Traveling Waves for Delayed Cellular Neural Networks with Nonmonotonic Output Functions

Abstract

This work investigates traveling waves for a class of delayed cellular neural networks with nonmonotonic output functions on the one-dimensional integer lattice ℤ. The dynamics of each given cell depends on itself and its nearest 𝑚 left or 𝑙 right neighborhood cells with distributed delay due to, for example, finite switching speed and finite velocity of signal transmission. Our technique is to construct two appropriate nondecreasing functions to squeeze the nonmonotonic output functions. Then we construct a suitable wave profiles set and derive the existence of traveling wave solutions by using Schauder's fixed point theorem.

1. Introduction

Recently, there have been an increasing activity and interest in the study of nervous systems, namely, in the study of equations modeling neural networks, which are applied to a broad scope of fields such as image and video signal processing, robotic and biological versions, and higher brain functions (see [1–6] for more details). The methodology of cellular neural networks (for short, CNN) was first proposed by Chua and Yang [7–9] as an achievable alternative to fully connected neural networks in electric circuit systems. The dynamic evolution of the network is governed by the assumed dynamics of the individual processing units and their reciprocal interactions. The infinite system of ordinary differential equations for CNN distributed in a one-dimensional integer lattice can be described by 𝑑𝑥_𝑛(𝑡)𝑑𝑡=−𝑥_𝑛(𝑡)+𝑧+^𝑚_𝑖=1𝑎_𝑖𝑓𝑥_𝑛−𝑖(𝑡)+𝛼𝑓𝑥_𝑛(𝑡+_𝑚𝑖=1)𝛽_𝑖𝑓𝑥_𝑛+𝑖(𝑡), (1) where 𝑓 is the piecewise-linear output function given by 𝑓(𝑥)=|𝑥|1,if𝑥≥1,𝑥,if≤1,−1,if𝑥≤−1. (2) The quantity 𝑧 is called a threshold or bias term and is related to independent voltage sources in electric circuits. The real constant coefficients 𝑎_𝑖, 𝛼, and 𝛽_𝑖 of the output function 𝑓 constitute the so-called space-invariant template that measures the synaptic weights of self-feedback and neighborhood interaction. In recent years, when the output function 𝑓 is defined as (2), some incisive mathematical analyses have subsequently been done; see [1–3, 7–9] and the references cited therein.

However, to be more realistic, the neural models should be incorporated into time delays, since the transmission of information from one neuron to another is not instantaneous. In models of electronic neural networks, the dynamics of each given cell depends on itself and its nearest left or right neighbors where delays exist in left or right neighborhood interactions due to, for example, finite switching speed and finite velocity of signal transmission (see [7]). Such assumptions can be also found in [1, 10–13]. In the past decades, there are many mathematical results on the study of dynamics for neural models. Particularly, the investigation of traveling wave solutions has attracted much attention due to its significant nature in biology, chemistry, epidemiology, and physics, such as reaction-diffusion equations [14–23] and lattice equations [2, 8, 24, 25]. For delayed CNN models (for short, DCNN) with the piecewise-linear output function (2), the existence of traveling waves has been investigated in [1, 11–13]. If the output function is nonlinear, for example, 𝑓(𝑥)=𝜋1,if𝑥≥1,sin2𝑥,if|𝑥|≤1,−1,if𝑥≤−1, (3) or 𝑓(𝑥)=1,if𝑥≥1,2𝑥−𝑥²,if0≤𝑥≤1,2𝑥+𝑥²,if−1≤𝑥≤0,−1,if𝑥≤−1, (4) the authors of [10] also derived the existence of traveling wave solutions. Recently, when the propagation of signal is very slow, authors of [26] considered CNN models with infinite time delays. More precisely, Yu et al. [26] investigated the existence and nonexistence of monotonic traveling waves for the following more general CNN models with infinite distributed time delay terms: 𝑥^′_𝑛(𝑡)=−𝑥_𝑛(𝑡)+^𝑚_𝑖=1𝑎_𝑖^𝜏₀𝐽_𝑖𝑦𝑓𝑥_𝑛−𝑖𝑡−𝑦𝑑𝑦+𝛼^𝜏₀𝐽_𝑚+1𝑦𝑓𝑥_𝑛+𝑡−𝑦𝑑𝑦_𝑙𝑗=1𝛽_𝑗^𝜏₀𝐽_𝑚+1+𝑗𝑦𝑓𝑥_𝑛+𝑗𝑡−𝑦𝑑𝑦, (5) for 𝑛∈ℤ, 𝑚,𝑙∈ℕ. Here 𝑎_𝑖(𝑖=1,…,𝑚), 𝛼, and 𝛽_𝑗(𝑗=1,…,𝑙) are nonnegative constants; 𝜏 may be a positive constant or infinity; and the output function 𝑓(𝑢) is nondecreasing. Here the dynamics of each given cell depends on itself and its nearest 𝑚 left or 𝑙 right neighborhood cells with distributed delay due to, for example, finite switching speed and finite velocity of signal transmission. Moreover, we point out that the CNN models with discrete time delays or no delays case can be included in this model by choosing suitable kernel functions (see [26]).

Note that some known results on the existence of traveling waves for (5) can be obtained only when the output functions are nondecreasing. But the output functions may not be nondecreasing. For example, let us consider the following two nonmonotonic output functions: 𝑓(𝑥)=1,if𝑥≥1,2sin5𝜋6𝑥,if|𝑥|≤1,−1,if𝑥≤−1,𝑓(𝑥)=1,if𝑥≥1,3𝑥−2𝑥²,if0≤𝑥≤1,3𝑥+2𝑥²,if−1≤𝑥≤0,−1,if𝑥≤−1. (6)(7) It is obvious that the previous results in [2, 3, 10, 12, 27] cannot be applied for CNN or DCNN models with nonmonotonic output functions like (6) or (7). Therefore, it gives us the motivation to consider the existence of traveling wave solutions for DCNN model (5) with nonmonotonic output functions.

Throughout this paper, we assume that there exists a 𝑏>0 such that 𝑓 and 𝐽 satisfy the following assumptions.(F1)𝑓∈𝐶([0,𝑏],[0,𝑏/(𝑎+𝛼+𝛽)]) is an odd function, 𝑓(0)=0, and there exist 𝐾>0 with 𝐾≤𝑏 and 𝐿>0 such that 𝑓(𝐾)𝑓(𝑢)(𝑣||,[],𝑎+𝛼+𝛽=𝐾,−𝑓)≤𝐿𝑢−𝑣for𝑢,𝑣∈0,𝑏 (8)where 𝑎_𝑖≥0, 𝑖=1,…,𝑚, 𝛼≥0, and 𝛽_𝑗≥0,𝑗=1,…,𝑙, 𝑎∶=^𝑚_𝑖=1𝑎_𝑖,𝛽∶=^𝑙_𝑗=1𝛽_𝑗. (9)(F2)𝑓^′(0)𝑢≥𝑓(𝑢)>0, (𝑎+𝛼+𝛽)𝑓(𝑢)>𝑢 for 𝑢∈(0,𝐾), and (𝑎+𝛼+𝛽)𝑓(𝑢)<𝑢 for 𝑢∈(𝐾,𝑏].(F3)There exists 𝜎∈(0,1] such that limsup_𝑢→0+𝑓^′(0)−𝑓(𝑢)𝑢𝑢^−𝜎<+∞. (10)(F4)(𝑎+𝛼+𝛽)𝑓(𝑢)<2𝐾−𝑢 for 𝑢∈(0,𝐾) and (𝑎+𝛼+𝛽)𝑓(𝑢)>2𝐾−𝑢 for 𝑢∈(𝐾,𝑏].(J)𝐽_𝑖∶[0,𝜏]→[0,+∞) are piecewise continuous functions satisfying ^𝜏₀𝐽_𝑖𝑦𝑑𝑦=1,𝑖=1,…,𝑚+𝑙+1, (11)where 0<𝜏<∞. If 𝜏=∞, we further assume that ^∞₀𝐽_𝑖𝑦𝑒^𝜅𝑦𝑑𝑦<∞forany𝜅∈[0,∞),𝑖=1,…,𝑚+𝑙+1. (12)

