on the semilinear integro-differential nonlocal cauchy - DML-PL

Transkrypt

Discussiones Mathematicae
Differential Inclusions, Control and Optimization 25 (2005 ) 5–18
ON THE SEMILINEAR INTEGRO-DIFFERENTIAL
NONLOCAL CAUCHY PROBLEM
Piotr Majcher
Faculty of Mathematics and Computer Science
A. Mickiewicz University
Umultowska 87, 61–614 Poznań, Poland
e-mail: [email protected]
and
Magdalena Roszak
Institute of Mathematics and Physics
University of Technology and Agriculture
Al. Prof. S. Kaliskiego 7, 85–796 Bydgoszcz
Abstract
In this paper, we prove an existence theorem forR the pseudo-nont
local Cauchy problem x0 (t) + Ax(t) = f (t, x(t), t0 k(t, s, x(s))ds),
x0 (t0 ) = x0 − g(x), where A is the infinitesimal generator of a C0 semigroup of operator {T (t)}t>0 on a Banach space. The functions f, g are
weakly-weakly sequentially continuous and the integral is taken in the
sense of Pettis.
Keywords: integro-differential equations, measure of weak non-compactness, non-local problem.
2000 Mathematics Subject Classification: 45J05, 34G20, 47H09.
1.
Introduction
The main goal of this paper is to prove some existence results for the pseudo
non-local problem

Z t

0
 x (t) + Ax(t) = f (t, x(t),
k(t, s, x(s))ds), t ∈ I = [t0 , t0 + a],
t0
(1)


