On locally condensing operators

Nonlinear Analysis 75 (2012) 3552–3557

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On locally condensing operators Nina A. Erzakova Moscow State Technical University of Civil Aviation, 125993 Moscow, Russian Federation

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Article history: Received 18 August 2011 Accepted 12 January 2012 Communicated by S. Carl MSC: primary 47H08 secondary 47H10 47H20 47J05 47J15

abstract We introduce the notion of a locally strongly condensing operator. This is a modification of the definition given by Nussbaum. Such operators form a linear space and a near-ring. We discuss some applications of the notion to nonlinear equations and the theory of bifurcation points. © 2012 Elsevier Ltd. All rights reserved.

Keywords: Measure of non-compactness Condensing map Near-ring Bifurcation point Nonlinear equation Fréchet derivative

1. Introduction We first recall basic definitions and notation [1, 1.2.1] in a form convenient for us. Let ψ be a scalar function defined on the set of all subsets of a Banach space E. We say that ψ is a measure of noncompactness on E if ψ(coU ) = ψ(U ) for all U ⊂ E where coU is the convex closure of U. Let E and E1 be Banach spaces. A continuous (not necessarily linear) operator f : G ⊆ E → E1 is called condensing [1, 1.5.1] if ψE1 (f (U )) < ψE (U ) for any bounded set U ⊆ G with non-compact closure. A continuous operator f : G ⊂ E → E1 is called (k, ψ)-bounded if there exists a constant k > 0 such that ψE1 (f (U )) 6 kψE (U ) for all U ⊆ G. A continuous map f : M → E, where M is an open subset of the Banach space E, is called locally strictly ψ -condensing (see Nussbaum’s paper [2] and [1, 1.8.12]) if for any point x ∈ M there exist a neighborhood Vx of x and a number kx < 1 such that the restriction f |Vx is (kx , ψ)-bounded. Denote by B(x, r ) the open ball of radius r with the center at the point x and by R+ the set of all positive real numbers. We say that a continuous map f : M ⊆ E → E1 is locally strongly ψ -condensing on M if there exists a function λM ,f : R+ → R+ , lim λM ,f (r ) = 0,

r →0

(1)

such that

ψE1 (f (U )) 6 λM ,f (r )ψE (U ) for any x ∈ M and U ⊆ B(x, r ) ∩ M. E-mail address: [email protected]. 0362-546X/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2012.01.014

(2)

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In the propositions below a measure of non-compactness satisfies some of the following properties [1, 1.1.4]: (1) semi-homogeneity: ψE (tU ) = |t |ψE (U ) (t is a number); (2) algebraic semi-additivity: ψE (U + V ) ≤ ψE (U ) + ψE (V ) where U + V = {u + v : u ∈ U , v ∈ V }; (3) continuity with respect to the Hausdorff metric ρ :

∀(U , ε > 0)∃(δ > 0)∀(U1 )[ρ(U , U1 ) < δ ⇒ |ψE (U ) − ψE (U1 )| < ε], where ρ(U , U1 ) = inf{ε > 0 : U1 ⊂ U + ε B, U ⊂ U1 + ε B}, B = {x ∈ E : ∥x∥ 6 1}; (4) regularity: ψE (U ) = 0 if and only if the closure of U is compact; (5) semi-additivity: ψE (U ∪ V ) = max{ψE (U ), ψE (V )}; (6) invariance under translations: ψE (U + u) = ψE (U ) (u ∈ E ). We say that a measure of non-compactness ψ satisfies the property of boundedness if ψE (U ) < ∞ if and only if U ⊆ E is bounded in E. Lemma 1. Let f : M ⊆ E → E1 be a locally strongly ψ -condensing operator on M. Let the measure of non-compactness ψ satisfy the property of boundedness. Then for all bounded sets U ⊆ M, sets f (U ) are bounded too. Proof. Let U be a bounded subset of M. Then U ⊆ (B(x, r ) ∩ M ) for some ball B(x, r ). By the property of boundedness, ψE (U ) < ∞. Hence (2) implies that ψE1 (f (U )) < ∞. Therefore f (U ) is bounded in E1 by the property of boundedness.  2. The near-ring of locally strongly condensing operators We recall [3, p. 7] that a set N with two binary operations + (called addition) and · (called multiplication) is called a (right) near-ring if: A1: N is a group (not necessarily Abelian) under addition; A2: multiplication is associative (so N is a semigroup under multiplication); and A3: multiplication distributes over addition on the right: for any f , g and h in N, it holds that (g + h)f = gf + hf . Theorem 1. Let ψ be a measure of non-compactness satisfying the properties of boundedness, semi-homogeneity, and algebraic semi-additivity. Then the set Λ of all (not necessarily linear) operators f : E → E locally strongly ψ -condensing on every bounded set M ⊂ E forms a linear space, a semigroup with respect to multiplication, and a right near-ring. Proof. Let M be an arbitrary bounded subset of E. We prove that any finite real linear combination of locally strongly ψ -condensing operators on M is a locally strongly ψ -condensing operator on M again. Indeed, let f and g be locally strongly ψ -condensing operators on M. Then for any x ∈ M and U ⊆ B(x, r ) ∩ M we have

