Skip to content
BY 4.0 license Open Access Published by De Gruyter February 8, 2018

Solving Becker’s Problem on Periodic Solutions of Parabolic Evolution Equations

Giovanni Vidossich

Abstract

We present existence and multiplicity theorems for periodic mild solutions to parabolic evolution equations. Their peculiarity is a link with the spectrum of the generator of the semigroup rather than with the spectrum of the linearized periodic BVP for the evolution equation. They provide a positive solution to the open problem risen by Becker [3], they extend some results of Castro and Lazer [5] from scalar to systems of parabolic equations, and they are new even for finite-dimensional ODEs.

MSC 2010: 47D60; 35K55; 34B15

1 Introduction

This paper is devoted to periodic solutions of evolution equations of the type

(1.1) u = A u + B ( t , u ) u + f ( t , u ) ,

where A is a symmetric linear operator generating a compact semigroup on a given Hilbert space H, each B(t,u) is a bounded symmetric linear operator on H, and B and f are p-periodic in t.

Becker rose in [3, §4] the following problem: Does (1.1) have a p-periodic mild solution when there exist two successive eigenvalues λ,λ′′ of A such that

λ + λ ′′ 2 I + B ( t , ) < β ,

where β is less than the distance from (λ+λ′′)/2 to the nearest eigenvalue of the periodic BVP for u=Au?

Partial answers to this question have been given by Becker [3], Sager [10] and Vidossich [12]. The idea to involve eigenvalues of A to get periodic solutions of nonlinear parabolic equations is not new; it has been firstly considered in Brézis–Nirenberg [4, §V.1] for the resonance case and later in Castro–Lazer [5] for the first eigenvalue.

Lemma 2.2 below shows that the spectrum of the periodic BVP for u=Au is -σ(A). Therefore, Becker’s problem can be rephrased as the problem of finding periodic mild solutions of (1.1) when the spectrum of each B(t,u) is included between two consecutive members of -σ(A). This is exactly the version that is considered here. By a detailed computations of the Leray–Schauder degree made for the perturbation of the identity by the completely continuous operator introduced by Ward [14] in connection with the periodic problem of evolution equations, we prove two existence theorems (one for higher eigenvalues and one for the first one) and a multiplicity theorem. These results provide the solution to Becker’s problem and, in addition, have two original applications:

  1. They provide extensions to systems of parabolic PDEs of some theorems proved in Castro–Lazer [5] for scalar parabolic equations by means of the sub-supersolution method.

  2. In the finite-dimensional case, they answer the questions that follow. If H=N and A is a symmetric square matrix, it is well known that the ODE u=Au+f(t,x) has a p-periodic solution when u=Au has only the trivial one and

    lim u f ( t , u ) u = 0 .

    Now, what happens when this limit is not 0 and it is between two consecutive eigenvalues of A? Also, what happens when the limit does not exists at all and the spectrum of the linearization at infinity ranges in a closed interval contained in σ(A)?

The paper is divided in two parts. The first one is devoted to general results and the other to their applications to parabolic systems. Both parts start with a collection of specific standing notations and assumptions in order to avoid tedious repetitions and to focus on the key points of the statements.

2 General Results

This part contains first a series of lemmas that will be used later on for the computation of the topological degree. Then, using these lemmas, we present and prove the main theorems.

2.1 Notations, Standing Assumptions and Preliminaries

We shall use the following notations:

  1. H := ( H , ( | ) , ) is a real Hilbert space.

  2. A 1 A 2 denotes the standard order relation in the space of linear symmetric operators HH (whose cone of positive elements is made of positive semi-definite operators). In other words,

    A 1 A 2 ( A 1 x | x ) ( A 2 x | x ) for all  x H .

  3. A is a symmetric, densely defined, linear operator on H which generates a compact semigroup in H.

  4. λ n denotes the nth eigenvalue of A when n1 or + when n=0, assuming that the eigenvalues of A are ordered in the decreasing way, counting multiplicities, i.e.,

    λ n λ n - 1 λ 1 < λ 0 = + .

  5. ( e n ) n is a Hilbert basis of H consisting of the eigenvectors of A, with en corresponding to λn.

  6. E n is the subspace of H spanned by the union of the eigenspaces corresponding to λ1,,λn when n1 or En={0} when n=0. Thus, dim(En)=n when λn<λn+1.

  7. P n is the orthogonal projection HEn.

  8. Q n := I - P n .

  9. S := ( S ( t ) ) t 0 is the compact semigroup generated by A in H.

  10. X := ( X , X ) is a Banach space such that

    1. X is a vector subspace of H and the inclusion XH is continuous,

    2. ( S ( t ) | X ) t 0 is a compact semigroup on X.

  11. ( Z ) denotes the space of bounded linear operators ZZ when Z is a Banach space.

  12. I is the identity mapping.

Further assumptions on these objects are mentioned when needed.

The appearance of the Banach space XH is motivated by the following remark. The statement

lim u L 2 0 f ( u ) - B 0 u L 2 u L 2 = 0 ,

with B0(L2(Ω)), implies that f:L2(Ω)L2(Ω) is Fréchet differentiable at the origin and consequently f is an affine map by [1, Proposition 2.8]. This would contradict our wish to consider genuine nonlinearities in the multiplicity Theorem 2.13 below.

To simplify references and readings, the next lemma collects some results from Ward [14] that are essential below (they are deduced from Lemmas 2.3 and 2.4, and a portion of the proof of Corollary 3.3 in [14]).

Lemma 2.1.

Let Y be a Banach space, let T:=(T(t))t0 be a compact semigroup on it with generator C and let g:R+×YY be a Carathéodory map which is p-periodic in t and satisfies the condition

g ( t , x ) h ( t ) + const x for a.e.  t and all  x ,

with hL1([0,p]). If L:=I-T(p) is an invertible linear operator YY, then for any mild solution u of

u = C u + g ( t , u ) ,

the following statements are equivalent:

  1. u is p -periodic,

  2. u ( 0 ) = u ( p ) ,

  3. for every t [ 0 , p ] , we have

    u ( t ) = T ( t ) L - 1 0 p T ( p - s ) g ( s , u ( s ) ) 𝑑 s + 0 t T ( t - s ) g ( s , u ( s ) ) 𝑑 s .

In addition, the operator F:C0([0,p],Y)C0([0,p],Y), defined by

F ( u ) ( t ) := T ( t ) L - 1 0 p T ( p - s ) g ( s , u ( s ) ) 𝑑 s + 0 t T ( t - s ) g ( s , u ( s ) ) 𝑑 s ,

is completely continuous.

