SOLVABILITY OF NONLINEAR EQUATIONS
PETRONELA CATAN ˘A
Using some numerical characteristics for nonlinear operatorsFacting between two Banach spacesXandY, we discuss the solvability of a linear ecuationλx−Lx= y, y∈X.We extend the spectral sets defined by means of lower characteristics to discuss the solvability of the nonlinear equationλx−F(x) =y, y∈X.
AMS 2000 Subject Classification: 47J10.
Key words: measure of noncompactness, epi andk-epi operators, measure of solva- bility and stable solvability, nonlinear integral equation.
1. INTRODUCTION
We use some numerical characteristics for nonlinear operators F be- tween two Banach spaces X and Y over K (see [3], [2]), to describe mapping properties of F, such as compactness, Lipschitz continuity or quasibounded- ness. We consider several subsets ofK by means of the lower characteristics [F]Lip,[F]q,[F]b and [F]a,defined below, because these give us information on the solvability of the linear equation λx−Lx=y,y ∈X. Our idea is to use these sets to provide information on the solvability of the nonlinear equation λx−F(x) =y,y∈X.
We will consider a more general problem of the formλJ(x)−F(x) =y, y∈X,whereF andJ are continuous nonlinear operators between the Banach spaces X and Y. Using the measure of solvability of F and the homotopy property of k-epi operators, we give a result for stably solvable operators, which can be proved as a direct consequence of the Rouch´e type estimate for stably solvable operators, involving the measure of stable solvability ofF.
These results are ilustrated by means of applications to nonlinear integral equation. We namely consider a Hammerstein integral equation and a Uryson integral equation of second kind.
2. PRELIMINARIES
LetX andY be two Banach spaces overKandF :X→Y a continuous operator. We recall a useful topological characteristic in the theory and appli- cations of both linear and nonlinear analysis. The measure of noncompactness
MATH. REPORTS9(59),3 (2007), 249–256
of a bounded subsetM of X is defined by
(2.1) α(M) = inf{: >0, M has a finite -net in X}.
Here, by a finite-net forM we understand a finite set{x1, . . . , xn} ⊂Xwith the property thatM ⊆[x1+B(θ)]∪ · · ·∪[xn+B(θ)],for the closed ball with centreθ and radius >0 in X.
Given the set F ∈ C(X, Y) of all continuous operators from X into Y, we define (see [2])
(2.2) [F]Lip= sup
x=y
F(x)−F(y)
x−y and [F]Lip= inf
x=y
F(x)−F(y) x−y
and write F ∈Lip(X, Y) if [F]Lip <∞; [F]Lip= 0 means that F is constant.
We also consider another characteristics (2.3) [F]Q = lim
x→∞supF(x)
x and [F]q= lim
x→∞infF(x) x
ofF ∈C(X, Y),and we writeF ∈Q(X, Y) if [F]Q<∞and call the operator F quasibounded; [F]Q= 0 means that F has strictly sublinear growth, more precisely,
F(x)=o(x) asx → ∞.
We also consider
(2.4) [F]B= sup
x=0
F(x)
x and [F]b = inf
x=0
F(x) x
and writeF ∈B(X, Y) if [F]B<∞and call the operatorF linearly bounded;
[F]B impliesF = Θ.
Let X and Y be two infinite dimensional Banach spaces. Recall that a continuous operatorF :X→Y is said to beα-Lipschitz if there existsk >0 such thatα(F(M))≤kα(M) for any bounded subset M ⊂X. Set
(2.5) [F]A= inf{k:k >0, α(F(M))≤kα(M)}).
We say that [F]A is the measure of noncompactness ofF or theα-norm ofF; if [F]A ≤1 the operator F is called α-nonexpansive and α-contractive if the inequality is strict. We also introduce the lower characteristic
(2.6) [F]a= sup{k:k >0, α(F(M))≥kα(M)}.
As in the linear case, equivalent representation, in infinite dimensional spaces are useful:
(2.7) [F]A= sup
α(M)>0
α(F(M))
α(M) and [F]a= inf
α(M)>0
α(F(M)) α(M) .