ψ((c1 f + c2 g )(U )) = ψ(c1 f (U ) + c2 g (U )) 6 |c1 |λM ,f (r )ψ(U ) + |c2 |λM ,g (r )ψ(U ) = (|c1 |λM ,f (r ) + |c2 |λM ,g (r ))ψ(U ). Taking λM ,c1 f +c2 g (r ) = |c1 |λM ,f (r ) + |c2 |λM ,g (r ), we obtain a function satisfying (1) and (2). Therefore, the set Λ of all the operators f : E → E locally strongly ψ -condensing on every bounded set M ⊂ E forms a linear space over a field of real numbers and, in particular, an Abelian group with respect to addition. Now we show that Λ is a semigroup with respect to multiplication. Let f , g ∈ Λ. For any bounded set M there exists R > 0 such that M ⊆ B(θ , R) where θ is a zero element of E. By the assumption, g is locally strongly ψ -condensing on every bounded set and, in particular, on B(θ , R). Thus for any x ∈ E, r > 0, and U ⊆ B(x, r ) ∩ M ⊆ B(x, r ) ∩ B(θ , R) we have

ψ(g (U )) 6 λB(θ,R),g (r )ψ(U ).

(3)

Recall that ψ satisfies the property of boundedness. Then, by Lemma 1, g (M ) is bounded in E. Thus g (M ) ⊂ B(θ , R1 ) for some R1 > 0. By the assumption, f is a locally strongly ψ -condensing operator on every bounded set in E too and, in particular, on B(θ, R1 ). Thus ψ(f (g (U ))) 6 λB(θ,R1 ),f (R1 )ψ(g (U )) where λB(θ,R1 ),f satisfies (1) and (2). Then

ψ((fg )(U )) = ψ(f (g (U ))) 6 λB(θ,R1 ),f (R1 )ψ(g (U )) 6 λB(θ,R1 ),f (R1 )λB(θ,R),g (r )ψ(U ). Therefore, fg ∈ Λ with

λM ,fg (r ) =

inf

R,R1 :M ⊆B(θ,R),g (M )⊆B(θ,R1 )

λB(θ,R1 ),f (R1 )λB(θ,R),g (r ).

Let f , g , h ∈ Λ. Then (g + h)f = gf + hf . Hence Λ is a right near-ring.



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Remarks. (1) The product of a (k, ψ)-bounded operator E1 → E2 and a locally strongly ψ -condensing operator E → E1 is a locally strongly ψ -condensing operator E → E2 . (2) If f · (g + h) = fg + fh for all f , g , h ∈ Λ, then Λ is a ring. (3) One can construct examples in which a sum of locally strictly condensing operators (in Nussbaum’s sense) is not locally strictly condensing and a product of a (k, ψ)-bounded operator and a locally strictly condensing operator is not locally strictly condensing either. 3. Examples of locally strongly condensing operators Let Ω ⊂ Rn , µ(Ω ) < ∞, where µ is a continuous measure on Ω , i.e., for every D ⊂ Ω , µ(D) > 0, there exist D1 , D2 ⊂ D, µ(D) D1 ∪ D2 = D, D1 ∩ D2 = ∅, µ(D1 ) = µ(D2 ) = 2 . (We follow [4] here.) Considerthe spaces Lp = Lp (Ω ), 1 6 p < ∞. For every measurable D ⊆ Ω denote by PD : Lp → Lp the operator

(PD u)(x) =

u(x), 0,

x ∈ D, x ̸∈ D.

