ASYMPTOTIC BEHAVIOUR OF SOLUTIONS OF DIFFERENCE EQUATIONS IN BANACH SPACES

ASYMPTOTIC BEHAVIOUR OF SOLUTIONS OF DIFFERENCE EQUATIONS IN BANACH SPACES

Anna Kisio lek Technical University of Pozna´ n Piotrowo 3, PL–60–965 Pozna´ n, Poland

e-mail: akisiolek@wp.pl

Abstract

In this paper we consider the first order difference equation in a Banach space

∆x

=

∞

X

i=0

a

f (x

).

We show that this equation has a solution asymptotically equal to a.

As an application of our result we study the difference equation

∆x

=

∞

X

i=0

a

g(x

) +

∞

X

i=0

b

h(x

) + y

and give conditions when this equation has solutions.

In this note we extend the results from [8, 9]. For example, in [9]

the function f is a real Lipschitz function. We suppose that f has values in a Banach space and satisfies some conditions with respect to the measure of noncompactness and measure of weak noncompactness.

Keywords: Banach space, difference equation, fixed point, measure of noncompactness, asymptotic behaviour of solutions.

2000 Mathematics Subject Classification: 39A10, 47N99.

1. Introduction

Let c be the set of complex numbers. Let R be the set of real numbers and

l 1 (C) the space of complex valued sequences (c n ) such that:

k(c n )k 1 :=

∞

X

i=1

|c n | < ∞.

Let (X, k · k) be a complex or real Banach space and l ∞ (X) denote the space of bounded sequences x = (x _n ) in X with the norm

kxk ∞ = k(x _n )k ∞ = sup

n

kx _n k.

With this norm l ∞ (X) is a Banach space.

Consider the first order difference equation:

(∗) ∆x _n =

∞

X

i=0

a ⁱ _n f (x _n+i ),

where ∆x _n = x _n+1 − x _n denotes the difference operator, the coefficients a ⁱ _n are complex numbers and f is a function from X to X.

By a solution of equation (∗) we understand a sequence x = (x n ) in l _∞ (X) which satisfies the equation (∗).

The first order difference equation (∗) was considered by C. Gonzalez and A. Jimenez-Melado [9]. They gave sufficient conditions on the coeffi- cients a ⁱ _n and on the function f so that this equation has solutions asymp- totically constant. In [9] f : X → X was a Lipschitz function.

In this note, the results from [9] are extended, the Lipschitz function is replaced by a continuous mapping, which is condensing with respect to the measure of noncompactness.

2. Main results

We give necessary and sufficient conditions for the existence of asymptoti- cally constants solutions.

For each set of coefficients {a ⁱ _n } _n,i∈N we can create a new set {α ^j n } _n,j∈N such that

α ^j _n =

j

X

k=0

a ^j−k _n+k = a ⁰ _n+j + a ¹ _n+j−1 + a ² _n+j−2 + . . . + a ^j _n , n, j ∈ N.

Our result will be proved by the following fixed point theorem

Theorem 1.1 [5]. Let D be a nonempty, closed, convex and bounded subset of a Banach space.

Let F : D → D be a continuous mapping, which is condensing with respect to the measure of noncompactness α, i.e.

(∗∗) α(F (V )) ≤ Lα(V ), L < 1.

Then F has a fixed point, where α is the Kuratowski’s measure of noncom- pactness (see [4]).

We shall use the following properties of the measure of noncompactness α.

Theorem 1.2. Let A, B be bounded subsets of a Banach space X. Then (a) α(A) = 0 ⇒ A is a relatively compact subset of X

(b) A ⊂ B ⇒ α(A) ≤ α(B) (c) α(k · A) = |k| · α(A)

(d) α(A ∪ B) = max(α(A), α(B)) (e) α(A + B) ≤ α(A) + α(B) (f) α(conv A) = α(A).

For the properties of the measure of noncompactness, see [3] and [4].

Theorem 1.3. Let V ⊂ C(N ⁺ , X) be a family of functions. Then α(V ) = α(V (N ⁺ )) = sup{α(V (i)) : i ∈ N ⁺ } where α(V ) denotes the measure of noncompactness in C(N ⁺ , X).

A similar theorem as Theorem 1.1 was proved by O. Arino, S. Gautier, J.P.

Penot [1] (see also [10]) when f is weakly-weakly sequentially continuous, i.e., if x n

−→x w 0 (weakly), then f (x n ) −→f (x ^w 0 ) for each sequence (x n ).

