Mem. Differential Equations Math. Phys. 34 (2005), 149–152
I. Kiguradze and B. P˚uˇza
ON THE WELL-POSEDNESS OF NONLINEAR BOUNDARY VALUE PROBLEMS FOR FUNCTIONAL DIFFERENTIAL EQUATIONS
(Reported on October 25, 2004)
Let−∞< a < b <+∞,I= [a, b],nbe a natural number, and letf :C(I;Rn)→ L(I;Rn) andh:C(I;Rn)→Rnbe continuous operators. Consider the boundary value problem
dx(t)
dt =f(x)(t), (1)
h(x) = 0, (2)
by a solution of which we mean an absolutely continuous vector functionx :I→Rn satisfying both the system (1.1) almost everywhere onIand the condition (1.2).
The well-posedness of this problem is more or less satisfactorily investigated only in the cases when f is either the linear, or the Nemytski operator (see, e.g., [1]–[9] and the references therein). In a general case to which we propose the present paper, the well-posedness of the problem (1), (2) remains still little studied.
In what follows, the following notation will be used.
Ris the set of real numbers,R+= [0,+∞[ ;
Rnis the space ofn-dimensional vectors x= (xi)ni=1with componentsxi∈R(i= 1, . . . , n) and the norm
kxk=
n
X
i=1
|xi|;
C(I;Rn) is the space of continuous vector functionsx:I→Rnwith the norm kxkC = max
kx(t)k: t∈I ;
L(I;Rn) is the space of vector functionsx:I→Rnwith Lebesgue integrable compo- nents and the norm
kxkC =
b
Z
a
kx(t)kdt;
L(I;R+) =
x∈L(I;R) : x(t)≥0 fort∈I ;
M(I×R+;R+) is the set of nondecreasing in the second argument functions ω : I×R+→R+such thatω(·, ρ)∈L(I;R+) forρ∈R+andω(t,0) = 0 fort∈I.
Ifx0 ∈C(I;Rn),ρ∈]0,+∞[ ,η∗∈L(I;R+) andη:C(I;Rn)→L(I;Rn), then we put
U(x0;ρ) =
x∈C(I;Rn) : kx−x0k< ρ
and denote byUη,η∗(x0;ρ) the set of absolutely continuous vector functionsx∈ U(x0;ρ) such that
x0(t)−η(x0)(t)
≤η∗(t) for almost all t∈I.
2000Mathematics Subject Classification.34K10.
Key words and phrases. Nonlinear functional differential equation, nonlinear bound- ary value problem, well-posedness.
150
Along with (1), (2) we will consider the perturbed problem dx(t)
dt =f(x)(t) +η(x)(t), (3)
h(x) +γ(x) = 0, (4)
whereη:C(I;Rn)→L(I;Rn) andγ:C(I;Rn)→Rnare continuous operators.
Letx0be a solution of the problem (1), (2), and letρbe a positive constant. Introduce the following definitions.
Definition 1.The problem (1), (2) is said to be (x0;ρ)-well-posedif for anyε∈]0, ρ[ , ρ∗∈]0,+∞[ ,η∗∈ L(I;R+) and ω∈M(I×R+;R+) there existsδ >0 such that no matter how are the continuous operatorsη:C(I;Rn)→L(I;Rn) andγ:C(I;Rn)→Rn, satisfying the conditions
η(x)(t)−η(y)(t)
≤ω t,kx−ykC
, kγ(x)k ≤ρ for t∈I, xandy∈ U(x0;ρ),
t
Z
a
η(x)(s)ds
≤δ, kγ(x)k< δ for t∈I, x∈ Uη,η∗(x0;ρ),
the perturbed problem (3), (4) has at least one solution contained in the ballU(x0;ρ), and each of such solutions belongs also to the ballU(x0;ε).
Definition 2. The problem (1), (2) is said to bewell-posedif it is (x0;ρ)-well-posed for an arbitraryρ >0.
Definition 3. The pair (p, `) of continuous operators p : C(I;Rn)×C(I;Rn)→ L(I;Rn) and`:C(I;Rn)×C(I;Rn)→Rnis said to beconsistentif:
(i) for anyx ∈ C(I;Rn), the operatorsp(x,·) : C(I;Rn) → L(I;Rn) and`(x,·) : C(I;Rn)→Rnare linear;
(ii) there exist an integrable in the first argument and nondecreasing in the second argument functionα:I×R+→R+ and a nondecreasing functionα0:R+ →R+such that for arbitraryxandy∈C(I;Rn) and for almost allt∈Ithe inequalities
p(x, y)(t)
≤α t,kxkC
kykC, k`(x, y)k ≤α0(kxkC)kykC
are fulfilled;
(iii) there exists a positive constantβ such that for anyx∈ C(I;Rn),q∈L(I;Rn) andc0∈Rn, an arbitrary solutionyof the boundary value problem
dy(t)
dt =p(x, y)(t) +q(t), `(x, y) =c0
admits the estimate
kykC ≤β kc0k+kqkL
.
