DSpace at VNU: Periodic solutions of evolution equations

Anti-periodic solutions for evolution equations with mappings in the class S+ Yu Qing Chen∗1, Yeol Je Cho∗∗2, and Donal O’Regan∗∗∗3 1 Department of Mathematics, Foshan University, Foshan

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Anti-periodic solutions for evolution equations with mappings in the class (S+)

Yu Qing Chen∗1, Yeol Je Cho∗∗2, and Donal O’Regan∗∗∗3

1

Department of Mathematics, Foshan University, Foshan, Guangdong 528000, P R China

2 Department of Mathematics Education and the RINS, College of Education, Gyeongsang National University,

Chinju 660-701, Korea

3 Department of Mathematics, National University of Ireland, Galway, Ireland

Received 1 June 2004, accepted 3 September 2004

Published online 31 January 2005

Key words Evolution equation, anti-periodic solution, a mapping of class(S+)

MSC (2000) Primary: 34C25, 34G20; Secondary: 47H05, 47H10

In this paper, we study the existence of anti-periodic solutions for the first order evolution equation

u (t) + ∂Gu(t) + f(t) = 0 , t ∈ R ,

u(t + T ) = −u(t) , t ∈ R ,

in a Hilbert spaceH, where G : H → R is an even function such that ∂G is a mapping of class (S+) and

f : R → R satisfies f(t + T ) = −f(t) for any t ∈ R with f(·) ∈ L2(0, T ; H).

c

2005 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim

1 Introduction

Okochi [14], [15] initiated the study of anti-periodic solutions for the following nonlinear evolution equation

u (t) + ∂φ(u(t)) f (t) , a e t ∈ R ,

in Hilbert spaces, where ∂φ is the subdifferential of an even function φ on a real Hilbert space H and f is a

T -anti-periodic function It was shown in [15], by applying a fixed point theorem for a non-expansive mapping,

that the problem (E 1.1) has a solution Following Okochi’s work, Haraux [13] proved some existence and uniqueness theorems for anti-periodic solutions of gradient type equations by using Brouwer’s and Schauder’s fixed point theorem Later, Aftabizadeh, Aizicovici and Pavel [1], [2], [3] studied the anti-periodic solutions for first and second order evolution equations in Hilbert and Banach spaces by using maximal monotone or

m-accretive operator theory We refer the reader to Aizicovici, MacKibben and Reich [4], Chen, Cho [8], [9],

[12], Souplet [19], [20] for other works on anti-periodic solutions

In this paper, we will study the existence problems of anti-periodic solution for the first order evolution equa-tion

u (t) + ∂Gu + f (t) = 0 , a e t ∈ R ,

in a real separable Hilbert space H, where G : H → R is an even function such that ∂G is a mapping of class (S+)and f : R → R satisfies f (t + T ) = −f (t) for t ∈ R and f (·) ∈ L2(0, T ; H) This equation is still of gradient type, but we do not require any Lipschitz condition on ∂G which is required in [9] and [13].

∗ e-mail: yqchen@foshan.net

∗∗ Corresponding author: e-mail: yjcho@gsnu.ac.kr

∗∗∗ e-mail: donal.oregan@nuigalway.ie

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2 The existence results

In this section, assume that H is a real Hilbert space, G : D(G) ⊆ H → R is an even function, and f (t) : R → H

is a function satisfies f (t + T ) = −f (t) Consider the following anti-periodic problem

u (t) + ∂Gu(t) + f (t) = 0 , a e t ∈ R ,

Definition 2.1 A function u(·) is called a weak anti-periodic solution of the problem (E 2.1) if u(t + T ) =

−u(t) for t ∈ R and the weak derivative u (t) satisfies

u (t) + ∂Gu(t) + f (t) = 0

for almost all t ∈ R.

Lemma 2.2 ([12]) If u, u ∈ L2(0, T ; H) and u(t + T ) = −u(t) for any t ∈ R, then

|u| ∞ ≤

√ T

2

T

0 |u (s)|2ds

1

.

In this paper, we shall use different assumptions and different methods from those papers mentioned in

Sec-tion 1 For completeness, let us recall the definiSec-tion of mappings in the class (S+)in a Hilbert space.

Definition 2.3 Let T : D(T ) ⊆ H → H be a mapping.

(1) If{x n } ⊂ D(T ), x n x0and lim supn→∞ (T xn , x n − x0 ≤ 0 imply that x n → x0, then we call T a mapping of class (S+),

(2) T is said to be demi-continuous if xn → x0implies that T xn T x0.

For mappings of class (S+)in reflexive Banach spaces and their applications, we refer the reader to [5]–[7], [9]–[11], [17] and [18]

Theorem 2.4 Let H be a real separable Hilbert space and let G : H → R be even and Fr´echet differentiable.

