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Bistable Traveling Waves for Monotone Semiflows with Applications

This paper is devoted to the study of traveling waves for monotone evolution systems of bistable type. Under an abstract setting, we establish the existence of bistable traveling waves for discrete and continuous-time monotone semiflows. This result is then extended to the cases of periodic habitat and weak compactness, respectively. We also apply the developed theory to four classes of evolution systems.


Introduction
In this paper, we study traveling waves for monotone (i.e., order-preserving) semiflows {Q t } t∈T with the bistability structure on some subsets of the space C := C(H, X ) consisting of all continuous functions from the habitat H(= R or Z) to the Banach lattice X , where T = Z + or R + is the set of evolution times. Here the bistability structure is generalized from a number of studies for various evolution equations. It means that the restricted semiflow on X admits two ordered stable equilibria, between which all others are unstable. We focus on the existence of traveling waves connecting these two stable equilibria, which are called bistable traveling waves. This setting allows us to study not only autonomous and timeperiodic evolution systems in a homogeneous habitat (media), but also those in a periodic habitat. Besides, the obtained results can be extended to the semiflows with weak compactness on some subsets of the space M consisting of all monotone functions from R to X .
To explain the concept of the bistability structure, we recall some related works on typical evolution equations. Fife and McLeod [19,20] proved the existence and global asymptotic stability of monotone traveling waves for the following reactiondiffusion equation: where a ∈ (0, 1). Clearly, the restriction of system (1.1) on X = R is the ordinary differential equation u ′ = u(1 − u)(u − a), which admits a unique unstable equilibrium between two ordered and stable ones. The same property is shared by the nonlocal dispersal equation in [4,16,46] and the lattice equations in [3,49,50]. Chen [13] studied a general nonlocal evolution equation u t = A(u(·, t)), which also possess the above bistability structure. Some related investigations on discrete-time equations can be found in [27,15]. For the time-periodic reaction-diffusion equation u t = u xx + f (t, u), the spatially homogeneous equation is a time-periodic ordinary differential equation. In this case, the equilibrium in the bistability structure should be understood as the time-periodic solution. Under such bistability assumption, Alikakos, Bates and Chen [1] obtained the existence of bistable time-periodic traveling waves. Recently, Yagisita [46] studied bistable traveling waves for discrete and continuous-time semiflows on the space consisting of all left-continuous and nondecreasing functions from R to X = R under the assumption that there is exactly one intermediate unstable equilibrium. It should be mentioned that the result in [46] for continuous-time semiflows requires an additional assumption on the existence of a pair of upper and lower solutions. Note that the restrictions on X = R of the afore-mentioned systems are all scalar equations, and hence, there is only one unstable equilibrium in between two stable ones. But in the case where X = R n , there may be multiple unstable equilibria. This is one of the main reasons why some ideas and techniques developed for scalar equations can not be easily extended to higher dimensional systems. Volpert [41] established the existence and stability of traveling waves for the bistable reactiondiffusion system u t = D∆u + f (u) by using topological methods, where D is a positive definite diagonal matrix. Fang and Zhao [18] further extended these results to the case where D is semi-positive definite via the vanishing viscosity approach.
Consider the following parabolic equation in a cylindrical domain Σ = R × Ω: u t = ∆u + α(y)u x + f (u), x ∈ R, y = (y 1 , · · ·, y n−1 ) ∈ Ω, t > 0, ∂u ∂ν = 0 on R × ∂Ω × (0, +∞), (1.2) where f is of the same type as the nonlinearity in (1.1) and Ω is a bounded domain with smooth boundary in R n−1 . Obviously, the restriction of the solution semiflow of (1.2) on X = C(Ω, R) gives rise to the following x-independent system: u t = ∆ y u + f (u), y ∈ Ω, t > 0, ∂u ∂ν = 0 on ∂Ω × (0, +∞). (1.3) One can see from Matano [30] (or Casten and Holland [12]) that any nonconstant steady state of (1.3) is linearly unstable when the domain Ω is convex. It follows that if Ω is convex, then (1.2) admits the bistability structure: its x-independent system has two (constant) linearly stable steady states, between which all others are linearly unstable. In such a case, Berestycki and Nirenberg [11] obtained the existence and uniqueness of bistable traveling waves. In the case where Ω is an appropriate dumbbell-shaped domain, Matano [30] constructed a counterexample to show that (1.3) has stable non-constant steady states, and Berestycki and Hamel [6] also proved the nonexistence of traveling waves connecting two stable constant steady states. For bistable traveling waves in time-delayed reaction-diffusion equations, we refer to [31,35,29,36]. For such an equation with time delay τ > 0, one can choose X = C([−τ, 0], R) so that its solution semiflow has the bistability structure.
Recently, there is an increasing interest in reaction diffusion equations in periodic habitats. A typical example is u t = (du x ) x + f (u), x ∈ R, t > 0, (1.4) where d ∈ C 1 (R, R) is a positive periodic function with period r > 0. Define Y := C([0, r], R) and C per (R, R) := {f ∈ C(R, R) : f (x + r) = f (x), ∀x ∈ R}. It is easy to see that C(R, R) = {f ∈ C(rZ, Y) : f (ri)(r) = f (r(i + 1))(0), ∀i ∈ Z} := K, and that any element in C per (R, R) is a constant function in K. Thus, the solution semiflow of (1.4) on C(R, R) can be regarded as a conjugate semiflow on K, and hence, the bistability structure should be understood as: the restriction of the solution semiflow of (1.4) on C per (R, R) has two ordered r-periodic steady states, between which all others are unstable. Assuming that the function f is of bistable type, Xin [42] obtained the existence of spatially periodic (pulsating) traveling wave as long as d is sufficiently close to a positive constant in a certain sense (see also [43,44]). However, whether the solution semiflow of (1.4) admits the bistability structure remains an open problem. We will give an affirmative answer in section 6.3 and further improve Xin's existence result. Meanwhile, a counterexample will be constructed to show that the solution semiflow of (1.4) has no bistability structure in the general case of varying d(x). More recently, Chen, Guo and Wu [14] proved the existence, uniqueness and stability of spatially periodic traveling waves for onedimensional lattice equations in a periodic habitat under the bistability assumption. There are also other types of bistable waves (see, e.g., [7,33]). For monostable systems in periodic habitats, we refer to [5,9,10,22,23,26] and references therein.
In general, there are multiple intermediate unstable equilibria in between two stable ones in the case where the space X is high dimensional. Meanwhile, it is possible for the given system to have intermediate unstable time-periodic orbits in X . These give more difficulties to the study of bistable semiflows than monostable ones, whose restricted systems on X have only one unstable and one stable equilibria. To overcome these difficulties, we will show that all these unstable equilibria and all points in these periodic orbits are unordered in X under some appropriate assumptions. With this in mind, a bistable system can be regarded as the union of two monostable systems although such a union is not unique. From this point of view, we establish a link between monostable subsystems and the bistable system itself, which plays a vital role in the propagation of bistable traveling waves. This link is stated in terms of spreading speeds of monostable subsystems (see assumption (A6)). For spreading speeds of various monostable evolution systems, we refer to [2,9,22,25,26,28,37,38,48] and references therein.
In our investigation, we consider seven cases: (I) T = Z + and H = R; (II) T = Z + and H = Z; (III) T = R + and H = R; (IV) T = R + and H = Z; (V) periodic habitat; (VI) weak compactness; (VII) time periodic. For the case (I), we combine the above observations for general bistable semiflows and Yagisita's perturbation idea in [46] to prove the existence of traveling waves. For the case (III), we use the bistable traveling waves φ ± (x+c ±,s ) of discrete-time semiflows {(Q s ) n } n≥0 to approximate the bistable wave of the continuous-time semiflow {Q t } t≥0 . This new approach heavily relies on an estimation of the boundedness of 1 s c ±,s as s → 0, which is proved by the bistability structure of the semiflow (see inequalities (3.9) and (3.10)). It turns out that our result does not require the additional assumption on the existence of a pair of upper and lower solutions as in [46]. In the case (II), both the evolution time T and the habitat H are discrete, a traveling wave ψ(i + cn) of {Q n } n≥0 cannot be well-defined in the usual way because the wave speed c and hence, the domain of ψ is unknown. So we define it to be a traveling wave of an associated mapQ. However,Q has much weaker compactness than Q. To overcome this difficulty, we establish a variant of Helly's theorem for monotone functions from R to X in the Appendix, which is also of its own interest. This discovery also enables us to study monotone semiflows in a periodic habitat and with weak compactness, respectively. Further, we can deal with the case (IV) by the similar idea as in the case (III) because now traveling waves in the case (II) are defined on R. Traveling waves for a time-periodic system can be obtained with the help of the discrete-time semiflow generated by the associated Poincaré map. Motivated by the discussions in [26,Section 5], we can regard a semiflow in a periodic habitat as a conjugate semiflow in a homogeneous discrete habitat, and hence, we can employ the arguments for the cases (II) and (IV) to establish the existence of spatially periodic bistable traveling waves.
The rest of this paper is organized as follows. In section 2, we present our main assumptions. Section 3 is focused on discrete-time, continuous-time, and timeperiodic compact semiflows on some subsets of C. In section 4, we extend our results to compact semiflows in a periodic habitat. In section 5, we further investigate semiflows with weak compactness. In section 6, we apply the abstract results to four classes of evolution systems: a time-periodic reaction-diffusion system, a parabolic system in a cylinder, a parabolic equation with periodic diffusion, and a time-delayed reaction-diffusion equation. A short appendix section completes the paper.