Remark 1. If 𝑓 satisfies the following hypotheses,(H1)𝑓 is a continuous odd function, 𝑓(0)=0, 𝑓(𝑥)=1 for 𝑥≥1, ∑^𝑚_𝑖=1𝑎_𝑖+∑^𝑙_𝑗=1𝛽_𝑗>0, 𝛼𝑓^′(0)≥1, and there exists 𝐿>0 such that |𝑓(𝑢)−𝑓(𝑣)|≤𝐿|𝑢−𝑣| for 𝑢,𝑣∈[0,1],(H2)𝑓^′(0)𝑢≥𝑓(𝑢)>0 for 𝑢∈(0,1]; in addition, 𝑓^′′(0) exists and (𝛼+𝑎+𝛽)𝑓(𝑢)>𝑢 for 𝑢∈(0,1),then it is obvious that (F1)–(F3) hold and 0, 𝐾=∑^𝑚_𝑖=1𝑎_𝑖+𝛼+∑^𝑙_𝑗=1𝛽_𝑗, and −𝐾 are three equilibria of (5).

Remark 2. If 𝑎+𝛼+𝛽>1, then the nonmonotonic output functions (6) and (7) satisfy assumptions (F1)–(F4).

Our purpose is to investigate the existence of traveling waves for (5) with nonmonotonic output functions. A traveling wave solution of (5) is a special translation invariant solution of (12) with the form 𝑥_𝑛(𝑡)=𝜙(𝑛−𝑐𝑡), 𝑖∈ℤ, and 𝑡∈ℝ for a wave profile 𝜙(𝜉), 𝜉=𝑛−𝑐𝑡∈ℝ, with a given wave speed 𝑐∈ℝ. Letting 𝜉=𝑛−𝑐𝑡, it follows that 𝜙 must be a solution of the following wave profile equation: −𝑐𝜙^′(𝜉)=−𝜙(𝜉)+^𝑚_𝑖=1𝑎_𝑖^𝜏₀𝐽_𝑖𝑦𝑓𝜙𝜉−𝑖+𝑐𝑦𝑑𝑦+𝛼^𝜏₀𝐽_𝑚+1𝑦𝑓𝜙+𝜉+𝑐𝑦𝑑𝑦_𝑙𝑗=1𝛽_𝑗^𝜏₀𝐽_𝑚+1+𝑗𝑦𝑓𝜙𝜉+𝑗+𝑐𝑦𝑑𝑦. (13) Under assumptions (F1)-(F2), it is easily seen that 0 and ±𝐾 are three equilibria of (13). We are mainly interested in the existence of traveling waves satisfying one of the following asymptotic boundary conditions: (BC1)lim_{𝜉→−∞}𝜙(𝜉)=0,liminf_𝜉→+∞𝜙(𝜉)()>0;BC2liminf_{𝜉→−∞}𝜙(𝜉)>0,lim_𝜉→+∞𝜙(𝜉)()=0;BC3lim_{𝜉→−∞}𝜙(𝜉)=0,limsup_𝜉→+∞𝜙(𝜉)()<0;BC4limsup_{𝜉→−∞}𝜙(𝜉)<0,lim_𝜉→+∞𝜙(𝜉)=0. (14)

Since the output functions are nonmonotonic, our main idea is to construct two appropriate nondecreasing functions to squeeze the nonmonotonic output functions. Then we can apply the results in [26] and Schauder’s fixed point theorem to derive the existence of traveling wave solutions. Now we state our main results.

Theorem 3. Assume (F1)–(F3) and (J) hold. If 𝛽>0 and (𝛼+𝛽)𝑓^′(0)>1, then there exists 𝑐^∗₁<0 such that the following statements hold.(1)For each 𝑐<𝑐^∗₁, (5) has a traveling wave 𝜙(𝑛−𝑐𝑡) with speed 𝑐 such that 𝜙∈𝐶(ℝ,[0,𝑏])/{0,𝐾} and satisfies (BC1). Moreover, lim_𝜉→+∞𝜙(𝜉)=𝐾𝑜𝑟0<liminf_𝜉→+∞𝜙(𝜉)<𝐾<limsup_𝜉→+∞𝜙(𝜉)≤𝑏,lim_{𝜉→−∞}𝜙(𝜉)𝑒^−𝜆1𝜉=1,lim_{𝜉→−∞}𝜙^′(𝜉)𝑒^−𝜆1𝜉=𝜆₁, (15)where 𝜆₁ is the smallest positive root of characteristic function of (13) at 0. If, in addition, (F4) holds, then lim_𝜉→+∞𝜙(𝜉)=𝐾.(2)For each 𝑐<𝑐^∗₁, (5) has a traveling wave 𝜓(𝑛−𝑐𝑡) with speed 𝑐 such that 𝜓∈𝐶(ℝ,[−𝑏,0])/{0,−𝐾} and satisfies (BC3). Moreover, lim_𝜉→+∞𝜓(𝜉)=−𝐾𝑜𝑟−𝑏≤liminf_𝜉→+∞𝜓(𝜉)<−𝐾<limsup_𝜉→+∞𝜓(𝜉)<0,lim_{𝜉→−∞}𝜓(𝜉)𝑒^−𝜆1𝜉=−1,lim_{𝜉→−∞}𝜓^′(𝜉)𝑒^−𝜆1𝜉=−𝜆₁. (16)If, in addition, (F4) holds, then lim_𝜉→+∞𝜓(𝜉)=−𝐾.