Denote by νLp (U ) the measure of nonequiabsolute continuity of norms of elements of U in Lp . In other words,

νLp (U ) = lim sup ∥PD u∥Lp . µ(D)→0 u∈U

Then νLp is a measure of non-compactness in Lp (Ω ). Example 1. Let b(·) ∈ Lp and a > 0. Consider a continuous operator F1 : Lq → Lp , q > p, defined by F1 (u)(s) = b(s) + a · sgn(u(s))|u(s)|q/p . Then F1 is a locally strongly ν -condensing operator on Lp with λLq ,F1 (r ) = ar q/p−1 . Proof. Consider U ⊆ BLq (u1 , r ) where BLq (u1 , r ) is a ball. We have

νLp (F1 U ) = lim sup ∥PD F1 (u)∥Lp µ(D)→0 u∈U

= lim sup ∥PD (b(·) + a · sgn(u(·))|u(·)|q/p )∥Lp µ(D)→0 u∈U

q/ p

= lim sup a∥PD |u|q/p ∥Lp = lim sup a∥PD |u| ∥Lq µ(D)→0 u∈U q/p

µ(D)→0 u∈U

= aνLq (U ) 6 ar q/p−1 νLq (U ).  We need: Lemma 2. Let U ⊆ Lp . Then νLp (U ) < ∞ if and only if U is bounded, i.e., ν satisfies the property of boundedness. Proof. If ∥u∥ 6 R for all u ∈ U, then νLp (U ) 6 R < ∞. Suppose that νLp (U ) < ∞. By the definition of νLp (U ),

∀ε > 0∃δ > 0 : µ(D) < δ ⇒ ∥PD (u)∥Lp < νLp (U ) + ε,

∀u ∈ U . I We can find subsets Di ⊆ Ω , 1 6 i 6 I, I > µ(Ω )/δ + 1, such that Ω = i=1 Di , Di ∩ Dj = ∅ for i ̸= j, and µ(Di ) < δ . Then ∥u∥pLp 6

I 

∥PDi (u)∥pLp < I (νLp (U ) + ε)p

i=1

for all u ∈ U.



Let U ⊆ E where E is a Banach space. The Hausdorff measure of non-compactness χ (U ) of the set U is the infimum of the numbers ε > 0 such that U has a finite ε -net in E (see [1, 1.1.2]). If U is a bounded subset of Lp , then νLp (U ) ≤ χLp (U ). If U is compact in measure, then νLp (U ) = χLp (U ) [5–8]. Here compactness in measure [1, 4.9.1] means compactness in the normed space S of all measurable, almost everywhere finite functions u, equipped with the norm

∥u∥ = inf {s + µ{t : |u(t )| > s}}. s >0

We say that K : Lp → Lq is a compact in measure operator if K (U ) is compact in measure for any bounded U ⊂ Lp . We say that F : Lp → Lq is comparable with G: Lp → Lq on a set U ⊆ Lp if |F (u)(s)| 6 |G(u)(s)| for all u ∈ U and almost all s ∈ Ω .

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Example 2. Let K : Lp → Lq , q > p, be a compact in measure (k, ν)-bounded operator. Let F : Lq → Lp be a continuous operator comparable on some ball BLq (u1 , R) with an operator G: Lq → Lp locally strongly ν -condensing on the same ball BLq (u1 , R). Then KF : Lq → Lq is a locally strongly χ -condensing operator on BLq (u1 , R). Proof. By assumption, G is a locally strongly ν -condensing operator on BLq (u1 , R). Thus νLp (G(U )) 6 λBLq (u1 ,R),G (r )νLq (U ) for any U ⊆ (BLq (x, r )∩ BLq (u1 , R)). Hence νLp (G(U )) 6 λBLq (u1 ,R),G (r )χLq (U ). By the definition of ν , inequality |F (u)(s)| 6 |Gu(s)| implies νLp (F (U )) 6 νLp (G(U )). Therefore, νLp (F (U )) 6 λBLq (u1 ,R),G (r )χLq (U ). Note that νLp (F (U )) < ∞ and, by Lemma 2, F (U ) is bounded in Lp . Moreover, χLq (K (F (U ))) = νLq (K (F (U ))) 6 kνLp (F (U )) since K is a compact in measure (k, ν)bounded operator. Hence

