Functional differential equations of mixed type in Banach spaces

R ^ENDICONTI

del

S ^EMINARIO M ^ATEMATICO

della

U NIVERSITÀ DI P ^ADOVA

C ^UI B ^AOTONG

Functional differential equations of mixed type in Banach spaces

Rendiconti del Seminario Matematico della Università di Padova, tome 94 (1995), p. 47-54

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of Mixed Type ^{in Banach} Spaces.

CUI

BAOTONG (*)

1. Introduction.

In the last few _year the fundamental

theory

^of functional differen- tial

equations

^{with a}

single delay

^{in a} Banach space has

undergone

^in-

tensive

development (see [1,2]).

^The fundamental

theory

of functional differential

equations

^{of mixed}

type, however,

is still in a initial

stage

^of

development ^[3-8].

^{In this} paper, ^we consider the functional differential

equation

^{of mixed}

type

^{of the form}

in a Banach _space B.

Let B ⁼

(B, () ’ 11)

^{be a Banach}

^0,

s &#x3E; 0 are constants and

- m

a {3 ~

^{+00. Assume}

that f : [ a, ~ ]

^{x B x B} x B - B satisfies the conditions:

(a) f (’,

^x, y,

z)

^is

strongly

measurable for all fixed _{x, y,} z E B and there exist xo , yo ,

zo E B

^{such that}

( b )

there exist

nonnegative

^{constants a, b and c such that}

(c)

^is continuous and r 5

i( t ) ~

^s.

(*) Indirizzo dell’A.:

Department

^of Mathematics, Binzhou Normal

College,

Shandong

256604,

People’s Republic

^{of China.}

48

We consider the

equation ⁽¹⁾

^{with two}

boundary

conditions:

where

is Bocher

integrable function},

is Bocher

integrable function} ,

DEFINITION. x:

[a -

⁺

s) ~ B

^{is a} solution of the Problem

(1) (2),

^if

^x(t)

^is continuous on

[a, P)

and satisfies

(1)

^and

(2).

Let

a =

[a -

s,

P

⁺

s)

is continuous on

[a, {3)

and satisfies

(2)},

It is _{easy to} see that x is a solution of the

problem (1), (2)

^{if and}

only

if x is a fixed

point

^{of the}

operato T,

^{defined on ci}

by

2. Existence and

uniqueness.

THEOREM 1

(Existence

^and

uniqueness).

^Let

(a),(b)

^and

(c)

hold and s ~ l ⁼

p -

^a. If there exists a ~, ~ 0 such that

then,

^{for each} xor=

ai

the iterates T :

ai - ai

is defined

by

(3),

converge in the metric _ez of ai defined

by

to a solution of the

problem ^{(1) (2),}

^{which is}

unique

ⁱⁿ tL,l.

PROOF. L et x ^E aA, ^then sup and

x(t)

is continu-

a t B

ous on

[a, fl)

and satisfies

(2).

^Hence

Tx(t)

is continuous on

[a, {3)

^and

satisfies

(2).

From

(3),

for each A &#x3E; 0 and

any t

^E

[a, /3),

^{we have that}

Note that x E

a., q5 and

^{are Bocher}

integrable functions,

^{so we can ob-}

tain that

For A ⁼

0, using ^(ac)

^{and x}

e t1o ,

^i.e. sup

~~ ^Tx(t) ~~ I

^{00, we have Tx E}

E So

T(aA)

^c aA for each A &#x3E;

50

Now we define

is continuous in

[a, fl),

and

then the

operator

^W maps the set 1B into itself. We can prove that for _any

y E 93

^{and each}

where

and

Thus from

(5)

^{we have that}

Hence T is contractive with

respect

to gi

by (4).

Since ai ^is

complete

ⁱⁿ

this

metric,

the Banach’s contractive

mapping

^theorem

implies

^{that the}

iterates _{xo E converge} in

( rz~, ,

^{to a}

unique

^fixed

point

^of

T,

i.e. to a

unique

^{solution of the}

problem (1), (2)

ⁱⁿ

a..

COROLLARY 1. Let

(a)

^and

( b ) hold,

^{and = r = const} &#x3E; 0. If there exists £ * 0 such that

then the

problem (1), (2)

^{has an}

unique

solution in

aA.

EXAMPLE. Consider the Lecornu’s

equations [4, 6]

where a and b are constants and

r(t)

^{= 1. The} conditions

(a)

^and

( b )

^are

verified.