Notations and assumptions
Throughout this paper, we assume that X is an ordered Banach space with the norm · X and the cone X + . Further, we assume that X is also a vector lattice with the following monotonicity condition: where |z| X := sup{z, −z}. Such a Banach space is called a Banach lattice. We use C(M, R d ) to denote the set of all continuous functions from the compact metric space M to the d-dimensional Euclidean space R d . We equip C(M, R d ) with the maximum norm and the standard cone consisting of all nonnegative functions. Then C(M, R d ) is a special Banach lattice, which will be used in this paper. For more general information about Banach lattices, we refer to the book [32].
Let the spatial habitat H be the real line R or the lattice rZ := {· · ·, −2r, −r, 0, r, 2r, · · ·} for some positive number r. For simplicity, we let r = 1. We say a function φ : H → X is bounded if the set { φ(x) X : x ∈ H} ⊂ X is bounded. Throughout this paper, we always use B to denote the set of all bounded functions from R to X , and C to denote the set of all bounded and continuous functions from H to X . Moreover, any element in X can be regarded as a constant function in B and C.
In this paper, we equip C with the compact open topology, that is, a sequence φ n converges to φ in C if and only if φ n (x) converges to φ(x) in X uniformly for x in any bounded subset of H. The following norm on C can induce such topology: We assume that Int(X + ) is not empty. For any u, v ∈ X , we write u ≥ v provided u − v ∈ X + , u > v provided u ≥ v but u = v, and u ≫ v provided u − v ∈ Int(X + ). A set E ⊂ X is said to be totally unordered if any two elements (if exist) are unordered. For any φ, ψ ∈ C, Let β ∈ Int(X + ) and Q be a map from C β to C β . Let E be the set of all fixed points of Q restricted on X β . Definition 2.1. For the map Q : X β → X β , a fixed point α ∈ E is said to be strongly stable from below if there exist a number δ > 0 and a unit vector e ∈ Int(X + ) such that Strong instability from below is defined by reversing the inequality (2.2). Similarly, we can define strong stability (instability) from above.
Given y ∈ H, define the translation operator T y on B by T y [φ](x) = φ(x − y). Assume that 0 and β are in E. We impose the following hypothesis on Q: (A2) (Continuity) Q : C β → C β is continuous with respect to the compact open topology.
(A3) (Monotonicity) Q is order preserving in the sense that (A4) (Compactness) Q : C β → C β is compact with respect to the compact open topology.
(A5) (Bistability) Two fixed points 0 and β are strongly stable from above and below, respectively, for the map Q : X β → X β , and the set E \ {0, β} is totally unordered.
Note that the above bistability assumption is imposed on the spatially homogeneous map Q : X β → X β . We allow the existence of other fixed points on the boundary of X β so that the theory is applicable to two species competitive evolution systems. The non-ordering property of E \ {0, β} can be obtained by the strong instability of all fixed points in this set if the semiflow is eventually strongly monotone. More precisely, a sufficient condition for hypothesis (A5) to hold is: (A5 ′ ) (Bistability) Q : X β → X β is eventually strongly monotone in the sense that there exists for all m ≥ m 1 whenever u > v in X β . Further, for the map Q : X β → X β , two fixed points 0 and β are strongly stable from above and below, respectively, and each α ∈ E \ {0, β} (if exists) is strongly unstable from both below and above. The following figures illustrate the bistability structures in (A5) and (A5 ′ ). Next we show that the assumption (A5 ′ ) implies (A5). In applications, however, one may find other weaker sufficient conditions than (A5 ′ ) for (A5) to hold. Proposition 2.1. If (A5 ′ ) holds, then for any α 1 , α 2 ∈ E \ {0, β}, we have α 1 < α 2 and α 2 < α 1 .
Note that in (A5) we do not require α ≫ 0 or α ≪ β. But (A5) is sufficient to guarantee that α and β can be separated by two neighborhoods in [α, β] X , and a similar claim is valid for 0 and α (see Lemma 3.1). In view of assumption (A5), we can find a positive number δ β > 0 and a unit vector e β ∈ Int(X + ) such that Choose a nondecreasing initial function φ − α ∈ C β with the property that It then follows from assumptions (A1)-(A2) and (A5) that and hence, there exits σ > 0 such that Define a sequence a n,σ of points in X as follows: (σ) = a 1,σ . By induction, we see that a n,σ is nondecreasing in n. Thus, assumption (A4) implies that a n,σ tends to a fixed point e with e ≥ a 1,σ ≫ β − δ β e β . Therefore, e = β.
By the above observation, we have n→∞ a n,σ = β, and hence, . We further claim that c * − (α, β) is independent of the choice of φ − α as long as φ − α has the property (2.5). Indeed, for any given φ with the property (2.5), we have , ∀x ∈ H. It then follows that for any c ∈ Λ(φ − α ) and ǫ > 0, which implies that c − ǫ ∈ Λ(φ) and hence, sup Λ(φ − α ) = sup Λ(φ). For convenience, we may call c * − (α, β) as the leftward asymptotic speed of propagation of φ − α . Following the above procedure, we can find δ 0 > 0, e 0 ∈ Int(X + ) such that (2.7) Here we emphasis that δ 0 , e 0 above and δ β , e β will play a vital role in the whole paper because they describe the local stability of fixed points 0 and β. Similarly, we can define θ + := sup{θ ∈ [0, 1] : θα ∈ [0, δ 0 e 0 ] X } and v + α := θ + α. Let φ + α ∈ C β be a nondecreasing initial function with the property that Due to the same reason, we can define the number which is called the rightward asymptotic speed of propagation of φ + α . As showed above, these two speeds are bounded below, but may be plus infinity. To better understand these two spreading speeds, we use Figure 2 (Left) to explain them. Now we are ready to state our last assumption on Q: (A6) (Counter-propagation) For each α ∈ E \ {0, β}, c * − (α, β) + c * + (0, α) > 0. Assumption (A6) assures that two initial functions in the left hand side of Figure  2 will eventually propagate oppositely although one of these two speeds may be negative. It is interesting to note that assumption (A6) is nearly necessary for the propagation of a bistable traveling wave. Indeed, if a monotone evolution system admits a bistable traveling wave, then it is usually unique (up to translation) and globally attractive (see, e.g., Remark 6.2). This implies that the solution starting from the initial data 1 2 (φ + α + φ − α ) converges to a phase shift of the bistable wave. If c * − (α, β) + c * + (0, α) < 0, then the comparison principle would force the solutions starting from φ ± α to split the bistable wave. Comparing with the definition of spreading speeds (short for asymptotic speeds of spread/propagation) for monostable semiflows (see, e.g., [2,26]), one can find that the leftward spreading speed of the monostable subsystem {Q n } n≥0 restricted on [α, β] C is shared by a large class of initial functions, and in many applications, it equals c * − (α, β). A similar observation holds for c * + (0, α). Thus, for a specific bistable system, the assumption (A6) can be verified by using the properties of spreading speeds for monostable subsystems.