χLq (K (F (U ))) = νLq (K (F (U ))) 6 kνLp (F (U )) 6 kλBLq (u1 ,R),G (r )νLq (U ) 6 kλBLq (u1 ,R),G (r )χLq (U ) for any U ⊆ (BLq (x, r ) ∩ BLq (u1 , R)).



Remarks. (1) The measure of non-compactness χ satisfies all the conditions from the list in the introduction (see also [1, 1.1.4]). The measure of non-compactness ν is not regular, i.e., νLp (U ) = 0 does not imply that closure of U is compact. However ν satisfies all the other conditions from that list. (2) By Example 2, Hammerstein operators KF are locally strongly χ -condensing. Here K : Lp → Lq (1 6 p < q < ∞) is a linear regular bounded integral operator and F : Lq → Lp is a superposition operator. We use the fact that any linear bounded operator K is (∥K ∥, ν)-bounded. One can analogously construct other locally strongly condensing operators using operators and spaces considered in [7,8]. (3) Not all locally strongly ψ -condensing operators are (k, ψ)-bounded, since λ(r ) may increase without bound as r → ∞. Indeed, consider a locally strongly ν -condensing operator F (u)(s) = sgn(u(s))|u(s)|q/p ,

q > p,

from Example 1. The property of (k, ν)-boundedness implies sup

U :νLq (U )̸=0

νLp (F (U )) 6 k. νLq (U )

Let Ω = (0, 1). Denote by U the set of functions ui ( s ) =

 1/q i 0

for for

s ∈ (0, 1/i), s ̸∈ (0, 1/i),

i = 1, 2, . . . .

Then F (ui )(s) =



|ui (s)|q/p 0

for for

s ∈ (0, 1/i), = s ̸∈ (0, 1/i),

 1/p i 0

for for

s ∈ (0, 1/i), s ̸∈ (0, 1/i),

for i = 1, 2, . . . . Let r > 0. The measure of the supports of ui and F (ui ) tends to 0 as i → ∞. Hence

νLq (rU ) = lim ∥rui ∥Lq = r , i→∞

νLp (F (rU )) = lim ∥F (rui )∥Lp = lim ∥r q/p |ui |q/p ∥Lp = r q/p , i→∞

i→∞

and

νLp (F (rU )) = r q/p−1 . νLq (rU ) Therefore, F is not (k, ν)-bounded, since r q/p−1 → ∞ as r → ∞. 4. Properties and applications In [1, Theorem 1.5.9] it was shown that if a (k, χ )-bounded operator f has a Fréchet derivative f ′ , then f ′ is a (k, χ )bounded operator too. Here we repeat almost verbatim the proof of that theorem and show that if f is a locally strongly ψ -condensing operator with the continuous Fréchet derivative f ′ , then f ′ is compact. Theorem 2. Let ψ be a semi-homogeneous algebraically semi-additive continuous regular measure of non-compactness on a Banach space E. Let f : E → E be an operator that is locally strongly ψ -condensing on a ball B(u1 , r ), r > 0. Suppose that there exists a continuous Fréchet derivative f ′ (u1 ) of f at u1 . Then f ′ (u1 ) is compact.

