On a Nonlinear Nonlocal Cauchy Problem

(1)

On a Nonlinear Nonlocal Cauchy Problem

Sergiu Aizicovici

Abstract -We prove the existence of integral solutions to a nonlinear time-dependent functional differential equation with a nonlocal initial condition. The approach relies on the theory of m-accretive operators and compactness methods.

Index Terms-Compact evolution operator, evolution equation, m-accretive operator, nonlocal Cauchy problem.

I. INTRODUCTION

We are concerned with the existence of solutions to the nonlocal Cauchy problem

) ( ) 0 (

], , 0 [ ),

)( ( ) ( ) ( ) ( '

u g u

T I t t u F t u t A t u

=

= ∈ ∋

+

(1)

in a real Banach space X . Here {A(t):t∈I}are

m-accretive operators in X , while F ,gare functionals

defined on C(I;X)with values in L1(I;X)and X , respectively. Such problems arise in physics and engineering, in particular in the mathematical modeling of heat or diffusion processes, or in the study of atomic reactors; see [1]-[3].

The study of abstract evolution equations with nonlocal initial conditions was initiated by

Byszewski [4], who studied a problem of the form (1) whereA(t)=A,linear and independent of time,

)), ( , ( ) )(

(Fu t = f t u t f :I×X →X,and

), ( )

( 1

i p

i iu t c u

g

∑

=

= with0<t1 <...<tp ≤T

and ci∈R.

Manuscript received March 1, 2009.

S.A. Author is with the Department of Mathematics, Ohio University, Athens, OH 45701 USA (phone: 740-593-1272; fax: 740-593-9805; e-mail: [email protected]).

For some recent results on fully nonlinear nonlocal Cauchy problems, see [5]-[7]. In general, the existing results require a Lipschitz condition on F org,or a compactness assumption on g.In [8], the

authors remove these restrictions for a linear,

autonomous version of (1). It is our goal to generalize their approach to the fully nonlinear, time-dependent case.

II. PRELIMINARIES

In this section, we collect some basic facts on m-accretive operators, evolution operators and non-autonomous evolution equations; see [9], [10] for

details. Let ( X, .)be a real Banach space, of dual

). . *,

(X _* The duality mapping J:X →X*is defined

by

, },

* )

( * : * * { )

(x x X x x x 2 x _*2 x X

J = ∈ = = ∀ ∈

while the so-called upper semi-inner product is given by

. , )}, ( * : ) ( * sup{

,x x y x J x x y X

y > = ∈ ∀ ∈

< ₊

It can be shown that <.,.>₊is upper semicontinuous on X×X._{Let A be a multivalued operator on}X,of

domain D(A)and range R( A).We say that A is

accretive if < y'−y,x'−x>₊≥0for all x,x'∈D(A) and ally∈Ax,y'∈Ax'. If alsoR(Id+λA)= Xfor all

, 0 >

λ where Iddenotes the identity on X,then A is called m-accretive.

Let {A(t):t∈I}be a family of m-accretive

operators inX,of domains D(A(t)), with

)) ( (At

D = D (independent of t), which satisfy the

(2)

)

(H_A_(t₎ There exist two continuous functions

)) , 0 [ ( : , : ₂

1 I→X m R+→R+ R+ = ∞

m such that

}), , (max{ ) ( ) ( , 2 1 2 2 1 2 1 2 1 2 1 x x m x x s m t m x x y y − − − ≥ > − − < + . 0 , ) ( )), ( ( , ) ( )), ( ( 2 2 2 1 1 1 T t s x s A y s A D x x t A y t A D x ≤ ≤ ≤ ∈ ∈ ∈ ∈ ∀

If (H_A_(t₎)holds, then the family {A(t):t∈I}

gives rise to an evolution operator U( st, )on D via

the formula x n s t i s A n s t Id x s t U n i n 1 1 )) ( ( lim ) , ( − = ∞ → − + − +

=

∏

, (2)

for all x∈D,0≤s≤t≤T.