Theorem 4. Assume (F1)–(F3) and (J) hold. If 𝑎>0 and (𝑎+𝛼)𝑓^′(0)>1, then there exists 𝑐^∗₂>0 such that the following statements hold.(1)For each 𝑐>𝑐^∗₂, (5) has a traveling wave 𝜙(𝑛−𝑐𝑡) with speed 𝑐 such that 𝜙∈𝐶(ℝ,[0,𝑏])⧵{0,𝐾} and satisfies (BC2). Moreover, lim_{𝜉→−∞}𝜙(𝜉)=𝐾𝑜𝑟0<liminf_{𝜉→−∞}𝜙(𝜉)<𝐾<limsup_{𝜉→−∞}𝜙(𝜉)≤𝑏,lim_𝜉→+∞𝜙(𝜉)𝑒^−𝜆2𝜉=1,lim_𝜉→+∞𝜙^′(𝜉)𝑒^−𝜆2𝜉=𝜆₂, (17) where 𝜆₂<0 is the largest negative root of characteristic function of (13) at 0. If, in addition, (F4) holds, then lim_{𝜉→−∞}𝜙(𝜉)=𝐾.(2)For each 𝑐>𝑐^∗₂, (5) has a traveling wave 𝜓(𝑛−𝑐𝑡) with speed 𝑐 such that 𝜓∈𝐶(ℝ,[−𝑏,0])⧵{0,−𝐾} and satisfies (BC4). Moreover, lim_{𝜉→−∞}𝜓(𝜉)=−𝐾𝑜𝑟−𝑏≤liminf_{𝜉→−∞}𝜓(𝜉)<−𝐾<limsup_{𝜉→−∞}𝜓(𝜉)<0,lim_𝜉→+∞𝜓(𝜉)𝑒^−𝜆2𝜉=−1,lim_𝜉→+∞𝜓^′(𝜉)𝑒^−𝜆2𝜉=−𝜆₂. (18)If, in addition, (F4) holds, then lim_{𝜉→−∞}𝜓(𝜉)=−𝐾.

Theorem 5. Assume (F1)–(F3) and (J) hold. If 𝛽=0 and (𝑎+𝛼)𝑓^′(0)>1, then the following statements hold.(1)For any 𝑐<0, (5) has a traveling wave 𝜙(𝑛−𝑐𝑡) with speed 𝑐 such that 𝜙∈𝐶(ℝ,[0,𝑏])⧵{0,𝐾} and satisfies (BC3). Moreover, lim_𝜉→+∞𝜙(𝜉)=𝐾𝑜𝑟0<liminf_𝜉→+∞𝜙(𝜉)<𝐾<limsup_𝜉→+∞𝜙(𝜉)≤𝑏,lim_{𝜉→−∞}𝜙(𝜉)𝑒^−𝜆3𝜉=1,lim_{𝜉→−∞}𝜙^′(𝜉)𝑒^−𝜆3𝜉=𝜆₃, (19)where 𝜆₃>0 is a unique real root of characteristic function of (13) at 0. If, in addition, (F4) holds, then lim_𝜉→+∞𝜙(𝜉)=𝐾. (2)For any 𝑐<0, (5) has a traveling wave 𝜓(𝑛−𝑐𝑡) with speed 𝑐 such that 𝜓∈𝐶(ℝ,[−𝑏,0])⧵{0,−𝐾} and satisfies (BC3). Moreover, lim_𝜉→+∞𝜓(𝜉)=−𝐾𝑜𝑟−𝑏≤liminf_𝜉→+∞𝜓(𝜉)<−𝐾<limsup_𝜉→+∞𝜓(𝜉)<0,lim_{𝜉→−∞}𝜓(𝜉)𝑒^−𝜆3𝜉=−1,lim_{𝜉→−∞}𝜓^′(𝜉)𝑒^−𝜆3𝜉=−𝜆₃. (20)If, in addition, (F4) holds, then lim_𝜉→+∞𝜓(𝜉)=−𝐾.

Theorem 6. Assume (F1)–(F3) and (J) hold. If 𝑎=0 and (𝛼+𝛽)𝑓^′(0)>1, then the following statements hold.(1)For any 𝑐>0, (5) has a traveling wave 𝜙(𝑛−𝑐𝑡) with speed 𝑐 such that 𝜙∈𝐶(ℝ,[0,𝑏])⧵{0,𝐾} and satisfies (BC2). Moreover, lim_{𝜉→−∞}𝜙(𝜉)=𝐾𝑜𝑟0<liminf_{𝜉→−∞}𝜙(𝜉)<𝐾<limsup_{𝜉→−∞}𝜙(𝜉)≤𝑏,lim_𝜉→+∞𝜙(𝜉)𝑒^−𝜆4𝜉=1,lim_𝜉→+∞𝜙^′(𝜉)𝑒^−𝜆4𝜉=𝜆₄, (21)where 𝜆₄<0 is a unique real root of characteristic function of (13) at 0. If, in addition, (F4) holds, then lim_{𝜉→−∞}𝜙(𝜉)=𝐾. (2)For any 𝑐>0, (5) has a traveling wave 𝜓(𝑛−𝑐𝑡) with speed 𝑐 such that 𝜓∈𝐶(ℝ,[−𝑏,0])⧵{0,−𝐾} satisfies (BC4). Moreover, lim_{𝜉→−∞}𝜓(𝜉)=−𝐾𝑜𝑟−𝑏≤liminf_{𝜉→−∞}𝜓(𝜉)<−𝐾<limsup_{𝜉→−∞}𝜓(𝜉)<0,lim_𝜉→+∞𝜓(𝜉)𝑒^−𝜆4𝜉=−1,lim_𝜉→+∞𝜓^′(𝜉)𝑒^−𝜆4𝜉=−𝜆₄. (22)If, in addition, (F4) holds, then lim_{𝜉→−∞}𝜓(𝜉)=−𝐾.

Remark 7. In the above theorems, we obtain that the traveling waves either converge to the nontrivial equilibrium (𝐾 or −𝐾) or oscillate on the nontrivial equilibrium at infinity. Furthermore, we give a sufficient condition for the convergence of traveling waves to the nontrivial equilibrium. But under what conditions the wave will oscillate on the nontrivial equilibrium at infinity is an interesting problem. On the other hand, similarly to the proofs of nonexistence of traveling waves in [26], we can also obtain the same conclusions for nonmonotonic DCNN models.

Remark 8. The different results can be obtained through the choice of the signs for 𝑎 and 𝛽 and (5) reduces to kinds of CNN model with mixed delays by choosing suitable kernel functions. Please refer to [26].

The remainder of this paper is organized as follows. In Section 2, we recall some properties of the characteristic function of (13). Then, in Section 3, we construct two appropriate nondecreasing functions to squeeze the nonmonotonic output functions. Based on the construction of two nondecreasing functions, we devote Section 4 to proofs of the existence of traveling waves for (5). Our approach is to squeeze the nonmonotonic output functions and Schauder’s fixed point theorem in a suitable Banach space. Furthermore, according to the construction of a wave profiles set, we can obtain the exponential asymptotic behavior of traveling waves in the infinity.

2. Properties of the Characteristic Function

In this section, we recall some properties of the characteristic function for (13). The characteristic function arises from the linearized equation of (13) at the equilibrium solution 0 and is given by Δ(𝜆,𝑐)∶=−𝑐𝜆+1−𝑓^′(0)^𝑚_𝑖=1𝑎_𝑖^𝜏₀𝐽_𝑖𝑦𝑒^{𝜆(−𝑖+𝑐𝑦)}𝑑𝑦−𝑓^′(0)𝛼^𝜏₀𝐽_𝑚+1𝑦𝑒^𝜆𝑐𝑦𝑑𝑦−𝑓^′(0)^𝑙_𝑗=1𝛽_𝑗^𝜏₀𝐽_𝑚+1+𝑗𝑦𝑒^{𝜆(𝑗+𝑐𝑦)}𝑑𝑦. (23) The main properties of the characteristic function are stated in the following lemmas, and all the proofs can be found in our recent work [26].