Semiflows in a homogeneous habitat
We say a habitat is homogeneous for the semiflow In this section, we will establish the existence of bistable traveling waves for the semiflow {Q t } t∈T on E in the following order: discrete-time semiflows in a continuous habitat, discrete-time semiflows in a discrete habitat, time-periodic semiflows, continuous-time semiflows in a continuous habitat, and continuous-time semiflows in a discrete habitat.

Discrete-time semfilows in a continuous habitat
In this case, time T is discrete and habitat H is continuous: T = Z + and H = R. For convenience, we use Q to denote Q 1 , and consider the semiflow {Q n } n≥0 , where Q n is the n-th iteration of Q.
Definition 3.1. ψ(x + cn) with ψ ∈ C is said to be a traveling wave with speed We first show that 0 and β are two isolated fixed points of Q in X β if (A5) holds. Proof. Assume, for the sake of contradiction, that 0 = α ∈ E ∩ X δ 0 e 0 . Define the numberδ ∈ (0, δ 0 ] byδ Then it follows that α ≤δe 0 but α ∈ [0,δe 0 ]] X . However, by the monotonicity of Q and the fact that 0 is strongly stable, we have This contradicts α ∈ [0,δe 0 ]] X . And hence, Choose δ > 0 such that Assume that ψ andψ are two nondecreasing functions in C(R, X β ) with the properties that Clearly ψ ≤ψ. And we have the following observation.
. Then for each n ∈ N, G n := Q • A κn has a fixed point φ n in C β such that φ n is nondecreasing and ψ n ≤ φ n ≤ψ n .
Proof. We first show that ψ n ≤ G n [ψ n ]. Indeed, when x < n we have Similarly, we haveψ n ≥ G n [ψ n ]. It then follows that Since G n is order preserving and ψ n (x) is nondecreasing in x, we know that G k n [ψ n ](x) is nondecreasing both in k and x. Recall that G n is compact due to assumption (A4). It then follows that G k n [ψ n ] converges in C β . Denote the limit by φ n . By inequality (3.2), we also get ψ n ≤ φ n ≤ψ n . Moreover, φ n (x) is also nondecreasing due to Proposition 7.1 (2). And obviously, This completes the proof.
The following lemma reveals a relation between the wave speeds of monostable traveling waves in the sub-monostable systems and the numbers defined in (2.6) and (2.8).
Proof. We only prove the statement (1) since the proof for (2) is similar. In view of Lemma 3.1, we see that α and β can be separated by balls N (α, γ) and N (β, γ) with radius γ < δ β /2 in the metric space [α, β] X . Then α < u, ∀u ∈ N (β, γ). We a contradiction. Thus, we have c ≥ c * − (α, β). Now we are ready to prove the main result of this subsection.
Theorem 3.1. Assume that Q satisfies (A1)-(A6). Then there exists c ∈ R such that the discrete semiflow {Q n } n≥1 admits a non-decreasing traveling wave with speed c and connecting 0 to β .
Now we define a n , b n ∈ R as follows: It then follows that −1 − (n +c) ≤ a n ≤ b n ≤ 1 + (n +c) Define φ −,n (x) := φ n (x + a n ) and φ +,n (x) := φ n (x + b n ). Then Obviously, φ − (0) = lim n→∞ φ n (a n ) and φ + (0) = lim n→∞ φ n (b n ). By the definitions of a n and b n , we immediately have φ − (0) = 0 and φ + (0) = β, and hence 0 < because the proof of the other one is similar. Note that the following limit is uniform for x in any bounded subset M ⊂ R lim n→∞ κ n (x + a n ) − a n = lim It then follows that for any ∈ R, we have where the last equality is obtained from Proposition 7.2(2) and the continuity of Q. Secondly, we prove that φ ± (x) obtained in the first step have the following properties: (ii) φ − (+∞) and φ + (−∞) are ordered.
Indeed, let x n → +∞ be an increasing sequence in R.
which, together with the choice of δ, implies that φ − (−∞) = 0 and φ + (+∞) = β. Further, since any two real numbers are ordered, we see that there exist sequences Define Γ 1 := {k ∈ N : k + a n km ≤ −k + b n km , ∀m ≥ 1} and Γ 2 := N \ Γ 1 . Then either Γ 1 or Γ 2 has infinitely many elements. If Γ 1 does, then there holds Finally, we prove that either φ − or φ + connects 0 to β. Indeed, we have shown in the second step that φ − (+∞) and φ + (−∞) are ordered. It then follows from the bistability assumption (A5) that there are only three possibilities: We further claim that the possibility (iii) cannot happen. Otherwise, Lemma 3.4 which contradicts assumption (A6). Thus, either (i) or (ii) holds, and hence, we complete the proof.

