Download Existence and uniqueness of solutions for partial differential

Serdica Math. J. 23 (1997), 203-210
EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR
PARTIAL DIFFERENTIAL-FUNCTIONAL EQUATIONS OF
THE FIRST ORDER WITH DEVIATING ARGUMENT OF
THE DERIVATIVE OF UNKNOWN FUNCTION
A. Augustynowicz
Communicated by Il. D. Iliev
Abstract. We consider the existence and uniqueness problem for partial
differential-functional equations of the first order with the initial condition
for which the right-hand side depends on the derivative of unknown function
with deviating argument.
1. Introduction. We consider the following Cauchy problem
(1)

 Dx z(x, y) = f (x, y, z(·), Dy z(α(x, y), β(x))),
 z(0, y) = v(y),
(x, y) ∈ E
y ∈ [0, b],
where E = [0, a]×[0, b], a, b > 0, f ∈ C(E×C(E; Rn )×Rn ; Rn ), v ∈ C([0, b]; Rn ),
α ∈ C(E; [0, a]), β ∈ C([0, a]; [0, b]) and Dp g means the partial derivative of g
with respect to p.
1991 Mathematics Subject Classification: 35F25
Key words: differential-functional equations, partial of the first order, existence and uniqueness, deviating argument of derivative
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A. Augustynowicz
The problem Dx z(x, y) = f (x, y, z(·), Dy z(x, y)), z(0, y) = v(y) has been
investigated intensively in the literature (see [5], [6], [7] for references). But there
are only few results concerning equations with Dy z depending on a deviating
argument (see [1]–[3], [8]–[10]). In the present paper we consider such an equation.
But notice that the deviating argument we consider is of peculiar type and it is
not a generalization of not deviating argument (x, y).
The proof of our result is based on Bielecki method of changing norm (see
[4]).
2. Notations and assumptions. We denote Ex = {(s, y) ∈ E : 0 ≤
s ≤ x}, C = C(E; Rn ),
kzk0,x = sup{|z(s, y)| : (s, y) ∈ Ex },
kzkK = sup{|z(x, y)|e−Kx : (x, y) ∈ E}
for z ∈ C, x ∈ [0, a], K ∈ R, where | · | means a fixed norm in Rn . Notice that
k · kK is a norm equivalent to the supremum norm and C is a Banach space with
this norm.
Dq f means the partial derivative of f with respect to the fourth argument
We need the following assumptions.
(H1 ) The function f satisfies the following Volterra condition: for all (x, y) ∈ E,
u, v ∈ C, q ∈ Rn if u(s, y) = v(s, y) for (s, y) ∈ Ex , then f (x, y, u(·), q) =
f (x, y, v(·), q).
(H2 ) The functions f , Dy f , Dq f , α, Dy α, β, v, Dy v are continuous and there
exists L > 0 such that
|f (x, y, p, q) − f (x, y, P, Q)| ≤ L(kp − P k0,x + |q − Q|),
|Dy f (x, y, p, q) − Dy f (x, y, P, Q)| ≤ L(kp − P k0,x + |q − Q|),
|Dq f (x, y, p, q) − Dq f (x, y, P, Q)| ≤ L(kp − P k0,x + |q − Q|),
for (x, y) ∈ E, p, P ∈ C, q, Q ∈ Rn .
(H3 ) α(x, y) ≤ x for (x, y) ∈ E and
l = sup{|Dy α(x, y)Dq f (x, y, p, q)| : (x, y) ∈ E, p ∈ C, q ∈ Rn } < 1
205
Existence and uniqueness of solutions . . .
Notice that (H3 ) implies that the growth of f (x, y, p, ·) is at most linear
except for Dy α(x, y) = 0.
3. The result. Now we state
Theorem.
Suppose that assumptions (H1 ), (H2 ), (H3 ) are satisfied,
then there exists exactly one solution ẑ of (1) such that Dx Dy ẑ and Dy Dx ẑ exist
and they are continuous functions.
P r o o f. Let us define
F (U, u)(x, y) =
Zx Zy
u(s, t)dtds + v(y)+
0 0
+
Zx f s, 0, U, v ′ (β(s)) +
0
G(U, u)(x, y) = Dy f x, y, U, v ′ (β(x)) +
α(s,y)
Z
0
α(x,y)
Z
u(σ, β(s)) dσ ds
u(s, β(x)) ds +
0
α(x,y)
Z
+Dy α(x, y)Dq f x, y, U, v ′ (β(x)) +
u(s, β(x)) ds u(α(x, y), β(x))
0
for U, u ∈ C, (x, y) ∈ E. We prove that there exists exactly one pair (Û , û) of
continuous functions, which are solutions of the equations
(2)
U = F (U, u) and u = G(U, u).
