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Fixed Point Theory, 17(2016), No. 2, 275-288

http://www.math.ubbcluj.ro/∼nodeacj/sfptcj.html

SEMILINEAR EVOLUTION SYSTEMS

WITH NONLINEAR CONSTRAINTS

OCTAVIA BOLOJAN AND RADU PRECUP

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)⊆Xi→Xigenerates

a strongly continuous semigroup of contractions {Si(t), t ≥0}on the Banach space

Xi,|.|Xi, fi: [0, T ]×X1×X2→Xiis 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)|Xie−θ(t−a).

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(t−s)f1(s, x(s), y(s))ds

y(t) = S2(t)α2[x, y] + Zt

0

S2(t−s)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(t−s)f1(s, x(s), y(s))ds,

N2(x, y)(t) = S2(t)α2[x, y] + Zt

0

S2(t−s)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×X→Rn

+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 r≤sone means that ri≤sifor 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:X→Rn

+with kxk= 0 only for x= 0; kλxk=|λ| kxkfor x∈X, 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) := kx−yk,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:X→Xis any mapping, we say that

Nis a generalized contraction (in Perov’s sense) provided that a (Lipschitz) matrix

M∈Mn×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:X→Xwith the Lipschitz matrix Mhas a unique ﬁxed

point x∗,and

d(Nk(x), x∗)≤Mk(I−M)−1d(x, N (x)),

for all x∈Xand k∈N.

There are known several characterizations of matrices like in Perov’s theorem (see

[25] and [30, pp 12, 88]). More exactly, for a matrix M∈Mn×n(R+),the following

statements are equivalent:

(a) Mk→0 as k→ ∞;

(b) I−Mis nonsingular and (I−M)−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) I−Mis nonsingular and inverse-positive, i.e. (I−M)−1has nonnegative

entries.

Remark 1.2. Recal that in case n= 2,a matrix M∈M2×2(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 A∈Mn×n(R+)is a matrix with ρ(A)<1,then ρ(A+B)<1for

every matrix B∈Mn×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],0≤a≤Tis 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)|x−x|X1+ai2(t)|y−y|X2,if t∈[0, a]

bi1(t)|x−x|X1+bi2(t)|y−y|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]|Xi≤Ai1|x−x|C([0,a],X1)+Ai2|y−y|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 z∈C([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)kX≤Mθku−ukX,

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(t−s)k |f1(s, x(s), y(s)) −f1(s, x(s), y(s))|X1ds

≤C1A11 |x−x|C([0,a],X1)+A12 |y−y|C([0,a],X2)+

+C1Za

0a11(s)|x(s)−x(s)|X1+a12 (s)|y(s)−y(s)|X2ds

≤C1A11 +|a11|L1[0,a]|x−x|C([0,a],X1)

+C1A12 +|a12|L1[0,a]|y−y|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 |x−x|C([0,a],X1)+m12 |y−y|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(t−s)k |f1(s, x(s), y(s)) −f1(s, x(s), y(s))|X1ds

+Zt

a

kS1(t−s)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(t−s)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)|X1e−θ(s−a)eθ(s−a)ds

+Zt

a

b12(s)|y(s)−y(s)|X2e−θ(s−a)eθ(s−a)ds

≤C1|x−x|Cθ([a,T ],X1)Zt

a

b11(s)eθ(s−a)ds

+|y−y|Cθ([a,T ],X2)Zt

a

b12(s)eθ(s−a)ds.

Now H¨older’s inequality gives

Zt

a

b1j(s)eθ(s−a)ds ≤|b1j|Lp[a,T ]

(qθ)1/q eθ(t−a), j = 1,2.

Then (2.7) yields

Zt

a

kS1(t−s)k |f1(s, x(s), y(s)) −f1(s, x(s), y(s))|X1ds

≤C1

(qθ)1/q |b11 |Lp[a,T ]|x−x|Cθ([a,T ],X1)+|b12|Lp[a,T ]|y−y|Cθ([a,T ],X2)eθ(t−a).