It follows that

U(t,t)=Idand U(t,s)x−U(t,s)y ≤ x−y,

for all x,y∈D,and all 0≤s≤t≤T.The evolution

operator U is said to be compact if U( st, )maps

bounded subsets of D into relatively compact subsets of D for all 0≤s<t≤T._{In the special case when}

A t

A()= is a time-independent m-accretive operator,

) ( ) 0 , (t St

U = is the contraction semigroup generated

by A− on D( A).

Next, consider the Cauchy problem

, ) 0 ( , ), ( ) ( ) ( ) ( ' 0 u u I t t f t u t A t u = ∈ ∋ + (3)

where {A(t):t∈I}satisfy (H_A₍_t₎),f ∈L1(I;X),and

. 0 D u ∈

Definition 1. An integral solution of problem (3) is a function u∈C( DI; )satisfying u(0)=u₀and the

inequality , ] ) ( ) ( ) ( ) ( , ) ( [ 2 ) ( ) ( 1 1 2 2

τ

θ

τ

d m m x u M x u y f x s u x t u t s − − + > − − < ≤ − − −

∫

+ , ) ( )), ( ( ], , 0 [ ,

0≤s≤t≤T θ∈ T x∈D Aθ y∈Aθ x

∀ and }). , (max{ ) ; ( 2 x u _C_I _X m

M =

It is well-known that (3) has a unique integral

solution for each u₀∈Dand f∈L1(I;X),provided that (HA(t))is satisfied. In particular, U(t,0)u0is the

integral solution of (3) whenf ≡0.

Proposition 2. Let (H_A_(t₎)be satisfied, and let

v

u, be integral solutions of (3), corresponding to

) ,

(u₀ f and (v₀,g),respectively (with u₀,v₀∈D

and f,g∈L1(I;X)). Then

. 0 , ) ( ) ( ) ( ) ( ) ( ) ( T t s d g f s v s u t v t u t

s − ∀ ≤ ≤ ≤

+ − ≤ −

∫

τ

(4)

III. MAIN RESULTS

For a fixed finite r>0,we set

}. , ) ( : ) ; ( { }, : { I t B t X I C K r x X x B r r r ∈ ∀ ∈ ∈ = ≤ ∈ =

φ

We assume that:

} ); ( { )

(H1 At t∈I satisfy(HA(t)),and the

corresponding evolution operatorU (given by (2)) is compact;

)

(H₂ The operator F:C(I;X)→L1(I;X)is

continuous, and there exists α=α_r∈L1(I;R₊)such that

F(u)(t) ≤α(t),a.e. on ,I for all u∈K_r;

)

(H₃ The function g:C(I;X)→Dis continuous

and maps K_rinto a bounded set;

)

(H₄ There exists δ ∈( T0, )such that

) ( ) ( ), ( )

(u F v gu g v

F = = for any u,v∈K_rwith

]; , [ ), ( )

(s vs s T

(3)

)

(H₅ +

∫

≤

∈ ∈

T

K I t

r d g

t U

r 0

,

. ) ( )

( ) 0 , (

sup φ α τ τ

φ

Definition 3. A function u∈C( DI; )is called an

integral solution of problem (1), if it is an integral solution, in the sense of Definition 1, of problem (3) with F(u)(t)in place of f(t),and g(u)in place of

. 0 u

Theorem 4. Let (H₁)−(H₅)be satisfied. Then

problem (1) has at least one integral solution.

This result does not cover the case when ))

( , ( ) )(

(u t f t u t

F = for a given function

, :I X X

f × → since (H₄)is not satisfied. We now

replace (H₄)by the following condition:

)

(H₄# lim_ε_↓₀ g(φ)−g(φε) =0,uniformly for all ,

r K ∈

φ with

  

≤ ≤

≤ ≤ =

. ),

(

, 0 ), ( ) (

T t t

t t

ε φ

ε ε

φ φε

Theorem 5. Let (H1)−(H3),(H4#)and (H5)be

satisfied. Then problem (1) has at least one integral solution.