2.2 Technical Lemmas

This subsection contains a series of lemmas needed for the computation of the topological degree used in the proofs of the theorems below.

The first lemma clarifies Becker’s question, justifying Theorem 2.8 below as an answer. In addition, this will be used in the proof of Theorem 2.13 below, and it shows that u=Auhas a non-trivial periodic solution if and only if λ=0 is an eigenvalue of A. This lemma is nothing else than the infinite-dimensional version of the well-known proposition saying that a linear homogeneous ODE in N has a non-trivial periodic solution if and only if 1 is a Floquet multiplier.

Lemma 2.2.

Let CL(H) be symmetric. The number λ is an eigenvalue of the BVP

(2.1) { u = ( A + C ) u + λ u , u is  p -periodic,

if and only if -λσ(A+C). In addition, the multiplicity of -λ as eigenvalue of A+C equals the multiplicity of λ as eigenvalue of (2.1).

Of course, we consider only mild solutions to (2.1).

Proof.

If -λσ(A+C), then the eigenvectors of -λ are the constant values of non-trivial periodic solutions to the evolution equation in (2.1), so that λ is an eigenvalue of (2.1). Consequently, the multiplicity of -λ as eigenvalue of A+C is less than or equal to the multiplicity of λ as eigenvalue of (2.1).

To prove the converse, we need first to state some auxiliary results. By [9, Theorem 4.2], A+C is the generator of a compact semigroup T:=(T(t))t0 in H. Each resolvent operator Rμ of A+C is compact by [9, Theorem 3.3], and so its eigenspaces have finite dimension and its eigenvalues form a sequence. In addition, Rμ is symmetric by the symmetry of A+C. Since Rμ has dense range (as A+C has dense domain by virtue of the Hille–Yosida theorem), the union of its eigenspaces is dense in H. Since A+C and Rμ have the same eigenvectors while their eigenvalues are in a one-to-one correspondence, we deduce that the eigenvalues of A+C are countable, and the corresponding eigenspaces are finite-dimensional and their union is dense in H. Consequently, the following setting is allowed:

  1. λ ~ n is the nth eigenvalue of A+C, assuming that they are in a decreasing order counting multiplicities.

  2. E ~ n is the subspace of H spanned by the union of the eigenspaces corresponding to λ~1,,λ~n.

  3. P ~ n is the orthogonal projection of HE~n.

  4. ( e ~ n ) n is a Hilbert basis of H consisting of eigenvectors of A+C, with e~n corresponding to λ~n.

In addition, we have P~nI pointwise.

With these results in mind, let u be a non-trivial mild solution of (2.1). Since the evolution equation is linear, x:=u(0)0. Consider two sequences (vn)n,(wn)n satisfying the finite-dimensional Cauchy problems

{ v n = ( A + C ) P ~ n v n , v n ( 0 ) = P ~ n x and { w n = ( A + C + λ ) P ~ n w n , w n ( 0 ) = P ~ n x

in E~n, n1. By [13, Lemma 2] and the uniqueness of mild solutions to linear evolution equations, we have vnT()x and wnu uniformly on [0,p]. Since vn and wn are continuously differentiable, and linear Cauchy problems enjoy uniqueness of solutions, a direct computation shows that

w n ( t ) = e λ t v n ( t ) , n 1 ,  0 t p .

Consequently,

u ( t ) = e λ t T ( t ) x , 0 t p .

Therefore, x=u(p)=eλpT(p)x. In the Fourier series of the eigenfunctions of A+C, the identity x=eλpT(p)x reads

n x ^ n e ~ n = e λ p n x ^ n e λ ~ n p e ~ n ,

by virtue of [2, formula (4.4.6)]. Therefore,

x ^ n = x ^ n e ( λ ~ n + λ ) p , n 1 ,

by the uniqueness of Fourier coefficients. Since x0, there exists n0 such that x^n00. Consequently, e(λ~n0+λ)p=1 and so λ~n0+λ=0, hence -λσ(A+C). Since x^n0 if and only if λ~n=-λ, hence if and only if λ~n=λ~n0, we have that x belongs to the eigenspace of -λ as eigenvalue of A+B. Therefore, the multiplicity of λ as eigenvalue of (2.1) is less than or equal to the multiplicity of -λ as eigenvalue of A+C. Combining this with the above, we conclude that λ and -λ have the same multiplicity as eigenvalues of (2.1) and A+C, respectively. ∎

Lemma 2.3.

Assume that CL1([0,p],L(H)), μ,νR and n0 satisfy the following conditions:

  1. C ( t ) is symmetric for a.e. t,

  2. - λ n < μ ν < - λ n + 1 ,

  3. μ I C ( t ) ν I for a.e. t.

Then

( A x + C ( t ) x | Q n x - P n x ) max { - λ n - μ , λ n + 1 + ν } x 2

for a.e. t and all xm>nEm{0}.

Proof.

Fix m>n and xEm{0}. We distinguish two cases. Case 1: n>0. Let x^k be the Fourier coefficient of x corresponding to ek. In view of the identities

x = P n x + Q n x , P n x = k n x ^ k e k , Q n x = n < k m x ^ k e k

and the continuity of A|Em, for a.e. t we have

( A x + C ( t ) x | Q n x - P n x ) = ( A P n x + A Q n x + C ( t ) P n x + C ( t ) Q n x | Q n x - P n x )
= ( A P n x | Q n x ) - ( A P n x | P n x ) + ( A Q n x | Q n x ) - ( A Q n x | P n x )
+ ( C ( t ) P n x | Q n x ) - ( C ( t ) P n x | P n x ) + ( C ( t ) Q n x | Q n x ) - ( C ( t ) Q n x | P n x )
= - ( A P n x | P n x ) + ( A Q n x | Q n x ) - ( C ( t ) P n x | P n x ) + ( C ( t ) Q n x | Q n x )
(by the symmetry of  A  and  C ( t ) )
= - k n λ k x ^ k 2 + n < k m λ k x ^ k 2 - γ x ( t ) k n x ^ k 2 + γ x ′′ ( t ) n < k m x ^ k 2
(where  γ x ( t )  and  γ x ′′ ( t )  are suitable numbers between  μ  and  ν )
( - λ n - γ x ( t ) ) k n x ^ k 2 + ( λ n + 1 + γ x ′′ ( t ) ) n < k m x ^ k 2
(as the sequence of the  λ k ’s is decreasing)
max { - λ n - μ , λ n + 1 + ν } k n λ k x ^ k 2 + max { - λ n - μ , λ n + 1 + ν } n < k m x ^ k 2
= max { - λ n - μ , λ n + 1 + ν } x 2 (since  x E m ) .