Discrete-time semiflows in a discrete habitat
In this case, both time T and habitat H are discrete: T = Z + and H = Z. Without confusion, we consider the semiflow {Q n } n≥0 in a metric space E ⊂ C. Since the habitat is discrete, we cannot use the definition of traveling waves with a unknown speed as in Definition 3.1. This is because the wave profile ψ(x) may not be welldefined for all x ∈ R. So we start with the modification of the definition of traveling waves in a discrete habitat.
By Definition 3.2 and Proposition 7.3, it follows that there exists Then, with a little abuse of notation, we have Q n [φ](i) = φ(i + cn), ∀i ∈ Z, n ≥ 0. Such a definition of traveling waves is motivated by the idea employed in the proof of Theorem 3.2.
Let β ≫ 0 be a fixed point of Q. DefineQ : Then we see from [25, Lemma 2.1] thatQ satisfies (A1)-(A3) and (A5) with Q =Q and we have similar results as in Lemma 3.2 and 3.3.
Lemma 3.5. Assume that Q satisfies (A1)-(A3) and (A5). Then there exists a positive rational numberc such that for any c ≥c, we havẽ Lemma 3.6. Assume that Q satisfies (A1)-(A5). Then for each n ∈ N,G n := Q • A κn has a fixed pointφ n in B β such thatφ n is nondecreasing and ψ n ≤φ n ≤ψ n .
Proof. By the same arguments as in the proof of Lemma 3.3, we can obtain a similar inequality as (3.2): Define w n,1 := ψ n and w n,k+1 :=G n [w n,k ], k ≥ 1. Then Note that Q[C β ] is compact and w n,k is nondecreasing in k. It then follows that for any fixed x ∈ R, w n,k (x) converges in X β . Denote the limit byφ n (x). Thenφ n (x) is nondecreasing in x ∈ R and ψ n ≤φ n ≤ψ n . Taking k → ∞ in (3.5), we arrive at This completes the proof.
To overcome the difficulty due to the lack of compactness forQ, we will use the properties of monotone functions established in the Appendix to show the convergence of a sequence inQ[B β ].
Theorem 3.2. Assume that X = C(M, R d ) and Q satisfies (A1)-(A6). Then there exists c ∈ R such that the semiflow {Q n } n≥1 on C β admits a nondecreasing traveling wave ψ(x + cn) with speed c and connecting 0 to β. Further, ψ is either left or right continuous.
Proof. As in the proof of Theorem 3.1, we definẽ a n := sup Then −1 − (n +c) ≤ã n ≤b n ≤ 1 + (n +c). Note that for any x ∈ R, we havẽ 0) is precompact in X β , it then follows that for any x ∈ R,φ n (x − ) := lim y↑xφn (y) andφ n (x + ) := lim y↓xφn (y) both exist. And hence, by the definitions of a n andb n , we haveφ . Let Q be the set of all rational numbers, and {x l } l≥1 ⊂ Q be an increasing sequence converging to x. Using φ n =G n [φ n ] again, we see that for any i ∈ Z and l ≥ 1, and that Q is countable. It then follows that there exists a subindex (still denoted by {n}) and ξ − ≤ ξ + ∈ R such that lim n→∞ãn /n = ξ − , lim n→∞bn /n = ξ + and that for any x ∈ Q, i ∈ Z and l ≥ 1, sequencesφ ±,n (x),φ n (κ n (i +ã n + x l )) and φ n (κ n (i +b n + x l )) converge in X β . And hence, the following limits Clearly,φ ± are nondecreasing functions in B β and for any x ∈ R \ Q,φ ± (x ± ) all exist. Hence, we see from Theorem 7.1, that there exists a countable subset Thus,φ − (x) is left continuous andφ + (x) is right continuous. Note thatφ ± (x) = φ ± (x) for all x ∈ R\Q. It then follows thatφ ±,n (x) converges toφ ± (x) for x ∈ R\Γ 2 , where Γ 2 := Q ∪ Γ 1 is also countable.
Let y k ∈ R \ Γ 2 be an increasing sequence converging to 0 and z k ∈ R \ Γ 2 be an increasing sequence converging to 1, respectively. Note that It then follows that In view of Proposition 7.5, we obtain thatφ A similar result also holds forφ + . Now, the same argument as in the proof of Theorem 3.1 completes the proof.

Time-periodic semiflows
Let ω ∈ T be a positive number, where T = R + or Z + . Recall that a family of mappings {Q t } t∈T is said to be an ω-time periodic semiflow on a metric space E ⊂ C provided that it has the following properties: The mapping Q ω is called the Poincaré map associated with this periodic semiflow.
(ii) In the case where H = Z, U(t, x + ct) is said to be an ω-time periodic traveling wave with speed c of the semiflow {Q t } t∈T if there exists a countable subset Theorem 3.3. Let β(t) be a strongly positive ω-time periodic orbit of {Q t } t∈T restricted on X . Assume that Q := Q ω satisfies hypotheses (A1)-(A6) with β = β(0). Then {Q t } t∈T admits a traveling wave U(t, x + ct) with U(t, −∞) = 0 and U(t, +∞) = β(t) uniformly for t ∈ T . Furthermore, U(t, x) is nondecreasing in x ∈ R.