If ∂G is a demi-continuous bounded mapping of class (S+) and f : R → H is in L2(0, T ; H) and satisfies

f (t + T ) = −f (t) for all t ∈ R, then the equation

u + ∂Gu(t) + f (t) = 0 , a e t ∈ R ,

has a weak solution.

P r o o f Since H is separable, there exists an orthogonal basis {e1, e2, } of H Set

H n = Span{e1, e2, , e n }

for n = 1, 2, and let Pn : H → Hnbe the projection We consider the equation

u (t) + Pn ∂Gu(t) + P n f (t) = 0 , a e t ∈ R ,

For each n = 1, 2, , set

W n = {u : R → H n is continuous, u(t + T ) = −u(t)} ,

and

W 1,2 n =

u ∈ W n :

T

0 |u (t)|2dt < ∞

.

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Then Wnis a Banach space under the norm|u| ∞= maxt∈[0,T ] |u(t)|, and we may define a norm on W 1,2

n by

u W 1,2 = |u| ∞+

T

0 |u (t)|2dt

1

,

where| · | is the norm in H.

For each v(·) ∈ Wn, consider the following equation

u (t) + Pn ∂Gv(t) + P n f (t) = 0 , a e t ∈ R ,

It is easy to check that

u(t) = −

t

0 [Pn ∂Gv(s) + P n f (s)] ds +1

2

T

0 [Pn ∂Gv(s) + P n f (s)] ds

is the unique solution of the problem (E 2.4)

Next, we define a mapping L : Wn → W n as follows: For each v(·) ∈ Wn , Lv is the solution of the problem

(E 2.4)

We prove that L is continuous Suppose vj(·) ∈ W n and vj(·) → v0 ·) in W n Then|v j(·) − v(·)| ∞ → 0 as

j → ∞ Since

(Lvj (t)) − (Lv0(t)) + (Pn ∂Gv j (t) − Pn ∂Gv0(t)) = 0 , a e t ∈ R , (2.1)

if we multiply both sides of (2.1) by (Lvj(t) − Lv0(t)) and integrate over (0, T ), we get

T

0 |(Lv j(t) − Lv0(t)) |2dt +

T

0 (Pn ∂Gv j (t) − Pn ∂Gv0(t))(Lvj(t) − Lv0(t)) ) dt = 0 Since Pn ∂G is continuous in H n , we have

T

0 |(Lv j(t) − Lv0(t)) |2dt

1

≤ √ T |P n ∂Gv j(·) − P n ∂Gv0 ·)| ∞ −→ 0 as j −→ ∞

This with Lemma 2.2 guarantees that L is continuous For each v(·) ∈ Wn, again by the problem (E 2.4), we get

T

0 |(Lv(t)) |2dt +

T

0 (Pn ∂Gv(t), (Lv(t)) ) dt +

T

0 (Pn f (t), (Lv(t)) ) dt = 0

Thus it follows that

T

0 |(Lv(t)) |2dt

1

≤

T

0 |P n Gv(t)|2dt

1

+

T

0 |f(t)|2dt

1

From (2.2), the continuity of Pn ∂G in H n and Lemma 2.2, we know that L maps bounded sets of Wnto bounded

sets in Wn The compact embedding of W 1,2 n into Wn together with L : Wn → W 1,2

n continuous guarantees that

L : W n → W nis a compact mapping

nwith

|v(·)| ∞ >

√ T

2

T

0 |f(t)|2dt and λ ≥ 1 If this is not true, then there exist λ0≥ 1, v0 ·) ∈ W nwith|v(·)| ∞ > √2T T

0 |f(t)|2dt

1

such that

Lv0 ·) = λ0v0 ·), i.e.,

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Multiply both sides of (2.3) by v 0(t) and integrate over [0, T ] to obtain

T

0 (λ0v 0(t) + Pn ∂Gv0(t) + Pn f (t), v 0(t)) dt = 0 (2.4) Notice that

T

0 (Pn ∂Gv0(t), v0 (t)) dt =

T

0 (∂Gv0(t), v0 (t)) dt = 0

and so we have

T

0 |v

0(t)|2dt ≤

T

0 |f(t)|2dt

1

,

which implies from Lemma 2.2 that

|v(·)| ∞ ≤

√ T

2

T

0 |f(t)|2dt

1

,

which is a contradiction

Now, if we take r0> √2T T

0 |f(t)|2dt

1

, by the above argument and the homotopy invariance property of the Leray-Schauder degree, we know that

deg(I − L, B(0, r0), 0) = deg(I, B(0, r0), 0) = 1 ,

where B(0, r0 is the open ball centered at 0 with radius r0in Wn Therefore, L has a fixed point in B(0, r0 ,

i.e., there exists vn(·) ∈ W n such that Lvn(·) = v n(·) Hence problem (E 2.3) has a solution v n(·).