x(t0 ) = x0 − g(x), x0 ∈ E, a > 0
6
P. Majcher and M. Roszak
where A is the infinitesimal generator of a C0 semigroup of operators
{T (t)}t>0 on a Banach space E (see [19]). The functions f, g are weaklyweakly sequentially continuous and the integral is taken in the sense of
Pettis [2].
In the proof of the theorem, we will apply the measure of weak noncompactness ([4, 15]) and a fixed point theorem of Darbo type ([4, 14]).
The theory of differential equations described by non-local conditions is
a very interesting and important theory ([1, 7–10, 12–13, 16, 18]) as it can
be applied in many real world problems [20], for example in physics.
Byszewski [6] studies semilinear non-local evolution equations. In [7, 8]
he uses the theory of a strongly continuous semigroup to study the existence
and some properties of solutions of (1) (cf. also [9]).
2.
Definition and notation
Throughout this paper E will denote an infinite dimensional Banach space
with the norm k · k. Denote by (E, ω) the space E with its weak topology
σ(E, E ∗ ). Let Br = {x ∈ E : kxk ≤ r}.
Definition 1. We say that the function f : E1 → E2 , where E1 , E2 are
Banach spaces, is weakly-weakly sequentially continuous if for every weakly
convergent sequence (xn ), xn ∈ E1 the sequence (f (xn )) is convergent to
f (x).
A weakly-weakly continuous function is weakly-weakly sequentially continuous, but the converse is not true (a weak topology is not metrizable). Some
comparison results between such type of continuity can be found in [3].
A semigroup of operators plays an essential role in the construction
of solutions to our problem (1). We say that a family of continuous linear
operators T (t), t ∈ (0, ∞) constitutes a C0 -semigroup if T (t+s) = T (t)T (s),
(s, t ∈ (0, ∞)) and for all x ∈ E limt→0+ T (t)x = x, T (0) = Id. The theory
of semigroups of linear operators can be found in several books ([7, 19, 20]
for instance).
In this paper, we assume that A is the infinitesimal generator of a C0
semigroup {T (t)}t≥0 of operators on E, M = supt∈[0,a] kT (t)kL(E) , X =
C(I, E), where k · kL(E) is a norm in space of all continuous linear operators
on E. For simplicity, the norm k · kL(E) will be denoted again by k · k.
On the semilinear integro-differential ...
7
Definition 2 ([5, 15]). Given a bounded subset A ⊂ E, we define the
measure of weak non-compactness ω(A) as follows
ω(A) = inf{ε > 0 : A ⊂ C + B(0, ε), C ⊂ K ω },
where K ω is a family of all weakly compacted subsets of E.
For the properties of ω, see [4, 5], for instance.
Definition 3 ([4, 5]). Let ℵ denote the set of all bounded subsets of E. A
function Φ : ℵ → [0, ∞) satisfying for each A, B ∈ ℵ the following conditions:
1◦ Φ(A) = 0 if and only if Ā[Āω ] is an compact set;
2◦ if A ⊂ B, then Φ(A) ≤ Φ(B);
3◦ Φ(A ∪ {x}) = Φ(A), x ∈ E;
4◦ Φ(λA) = |λ|Φ(A), λ ∈ R;
5◦ Φ(A ∪ B) = max{Φ(A), Φ(B)};
6◦ Φ(A + B) ≤ Φ(A) + Φ(B);
7◦ Φ(convA) = Φ(A).
will be called an axiomatic measure of [weak] non-compactness.
We will use the following lemma.
Lemma 4 ([17]). Let V ⊂ Cω (I, E) be a family of strongly equicontinuous
functions. Then a function t 7→ Φ(V (t)) is continuous and
Φ (V (I)) = sup Φ (V (t)) ,
t∈I
where V (t) = {v(t) : v ∈ V }, V (I) = {v(t) : v ∈ V, t ∈ I}.
Now, we give a fixed point theorem of Darbo type.
Theorem 5 (Darbo [14], Banaś, Rivero [5]). Let X be a nonempty, closed,
convex and bounded subset of a Banach space E, and let F : X → X be
weakly-weakly sequentially continuous. If there exists a constant L < 1,
such that for each bounded set V ⊂ X
ω(F (V )) ≤ L · ω(V ),
where ω denotes some measure of non-compactness, then F has a fixed point.
8
3.
Assumptions
In this paper we assume the following assumptions:
1. For each continuous function x : I →RE, f : I × E 2 → E and each t ∈ I,
s
the function s 7→ T (t − s)f (s, x(s), t0 k(s, v, x(v))dv) is integrable in
the sense of Pettis.
2. For each t ∈ I, z ∈ E 2 the function z 7→ f (t, z) is weakly-weakly
sequentially continuous.
3. For each x∗ ∈ E ∗ , kx∗ k ≤ 1 there exists a function p∗ ∈ L1 (I, R), such
that for each z ∈ E 2 and almost all t ∈ I
|x∗ (f (t, z))| ≤ p∗ (t).
Moreover, there exists a constant P > 0, such that for each x∗ ∈ E ∗ ,
kp∗ k1 < P.
4. For each bounded set A, B ⊂ E and almost all t ∈ I
ω(f (I × A × B)) ≤ c3 max {ω(A), ω(B)} ,
where ω is the measure of weak non-compactness.
5. The function g is a weakly-weakly sequentially continuous defined into
C(I, E), taking values in space E.
6. For each x ∈ C(I, E)
kg(x)k ≤ L, L − constant.
7. For each bounded and equicontinuous set V ⊂ C(I, E)
ω(g(V )) ≤ C · ω(V ),
where C · M < 1 and 0 < C <
1−ac3 M
.
M
8. The function k : I 2 ×E → E is a weakly-weakly sequentially continuous.
9. For each continuous function x : I → E, functions: k(·, ·, x(·)) and
R (·)
f (·, x(·), t0 k(·, s, x(s))ds are integrable in the sense of Pettis.
4.
9
Pseudo-solutions
Suppose that functions f and k are weakly-weakly sequentially continuous,
g : C(I, E) → E, g(B) ⊂ D(A), B ⊂ E, D is the domain of the operator A,
x0 ∈ E and the integral from the above equality is the integral in the sense
of Pettis. Let x∗ ∈ E ∗ . Consider the following problem

Z t

 (x∗ x)0 (t)+A∗ x∗ x(t) = x∗ f (t, x(t), k(t, s, x(s))ds), t ∈ I
t0
(2)