IfA(t)= A(independent of time), then the

evolution operator Uis replaced by the contraction

semigroup S(t)generated by A− (cf. Section II). The

corresponding autonomous initial-value problem (3) has a unique integral solution (that is, a function

) ( ; (I D A C

u∈ ) satisfying the inequality in Definition

1 with M =0)for any f∈L1(I;X)and u₀∈D(A).

Theorem 5 now yields the following result:

Corollary 6. Let A be an m-accretive operator inX,such that S(t),the semigroup generated by

A

− on D( A), is compact for t>0.If also

) ( ),

(H₂ H₃ (with D=D( A)), (H₄#)and (H₅)(with

) (t

S in place of U(t,0)) are satisfied, then there

exists an integral solution of the problem

). ( ) 0 (

, ), )( ( ) ( ) ( '

u g u

I t t u F t Au t u

=

∈ ∋

+

(5)

Remark 7. In the special case when

F(u)(t)= f(t,u(t)), f :I×X →X,

where f satisfies Caratheodory type conditions,

Corollary 6 is comparable to Theorem 4.3 in [7], which was proved by a different method.

IV. PROOF OF THEOREM 4

We sketch the proof of Theorem 4, only. The proof of Theorem 5 can be carried out by using Theorem 4, and adapting the approximating procedure of [8]. The details will appear elsewhere.

Set

]}. , [ , ) ( : ) ]; , ([ { )

( u C T X ut r t T

K_r δ = ∈ δ ≤ ∀ ∈δ

For any u∈K_r(δ),let u~∈K_rbe given by

  

≤ ≤

≤ ≤ =

. ),

(

, 0 ), ( ) ( ~

T t t u

t u

δ δ δ

Also, define

). ~ ( ) ( ~ ; )), ( ~ ( ) )( ( ~

u g u g I t t u F t u

F = ∈ = (6)

By (H₂)−(H₄)and (6), it follows that F~and g~are

continuous from K_r(δ)to L1(I;X)and _D,

respectively. In addition, we have

F~(u)(t) ≤α(t),a.e. on I,∀u∈K_r(δ), (7)

. ) ( ) 0 , ( sup ) ( ~ ) 0 , ( sup

, )

( ,

∞ < =

∈ ∈ ∈

∈Iu Kr U t g u t Iv KrU t g v

t δ

(8)

Define the map Ψ:Kr(δ →) C([δ,T];X)by

), ( ],

, [ ), ( ) )(

(w t =uw t t∈ δ T w∈Kr δ

Ψ where u_wis

the unique integral solution of

) ( ~ ) 0 (

, ), )( ( ~ ) ( ) ( ) (

w g u

I t t w F t u t A t u dt d

w

w w

=

∈ ∋

+

(4)

From (4) we infer that

() (,0)~( ) ~( )( ) . 0 F w s ds w

g t U t

u_w − ≤

∫

t

This, (7), (8) and (H5)lead to

, , ) (

) ( ) 0 , ( sup )

(

0 ,

I t r ds s

v g t U t

u

T K v I t w

r

∈ ∀ ≤

+ ≤

∫

∈ ∈

α (10)

so that Ψmaps Kr(δ)into itself. Moreover, on

account of (4), (6), (H2),(H3),Definition 1 and the

upper semicontinuity of <.,.>₊, it is easily seen that Ψis continuous. Next, employing (6), (7),

) ( )

(H1 − H3 and the theory of [11], we deduce that

)) ( (Kr δ

Ψ is relatively compact in C([δ,T];X). Applying Schauder’s fixed point theorem, we

conclude that Ψhas a fixed point w*∈Kr(δ).

Let u(t)=uw*(t),t∈Iand remark that

]. , [ ), ( ) ( ) *)( ( ) (

* t w t u * t u t t T

w =Ψ = w = ∀ ∈δ This,

together with (H4), (9) and (10), implies that uis an

integral solution of (1), as desired. The proof is complete.