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Proof. Define A = f ′ (u1 ). Then Ah = f (u1 + h) − f (u1 ) + ω(h), h ∈ E, where ∥ω(h)∥/∥h∥ → 0 as ∥h∥ → 0. Linearity of A implies A(U ) =

1

ρ

A(ρ U ) ⊆

1

ρ

[f (u1 + ρ U ) − f (u1 ) + ω(ρ U )]

for any bounded U and ρ > 0. Note that 1

ρ

ψE (f (u1 )) = 0

since the set {f (u1 )} consists of one point and is compact. Hence semi-homogeneity and algebraic semi-additivity of ψE imply

ψE (A(U )) ≤

1

ρ

ψE (f (u1 + ρ U )) + ψE



1

ρ

 ω(ρ U ) .

Let ρ > 0 be small enough. Then u1 + ρ U ⊂ B(u1 , r ). Recall that f : E → E is a locally strongly ψ -condensing operator on B(u1 , r ). Thus the equality ψE (u1 ) = 0 and algebraic semi-additivity of ψE imply

ψE (f (u1 + ρ U )) 6 λB(u1 ,r ),f (r )ψE (u1 + ρ U ) 6 λB(u1 ,r ),f (r )ψE (ρ U ). Finally we apply semi-homogeneity of ψ and obtain

ψE (A(U )) 6 λB(u1 ,r ),f (r )ψE (U ) + ψE



1

ρ

 ω(ρ U ) .

Consider ρ → 0. Then the last term vanishes since ∥ω(h)∥/∥h∥ → 0 as ∥h∥ → 0 and ψ is continuous with respect to the Hausdorff metric. Hence

ψE (A(U )) 6 λB(u1 ,r ),f (r )ψE (U ). By virtue of (1), ψE (A(U )) = 0. Thus the closure of A(U ) is compact since ψ is regular.



Theorem 3. Let ψ be a measure of non-compactness on a Banach space E satisfying the properties of semi-homogeneity, continuity, regularity, semi-additivity, invariance under translations, and boundedness. Let f : E → E be an operator that is locally strongly ψ -condensing on some ball B(u0 , r ), r > 0. Then there exists α0 > 0 such that for all |α| 6 α0 the equation u = u0 + α f (u) has at least one solution u ∈ B(u0 , r ). Proof. By the assumption, f is a locally strongly ψ -condensing operator on B(u0 , r ) with λB(u0 ,r ),f satisfying (1), (2). Choose 0 < r0 6 r such that λB(u0 ,r ),f (r0 ) < 1. Note that ψE (U ) < ∞ if and only if U ⊆ E is bounded in E. Thus there exists M > 0 such that ∥f (u)∥E 6 M for all u ∈ B(u0 , r0 ). Choose 0 < α0 6 min{r0 /M , 1} such that u0 + α f (B(u0 , r0 )) ⊆ B(u0 , r0 ) for all |α| 6 α0 . Note that ψE (u0 + α f (U )) = ψE (α f (U )) (invariance under translations) and ψE (α f (U )) = |α|ψE (f (U )) (semi-homogeneity). Hence inequalities (2) and αλB(u0 ,r ),f (r0 ) < 1 for all |α| 6 α0 imply

ψE (u0 + α f (U )) 6 |α|λB(u0 ,r ),f (r0 )ψE (U ) < ψE (U ) for all U ⊆ B(u0 , r0 ) such that ψE (U ) ̸= 0. Therefore, u0 + α f (u) is condensing on B(u0 , r0 ) with respect to ψ . By Akhmerov et al. [1, 1.5.11, 1.5.12], u0 + α f (u) has a least one fixed point.  Let f : E → E be a continuous operator on a Banach space E such that f (θ ) = θ . Recall that α0 is a bifurcation point of f if for any ε > 0, δ > 0 there exists α ∈ (α0 − ε, α0 + ε) such that u = α f (u),

(4)

has at least one nonzero solution in B(θ , δ). Theorem 4. Suppose that the assumptions of Theorem 2 hold on some ball B(θ , r ), r > 0, f (θ ) = θ , and there exists a continuous Fréchet derivative f ′ (θ ) of f at θ . Then α ∈ R can be a bifurcation point of f only if α1 is an eigenvalue of the linear operator f ′ (θ). Proof. By Theorem 2, f ′ (θ ) is a compact operator. Suppose α1 is not an eigenvalue. Then (I −α0 f ′ (θ ))−1 is continuous. Define

ε=

1 3∥(I − α0

f ′ (θ ))−1 ∥

∥f ′ (θ )∥

.