Case 2: n=0. In this case, Qn-Pn=Q0-P0=I and so

( A x + C ( t ) x | Q 0 x - P 0 x ) = ( A x | x ) + ( C ( t ) x | x ) λ 1 x 2 + γ x ( t ) x 2

for a suitable γx(t)[μ,ν]. Hence, the conclusion is the same as in the previous case. ∎

Lemma 2.4.

If P:HH is a symmetric projection, then the following inequalities hold for every x,yH:

2 ( x - y | - P x ) P y 2 - P x 2 , 2 ( x - y | P y ) P x 2 - P y 2 .

Proof.

We have

0 P y - P x 2 = P y 2 + P x 2 - 2 ( P y | P x ) ± P x 2
= 2 ( P x - P y | P x ) + P y 2 - P x 2
= 2 ( x - y | P x ) + P y 2 - P x 2     (by the symmetry of  P  and  P 2 = P ),

which implies the first inequality. The other inequality follows similarly from

0 P x - P y 2 = P x 2 + P y 2 - 2 ( P x | P y ) ± P y 2 .

Note that there is no periodicity hypothesis in the next lemma (although its conclusion is related to periodic solutions by virtue of Lemma 2.1).

Lemma 2.5.

Let CL1([0,p],L(H)) be symmetric and let σ(t,x) be the value at t of the unique mild solution to the linear Cauchy problem

{ u = A u + C ( t ) u , u ( 0 ) = x .

If μ,νR and n0 satisfy the conditions

- λ n < μ ν < - λ n + 1 , μ I C ( t ) ν I for a.e.  t 0 ,

then σ(t,x)x for every x0 and t>0.

Proof.

Fix x0 and t>0. For each m>n, let um be the unique mild solution to the following linear Cauchy problem in Em:

{ u = A P m u + P m C ( t ) P m u , u ( 0 ) = P m x .

Since Em is invariant by A, and PmI pointwise, [13, Lemma 2] implies that

(2.2) lim m u m = σ ( , x )

uniformly on compacta. In addition, we have

u m = A u m + P m C ( t ) u m .

Being a solution to a finite-dimensional ODE, um is absolutely continuous, hence

d d s P n u m ( s ) = P n u m ( s ) = P n ( A u m ( s ) + P m C ( s ) u m ( s ) )

a.e. on [0,p]. Multiplying this identity by Pnum(s) in the inner product of H and integrating the result, we get (by the symmetry of Pn and the fact that Pn2=Pn)

0 = 0 t 1 2 d d s P n u m ( s ) 2 d s - 0 t ( P n ( A u m ( s ) + P m C ( s ) u m ( s ) ) | P n u m ( s ) ) d s
= 1 2 P n u m ( t ) 2 - 1 2 P n u m ( 0 ) 2 - 0 t ( A u m ( s ) + P m C ( s ) u m ( s ) | P n u m ( s ) ) d s
= 1 2 P n u m ( t ) 2 - 1 2 P n x 2 - 0 t ( A u m ( s ) + P m C ( s ) u m ( s ) | P n u m ( s ) ) d s ,

as um(0)=Pmx and PnPm=Pn. This (together with Lemma 2.4 and the fact that PnPm=Pn) is used to deduce the following inequality:

( u m ( t ) - P m x | - P n u m ( t ) ) + 0 t ( A u m ( s ) + P m C ( s ) u m ( s ) | P n u m ( s ) ) d s
1 2 P n x 2 - 1 2 P n u m ( t ) 2 + 0 t ( A u m ( s ) + P m C ( s ) u ( s ) | P n u m ( s ) ) d s = 0 .

Similarly (by replacing Pn with Qn in the above arguments), we get

( u m ( t ) - P m x | Q n P m x ) - 0 t ( A u m ( s ) + P m C ( s ) u m ( s ) | Q n u m ( s ) ) d s
1 2 Q n u m ( t ) 2 - 1 2 Q n P m x 2 - 0 t ( A u m ( s ) + P m C ( s ) u m ( s ) | Q n u m ( s ) ) d s = 0 .

Adding side-by-side the two inequalities yields

( u m ( t ) - P m x | Q n P m x - P n u m ( t ) ) 0 t ( A u m ( s ) + P m C ( s ) u m ( s ) | Q n u m ( s ) - P n u m ( s ) ) d s .

By virtue of the uniqueness of mild solutions to linear finite-dimensional Cauchy problems, we have um(s)0 for every s and every m large enough so that Pmx0. In addition, we have

( P m C ( s ) u m ( s ) | Q n u m ( s ) - P n u m ( s ) ) = ( C ( s ) u m ( s ) | P m Q n u m ( s ) - P m P n u m ( s ) )
= ( C ( s ) u m ( s ) | Q n u m ( s ) - P n u m ( s ) )     (as  u m ( s ) E m ).

Substituting this in the previous inequality and using Lemma 2.3 yields

( u m ( t ) - P m x | Q n P m x - P n u m ( t ) ) max { - λ n - μ , λ n + 1 + ν } 0 t u m ( s ) 2 d s

for m sufficiently large. Taking the limit as m, we get

( σ ( t , x ) - x | Q n x - P n σ ( t , x ) ) max { - λ n - μ , λ n + 1 + ν } 0 t σ ( s , x ) 2 d s ,

by virtue of (2.2), the fact that Pmxx and the continuity of Pn and Qn. Since σ(0,x)0 and σ(,x) is continuous by (2.2), the right-hand side of this inequality is negative. Thus, σ(t,x)x, as desired. ∎

Lemma 2.6.

Let B:R+×XL(X) and f:R+×XX be p-periodic in t and assume that they satisfy the Carathéodory conditions, with B uniformly bounded and

f ( t , x ) h ( t ) + const x for a.e.  t and all  x ,

where hL1([0,p]). If

lim x X f ( t , x ) x X = 0

uniformly in t and there exist μ,ν,ρR and nN such that

  1. - λ n < μ ν < - λ n + 1 ,

  2. B ( t , x ) is symmetric for a.e. t and all x satisfying x X ρ ,

  3. μ I B ( t , x ) ν I for a.e. t and all x satisfying x X ρ ,

then there exists an upper bound in C0([0,p],X) for the p-periodic mild solutions of

u = A u + τ { B ( t , u ) u + f ( t , u ) } + ( 1 - τ ) μ u ,

which is independent of τ[0,1].

Proof.

Suppose, by contradiction, that the conclusion of the lemma is false. Then there exist τk[0,1] and ukC0([0,p],X) such that τkτ for a suitable τ, uk and uk is a p-periodic mild solution to

u k = A u k + τ k { B ( t , u k ) u k + f ( t , u k ) } + ( 1 - τ k ) μ u k .

