Integr. equ. oper. theory 49 (2004), 405–418 0378-620X/030405-14, DOI 10.1007/s00020-002-1268-7 c 2004 Birkh¨ auser Verlag Basel/Switzerland
Integral Equations and Operator Theory
(Lp, Lq )-Admissibility and Exponential Dichotomy of Evolutionary Processes on the Half-line P. Preda, A. Pogan and C. Preda Abstract. A new characterization of exponential dichotomy for evolutionary processes in terms of (Lp , Lq )-admissibility is presented, using a direct treatment, without the so-called evolution semigroup. Mathematics Subject Classification (2000). 34D05, 47D06, 93D20. Keywords. Admissibility, evolutionary processes, exponential dichotomy.
1. Introduction It is well known that the analysis of exponential dichotomy has developed rapidly since their origins was founded by Oscar Perron in his paper “Die Stabilit¨ atsfrage bei Differentialgleichungen” appeared in 1930. Many results on the case of differential equations can be found in the monographs due to Massera-Sch¨ affer [10], Hartman [5] , Daleckij-Krein [4], Coppel [3], Chicone-Latushkin [2]. The main work still focuses on the case of evolutionary-processes. We have different characterization of exponential dichotomy for a strongly continuous, exponentially bounded evolution family (or more general in the case of evolutionaryprocesses) in the papers due to N. van Minh [12,13], F. Rabiger [12], Y. Latushkin [8,9], P. Randolph [9], R. Schnaubelt [9,12,20], P. Preda [15,16,17] and M. Megan [11,15]. Arguments in these papers again illustrate the general philosophy of “autonomization” of nonautonomous problems by passing from evolution families to associated evolution semigroups. In contrast to this “philosophy”, the present paper shows that we can characterize the exponential dichotomy in terms of (Lp , Lq )-admissibility in a direct way, without the so-called evolution semigroup. So the aim of this paper is to establish the connection between (Lp , Lq )-admissibility (1 ≤ p ≤ q ≤ ∞ and p1 + 1q is not
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necessarily 1) and exponential dichotomy for evolutionary processes on half-line and also we give directly proofs for our results.
2. Preliminaries Let X be a real or complex Banach space, B(X)-the space of all bounded linear operators of X into itself and IR+ = [0, ∞). We denote by ||·|| the norms of vectors in X and of operators in B(X). The classical result of O.Perron stands that the differential system (A)
x(t) ˙ = A(t)x(t), t ≥ 0.
is exponential dichotomic if and only if for all continuous and bounded f : IR+ → X there exists a bounded solution of the equation (A, f )
x(t) ˙ = A(t)x(t) + f (t), t ≥ 0.
where A is an operator valued function locally Bochner integrable with t+1 ||A(τ )||dτ < ∞ sup t≥0
t
and X a finite dimensional space. This result was extended to the case of infinite dimensional Banach spaces in a natural way. The Cauchy problem associated to the equation (A, f ) has a solution given by t U (t, τ )f (τ )dτ. x(t) = U (t, t0 )x(t0 ) + t0
U is the evolutionary process generated by the equation (A), U (t, t0 ) = Φ(t)Φ−1 (t0 ) where Φ is the unique solution of the Cauchy Problem Φ (t) = A(t)Φ(t) Φ(0) = I The case where {A(t)}t≥0 is a family of unbounded linear operators, impose another “kind” of solution for (A, f ). So we have to deal with the so-called mild solution for (A, f ) given by t U (t, τ )f (τ )dτ. x(t) = U (t, t0 )x0 + t0