By

^the

Corollary

^{1 and}

(6),

if there exists a ~, ~ 0 such that

I 1

, then the

problem (7), (2)

^{has a}

unique

solution in ai ^{for r = s = 1.}

COROLLARY 2. Let the

hypotheses

of the Theorem 1 hold and

~i

00, then the

problem (1), ⁽²⁾

^{has a} bounded solution on

[a,

PROOF. Since

fl

^{00, then}

ai = t1o

for all Â. ~ 0. Hence the

problem (1), (2)

^{has a}

unique

solution in

t1o by

the Theorem

1,

and the solution is bounded.

3.

Dependence

^{on the}

boundary equations.

THEOREM 2

(Dependence).

^{Let all} conditions of the Theorem 1 hold.

Suppose

and wn (a) - o(a)

^Let be the solution of the

equation (1)

^{with the}

boundary

^conditions

n ⁼

1, 2,

^{.... Then the} sequence

(t))

of the solutions of the

problem (1), ( 8 )n ( n

⁼

1, 2, ...)

^{has the}

following properties:

(i) if fl

00, then ^{as n ~ ~}

uniformly

^on

[a, #);

(ii) if fl

^{= 00, then}

xn ( t )

^~

x(t) uniformly

^{on the com-}

pact

^{intervals of}

[a, {3).

52

PROOF. Let

is continuous on

[ a, P)

and satisfies

( 8 )n ~ ,

and we define the

operators Tn :

for

n = 1, 2, .... Similarly

^{to the}

proof

of the Theorem

1,

^{we can} prove that c

ani

^{and the}

operator Tn

^{has a}

unique

^fixed

point

^{for each}

n . Hence for each n the

problem (1), ( 6 )n

^{has a}

unique

solution in

anA.

Let x E ai be the fixed

point

^{of T}

and xn

^E

ani

be the fixed

point

^of

Tn

^for

each n,

^{then we have that}

where

Mn

⁼ Const. &#x3E; 0 and lim

Mn

⁼ 0 . Thus from

n - oo

it is _{easy to prove} our results.

4. Remarks.

REMARK 1.. The above results can be estended

naturally

^{to the}

problem involving

^several

arguments

where &#x3E; 0

( i

⁼

1, 2, ... , m )

^are

continuous,

and the exist _nonnega- tive

constants ri and si ( i

⁼

1, 2,

^{... ,}

^{m )}

^{such that}

0 ~ ri ~

^and

also to functional differential

equation

^{of mixed}

type

of the form

where p and q

^are

positive numbers, rj (t)

^and

hj (t)

^are

nonnegative

continuous functions.

REMARK 2. The

equations

and

where g, h, gi ^{( i}

⁼

^{1, 2,}

^{... ,}

p) and hj ^{( j}

⁼

^{1, 2 ,}

^...,

q )

^are

continuous, andg&#x3E; 1,0h 1,0h~

¹ for each i

and,

^{are of}

type (10)

too.

In

fact,

^{let then we} have that from

(11)

where &#x3E;

0,

^{0 .}

Acknowledgment.

The author is

grateful

^to the referee for

helpful suggestions.

REFERENCES

[1] V. LAKSHMIKANTHAM - S. LEELA, ^Nonlinear

Differential Equations

^{in Ab-}

stract

Spaces, Pergamon

Press, ^{New York} (1981).

[2] ^{F. J.} BURKOWSKI, ^{Existence and}

uniqueness

^theorems

for differntial

equa- tions with

deviating

arguments

of mixed

type, ⁱⁿ

Delay

and Functional

Dif- ferential Equations

^{and Their}

Applications

(Edited

by

^Klaus Schmitt), ^Aca-

demic Press, ^{New York} (1972), pp. 309-311.

[3] CUI BAOTONG - XIA DAFENG, Existence and

uniquenss of analytic

^solutions

of functional differential equations, Fuyang Shiyuan

Xuebao, ² (1989), pp.

87-93.

54

[4] ^ZHENG ZUXIU, Functional

Differential Equations,

^Xi’an

Jiaotong University

Press (1989).

[5] ^CUI BAOTONG,

Asymptotic stability of

the solutions

of

^nonlinear

functional differential equations of neutral

type, J. Math. Res. &#x26;

Exposition,

¹¹ (1991),

p. 423.

[6] ^J. HALE,

Theory of

Functional

Differential Equations, Springer-Verlag,

New York (1977).

[7] CUI BAOTONG,

Asymptotic stability of large

systems

of

^neutral type, ^Mathe-

matica

Applicata,

3(3) (1989), pp. 81-83.

[8] ^CUI BAOTONG, Oscillation

criteria for

n-th order

delay differential equations,

Ann. Diff.

Equs.,

⁷ (1991), pp. 1-4.

Manoscritto pervenuto ^{in redazione il 12 novembre 1991}

e, in forma revisionata, ^{il 17}

gennaio

^1994.