Let K > L be fixed. For every U1 , U2 , u ∈ C assumption (H2 ) gives
|F (U1 , u)(x, y) − F (U2 , u)(x, y)| ≤
Zx
LkU1 − U2 k0,s ds ≤
0
≤L
Zx
0
kU1 − U2 kK eKs ds ≤
L Ks
e kU1 − U2 kK ,
K
so
kF (U1 , u) − F (U2 , u)kK ≤
L
kU1 − U2 kK .
K
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A. Augustynowicz
Therefore F (·, u) is a contraction and for each u ∈ C there exists exactly one
fixed point U (u) of F (·, u). Moreover
|U (u1 )(x, y) − U (u2 )(x, y)| = |F (U (u1 ), u1 )(x, y) − F (U (u2 ), u2 )(x, y)| ≤
≤ |F (U (u1 ), u1 )(x, y) − F (U (u2 ), u1 )(x, y)|+
+|F (U (u2 ), u1 )(x, y) − F (U (u2 ), u2 )(x, y)| ≤
L
≤ kU (u1 ) − U (u2 )kK eKx +
K
Zx Zy
|u1 (s, t) − u2 (s, t)| ds dt+
0 0
Zx α(s,t)
Z
+L
|u1 (σ, β(s)) − u2 (σ, β(s))| dσ ds ≤
0
≤
L
kU (u1 ) − U (u2 )kK eKx + b
K
Zx
0
ku1 − u2 kK eKs ds+
0
+L
Zx Zs
ku1 − u2 kK eKσ dσds ≤
0 0
L
b
aL
≤ ( kU (u1 ) − U (u2 )kK + ( +
)ku1 − u2 kK )eKx ,
K
K
K
hence
kU (u1 ) − U (u2 )kK ≤ lK ku1 − u2 kK ,
where lK = (K − L)−1 (b + aL). For sufficiently large K let us define
WK = {u ∈ C : kukK ≤ M },
where
M = (1 − (LlK +
L
+ l))−1 P, P = 1 + max |Dy f (x, y, U (Θ), v ′ (β(x)))|
K
(x,y)∈E
and Θ(x, y) = 0 for (x, y) ∈ E. Denote also GU (u) = G(U (u), u). We prove that
L
GU (WK ) ⊂ WK for K > 0 such that LlK +
+ l < 1.
K
If u ∈ WK then
207
Existence and uniqueness of solutions . . .
|G(U (u), u)(x, y)| ≤
α(x,y)
Z
′
≤ Dy f x, y, U (u), v (β(x))+
u(s, β(x)) ds − Dy f x, y, U (Θ), v ′ (β(x)) +
0
+|Dy f x, y, U (Θ), v ′ (β(x)) | + l|u(α(x, y), β(x))| ≤
≤ LkU (u) − U (Θ)k0,x + L
α(x,y)
Z
|u(s, β(x))| ds + P + l|u(α(x, y), β(x))| ≤
0
L
≤ (LkU (u) − U (Θ)kK + kukK + P + lkukK )eKx ≤
K
L
≤ (LlK +
+ l)kukK eKx + P eKx ,
K
therefore
L
kGU (u)kK ≤ (LlK +
+ l)M + P = M.
K
We have just showed that GU (WK ) ⊂ WK for sufficiently large K. We
prove that GU is a contraction on WK with respect to the norm k · kJ for sufficiently large J and K.
For u, ū ∈ WK , c = max(x,y)∈E |Dy α(x, y)| we have
|G(U (u), u)(x, y) − G(U (ū), ū)(x, y)| ≤
≤ LkU (u) − U (ū)k0,x + L
α(x,y)
Z
|u(s, β(x)) − ū(s, β(x))| ds +
0
α(x,y)
Z
′
+|Dy α(x, y)|Dq f x, y, U (u), v (β(x)) +
u(s, β(x)) ds −
0
′
−Dq f x, y, U (ū), v (β(x)) +
α(x,y)
Z
0
ū(s, β(x)) ds |u(α(x, y), β(x))|
+|Dy α(x, y)||Dq f x, y, U (ū), v ′ (β(x)) +
α(x,y)
Z
ū(s, β(x)) ds | ×
0
×|u(α(x, y), β(x)) − ū(α(x, y), β(x))| ≤
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A. Augustynowicz
≤ LkU (u) − U (ū)kJ eJx +
+cL kU (u) − U (ū)k0,x +
L
ku − ūkJ eJx +
J
α(x,y)
Z
|u(s, β(x)) − ū(s, β(x))| ds M eKx +
0
+lku − ūkJ eJx
≤ (LlJ +
L
1
+ cL(lJ + )M eKx + l)eJx ku − ūkJ
J
J
so
kGU (u) − GU (ū)kJ ≤ (LlJ +
L
1
+ cLM (lJ + )M eKa + l)ku − ūkJ .