Definition 2.1. A family U = {U (t, s)}t≥s≥0 of bounded, linear operators acting on X is called an evolutionary process if the following statements hold: e1 ) U (t, t) = I (where I is the identity operator on X), for all t ≥ 0; e2 ) U (t, s)U (s, r) = U (t, r), for all t ≥ s ≥ r ≥ 0; e3 ) U (·, s)x is continuous on [s, ∞) for all s ≥ 0, x ∈ X; U (t, ·)x is continuous on [0, t] for all t ≥ 0, x ∈ X;
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e4 ) there exist M, ω > 0 such that ||U (t, s)|| ≤ M eω(t−s) , for all t ≥ s ≥ 0. For I a real interval we denote: Lp (I, X) = {f : I → X | f − measurable and I f (t)p dt < ∞}, for all p ∈ [1, ∞) and by: L∞ (I, X) = {f : I → X | f − measurable and ess supt≥0 f (t) < ∞}. p L (I, X) and L∞ (I, X) are Banach spaces endowed with the respectively norms: p1 f p = f (t)p I
and f ∞ = ess sup f (t). t≥0
For the simplicity of notations we will also use the notation Lp for Lp (I, X) where p ∈ [1, ∞]. Also, we set C = {f : IR+ → X | f − continuous and bounded}. Definition 2.2. An application P : IR+ → B(X) is said to be a dichotomy projection family if: p1 ) P 2 (t) = P (t), for all t ≥ 0; p2 ) P (·)x ∈ C, for all x ∈ X. We set Q(t) = I − P (t), t ≥ 0. Definition 2.3. An evolutionary process U is said to be: a) compatible with the dichotomy projection family P : IR+ → B(X) if d1 ) U (t, s)P (s) = P (t)U (t, s), for all t ≥ s ≥ 0; d2 ) U (t, s) : KerP (s) → KerP (t) is an isomorphism for all t ≥ s ≥ 0; b) uniformly exponentially dichotomic (u.e.d) if there exist P a dichotomy projection family with which U is compatible and two constants N , ν > 0 such that the following conditions hold: d3 )||U (t, s)x|| ≤ N e−ν(t−s) ||x||, for all x ∈ ImP (s), t ≥ s ≥ 0; 1 d4 ) ||U (t, s)x|| ≥ eν(t−s) ||x||, for all x ∈ KerP (s), t ≥ s ≥ 0. N In what follows we will consider the evolutionary processes U for which exists P a dichotomy projection family compatible to U (the conditions d1 ), d2 ) are satisfied). In this case we will denote U1 (t, s) = U (t, s)|Im p
P (s) ,
U2 (t, s) = U (t, s)|Ker
P (s) .
Definition 2.4. The pair (L , L ) is said to be admissible for U if for all f ∈ Lp we have: a1 ) U2−1 (·, t)Q(·)f (·) ∈ L1[t,∞) (X), for all t ≥ 0; t ∞ a2 ) xf : IR+ → X, xf (t) = U1 (t, s)P (s)f (s)ds− U2−1 (s, t)Q(s)f (s)ds, lies in Lq .
q
0
t
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Lemma 2.5. With our assumption we have that: i) U2−1 (·, t0 )Q(·)x is continuous on [t0 , ∞), for all (t0 , x) ∈ IR+ × X. ii) U2−1 (·, T0 )Q(·)f (·) is strongly measurable for every strongly measurable f and all t0 ≥ 0. Proof. Let t ≥ t0 ≥ 0, h ∈ (0, 1), x ∈ X. Then U2 (t + 1, t0 ) = U2 (t + 1, r)U2 (r, t0 ), for all r ∈ [t, t + 1], and so U2−1 (t + h, t0 ) = U2−1 (t + 1, t0 )U2 (t + 1, t + h) U2−1 (t, t0 ) = U2−1 (t + 1, t0 )U2 (t + 1, t). It results that ||U2−1 (t + h, t0 )Q(t + h)x − U2−1 (t, t0 )Q(t)x|| = ≤ =
||U2−1 (t + 1, t0 )[U2 (t + 1, t + h)Q(t + h)x − U2 (t + 1, t)Q(t)x]||
||U2−1 (t + 1, t0 )|| ||U2 (t + 1, t + h)Q(t + h)x − U2 (t + 1, t)Q(t)x||
||U2−1 (t + 1, t0 )|| ||U (t + 1, t + h)Q(t + h)x − U (t + 1, t)Q(t)x||
≤
||U2−1 (t + 1, t0 )|| [||U (t + 1, t + h)(Q(t + h))x − Q(t)x|| +||U (t + 1, t + h)Q(t)x − U (t + 1, t)Q(t)x||]
≤
||U2−1 (t + 1, t0 )||[M eω(1−h) ||Q(t + h)X − Q(t)x|| +||U (t + 1, t + h)Q(t)x − U (t + 1, t)Q(t)x||].