Continuous-time semiflows in a continuous habitat
In this subsection, we consider continuous-time semiflows in the continuous habitat H = R. Recall that a family of mappings {Q t } t≥0 is said to be a semiflow on a metric space E ⊂ C provided that Q t : E → E satisfies the following properties: Before moving to the study of traveling waves of the semiflow {Q t } t≥0 , we first investigate the spatially homogeneous system, that is, the system restricted on X . Let β ≫ 0 be an equilibrium in X . For each t > 0, we use Σ t to denote the set of all fixed points of the map Q t restricted on X β . Clearly, the equilibrium set of the semiflow is Σ := ∩ t>0 Σ t , which is a subset of Σ t for any t > 0.  Proof. Let t 0 > 0 be given. We first show that any two points u ∈ Σ \ {0, β} and v ∈ Σ t 0 \ {0, β} are unordered. Assume, for the sake of contradiction, that u and v are ordered. Without loss of generality, we also assume that u < v. Then the eventual strong monotonicity implies that u ≪ v. Since u is strongly unstable from above, there exist a unit vector e ∈ Int(X + ) and a number δ > 0 such that is eventually strongly increasing and converges to some α ∈ Σ. Note that α ∈ [[u, v] X is strongly unstable from below. Hence, by the same arguments as in the proof of Proposition 2.1, we obtain a contradiction.
Next we show the set Σ t 0 \ Σ is unordered. For this purpose, we see from the first step that it suffices to prove that for any two ordered elements u < v in Σ t 0 \ Σ, [u, v] X ∩ Σ = ∅. Indeed, by the eventual strong monotonicity, we have u ≪ v. Then, we can choose a sequence {u n } on the segment connecting u and v such that u ≪ u n ≪ u n+1 ≪ v, ∀n ≥ 1. By [34,Theorem 1.3.7] Note that ∪ n≥1 ω(u n ) is contained in the compact set Q t 0 [X β ]. In the compact metric space consisting of all nonempty compact subsets of Q t 0 [X β ] with Hausdorff distance d H , the sequence {ω(u n ) : n ≥ 1} has a convergent subsequence. Without loss of generality, we assume that for some nonempty compact set ̟ ⊂ Q t 0 [X β ], lim n→∞ d H (ω(u n ), ̟) = 0. Since each ω(u n ) is invariant for the semiflow {Q t } t≥0 , so is the compact set ̟, that is, Q t ̟ = ̟, ∀t ≥ 0. For any given x, y ∈ ω, there exist two sequences of points x n , y n ∈ ̟(u n ) such that x n → x and y n → y as n → ∞. Since ω(u n ) ≤ ω(u n+1 ), we have x n ≤ y n+1 and y n ≤ x n+1 , ∀n ≥ 1. Letting n → ∞, we then have x ≤ y and y ≤ x, and hence x = y. This implies that ̟ is a singleton, that is, ̟ = {α}. By the invariance of ̟ for the semiflow, we see that α is an equilibrium.
For a continuous-time semiflow {Q t } t≥0 , we need the following definition of traveling waves.
Theorem 3.4. Assume that for each t > 0, the map Q t satisfies assumptions (A1),(A3)-(A5) with E = Σ t , and the time-one map Q 1 satisfies (A6) with E = Σ. Then there exists c ∈ R such that {Q t } t≥0 admits a non-decreasing traveling wave with speed c and connecting 0 to β .
Proof. Let e 0 , e β and δ be chosen as in 2.7, 2.3 and 3.1, respectively. We proceeds with three steps.
Firstly, we show that there exists s k ↓ 0 such that each discrete semiflow {Q n s k } n≥0 admits two nondecreasing traveling waves ψ ±,s k (x + c ±,s k t) with c −,s k ≥ c +,s k and ψ ±,s k has the following properties: Indeed, since for each s > 0 the map Q s satisfies (A1)-(A5), from the first two steps of the proof for Theorem 3.1, we see that for the discrete semiflow {(Q s ) n } n≥0 , there exists two nondecreasing traveling waves φ ±,s (x+c ±,s t) with the following properties: (1) φ −,s connects 0 to some α −,s ∈ E s \{0} and φ +,s connects some α +,s ∈ E s \{β} to β; (2) α −,s and α +,s are ordered and c −,s ≥ c +,s .
By a similar argument as in [47,Theorem 1.3.7], it then follows that both α ±,s have a subsequence α ±,s k which tends to an equilibrium of the semiflow as s k → 0, say the limit α − and α + , respectively. Since α −,s and α +,s are ordered, it follows from Proposition 3.1 that there are only three possibilities for the relation of α − and α + : If α − = β, then for sufficiently large k we can define And hence, ψ −,s (x) := ψ s (x+a s ) and ψ +,s (x) := ψ s (x+b s ) are the required traveling waves. If α + = 0, then for sufficient large k we can define And hence, ψ −,s k (x) := φ +,s k (x + a s k ) and ψ +,s k (x) := φ +,s k (x + b s k ) are the required traveling waves.
To prove that 1 s k c ±,s k have convergent subsequences, we only need to prove that 1 s k c −,s k is bounded above and 1 s k c +,s k is bounded below because c −,s k ≥ c +,s k . Assume, for the sake of contradiction, that some subsequence, still say 1 s k c −,s k , tends to +∞. Note that for each s > 0 there exists n s ∈ Z + such that the integer part of Thus, using the first observation in (3.8), we have which contradicts the fact that Q 1 [δe 0 ] ≪ δe 0 . Similarly, if 1 s k c +,s k → −∞, then the second observation in (3.8) implies that which contradicts the fact that Finally, we show that either ψ − (x + c − t) or ψ + (x + c + t) established in the second step is a traveling wave connecting 0 to β. Indeed, for any t > 0, there exists m k ∈ Z and r k ∈ [0, s k ) such that t = m k s k − r k . Clearly, r k → 0 as k → ∞. Then we have where the last equality follows from Proposition 7.2(2). From the equality Q t [ψ ± ] = ψ ± (· + ct), ∀t ≥ 0, we see that ψ(±∞) are equilibria. Recall that ψ − (−∞) ≤ δe 0 ≤ ψ(+∞) and ψ + (+∞) ≥ β − δe β ≥ ψ + (−∞). It then follows that ψ − (−∞) = 0, ψ + (+∞) = β, and there are only three possibilities for ψ − (+∞) and ψ + (−∞): Since the time-one map Q 1 satisfies (A6) with E = Σ, we can employ the same arguments as in the proof of Lemma 3.4 to exclude the possibility (iii). Thus, either (i) or (ii) holds, and hence, we complete the proof.

