We prove an existence result for T-periodic solutions to nonlinear evolution equations of the form x(t)+A(t.x(t))=f(t.x(t)). O<t<T.

Here VHV* is an evolution triple, A :I×V→V* is a uniformly monotone operator, and f :I×H→V* is a Caratheodory mapping which is Hölder continuous with respect to x in H and exponent 0<1. For illustration, an example of a quasi-linear parabolic differential equation is worked out in detail

P., Sattayatham

S., Tangmanee

Wei, Wei

We prove an existence result for T-periodic solutions to nonlinear evolution equations of the form x(t)+A(t.x(t))=f(t.x(t)). O<t<T.

Here VHV* is an evolution triple, A :I×V→V* is a uniformly monotone operator, and f :I×H→V* is a Caratheodory mapping which is Hölder continuous with respect to x in H and exponent 0<1. For illustration, an example of a quasi-linear parabolic differential equation is worked out in detail

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J. Math. Anal. Appl. 276 (2002) 98–108www.elsevier.com/locate/jmaaOn periodic solutions of nonlinear evolutionequations in Banach spaces✩P. Sattayatham,∗ S. Tangmanee, and Wei WeiSchool of Mathematics, Suranaree University of Technology, Nakhon Ratchasima, 30000, ThailandReceived 30 October 2000Submitted by J. DiestelAbstractWe prove an existence result forT -periodic solutions to nonlinear evolution equationsof the formẋ(t)+A(t, x(t))= f (t, x(t)), 0< t < T.HereV ↪→ H ↪→ V ∗ is an evolution triple,A : I × V → V ∗ is a uniformly monotoneoperator, andf : I ×H → V ∗ is a Caratheodory mapping which is Hölder continuous withrespect tox in H and exponent 0< α  1. For illustration, an example of a quasi-linearparabolic differential equation is worked out in detail. 2002 Elsevier Science (USA). All rights reserved.Keywords:Evolution equation; Time-periodic solution; Quasi-linear parabolic differential equation1. IntroductionIn this paper, we establish an existence result of periodic solutions for a classof nonlinear evolution equations in Banach spaces. Our approach will be basedon techniques and results of the theory of monotone operators and the Leray–Schauder fixed point theorem.✩ This work was supported by National Research Council, Thailand.* Corresponding author.E-mail address:pairote@ccs.sut.ac.th (P. Sattayatham).0022-247X/02/$ – see front matter 2002 Elsevier Science (USA). All rights reserved.PII: S0022-247X(02)00378-5P. Sattayatham et al. / J. Math. Anal. Appl. 276 (2002) 98–108 99The problem of existence of periodic solutions for nonlinear evolution equa-tions has been studied by several authors. However, most of the works concen-trated on semilinear systems. We refer to the works of Browder [2], Prüss [7], andBecker [1]. The first fully nonlinear existence results for the periodic problem,were obtained by Vrabie [9] and Hirano [4]. Vrabie assumes that the nonlineartime invariant operatorA monitoring the evolution equation is such thatA−λI ism-accretive