Lemma 9. Assume that 𝛽>0 and (𝛼+𝛽)𝑓^′(0)>1. Then there exists a unique 𝑐^∗₁<0 such that (1)if 𝑐≤𝑐^∗₁, then there exist two positive numbers 𝜆₁(𝑐) and 𝜆₁(𝑐) with 𝜆₁∶=𝜆₁(𝑐)≤𝜆₁∶=𝜆₁(𝑐) such that Δ(𝜆₁,𝑐)=Δ(𝜆₁,𝑐)=0;(2)if 𝑐=𝑐^∗₁, then 𝜆₁=𝜆₁=𝜆^∗₁, and if 𝑐<𝑐^∗₁, then 𝜆₁<𝜆^∗₁<𝜆₁, Δ(⋅,𝑐)>𝑖𝑛𝜆₁,𝜆₁,Δ(⋅,𝑐)<𝑖𝑛ℝ⧵𝜆₁,𝜆₁; (24)(3)if 𝑐>𝑐^∗₁, then Δ(𝜆,𝑐)<0 for all 𝜆≥0.

Lemma 10. Assume that 𝑎>0 and (𝑎+𝛼)𝑓^′(0)>1. Then there exists a unique 𝑐^∗₂>0 such that (1)if 𝑐≥𝑐^∗₂, then there exist 𝜆₂<0 and 𝜆₂<0 with Δ(𝜆₂,𝑐)=Δ(𝜆₂,𝑐)=0;(2)if 𝑐=𝑐^∗₂, then 𝜆₂=𝜆₂=𝜆^∗₂, and if 𝑐>𝑐^∗₂, then 𝜆₂<𝜆^∗₂<𝜆₂, Δ(⋅,𝑐)>𝑖𝑛𝜆₂,𝜆₂,Δ(⋅,𝑐)<𝑖𝑛ℝ⧵𝜆₂,𝜆₂; (25)(3)if 𝑐<𝑐^∗₂, then Δ(𝜆,𝑐)<0 for all 𝜆≤0.

Lemma 11. Assume that 𝛽=0 and (𝑎+𝛼)𝑓^′(0)>1. Then we have the following:(1)for any 𝑐<0, there exists a unique real root 𝜆₃∶=𝜆₃(𝑐)>0 such that Δ(𝜆₃,𝑐)=0. Moreover, for 𝑐<0, there exists 𝜖>0 such that Δ(𝜆₃+𝜖,𝑐)>0 for 𝜖∈(0,𝜖) and Δ(𝜆,𝑐)>0 for 𝜆∈(𝜆₃,𝜆₃+𝜖);(2)for 𝑐≥0, Δ(𝜆,𝑐)<0 for all 𝜆>0.

Lemma 12. Assume that 𝑎=0 and (𝛼+𝛽)𝑓^′(0)>1. Then we have the following:(1)for any 𝑐>0, there exists a unique real root 𝜆₄∶=𝜆₄(𝑐)<0 such that Δ(𝜆₄,𝑐)=0. Moreover, for any 𝑐>0, there exists 𝜖̂>0 such that Δ(𝜆₄−𝜖,𝑐)>0 for 𝜖∈(0,𝜖)̂ and Δ(𝜆,𝑐)>0 for 𝜆∈(𝜆₄−𝜖,𝜆₄);(2)for 𝑐≤0, Δ(𝜆,𝑐)<0 for all 𝜆<0.

According to the results of the above lemmas, in Section 4, we can see that the roots of characteristic equation play crucial roles in studying the behavior solutions of (13) near the equilibrium 0.

3. Construction of Nondecreasing Functions

In this section, we will construct two nondecreasing functions such that 𝑓 lies between the two functions. Based on the construction of the nondecreasing functions, we can apply the results in [26] to derive the existence results.

Let us define functions 𝑓⁻(𝑢) and 𝑓⁺(𝑢) by 𝑓⁻(𝑢)=inf_{𝑢≤𝑣≤𝑏}𝑓(𝑣,𝑓₊)(𝑢)=min𝑓^′(0)𝑏𝑢,[].𝑎+𝛼+𝛽for𝑢∈0,𝑏 (26) Then 𝑓⁻(𝑢) and 𝑓⁺(𝑢) satisfy the following properties.

Lemma 13. Assume that (F1)–(F3) hold.(1)𝑓⁻(𝑢) and 𝑓⁺(𝑢) are nondecreasing and Lipschitz continuous on [0,𝑏].(2)𝑓⁻(𝑢)≤𝑓(𝑢)≤𝑓⁺(𝑢) for all 𝑢∈[0,𝑏].(3)𝑓^′(0)𝑢≥𝑓⁻(𝑢)>0 and 𝑓^′(0)𝑢≥𝑓⁺(𝑢)>0 for all 𝑢∈(0,𝑏].(4)𝑓⁺(0)=(𝑎+𝛼+𝛽)𝑓⁺(𝑏)−𝑏=0 and there exists an 𝑢^∗₋ with 0<𝑢^∗₋≤𝐾 such that 𝑓⁻(0)=(𝑎+𝛼+𝛽)𝑓⁻(𝑢^∗₋)−𝑢^∗₋=0. Moreover, we have (𝑎+𝛼+𝛽)𝑓⁻(𝑢)>𝑢 for any 𝑢∈(0,𝑢^∗₋) and (𝑎+𝛼+𝛽)𝑓⁺(𝑢)>𝑢 for any 𝑢∈(0,𝑏).(5)There exists 𝜎∈(0,1] such that limsup_𝑢→0+𝑓^′(0)−𝑓⁻(𝑢)𝑢𝑢^−𝜎<∞. (27)

Proof. It is obvious that the assertions of (1), (3), and (5) hold. Therefore, we only prove the assertions of parts (2) and (4).
(2) By the definition of 𝑓⁻(𝑢) and (F2), we have 𝑓⁻(𝑢)≤𝑓(𝑢) for 𝑢∈[0,𝑏]. According to (F1)-(F2), it is obvious that 𝑓(𝑢)≤𝑓^′(0)𝑢 and 𝑓(𝑢)≤𝑏/(𝑎+𝛼+𝛽) for 𝑢∈[0,𝑏]. Thus, 𝑓(𝑢)≤𝑓⁺(𝑢) for 𝑢∈[0,𝑏].
(4) Obviously, 𝑓⁻(0)=𝑓⁺(0)=0. Since (𝑎+𝛼+𝛽)𝑓^′(0)>1, it follows that (𝑎+𝛼+𝛽)𝑓⁺(𝑏)=𝑏 and there exists 𝜅>0 such that (𝑎+𝛼+𝛽)𝑓⁻(𝜅)>𝜅. By the assertion of part (2), we have 𝑎+𝛼+𝛽𝑓⁻(𝐾)≤𝑎+𝛼+𝛽𝑓(𝐾)=𝐾. (28) Thus, there exists an 𝑢^∗₋∈(𝜅,𝐾] such that 𝑎+𝛼+𝛽𝑓⁻𝑢^∗₋=𝑢^∗₋,𝑎+𝛼+𝛽𝑓⁻(𝑢)>𝑢forany𝑢∈0,𝑢^∗₋. (29) On the other hand, since (𝑎+𝛼+𝛽)𝑓^′(0)𝑢>𝑢 and 𝑏>𝑢 for any 𝑢∈(0,𝑏), by the definition of 𝑓⁺(𝑢), we have (𝑎+𝛼+𝛽)𝑓⁺(𝑢)>𝑢 for any 𝑢∈(0,𝑏). This completes the proof.

