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Semilinear evolution systems with nonlinear constraints

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Abstract

The purpose of the present paper is to study the existence of solutions to semilinear evolution systems with nonlinear constraints. We establish new existence results using the fixed point principles of Perov and Schauder, combined with the technique that uses matrices with the spectral radius less than one and vector-valued norms. This vectorial approach is fruitful for the treating of systems in general and allows the system nonlinearities to behave independently as much as possible. Moreover, the constants from the Lipschitz or growth conditions are put into connection with the support of the nonlinear operators expressing the constraints. The paper extends and complements previous results from the literature.
Fixed Point Theory, 17(2016), No. 2, 275-288
http://www.math.ubbcluj.ro/nodeacj/sfptcj.html
SEMILINEAR EVOLUTION SYSTEMS
WITH NONLINEAR CONSTRAINTS
Department of Mathematics, Babe¸s-Bolyai University
Cluj-Napoca, Romania
E-mail: r.precup@math.ubbcluj.ro
Abstract. The purpose of the present paper is to study the existence of solutions to semilinear
evolution systems with nonlinear constraints. We establish new existence results using the ﬁxed
point principles of Perov and Schauder, combined with the technique that uses matrices with the
spectral radius less than one and vector-valued norms. This vectorial approach is fruitful for the
treating of systems in general and allows the system nonlinearities to behave independently as much
as possible. Moreover, the constants from the Lipschitz or growth conditions are put into connection
with the support of the nonlinear operators expressing the constraints. The paper extends and
complements previous results from the literature.
Key Words and Phrases: Abstract evolution equation, nonlocal condition, ﬁxed point, vector-
valued norm, spectral radius of a matrix.
2010 Mathematics Subject Classiﬁcation: 34G20, 34B10, 47J35, 47H10.
1. Introduction and preliminaries
We are concerned with the existence and the uniqueness of solutions to the semi-
linear system of abstract evolution equations with constraints
x0(t) + A1x(t) = f1(t, x (t), y(t))
y0(t) + A2y(t) = f2(t, x (t), y(t))
g1(x, y) = 0
g2(x, y) = 0.
(0 < t < T ) (1.1)
Here, for each i= 1,2,the linear operator Ai:D(Ai)XiXigenerates
a strongly continuous semigroup of contractions {Si(t), t 0}on the Banach space
Xi,|.|Xi, fi: [0, T ]×X1×X2Xiis a given function and gi:C([0, T ], X1)×
C([0, T ], X2)Xiis a nonlinear operator.
It is convenient that the constraints are written equivalently under the form of
nonlocal conditions
x(0) = α1[x, y], y (0) = α2[x, y],(1.2)
where
α1[x, y] = g1(x, y) + x(0) , α2[x, y ] = g2(x, y) + y(0) .
In the literature, it has been given a particular attention and interest to diﬀerent
classes of problems with nonlocal conditions. In this matter, we mention the papers
275
276 OCTAVIA BOLOJAN AND RADU PRECUP
[6]-[8], [15], [16]-[22], [28], [35], [38] and references therein. This type of problems
arise in the study of mathematical modeling of real processes, such as heat, ﬂuid,
chemical or biological ﬂow, where the nonlocal conditions can be seen as feedback
controls. Other discussions about the importance of nonlocal conditions in diﬀerent
areas of applications, examples of evolution equations and systems with or without
delay, subjected to nonlocal initial conditions and references to other works dealing
with nonlocal problems can be found in [9], [21], [23], [26], [31], [33], [34].
The constraints (1.2) can be either of discrete type (or multi-point conditions), or
of continuous type expressed by continuous operators. Also, the nonlocal conditions
can be linear or nonlinear. For the case of nonlinear nonlocal conditions we mention
the recent paper [5], as well as [11], [13], [14], [24] and references therein.
In all cases, it is important to take into consideration the support of the nonlocal
conditions, that is the smallest interval [0, a][0, T ] on which those conditions act
in the sense that whenever x, x C([0, T ], X1) and y, y C([0, T ], X2),
x|[0,a]=x|[0,a]and y|[0,a]=y|[0,a]imply αi[x, y] = αi[x, y], i = 1,2.