∗
∗
x x(t0 ) = x (x0 − g(x)) .
We will accept the following definition of the pseudo-solution to the non-local
Cauchy problem (1).
Definition 6. The function x satisfying the following equality
(3)
x(t) = T (t − t0 )x0 − T (t − t0 )g(x)
Z t
Z s
+
T (t − s)f (s, x(s),
k(s, v, x(v))dv)ds,
t0
t0
will be called a pseudo-solution to the non-local Cauchy problem (1).
Now we can formulate our main result.
Theorem 7. Assume 1–9 hold and there exists an integrable function c4 :
I → R+ such that for each t ∈ I, ε > 0 and for each bounded subset X ⊂ E
there exists a closed subset Iε ⊂ I such that mes(I \ Iε ) < ε and
ω(k({s, v} × X)) ≤ sup c4 (s)ω(X),
s∈Iε
for v ∈ Iε .
Then there exists at least one pseudo-solution to the Cauchy problem (1).
P roof. Let us define an operator G : C(I) → C(I) by
Z t
Z s
Gx (t) = T (t−t0 )x0 −T (t−t0 )g(x)+ T (t−s)f (s, x(s),
k(s, v, x(v))dv)ds.
t0
Let
t0
e = {x ∈ C(I) :k x k≤ b, M · (kx0 k + L) + a · M · P = b} .
B
10
e → B
e and is weakly-weakly sequentially
Notice that the operator G : B
continuous.
It follows from:
e we have
(i) for each x∗ ∈ E ∗ , such that k x∗ k≤ 1 and for each x ∈ B,
|x∗ Gx (t)|
¯ µ
Z t
¯
= ¯¯x∗ T (t − t0 )x0 − T (t − t0 )g(x) +
T (t − s)f (s, x(s),
t0
Z
¶¯
s
¯
k(s, v, x(v))dv)ds ¯¯
t0
¯
Z t
Z
¯
≤ ¯¯x∗ T (t − t0 ) (x0 − g(x)) +
x∗ T (t − s)f (s, x(s),
t0
s
t0
¯
¯
k(s, v, x(v))dv)ds¯¯
¯
Z t¯
Z s
¯ ∗
¯
∗
≤ |x T (t − t0 )(x0 − g(x))| + ¯¯x T (t − s)f (s, x(s), k(s, v, x(v))dv)¯¯ ds
t0
t0
Z t¯
Z
¯ ∗
∗
¯
≤ M · (kx0 k + L) +
¯T (t − s)x f (s, x(s),
t0
s
t0
¯
¯
k(s, v, x(v))dv)¯¯ ds
≤ M · (kx0 k + L) + a · M · P = b
hence
| x∗ Gx (t) |≤ b.
Then
kGx k = sup {| x∗ Gx (t) |: x∗ ∈ E ∗ , k x∗ k≤ 1} ≤ b
e
which means that Gx ∈ B.
e is strongly
(ii) From the following inequality we get that the set G(B)
equicontinuous. Let τ < t, so we obtain
11
|x∗ (Gx (t) − Gx (τ ))|
¯
Z t
Z
¯ ∗
¯
= ¯x (T (t − t0 )(x0 − g(x)) +
T (t − s)f (s, x(s),
t0
Z
− T (τ − t0 )(x0 − g(x)) −
s
k(s, v, x(v))dv)ds
t0
Z
τ
s
T (τ − s)f (s, x(s),
t0
t0
¯
¯
k(s, v, x(v))dv)ds)¯¯
¯
Z τ
¯ ∗
¯
= ¯x ((T (t − t0 ) − T (τ − t0 ))(x0 − g(x)) +
(T (t − s)
t0
Z
s
− T (τ − s))f (s, x(s),
k(s, v, x(v))dv)ds
t0
Z
Z
t
+
s
T (t − s)f (s, x(s),
τ
t0
¯
¯
k(s, v, x(v))dv)ds)¯¯
≤ |x∗ ((T (t − t0 ) − T (τ − t0 ))(x0 − g(x)))|
Z
τ
+
t0
¯ µ
Z
¯ ∗
¯
|(T (t − s) − T (τ − s)) | · ¯x f (s, x(s),
∗
s
t0
Z t¯ µ
Z
¯ ∗
¯
+M·
¯x f (s, x(s),
τ
s
