ON THE PICARD PROBLEM FOR HYPERBOLIC DIFFERENTIAL EQUATIONS IN BANACH SPACES

ON THE PICARD PROBLEM FOR HYPERBOLIC DIFFERENTIAL EQUATIONS IN BANACH SPACES

Antoni Sadowski

Institute of Mathematics and Physics Technical University of Warsaw, Branch PÃlock

ul. ÃLukasiewcza 17, 09–400 PÃlock, Poland

Abstract

B. Rzepecki in [5] examined the Darboux problem for the hy- perbolic equation z xy = f (x, y, z, z xy ) on the quarter-plane x ≥ 0, y ≥ 0 via a fixed point theorem of B.N. Sadovskii [6]. The aim of this paper is to study the Picard problem for the hyperbolic equation z xy = f (x, y, z, z x , z xy ) using a method developed by A. Ambrosetti [1], K. Goebel and W. Rzymowski [2] and B. Rzepecki [5].

Keywords: boundary value problem, fixed point theorem, functional- integral equation, hyperbolic equation, measure of noncompactness.

2000 Mathematics Subject Classification: 35L70.

1. Notations and formulations

By (E, k·k) we shall denote a real Banach space. The symbol (R ^k , k·k) is reserved for n-dimensional Euclidean space. We introduce the notion

R ₊ =< 0, ∞), Q = R ₊ × R ₊ ⊂ R ² and

Ω = Q × E × E × E.

Let B be the family of bounded sets of E. Then α : B → R ₊ , defined by

α(B) = inf{d > 0 : B admits a finite cover by sets of diameter ≤ d},

B ∈ B, is called Kuratowski’s measure of noncompactness.

Let σ, τ : R ₊ → E be two functions such that τ (0) = σ(β(0)), where β : R ₊ → R ₊ is a given function.

We shall consider the following problem

(P.P )

z _xy (x, y) = f (x, y, z(x, y), z _x (x, y), z _xy (x, y)) z(x, 0) = σ(x)

z(β(y), y) = τ (y) where f : Ω → E is a given function.

The above (P.P ) problem is usually called the Picard problem for hyperbolic equations.

2. The main result The aim of this paper is to prove the following theorem

Theorem 2.1. Assume that σ, τ : R ₊ → E are C ¹ -mappings such that τ (0) = σ(β(0)), where β : R ₊ → R ₊ is a function of class C ¹ satisfying the following condition β : [0, M _n ] → [0, M _n ], where (M _n ), n ∈ N is an increasing and unbounded sequence.

Assume further that f : Ω → E is uniformly continuous on bounded subsets of Ω and

(2.1) kf (x, y, u, v, w)k ≤ G(x, y, kuk , kvk , kwk)

for (x, y, u, v, w) ∈ Ω. Suppose that for each bounded subset P of Q there exist nonnegative constants k(P ) and L(P ) < ¹ ₂ such that

(2.2) α(f (x, y, U, V, w)) ≤ k(P )(α(U ) + α(V )) and

(2.3) kf (x, y, u, v, w ₁ ) − f (x, y, u, v, w ₂ )k ≤ L(P ) kw ₁ − w ₂ k

for all (x, y) ∈ P, u, v, w ₁ , w ₂ ∈ E. For any nonempty bounded subsets U, V

of E, let α denote Kuratowski’s measure of noncompactness in E.

Assume in addition that the function (x, y, r, s, t) → G(x, y, r, s, t) is nonde- creasing for each (x, y) ∈ Q (i.e. 0 ≤ r ₁ ≤ r ₂ , 0 ≤ s ₁ ≤ s ₂ and 0 ≤ t ₁ ≤ t ₂ implies G(x, y, r ₁ , s ₁ , t ₁ ) ≤ G(x, y, r ₂ , s ₂ , t ₂ )) and the scalar inequality

(2.4) G Ã

x, y,

¯ ¯

¯ ¯

¯ Z _x

β(y)

Z _y

0

g(s, t)dsdt

¯ ¯

¯ ¯

¯ , Z _y

0

g(x, t)dt, g(x, y)

!

≤ g(x, y)

has a locally bounded solution g ₀ on Q.

Under these assumptions, (P.P ) has at least one solution on Q.

For the proof we need the following two lemmas.

Lemma 2.1. Let (M, d) be a metric space and let A ₁ , A ₂ be transformations mapping bounded sets of M into bounded sets of M. Assume that

F : A ₁ (M ) × A ₂ (M ) × M → M is a mapping such that

d(F (A ₁ x, A ₂ y, z ₁ ), F (A ₁ x, A ₂ y, z ₂ )) ≤ Ld(z ₁ , z ₂ ) for (x, y, z ₁ , z ₂ ∈ M, L ≥ 0)

and

α(F (A ₁ X × A ₂ X × {z})) ≤ ψ ₁ (α(A ₁ X)) + ψ ₂ (α(A ₂ X)) for z ∈ M, X being a bounded subset of M, and ψ _i : R ₊ → R ₊ ; i = 1, 2.