The notion of support plays an essential role in the existence results for the nonlocal
problems, as shown in the paper [8]. More exactly, it was shown that it is necessary
to impose stronger conditions on the nonlinearities on the subinterval [0, a],compared
to those required on the rest of the interval [0, T ].As shown in the previous works
[2]-[5], [8], [18], [19], the integral system equivalent to the nonlocal problem is of
Fredholm type on the support [0, a] and of Volterra type on [a, T ].From a physical
point of view, on the time interval [0, a],the evolution process is subjected to some
constraints, and after ”the moment a” it becomes free of any constraints.
In connection with the support of the nonlocal conditions, we shall consider a
special norm on C([0, T ], Xi) (i= 1,2) ,namely
|x|= max n|x|C([0,a],Xi),|x|Cθ([a,T ],Xi)o,
where |.|C([0,a],Xi)is the usual max norm on C([0, a], Xi),
|x|C([0,a],Xi)= max
t[0,a]|x(t)|Xi,
while for any θ > 0,|x|Cθ([a,T ],Xi)is the Bielecki norm on C([a, T ], Xi),
|x|Cθ([a,T ],Xi)= max
t[a,T ]|x(t)|Xieθ(ta).
In what follows, we look for global mild solutions on the interval [0, T ],i.e. a pair
(x, y)C([0, T ], X1)×C([0, T ], X2) satisfying the integral system
x(t) = S1(t)α1[x, y] + Zt
0
S1(ts)f1(s, x(s), y(s))ds
y(t) = S2(t)α2[x, y] + Zt
0
S2(ts)f2(s, x(s), y(s))ds.
(1.3)
This system can be viewed as a ﬁxed point problem in C([0, T ], X1)×C([0, T ], X2) for
the nonlinear operator N= (N1, N2) : C([0, T ], X1)×C([0, T ], X2)C([0, T ], X1)
SEMILINEAR EVOLUTION SYSTEMS 277
×C([0, T ], X2) deﬁned by
N1(x, y)(t) = S1(t)α1[x, y] + Zt
0
S1(ts)f1(s, x(s), y(s))ds,
N2(x, y)(t) = S2(t)α2[x, y] + Zt
0
S2(ts)f2(s, x(s), y(s))ds.
(1.4)
Basic notions and results from the semigroup theory that are frequently used in
our work can be found, for example, in [10], [12] and [32]. Next, we recall some basic
notions that are used in our vectorial approach. Details can be found in [1], [25]-[27],
[29] and [30].
By a vector-valued metric on a set Xwe mean a mapping d:X×XRn
+such
that (i) d(x, y) = 0 if and only if x=y; (ii) d(x, y) = d(y, x) for all x, y Xand
(iii) d(x, y)d(x, z) + d(z, y) for all x, y, z X. Here by we mean the natural
componentwise order relation of Rn,more exactly, if r, s Rn, r = (r1, r2, ..., rn),
s= (s1, s2, ..., sn),then by rsone means that risifor i= 1,2, ..., n.
A set Xtogether with a vector-valued metric dis called a generalized metric space.
For such a space, the notions of Cauchy sequence, convergence, completeness, open
and closed set are similar to those in usual metric spaces.
Similarly, we speak about a vector-valued norm on a linear space X, as being a
mapping k.k:XRn
+with kxk= 0 only for x= 0; kλxk=|λ| kxkfor xX, λ R,
and kx+yk≤kxk+kykfor every x, y X. To any vector-valued norm k.kone can
associate the vector-valued metric d(x, y) := kxyk,and one says that (X, k.k) is a
generalized Banach space if Xis complete with respect to d.
If (X, d) is a generalized metric space and N:XXis any mapping, we say that
Nis a generalized contraction (in Perov’s sense) provided that a (Lipschitz) matrix
MMn×n(R+) exists such that its powers Mktend to the zero matrix 0 as k→ ∞,
and
d(N(x), N (y)) Md(x, y) for all x, y X.
