΦ·, x0is continuous atλ0, we may further findδ>0 such that
pΦλ, x0≤ ε 2C,
pTλ, x0−x0≤CpΦλ, x0≤ ε 2
3.8
for|λ−λ0| ≤δ. Setδ:min{ε, δ}we have
pTλ, x−x0≤pTλ, x−Tλ, x0 pTλ, x0−x0≤ ε 2
ε
2 ε 3.9
for|λ−λ0| ≤δandx∈x0Uε, q. This shows that
Tλ,·x0Uε, q⊆x0Uε, q 3.10
for eachλsuch that|λ−λ0| ≤ δ. Then, by7, Theorem 1.1, when|λ−λ0| ≤ δ, the operator
Tλ,·has a unique fixed pointxλ∈x0Uε, q, which is obviously a solution of3.1.
4. An Application
As an example of application of our main result, we study the stability of the solutions of an operator equation with respect to a parameter.
Consider inXthe Hammerstein equation
xλKFx, 4.1
containing a parameterλ ∈ Λ. In our case Kis a continuous linear operator onX andF :
X → Xis the so-called superposition operator. We have the following theorem.
Theorem 4.1. LetK beP-bounded. Suppose that for eachx ∈ Xthere existsq ∈ P such that the operatorFsatisfies the Lipschitz condition
for anyp ∈Pandx1, x2 ∈xUr, q, wherelimr→0ωr 0. Ifx0 ∈Xis a solution of 4.1for λλ0, then there existε >0 andδ >0such that, for eachλ∈Λwith|λ−λ0| ≤δ,4.1has a unique solutionxλ∈x0Uε, q.
Proof. Since the linear operatorKisP-bounded, we can find a constantC >0 such that
pKx≤Cpx, x∈X, p∈P. 4.3
Ifλ 0, thenx0 0 is clearly a solution of4.1. Consider the operator Φ0 : Λ×X → X
defined by
Φ0λ, x x−λKFx, 4.4
and setL0λx xfor anyλ ∈Λandx∈X. Clearly the operatorΦ·,0is continuous at 0. By the hypothesis made on the operatorF, there existsq∈P such that
pΦ0λ, x1−Φ0λ, x2−L0λx1−x2≤Cρωrpx1−x2 4.5
for anyp ∈ P; when|λ| ≤ ρandx1, x2 ∈ Ur, q, the operatorsΦ0 andL0 areP-osculating
at0,0. Moreover, for eachλ ∈Λ, we haveL0λ−1 L0λandpL0λ−1x pxfor any
x ∈ X andp ∈ P. Then the result follows byTheorem 3.1. Now assume that x0 ∈ X is a solution of4.1for someλ0/0. LetΦ:Λ×X → Xbe defined by
Φλ, x xλ −KFx, 4.6
and setLλxx/λfor anyλ∈Λandx∈X. The operatorΦ·, x0is continuous atλ0and there existsq∈Psuch that
pΦλ, x1−Φλ, x2−Lλx1−x2≤Cωrpx1−x2 4.7
for any p ∈ P, when λ ∈ Λ and x1, x2 ∈ x0 Ur, q. So the operatorsΦ and L are P -osculating at λ0, x0. Further, assuming |λ−λ0| ≤ a for somea > 0, we can find b > 0 such thatpLλ−1x≤bpxfor anyp∈Pandx∈X. As before, the proof is completed by appealing toTheorem 3.1.
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