It is easy to see that U2−1 (·, t0 )Q(·)x is right-handed continuous on [t0 , ∞). ii) it follows easily from i) . Lemma 2.6. If 1 ≤ p ≤ q ≤ ∞, ν > 0, h ∈ Lp (IR+ , IR+ ), g1 , g2 : IR+ → IR+ given by t ∞ g1 (t) = 0 e−ν(t−s) h(s)ds, g2 (t) = t e−ν(t−s) h(s)ds then g1 , g2 ∈ Lq (IR+ , IR+ ). Proof. Let h1 , k1 , k2 : IR → IR the functions defined by h(t), t ≥ 0 h1 (t) = 0, t < 0. −νt e , t≥0 k1 (t) = 0, t < 0. 0, t ≥ 0 h1 (t) = eνt , t < 0. From the fact that p ≤ q it follows that there exists r ≥ 1 such that p1 + 1r = + 1. Also, it is easy to check that k1 , k2 ∈ Lr (IR, IR) and using the Young’s inequality (see for instance [1, Proposition 1.3.2, page 22] we have that k1 ∗ h1 , k2 ∗ h1 ∈ Lq (IR, IR). By the other hand we have that 1 q
g1 (t) = (k1 ∗ h1 )(t), and so the conclusion follows.
g2 (t) = (k2 ∗ h1 )(t),
for all t ≥ 0,
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3. The main result Lemma 3.1. If the pair (Lp , Lq ) is admissible to U then there is K > 0 such that xf ≤ Kf ,
for all f ∈ Lp .
Proof. Let Λt : Lp → L1 ([t, ∞), X), Λt f = U2−1 (·, t)Q(·)f (·) for t ≥ 0. It is obvious that Λt is a linear operator for all t ≥ 0. Consider t ≥ 0, {fn }n≥1 ⊂ Lp , f ∈ Lp g ∈ L1 ([t, ∞), X) such that Lp
Lq
fn → f, Λt fn → g. Then there exists a subsequence {fnk }k≥1 of {fn }n≥1 such that: fnk →f, Λt fnk →g a.p.t. But (Λt fnk )(s) − (Λt f )(s) ≤ U2−1 (s, t)Q(s) fnk (s) − f (s), for all k ≥ 1 and all s ≥ t and so Λt fnk → Λt f a.p.t. It follows easily that Λt is a bounded operator for all t ≥ 0. Let us define the linear operator T : Lp → Lq t ∞ (T f )(t) = U1 (t, s)P (s)f (s)ds − U2−1 (s, t)Q(s)f (s)ds. 0
t
Lp
Lq
If {gn }n≥1 ⊂ Lp , g ∈ Lp , h ∈ Lq , gn → g, T gn → h then t (T gn )(t) − (T g)(t) ≤ U1 (t, s)P (s)(gn(s) − g(s))ds 0 ∞ U2−1 (s, t)Q(s)(gn (s) − g(s))ds + t t U1 (t, s) P (s) gn (s) − g(s)ds + Λt (gn − g) ≤ 0
≤ M eωt sup P (s)t s≥0
p−1 p
||gn − g||p + Λt (gn − g),
for all t ≥ 0 and all n ∈ N ∗ . It follows that T g = h, hence T is bounded. So xf = T f ≤ T f , for all f ∈ Lp .