for someλ > 0, while Hirano [4] considers time invariant nonlinearoperatorA which is a subdifferential of a convex function defined in a Hilbertspace. Moreover, Hu and Papageorgiou [5] consider evolution inclusions definedin an evolution triple of Hilbert spaces. They proved the existence of a periodicsolution for a problem with a Caratheodory multivalued perturbationF(t, x) de-fined onI ×H intoH .The time-dependent systems for the periodic problem were investigatedrecently by Kandilakis and Papageorgiou [6] and Shioji [8]. Kandilakis andPapageorgiou consider multivalued perturbations while Shioji consider single-valued perturbation. However, the assumptions on the monitoring operatorA ndon the perturbationF(t, x) of both papers imply that the operatorA + F is apseudo-monotone operator.In this paper, we also consider a time-dependent systems with a single-valued perturbation. Here, the perturbationf (t, x) is assumed to be continuousand defined onI × H with values inV ∗. Our assumptions on the monitoringoperatorA and on the perturbationf (t, x) do not imply thatA + f is pseudo-monotone.2. System descriptionThe mathematical setting of our problem is the following. LetH be a realseparable Hilbert space,V be a dense subspace ofH having structure of areflexive Banach space, with the continuous embeddingV ↪→ H ↪→ V ∗, whereV ∗ is the topological dual space ofV . The system model considered here is basedon this evolution triple. Let the embeddingV ↪→ H be compact.Let 〈x, y〉 denote the pairing of an elementx ∈ V ∗ and an elementy ∈ V . Ifx, y ∈H , then〈x, y〉 = (x, y), where(x, y) is the scalar product onH . The normin any Banach spaceX will be denoted by‖·‖X . LetT be a fixed positive constantandI = (0, T ). Letp,q  1 be such thatp−1 + q−1 = 1 and 2 p <+∞.We denoteLp(I,V ) by X. Then the dual space ofX is Lq(I,V ∗) and isdenoted byX∗. Forp,q satisfy the above conditions, it follows from reflexivityof V that bothX andX∗ are reflexive Banach spaces (see Zeidler [10, p. 411]).DefineWpq = {x: x ∈X, ẋ ∈X∗},100 P. Sattayatham et al. / J. Math. Anal. Appl. 276 (2002) 98–108where the derivative in this definition should be understood in the sense ofdistribution. Furnished with the norm‖x‖Wpq = ‖x‖X + ‖ẋ‖X∗ , the space(Wpq,‖ · ‖) becomes a Banach space which is clearly reflexive and separable.Moreover, Wpq embeds intoC(I,H) continuously (see Proposition 23.23of [10]). So every element inWpq has a representative inC(I ,H). Because ofthe embeddingV ↪→ H is compact, the embeddingWpq ↪→ Lp(I,H) is alsocompact (see Problem 23.13 of [10]). The pairing betweenX andX∗ is denotedby 〈〈 , 〉〉.Consider the following equation{ẋ(t)+A(t, x(t))= f (t, x(t)), t ∈ I ,x(0)= x(T ), (2.1)where the operatorsA : I × V → V ∗ andf : I × H → V ∗. By a solutionx ofproblem (2.1), we mean a functionx ∈ {x ∈Wpq : x(0)= x(T )} such that〈ẋ(t), v〉+ 〈A(t, x), v〉= 〈f (t, x), v〉for all v ∈ V and almost allt ∈ I .We need the following hypotheses on the data problem (2.1).