We now introduce several subsets of K by means of the lower charac- teristics [F]Lip,[F]q,[F]b and [F]a (see [2]):
(2.8) σLip(F) ={λ∈K: [λI−F]Lip= 0}, σq(F) ={λ∈K: [λI−F]q= 0}, σb(F) ={λ∈K: [λI−F]b= 0}, σa(F) ={λ∈K: [λI−F]a= 0}.
For F ∈ L(X), these subspectra give information about the solvability of the linear equation
(2.9) λx−Lx=y, y∈X.
Our idea is to use the spectral sets to provide information on the solv- ability of the nonlinear equation
(2.10) λx−F(x) =y, y∈X.
Ifλ∈σLip(F) then the operator λI−F is injective and equation (2.10) has at most one solution for a fixedy.The relationλ∈ {σLip(F), σq(F), σb(F)}
does not imply the surjectivity of the operator λI −F, not even in the lin- ear case.
3. THE MEASURE OF SOLVABILITY OF F
We consider a general problem of the form (3.1) λJ(x)−F(x) =y, y ∈Y,
whereF andJ are continuous nonlinear operators between two Banach spaces X and Y.
Definition 3.1 (see [2]). Let X and Y be Banach spaces over K. Denote by F(X) the family of all open, bounded, connected subsets Ω of X with θ ∈ Ω.A continuous operator F : Ω → Y is called epi on Ω if F(x) =θ on
∂Ω and, for any compact operator G: Ω→Y satisfying G(x)≡0 on∂Ω,the equationF(x) =G(x) has a solution x∈Ω.More generaly, we call F a k-epi operator on Ω,k≥0 if the property mentioned before holds for all operators with [G]A≤k(not only for compact operators).
ForF : Ω→Y and Ω∈F(X) as before, we introduce (3.2) νΩ(F) = inf{k:k >0, F is not k-epi on Ω}
(3.3) ν(F) = inf
Ω∈F(X)νΩ(F), whereν(F) stands for the measure of solvability of F.
The homotopy property gives a continuation principle for epi and k-epi operators. It may be compared with its analogue property of the topological degree. We recall the homotopy property. Suppose thatF0 : Ω→Y isk0-epi on Ω for somek0 ≥0, that H: Ω×[0,1]→Y is continuous with H(x,0)≡θ
and α(H(M ×[0,1])) ≤ kα(M), M ⊆ Ω for some k ≤ k0. Let S = {x ∈ Ω :F0(x) +H(x, t) =θ for some t∈ [0,1]}. If S∩∂Ω =∅ then the operator F1 =F0+H(·,1) isk1-epi on Ω for k1 ≤k0−k.
Theorem 3.1. Let F ∈ H(X, Y) and J : X → Y with ν(J) > 0. Fix λ∈Kwith|λ|ν(J)>[F]A and let
(3.4) S={x∈X:λJ(x) =tF(x) for some t∈(0,1]}.
Then eitherS is bounded, or the operatorλJ−F isk-epi onΩ for some Ω∈F(X) and every k≤ |λ|ν(J)−[F]A.
Proof. Applying the homotopy property ofk-epi operators to the opera- torG=λJ and the homotopy H(x, t) =−tF(x), we getα(H(M ×[0,1])) ≤ α(co(F(M)∪ {θ})) =α(F(M))≤[F]Aα(M), M ⊆Ω.
If S is bounded, we may find Ω∈F(X) such that S∩∂Ω = ∅. Again, from homotopy, we conclude that the operatorG+H(·,1) =λJ−F is k-epi on Ω,for 0≤k≤ |λ|ν(J)−[F]A.
Using a Rouch´e type estimate, one can show that
|λ| sup
x∈∂ΩJ(x)< inf
x∈∂ΩF(x), Ω∈F(x), without using the setS.
Definition 3.2 (see [6]). We call stably solvable a continuous operator F : X → Y if, given any compact operator G :X → Y with [G]Q = 0, the equationF(x) =G(x) has a solution x∈X.
Remark.Every stable solvable operator is surjective, takeG(X)≡y,but the converse is not true.