Then

α(F (A ₁ X × A ₂ X × X)) ≤ 2Lα(X) + ψ ₁ (α(A ₁ X)) + ψ ₂ (α(A ₂ X)) for any bounded subset X of M.

The proof of this Lemma is similar to that in [4].

Lemma 2.2. If W is a bounded equicontinuous subset of a Banach space of continuous E-valued functions defined on a compact subset P = [a ₁ , a ₂ ] × [b ₁ , b ₂ ] of Q, then

α µZ _a

₂

a

1

Z _b

₂

b

1

W (s, t)dsdt

¶

≤ Z _a

₂

a

1

Z _b

₂

b

1

α(W (s, t))dsdt.

Lemma 2.2 is an adaptation of the corresponding result of Goebel and Rzymowski [2].

P roof of T heorem 2.1. Without loss of generality, we may assume that σ = 0 and τ = 0 (see [3]). Problem (P.P ) is equivalent to the functional- integral equation

(2.5) w(x, y) = f Ã

x, y, Z _x

β(y)

Z _y

0

w(s, t)dsdt, Z _y

0

w(x, t)dt, w(x, y)

!

Denote by C(Q, E) the space of all continuous functions from Q to E (C(Q, E) is a Frechet space whose topology is introduced by seminorms of uniform convergence on compact subsets of Q), and by X the set of all w

∈ C(Q, E) with

(2.6) kw(x, yk ≤ g ₀ (x, y)

(x, y) ∈ Q. Let P be a bounded subset of Q. From the uniform continuity of f on bounded subsets of Ω there follows the existence of a function δ _P : (0, ∞) → (0, ∞) such that

(2.7)

° °

° °

° °

° °

° °

f (x

⁰

, y

⁰

, Z _x

⁰

β(y

⁰

)

Z _y

⁰

0

w(s, t)dsdt, Z _y

⁰

0

w(x

⁰

, t)dt, w(x

⁰

, y

⁰

))

−f (x

⁰⁰

, y

⁰⁰

, Z _x

⁰⁰

β(y

⁰⁰

)

Z _y

⁰⁰

0

w(s, t)dsdt, Z _y

⁰⁰

0

w(x

⁰⁰

, t)dt, w(x

⁰⁰

, y

⁰⁰

))

° °

° °

° °

° °

° °

< ε

w ∈ X; (x

⁰

, y

⁰

) and (x

⁰⁰

, y

⁰⁰

) ∈ P y satisfy the relations |x

⁰

− x

⁰⁰

| < δ _P (ε) and

|y

⁰

− y

⁰⁰

| < δ _P (ε).

Consider the set X ₀ ⊂ X possessing the following property: for each bounded subset P ⊂ Q, ε > 0 and |x

⁰

− x

⁰⁰

| < δ _P (ε), |y

⁰

− y

⁰⁰

| < δ _P (ε), (x

⁰

, y

⁰

), (x

⁰⁰

, y

⁰⁰

) ∈ P , the inequality

(2.8)

° °

°w(x

⁰

, y

⁰

) − w(x

⁰⁰

, y

⁰⁰

)

° °

° ≤ (1 − L(P )) ⁻¹ ε

holds for every w ∈ X ₀ .

The set X ₀ is a closed, convex and almost equicontinuous subset of C(Q, E).

To apply the fixed point theorem of B.N. Sadovskii [6] we define the contin- uous mapping

T : C(Q, E) → C(Q, E) by the formula

(2.9) (T w)(x, y) = f Ã

x, y, Z _x

β(y)

Z _y

0

w(s, t)dsdt, Z _y

0

w(x, t)dt, w(x, y)

! .

Let w ∈ X ₀ . Then k(T w)(x, y)k

(2.10) ≤ G Ã

x, y,

¯ ¯

¯ ¯

¯ Z _x

β(y)

Z _y

0

kw(s, t)k dsdt

¯ ¯

¯ ¯

¯ , Z _y

0

kw(x, t)k dt, kw(x, y)k

!

≤ G Ã

x, y,

¯ ¯

¯ ¯

¯ Z _x

β(y)

Z _y

0

g ₀ (s, t)dsdt

¯ ¯

¯ ¯

¯ , Z _y

0

g ₀ (x, t)dt, g ₀ (x, y)

!

≤ g ₀ (x, y).