4. Existence of Traveling Wave Solutions

Now, applying the techniques developed in [26] to the functions 𝑓⁻(𝑢) and 𝑓⁺(𝑢), we prove the main theorems in the sequel.

4.1. Traveling Waves with (BC1) and (BC3)

First, we consider the existence of traveling wave solutions satisfying condition (BC1). Once there exist traveling waves with (BC1), then we can easily obtain that (5) admits traveling wave solutions satisfying condition (BC3).

Let us define the operator 𝑇∶𝐶(ℝ,[0,𝑏])→𝐶(ℝ,[0,𝑏]) by 𝑇𝜙(𝜉)=𝑒^{−𝛾𝜉𝜉}_−∞𝑒^𝛾𝑦𝐹𝜙𝑦𝑑𝑦, (30) where 𝛾>−1/𝑐>0 is a constant and 𝐹𝜙(𝜉)=1𝛾+𝑐𝜙(𝜉)−1𝑐^𝑚_𝑖=1𝑎_𝑖^𝜏₀𝐽_𝑖𝑦𝑓𝜙−𝛼𝜉−𝑖+𝑐𝑦𝑑𝑦𝑐^𝜏₀𝐽_𝑚+1𝑦𝑓𝜙−1𝜉+𝑐𝑦𝑑𝑦𝑐^𝑙_𝑗=1𝛽_𝑗^𝜏₀𝐽_𝑚+1+𝑗𝑦𝑓𝜙𝜉+𝑗+𝑐𝑦𝑑𝑦. (31) According to (F1) and 𝑐<0, 𝑇 is well defined. It is easy to see that a fixed point 𝜙 of 𝑇 or a solution of the equation 𝜙(𝜉)=𝑇𝜙(𝜉),𝜉∈ℝ, (32) is a traveling wave solution of (5).

Let 𝑇^± be defined as in (30) with 𝑓 replaced by 𝑓^±; that is, 𝑇^±𝜙(𝜉)=𝑒^{−𝛾𝜉𝜉}_−∞𝑒^𝛾𝑦𝐹^±𝜙𝑦𝑑𝑦, (33) where 𝐹^±𝜙(𝜉)=1𝛾+𝑐𝜙(𝜉)−1𝑐^𝑚_𝑖=1𝑎_𝑖^𝜏₀𝐽_𝑖𝑦𝑓^±𝜙−𝛼𝜉−𝑖+𝑐𝑦𝑑𝑦𝑐^𝜏₀𝐽_𝑚+1𝑦𝑓^±𝜙−1𝜉+𝑐𝑦𝑑𝑦𝑐^𝑙_𝑗=1𝛽_𝑗^𝜏₀𝐽_𝑚+1+𝑗𝑦𝑓^±𝜙𝜉+𝑗+𝑐𝑦𝑑𝑦. (34) According to Lemma 13, it is easily seen that 𝐹^± and 𝑇^± are nondecreasing on 𝐶(ℝ,[0,𝑏]) and 𝑇⁻𝜙≤𝑇𝜙≤𝑇⁺𝜙,for𝜙∈𝐶(ℝ,[0,𝑏]). (35)

Now we define the functions 𝜙⁺(𝜉) and 𝜙⁻(𝜉) by 𝜙⁺(𝜉)=∶min𝑏,𝑒^𝜆1𝜉,𝜙⁻(𝜉)=∶min𝑢^∗₋,𝑒^𝜆1𝜉for𝜉∈ℝ. (36) Since 𝑢^∗₋≤𝑏, it is obvious that 𝜙⁻(𝜉)≤𝜙⁺(𝜉). Similar to the proof of Lemma 3.1 in [26], it is easily seen that the following lemma holds.

Lemma 14. Assume that (F1)–(F3) hold. Then, for any 𝑐<𝑐^∗₁, 𝑇⁺𝜙⁺(𝜉)≤𝜙⁺(𝜉),𝑇⁻𝜙⁻(𝜉)≤𝜙⁻(𝜉),∀𝜉∈ℝ. (37)

Since 𝑓⁻ is nondecreasing and satisfies the assumptions illustrated in [26], then we obtain the following result.

Lemma 15. Assume that (F1)–(F3) hold. Then, for any 𝑐<𝑐^∗₁, there exists a fixed point 𝜙⁻ of 𝑇⁻ such that 𝑇⁻(𝜙⁻)(𝜉)=𝜙⁻(𝜉) and 𝜙⁻(𝜉)≤𝜙⁻(𝜉) for all 𝜉∈ℝ. Moreover, lim_{𝜉→−∞}𝜙⁻(𝜉)𝑒^−𝜆1𝜉=1,lim_𝜉→+∞𝜙⁻(𝜉)=𝑢^∗₋. (38)

Proof. The proof is the same as that of Theorem 1.1 of [26], and it is omitted.

By Lemmas 13–15 and (33), we easily verify that 𝜙⁻(𝜉)=𝑇⁻𝜙⁻(𝜉)≤𝑇⁻𝜙⁻(𝜉)≤𝑇⁺𝜙⁻(𝜉)≤𝑇⁺𝜙⁺(𝜉)≤𝜙⁺(𝜉). (39) For a given number 𝜆>0, let 𝑋_𝜆∶=𝜙∈𝐶(ℝ,ℝ)∣sup_𝜉∈ℝ𝜙(𝜉)𝑒^−𝜆𝜉,𝜙<∞_𝑋𝜆=sup_𝜉∈ℝ𝜙(𝜉)𝑒^−𝜆𝜉. (40) It is easy to see that (𝑋_𝜆,‖⋅‖_𝑋𝜆) is a Banach space. Then we define the following set: Γ∶=𝜙∈𝑋_𝜆∣𝜙⁻(𝜉)≤𝜙(𝜉)≤𝜙⁺(𝜉),for𝜉∈ℝ, (41) where 𝜆∈(0,𝜆₁). By (39) and the definitions of 𝜙⁻, 𝜙⁺, we easily check that Γ is a nonempty closed convex subset of 𝑋_𝜆. Thus, we have the following assertion.

Lemma 16. Assume that (F1)-(F2) hold. Then we have(1)𝑇(Γ)⊂Γ;(2)𝑇∶Γ→Γ is completely continuous with respect to the norm ‖⋅‖_𝑋𝜆 in 𝑋_𝜆.