V. AN APPLICATION

For simplicity, we restrict ourselves to an example that illustrates Corollary 6. Consider the initial-boundary value problem

, ), , ( )

( ) , 0 (

, )

, ( )), , ( ( ) , (

, ) , ( , ) , ( ) (

)) , ( , ( ) , ( ) , (

1 0

0

Ω ∈ +

=

Ω ∂ × ∈ ∈

∂ ∂ −

Ω × ∈ −

+ =

∆ −

∑

∫

=

x x t u c x u x u

I x t x t u x t n u

I x t ds x s u s t k

x t u t h x t u x t u

p

i i i t

t

β (11)

where Ωis a bounded domain in RNwith a smooth boundary ∂Ω,βis an m-accretive operator on R with

n ∂

∂ ∈ (0),

0 β denotes the outward normal derivative,

, ...

0<t₁< <t_p ≤T ci,i=1,...pare given constants,

, :

), ( 1

R R I h I L

k∈ × → and u0∈L2(Ω).This

problem can be written in the form (1) in the

spaceX =L2(Ω),by setting

), ( :

) ( { ) (

, 2 u

n u H

u A D

A ∈β

∂ ∂ − Ω ∈ = ∆

−

= a.e. on

}, Ω ∂

∫

−

+

=ht ut x tkt s u s x ds x

t u F

0 ( ) ( , ) , ))

, ( , ( ) )( )(

( (12)

= ) )( (u x

g ( ) ( , ),(, ) .

1

0 +

∑

∈ ×Ω

=

I x t x t u c x u

p

i i i

It is well-known [10] that A is m-accretive in X with D(A)= X,and that the semigroup S(t)

generated by A− on X is compact (for t>0), with .

0 , 0 0 )

(t = ∀t≥

S Assume that

)

(H6 t→h( yt, )is measurable in tfor all y∈R,and

continuous in yfor a.a. t∈I;

)

(H7 There exist 0, ( ; ) 1

+

∈ > b L I R

a such that

), ( )

,

(t y ay bt

h ≤ +

for almost all t∈Iand all y∈R..

Then, it is easily verified that F and g,as given in

(12), satisfy (H2),and respectively (H3)and ( ). # 4 H

Finally, in this set-up, condition (H5)reduces to

) (H8

, ) (

) (

2 / 1 ) (

) ( 1

) ( 0

1

1 2

r b

k T c a r u

I L

I L p

i i L

≤ Ω

+ +

+

∑

= Ω

µ

where µdenotes the Lebesgue measure.

(5)

Corollary 8. Ifu₀∈L2(Ω),k∈L1(I)and

) ( )

(H6 − H8 hold, then problem (1) has at least one

integral solution u∈C(I;L2(Ω)).

REFERENCES

[1] J. Chadam and H. M. Yin, Determination of an unknown function in a parabolic equation with an overspecified condition, Math. Meth. Appl. Sci. 13 (1990), 421-430.

[2] K. Deng, Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions, J. Math. Anal. Appl. 179 (1993), 630-637.

[3] Y. Lin, Analytical and numerical solutions for a class of nonlocal nonlinear parabolic differential equations, SIAM J. Math. Anal. 25 (1994), 1577-1596.

[4] L. Byszewski, Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl. 162 (1991), 494-505.

[5] S. Aizicovici and H. Lee, Existence results for nonautonomous evolution equations with nonlocal initial conditions, Commun. Appl. Anal. 11 (2007), 285-297.

[6] S. Aizicovici and V. Staicu, Multivalued evolution equations with nonlocal initial conditions in Banach spaces, NoDEA, Nonlinear Differential Equations Appl. 14 (2007), 361-376.

[7] X. Xue, Nonlinear differential equations with nonlocal conditions in Banach spaces, Nonlinear Anal. 63 (2005), 575-586.

[8] Q. Liu and R. Yuan, Existence of mild solutions for semilinear evolution equations with nonlocal initial conditions, Nonlinear Anal., to be published.

[9] N. H. Pavel, Nonlinear Evolution Operators and Semigroups, Lecture Notes in Math., vol. 1260, Berlin: Springer-Verlag, 1987.

[10] I. I. Vrabie, Compactness Methods for Nonlinear Evolutions, Pitman Monographs Surveys Pure Appl. Math., vol. 75, Harlow: Longman Scientific & Technical, 1995.