(A1) A : I × V → V ∗ is an operator such that(1) t →A(t, x) is measurable;(2) For eacht ∈ I , the operatorA(t) :V → V ∗ is uniformly monotone andhemicontinuous, that is, there exists a constantC1  0 such that〈A(t, x1)−A(t, x2), x1 − x2〉 C1‖x1 − x2‖pV ,for all x1, x2 ∈ V,and the maps → 〈A(t, x + sz), y〉 is continuous on[0,1] for allx, y, z ∈ V ;(3) Growth condition: There exists a constantC2 > 0 and a nonnegativefunctiona1(·) ∈ Lq(I) such that∥∥A(t, x)∥∥V ∗  a1(t)+C2‖x‖p−1V , for all x ∈ V, a.e. onI ;(4) Coerciveness: There exists a constantC3 > 0 such that〈A(t, x), x〉 C3‖x‖PV , for all x ∈ V, a.e. onI.Without loss of generality, we can assume thatA(t,0)= 0 for all t ∈ I .(F1) f : I ×H → V ∗ is an operator such that(1) t → f (t, x) is measurable;(2) x → f (t, x) is continuous;(3) There exists a nonnegative functionh1(·) ∈ Lq(I) and a constantC4 > 0such that∥∥f (t, x)∥∥V ∗  h1(t)+C4‖x‖k−1H , for all x ∈ V, t ∈ I,where 1 k < p is constant;P. Sattayatham et al. / J. Math. Anal. Appl. 276 (2002) 98–108 101(4) f (t, x) is Hölder continuous with respect tox with exponent 0< α  1in H and uniformly int . That is, there is a constantL such that∥∥f (t, x1)− f (t, x2)∥∥V ∗  L‖x1 − x2‖αH ,for all x1, x2 ∈H and for allt ∈ I.It is convenient to rewrite system (2.1) as an operator equation inWpq(T )={x ∈Wpq : x(0)= x(T )}.Forx ∈X, we setA(x)(t)=A(t, x(t)), F (x)(t)= f (t, x(t)), t ∈ I.It follows from Theorem 30.A of Zeidler [10] that the operatorA :X → X∗ isbounded, uniformly monotone, hemicontinuous, and coercive. By using the sametechnique, one can show that the operatorF :Lp(I,H) → X∗ is bounded andsatisfies∥∥F(u)∥∥X∗ M1 +M2‖u‖(k−1)Lp(I,H), for all u ∈Lp(I,H).Lemma 1. Under assumption(F1), the operatorF :Lp(I,H) → X∗ is Höldercontinuous with exponentα, 0 < α  1, and F(xn) → F(x) in X∗ wheneverxnW→ x in Wpq .Proof. Let x1, x2 ∈ Lp(I,H). By hypothesis (4) of (F1) and Hölder inequality,we get∥∥F(x1)− F(x2)∥∥X∗=( T∫0∥∥f (t, x1(t))− f (t, x2(t))∥∥qV ∗ dt)1/qL( T∫0∥∥x1(t)− x2(t)∥∥qαH dt)1/qL( T∫0∥∥x1(t)− x2(t)∥∥pH dt)α/p( T∫011−p/qα dt)α/(qα−1)L′‖x1 − x2‖αLp(I,H)for some constantL′. This proves thatF is Hölder continuous with exponentα inLp(I,H). HenceF is continuous onLp(I,H).102 P. Sattayatham et al. / J. Math. Anal. Appl. 276 (2002) 98–108Since the embeddingV ↪→ H is compact, the embeddingWpq ↪→ Lp(I,H)is compact. That is,xn → x in Lp(I,H) whenever xn W→ x in Wpq.By using the above relation and the continuity ofF , we haveF(xn)→ F(x) in X∗ whenever xn W→ x in Wpq. ✷Moreover, we observe that the original problem (2.1) is equivalent to thefollowing operator equation:{ẋ +A(x)= F(x),x ∈Wpq(T ). (2.2)Lemma 2. Assume(A1) and (F1)are satisfied. Then the setS1= {x ∈Wpq(T ) | ẋ +A(x)= σF(x), for someσ ∈ [0,1]} (2.3)is bounded inWpq . Moreover, there exists a positive constantM such that∥∥A(x)∥∥X∗ M and maxt∈I∥∥x(t)∥∥HMfor all x ∈ S1.Proof. Let x ∈ S1, then we have〈〈ẋ, x〉〉 + 〈〈A(x), x〉〉= 〈〈σF(x), x〉〉.SinceA is coercive (hypothesis (A1)) thenC3‖x‖pX 〈〈σF(x), x〉〉− 〈〈ẋ, x〉〉. (2.4)By using integration by part and the relationx(0) = x(T ), one can see that thesecond term on the right-hand side of (2.4) is equal to zero. Hence, inequality (2.4)reduces toC3‖x‖pX  σ( T∫0∥∥f (t, x)∥∥qV ∗ dt)1/q( T∫0‖x‖pV dt)1/p σ( T∫0(h1(t)+ ‖x‖k−1H)qdt)1/q‖x‖X α1‖x‖X + α2‖x‖kX (2.5)for some constantsα1 > 0 andα2 > 0. Since 1 k < p, thus, by virtue of theinequality (2.5), we can find a constantM1 > 0 such that‖x‖X M1 (2.6)for all x ∈ S1.P. Sattayatham et al. / J. Math. Anal. Appl. 276 (2002) 98–108 103From (2.6), the boundedness of operatorsA and F , and the continuousembeddingX ↪→ Lp(I,H), we obtain∥∥A(x)∥∥X∗ M2 and∥∥F(x)∥∥X∗ M2, (2.7)for some constantM2 > 0 and allx ∈ S1. Therefore,‖ẋ‖X∗ ∥∥A(x)∥∥X∗ +∥∥F(x)∥∥X∗  2M2, for all x ∈ S1. (2.8)It follows from (2.6) and (2.8) that‖x‖Wpq M1 + 2M2.Hence, S1 is a bounded subset ofWpq .Finally, we note thatWpq ↪→ C(I,H); then‖x‖C(I,H)  α‖x‖Wpq ,and hencemaxt∈I∥∥x(t)∥∥HM3for some positive constantsα, M3, and for all x ∈ S1. ChoosingM =max(M2,M3), the assertion follows. ✷Theorem 3. Under assumptions(A1) and (F1), Eq. (2.2)has a solutionx ∈Wpq(T ).Proof. We denoteS = LP (I,H). DefineG :S × [0,1] → S by G(u,σ) = v,wherev is the solution of the following problem:{v̇ +A(v)= σF(u),v(0)= v(T ). (2.9)SinceA is uniformly monotone, thenA is strictly monotone. By Theorem 32.Dof [10], for anyu ∈ S, problem (2.9) has a unique solutionv ∈Wpq ⊂ S. SoG iswell defined.(1) We now assert thatG :S × [0,1] → S is continuous and compact.In fact, for any sequence(un, σn)⊂ S × [0,1] such that(un, σn)→ (u,σ ) in S × [0,1],we denotevn to be the solution of the problem{v̇n +A(vn)= σnF(un),vn(0)= vn(T ),andv to be the solution of the problem{v̇ +A(v)= σF(u),v(0)= v(T ).104 P. Sattayatham et al. / J. Math. Anal. Appl. 276 (2002) 98–108Therefore,〈〈v̇n − v̇, vn − v〉〉 +〈〈A(vn)−A(v), vn − v〉〉= 〈〈σnF(un)− σF(u), vn − v〉〉. (2.10)Using integration by parts and the monotonicity of the operatorA, we obtainfrom (2.10) that12(∥∥vn(T )− v(T )∥∥2H − ∥∥vn(0)− v(0)∥∥2H )+C1‖vn − v‖pX〈〈σnF(un)− σF(u), vn − v〉〉T∫0∥∥σnf (t, un)− σf (t, u)∥∥V ∗∥∥vn(t)− v(t)∥∥V dt εp‖vn − v‖pX +ε−q/pq∥∥σnF(un)− σF(u)∥∥qX∗,for all ε > 0. (2.11)Since F(u) is Hölder continuous with exponentα, F :Lp(I,H) → X∗ isbounded, by choosingε in (2.11) small enough, thenM1‖vn − v‖pX M2∥∥σnF(un)− σF(u)∥∥qX∗=M2∥∥σnF(un)− σnF(u)+ σnF(u)− σF(u)∥∥qX∗M3(∥∥F(un)− F(u)∥∥qX∗ + |σn − σ |q∥∥F(u)∥∥qX∗)M4‖un − u‖qαS +M5|σn − σ |q(1+ ‖u‖(k−1)S)q,for some positive constantsM1,M2,M3,M4, andM5. Noting that the embeddingLp(I,V ) ↪→ Lp(I,H) is continuous, we have‖vn − v‖S M ′(‖un − u‖qα/pS + |σn − σ |q/p(1+ ‖u‖(k−1)S )q/p),for some constantM ′ > 0. Hence,G :S × [0,1] → S is continuous.Let v be the solution of problem (2.9) with‖u‖S < b1 for someσ ∈ [0,1],whereb1 > 0 is a constant. Similar to the proof of Lemma 2, one can show thatthere exists constantb2 > 0 such that‖v‖Wpq  b2.Hence,G maps bounded sets inS × [0,1] into bounded sets inWpq . Since theembeddingWpq ↪→ S is compact,G :S × [0,1] → S is also compact.