Proof. (1) For 𝜙∈Γ, by (39) and Lemmas 14–15, it follows that 𝜙⁻(𝜉)=𝑇⁻𝜙⁻(𝜉)≤𝑇⁻𝜙(𝜉)≤𝑇𝜙(𝜉)≤𝑇⁺𝜙(𝜉)≤𝑇⁺𝜙⁺(𝜉)≤𝜙⁺(𝜉), (42) which implies that 𝑇(Γ)⊂Γ.
(2) For any 𝜙,𝜓∈Γ, it follows from (F1) that 𝐹𝜙(𝜉)𝜓(𝜉≤1−𝐹)𝛾+𝑐𝜙(𝜉)(𝜉−𝛼−𝜓)𝑐^𝜏₀𝐽_𝑚+1𝑦𝑓𝜙𝜓−1𝜉+𝑐𝑦−𝑓𝜉+𝑐𝑦𝑑𝑦𝑐^𝑚_𝑖=1𝑎_𝑖^𝜏₀𝐽_𝑖𝑦×𝑓𝜙𝜓−1𝜉−𝑖+𝑐𝑦−𝑓𝜉−𝑖+𝑐𝑦𝑑𝑦𝑐^𝑙_𝑗=1𝛽_𝑗^𝜏₀𝐽_𝑚+1+𝑗𝑦×𝑓𝜙𝜓≤1𝜉+𝑗+𝑐𝑦−𝑓𝜉+𝑗+𝑐𝑦𝑑𝑦𝛾+𝑐𝜙(𝜉)(𝜉−−𝜓)𝐿𝛼𝑐^𝜏₀𝐽_𝑚+1𝑦𝜙−𝐿𝜉+𝑐𝑦−𝜓𝜉+𝑐𝑦𝑑𝑦𝑐^𝑚_𝑖=1𝑎_𝑖^𝜏₀𝐽_𝑖𝑦𝜙−𝐿𝜉−𝑖+𝑐𝑦−𝜓𝜉−𝑖+𝑐𝑦𝑑𝑦𝑐^𝑙_𝑗=1𝛽_𝑗^𝜏₀𝐽_𝑚+1+𝑗𝑦×𝜙𝜉+𝑗+𝑐𝑦−𝜓𝜉+𝑗+𝑐𝑦𝑑𝑦. (43) Thus, we have 𝑇(𝜙)−𝑇(𝜓)_𝑋𝜆=sup_𝜉∈ℝ𝑇𝜙(𝜉)−𝑇𝜓(𝜉)𝑒^−𝜆𝜉≤sup_𝜉∈ℝ𝑒^{−(𝛾+𝜆)𝜉𝜉}_−∞𝐹𝜙(𝑠)−𝐹𝜓(𝑠)𝑒^𝛾𝑠≤1𝑑𝑠1𝜆+𝛾𝛾+𝑐−𝐿𝑐^𝑚_𝑖=1𝑎_𝑖^𝜏₀𝐽_𝑖𝑦𝑒^{𝜆(𝑐𝑦−𝑖)}−𝑑𝑦𝐿𝛼𝑐^𝜏₀𝐽_𝑚+1𝑦𝑒^𝑐𝜆𝑦−𝐿𝑑𝑦𝑐^𝑙_𝑗=1𝛽_𝑗^𝜏₀𝐽_𝑚+1+𝑗𝑦𝑒^{𝜆(𝑐𝑦+𝑗)}×𝑑𝑦𝜙−𝜓_𝑋𝜆, (44) which implies that 𝑇∶Γ→Γ is continuous. On the other hand, for any 𝜙∈Γ, 𝜉∈ℝ, we have 𝐹(𝜙)(𝜉)≤𝛾𝑏, and 𝑇𝜙𝜉₁−𝑇𝜙𝜉₂≤𝛾𝑏𝑒^{−𝛾𝜉1𝜉1}_−∞𝑒^𝛾𝑠𝑑𝑠−𝑒^{−𝛾𝜉2𝜉2}_−∞𝑒^𝛾𝑠𝑒𝑑𝑠≤𝛾𝑏^{−𝛾𝜉1𝜉1}_𝜉2𝑒^𝛾𝑠𝑑𝑠+𝑒^−𝛾𝜉2−𝑒^{−𝛾𝜉1𝜉2}_−∞𝑒^𝛾𝑠𝑑𝑠=2𝑏1−𝑒^{−𝛾(𝜉1−𝜉2)} (45) for 𝜉₁≥𝜉₂, 𝜉₁,𝜉₂∈ℝ. Since lim_𝜉→0(1−𝑒^−𝛾𝜉)=0, it follows that the family of functions {𝑇(𝜙)(𝜉)∶𝜙∈Γ} is uniformly bounded and equicontinuous in 𝜉∈ℝ. Thus, by Arzela-Ascoli theorem, for any given sequence {𝜓_𝑛}_𝑛∈ℕ⁺ in 𝑇(Γ), there exist 𝑛_𝑘→∞ and 𝜓∈𝐶(ℝ,ℝ) such that lim_𝑘→∞𝜓_𝑛𝑘(𝜉)=𝜓(𝜉) uniformly for 𝜉 in any compact subset of ℝ. Since 𝜙⁻(𝜉)≤𝜓_𝑛𝑘(𝜉)≤𝜙⁺(𝜉) for any 𝜉∈ℝ, we have 𝜙⁻(𝜉)≤𝜓(𝜉)≤𝜙⁺(𝜉) for any 𝜉∈ℝ, and therefore 𝜓(𝜉)∈Γ.
Next, we claim that lim_𝜉→±∞𝜙⁺(𝜉)−𝜙⁻(𝜉)𝑒^−𝜆𝜉=0. (46) In fact, since lim_𝜉→+∞𝜙⁺(𝜉)=𝑏 and lim_𝜉→+∞𝜙⁻(𝜉)=𝑢⁻_∗, we have lim_𝜉→+∞𝜙⁺(𝜉)−𝜙⁻(𝜉)𝑒^−𝜆𝜉=lim_𝜉→+∞𝜙⁺(𝜉)𝑒^−𝜆𝜉−lim_𝜉→+∞𝜙⁻(𝜉)𝑒^−𝜆𝜉=0. (47) Since lim_{𝜉→−∞}𝜙⁻(𝜉)𝑒^−𝜆1𝜉=1 and 0<𝜆<𝜆₁, it follows that lim_{𝜉→−∞}𝜙⁺(𝜉)−𝜙⁻(𝜉)𝑒^−𝜆𝜉=lim_{𝜉→−∞}𝑒^{(𝜆1−𝜆)𝜉}−lim_{𝜉→−∞}𝜙⁻(𝜉)𝑒^−𝜆1𝜉𝑒^{(𝜆1−𝜆)𝜉}=0. (48) Thus, it is easy to see that 𝑇∶Γ→Γ is compact with respect to the norm ‖⋅‖_𝑋𝜆 in 𝑋_𝜆. The proof is complete.

Now we start the proofs of Theorems 3 and 5.