ForF ∈C(X, Y) is called the number
(3.5) µ(F) = inf{k:k≥0, F is not k-stably solvable}
the measure of stable solvability of F. On account of this definition we may use a Rouch´e type inequality, namely,
(3.6) µ(F+G)≥µ(F)−max{[G]A,[G]Q} forF, G∈C(X, Y) and the following result holds.
Lemma 3.1. Let F, G ∈ C(X, Y). If F is k-stably solvable with k ≥ [G]A and k ≥ [G]Q, then F +G is k-stably solvable for 0 ≤ k ≤ k − max{[F]A,[F]Q}.
We next have
Lemma 3.2. Suppose thatF : Ω→ Y is strictly epi on Ωand G: Ω→Y satisfies supx∈∂ΩG(x)<dist(θ, F(∂Ω)) and [G|Ω]< νΩ(F). Then F +G is strictly epi on Ω.
Theproof of Lemma 3.2 (see[2]) shows that the Rouch´e type inequality (3.7) ν(F +G)≥ν(F)−[G]A
holds for the characteristicν(F),thus paralleling (3.6).
The next lemma connects the measure of solvability and the measure of stable solvability ofF.
Lemma 3.3. For anyF ∈C(X, Y) we have µ(F)≤ν(F).
The next theorem gives a similar result for stably solvable operators and can be proved using the Rouch´e type estimate (3.6) for stably solvable operators.
Theorem 3.2. Suppose that F ∈H(X, Y)∩Q(X, Y) and J : X → Y satisfies µ(J) > 0. Fix λ ∈ K with |λ|µ(J) > max{[F]A,[F]Q}. Then the operator λJ−F is k-stably solvable for every k≤ |λ|µ(J)−max{[F]A,[F]Q}.
In particular, equation (3.1) has a solution x∈X for every y∈Y.
4. SOME APPLICATIONS INVOLVING NONLINEAR INTEGRAL EQUATION
(I) Let us first consider a Hammerstein integral equation of the form (4.1) λx(s)−
1
0 k(s, t)f(t, x(t)) dt=y(s), 0≤s≤1. The nonlinear Hammerstein operator
(4.2) H(x)(s) = 1
0 k(s, t)f(t, x(t)) dt
can be used as a compositionH =KF of the nonlinear Nemytskij operator (4.3) F(x)(t) =f(t, x(t))
generated by the nonlinearity off and the linear integral operator
(4.4) ky(s) =
1
0 k(s, t)y(t) dt generated by the kernel functionk.
Assume thatk: [0,1]×[0,1]→Ris continuous while f : [0,1]×R→R satisfies a Carath´eodory condition and a growth condition of the form
(4.5) |f(t, u)| ≤a(t) +b(t)|u|, 0≤t≤1, u∈R, with functionsa, b∈L1[0,1].
We writex1 for theL1-norm and define a scalar functionh by
(4.6) h(t) = max
0≤s≤1|k(s, t)|, 0≤t≤1.
Proposition 4.1(see [2]).Suppose that |λ|>hb1.Then equation(4.1) has a solution x ∈C[0,1] for y(s) ≡0. Moreover, if a(t) ≡0 in (4.5), then equation (4.1) has a solution x∈C[0,1] for every y∈C[0,1].
Proof.We apply Theorem 3.2 withX=Y =C[0,1], J =I. The nonlin- ear Hammerstein operator (4.2) being compact inX, we see that [λI−H]a =
|λ|>0 and distinguish two cases forλ.
First, suppose thatλ /∈σb(H) i.e. [λI−H]b>0.Consider the set (4.7) S ={x∈X:λx=tH(x) for some t∈(0,1]}.
Forx∈S we have
|λ|x∞≤ H(x)∞≤ ha1+hb1x∞, hence x∞≤ ha1
|λ| − hb1. So, the setS defined by (4.7) is bounded and Theorem 3.2 implies that the operator λI −H is k-epi on X for k < |λ|, i.e., ν(λI −H) > 0. The assumption [λI−H]b >0 implies thatλ∈ρF(H),so the equation H(x) =λx has a solution.
Second, suppose thatλ∈σb(H), i.e., [λI−H]b = 0.Then we can find a sequence{xn} ∈X such that
λxn−H(xn)∞≤ 1 nxn∞ and
|λ|x∞− ha1− hb1x∞≤ 1
nxn∞.