Setting

z k := u k u k , B k ( t ) := { B ( t , u k ( t ) ) if  u k ( t ) X > ρ , μ I if  u k ( t ) X ρ ,
g k ( t ) := f ( t , u k ( t ) ) u k , f k ( t ) := { 0 if  u k ( t ) X > ρ , - μ z k ( t ) + B ( t , u k ( t ) ) z k ( t ) if  u k ( t ) X ρ ,

the above equation can be rewritten as

(2.3) z k = ( A + μ I ) z k + τ k { B k ( t ) z k + f k ( t ) + g k ( t ) - μ z k } .

This form will allow us to use Lemma 2.1. The operator A+μI is the generator of a compact semigroup T:=(T(t))t0 on X, by virtue of [9, Theorem 4.2]. Set L:=I-T(p). By virtue of the previous lemma, ker(L)={0}, and so L has a bounded inverse by Fredholm’s alternative. Then, from Lemma 2.1 and the above equation, we get

z k ( t ) = T ( t ) L - 1 0 p T ( p - s ) τ k { B k ( s ) z k ( s ) + f k ( s ) + g k ( s ) - μ z k ( s ) } 𝑑 s
(2.4) + 0 t T ( t - s ) τ k { B k ( s ) z k ( s ) + f k ( s ) + g k ( s ) - μ z k ( s ) } 𝑑 s .

Since B has bounded range in (X), the sequence (Bk)k is bounded in L2([0,p],(H)). Then, passing to a subsequence if necessary, we can assume that BkBweakly in L2([0,p],(H)). A well-known theorem of Mazur guarantees that a sequence of convex combinations of the Bk’s converges strongly to B in L2([0,p],(H)). Consequently,

μ I B ( t ) ν I for a.e.  t .

Since B has bounded range in (X), we have

lim k g k ( t ) = 0 = lim k f k ( t ) in  C 0 ( [ 0 , p ] , X )

and, in addition, the quantities inside the integrals in (2.4) are uniformly bounded in X with respect to k. Consequently, (zk)k has a convergent subsequence in C0([0,p],X), by virtue of the compactness on bounded sets of the operator on the right-hand side of (2.4) granted by Lemma 2.1. To simplify notations, assume that zkz in C0([0,p],X), hence also in C0([0,p],H), for a suitable z.

As T is uniformly bounded on [0,p],

( C , u ) 0 t T ( t - s ) C ( s ) u ( s ) 𝑑 s

is a continuous bilinear map L2([0,p],(H))×C0([0,p],H)H for each t. Consequently,

0 t T ( t - s ) B k ( s ) τ k z k ( s ) 𝑑 s 0 t T ( t - s ) B ( s ) τ z ( s ) 𝑑 s

in the weak topology of H for each t.

We can rewrite (2.3) also as

z k ( t ) = T ( t ) z k ( 0 ) + 0 t T ( t - s ) τ k { B k ( s ) z k ( s ) + f k ( s ) + g k ( s ) - μ z k ( s ) } 𝑑 s .

Taking limits here in the weak topology of H yields

z ( t ) = T ( t ) z ( 0 ) + 0 t T ( t - s ) τ { B ( s ) z ( s ) - μ z ( s ) } 𝑑 s

for each t. This means that z is a p-periodic mild solution of

z = A z + τ B ( t ) z + ( 1 - τ ) μ z .

As μIτB(t)+(1-τ)μIνI for a.e. t, z=1 and z(0)=z(p), we have contradicted the previous lemma and so we are done. ∎

Mutatis mutandi the above argument also gives the following lemma.

Lemma 2.7.

Assume that B and f are as in the previous lemma. If

lim x X 0 f ( t , x ) x X = 0

uniformly in t and there exist constants μ,ν,ρ and an integer n such that

  1. - λ n < μ ν < - λ n + 1 ,

  2. B ( t , x ) is symmetric for a.e. t and all x satisfying x X ρ ,

  3. μ I B ( t , x ) ν I for a.e. t and all x satisfying x X ρ ,

then there exists a lower bound in C0([0,p],X) for the p-periodic mild solutions u of

u = A u + τ { B ( t , u ) u + f ( t , u ) } + ( 1 - τ ) μ u ,

which is independent of τ[0,1].

2.3 Main Results

The above lemmas are used to prove the three theorems that follow.

Theorem 2.8.

Let B:R+×HL(H) and f:R+×HH be p-periodic in t and assume that they satisfy the Carathéodory conditions, with B bounded on bounded sets and

f ( t , x ) h ( t ) + const x for a.e.  t and all  x ,

where hL1([0,p]). If

lim x f ( t , x ) x = 0

uniformly in t and there exist μ,ν,ρR and nN such that

  1. - λ n < μ ν < - λ n + 1 ,

  2. B ( t , x ) is symmetric for a.e. t and all x satisfying x ρ ,

  3. μ I B ( t , x ) ν I for a.e. t and all x satisfying x ρ ,

then (1.1) has a p-periodic mild solution.

Proof.

Since the formula

C = sup y = 1 | ( C y | y ) |

defines a norm for symmetric operators, and B is bounded on bounded sets, the range of B is bounded in (H). Thus, we are in the framework of Lemma 2.6 if we choose X=H. As seen in the proof of Lemma 2.6, A+μI is the generator of a compact semigroup T:=(T(t))t0 on H and the operator L:=I-T(p) has a bounded inverse. Therefore, for each τ[0,1], we can define an operator Fτ:C0([0,p],H)C0([0,p],H) by

F τ ( u ) ( t ) := T ( t ) L - 1 0 p T ( p - s ) τ { B ( s , u ( s ) ) u ( s ) + f ( s , u ( s ) ) - μ u ( s ) } 𝑑 s
+ 0 t T ( t - s ) τ { B ( s , u ( s ) ) u ( s ) + f ( s , u ( s ) ) - μ u ( s ) } 𝑑 s .

By virtue of Lemma 2.1, Fτ is completely continuous on C0([0,p],H), and u=Fτ(u) means that u is a p-periodic mild solution to

u = A u + τ { B ( t , u ) u + f ( t , u ) } + ( 1 - τ ) μ u .

Then Lemma 2.6 implies the existence of ε>0 such that

F τ ( u ) u , u ε ,  0 τ 1 .

Consequently, calling U an open ball centered at the origin with radius greater than ε, we have

deg ( I - F 1 , U , 0 ) = deg ( I , U , 0 ) = 1 ,

by the homotopy invariance property of the Leray–Schauder degree, and F00. Then F1 has a fixed point u0 by the solution property of the degree. In view of the above, u0 is a p-periodic mild solution of (1.1). ∎

Two simple consequences of Theorem 2.8 follow.