Proof of Theorem 3. (1) By Lemma 3.4, Schauder’s fixed point theorem implies that there exists 𝜙∈Γ such that 𝜙=𝑇(𝜙). Since 0≤𝜙⁻(𝜉)≤𝜙(𝜉)≤𝜙⁺(𝜉)≤𝑏for𝜉∈ℝ, (49) we can easily obtain that, for 𝑐<𝑐^∗₁, lim_{𝜉→−∞}𝜙(𝜉)=0,lim_{𝜉→−∞}𝜙(𝜉)𝑒^−𝜆1𝜉=1,lim_{𝜉→−∞}𝜙^′(𝜉)𝑒^−𝜆1𝜉=𝜆₁. (50) Moreover, we have 𝑢^∗₋=lim_𝜉→+∞𝜙⁻(𝜉)≤liminf_𝜉→+∞𝜙(𝜉)∶=𝑙₁≤limsup_𝜉→+∞𝜙(𝜉)∶=𝑙₂≤lim_𝜉→+∞𝜙⁺(𝜉)=𝑏. (51) Next, we prove that 𝑙₁=𝑙₂=𝐾or𝑢^∗₋≤𝑙₁<𝐾<𝑙₂≤𝑏. (52) In fact, if 𝑙₁=𝑙₂, then lim_𝜉→+∞𝜙(𝜉)=𝑙₁ exists. Taking 𝜉→+∞ and applying L’Hospital’s rule to (32), we can obtain (𝑎+𝛼+𝛽)𝑓(𝑙₁)=𝑙₁ which yields 𝑙₁=𝑙₂=𝐾.
Now let us consider 𝑙₁<𝑙₂. Since it is impossible that 𝜙(𝜉) is eventually monotone, 𝜙(𝜉) is oscillating on positive infinity and then there exist two sequences {𝜁_𝑛}_𝑛∈ℕ with 𝜁_𝑛→∞ as 𝑛→∞ and {𝜉_𝑘}_𝑘∈ℕ with 𝜉_𝑘→∞ as 𝑘→∞ such that 𝜙𝜁_𝑛⟶𝑙₁,𝜙^′𝜁_𝑛𝜙𝜉⟶0as𝑛⟶∞,_𝑘⟶𝑙₂,𝜙^′𝜉_𝑘⟶0as𝑘⟶∞. (53) Since 𝑓 is continuous, for 𝜖>0, there exists a 𝛿>0 such that max𝑓(𝑢)∣𝑢∈𝑙₁−𝛿,𝑙₂𝑓(𝑢)𝑙+𝛿<max∣𝑢∈₁,𝑙₂+𝜖. (54) For such a 𝛿>0, there exists a large enough number 𝑀>0 such that 𝜙(𝜉)∈𝑙₁−𝛿,𝑙₂+𝛿,∀𝜉≥𝑀. (55) If 𝜏<+∞, we choose a large enough integer 𝑁₁>0 such that 𝜉_𝑘>𝑀+𝑚−𝑐𝜏 for all 𝑘>𝑁₁. Thus, for 𝜉_𝑘>𝑀+𝑚−𝑐𝜏, we have −𝑐𝜙^′𝜉_𝑘+𝜙𝜉_𝑘=^𝑚_𝑖=1𝑎_𝑖^𝜏₀𝐽_𝑖𝑦𝑓𝜙𝜉_𝑘−𝑖+𝑐𝑦𝑑𝑦+𝛼^𝜏₀𝐽_𝑚+1𝑦𝑓𝜙𝜉_𝑘++𝑐𝑦𝑑𝑦_𝑙𝑗=1𝛽_𝑗^𝜏₀𝐽_𝑚+1+𝑗𝑦×𝑓𝜙𝜉_𝑘≤+𝑗+𝑐𝑦𝑑𝑦_𝑚𝑖=1𝑎_𝑖+𝛼+^𝑙_𝑗=1𝛽_𝑗×max𝑓(𝑢)∣𝑢∈𝑙₁−𝛿,𝑙₂<𝑓(𝑢)𝑙+𝛿𝑎+𝛼+𝛽×max∣𝑢∈₁,𝑙₂+𝜖. (56) If 𝜏=+∞, for the above 𝜖>0, there exists a sufficiently large number 𝑀₁>0 such that ^𝑚_𝑖=1𝑎_𝑖𝑀^+∞₁𝐽_𝑖𝑦𝑓𝜙𝜉_𝑘𝛼−𝑖+𝑐𝑦𝑑𝑦<𝜖,_𝑀+∞1𝐽_𝑚+1𝑦𝑓𝜙𝜉_𝑘+𝑐𝑦𝑑𝑦<𝜖,_𝑙𝑗=1𝛽_𝑗𝑀^+∞₁𝐽_𝑚+1+𝑗𝑦𝑓𝜙𝜉_𝑘+𝑗+𝑐𝑦𝑑𝑦<𝜖. (57) Now let us choose a large enough integer 𝑁₂>0 such that 𝜉_𝑘>𝑀+𝑚−𝑐𝑀₁ for all 𝑘>𝑁₂. Therefore, for 𝜉_𝑘>𝑀+𝑚−𝑐𝑀₁, we have −𝑐𝜙^′𝜉_𝑘+𝜙𝜉_𝑘=^𝑚_𝑖=1𝑎_𝑖^𝜏₀𝐽_𝑖𝑦𝑓𝜙𝜉_𝑘−𝑖+𝑐𝑦𝑑𝑦+𝛼^𝜏₀𝐽_𝑚+1𝑦𝑓𝜙𝜉_𝑘++𝑐𝑦𝑑𝑦_𝑙𝑗=1𝛽_𝑗^𝜏₀𝐽_𝑚+1+𝑗𝑦×𝑓𝜙𝜉_𝑘+𝑗+𝑐𝑦𝑑𝑦<3𝜖+^𝑚_𝑖=1𝑎_𝑖+𝛼+^𝑙_𝑗=1𝛽_𝑗×max𝑓(𝑢)∣𝑢∈𝑙₁−𝛿,𝑙₂<𝑓(𝑢)𝑙+𝛿𝑎+𝛼+𝛽×max∣𝑢∈₁,𝑙₂+4𝜖. (58)
According to the above argument and taking the limit as 𝑘→+∞, we obtain 𝑙₂≤𝑎+𝛼+𝛽max𝑓(𝑢)∣𝑢∈𝑙₁,𝑙₂+𝜖. (59) Since 𝜖 is arbitrary, it follows that 𝑙₂≤𝑎+𝛼+𝛽max𝑓(𝑢)∣𝑢∈𝑙₁,𝑙₂. (60) Similarly, we can obtain 𝑙₁≥𝑎+𝛼+𝛽min𝑓(𝑢)∣𝑢∈𝑙₁,𝑙₂. (61) If 𝑙₁<𝑙₂≤𝐾, then (61) and (F2) imply that 𝑙₁≥𝑎+𝛼+𝛽min𝑓(𝑢)∣𝑙₁≤𝑢≤𝑙₂>𝑙₁, (62) which leads to a contradiction. On the other hand, if 𝐾≤𝑙₁<𝑙₂, then (60) and (F2) also imply that 𝑙₂≤𝑎+𝛼+𝛽max𝑓(𝑢)∣𝑙₁≤𝑢≤𝑙₂<𝑙₂, (63) which also gives a contradiction. Hence we conclude that 𝑙₁<𝐾<𝑙₂.
If, in addition, (F4) holds, then we claim that lim_𝜉→+∞𝜙(𝜉)=𝐾. Suppose the claim is false; that is, 𝑙₁<𝑙₂. Let 𝑘₁,𝑘₂∈[𝑙₁,𝑙₂] such that 𝑓𝑘₁=max𝑓(𝑢)∣𝑙₁≤𝑢≤𝑙₂,𝑓𝑘₂=min𝑓(𝑢)∣𝑙₁≤𝑢≤𝑙₂. (64) Then we consider the following three cases.
Case 1. 𝐾≤𝑘₁≤𝑙₂.
If 𝑘₁=𝑙₂, according to (60), we have 𝑙₂≤(𝑎+𝛼+𝛽)𝑓(𝑙₂) since 𝑙₂>𝐾. Therefore 𝑘₁<𝑙₂. Then, according to (60) and 𝑘₁≥𝐾, we have 𝑙₂≤(𝑎+𝛼+𝛽)𝑓(𝑘₁)≤𝑘₁<𝑙₂, which leads to a contradiction.
Case 2. 𝑙₁≤𝑘₂≤𝐾.
Similar to the discussion of Case 1, the case cannot happen.
Case 3. 𝑘₁<𝐾<𝑘₂.
By (60)-(61) and (F4), it follows that 𝑙₂−𝑙₁≤𝑎+𝛼+𝛽𝑓𝑘₁−𝑎+𝛼+𝛽𝑓𝑘₂<2𝐾−𝑘₁−2𝐾−𝑘₂=𝑘₂−𝑘₁, (65) which is impossible.
Thus, 𝑙₁=𝑙₂; that is, the limit lim_𝜉→+∞𝜙(𝜉)=𝑙₁∈[𝑢^∗₋,𝑏] exists. Taking 𝜉→+∞ and applying L’Hospital’s rule to (32), we obtain that 𝑙₁=𝑙₂=𝐾. Hence, the assertion of part (i) follows.
(2) Since 𝑓 is an odd function, letting 𝜓=−𝜙, then (13) reduces to −𝑐𝜓^′(𝜉)=−𝜓(𝜉)+^𝑚_𝑖=1𝑎_𝑖^𝜏₀𝐽_𝑖𝑦𝑓𝜓𝜉−𝑖+𝑐𝑦𝑑𝑦+𝛼^𝜏₀𝐽_𝑚+1𝑦𝑓𝜓+𝜉+𝑐𝑦𝑑𝑦_𝑙𝑗=1𝛽_𝑗^𝜏₀𝐽_𝑚+1+𝑗𝑦𝑓𝜓𝜉+𝑗+𝑐𝑦𝑑𝑦. (66) It is easy to check that (66) has the same form as (13) and hence we can obtain the result of part (ii). The proof is complete.