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By the definition of the Fréchet derivative, f ′ (θ )h = f (θ + h) − f (θ ) + ω(h) where ∥ω(h)∥E /∥h∥E → 0 as ∥h∥E → 0. Choose δ > 0 such that

∥ω(u)∥E <

∥ u∥ E 3∥(I − α0 f ′ (θ ))−1 ∥(|α0 | + ε)

.

for all u ∈ B = B(θ , δ). Consider any α such that |α − α0 | < ε . We claim that the solution of Eq. (4) in B is zero and therefore α0 is not a bifurcation point. Indeed, suppose that u = α f (u). Then α f ′ (θ )u = α f (u) + αω(u) = u + αω(u). In other words, u = α f ′ (θ )u − αω(u). The equality u = (I − α0 f ′ (θ ))−1 (u − α0 f ′ (θ )u) implies

∥u∥E 6 ∥(I − α0 f ′ (θ ))−1 ∥ ∥u − α0 f ′ (θ )u∥E = ∥(I − α0 f ′ (θ ))−1 ∥ ∥α f ′ (θ )u − αω(u) − α0 f ′ (θ )u∥E 6 |α − α0 |∥(I − α0 f ′ (θ ))−1 ∥ ∥f ′ (θ )∥ ∥u∥E + |α|∥(I − α0 f ′ (θ ))−1 ∥ ∥ω(u)∥E 6 (2/3)∥u∥E . Hence u = 0.



In the proof of Theorem 4 we have partially used some arguments from the proof of [9, Chapter 4, Lemma 2.1]. In contrast to the case for [10, Theorem 17.2], operator f considered in Theorem 4 is not necessarily compact. In Theorems 2–4 we generalize analogous results from [7, Theorems 1 and 2] and [8, Theorems 2 and 3] to the case of arbitrary locally strongly ψ -condensing operators. Equations of the form u = f (u; λ) considered in this paper arise in many problems of nonlinear mechanics, such as determining critical loads and forms of stability loss in elastic systems, and studying self-oscillation processes and birth processes of waves and moving fluids, etc. In such problems the role of the parameter λ can be played by a load, the frequency of self-oscillations, the velocity of a fluid, etc [4, Section 21.4]. References [1] R.R. Akhmerov, M.I. Kamenskii, A.S. Potapov, A.E. Rodkina, B.N. Sadovskii, Measures of Noncompactness and Condensing Operators, Birkhauser Verlag, Basel, Boston, Berlin, 1992. [2] R.D. Nussbaum, The fixed point index for local condensing maps, Ann. Mat. Pura ed. Appl. 89 (1971) 217–258. [3] Pilz G. Near-Rings, The Theory and its Applications, North-Holland, Amsterdam, 1983. [4] M.A. Krasnosel’ski˘ı, P.P. Zabre˘ıko, E.I. Pustyl’nik, P.E. Sobolevski˘ı, Integral operators in spaces of summable functions, in: Monographs and Textbooks on Mechanics of Solids and Fluids, Mechanics: Analysis, Noordhoff International Publishing, Leiden, 1976, p. xv+520 pp. [5] N.A. Yerzakova, On measures of non-compactness in regular spaces, Zeitshcr. Anal. Anw. 15 (2) (1996) 299–307. [6] N.A. Erzakova, Compactness in measure and measure of noncompactness, Siberian Math. J. 38 (5) (1997) 926–928. [7] N.A. Erzakova, Solvability of equations with partially additive operators, Funct. Anal. Appl. 44 (3) (2010) 216–218. [8] N.A. Erzakova, On measure compact operators, Russian Mathematics (Iz. VUZ) 55 (9) (2011) 37–42. [9] M.A. Krasnosel’skii, Topological Methods in the Theory of Nonlinear Integral Equations, A Pergamon Press Book The Macmillan Co., New York, 1964, xi+395 pp. [10] M.M. Vainberg, Variational Methods for the Study of Nonlinear Operators, Holden-Day, Inc., San Francisco, CA, London, Amsterdam, 1964, x+323 pp.