Corollary 2.9.

Let g:R+×HH be bounded on bounded sets and p-periodic in t, and assume that it satisfies the Carathéodory conditions. If there exist BL2(R+,L(H)), μ,νR and nN such that

  1. - λ n < μ ν < - λ n + 1 ,

  2. B is p -periodic and B ( t ) is symmetric for a.e. t,

  3. μ I B ( t ) ν I for a.e. t,

  4. lim x g ( t , x ) - B ( t ) x / x = 0 uniformly in t,

then u=Au+g(t,u) has at least one p-periodic mild solution.

Proof.

Apply the previous theorem with BB and f:=g-B. ∎

Corollary 2.10.

Let A be a symmetric N×N-matrix and g,h:R×RNRNp-periodic in t, with g continuously differentiable and h continuous. If there exist μ,ν,ρR and nN such that

  1. - λ n < μ ν < - λ n + 1 ,

  2. g x ( t , x ) is symmetric for all t and x satisfying x ρ ,

  3. μ I g x ( t , x ) ν I for all t and x satisfying x ρ ,

and, in addition,

lim x h ( t , x ) x = 0

uniformly in t, then

u = A u + g ( t , u ) + h ( t , u )

has a p-periodic mild solution.

Proof.

Let RN×N be the space of N×N-matrices. Defining B:×NN×N by

B ( t , x ) := 0 1 g x ( t , ξ x ) 𝑑 ξ ,

it is easily verified that the assumptions of the previous theorem are fulfilled with ρ replaced by a larger constant. ∎

When the first eigenvalue of A is involved and is negative, the next theorem provides a better result than Theorem 2.8 and generalizes [12, Theorem 5] to continuous functions f.

Theorem 2.11.

If λ1<0, f:R+×HH is continuous, p-periodic in t and bounded on bounded sets, and there exist μ]0,-λ1[ and ρ>0 such that

( f ( t , x ) | x ) x 2 μ , t 0 , x ρ ,

then (1.1) has p-periodic mild solutions.

Proof.

Consider the family of evolution equations

(2.5) u = A u + τ f ( t , u ) , 0 τ 1 .

The well-known estimate

S ( t ) e λ 1 t

implies that S(p)xx when x0, and so L:=I-S(p) is invertible by Fredholm’s alternative. Then we can define Fτ:C0([0,p],H)C0([0,p],H) by

F τ ( u ) ( t ) := S ( t ) L - 1 τ 0 p S ( p - s ) f ( s , u ( s ) ) 𝑑 s + τ 0 t S ( t - s ) f ( s , u ( s ) ) .

This operator is completely continuous and its fixed points are p-periodic mild solutions of (2.5), by virtue of Lemma 2.1.

To find an a priori bound for the p-periodic mild solution of (2.5) which is independent of τ, we fix τ and consider any p-periodic mild solution uτ of (2.5) with uτ>1 (otherwise there nothing to prove). By [8, Lemma 1], there exists a sequence of locally Lipschitz functions fk:[0,+[×HH converging uniformly to f (because f can be extended continuously to the Banach space ×H by virtue of the Dugundji extension theorem). Setting

f k , τ ( t , x ) := f k ( t , x ) + f ( t , u τ ( t ) ) - f k ( t , u τ ( t ) )  for all  t , x , k ,

we get a sequence of locally Lipschitz functions such that uτ is a mild solution to

u = A u + τ f k , τ ( t , u ) , k 1 ,

and, in addition,

lim k f k , τ ( t , x ) = f ( t , x ) uniformly .

Fix μ0]μ,-λ1[. From

( f k , τ ( t , x ) | x ) x 2 = ( f k , τ ( t , x ) - f ( t , x ) | x ) x 2 + ( f ( t , x ) | x ) x 2
f k , τ ( t , x ) - f ( t , x ) x + ( f ( t , x ) | x ) x 2 ,

we deduce the existence of ρ0ρ and k01 such that

( f k 0 , τ ( t , x ) | x ) x 2 < μ 0 , t 0 , x ρ 0 .

There exists ϵ>0 such that

f k 0 , τ ( t , x ) ϵ , 0 t 2 p , x ρ 0 ,

as f is bounded on bounded sets.

Let um,τ be the unique solution to the finite-dimensional Cauchy problem

{ u = A u + τ P m f k 0 , τ ( t , u ) , u ( 0 ) = P m u τ ( 0 )

in Em, which makes sense since Em is invariant by A and A|Em:EmEm is continuous. In addition, um,τ is continuously differentiable. Since PmI pointwise and fk0,τ is locally Lipschitz, [13, Lemma 2] implies that

lim m u m , τ = u τ

uniformly on compacta. We claim that

  1. for m large, there exists tm,τ[0,p] such that um,τ(tm,τ)ρ0.

For, if um,τ(t)ρ0 for all t[0,p] and infinitely many m, then for these m, we have

d d t u m , τ ( t ) 2 = 2 ( u m , τ ( t ) | u m , τ ( t ) )
= 2 ( A u m , τ ( t ) + τ P m f k 0 , τ ( t , u m , τ ( t ) ) | u m , τ ( t ) )
= 2 ( A u m , τ ( t ) | u m , τ ( t ) ) + τ ( f k 0 , τ ( t , u m , τ ( t ) ) | u m , τ ( t ) ) (as  P m  is symmetric and  u m , τ ( t ) E m )
2 ( λ 1 + μ 0 ) u m , τ ( t ) 2 .

Setting

μ 1 := λ 1 + μ 0 ,

integrating the above inequality and using Gronwall’s lemma on the result, we get

u m , τ ( t ) 2 P m u τ ( 0 ) 2 e 2 μ 1 t , 0 t p .

Taking limits produces

u τ ( t ) 2 u τ ( 0 ) 2 e 2 μ 1 t , 0 t p .

Since μ1<0, we deduce

u τ ( p ) 2 u τ ( 0 ) 2 e 2 μ 1 p < u τ ( 0 ) 2 = u τ ( p ) 2 ,

which is a contradiction, showing that () holds true.

Now set

J m , τ := { t 0 : u m , τ ( t ) ρ 0 } ,

which is non-empty for m large enough by virtue of (). We have

d d t u m , τ ( t ) 2 = 2 ( A u m , τ ( t ) + τ P m f k 0 , τ ( t , u m , τ ( t ) ) | u m , τ ( t ) )
2 { λ 1 u m , τ ( t ) 2 + ϵ u m , τ ( t ) if  t J m , τ , μ 1 u m , τ ( t ) 2 otherwise,    (as  P m  is symmetric and  u m , τ ( t ) E m )
2 μ 1 u m , τ ( t ) 2 + 2 ϵ u m , τ ( t )
2 ϵ u m , τ ( t )    (as  μ 1 < 0 ) .