Furthemore, for ε > 0 and (x

⁰

, y

⁰

), (x

⁰⁰

, y

⁰⁰

) ∈ P such that |x

⁰

− x

⁰⁰

| < δ _P (ε),

|y

⁰

− y

⁰⁰

| < δ _P (ε) we have (see (2.3),(2.7) and (2.8))

° °

°(T w)(x

⁰

, y ⁰ ) − (T w)(x

⁰⁰

, y

⁰⁰

)

° °

°

≤

° °

° °

° °

° °

° °

°

f (x

⁰

, y

⁰

, Z _x

⁰

β(y

⁰

)

Z _y

⁰

0

w(s, t)dsdt, Z _y

⁰

0

w(x

⁰

, t)dt, w(x

⁰

, y

⁰

))+

−f (x

⁰

, y

⁰

, Z _x

⁰

β(y

⁰

)

Z _y

⁰

0

w(s, t)dsdt, Z _y

⁰

0

w(x

⁰

, t)dt, w(x

⁰⁰

, y

⁰⁰

))

° °

° °

° °

° °

° °

° (2.11)

+

° °

° °

° °

° °

° °

f (x

⁰

, y

⁰

, Z _x

⁰

β(y

⁰

)

Z _y

⁰

0

w(s, t)dsdt, Z _y

⁰

0

w(x

⁰

, t)dt, w(x

⁰⁰

, y

⁰⁰

)+

−f (x

⁰⁰

, y

⁰⁰

, Z _x

⁰⁰

βy

⁰⁰

)

Z _y

⁰⁰

0

w(s, t)dsdt, Z _y

⁰⁰

0

w(x

⁰⁰

, t)dt, w(x

⁰⁰

, y

⁰⁰

))

° °

° °

° °

° °

° °

≤ L(P )

° °

°w(x

⁰

, y

⁰

) − w(x

⁰⁰

, y

⁰⁰

)

° °

° + ε ≤ (1 − L(P ) ⁻¹ ε .

Thus, the inclusion T(X ₀ ) ⊂ X ₀ holds.

Let n be a positive integer and let W be a nonempty subset of X ₀ . Put P _n = [0, M _n ] × [0, M _n ], k _n = k(P _n ) and L _n = L(P _n ). Now we shall show the basic inequality (see [5]):

(2.12)

sup

P

n

exp(−p _n y)α(T (W (x, y)))

≤ ¡

p ⁻¹ _n k _n (M _n + 1) + 2L _n ¢ sup

P

n

exp(−p _n y)α(W (x, y)) where p _n > 0 (n = 1, 2, . . .).

By Lemma 2.2 (see [5]), we obtain for a fixed (x, y) ∈ P _n the following inequality

(2.13)

α ÃZ _x

β(y)

Z _y

0

W (s, t)dsdt

!

≤

¯ ¯

¯ ¯

¯ Z _x

β(y)

Z _y

0

exp(−p _n t) exp(p _n t)α(W (s, t))dsdt

¯ ¯

¯ ¯

¯

≤ Z _M

_n

0

Z _y

0

exp(−p _n t) exp(p _n t)α(W (s, t))dsdt

≤ sup

P

n

(exp(−p _n t)α(W (s, t))) Z _M

_n

0

Z _y

0

exp(p _n t)dsdt

≤ p ⁻¹ _n M _n exp(p _n y)sup

P

n

(exp(−p _n t)α(W (s, t))).

Analogously, we have

(2.14)

α µZ _y

0

W (x, t)dt

¶

≤ Z _y

0

α(W (x, t))dt

= Z _y

0

exp(−p _n t) exp(p _n t)α(W (x, t))dt

≤ sup

P

n

(exp(−p _n t)α(W (s, t))) Z _y

0

exp(p _n t)dt

≤ p ⁻¹ _n exp(p _n y) sup

P

n

exp(−p _n t)α(W (s, t)).

The inequality (2.12) is a simple consequence of (2.13), (2.14) and Lemma 2.1.

Let p _n > (1 − 2L _n ) ⁻¹ k _n (M _n + 1) (n = 1, 2, . . .). Define Φ(W ) = (sup

P

1

exp(−p ₁ y)α(W (x, y)), sup

P

2

exp(−p ₂ y)α(W (x, y)), . . .) for any nonempty subset W of X ₀ .

By Ascoli’s theorem, the properties of α and inequality (2.12) it follows that all assumptions of B.N. Sadovskii’s fixed point theorem are satisfied.

Consequently, the mapping T has a fixed point in X ₀ . The proof of the Theorem is complete.

References

[1] A. Ambrosetti, Un teorema di essistenza per le equazioni differenziali nagli spazi di Banach, Rend. Sem. Mat. Univ. Padova 39 (1967), 349–360.

[2] K. Goebel, W. Rzymowski, An existence theorem for the equations x

⁰

= f (t, x) in Banach space, Bull. Acad. Polon. Sci., S´er. Sci. Math. 18 (1970), 367–370.

[3] P. Negrini, Sul problema di Darboux negli spazi di Banach, Bolletino U.M.I.

(5) 17-A (1980), 156–160.

[4] B. Rzepecki, Measure of Non-Compactness and Krasnoselskii’s Fixed Point Theorem, Bull. Acad. Polon. Sci., S´er. Sci. Math. 24 (1976), 861–866.

[5] B. Rzepecki, On the existence of solution of the Darboux problem for the hyper- bolic partial differential equations in Banach Spaces, Rend. Sem. Mat. Univ.

Padova 76 (1986).

[6] B.N. Sadovskii, Limit compact and condensing operators, Russian Math.

Surveys 27 (1972), 86–144.

Received 25 April 2003