It is easy to show that

T

0 |v

n (t)|2dt ≤

T

0 |P n f (t)|2dt

1

From (2.5) and Lemma 2.2, it follows that

|v n(·)| ∞ ≤

√ T

2

T

0 |P n f (t)|2dt

1

From (2.5), we may assume that v k ·) y(·) ∈ L2(0, T ; H) and, in view of (2.6), we may also assume that

v n (0) v0∈ H (otherwise, take a subsequence) Since v n(t) = vn(0) + t

0v n (s) ds, we have vn(t) v0(t) =

v0+ t

0y(s) ds, where t

0y(s) ds is the weak integral.

Multiply (E 2.3) by vn(t) − v0(t) and integrate over [0, T ] to get

T

0 [(v n (t), vn (t) − v0(t)) + (∂Gvn(t), vn (t) − Pn v0(t)) + (Pn f (t), v n(t) − v0(t))] dt = 0 Let n → ∞ to obtain

lim

n→∞

T

0 (∂Gvn (t), vn (t) − Pn v0(t)) dt = 0

We claim that limn→∞ T

0 (∂Gvn(t), v0(t) − Pn v0(t)) dt = 0 In fact, we have

|(∂Gv n(t), v0(t) − Pn v0(t))| ≤ |∂Gvn(t)||v0(t) − Pn v0(t)| −→ 0 as n −→ ∞

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for t ∈ [0, T ], and it follows from (2.6) and the boundedness of ∂G that |∂Gvn(t)||v0(t) − Pn v0(t)| is bounded

by some integrable function, thus by Lebesgue’s dominated convergence theorem it follows that

lim

n→∞

T

0 (∂Gvn (t), v0(t) − Pn v0(t)) dt = 0

Thus we have

lim

n→∞

T

However ∂G is a mapping of class (S+), and we claim that

lim inf

n→∞ (∂Gvn(t), vn(t) − v0(t)) ≥ 0 , t ∈ [0, T ]

Assume that it is false Then there exists t0∈ [0, T ] such that

lim inf

n→∞ (∂Gvn(t0), vn(t0 − v0(t0)) < 0 ,

but ∂G is a mapping of class (S+), so we have vn (t0 → v0(t0 , and this with the demi-continuity of ∂G implies

that

(∂Gvn (t0), vn(t0 − v0(t0)) −→ 0 as n → ∞ ,

which is a contradiction This and (2.7) imply that{(∂Gv n(t), vn(t)−v0(t))} ∞ n=1converges to zero in measure as

n → ∞ and so {(∂Gv n(t), vn(t) − v0(t))} ∞ n=1has a subsequence{(∂Gv n k (t), vn k (t) − v0(t))} ∞ k=1converging

to zero as k → ∞ for almost all t ∈ [0, T ] Since ∂G is a mapping of class (S+), we get vn k (t) → v0(t) for almost all t ∈ [0, T ].

From the demi-continuity of ∂G, we have ∂Gvn k (t) ∂Gv0(t) for almost all t ∈ [0, T ] It is obvious that the weak derivative v 0(t) = ∂Gv0(t) + f (t) Therefore, equation (E 2.2) has a weak solution.

3 An example

In this section, we give an application of our results to anti-periodic solutions for partial differential equations

Example 3.1 Let a : R → R be a function satisfying the following conditions:

(i) a is continuous and|a (x)| ≤ L |x| + C for x ∈ R, where L > 0, C > 0 are constants,

(ii) (a (x) − a (y))(x − y) ≥ α(x − y)2for x, y ∈ R, where α > 0 is a constant.

Consider the anti-periodic problem

⎧

⎪

u t(t, x) = a (ux(t, x))uxx(t, x) + (1 + x4) sin3t ,

u(t + π, x) = −u(t, x) ,

u(t, 0) = 0 , u(t, 1) = 0

(E 3.1)

for any (t, x) ∈ R × (0, 1) and t ∈ R We call u(t, x) as a generalized solution of (E 3.1) if u(t + π) = −u(t)

and

1

0 u t(t, x)v(x) dx = −

1

0 a (ux(t, x))v (x) +

1

0 (1 + x4) sin3tv(x) dx for all v(·) ∈ H1((0, 1)) and almost all t ∈ R.