Hence (|λ| − hb1 −n1)xn∞ ≤ ha1, i.e., {xn} is bounded because |λ| >
hb1.Consequently, λxn−H(xn)∞→0 as n→ ∞.LetM :={x1, x2, . . .}
and [λI −H]aα(M) ≤ α((λI −H)(M)) = 0. Then {xn} has a convergent subsequence and its limit is a solution of the equationH(x) =λx.
Now, assume that a(t) ≡ 0. Then Feng’s spectral radius defined by rF(H) = sup{|λ|:λ∈ρF(H)}, whereρF(H) ={λ∈K:λI−H is F-regular} is the Feng resolvent set, satisfies
rF(H)≤max{[H]A,[H]B}= sup
x=θ
H(x)∞
x∞ ≤ hb1, soλ∈ρF(H) for |λ|>hb1.
(II) Another application refers to Uryson integral equation of the second kind
(4.8) λx(s)− 1
0 k(s, t, x(t)) dt= 0, 0≤s≤1.
We shall study the nonlinear Uryson operator
(4.9) U(x)(s) =
1
0 k(s, t, x(t)) dt generated by (4.8) in the spaceL2[0,1].
About the continuous nonlinear kernel functionk: [0,1]×[0,1]→R we make the following assumptions:
(4.10) sup
|u|≤r|k(s, t, u)| ≤βr(s, t) withMr = sup
0≤s≤1
1
0 βr(s, t) dt <∞, (4.11) sup
|u|≤r|k(s, t, u)−k(σ, t, u)| ≤γr(s, σ, t) with lim
s→σ
1
0 γr(s, σ, t) dt= 0, (4.12) |k(s, t, u)| ≤Ψ(s, t)(1 +|u|) withM =
1
0 Ψ(s, t)2dtds <∞.
Proposition 4.2 (see [2]). Suppose that|λ|>4M. Then equation (4.8) has a solution x∈L2[0,1].
Proof. We apply Theorem 3.2 with X = Y = L2[0,1] and J = I.
The nonlinear Uryson operator (4.9) is compact inX,under the assumptions (4.10)–(4.12). For any x∈X we have
|U(x)(s)|2 = 1
0 k(s, t, x(t)) dt 2
≤ 1
0 Ψ(s, t)(1 +|x(t)|) dt 2
≤
≤ 1
0 Ψ(s, t)2dt 1
0 (1 +|x(t)|)2dt
≤4 1
0 Ψ(s, t)2dt
(1 +x2). (We have used the fact that (a+b)p≤2p(ap+bp) for a, b≥0 andp≥1.) So, we have
U(x)2 ≤4
1
0 Ψ(s, t)2dtds
(1 +x2)≤4M(1 +x2). Again, we can distinguish two cases: [λI−U]b >0 and [λI−U]b = 0 In the first case, the set S = {x ∈ X : λx = tU(x) fort ∈ (0,1]} is bounded because for everyx∈S we have
|λ|2x2 ≤ U(x)2 ≤4M(1 +x2), hence
x2 ≤ 4M
|λ|2−4M.
By Theorem 3.2, again, the operator λI −U is k-epi for k < |λ|, so that λ∈ρF(U).
In the second case, we can find a sequence{xn} inX such thatλxn− U(xn) ≤ xn/n. As before, the sequence{xn} is bounded and the estimate
1
nxn ≥ |λ|xn − U(xn) ≥ |λ|xn −2√
M
1 +xn2 implies that
|λ| −2√ M
1 xn2 + 1
1
2 ≤ 1
n.
Letting n→ ∞, the unboundedness of {xn} would give |λ| ≤ 2√
M, contra- dicting the choice ofλ.So, we proved that{xn}is bounded and the proof can be completed as in Proposition 4.1.
Acknowledgements. The author wants to express her gratitude to Professor Dan Pascali for his support in the preparation of this paper.
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Received 6 January 2007 “Carmen Sylva” High-School
Eforie Sud, Constant¸a, Romania petronela catana@yahoo.com