(2) Next, we must show that the set{u ∈ S | u=G(u,σ) for some 0 σ  1}is bounded inS.P. Sattayatham et al. / J. Math. Anal. Appl. 276 (2002) 98–108 105Assume thatu ∈ S andu=G(u,σ). Thenu ∈Wpq and satisfies the problem{u̇+A(u)= σF(u),u(0)= u(T ).By Lemma 2, we get‖u‖Wpq  M. Again, since the embeddingWpq ↪→ S iscompact, we get‖u‖S  Bfor some constantB > 0.(3)G(u,0)= 0, for anyu ∈ S,For anyu ∈ S, setG(u,0)= v0, wherev0 satisfies{v̇0 +A(v0)= 0,v0(0)= v0(T ). (2.12)By uniqueness of the solution of Eq. (2.12), we get fromA(0)= 0 thatv0 = 0, in Wpq.Since the imbeddingWpq ↪→ S is continuous, we getv0 = 0, in S,that is,G(u,0)= 0, for anyu ∈ S.(4) Applying Leray–Schauder fixed point theorem (see [3, p. 231]) in thespaceS, there is one fixed pointx ∈ S such thatx =G(x,1)andx ∈ S∩Wpq . That is,x is a solution of problem (2.2). Since the problem (2.1)is equivalent to the problem (2.2), there exists a periodic solution for nonlinearevolution equation (2.1). ✷3. ExamplesIn this section, to illustrate the applicability of our work, we prove the existenceof a periodic solution for a quasi-linear parabolic partial differential equations oforder 2m.LetΩ be a bounded domain inRn with smooth boundary∂Ω , QT = (0, T )×Ω , 0< T <∞. Let α = (α1, α2, . . . , αn) be a multi-index with{αi} nonnegativeintegers and|α| =∑ni αi . Supposep  2 andq = p/(p − 1), Wm,p(Ω) denotesthe standard Sobolev space with the usual norm:‖ϕ‖Wm,p =( ∑|α|m‖Dαϕ‖pLp(Ω))1/p, m= 0,1,2, . . . .106 P. Sattayatham et al. / J. Math. Anal. Appl. 276 (2002) 98–108Let Wm,p0 (Ω) = {ϕ ∈ Wm,p | Dβϕ|∂Ω = 0, |β|  m− 1}. It is well known thatC∞0 (Ω) ↪→ Wm,p0 (Ω) ↪→ L2(Ω) ↪→ W−m,p(Ω) and the embeddingWm,p0 (Ω)↪→ L2(Ω) is compact. DenoteV ≡ Wm,p0 (Ω) and H ≡ L2(Ω); then V ∗ ≡W−m,q(Ω).Example 1. We consider the time-periodic solutions for 2m-order quasi-linearparabolic equations∂∂tx(t, z)+∑|α|m(−1)|α|DαAα(t, z, η(x)(t, z))=∑|α|m(−1)|α|Dαfα(t, z, x(t, z)) onQT ,Dβx(t, z)= 0 on[0, T ] × ∂Ω for all β: |β| m− 1,x(0, z)= x(T , z) onΩ,(3.1)whereη(x)≡ {(Dγ x), |γ | m} andM = (n+m)!/(n!m!).We need the following hypotheses on the data of (3.1):(A′) The functionsAα(|α| m) :QT ×RM → R are functions such that(1) (t, z) → Aα(t, z, η) is measurable onQT for η ∈ RM , η → Aα(t, z, η)is continuous onRM for almost all(t, z) ∈QT ;(2) |Aα(t, z, η)|  a1(t, z)+ c1(z)∑|γ |m |ηγ |p−1 with a1(· , ·) ∈ Lq(QT )andc1(·) ∈ L∞(Ω) for almost allt ∈ (0, T );(3)∑|α|m(Aα(t, z, η)−Aα(t, z, η̃))(ηα − η̃α) c2(z)∑|γ |m |ηγ − η̃γ |pwith c2(·) ∈L∞(Ω) for almost allt ∈ (0, T );(4) Aα(t, z,0)= 0 for all (t, z) ∈QT .