Theorem 3.2. The pair (L1 , L∞ ) is admissible to U if and only if there exists N > 0, such that U1 (t, t0 )x ≤ N x, for all t ≥ t0 ≥ 0 and x ∈ Im P (t0 ) U2−1 (t, t0 )x ≤ N x, for all t0 ≥ t ≥ 0 and x ∈ Ker P (t).
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Proof. Sufficiency. Let f ∈ L1 (IR+ , X). From Lemma 2.1. it follows that the map U2−1 (·, t0 )Q(·)f (·) is strongly measurable and ||U2−1 (s, t0 )Q(s)f (s)|| ≤ N sup ||Q(v)||||f (s)||, v≥0
for all s ∈ [t0 , ∞) and t0 ≥ 0, which implies that U2−1 (·, t0 )Q(·)f (·) ∈ L1 ([t0 , ∞)) for all t0 ≥ 0. Also, t ||xf (t)|| ≤
∞ N sup ||P (v)||||f (s)||ds + v≥0
0
N sup ||Q(v)||||f (s)||ds t
v≥0
≤ N (sup ||P (v)|| + sup ||Q(v)||)||f ||1 v≥0
v≥0
for all t ≥ 0.
Necessity. Let x ∈ X, δ > 0, t0 ≥ 0 and f : IR+ → X U (t, t0 )x, t ∈ [t0 , t0 + δ], f (t) = 0, t ∈ IR+ \ [t0 , t0 + δ]. t0 +δ It is easy to see that f ∈ L1 , f 1 = U (s, t0 )xds and − xf (t) =
t0
t0 +δ
t0
U2−1 (s, t)Q(s)f (s)ds, 0 ≤ t ≤ t0 ,
t0 +δ t0
U1 (t, s)P (s)f (s)ds, t ≥ t0 + δ.
If x ∈ ImP (t0 ) and t ≥ t0 + δ then, t0 +δ xf (t) = U1 (t, s)U1 (s, t0 )xds = δU1 (t, t0 )x, t0
which implies that U1 (t, t0 )x ≤
K δ
t0 +δ
t0
U (s, t0 )xds.
Making δ → 0 we obtain that: U1 (t, t0 )x ≤ Kx, for all t ≥ t0 ≥ 0 and all x ∈ ImP (t0 ). If x ∈ KerP (t0 ) then f (s) ∈ KerP (s), for all s ∈ [t0 , t0 + δ] and t0 +δ xf (t) = − U2−1 (s, t)U2 (s, t0 )xds t0 t0 +δ
U2−1 (t0 , t)U2−1 (s, t0 )U2 (s, t0 )xds t0 −δU2−1 (t0 , t)x
= − =
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It results that
K t0 +δ ≤ U (s, t0 )x||ds, δ t0 and by making again δ → 0 we obtain that ||U2−1 (t0 , t)x
U2−1 (t0 , t)x ≤ Kx, for all t0 ≥ t ≥ 0 and all x ∈ KerP (t0 )
Lemma 3.3. If h is a positive function from Lq (IR+ , IR), such that the following statement holds: h(r) ≤ ah(t) + b, for all r ≥ t ≥ 0 with r − t ≤ 1, then h ∈ L∞ (IR+ , IR). Proof. With our hypothesis we have that h(n + 1) ≤ ah(s) + b, for all n ∈ N, and all s ∈ [n, n + 1] and so
n+1
h(n + 1) ≤ a
h(s)ds + b ≤ ahq + b for all
n∈N
n
which implies that c := sup h(n) < ∞. n∈N
Using again the property of h from our hypothesis we obtain that h(t) ≤ ah(n) + b ≤ ac + b, for all n ∈ N, and all t ∈ [n, n + 1). Lemma 3.4. If the pair (Lp , Lq ) is admissible to U then i) For all f ∈ Lp t xf (t) = U (t, s)f (s)ds + U (t, 0)xf (0), for all t ≥ 0; 0
p
ii) For all f ∈ L there exist a, b > 0 such that xf (r) ≤ axf (t) + b, for all r ≥ t ≥ 0 with r − t ≤ 1; iii) The pair(Lp , L∞ ) is admissible to U. Proof. i) It is a simple computation. ii) r xf (r) = U (r, s)f (s)ds + U (r, 0)xf (0) 0 r t U (r, t)U (t, s)f (s)ds + U (r, t)U (t, 0)xf (0) + U (t, s)f (s)ds = 0 t r U (r, s)f (s)ds for all r ≥ t ≥ 0. = U (r, t)xf (t) + t
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It results that xf (r) ≤ M eω(r−t) ||xf (t) + ω
ω
≤ M e xf (t) + M e
r t
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M eω(r−s) f (s)ds
t+1 t
f (s)ds
≤ M eω xf (t) + M eω ||f p for all r ≥ t ≥ 0 with r − t ≤ 1. iii) It follows from ii) and Lemma 3.2.