Definition 4. A solutionx0of the problem (1), (2) is said to bestrongly isolated in radiusρ0, if there exist a consistent pair (p, `) of continuous operatorsp:C(I;Rn)× C(I;Rn)→L(I;Rn) and` :C(I;Rn)×C(I;Rn)→Rnand continuous operators q: C(I;Rn)→L(I;Rn) andc0:C(I;Rn)→Rnsuch that
sup
kq(x)(·)k: x∈C(I;Rn) ∈L(I;R+), sup
kc0(x)k: x∈C(I;Rn) <+∞, (5) f(x)(t) =p(x, x)(t) +q(x)(t), h(x) =`(x, x)−c0(x) for x∈ U(x0;ρ), and the boundary value problem
dx(t)
dt =p(x, x)(t) +q(x)(t), `(x, x) =c0(x) (6) has no solution, different fromx0.
Theorem 1. If the problem (1),(2) has a solutionx0 which is strongly isolated in radiusρ >0, then this problem is(x0;ρ)-well-posed.
151
Corollary 1. Let there exist a solutionx0 of the problem(1),(2), constantsρ0>0, α0 >0, a functionα∈L(I;R+)and continuous operatorsp:U(x0;ρ0)×C(I;Rn)→ L(I;Rn) and ` : U(x0;ρ0)×C(I;Rn) → Rn such that for arbitrary x ∈ U(x0;ρ0), y∈C(I;Rn)and for almost allt∈Ithe conditions
p(x, y)(t)
≤α(t)kykC, k`(x, y)k ≤α0kykC,
f(x)(t)−f(x0)(t) =p(x, x−x0)(t), h(x)−h(x0) =`(x, x−x0)
are fulfilled. Let, moreover, for an arbitraryx∈ U(x0;ρ)the operatorsp(x,·) :C(I;Rn)→ L(I;Rn)and`(x,·) :C(I;Rn)→Rnbe linear and the homogeneous problem
dy(t)
dt =p(x0, y)(t), `(x0, y) = 0
have only a trivial solution. Then for sufficiently small ρ >0 the problem (1),(2)is (x0;ρ)-well-posed.
Corollary 2. Letp :C(I;Rn)×C(I;Rn) →L(I;Rn), q :C(I;Rn) →L(I;Rn),
` :C(I;Rn)×C(I;Rn) →Rn and c0 :C(I;Rn) →Rn be continuous operators such that the pair (p, `)is consistent and the conditions (5)are fulfilled. Then the unique solvability of the problem(6)guarantees its well-posedness.
For an arbitrary natural numberk, we consider now the boundary value problem dx(t)
dt =f(x)(t) +ηk t, ζ(x)(t)
, (7k)
h(x) +γk(x) = 0, (8k)
whereηk:I×Rm→Rnis a vector function satisfying the local Carath´eodory conditions, whileζ:C(I;Rn)→C(I;Rm) andγk:C(I;Rn)→Rnare continuous operators, andζ andmare independent ofk.
ByXk(x0;ρ) we denote the set of solutions of the problem (7k), (8k) contained in the ballU(x0;ρ).
Theorem 2. Let the problem(1),(2)have a solutionx0 which is strongly isolated in radiusρ >0, and let there existρ0>0,ω∈M(I×R+;R+)and a continuous function ω0:R+→R+ such thatω0(0) = 0,
kζ(x)kC ≤ρ0,
ζ(x)−ζ(x)
C ≤ω0 kx−xkC
, γk(x)−γk(x)
≤ω0 kx−xkC
for xandx∈ U(x0;ρ) and
ηk(t, z)−ηk(t, z)
≤ω t,kz−zk
for t∈I, kzk ≤ρ0, kz0k ≤ρ0. Let, moreover,
k→+∞lim γk(x) = 0 for x∈ U(x0;ρ),
sup
t
Z
a
ηk(s, z)ds
: t∈I, z∈Rm,kzk ≤ρ0
→0 as k→+∞.
Then there exists a natural numberk0 such thatXk(x0;ρ)6=∅fork≥k0and sup
kx−x0k: x∈Xk(x0;ρ) →0 as k→+∞.
Acknowledgement The work was supported by GRDF (Grant No. 3318).
152
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Authors’ addresses:
I. Kiguradze
A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, Aleksidze St., Tbilisi 0193 Georgia
E-mail: [email protected] B. P˚uˇza
Masaryk University Faculty of Science
Department of Mathematical Analysis Jan´aˇckovo n´am. 2a, 662 95 Brno Czech Republic
E-mail: [email protected]