Then

u m , τ ( t ) 2 - u m , τ ( t m , τ ) 2 2 ϵ t m , τ t u m , τ ( ξ ) d ξ ,

hence

u m , τ ( t ) 2 ρ 0 2 + 2 ϵ t m , τ 2 p u m , τ ( ξ ) d ξ .

Taking limits, we get

u τ ( t ) 2 ρ 0 2 + 2 ϵ 0 2 p u τ ( ξ ) d ξ

and, consequently,

u τ 2 ρ 0 2 + 4 p ϵ u τ .

This and uτ1 imply

u τ ρ 0 2 + 4 p ϵ ,

and so we have an upper bound for the p-periodic solutions of (2.5), which is independent of τ.

If U is an open ball centered at the origin with radius greater than ρ02+4pϵ, then

deg ( I - F 1 , U , 0 ) = deg ( I , U , 0 ) = 1 ,

by the homotopy invariance property of the Leray–Schauder degree, and F00. Then the solution property of the degree implies the existence of a fixed point of F1, hence of a p-periodic mild solution of (1.1), by virtue of Lemma 2.1. ∎

The following corollary is a direct consequence of Theorem 2.11.

Corollary 2.12.

If A is a symmetric N×N-matrix whose largest eigenvalue λ1 is negative, then u=Au+f(t,u) has a p-periodic solution whenever f:R×RNRN is continuous, p-periodic in t and satisfies

( f ( t , x ) | x ) x 2 μ , t 0 , x ρ ,

where μ]0,λ1[ and ρ>0.

The next result is a multiplicity theorem.

Theorem 2.13.

Let B:R+×XL(X) and f:R+×XX be p-periodic in t, satisfy the Carathéodory conditions, with B uniformly bounded and

f ( t , x ) h ( t ) + const x for a.e.  t and all  x ,

where hL1([0,p]). Assume that the following hold:

  1. λ 1 < 0 .

  2. lim x X 0 f ( t , x ) / x X = 0 uniformly in t and there exist μ 0 , ν 0 , ρ 0 and n 0 such that ρ 0 > 0 and

    1. - λ n 0 < μ 0 ν 0 < - λ n 0 + 1 ,

    2. B ( t , x ) is symmetric for a.e. t and all x satisfying x X ρ 0 ,

    3. μ 0 I B ( t , x ) ν 0 I for a.e. t and all x satisfying x X ρ 0 .

  3. lim x X f ( t , x ) / x X = 0 uniformly in t and there exist μ , ν , ρ and n such that ρ > ρ 0 and

    1. - λ n < μ ν < - λ n + 1 ,

    2. B ( t , x ) is symmetric for a.e. t and all x satisfying x ρ ,

    3. μ I B ( t , x ) ν I for a.e. t and all x satisfying x ρ .

  4. The two integers n 0 , n have opposite parity, i.e., one is odd and the other even.

Then (1.1) has a non-trivial p-periodic mild solution.

Proof.

Since λ1<0, the operator L:=I-S(p) is invertible as seen at the beginning of the proof of the previous theorem. Consequently, for i=0, and 0τ1, we can define the operator Fτi:C0([0,p],X)C0([0,p],X) by

F τ i ( u ) ( t ) := S ( t ) L - 1 0 p S ( p - s ) { τ B ( s , u ( s ) ) u ( s ) + τ f ( s , u ( s ) ) + ( 1 - τ ) μ i u ( s ) } 𝑑 s
+ 0 t S ( t - s ) { τ B ( s , u ( s ) ) u ( s ) + τ f ( s , u ( s ) ) + ( 1 - τ ) μ i u ( s ) } 𝑑 s ,

which is completely continuous and whose fixed points are p-periodic mild solutions of

u = A u + τ { B ( t , u ) u + f ( t , u ) } + ( 1 - τ ) μ i u ,

by virtue of Lemma 2.1. By Lemma 2.7, there exists ε0]0,ρ0] such that

F τ 0 ( u ) u , 0 τ 1 , u X = ε 0 .

Then, from the homotopy invariance of the Leray–Schauder degree, we deduce that

deg ( I - F 1 0 , U 0 , 0 ) = deg ( I - F 0 0 , U 0 , 0 ) ,

where U0 is the open ball in C0([0,p],X) centered at the origin with radius ε0. The fixed points of F00 are p-periodic solutions of

u = A u + μ 0 u .

Lemma 2.5 says that this equation has only the trivial p-periodic solution, and so I-F00 is an invertible operator by Fredholm’s alternative. Then the Leray–Schauder theorem implies that

deg ( I - F 0 0 , U 0 , 0 ) = ( - 1 ) m 0 ,

where m0 is the sum of the multiplicities of the eigenvalues of F0 which are larger than 1. In view of Lemma 2.2, we have m0=n0. We conclude that

deg ( I - F 1 0 , U 0 , 0 ) = ( - 1 ) n 0 .

Similarly, with μ in place of μ0 and Lemma 2.6 in place of Lemma 2.7, we deduce

deg ( I - F 1 , U , 0 ) = ( - 1 ) n ,

where U is an open ball in C0([0,p],X), centered at the origin, with a suitable radius ε>ε0.

As (-1)n0(-1)n and F10=F1, the additivity property of the degree ensures that

deg ( I - F 1 , U U ¯ 0 , 0 ) 0 .

Consequently, the solution property of the degree implies that F1 has a fixed point in UU¯0. This corresponds to a p-periodic mild solution of (1.1). ∎

Corollary 2.14.

Let g:R+×XX be bounded on bounded sets and p-periodic in t, and assume that it satisfies the Carathéodory conditions. If, for i=0,, there exist BiL2(R+,L(X)) and μi,νiR, niN such that

  1. - λ n i < μ i ν i < - λ n i + 1 ,

  2. B i is p -periodic with B i ( t ) symmetric for a.e. t,

  3. μ i I B i ν i I for a.e. t,

  4. lim x i g ( t , x ) - B ( t ) x X / x X = 0 uniformly in t,

  5. the two integers n 0 and n have opposite parity,

then u=Au+g(t,u) has at least one p-periodic mild solution.

Proof.