Next (2.6) implies that

|N1(x, y)(t)−N1(x, y)(t)|X1

≤m11 |x−x|C([0,a],X1)+m12 |y−y|C([0,a],X2)

+C1

(qθ)1/q |b11 |Lp[a,T ]|x−x|Cθ([a,T ],X1)+|b12|Lp[a,T ]|y−y|Cθ([a,T ],X2)eθ(t−a).

Dividing by eθ(T−a)and taking the supremum when t∈[a, T ] we obtain

|N1(x, y)−N1(x, y)|Cθ([a,T ],X1)≤m11 |x−x|C([0,a],X1)+m12 |y−y|C([0,a],X2)

+C1

(qθ)1/q |b11 |Lp[a,T ]|x−x|Cθ([a,T ],X1)+|b12|Lp[a,T ]|y−y|Cθ([a,T ],X2).(2.8)

Thus, from (2.4), (2.8), we have that

|N1(x, y)−N1(x, y)|∗≤(m11 +n11 )|x−x|∗+ (m12 +n12)|y−y|∗(2.9)

where

n1j=C1

(qθ)1/q |b1j|Lp(a,T ), j = 1,2.(2.10)

SEMILINEAR EVOLUTION SYSTEMS 281

Similarly

|N2(x, y)−N2(x, y)|∗≤(m21 +n21 )|x−x|∗+ (m22 +n22)|y−y|∗,(2.11)

where

m2j=C2A2j+|a2j|L1[0,a], n2j=C2

(qθ)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"|x−x|∗

|y−y|∗#

or equivalently,

kN(u)−N(u)kX≤Mθku−ukX,

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]|Xi≤Ai1|x|C([0,a],X1)+Ai2|y|C([0,a],X2)+Ai3,(2.14)

for all x∈C([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 B⊂Xsuch 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(t−s)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(t−s)k |f1(s, x(s), y(s))|X1ds

+Zt

a

kS1(t−s)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(t−s)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)|X1e−θ(s−a)eθ(s−a)ds

+C1Zt

a

b12(s)|y(s)|X2e−θ(s−a)eθ(s−a)ds +C1Zt

a

b13(s)ds

≤C1|x|Cθ([a,T ],X1)Zt

a

b11(s)eθ(s−a)ds +C1|y|Cθ([a,T ],X2)Zt

a

b12(s)eθ(s−a)ds

+C1|b13|L1(a,T ).

Now, using H¨older’s inequality, we deduce that

Zt

a

kS1(t−s)k |f1(s, x(s), y(s))|X1ds

≤C1

(qθ)1/q |b11 |Lp(a,T )|x|Cθ([a,T ],X1)+|b12|Lp(a,T )|y|Cθ([a,T ],X2)eθ(t−a)

+C1|b13|L1(a,T ).

SEMILINEAR EVOLUTION SYSTEMS 283

Thus, (2.17) becomes

|N1(x, y)(t)|X1≤m11 |x|C([0,a],X1)+m12 |y|C([0,a],X2)+b

C1

+C1

(qθ)1/q |b11 |Lp(a,T )|x|Cθ([a,T ],X1)+|b12|Lp(a,T )|y|Cθ([a,T ],X2)eθ(t−a)

+C1|b13|L1(a,T ).

Dividing by eθ(T−a)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

(qθ)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 I−Mθis invertible and its inverse (I−Mθ)−1has nonnegative

elements and thus "R1

R2#≥(I−Mθ)−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, d2∈Rn;ai, bi∈C([0, T ],R+) ; mi, ni∈R+,for i= 1,2.

In this case, for each i= 1,2,we may take

Xi=Lri(Rn), ri≥1,

and Aithe directional derivative operator along direction di,

Aiu=di· ∇u=

n

X

j=1

dij

∂u

∂xj

,

D(Ai) = {u∈Lri(Rn) : di· ∇u∈Lri(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(x−tdi), 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, d2∈Rn;ai, bi, ci∈C([0, T ],R+) ; mi, ni∈R+,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