t0
¶¯
¯
k(s, v, x(v))dv) ¯¯ ds
¶¯
¯
≤ x∗ ((T (t − t0 ) − T (τ − t0 ))(x0 − g(x)))
Z
+ sup |(T (t − s) − T (τ − s)) | ·
∗
s∈I
τ
t0
Z t¯ µ
Z
¯ ∗
¯x f (s, x(s),
+M·
¯
τ
¯ µ
¶¯
Z s
¯ ∗
¯
¯x f (s, x(s), k(s, v, x(v))dv) ¯ ds
¯
¯
t0
¶¯
s
¯
t0
≤ max{|x∗ (x0 − g(x))|, a · P } · sup |(T (t − s) − T (τ − s))∗ |
s∈I
+ M · P · (t − τ ) → 0, as t → τ.
12
(iii) Let xn → x in (C(I, E), ω). Then we get
|x∗ [Gxn (t) − Gx (t)]|
¯ ·
Z t
¯ ∗
¯
= ¯x T (t − t0 )x0 − T (t − t0 )g(xn ) +
T (t − s)f (s, xn (s),
t0
Z
s
t0
k(s, v, xn (v))dv)ds − T (t − t0 )x0 + T (t − t0 )g(x)
Z
Z
t
−
s
T (t − s)f (s, x(s),
t0
t0
¸¯
¯
k(s, v, x(v))dv)ds ¯¯
¯ ·
Z t
¯ ∗
¯
≤ ¯x T (t − t0 )(g(xn ) − g(x)) +
T (t − s) (f (s, xn (s),
t0
Z
Z
s
t0
k(s, v, xn (v)dv) − f (s, x(s),
s
t0
¸¯
¯
k(s, v, x(v))dv))ds ¯¯
¯
Z t ·
¯
≤ M · ¯¯x∗ (g(xn ) − g(x)) + M ·
x∗ f (s, xn (s),
t0
Z
Z
s
t0
k(s, v, xn (v)dv) − f (s, x(s),
¯
Z
¯ ∗
¯
≤ M · ¯x (g(xn ) − g(x)) + M ·
s
t0
t0 +a
t0
Z
Z
s
t0
k(s, v, xn (v)dv) − f (s, x(s),
¯
Z
¯
≤ M · ¯¯x∗ (g(xn ) − g(x)) + M ·
s
t0
t0 +a
t0
Z
Z
s
t0
k(s, v, xn (v)dv) − f (s, xn (s),
¸¯
¯
k(s, v, x(v))dv) ¯¯ds
·
x f (s, xn (s),
∗
¸¯
¯
·
x∗ f (s, xn (s),
s
t0
¸¯
¯
Z
+M·
¯ ·
Z
¯x∗ f (s, xn (s),
¯
t0 +a ¯
t0
13
s
k(s, v, x(v)dv)
t0
Z
s
− f (s, x(s),
t0
¸¯
¯
k(s, v, x(v))dv) ¯¯ ds .
Since f, g and k are weakly-weakly sequentially continuous functions thus
(xn ) tends weakly to x. Hence
| x∗ [Gxn (t) − Gx (t)] |→ 0.
It follows that
sup {x∗ [Gxn (t) − Gx (t)] : x∗ ∈ E ∗ , k x∗ k≤ 1} → 0.
e→B
e is weakly-weakly sequenBy conditions (i) and (ii) we have that G : B
tially continuous.
Divide the interval I = [t0 , t0 + a] with points t0 < t1 < ... < tn = t0 + a
and denote Ti = Di ∩ Iε , where Di = [ti−1 , ti ].
Let t ∈ I and ε > 0. From the L
Ã uzin Theorem we deduce that there
exists a compact subset Iε ⊂ I such that mes(I \ Iε ) < ε and s → c4 (s) is a
continuous function.
e Since V is an equicontinuous set so t 7→ V (t) = ω(V (t)) is
Let V ⊂ B.
continuous and by the lemma of Mitchell, Smith [17]:
ω (V (Ti )) = sup ωV (t) = ωV (ti ) = v(ti ).
t∈Ti
Hence for every ε > 0, we get kc4 (s)v(r) − c4 (u)v(z)k < ε, for s, r, u, z ∈ Ti .
From properties of the measure of non-compactness ω and by the mean value
theorem for the Pettis integral, we have
14
µZ
Z
ω
k(z, ξ, V (ξ))dξ)dz
Iε
≤ω
µX
n
¶
z
f (z, V (z),
t0
mesTi convf (Ti , V (Ti ) ,
n
X
µ
¶
n
X
mesTi ω f (Ti , V (Ti ),