For such kind of mappings the following generalization of Banach’s contraction prin-
ciple holds:
Theorem 1.1 (Perov).If (X, d)is a complete generalized metric space, then any
generalized contraction N:XXwith the Lipschitz matrix Mhas a unique ﬁxed
point x,and
d(Nk(x), x)Mk(IM)1d(x, N (x)),
for all xXand kN.
There are known several characterizations of matrices like in Perov’s theorem (see
[25] and [30, pp 12, 88]). More exactly, for a matrix MMn×n(R+),the following
statements are equivalent:
(a) Mk0 as k→ ∞;
(b) IMis nonsingular and (IM)1=I+M+M2+... (where Istands for
the unit matrix of the same order as M);
(c) the eigenvalues of Mare located inside the unit disc of the complex plane,
i.e. ρ(M)<1,where ρ(M) is the spectral radius of M;
278 OCTAVIA BOLOJAN AND RADU PRECUP
(d) IMis nonsingular and inverse-positive, i.e. (IM)1has nonnegative
entries.
Remark 1.2. Recal that in case n= 2,a matrix MM2×2(R+)satisﬁes ρ(M)<1
if and only if
tr M < min {2,1 + det M}.
The following almost obvious lemma will be used in the sequel.
Lemma 1.3. If AMn×n(R+)is a matrix with ρ(A)<1,then ρ(A+B)<1for
every matrix BMn×n(R+)whose elements are small enough.
The role of matrices with spectral radius less than one in the study of semilinear
operator systems was pointed out in [27], also in connection with other abstract
principles from nonlinear functional analysis.
Besides Perov’s ﬁxed point theorem, in this work we shall also use the well-known
Schauder’s ﬁxed point theorem.
2. Main results
Throughout this section, we assume that Xiand Aiare as in the previous section,
and that the interval [0, a],0aTis the support of the nonlocal conditions.
Our ﬁrst result is an existence and uniqueness theorem for the case where the
nonlinearities f1, f2and mappings α1, α2are continuous and satisfy the Lipschitz
conditions
|fi(t, x, y)fi(t, x, y)|Xi(ai1(t)|xx|X1+ai2(t)|yy|X2,if t[0, a]
bi1(t)|xx|X1+bi2(t)|yy|X2,if t[a, T ]
(2.1)
for all (x, y),(x, y)X1×X2and some aij L1([0, a],R+), bij Lp([a, T ],R+)
and p > 1,and
|αi[x, y]αi[x, y]|XiAi1|xx|C([0,a],X1)+Ai2|yy|C([0,a],X2),(2.2)
for all x, x C([0, T ], X1) and y, y C([0, T ], X2) and some Aij R+(i, j = 1,2) .
Denote
Ci= sup {kSi(t)k:t[0, T ]}(i= 1,2) ,
where k·k stands for the norm of a continuous linear operator.
Assuming a vectorial condition involving the coeﬃcients aij and Aij ,but indepen-
dent on the coeﬃcients bij,we have the following result:
Theorem 2.1. Assume that the conditions (2.1) and (2.2) hold. In addition assume
that the spectral radius of the matrix
M0:= hCiAij +|aij |L1[0,a]ii,j=1,2(2.3)
is less than one. Then the problem (1.1) has a unique mild solution on [0, T ].
SEMILINEAR EVOLUTION SYSTEMS 279
Proof. We shall apply Perov’s theorem in X=C([0, T ], X1)×C([0, T ], X2) endowed
with the vector norm k.kXdeﬁned by
kukX= (|x|,|y|),
where u= (x, y) and for any zC([0, T ], Xi),
|z|= max n|z|C([0,a],Xi),|z|Cθ([a,T ],Xi)o.