Continuous-time semiflows in a discrete habitat
In this case, time T = R + and habitat H = Z. Let β ≫ 0 be an equilibrium of the semiflow {Q t } t≥0 . We start with the definition of traveling waves for this case.
Lemma 3.7. {Q t } t≥0 has the following properties: We combine the ideas in the proofs of Theorems 3.2 and 3.4 to prove the following result for continuous-time semiflows in a discrete habitat Z.
Theorem 3.5. Let X = C(M, R d ). Assume that for each t > 0, the map Q t satisfies (A1), (A3)-(A5) with E = Σ t , and that the time-one map Q 1 satisfies (A6) with E = Σ. Then there exists c ∈ R such that {Q t } t≥0 admits a non-decreasing traveling wave with speed c and connecting 0 to β.
Firstly, since for any s > 0 the map Q s satisfies assumptions (A1)-(A5), it then follows from the proof of Theorems 3.2 and 3.4 that there exists s k ↓ 0 such that {(Q s k ) n } n≥0 admits two nondecreasing traveling wavesψ ±,s k (x + c ±,s k n) with c −,s k ≥ c +,s k , that is, there exists countable subset Θ k such that Furthermore,ψ −,s k is left continuous andψ +,s k is right continuous with the following properties: Secondly, we show that for the above sequence s k , there exist a countable set Γ ⊂ R and a subsequence, still denoted by s k , such that 1 s k c ±,s k → c ± ∈ R and that ψ ±,s k (x) converges in X for all x ∈ R \Γ. Indeed, let Θ = ∪ ∞ k=1 Θ k . Hence, Θ is countable andQ Further,ψ ± (x ± ) exist for all x ∈ R \ Γ. Hence, it follows from Theorem 7.1 that there exists a countable subsetΓ of R such that ψ ±,s k (x) →ψ ± (x), ∀x ∈ R \Γ. (3.11) By similar arguments as in the second step of the proof of Theorem 3.4, we can show that 1 s k c ±,s k are bounded. Finally, we prove that eitherψ − (x + c − t) orψ + (x + c + t) is a nondecreasing traveling wave connecting 0 to β. Indeed, from (3.11) and Proposition 7.3, we see that there exists a countable subset Γ 1 of R such that Hence, for any x ∈ R \ Γ 1 and t > 0, we havẽ In the case where c − = 0, we can choose x 0 such that In the case where c − = 0, we know that there exists a countable subset Γ 2 of R such that Without loss of generality, we assume that c − > 0. For any y ∈ R, we can choose x 0 ∈ R and t 0 ≥ 0 such that x 0 +c − t 0 = y andψ − (x) is continuous at x = x 0 +i for all i ∈ Z. Now one can find x ±,k ∈ R\Γ 1 and t ±,k → t 0 with y ±,k := x ± +c − t ±,k ∈ R\Γ 2 such that y −,k ↑ y and y +,k ↓ y. Note that Thus,ψ − (x) is continuous in x ∈ R. And hence, again by Proposition 7.5 and the equality (3.12), is a traveling wave connecting 0 to some α − ∈ Σ \ {0}. Similarly, we can construct the traveling waveψ + (x + c + t) connecting some α + ∈ Σ \ {β}. Besides, α − and α + are ordered. Now the rest of the proof is essentially the same as in the proof of Theorem 3.4.

Semiflows in a periodic habitat
A typical example of evolution systems in a periodic habitat is Under the assumption that f has exactly three ordered zeros 0 < a < 1 and f ′ (0) < 0, f ′ (a) > 0, f ′ (1) < 0, Xin [42] employed perturbation methods to obtain the existence of spatially periodic traveling wave V (x + ct, x) with V (−∞, ·) = 0 and V (+∞, ·) = 1 provided that d(x) is sufficiently closed to a positive constant in certain sense (see also [44]). For a general positive periodic function d(x), the existence of such a traveling wave remains open. We will revisit this problem in subsection 6.3. A map Q : E → E ⊂ C is said to be spatially periodic with a positive period (ii) V (x+ct, x) is said to be a spatially r-periodic traveling wave with speed c of the It is easy to see that For any φ ∈ C, defineφ ∈ S bỹ φ(ri)(y) = φ(ri + y), ∀i ∈ Z, y ∈ [0, r] H .
Then we have the following observation.
Proof. For any x ∈ H, we can find i ∈ Z and y ∈ [0, r] H such that x = ri + y. It is easy to see that such decomposition of x is unique when x ∈ H \ rZ and is in two possible ways when x ∈ rZ. More precisely, when x ∈ rZ, it can be decomposed into either x = r(i+1)+0 or x = ri+r for some i ∈ Z. Note that f (r(i+1))(0) = f (ri)(r). It then follows that φ f (x) = φ f (ri + y) := f (ri)(y) is a well-defined function in C. Clearly If we define F : C → K by F (φ) =φ, then F is a homeomorphism between C and K. Let β(x) be a strongly positive r-periodic steady state of the semiflow {Q t } t≥0 . With a little abuse of notation, we use C β to denote the set {φ ∈ C : 0 ≤ φ ≤ β}. Now we can define a semiflow {P t } t∈T on Kβ := {f ∈ K : 0 ≤ f ≤β} by Clearly, P t • F = F • Q t , ∀t ∈ T , which implies that semiflows {Q t } t∈T and {P t } t∈T are topologically conjugate. Moreover, {P t } t∈T is spatially homogeneous andβ is its equilibrium. Thus, we see that the semiflow {Q t } t∈T on C β has a spatially r-periodic traveling wave if the semiflow {P t } t∈T on Kβ has a traveling wave. Before stating the main result, we first introduce the bistability assumption. Let β(x) ≫ 0 be an r-periodic steady state of the semiflow {Q t } t∈T . Assume that 0 is a trivial steady state. Define As in Definition 2.1, we can define the strong stability of periodic steady states for a map Q in the space of periodic functions.

Definition 4.2.
A steady state α ∈ Π β is said to be strongly stable from below for the map Q : Π β → Π β if there exist a positive number δ + α and a strongly positive element e + α ∈ Π β such that The strong instability from below is defined by reversing the inequality (4.3). Similarly, we can define strong stability (instability) from above.
We need the following bistability assumption on the spatially r-periodic map Q.
(A5 ′′ ) (it Bistability) 0 and β ≫ 0 are two strongly stable r-periodic steady states from above and below, respectively, for Q : Π β → Π β , and the set of all intermediate r-periodic steady states are totally unordered in Π β .
We note that a sufficient condition for the non-ordering property of all intermediate r-periodic steady states is: Q : Π β → Π β is eventually strongly monotone and all intermediate fixed points are strongly unstable from both above and below. Assume that for any t > 0, the map Q t satisfies (A2)-(A4) and the bistability assumption (A5 ′′ ). Further, assume that the map P 1 := F Q 1 F −1 satisfies assumption (A6) with C and β replaced by K andβ, respectively. Then the spatially r-periodic semiflow {Q t } t∈T admits an r-periodic traveling wave V (x, x + ct). Besides, V (x, ξ) is nondecreasing in ξ and connecting 0 to β(x).
Proof. Let t ≥ 0 be fixed and P t be defined as in (4.2). Then it is easy to see that P t satisfies (A1)-(A5) with C β replaced by Kβ. From Theorems 3.2 and 3.5, we see that {P t } t∈T admits a traveling waves U(x + ct) with U connecting 0 toβ. By the definitions of traveling waves in a discrete habitat (see Definitions 3.2 and 3.5), we can find x 0 ∈ R such that g := U(·+x 0 ) ∈ Kβ and P t [g](ri) = U(ri+ct+x 0 ), ∀i ∈ Z. By Lemma 4.1, we can find ψ, h t ∈ C such thatψ = g andh t = U(· + ct + x 0 ), and hence, P t [ψ] =h t . By the topological conjugacy of Q t and P t , we have Q t [ψ] = h t . Note thatψ = g = U(· + x 0 ) =h 0 . It then follows from Lemma 4.1 that ψ = h 0 . If c = 0, then we obtain Q t [ψ] = h t ≡ h 0 = ψ, which implies that ψ is a traveling wave with speed zero. If c = 0, then we define V (ξ, x) := hξ−x c (x). Consequently, This completes the proof.
To finish this section, we remark that the bistability structure can be obtained for equation (4.1) under appropriate conditions so that the existence result in [42,44] is improved (see the details in subsection 6.3). Further, Theorem 4.1 with H = Z and X = R can be used to rediscover the existence result in [14] for one dimensional lattice equation under the bistability assumption.