mesTj convk(Tj , Tj , V (Tj ))
i=1
≤
j=1
n
X
½
µX
¶¾
n
mesTi c3 max ω(V (Ti )), ω
mesTj convk(Tj , Tj , V (Tj ))
i=1
≤
¶
mesTj convk (Tj , Tj , V (Tj ))
j=1
i=1
≤
n
X
j=1
n
X
½
¾
n
X
mesTi c3 max ω(V (Ti )),
mesTj ω(k(Tj , Tj , V (Tj )))
i=1
j=1
½
¾
n
n
X
X
≤
mesTi c3 max ω(V ),
mesTj sup c4 (s) · ω(V (Tj ))
i=1
≤
s∈Tj
j=1
n
X
mesTi c3 ω(V ) +
i=1
n
X
n
X
mesTi c3
i=1
= ac3 ω(V ) + ac3
n
X
= ac3 ω(V ) + ac3
= ac3 ω(V ) +
n
X
mesTj sup c4 (sj ) · ω(V (Tj ))
s∈Tj
mesTj c4 (tj )v(tj ) +
j=1
then
s∈Tj
s∈Tj
j=1
·
j=1
j=1
n
X
n
X
¸
mesTi (c4 (sj )v(tj ) − c4 (tj )v(tj )) ,
j=1
µZ
ω
Z
f (z, V (z),
I
k(z, s, V (s))ds)dz
t0
Z
≤ ac3
¶
z
t
t0
c4 (s)v(s)ds + a2 c3 ε + ac3 ω(V ).
15
µ
ω(GV (t)) = ω T (t − t0 )x0 − T (t − t0 )g(V )
Z
Z
t
+
¶
k(s, v, V (v))dv)ds
s
T (t − s)f (s, V (s),
t0
t0
≤ ω(T (t − t0 )x0 − T (t − t0 )g(V ))
·
Z
+ M a · c3
t
t0
¸
c4 (s)v(s)ds + ε2 + ac3 ω(V )
·
≤ M · ω(g(V )) + M a · c3
Z
t
t0
¸
c4 (s)v(s)ds + ε2 + ac3 ω(V )
·
¸
Z t
≤ M · C · ω(V )) + M a · c3
c4 (s)v(s)ds + ε2 + ac3 ω(V )
t0
Z
= (M C + M ac3 )ω(V ) + M ac3
t
t0
c4 (s)v(s)ds + ε2 .
Since ε2 → 0, (ε2 = a2 c3 ε) as ε → 0 so
Z
ω (GV (t)) ≤ (M C + M ac3 ) ω (V ) + M ac3
t
t0
c4 (s)ω (V (s))ds.
Next, let us define a new axiomatic measure of non-compactness in the
following way
½
µ Z t
¶¾
ϕ(V ) = sup ω(V (t)) exp −r c4 (s)ds
,
t∈I
where r >
M ac3
1−(M C+M ac3 ) .
t0
Then we get
ω(GV (t))
≤ (M C + M ac3 )ω(V )
Z
+ M ac3
t
t0
½
µ Z s
¶
µZ s
¶¾
c4 (s) · exp −r c4 (τ )dτ exp r c4 (τ )dτ
ω(V (s))ds
t0
t0
16
Z
≤ (M C + M ac3 )ω(V ) + M ac3 · ϕ(V ) ·
t
t0
µ Z s
¶
c4 (s) · exp r c4 (τ )dτ ds
t0
µ
µ Z s
¶¶
1
= (M C + M ac3 )ω(V ) + M ac3 · ϕ(V ) ·
d exp r c4 (τ )dτ
t0 r
t0
Z
t
µ Z t
¶
1
= (M C + M ac3 )ω(V ) + M ac3 · · ϕ(V ) · exp r c4 (s)ds ,
r
t0
it follows that
¶
µ Z t
ω(GV (t)) exp −r c4 (s)ds
t0
µ Z t
¶
1
≤ (M C + M ac3 )ω(V ) · exp −r c4 (s)ds + M ac3 · · ϕ(V )
r
t0
and we deduce that
µ
¶
1
ϕ(GV ) ≤ M C + · ac3 + ac3 ϕ(V ).
r
By the assumption (7) and the condition that r >
M ac3
1−(M C+M ac3 ) ,
we have
µ
¶
1
M C + · ac3 + ac3 < 1.
r
In view of Theorem 5 we see that the operator G has a fixed point and the
proof is completed.
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York-Berlin 1983.
Received 1 January 2004

on the semilinear integro-differential nonlocal cauchy - DML-PL

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