Theorem 1.4. Let f : X → X be a bounded and continuous function. Let a ∈ X and for each n ∈ N, α _n ∈ l 1 (C) and kα _n k 1 → 0 as n → ∞. We assume that k 1 = sup _n≥n

{ P ∞

j=0 kα ^j n k}, kf (h n+j )k ≤ k, k 1 < _k ¹ for some n 0 ∈ N and n > n 0 . Moreover,

α(F (V )) ≤ kα(V )

for any bounded subset of X.

Then for each a ∈ X there exists a solution x = (x _n ) ∈ l ∞ (X) of the equation (∗) such that x n → a.

P roof. Define the operator T : D → D, where D = {y : y = (y 1 , y 2 , . . .), ky − bk ∞ ≤ kk 1 }, b = (a, a, a, . . .).

For h ∈ D, let T h = (T h) n be given by

(T h) n =



 

 

a if n ≤ n 0

a −

∞

X

j=0

α ^j _n f (h _n+j ) if n > n 0 . For n ≤ n 0 and for x = (x 1 , x 2 , . . .) = (a, a, a, . . .) we have

kT h − xk _X = 0.

For n > n 0 w have kT h _n − x _n k =

a −

∞

X

j=0

α ^j _n f(h _n+j ) − a =

∞

X

j=0

α ^j _n f (h _n+j )

≤

∞

X

j=0

kα ^j _n k kf (h _n+j )k ≤ k

∞

X

j=0

kα ^j _n k ≤ kk 1 . So the operator T : D → D.

Now, we show that the operator T is continuous. Let h n = (h ¹ _n , h ² _n , . . .), h 0 = (h ¹ ₀ , h ¹ ₀ , . . .) and h n → h 0 if n → ∞. Then

kT h _n − T h 0 k = sup

i

kT h ⁱ _n − T h ⁱ 0 k = 0 for n ≤ n 0 . For n > n 0 we have:

kT h n − T h 0 k = sup

i

kT h ⁱ _n − T h ⁱ 0 k

≤ sup

i

∞

X

j=0

kα ^j _n k kf (h ⁱ _n+j ) − f (h ⁱ _j )k

≤ sup

i s

X

j=0

kα ^j _n k kf (h ⁱ _n+j ) − f (h ⁱ _j )k

+ sup

i

∞

X

j=s+1

kα ^j _n k kf (h ⁱ _n+j )k + sup

i

∞

X

j=s+1

kα ^j _n k kf (h ⁱ _j )k.

Because

sup

n≥n

∞

X

j=0

kα ^j _n k = k 1 ,

f is continuous and bounded by k so T h _n → T h 0 if n → ∞.

Now, we will prove that T satisfies condition (∗∗) of Theorem 1.1.

Let V ⊂ D, where V = {v : v = (v 1 , v 2 , . . .)} and T (V ) = {T (v) : v ∈ V }.

Let V _k = {v _k : v ∈ V, v = (v 1 , v 2 , . . . , v _k , . . .)}. For n ≤ n 0 , we obtain:

α(T (V )) = α(a) = 0.

For n > n 0 , by Theorem 1.3 we have α(T (V )) = sup

n

α (

a −

∞

X

j=0

α ^j _n f (V n ) )!

≤ sup

n

"(

α(a) + α

∞

X

j=0

α ^j _n f (V _n )

!)#

≤ sup

n

α(a) + sup

n

α ( _∞

X

j=0

α ^j _n f (V n ) )!

≤ sup

n

∞

X

j=0

α(α ^j _n f (V n )) .

So:

α(T (V )) ≤ sup

n

∞

X

j=0

α(α ^j _n f (V _n )) ≤

∞

X

j=0

α(α ^j _n f (V ))

≤

∞

X

j=0

|α ^j _n |α(f (V )).

Using the inequality α(f (V )) ≤ kα(V ) we obtain α(T (V )) ≤ kα(V )

∞

X

j=0

|α ^j _n | ≤ kk 1 α(V ),

where kk 1 < 1.

By Theorem 1.1 T has a fixed point.

Now, we show that the fixed point of the operator T is asymptotically equal to a.

We know that T h = h.

For n > n 0 , we obtain

∆h _n = (T h) _n+1 − (T h) _n

= −

∞

X

j=0

α ^j _n f (h n+j ) +

∞

X

j=0

α ^j _n+1 f (h n+1+j ).