Set

g 0 ( t , u ) := g ( t , u ) - B 0 ( t ) u and g ( t , u ) := g ( t , u ) - B ( t ) u ,

and consider the function α: defined by

α ( t ) := { 0 , t 1 , t - 1 , 1 t 2 , 1 , t 2 ,

as well as the maps B:[0,p]×X(X) and f:[0,p]×XX defined by

B ( t , u ) := { 1 - α ( u X ) } B 0 ( t ) + α ( u X ) B ( t ) ,
f ( t , u ) := { 1 - α ( u X ) } g 0 ( t , u ) + α ( u X ) g ( t , u ) .

Obviously B and f satisfy the Carathéodory conditions, B is uniformly bounded, pointwise symmetric and

  1. lim u 0 , f ( t , u ) / u X = 0 uniformly in t,

  2. μ 0 I B ( t , u ) ν 0 I whenever uX1,

  3. μ I B ( t , u ) ν I whenever uX2.

In addition,

B ( t , u ) u + f ( t , u ) = ( 1 - α ( u X ) ) { g 0 ( t , u ) + B 0 ( t , u ) u } + α ( u X ) { g ( t , u ) + B ( t , u ) u } = g ( t , u ) .

Thus, an application of Theorem 2.13 yields the desired conclusion. ∎

The following result is a direct consequence of Theorem 2.13 (cf. the proof of Corollary 2.10).

Corollary 2.15.

Let A be a symmetric N×N-matrix with negative eigenvalues ordered in decreasing order counting multiplicities, i.e.,

λ N λ 1 < 0 .

Let f,g:R×RNRN be p-periodic in t with g continuously differentiable and f continuous satisfying

lim u 0 , f ( t , u ) u = 0

uniformly in t. Assume that, for i=0,, there exist μi,νi,ρiR and niN such that the following hold:

  1. 0 < ρ 0 < ρ .

  2. For each x satisfying x ρ 0 , we have

    1. - λ n 0 < μ 0 ν 0 - λ n 0 + 1 ,

    2. g x ( t , x ) is symmetric for all t,

    3. μ 0 I g x ( t , x ) ν 0 I for all t.

  3. For each x satisfying x ρ , we have

    1. - λ n < μ ν - λ n + 1 ,

    2. g x ( t , x ) is symmetric for all t,

    3. μ I g x ( t , x ) ν I for all t.

  4. n 0 and n have opposite parity.

Then

u = A u + g ( t , x ) + f ( t , u )

has a non-trivial p-periodic solution.

3 Application to Parabolic Systems

Here we apply the above theorems to the periodic BVP for parabolic systems

(3.1) { u t = Δ u + g ( t , x , u ) + h ( t , x , u ) in  [ 0 , + [ × Ω , u ( t , x ) = 0 in  [ 0 , + [ × Ω , u ( , x )  is  p -periodic for every  x ,

obtaining generalizations of some results proved in [5] by the sub-supersolution method. Precisely, the next two theorems generalize [5, Theorem 4], while Theorem 3.5 below generalizes [5, Theorem 6].

Of course, the Laplacian and the Dirichlet condition can be replaced by any differential operator subjected to a boundary condition on the space variable that produces a symmetric generator of a compact semigroup in L2(Ω,N).

3.1 Standing Notations and Assumptions

We shall use the following notation:

  1. Ω denotes a bounded domain of M.

  2. g : + × Ω × N N is a continuously differentiable, p-periodic in t map such that g(t,x,0)=0 whenever xΩ.

  3. h : + × Ω × N N is a continuous, p-periodic in t map such that h(t,x,0)=0 whenever xΩ.

  4. λ N , n denotes the nth eigenvalue of the following linear Dirichlet problem in N:

    (3.2) { Δ u = λ u in  Ω , u | Ω = 0 ,

    assuming that the eigenvalues are in the decreasing order counting multiplicities, with λN,0=+.

By saying Dirichlet problem inN, we mean that the ranges of the eigenfunctions are contained in N. In view of Theorem 3.5 below, there is an important connection between the eigenvalues of the scalar and of the vector Dirichlet problem to be pointed out, i.e., for the eigenvalues of (3.2) in and N, respectively. The two BVPs have the same eigenvalues but the eigenvalues have different multiplicities. If mn is the multiplicity of the eigenvalue λ1,n of (3.2) in , then mnN is the multiplicity of the eigenvalue λ1,n of (3.2) in N. This is due to the fact that L2(Ω,N) can be identified with the direct sum of N copies of L2(Ω). Consequently, in particular, we have

λ 1 , n < λ 1 , n + 1 λ N , n N = λ 1 , n and λ N , n N + 1 = λ 1 , n + 1 .

Thus, nothing changes in Theorems 3.1 and 3.3 if the eigenvalues of (3.2) in N substitute those in , while this is not so for Theorem 3.5.

Another point that needs explanation is the reason why in the sequel there is no smoothness assumption on Ω. If Ω is an arbitrary bounded domain of M, then Δ, together with the Dirichlet boundary condition, is the generator of a contraction semigroup on L2(Ω,N), by [6, §2.6.1], and this semigroup turns out to be compact, by [2, Corollary 4.4.2] (because the Poincaré inequality and the compactness of the imbedding W01,2(Ω)L2(Ω) hold for arbitrary bounded domains Ω, and so the Dirichlet form defines an equivalent inner product on W01,2(Ω,N) and the resolvents are compact as shown, e.g., in [7, §1.3]).

3.2 The Applications

In spirit, the theorems that follow are nothing else than the counterparts for (3.1) of the results in Section 2.3. However, the related technical details have to be settled.

Theorem 3.1.

Assume that

lim z h ( t , x , z ) z = 0

uniformly in (t,x). If ρ,μ,νR and nN satisfy the following conditions:

  1. - λ 1 , n < μ ν < - λ 1 , n + 1 ,

  2. g z ( t , x , z ) is a symmetric matrix and μ I g z ( t , x , z ) ν I whenever t 0 , x Ω and z ρ ,

then the parabolic BVP (3.1) has a mild solution.

Proof.

We plan to apply Theorem 2.8 with H:=L2(Ω,N). Since the formula

M = sup y = 1 | ( M y | y ) |

defines a norm for symmetric matrices, the hypotheses guarantee that gz<+. Set

α ( s ) := { 0 if  s ρ , s - ρ if  ρ s ρ + 1 , 1 if  s ρ + 1 ,

and define B:+×H(H) and G:+×HH by

( B ( t , u ) v ) ( x ) := 0 1 { α ( ξ u ( x ) ) g z ( t , x , ξ u ( x ) ) v ( x ) + ( 1 - α ( ξ u ( x ) ) ) μ v ( x ) } 𝑑 ξ ,
( G ( t , u ) ) ( x ) := 0 1 ( 1 - α ( ξ u ( x ) ) ) { g z ( t , x , ξ u ( x ) ) v ( x ) - μ u ( x ) } 𝑑 ξ + h ( t , x , u ( x ) ) .