Clearly, (X, k.kX) is a complete generalized metric space. Finding a mild solution
of problem (1.1) is equivalent to ﬁnding a ﬁxed point of the nonlinear operator N=
(N1, N2) deﬁned by (1.4). We have to prove that Nis a generalized contraction, more
exactly that
kN(u)N(u)kXMθkuukX,
for all u= (x, y), u = (x, y)Xand some matrix Mθwith ρ(Mθ)<1.To this end,
let u= (x, y), u = (x, y ) be any two elements of X. For t[0, a],using (2.1) and
(2.2), we have
|N1(u)(t)N1(u)(t)|X1≤ kS1(t)k |α1[x, y]α1[x, y]|X1
+Zt
0
kS1(ts)k |f1(s, x(s), y(s)) f1(s, x(s), y(s))|X1ds
C1A11 |xx|C([0,a],X1)+A12 |yy|C([0,a],X2)+
+C1Za
0a11(s)|x(s)x(s)|X1+a12 (s)|y(s)y(s)|X2ds
C1A11 +|a11|L1[0,a]|xx|C([0,a],X1)
+C1A12 +|a12|L1[0,a]|yy|C([0,a],X2).
Therefore, taking the supremum when t[0, a],we obtain
|N1(x, y)N1(x, y)|C([0,a],X1)
m11 |xx|C([0,a],X1)+m12 |yy|C([0,a],X2),(2.4)
where
m1j=C1A1j+|a1j|L1[0,a], j = 1,2.(2.5)
Next, for t[a, T ] and any θ > 0,we have
|N1(x, y)(t)N1(x, y)(t)|X1≤ kS1(t)k |α1[x, y ]α1[x, y]|X1(2.6)
+Za
0
kS1(ts)k |f1(s, x(s), y(s)) f1(s, x(s), y(s))|X1ds
+Zt
a
kS1(ts)k |f1(s, x(s), y(s)) f1(s, x(s), y(s))|X1ds.
280 OCTAVIA BOLOJAN AND RADU PRECUP
The sum of the ﬁrst two terms can be estimated according to (2.4). As concerns the
last term, we have
Zt
a
kS1(ts)k |f1(s, x(s), y(s)) f1(s, x(s), y(s))|X1ds (2.7)
C1Zt
ab11(s)|x(s)x(s)|X1+b12 (s)|y(s)y(s)|X2ds
=C1Zt
a
b11(s)|x(s)x(s)|X1eθ(sa)eθ(sa)ds
+Zt
a
b12(s)|y(s)y(s)|X2eθ(sa)eθ(sa)ds
C1|xx|Cθ([a,T ],X1)Zt
a
b11(s)eθ(sa)ds
+|yy|Cθ([a,T ],X2)Zt
a
b12(s)eθ(sa)ds.
Now H¨older’s inequality gives
Zt
a
b1j(s)eθ(sa)ds |b1j|Lp[a,T ]
()1/q eθ(ta), j = 1,2.
Then (2.7) yields
Zt
a
kS1(ts)k |f1(s, x(s), y(s)) f1(s, x(s), y(s))|X1ds
C1
()1/q |b11 |Lp[a,T ]|xx|Cθ([a,T ],X1)+|b12|Lp[a,T ]|yy|Cθ([a,T ],X2)eθ(ta).
Next (2.6) implies that
|N1(x, y)(t)N1(x, y)(t)|X1
m11 |xx|C([0,a],X1)+m12 |yy|C([0,a],X2)
+C1
()1/q |b11 |Lp[a,T ]|xx|Cθ([a,T ],X1)+|b12|Lp[a,T ]|yy|Cθ([a,T ],X2)eθ(ta).
Dividing by eθ(Ta)and taking the supremum when t[a, T ] we obtain
|N1(x, y)N1(x, y)|Cθ([a,T ],X1)m11 |xx|C([0,a],X1)+m12 |yy|C([0,a],X2)
+C1
()1/q |b11 |Lp[a,T ]|xx|Cθ([a,T ],X1)+|b12|Lp[a,T ]|yy|Cθ([a,T ],X2).(2.8)
Thus, from (2.4), (2.8), we have that
|N1(x, y)N1(x, y)|(m11 +n11 )|xx|+ (m12 +n12)|yy|(2.9)
where
n1j=C1
()1/q |b1j|Lp(a,T ), j = 1,2.(2.10)
SEMILINEAR EVOLUTION SYSTEMS 281
Similarly
|N2(x, y)N2(x, y)|(m21 +n21 )|xx|+ (m22 +n22)|yy|,(2.11)
where
m2j=C2A2j+|a2j|L1[0,a], n2j=C2
()1/q |b2j|Lp[a,T ], j = 1,2.(2.12)
Next, (2.9) and (2.11) can be put together under the vectorial form
"|N1(x, y)N1(x, y)|
|N2(x, y)N2(x, y)|#Mθ"|xx|
|yy|#
or equivalently,
kN(u)N(u)kXMθkuukX,
for all u= (x, y), u = (x, y )X. Clearly Mθcan be represented as Mθ=M0+M1,
where
M0= [mij ]i,j=1,2, M1= [nij ]i,j=1,2.