Lemma 3.5. Let g : ∆ → IR+ be a function such that 1) g(t, t0 ) ≤ g(t, s)g(s, t0 ), for all t ≥ s ≥ t0 ≥ 0, 2) there exist M, a > 0, b ∈ (0, 1) such that: g(t, t0 ) ≤ M, for all t0 ≥ 0 and all t ∈ [t0 , t0 + a]; g(t0 + a, t0 ) ≤ b, for all t0 ≥ 0. Then there are N , ν > 0 such that g(t, t0 ) ≤ N e−ν(t−t0 ) , for all t ≥ t0 ≥ 0.
0 . Then we have that Proof. Let t ≥ t0 ≥ 0 and n = t−t a g(t, t0 ) ≤ g(t, t0 + na)g(t0 + na, t0 ) ≤ g(t, t0 + na)bn ≤ ≤ M bn = M e−νna ≤ N e−ν(t−t0 ) where ν = − a1 ln b, N = M eνa .
Theorem 3.6. If q ∈ [1, ∞) then the pair (L1 , Lq ) is admissible to U if and only if U is u.e.d. Proof. Sufficiency. It follows from Definition 2.1., Lemma 2.1., Lemma 2.2. Necessity. If the pair (L1 , Lq ) is admissible to U then by Lemma 3.3. it follows that the pair (L1 , L∞ ) is admissible to U hence, there is N > 0 such that U1 (t, t0 ) ≤ N and U2−1 (t, t0 ) ≤ N, for all t ≥ t0 ≥ 0. Let t0 ≥ 0, x ∈ ImP (t0 ), δ > 0 and f : IR+ → X the function defined by U1 (t, t0 )x t ∈ [t0 , t0 + δ], f (t) = 0, t ∈ IR+ \ [t0 , t0 + δ]. Then f (t) ∈ ImP (t), for all t ≥ 0 and t U1 (t, s)P (s)f (s)ds, for all t ≥ 0. xf (t) = 0
It follows that xf (t) =
t0 +δ
t0
U1 (t, s)U1 (s, t0 )xds = δU1 (t, t0 )x for all t ≥ t0 + δ
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By Lemma 3.1. it results that q1 ∞ U1 (t, t0 )xq dt ≤ xf q ≤ Kf ||1 ≤ KN δx||. δ t0 +δ
For δ → 0 we obtain that
∞
t0
U1 (t, t0 )xq dt
q1
≤ N K||x||,
for all t0 ≥ 0 and all x ∈ KerP (t0 ). But t q1 1 q (t − t0 ) U1 (t, t0 )x = U1 (t, t0 )xq ds t0 t
≤
t0
≤ N
U1 (t, s)q U1 (s, t0 )xq ds
∞
t0
U1 (s, t0 )xq ds
q1
1q
≤ N 2 Kx||, for all t ≥ t0 ≥ 0 and all x ∈ ImP (t0 ). It is now clear that 1
(t − t0 ) q U1 (t, t0 ) ≤ N 2 K, for all t ≥ t0 ≥ 0. By Lemma 3.4. it follows that the condition d3) holds. If t0 ≥ 0, x ∈ KerP (t0 ) \ {0}, δ > 0 and g : IR+ → X is given by 1 U2 (t,t0 )x U2 (t, t0 )x, t ∈ [t0 , t0 + δ]. g(t) = 0, t ∈ IR+ \ [t0 , t0 + δ]. then g(t) ∈ KerP (t), for all t ≥ 0. For t ≤ t0 , ∞ U2−1 (s, t)Q(s)f (s)ds xg (t) = − t