Using the equation:

α ^j _n = a ⁰ _n+j + a ⁰ _n+j−1 + a ⁰ _n+j−2 + · · · + a ^j _n , n, j ∈ N we have:

∆h _n = h _n+1 − h _n = (T h) _n+1 − (T h) _n

= a ⁰ _n f (h n ) +

∞

X

j=0

α ^j+1 _n f (h n+1+j ) −

∞

X

j=0

α ^j _n+1 f (h n+1+j )

= a ⁰ _n f (h _n ) +

∞

X

j=0

(α ^j+1 _n − α ^j _n+1 )f (h _n+1+j )

= a ⁰ _n f (h _n ) +

∞

X

j=0

a ^j+1 _n f(h _n+j+1 )

=

∞

X

j=0

a ^j _n f (h n+j ),

which implies that h = (h n ) satisfies the difference equations (∗).

And by the definition of T the solution h _n → a.

We can generalize Theorem 1.1 for measure of the weak noncompactness β.

Let B = {x ∈ X : kxk ≤ 1} and let V be a bounded subset of X. The β(V ) measure of weak noncompactness β(V ) of V is defined by:

β(V ) = inf{t ≥ 0 : V ⊂ K + tB for some weakly compact K ⊂ X}.

Properties of the measure of weak noncompactness are analogously to prop-

erties of the measure of noncompactness [4, 7]. We shall use the following

properties of the measure of weak noncompactness β.

Theorem 1.5 (see [4, 7]). Let A, B be bounded subsets of X. Then (a) β(A) = 0 ⇒ A is a relatively weakly compact subset

(b) β(conv A) = β(A) (c) A ⊂ B ⇒ β(A) ≤ β(B) (d) β(kA) = kβ(A) for k ∈ (0, ∞)

(e) β({x 0 } ∪ A) = β(A).

Theorem 1.6 [11]. Let V ⊂ C(N ⁺ , X) be a family of functions. Then β(V ) = β(V (N ⁺ )) = sup{β(V (i)) : i ∈ N ⁺ },

where β(V ) denotes the measure of weak noncompactness in C(N ⁺ , X).

Theorem 1.7 (see [1, 10]). Let D be a nonempty, weakly closed, convex and bounded subset of a Banach space X. Let F : D → D be a weakly- weakly, sequentially continuous mapping, which is condensing with respect to the measure of weak noncompactness β, i.e.,

β(F (V )) ≤ kβ(V ) for β(V ) > 0, V ∈ D, then F has a fixed point.

Similary as Theorem 1.4, we can prove the following theorem:

Theorem 1.8. Let f be a bounded and weakly-weakly, sequentially contin- uous function. Let a ∈ X and kα _n k 1 → 0 as n → ∞, where (α _n ) ∈ l 1 (C).

We assume that k 1 = sup

n≥n

{

∞

P

j=0

kβ n ^j k} and kf (h _n+j )k ≤ k and k 1 < ¹ _k , for some n 0 ∈ N . Moreover,

β(f (V )) ≤ kβ(V ).

Then for each a ∈ X there exists a solution x = (x n ) ∈ l ∞ (X) of the equation (∗) such that x _n → a.

As an application of our result we consider the difference equation (see [8])

∆x n =

∞

X

i=0

a ⁱ _n g(x n+i ) +

∞

X

i=0

b ⁱ _n h(x n+i ) + y n ,

where P ∞

i=0 |a ⁱ _n | < ∞ and P ∞

i=0 |b ⁱ _n | < ∞ for i = 0, 1, 2, 3, . . . , r and P ∞ n=0 y n

is convergent.

Remark 1. We will show that for this equation the condition (∗∗) of Theorem 1.1 are satisfied. Assume that

G(x) =

∞

X

i=0

a ⁱ _n g(x _n+i ) and H(x) =

∞

X

i=0

b ⁱ _n h(x _n+i ).

If

kGx 1 − Hx 2 k ≤ kkx 1 − x 2 k then (see [3])

α(G(V )) ≤ kα(V ).

Let F = G + H, where G : C → X is a Lipschitz mapping and H : C → X is a compact operator and C is a nonempty, bounded set in a Banach space.

Because α(H(V )) = 0 so we obtain

α((G + H)(V )) = α(G(V ) + H(V )) ≤ α(G(V )) + α(H(V )) ≤ kα(V ).

So, G satisfies the condition (∗∗) of Theorem 1.1.

Remark 2. Observe that the class of continuous functions is different from the class of weakly-weakly sequentially continuous and weakly-weakly continuous functions.

There exist many important examples of mappings, which are weakly sequentially continuous but not weakly continuous.

The relationship between strong-weak and weak sequential continuity for mappings is studied in [2].

References

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Received 10 October 2003

Revised 3 January 2005