Semiflows with weak compactness
In assumption (A4) of section 2, we assume that Q : C β → C β is compact with respect to the compact open topology. In this section, we establish the existence of bistable waves under some weaker compactness assumptions.
Let τ > 0 be a fixed number. It is well known that the time-t solution map of time-delayed reaction-diffusion equations such as  , 0], R)). The first purpose of this section is to show that our results are still valid for this kind of evolution equations by introducing an alternative assumption (A4 ′ ).
In order to state this assumption, we need some notations for time-delayed evolution systems. Let τ ∈ T be a positive number, F be a Banach lattice with the positive cone F + having non-empty interior, β ∈ Int(F + ), and X β = C([−τ, 0], F β ). For any φ ∈ C β , we can regard it as an element in C([−τ, 0]×H, F + ). For any subset B of [−τ, 0] × H, we define φ| B as the restriction of φ on B.
Let T (0) = I, and for any t ∈ (0, τ ], let T (t) be the time-t map of the heat equation u t = ∆u. Then (5.1) can be written as the following form: and hence, Q t [φ](θ, x) = u(t + θ, x). Note that for any φ ∈ C β , T (t)φ → φ with respect to the compact open topology as t → 0. It then follows from the triangular inequality and the absolute continuity of integrals that for any compact subset Lemma 5.1. Let A ξ , ξ ≥ 1, be defined as in section 3 and β ∈ Int(F + ). Assume that Q : C β → C β satisfies (A4 ′ ). Then there exists an integer m 0 such that ∪ ξ∈ [1,1+δ] Proof. We only prove the case where H = R since the proof for H = Z is essentially similar. Let s and τ be defined in (A4 ′ ). For such s and τ , there exists m 0 ∈ N such that s ∈ ( 1 m 0 +1 τ, 1 m 0 τ ]. By assumption (A4 ′ )(i), we see that for any ξ ≥ 1 and This implies that for any ξ ≥ 1 and ǫ < s − 1 Since Q[C β ]| [−s+ǫ,0]×R is precompact, as assumed in (A4 ′ (ii)), it then follows that ∪ ξ∈ [1,2] Q•A ξ [C β ](0, ·) ⊂ C(R, Y β ) is precompact. By (A4 ′ )(iii) and similar arguments as above, we have This implies that ∪ ξ∈ [1,2] This implies that ∪ ξ∈ [1,2]  Proof. Following the proof of theses theorems, we only need to modify the parts where we use the compactness assumption (A4). At these parts, by Lemma 5.1 we can easily complete the proof.
Note that the solution maps of the integro-differential equation satisfy neither (A4) nor (A4 ′ ). The second purpose of this section is to modify our developed theory in such a way that it applies to these integro-differential systems. Let M denote the set of all nondecreasing functions from R to X and β ∈ X + . We equip M with the compact open topology. Assume that Q maps M β to M β . Let E denote the set of fixed point of Q restricted on X β . Suppose that 0 and β are in E. We impose the following assumptions on Q: (B4) (Weak Compactness) For any fixed x ∈ R, the set Q[M β ](x) is precompact in X β .
(B5) (Bistability) Fixed points 0 and β are strongly stable from above and below, respectively, for the map Q : X β → X β , and the set E \ {0, β} ⊂ X β is totally unordered.
Comparing assumptions (A1)-(A6) and (B1)-(B6), one can find that the assumptions of translation invariance, monotonicity, bistability and counter-propagation are the same. The difference lies in the assumptions of continuity and compactness. Clearly, compactness assumption (B4) is much weaker than (A4).
Theorem 5.2. Let X = C(M, R d ) and assume that Q : M β → M β satisfies (B1)-(B6). Then there exists c ∈ R and ψ ∈ M β connecting 0 to β such that Proof. Combining the proofs of Theorems 3.1 and 3.2, we can obtain the result. More precisely, one can repeat the proof of Theorem 3.1 except for the parts where the compactness assumption (A4) are used. For these parts, one use the idea in Theorem 3.2, whereQ has the same compactness property as Q.
In the rest of this section, we say {Q t } t≥0 is a semiflow on M β provided that Assume that {Q t } t≥0 is a semiflow on M β , and for any t > 0, the map Q t satisfies (B1) and (B3)-(B6). Then there exist c ∈ R and ψ ∈ M β connecting 0 to β such that Q t [ψ](x) = ψ(x + ct) for all x ∈ R.
Proof. As in the proof of Theorem 5.2, we can prove the conclusion by combing the proofs of Theorems 3.4 and 3.5.
Similarly, we can define ω-time periodic semiflows on M β and then obtain the following result.

Applications
In this section, we apply the obtained abstract results to four kinds of monotone evolution systems: a time-periodic reaction-diffusion system, a parabolic system in a cylinder, a parabolic equation with variable diffusion, and a nonlocal and timedelayed reaction-diffusion equation.
Let f ∈ C 1 (R + × R n , R n ). In order to apply Theorem 3.3 to system (6.1), we choose C := C(R, R n ), X := R n , and E ⊂ C to be the set of all bounded functions from R to R n . Using the solution maps {T (t)} t≥0 of the heat equation ∂u ∂t = A∆u, we write (6.1) as the following integral form: Define Q t [φ] := u(t; φ), ∀φ ∈ E. Let 0 and β ≫ 0 be two fixed points of the Poincaré map Q ω in X , and let E be the set of all spatially homogeneous fixed points of Q ω in X β . We impose the following assumptions: (C1) The Jacobian matrix D u f (t, u) is cooperative and irreducible for all t ≥ 0 and u ≥ 0.
(C2) The spatially homogeneous system u ′ = f (t, u) is of bistable type, that is, 0 and β are two stable fixed points of Q ω in the sense that s( d du Q ω [0]) < 0 and s( d du Q ω [β]) < 0, and any α ∈ E \ {0, β} is a unstable in the sense that is the stability modulus of the matrix M defined by s(M) = max{Reλ : λ is an eigenvalue}. Theorem 6.1. Assume that (C1)-(C2) hold, and let β(t) be the periodic solution of u ′ = f (t, u) with β(0) = β. Then there exists c ∈ R such that (6.1) admits a time-periodic traveling wave U(t, x + ct) connecting 0 to β(t).
Proof. It is easy to see that the discrete semiflow {Q n ω } n≥1 on C β satisfies (A1)-(A5) with Q = Q ω . Next we show that (A6) holds with Q = Q ω .