=
−
t0 +δ
t0 t0 +δ
=
−
t0
ds U −1 (s, t)U2 (s, t0 )xds U2 (s, t0 )x 2 ds U −1 (t0 , t)x. U2 (s, t0 )x 2
By Lemma 3.1. it results that t0 +δ t0 1q ds U2−1 (t0 , t)xq dt ≤ xg q ≤ Kg||1 ≤ Kδ U2 (s, t0 )x 0 t0 Making δ → 0 we obtain that t0 q1 U2−1 (t0 , t)xq dt ≤ Kx, 0
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for all t0 ≥ 0 and all x ∈ KerP (t0 ) \ {0}. If t ≥ t0 ≥ 0, x ∈ KerP (t), then t q1 1 −1 q (t − t0 ) U2 (t, t0 )x = U2−1 (t, t0 )xq ds t0 t
≤
≤ N
t0
U2−1 (s, t0 )q U2−1 (t, s)x||q ds
0
t
U2−1 (t, s)xq ds
1q
q1
≤ N Kx
which implies that 1
(t − t0 ) q U2−1 (t, t0 ) ≤ N K, for all t ≥ t0 ≥ 0. By Lemma 3.4. it follows that the condition d4 ) is satisfied.
Theorem 3.7. If p ∈ (1, ∞), then the pair (Lp , L∞ ) is admissible to U if and only if U is u.e.d. Proof. Sufficiency. It follows from Definition 2.1., Lemma 2.1., Lemma 2.2. Necessity. Let t0 ≥ 0, x ∈ ImP (t0 ) and f : IR+ → X, the function defined by −2ω(t−t ) 0 U1 (t, t0 )x, t > t0 e f (t) = 0, t ∈ [0, t0 ] It is clear that f (t) ∈ ImP (t), for all t ≥ 0 and ∞ p1 1 p1 f p ≤ M p e−ωps xp ds ≤M x ωp 0 xf (t) =
t
t0
U1 (t, s)(e−2ω(s−t0 ) U1 (s, t0 )x)ds = (1 −
1 −2ω(t−t0 ) )U1 (t, t0 )x, e 2ω
for all t ≥ t0 . Then 1 −2ω(t−t0 ) e U1 (t, t0 )x 2ω 1 M x ≤ xf ∞ + 2ω M ≤ Kf p + x 2ω MK M ≤ x, for all t ≥ t0 , and all x ∈ ImP (t0 ) + 2ω (ωp)1/p
U1 (t, t0 )x ≤ xf (t) +
and hence, U1 (t, t0 ) ≤ L, for all t ≥ t0 ≥ 0,
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MK M where L = ( (ωp) 1/p + 2ω ). Consider now t0 ≥ 0, x ∈ ImP (t0 ), δ > 0 and g : IR+ → X the function defined by U1 (t, t0 )x, t ∈ [t0 , t0 + δ] g(t) = 0, t ∈ IR+ \ [t0 , t0 + δ] It is easy to see that g(t) ∈ ImP (t), for all t ≥ 0 and t0 +δ p1 1 g||p ≤ U1 (t, t0 )xp dt ≤ Lδ p ||x||. t0
Then
xg (t) =
t0 +δ
t0
U1 (t, s)U1 (s, t0 )xds = δU1 (t, t0 )x, for all t ≥ t0 + δ.
