IN BANACH SPACES Stanis law Szufla

Differential Inclusions, Control and Optimization 29 (2009 ) 107–111

ON THE EXISTENCE OF SOLUTIONS OF AN INTEGRO-DIFFERENTIAL EQUATION

IN BANACH SPACES Stanis law Szufla

Adam Mickiewicz University, Pozna´ n, Poland Dedicated to Professor Micha l Kisielewicz

on the occasion of his 70th birthday

Consider the Cauchy problem

(1) x ^(m) (t) = f (t, x(t)) + Z t

0

g(t, s, x(s))ds,

(2) x(0) = 0, x ⁰ (0) = η 1 , . . . , x ^(m−1) (0) = η m−1

in a Banach space E, where m ≥ 1 is a natural number. We assume that D = [0, a], B = {x ∈ E : k x k≤ b} and f : D × B → E, g : D ² × B → E are bounded continuous functions. Let

m ₁ = sup{kf (t, x)k : t ∈ D, x ∈ B}

m ₂ = sup{kg(t, s, x)k : t, s ∈ D, x ∈ B}.

We choose a positive number d such that d ≤ a and (3)

m−1 X

j=1

k η _j k d ^j

j! + m ₁ d ^m

m! + m ₂ d ^m+1 m! ≤ b.

Let J = [0, d]. Denote by C = C(J, E) the Banach space of continuous functions z : J → E with the usual norm kzk C = max t∈J kz(t)k.

Let e B = {x ∈ C : kxk C ≤ b}. For t ∈ J and x ∈ e B put e g(t, x) =

Z t 0

g(t, s, x(s))ds.

Fix τ ∈ J and x ∈ e B. As the set J × x(J) is compact, from the continuity of g it follows that for each ε > 0 there exists δ > 0 such that

kg(t, s, x(s)) − g(τ, s, x(s))k < ε for t, s ∈ J with |t − τ | < δ.

In view of the inequality

keg(t, x) − eg(τ, x)k ≤ m 2 |t − τ | + Z τ

0

kg(t, s, x(s)) − g(τ, s, x(s))k ds, this implies the continuity of the function t → eg(t, x). On the other hand, the Lebesgue dominated convergence theorem proves that for each fixed t ∈ J the function x → eg(t, x) is continuous on e B . Moreover,

keg(t, x)k ≤ m ² t for t ∈ J and x ∈ e B.

Let α be the Kuratowski measure of noncompactness in E (cf. [1]).

The main result of the paper is the following

Theorem. Let w : IR ₊ 7→ IR ₊ be a continuous nondecreasing function such that w(0) = 0, w(r) > 0 for r > 0 and

Z

0+

dr

m

p

r ^m−1 w(r) = ∞.

If

(4) α (f (t, X)) ≤ w (α(X)) for t ∈ J and X ⊂ B,

and the set g(D ² × B) is relatively compact in E, then there exists at least one solution of (1)–(2) defined on J.

P roof. The problem (1)–(2) is equivalent to the integral equation x(t) = p(t) + 1

(m − 1)!

Z t 0

(t − s) ^m−1 [f (s, x(s)) + eg(s, x)]ds (t ∈ J), where p(t) = P m−1

j=1 η _j ^t _j!

. We define the mapping F by F (x)(t) = p(t) + 1

(m − 1)!

Z t 0

(t − s) ^m−1 [f (s, x(s)) + eg(s, x)]ds (t ∈ J, x ∈ e B).

Owing to (3), it is known (cf. [5]) that F is a continuous mapping e B 7→ e B and the set F ( e B) is equicontinuous. By the Mazur lemma the set W = S

0≤λ≤d λconvg(D ² × B) is relatively compact. Since {(t − s) ^m−1 eg(s, x) : x ∈ e B} ⊂ (t − s) ^m−1 W , we have α({(t − s) ^m−1 eg(s, x) : x ∈ e B}) ≤ (t − s) ^m−1 α(W ) = 0. Therefore, by the Heinz lemma [2]

(5)

α

 1

(m − 1)!

Z t 0

(t − s) ^m−1 eg(s, x)ds : x ∈ e B



≤ 2

(m − 1)!

Z t 0

α n

(t − s) ^m−1 eg(s, x) : x ∈ e B o

ds = 0.