Since ρ(M0)<1,from Lemma 1.3 we have that ρ(Mθ)<1 for a large enough θ > 0
making nij as small as necessary. Therefore, Perov’s theorem applies and gives the
conclusion.
If we assume that the operator Nis completely continuous, we can weaken condi-
tions (2.1) and (2.2) to at most linear growth conditions. In this case, we can apply
Schauder’s ﬁxed point theorem that guarantees the existence of the solution for our
studied problem, but not also the uniqueness.
Therefore, we give our second result assuming that the nonlinearities f1, f2and
the mappings α1, α2satisfy, instead of the Lipschitz conditions, some conditions of at
most linear growth, namely
|fi(t, x, y)|Xi(ai1(t)|x|X1+ai2(t)|y|X2+ai3(t),if t[0, a]
bi1(t)|x|X1+bi2(t)|y|X2+bi3(t),if t[a, T ](2.13)
for all (x, y)X1×X2and some aij L1([0, a],R+), bij Lp([a, T ],R+), p > 1,
and
|αi[x, y]|XiAi1|x|C([0,a],X1)+Ai2|y|C([0,a],X2)+Ai3,(2.14)
for all xC([0, T ], X1), y C([0, T ], X2) and some Aij R+(i= 1,2, j = 1,2,3) .
Theorem 2.2. Assume that conditions (2.13) and (2.14) are satisﬁed. If the spectral
radius of the matrix (2.3) is less than one, then problem (1.1) has at least one mild
solution.
Proof. In order to apply Schauder’s ﬁxed point principle, we need to ﬁnd a nonempty
closed bounded convex set BXsuch that
N(B)B. (2.15)
We shall look for the set Bunder the form B:= B1(0; R1)×B2(0; R2),where
Bi(0; Ri) is the closed ball of radius Ricentered in the origin of the space C([0, T ], Xi),
282 OCTAVIA BOLOJAN AND RADU PRECUP
i= 1,2.Thus we have to ﬁnd two positive numbers R1, R2such that (2.15) holds.
Let u= (x, y)X. For t[0, a],using (2.13) and (2.14), we have
|N1(u)(t)|X1≤ kS1(t)k |α1[x, y]|X1+Zt
0
kS1(ts)k |f1(s, x(s), y(s))|X1ds
C1A11 |x|C([0,a],X1)+A12 |y|C([0,a],X2)+A13+
+C1Za
0a11(s)|x(s)|X1+a12 (s)|y(s)|X2+a13(s)ds
C1A11 +|a11|L1(0,a)|x|C([0,a],X1)+C1A12 +|a12 |L1(0,a)|y|C([0,a],X2)+b
C1,
where b
C1:= C1A13 +|a13|L1(0,a).Taking the supremum when t[0, a],we obtain
|N1(x, y)|C([0,a],X1)m11 |x|C([0,a],X1)+m12 |y|C([0,a],X2)+b
C1,(2.16)
where m1j, j = 1,2 are given by (2.5). Next, for t[a, T ] and any θ > 0,we have
|N1(x, y)(t)|X1
≤ kS1(t)k |α1[x, y]|X1+Za
0
kS1(ts)k |f1(s, x(s), y(s))|X1ds
+Zt
a
kS1(ts)k |f1(s, x(s), y(s))|X1ds. (2.17)
Again, the sum of the ﬁrst two terms can be estimated according to (2.16). For the
last term, we obtain
Zt
a
kS1(ts)k |f1(s, x(s), y(s))|X1ds
C1Zt
ab11(s)|x(s)|X1+b12 (s)|y(s)|X2+b13(s)ds
C1Zt
a
b11(s)|x(s)|X1eθ(sa)eθ(sa)ds
+C1Zt
a
b12(s)|y(s)|X2eθ(sa)eθ(sa)ds +C1Zt
a
b13(s)ds
C1|x|Cθ([a,T ],X1)Zt
a
b11(s)eθ(sa)ds +C1|y|Cθ([a,T ],X2)Zt
a
b12(s)eθ(sa)ds
+C1|b13|L1(a,T ).