By Lemma 3.1. it results that 1
U1 (t, t0 )x ≤ LKδ p −1 x, for all t ≥ t0 + δ and hence U1 (t0 + a, t0 ) ≤ p
1 , for all t0 ≥ 0, 2
1 where a = ( 2KL ) p−1 . By Lemma 3.4. we obtain that the condition d3 ) holds. Let t ≥ 0, x ∈ KerP (t) \ {0} and h : IR+ → X, the function defined by U2 (s, t)x, s ∈ [t, t + 1] h(s) = 0, s ∈ IR+ \ [t, t + 1]
Then h(s) ∈ KerP (s) for all s ≥ 0 and t+1 xh (s) = − U2−1 (τ, s)U2 (τ, t)xdτ t
= = By Lemma 3.1. we have that U2−1 (t, s)x ≤ K(
−
t+1
t
U2−1 (t, s)xdτ
−U2−1 (t, s)x, for all s ≤ t. t
t+1
1
U2 (τ, t)xp dτ ) p ≤ M Keω x,
for all t ≥ s ≥ 0, and all x ∈ KerP (t) and hence U2−1 (t, s) ≤ M Keω , for all t ≥ s ≥ 0 Consider now t1 ≥ t0 ≥ 0, x ∈ KerP (t1 ) and v : IR+ → X the function defined by −1 U2 (t1 , s)x, s ∈ [t0 , t1 ] v(s) = 0, s ∈ IR+ \ [t0 , t1 ]
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It is easy to see that v(s) ∈ KerP (s), for all s ≥ 0 and t1 p1 1 U1−1 (t1 , s)xp ds ≤ M Keω (t1 − t0 ) p x vp = t0
A simple computation shows that t1 U2−1 (s, t)U2−1 (t1 , s)xds = − xv (t) = − =
t −(t1
− t)U2−1 (t1 , t)x,
t1
t
U2−1 (t1 , t)xds
for all t ∈ [t0 , t1 ]. By Lemma 3.1. we obtain that 1
U2−1 (t1 , t0 )x ≤ M K 2 eω (t1 − t0 ) p x, for all t1 ≥ t0 ≥ 0 and all x ∈ KerP (t1 ), which implies that U2−1 (t0 + a, t0 ) ≤
1 , for all t ≥ 0, 2
p p−1 where a = 2M K1 2 eω . By Lemma 3.4. it is now clear that the condition d4 ) holds.
Corollary 3.8. If 1 < p ≤ q < ∞ then the pair (Lp , Lq ) is admissible to U if and only if U is u.e.d. Proof. Sufficiency. It results by Definition 2.1., Lemma 2.1. and Lemma 2.2. Necessity. It follows from Theorem 3.3. and Lemma 3.2.
Remark 3.1. It is shown in [17] that U is u.e.d. if and only if the pair (C, C) is admissible to U. By the fact that C ⊂ L∞ and xf is a continuous function for all f ∈ L∞ and by Definition 2.1., Lemma 2.1., Lemma 2.2. we obtain that U is u.e.d. if and only if the pair (L∞ , L∞ ) is admissible to U. As a conclusion, we combine Theorem 3.2., Theorem 3.3., Corollary 3.1. and Remark 3.1. Theorem 3.9. 1 ≤ p ≤ q ≤ ∞ and (p, q) = (1, ∞) then the pair (Lp , Lq ) is admissible to U if and only if U is u.e.d. Acknowledgement The authors are indebted to the referee, whose comments and suggestions greatly improved the quality of the paper.
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P. Preda, A. Pogan and C. Preda Department of Mathematics West University of Timi¸soara Bd. V. Pˆ arvan, No 4 Timi¸soara 1900 Romania e-mail:
[email protected] [email protected] [email protected] Submitted: November 15, 2002 Revised: October 30, 2003
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