For any positive integer n put

v _n (t) =

 

 



 

 

p(t) if 0 ≤ t ≤ ^d _n

p(t) + _(m−1)! ¹

t−

Z

0

(t − s) ^m−1 [f (s, v _n (s)) + eg(s, v n )]ds if ^d _n ≤ t ≤ d.

Then, by (3), v _n ∈ e B and

(6) lim

n→∞ k v n − F (v n ) k C = 0.

Put V = {v _n : n ∈ N } and Z(t) = {x(t) : x ∈ Z} for t ∈ J and Z ⊂ C. As V ⊂ {v n − F (v n ) : n ∈ N } + F (V ) and V ⊂ e B, from (6) it follows that the set V is equicontinuous and the function t 7→ v(t) = α(V (t)) is continuous on J. Applying now the Heinz lemma and (5), we get

α(F (V )(t)) =

= α

 1

(m − 1)!

Z t 0

(t − s) ^m−1 [f (s, v _n (s)) + eg(s, v n )]ds : n ∈ N



≤ α

 1

(m − 1)!

Z t 0

(t − s) ^m−1 f(s, v n (s))ds : n ∈ N



+ α

 1

(m − 1)!

Z t 0

(t − s) ^m−1 e g(s, x)ds : x ∈ e B



= α

 1

(m − 1)!

Z t 0

(t − s) ^m−1 f (s, v _n (s))ds : n ∈ N



≤ 2

(m − 1)!

Z t 0

α 

(t − s) ^m−1 f (s, v n (s)) : n ∈ N
ds

≤ 2

(m − 1)!

Z t 0

(t − s) ^m−1 α(f (s, V (s))ds

≤ 2

(m − 1)!

Z t 0

(t − s) ^m−1 w(α(V (s)))ds.

On the other hand, from (6) and the inclusion

V (t) ⊂ {v n (t) − F (v n )(t) : n ∈ N } + F (V )(t) it follows that v(t) ≤ α (F (V )(t)) . Hence

v(t) ≤ 2 (m − 1)!

Z t 0

(t − s) ^m−1 w(v(s))ds for t ∈ J.

Putting h(t) = _(m−1)! ² R t

0 (t − s) ^m−1 w(v(s))ds, we see that h ∈ C ^m , v(t) ≤ h(t), h ^(j) (t) ≥ 0 for j = 0, 1, . . . , m , h ^(j) (0) = 0 for j = 0, 1, . . . , m − 1 and h ^(m) (t) = 2w(v(t)) ≤ 2w(h(t)) for t ∈ J. By Theorem 1 of [6], from this we deduce that h(t) = 0 for t ∈ J. Thus α(V (t)) = 0 for t ∈ J. Therefore for each t ∈ J the set V (t) is relatively compact in E, and by Ascoli’s theorem the set V is relatively compact in C. Hence we can find a subsequence (v n

) of (v n ) which converges in C to a limit u. As F is continuous, from (6) we conclude that u = F (u), so that u is a solution of (1)–(2).

Remark. It is known (cf. [7], Theorem 4) that under the assumptions of the Theorem the set of all solutions of (1)–(2) defined on J is a compact R δ

set in C(J, E).

References

[1] J. Bana´s, K. Goebel, Measures of noncompactness in Banach spaces, Marcel

Dekker, New York-Basel, 1980.

[2] H.P. Heinz, On the behaviour of measures of noncompactness with respect to differentiation and integration of vector-valued functions, Nonlinear Anal. 7 (1983), 1351–1371.

[3] M. Kisielewicz, Differential Inclusions and Optimal Control, PWN, Warszawa, Kluwer Academic Publishers, Dordrecht-Boston-London, 1991.

[4] M. Kisielewicz, Existence, uniqueness and continuous depedence of solutions of differential equations in Banach spaces, Ann. Polon. Math. 50 (1989), 117–128.

[5] S. Szufla, On Volterra integral equations in Banach spaces, Funkcial. Ekvac.

20 (1977), 247–258.

[6] S. Szufla, Osgood type conditions for an m-th order differential equation, Dis- cuss. Math. Diff. Inclusions 18 (1998), 45–55.

[7] S. Szufla, On the of structure of solutions sets of differential and integral equa- tions in Banach spaces, Ann. Polon. Math. 34 (1977), 165–177.

Received 9 June 2009