Put

2

1

0 |a(u (x))|2dx for u(·) ∈ H01((0, 1))

and

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f (t, x) =

1 + x4

sin3t, (t, x) ∈ R × (0, 1)

Then it is easy to check that

(∂Gu, v) =

1

0 a (u (x))v (x) dx , u(·), v(·) ∈ H01((0, 1)) (3.1)

From assumption (i) and (3.1), we know that ∂G is continuous and bounded on H1((0, 1)) and it is obvious that

f (t + π, x) = −f (t, x) for any (t, x) ∈ R × (0, 1) Therefore, (E 3.1) is equivalent to

u (t) = −∂Gu(t) + f (t, ·) , t ∈ R ,

Next, we check that ∂G is a mapping of class (S+)on H1(Ω) Assume that un u0in H1((0, 1)) and

lim supn→∞ (∂Gun , u n − u0 ≤ 0 Then we have

lim sup

n→∞

1

0 a (u n (x))(u n (x) − u 0(x)) dx ≤ 0

From assumption (ii), we know that

lim sup

n→∞

1

0 α |u n (x) − u 0(x)|2dx ≤ 0 and thus un → u0 in H1((0, 1)), as desired From Theorem 2.4, we know that (E 3.2) has a solution and so

(E 3.1) has a solution

Acknowledgements The second author was supported from the Korea Research Foundation Grant (KRF-2003-002-C00018)

References

[1] A R Aftabizadeh, S Aizicovici, and N H Pavel, On a class of second-order anti-periodic boundary value problems,

J Math Anal Appl 171, 301–320 (1992).

[2] A R Aftabizadeh, S Aizicovici, and N H Pavel, Anti-periodic boundary value problems for higher order differential

equations in Hilbert spaces, Nonlinear Anal 18, 253–267 (1992).

[3] S Aizicovici and N H Pavel, Anti-periodic solutions to a class of nonlinear differential equations in Hilbert spaces,

J Funct Anal 99, 387–408 (1991).

[4] S Aizicovici, M McKibben, and S Reich, Anti-periodic solutions to monotone evolution equations with discontinuous

nonlinearities, Nonlinear Anal 43, 233–251 (2001).

[5] F E Browder, Nonlinear Operators and Nonlinear Equations of Evolution in Banach Spaces, Proc Symp Pure Math

18, Part 2 (Amer Math Soc., Providence, R.I., 1976)

[6] S S Chang, Y Q Chen, and B S Lee, Some existence theorems for differential inclusions in Hilbert spaces, Bull

Austral Math Soc 54, 317–327 (1996).

[7] S S Chang, Y Q Chen, K K Tan, and X Z Yuan, On the existence of periodic solutions for nonlinear evolutions in

Hilbert spaces, Nonlinear Anal 44, 1019–1029 (2001).

[8] Y Q Chen, Note on Massera’s theorem on anti-periodic solution, Advances in Math Sci and Appl 9, 125–128 (1999) [9] Y Q Chen and Y J Cho, Anti-periodic solutions for semilinear evolution equations, J Concrete and Appl Math 1,

113–124 (2003)

[10] Y Q Chen, Y J Cho, and L Yang, Periodic solutions for nonlinear evolution equations, Dynamic Cont Discr Impul

Systems 9, 581–598 (2002).

[11] Y Q Chen and J K Kim, Existence of periodic solutions for first-order evolution equations without coercivity, J Math

Anal Appl 282, 801–815 (2003).

[12] Y Q Chen, X D Wang, and H X Xu, Anti-periodic solutions for semilinear evolution equations, J Math Anal Appl

273, 627–636 (2002).

[13] A Haraux, Anti-periodic solutions of some nonlinear evolution equations, Manuscripta Math 63, 479–505 (1989).

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[14] H Okochi, On the existence of periodic solutions to nonlinear abstract parabolic equations, J Math Soc Japan 40,

541–553 (1988)

[15] H Okochi, On the existence of anti-periodic solutions to a nonlinear evolution equation associated with odd

subdiffer-ential operators, J Funct Anal 91, 246–258 (1990).

[16] H Okochi, On the existence of anti-periodic solutions to nonlinear parabolic equations in noncylindrical domains,

Nonlinear Anal 14, 771–783 (1990).

[17] D Pascali and S Sburlan, Nonlinear Mappings of Monotone Type, Noordhoff, Leyden, 1978

[18] W V Petryshyn, Antipodes theorems forA-proper mappings of the modified type (S) or (S)+and to mappings with theP mproperty, J Funct Anal 71, 165–211 (1971).

[19] P Souplet, Uniqueness and nonuniqueness results for the antiperiodic solutions of some second-order nonlinear

evolu-tion equaevolu-tions, Nonlinear Anal 26, 1511–1525 (1996).

[20] P Souplet, Optimal uniqueness condition for the antiperiodic solutions of some nonlinear parabolic equations, Nonlinear

Anal 32, 279–286 (1998).

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