A reaction-diffusion-advection system in a cylinder
In this subsection, we consider the following system where A, B are positively definite diagonal n × n matrix, E is diagonal matrix of smooth functions of y, Ω is a bounded and convex open subset in R m−1 with smooth boundary ∂Ω, ∆ y = m−1 i=1 ∂ 2 /∂y 2 i , and ν is the outer unit normal vector to ∂Ω × R. The existence of bistable traveling waves for (6.8) with n = 1 was obtained in [11]. Here we extend this result to the case n ≥ 2. Assume that f ∈ C 1 (R n , R n ) satisfies the following two conditions: (D1) The Jacobian matrix Df (u) is cooperative and irreducible for all u ≥ 0.
Theorem 6.2. Assume that (D1)-(D2) hold. Then there exists c ∈ R such that system (6.8) admits a traveling wave connecting 0 to β with speed c.

A parabolic equation with periodic diffusion
In this subsection, we study the existence of spatially periodic traveling waves of the parabolic equation  (6.19) admits the bistability structure, then Proposition 3.1 implies (A5 ′′ ) and a similar argument as in the previous section shows that (A6) also holds. Thus, we focus on finding sufficient conditions on d(x) under which (6.19) admits the bistability structure.
Remark 6.1. By the proof above, it follows that the conclusion of Lemma 6.1 is valid for any f ∈ C 1 .
per . Proof. Clearly, Lemma 6.1 implies that Y = ∅. Let d * ∈ Y be given. We need to show that d * is an interior point of Y . Assume, for the sake of contradiction, that there is a sequence of points d n ∈ C 1 per \ Y such that d n → d * in C 1 per as n → ∞. Then (6.19) with d = d n admits a non-constant r-periodic steady state u n with the principle eigenvalue λ 1 (u n , d n ) ≤ 0. Using the transformation v n = d n (u n ) x , we see that (u n , v n ) is a periodic solution of the following ordinary differential system: By elementary phase plane arguments, it then follows that Thus, the sequence of functions ((u n ) x , (v n ) x ) is uniformly bounded and equicontinuous, and hence, (u n , v n ) has a uniformly convergent subsequence, still denoted by (u n , v n ). Let (u * , v * ) be the limiting function of (u n , v n ). Then u * is an r-periodic steady state of (6.19) with d = d * . It is easy to see from (6.23) that u * is not the constant function 0 or 1. Let φ n be the positive eigenfunction associated with λ 1 (u n , d n ). Then Dividing both sides of (6.24) by φ n and integrating from 0 to r, we obtain Since f ∈ C 1 and f ′ (a) > 0, we see that u * cannot be the constant a. Otherwise, the uniform convergence of u n to a implies that f ′ (u n (x)) > 0 for all x ∈ [0, r] and sufficiently large n, which contradicts (6.25). Thus, u * is a non-constant r-periodic function. Since d * ∈ Y , we have λ 1 (u * , d * ) > 0. Note that u n → u * in C(R, R) and d n → d * in C 1 per . By the variational characterization of the principal eigenvalue λ 1 (u n , d n ) (see, e.g., Eq. (5.2) of [8]), it then follows that 0 ≥ λ 1 (u n , d n ) → λ 1 (u * , d * ) > 0, a contradiction.
The following counter-example shows that the parabolic equation (6.19) admits no bistability structure in the general case of periodic function d(x).
Then there exists a positive function d ∈ C 1 per such that (6.19) admits a pair of linearly stable, non-constant, and r-periodic steady states.
Proof. We only consider the case where f (u) = u(1 − u 2 ) since the other one can be obtained under appropriate scalings. Our proof is based on the main result in [21,Theorem 3]. Without loss of generality, we assume that r = 4. In what follows, we use some notations of [21].
Let l ∈ (0, 1) be fixed and c 0 be the step function on [−1, 1] defined by closed to c 0 (in the sense that the distance between D and the graph of c is small enough), the following Neumann boundary problem u t = (cu x ) x + u(1 − u 2 ), x ∈ (−1, 1) u x (t, ±1) = 0 (6.27) admits an odd increasing steady state u c which is linearly stable. That is, there exist λ 1 < 0 and φ > 0 such that In particular, we can choose c such that c x (−1) = c x (1) = 0. Since c is even and f is odd, we see that v c (x) := u c (−x) is also a steady state, and λ 1 is the corresponding eigenvalue with the positive eigenfunction φ(−x). Now we can construct a linearly stable 4-periodic steady state of (6.19). Define two 4-periodic functions: and Then w 1 (x) is a 4-periodic steady state of (6.19) with d =d. Let the positive 4-periodic function ρ(x) be defined by It follows that λ 1 and ρ solve the following eigenvalue problem This implies that w 1 is a linearly stable periodic steady state of (6.19) with d =d. Similarly, so is w 2 (x) := w 1 (x + 2).
As a consequence of Theorem 4.1, together with Lemmas 6.1 and 6.2, we have the following result on the existence of bistable traveling waves for (6.19). Theorem 6.3. Letd be a given positive constant. Then there exists δ 0 > 0 such that for any d ∈ C 1 per with d −d C 0 < δ 0 , (6.19) admits a spatially periodic traveling wave solution u(t, x) := V (x + ct, x) with some speed c ∈ R and connecting 0 to 1. Besides, V (ξ, x) is nondecreasing in ξ.
We remark that Theorem 6.3 is a C 0 -perturbation result in C 1 per , and hence, it improves the existence result in [44, Theorem 3.1], where the H s -perturbation is used for some s > 2.

Appendix
In this appendix, we present certain properties of Banach lattices and countable subsets in R, and some convergence results for sequences of monotone functions, including an abstract variant of Helly's theorem.
Proposition 7.1. A Banach lattice X has the following properties: (1) For any u, v ∈ X with v ∈ X + , if −v ≤ u ≤ v, then u X ≤ v X .
(2) If u k → u and v k → v in X with u k ≥ v k , then u ≥ v.
Proposition 7.2. The space C has the following properties: (1) Let φ be a monotone function in C. If x k ∈ H nondecreasingly tends to x ∈ H ∪ {+∞} and lim k→∞ φ(x k ) = u ∈ X , then lim y↑x φ(y) = u. The similar result holds if x k nonincreasingly tends to x ∈ H ∪ {−∞}.