Now, using H¨older’s inequality, we deduce that
Zt
a
kS1(ts)k |f1(s, x(s), y(s))|X1ds
C1
()1/q |b11 |Lp(a,T )|x|Cθ([a,T ],X1)+|b12|Lp(a,T )|y|Cθ([a,T ],X2)eθ(ta)
+C1|b13|L1(a,T ).
SEMILINEAR EVOLUTION SYSTEMS 283
Thus, (2.17) becomes
|N1(x, y)(t)|X1m11 |x|C([0,a],X1)+m12 |y|C([0,a],X2)+b
C1
+C1
()1/q |b11 |Lp(a,T )|x|Cθ([a,T ],X1)+|b12|Lp(a,T )|y|Cθ([a,T ],X2)eθ(ta)
+C1|b13|L1(a,T ).
Dividing by eθ(Ta)and taking the supremum when t[a, T ],we obtain
|N1(x, y)|Cθ([a,T ],X1)m11 |x|C([0,a],X1)+m12 |y|C([0,a],X2)+b
C1
+C1
()1/q |b11 |Lp(a,T )|x|Cθ([a,T ],X1)+|b12|Lp(a,T )|y|Cθ([a,T ],X2)(2.18)
+C1|b13|L1(a,T ).
Therefore, from (2.16) and (2.18) we have
|N1(x, y)|(m11 +n11)|x|+ (m12 +n12)|y|+C1,(2.19)
where we have denoted C1:= b
C1+C1|b13|L1(a,T )and n1j, j = 1,2 are given by (2.10).
Similarly,
|N2(x, y)|(m21 +n21)|x|+ (m22 +n22)|y|+C2,(2.20)
where C2:= C2A23 +|a23|L1(0,a)+|b23 |L1(a,T)and m2j, n2j, j = 1,2 are given by
(2.12).
Now, (2.19) and (2.20) can be put together as
"|N1(x, y)|
|N2(x, y)|#Mθ"|x|
|y|#+"C1
C2#,
where Mθ=M0+M1is given by (2.3) and has the spectral radius less than one
according to the hypotheses of the theorem.
In what follows, we look for two positive numbers R1, R2such that if |x|
R1,|y|R2,then |N1(x, y)|R1,|N2(x, y)|R2.To this end, it is suﬃcient
that
Mθ"R1
R2#+"C1
C2#"R1
R2#.
Since ρ(M0)<1,from Lemma 1.3 we have that ρ(Mθ)<1 for a large enough θ > 0.
Then the matrix IMθis invertible and its inverse (IMθ)1has nonnegative
elements and thus "R1
R2#(IMθ)1"C1
C2#.(2.21)
Therefore we can take R1, R2the numbers corresponding to the equality in (2.21).
Thus Schauder’s ﬁxed point theorem can be applied and gives the conclusion.
284 OCTAVIA BOLOJAN AND RADU PRECUP
Suﬃcient conditions for the complete continuity of the operator N, as well as for
the mild solutions to be classical solutions can be found in the literature, for example
in [10], [32].
We note that Theorem 2.1 and Theorem 2.2 can be easily extended to the general n-
dimensional case, when the assumption about the spectral radius of the corresponding
matrix of order ncan be checked using computer algebra programs such as Maple,
Mathematica or Sage.
We conclude this paper by two examples illustrating our main results.
Example 2.1. Let us consider the semilinear transport system
u0(t) + d1· ∇u(t) = a1(t) sin u(t) + b1(t)v(t)
v0(t) + d2· ∇v(t) = a2(t)u(t) + b2(t) cos v(t)
u(0) = ZT/2
0
[m1u(s) + n1v(s)] ds
v(0) = ZT/2
0
[m2u(s) + n2v(s)] ds,
(2.22)
where d1, d2Rn;ai, biC([0, T ],R+) ; mi, niR+,for i= 1,2.