(F′) fα :QT ×R → R are functions such that(1) (t, z) → fα(t, z, x) is measurable onQT for x ∈ R, x → fα(t, z, x) iscontinuous onR for almost all(t, z) ∈QT ;(2) |fα(t, z, x)|  a2(t, z)+c4(z)|x|k−1 with 1  k < p, a2(· , ·) ∈Lq(QT ),andc4(·) ∈ L∞(Ω) for almost allt ∈ (0, T );(3) fα(t, z, x) is Hölder continuous with respectx and exponent 0< α  1;that is, there is a constantL∣∣fα(t, z, x1)− fα(t, z, x2)∣∣ L|x1 − x2|αfor anyx1, x2 ∈ R, (t, z) ∈QT .Forx, y ∈Wm,p0 , t ∈ I , we definea(t, x, y)=∫Ω∑|α|mAα(t, z, η(x)(t, z))Dαy dz.P. Sattayatham et al. / J. Math. Anal. Appl. 276 (2002) 98–108 107It is not difficult to verify that under the above assumption(A′), for eachx ∈ Vandt ∈ I , y → a(t, x, y) is a continuous linear form onV . Hence there exists anoperatorA : I × V → V ∗ such that〈A(t, x), y〉= a(t, x, y).Under the given assumption(A′), it is easy to verify thatA satisfies ourhypotheses (A1) of Section 2.Next, by using the time-varying Dirichlet formf : I ×H ×V →R defined byf (t, x, y)=∫Ω∑|β|mfβ(t, z, x(t, z))Dβy(z) dz.Then y → f (t, x, y) is a continuous linear form onV . Hence there exists anoperatorF : I ×H → V ∗ such thatf (t, x, y)= 〈F(t, x), y〉.Under the given hypotheses(F′), we obtain thatF satisfies our hypotheses (F1)of Section 2.Using the operatorsA andF as defined above, Eq. (3.1) can be written in anabstract form:{ẋ +A(t, x)= F(t, x),x(0)= x(T ). (3.2)So applying Theorem 3, we get the following theorem.Theorem 4. If hypotheses(A′) and(F′) hold, then there exists a periodic solutionx ∈ Lp(I,Wm,p0 (Ω)), ∂x/∂t ∈ Lq(I,W−m,q (Ω)) of Eq. (3.1).References[1] R.R. Becker, Periodic solutions of semilinear equations of evolution of compact type, J. Math.Anal. Appl. 82 (1981) 33–48.[2] F. Browder, Existence of periodic solutions for nonlinear equations of evolution, Proc. Nat. Acad.Sci. USA 53 (1965) 1110–1113.[3] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York, 1980.[4] N. Hirano, Existence of periodic solutions for nonlinear evolution equations in Hilbert spaces,Proc. Amer. Math. Soc. 120 (1994) 185–192.[5] S. Hu, N.S. Papageorgiou, On the existence of periodic solutions for a class of nonlinear evolutioninclusions, Boll. Un. Mat. Ital. 7B (1993) 591–605.[6] D. Kandilakis, N.S. Papageorgiou, Periodic solutions for nonlinear evolution inclusions, Arch.Math. (Brno) 32 (1996) 195–209.[7] J. Prüss, Periodic solutions of semilinear evolution equations, Nonlinear Anal. 3 (1979) 221–235.[8] N. Shioji, Existence of periodic solutions for nonlinear evolution equations with pseudo-monotone operators, Proc. Amer. Math. Soc. 125 (1997) 2921–2929.108 P. Sattayatham et al. / J. Math. Anal. Appl. 276 (2002) 98–108[9] I. Vrabie, Periodic solutions for nonlinear evolution equations in Banach space, Proc. Amer.Math. Soc. 109 (1990) 653–661.[10] E. Zeidler, Nonlinear Functional Analysis and Its Applications, II, Springer-Verlag, New York,1990.

On periodic solutions of nonlinear evolution equations in Banach spaces

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On periodic solutions of nonlinear evolution equations in Banach spaces

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