In view of

g ( t , x , z ) = 0 1 g z ( t , x , ξ z ) z 𝑑 ξ ,

solutions of our BVP correspond to solutions of u=Au+B(t,u)u+G(t,u). Clearly, each B(t,u) is symmetric. Applying coordinatewise the formula

( 0 1 ψ 𝑑 ξ ) 2 0 1 ψ 2 𝑑 ξ

(due to Jensen’s integral inequality, see [15, p. 122]) and Fubini’s theorem, for any uH, we have

Ω 0 1 ( 1 - α ( ξ u ( x ) ) ) { g z ( t , x , ξ u ( x ) ) u ( x ) - μ u ( x ) } d ξ 2 d x
Ω 0 1 | 1 - α ( ξ u ( x ) ) | 2 g z ( t , x , ξ u ( x ) ) u ( x ) - μ u ( x ) 2 d ξ d x
= 0 1 𝑑 ξ Ω 𝑑 x = 0 1 𝑑 ξ ( { ξ u ( x ) < ρ + 1 } 𝑑 x + { ξ u ( x ) ρ + 1 } 𝑑 x )
= 0 1 𝑑 ξ { ξ u ( x ) < ρ + 1 } 𝑑 x = 0 ε 𝑑 ξ { ξ u ( x ) < ρ + 1 } 𝑑 x + ε 1 𝑑 ξ { ξ u ( x ) < ρ + 1 } 𝑑 x
ε { g z + μ } 2 u L 2 2 + | Ω | { g z + μ } 2 ( ρ + 1 ) 2 ε 2 .

In view of the hypotheses, to each ε>0 correspond positive constants δε and Mε such that for all t[0,p] and xΩ, we have

h ( t , x , z ) z < ε when  z > δ ε    and    h ( t , x , z ) M ε when  z δ ε .

Then, for every uH,t[0,p] and ε>0, we have

h ( t , , u ( ) ) L 2 2 { u ( x ) 0 } h ( t , x , u ( x ) ) 2 u ( x ) 2 u ( x ) 2 𝑑 x
= { 0 < u ( x ) δ ε } + { u ( x ) > δ ε }
| Ω | M ε 2 + ε 2 u L 2 2 .

Since

lim u L 2 | Ω | 1 / 2 { g z + μ } ( ρ + 1 ) / ε + | Ω | 1 / 2 M ε u L 2 = 0 for each  ε ,

and

( i α i 2 ) 1 / 2 i α i whenever  α i 0  for all  i ,

the previous inequalities imply

lim u L 2 G ( t , u ) L 2 u L 2 = 0 .

On the other hand, from

( B ( t , u ) v | v ) L 2 = Ω ( 0 1 { α ( ξ u ( x ) ) g z ( t , x , ξ u ( x ) ) v ( x ) + ( 1 - α ( ξ u ( x ) ) ) μ v ( x ) } d ξ | v ( x ) ) d x
= Ω 0 1 ( α ( ξ u ( x ) ) g z ( t , x , ξ u ( x ) ) v ( x ) + ( 1 - α ( ξ u ( x ) ) ) μ v ( x ) | v ( x ) ) d ξ d x
= 0 1 d ξ Ω ( α ( ξ u ( x ) ) g z ( t , x , ξ u ( x ) ) v ( x ) + ( 1 - α ( ξ u ( x ) ) ) μ v ( x ) | v ( x ) ) d x
= 0 1 d ξ { { ξ u ( x ) < ρ } μ v ( x ) 2 d x + { ξ u ( x ) ρ } [ α ( ξ u ( x ) ) ( g z ( t , x , ξ u ( x ) ) v ( x ) | v ( x ) )
+ ( 1 - α ( ξ u ( x ) ) ) μ v ( x ) 2 ] d x }

and the hypotheses, we have

μ v L 2 2 ( B ( t , u ) v | v ) L 2 ν v L 2 2 , u , v H ,

i.e., μIB(t,u)νI for all uH. By the theorem of Krasnoselski on the continuity of Nemitski operators between LP-spaces, G and B are continuous and bounded on bounded subsets. Thus, Theorem 2.8 implies the existence of a p-periodic mild solution of u=Au+B(t,u)u+G(t,u). ∎

The next theorem shows that no differentiability assumption on g is needed in the scalar case.

Theorem 3.2.

Assume that N=1 and that

lim | z | h ( t , x , z ) | z | = 0

uniformly in (t,x). If the constants ρ,μ,ν and the integer n satisfy the condition

- λ 1 , n < μ g ( t , x , z ) z ν < - λ 1 , n + 1 , 0 t p , x Ω , | z | ρ ,

then the parabolic BVP (3.1) has a mild solution.

Proof.

With

B ( t , z ) := { g ( t , x , z ) z if  z 0 , 0 if  z = 0 ,

in place of 01gz(t,x,ξz)𝑑ξ, the proof is more or less the same as that of the previous theorem. ∎

Theorem 3.3.

Assume that the following hold:

  1. g z is uniformly bounded.

  2. There exist ρ > 0 and μ ] 0 , - λ 1 , 1 [ such that g z ( t , x , z ) is symmetric and g z ( t , x , z ) μ I whenever t 0 , xΩ and zρ.

  3. lim z h ( t , x , z ) / z = 0 uniformly in ( t , x ) .

Then the parabolic BVP (3.1) has a mild solution.

Proof.

Let H and A be as in the beginning of the proof of Theorem 3.1. Let f:[0,p]×HH be defined by

f ( t , u ) := g ( t , , u ( ) ) + h ( t , , u ( ) ) .

In view of

g ( t , x , y ) = 0 1 g z ( t , x , ξ y ) y 𝑑 ξ ,

we have

( f ( t , u ) | u ) L 2 = Ω ( 0 1 g z ( t , x , ξ u ( x ) ) u ( x ) d ξ | u ( x ) ) d x + Ω ( h ( t , x , u ( x ) ) | u ( x ) ) d x
= Ω 0 1 ( g z ( t , x , ξ u ( x ) ) u ( x ) | u ( x ) ) d ξ d x + Ω ( h ( t , x , u ( x ) ) | u ( x ) ) d x
= 0 1 d ξ Ω ( g z ( t , x , ξ u ( x ) ) u ( x ) | u ( x ) ) d x + Ω ( h ( t , x , u ( x ) ) | u ( x ) ) d x
= 0 1 d ξ { { ξ u ( x ) < ρ } ( g z ( t , x ,