In this case, for each i= 1,2,we may take
Xi=Lri(Rn), ri1,
and Aithe directional derivative operator along direction di,
Aiu=di· ∇u=
n
X
j=1
dij
∂u
∂xj
,
D(Ai) = {uLri(Rn) : di· ∇uLri(Rn)}.
Notice that here uis the gradient of uin the sense of distributions, and in (2.22) by
sin u(t) we mean the function x7→ sin (u(t) (x)) .The meaning of cos v(t) is similar.
According to [32, p. 88], the operator Aigenerates even a C0-group of isometries,
namely
Si(t)u(x) = u(xtdi), u Lri(Rn), t R, x Rn.
Hence Ci= 1.For this example, we have
f1(t, u(t), v(t)) = a1(t) sin u(t) + b1(t)v(t),
f2(t, u(t), v(t)) = a2(t)u(t) + b2(t) cos v(t),
α1[u, v] = ZT /2
0
[m1u(s) + n1v(s)] ds,
α2[u, v] = ZT /2
0
[m2u(s) + n2v(s)] ds.
Also, in this case a=T/2,while f1, f2satisfy (2.1) with
ai1=ai|[0,T/2] , ai2=bi|[0,T/2] ,
bi1=ai|[T/2,T], bi2=bi|[T /2,T ].
SEMILINEAR EVOLUTION SYSTEMS 285
In addition, the mappings α1, α2satisfy (2.2) with
A11 =T m1/2, A12 =T n1/2, A21 =T m2/2, A22 =T n2/2.
Then
M0=hCiAij +|aij |L1[0,a]ii,j=1,2(2.23)
="T m1/2 + |a1|L1[0,T/2] T n1/2 + |b1|L1[0,T/2]
T m2/2 + |a2|L1[0,T/2] T n2/2 + |b2|L1[0,T/2] #.
Therefore, according to Theorem 2.1, if the spectral radius of the matrix M0is less
than one, then the problem (2.22) has a unique mild solution in C([0, T ], Lr1(Rn))
×C([0, T ], Lr2(Rn)) .
The next example illustrates the existence result given by Theorem 2.2.
Example 2.2. Consider the semilinear transport system
u0(t) + d1· ∇u(t) = a1(t)u(t) sin (u(t) + v(t)) + b1(t)v(t) + c1(t)
v0(t) + d2· ∇v(t) = a2(t)u(t) + b2(t)v(t) cos (u(t) + v(t)) + c2(t)
u(0) = ZT/2
0
[m1u(s) + n1v(s)] ds
v(0) = ZT/2
0
[m2u(s) + n2v(s)] ds,
(2.24)
where d1, d2Rn;ai, bi, ciC([0, T ],R+) ; mi, niR+,for i= 1,2.
If the spectral radius of the matrix (2.23) is less than one, then the problem (2.24)
has at least one mild solution in C([0, T ], Lr1(Rn)) ×C([0, T ], Lr2(Rn)) .
Notice that the nonlinearities f1and f2from this example do not satisfy conditions
(2.1) and therefore Theorem 2.1 does not apply.
Acknowledgements. The ﬁrst author was supported by the Sectoral Operational
Programme for Human Resources Development 2007-2013, co-ﬁnanced by the Eu-
ropean Social Fund, under the project POSDRU/159/1.5/S/137750 - “Doctoral and
postoctoral programs - support for increasing research competitiveness in the ﬁeld
of exact Sciences”. The second author was supported by a grant of the Romanian
National Authority for Scientiﬁc Research, CNCS – UEFISCDI, project number PN-
II-ID-PCE-2011-3-0094.
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Received: August 1, 2015; Accepted: November 19, 2015.
Note. The paper was presented at the International Conference on Nonlinear
Operators, Diﬀerential Equations and Applications, Cluj-Napoca, 2015.
288 OCTAVIA BOLOJAN AND RADU PRECUP
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