J
J
Now it is clear that the operator GU is contractive in WK if K and J are
sufficiently large and this operator has exactly one fixed point û in WK . Since
every fixed point ū of GU in C satisfies |ū(0, y)| ≤ P − 1 for y ∈ [0, b] and M > P ,
then there exists K such that ū ∈ WK . We get from the above that the function
û is the unique fixed point of GU in C. Of course the pair (U (û), û) is the unique
solution of (2). Now we demonstrate that V = U (û) is a solution of (1). From
the definition of the operators F and G we get
V (0, y) = v(y),
Dy V (x, y) =
Zx
û(s, y) ds + v ′ (y),
0
d
f (x, y, V, v ′ (β(x)) +
û(x, y) = G(V, û)(x, y) =
dy
=
Dx V (x, y) =
Zy
α(x,y)
Z
û(s, β(x)) ds) =
0
d
f (x, y, V, Dy V (α(x, y), β(x))),
dy
û(x, t) dt + f (x, 0, V, Dy V (α(x, 0), β(x))) =
0
= f (x, y, V, Dy V (α(x, y), β(x))),
Existence and uniqueness of solutions . . .
209
so V is a solution of (1) and it is obvious that Dx Dy V = Dy Dx V = û, hence
these derivatives are continuous. On the other hand, if z is a solution of (1) and
Dx Dy z = Dy Dx z then differentiating equation (1) with respect to y we obtain
Dy Dx z(x, y) = Dy f (x, y, z, Dy z(α(x, y), β(x))) +
+ Dq f (x, y, z, Dy z(α(x, y), β(x))) Dx Dy z(α(x, y), β(x)) Dy α(x, y)
and
Dy z(x, y) =
Zx
Dx Dy z(s, y) ds + v ′ (y),
0
hence Dx Dy z = G(z, Dx Dy z) and it is easy to verify that z = F (z, Dx Dy z), so
z = U (Dx Dy z) and Dx Dy z = û. We get that z is a solution of (1) if and only if
(z, Dx Dy z) is a solution of (2). The proof is complete.
Remark. Under analogous assumptions we can prove a similar result
for equation (1) with y = (y1 , y2 , . . . , yk ) ∈ Rk .
REFERENCES
[1] A. Augustynowicz, H. Leszczyński. On the existence of analytic solutions to the Cauchy problem for first-order partial differential equations with
retarded variables. Comment. Math., to appear.
[2] A. Augustynowicz, H. Leszczyński. On x-analytic solutions to the
Cauchy problem for partial differential equations with retarded variables.
Z. Anal. Adwendungen 15, 2 (1996), 345-356.
[3] A. Augustynowicz, H. Leszczyński. Periodic solutions to the Cauchy
problem for PDEs with retarded variables. Preprint.
[4] A. Bielecki. Une remarque sur la methode de Banach-Cacciopoli-Tikhonov
dans la theorie des equations differentielles ordinaires. Bull. Acad. Polon. Sci.
Ser. Sci. Math. Astr. Phys. 4 (1956), 261-264.
[5] T. Czlapiński. On the existence of generalized solutions of nonlinear first
order partial differential-functional equations in two independent variables.
Czechoslovak Math. J. 41 (1991), 490-506.
210
A. Augustynowicz
[6] T. Czlapiński, Z. Kamont. Generalized solutions of quasi-linear hyperbolic systems of partial differential-functional equations. J. Math. Anal.
Appl., 172 (1993), 353-370.
[7] D. Jaruszewska-Walczak. Existence of solutions of first order partial
differential-functional equations. Boll. Un. Mat. Ital. B (7) 4-B (1990), 5782.
[8] H. Leszczyński. Fundamental solutions to linear first-order equations with
a delay at derivatives. Boll. Un. Mat. Ital. A (7) 10-A (1996), 363-375.
[9] W. Walter. Functional differential equations of the Cauchy-Kovalevsky
type. Aequationes Math. 28 (1985), 102-113.
[10] T. Yamanaka, H. Tamaki. Cauchy-Kovalevska theorem for functional partial differential equations. Comment. Math. Univ. St. Paul. 29, 1 (1980),
55-64.
Institute of Mathematics
Gdańsk University
ul.Wita Stwosza 57
80-952 Gdańsk
Poland
e-mail: [email protected]
Received March 10, 1992
Revised November 4, 1996