Proof of Theorem 5. Let us assume that 𝛽=0 and (𝑎+𝛼)𝑓^′(0)>1. Similar to the definitions of (36), for any 𝑐<0, we define the continuous functions as follows: 𝜙⁺(𝜉)=∶min𝑏,𝑒^𝜆3𝜉,𝜙⁻(𝜉)=∶min𝑢^∗₋,𝑒^𝜆3𝜉for𝜉∈ℝ, (67) where 𝜆₃ is defined in Lemma 11. Similar to the proofs of Lemmas 14–16, it is easily seen that 𝑇(Γ)⊂Γ and 𝑇∶Γ→Γ is compact with respect to the norm ‖⋅‖_𝑋𝜆 in 𝑋_𝜆, where 𝜆∈(0,𝜆₃). Hence, by the same arguments as those of the proof of Theorem 3, we can obtain the assertions of the theorem. This completes the proof.

4.2. Traveling Waves with (BC2) and (BC4)

By arguments similar to those of the proof of Theorem 3, in this section we show the existence of traveling wave solutions satisfying conditions (BC2) and (BC4).

For 𝛾>1/𝑐>0, we define the operator 𝑇∶𝐶(ℝ,[0,𝑏])→𝐶(ℝ,[0,𝑏]) by 𝑇𝜙(𝜉)=−𝑒^𝛾𝜉∞_𝜉𝑒^−𝛾𝑦𝐻𝜙𝑦𝑑𝑦, (68) where 𝐻𝜙(𝜉)=1𝑐𝜙(𝜉)−1−𝛾𝑐^𝑚_𝑖=1𝑎_𝑖^𝜏₀𝐽_𝑖𝑦𝑓𝜙−𝛼𝜉−𝑖+𝑐𝑦𝑑𝑦𝑐^𝜏₀𝐽_𝑚+1𝑦𝑓𝜙−1𝜉+𝑐𝑦𝑑𝑦𝑐^𝑙_𝑗=1^𝜏₀𝐽_𝑚+1+𝑗𝑦𝑓𝜙𝜉+𝑗+𝑐𝑦𝑑𝑦. (69) According to (F1) and (J), 𝑇 is well defined. It is obvious that a fixed point 𝜙 of 𝑇 is a traveling wave solution of (5). Similar to (33), we may define 𝑇^± through 𝑓^± in the same way.

First, we consider 𝑎>0 and (𝑎+𝛼)𝑓^′(0)>1. Let us define the continuous functions 𝑤^±(𝜉) by 𝑤⁺(𝜉)∶=min𝑏,𝑒^𝜆2𝜉,𝑤⁻(𝜉)∶=min𝑢^∗₋,𝑒^𝜆2𝜉,𝜉∈ℝ. (70) Since 𝑓⁻ is nondecreasing and satisfies the assumptions illustrated in [26], similar to Theorem 1.2 of [26], we can obtain the following results.

Lemma 17. Assume that (F1)–(F3) hold. Then, for any 𝑐>𝑐^∗₂, one has 𝑇⁺𝑤⁺(𝜉)≤𝑤⁺(𝜉),𝑇⁻𝑤⁻(𝜉)≤𝑤⁻(𝜉)∀𝜉∈ℝ. (71)

Lemma 18. Assume that (F1)–(F3) hold. Then, for any 𝑐>𝑐^∗₂, there exists a fixed point 𝑤⁻ of 𝑇⁻ such that 𝑇⁻(𝑤⁻)(𝜉)=𝑤⁻(𝜉) and 𝑤⁻(𝜉)≤𝑤⁻(𝜉) for all 𝜉∈ℝ. Moreover, lim_𝜉→+∞𝑤⁻(𝜉)𝑒^−𝜆2𝜉=1,lim_{𝜉→−∞}𝑤⁻(𝜉)=𝑢^∗₋. (72)

Now, fixing a number 𝜆∈(𝜆₂,0), we define the set Γ by Γ∶=𝜙∈𝑋_𝜆∣𝑤⁻(𝜉)≤𝜙(𝜉)≤𝑤⁺(𝜉),for𝜉∈ℝ. (73) It is easy to see that Γ is a nonempty, closed, and convex subset of 𝑋_𝜆. Similar to the proof of Lemma 16, we have the following lemma.

Lemma 19. Assume 𝑎>0 and (𝑎+𝛼)𝑓^′(0)>1. The following assertions hold.(1)𝑇(Γ)⊂Γ;(2)𝑇∶Γ→Γ is compact with respect to the norm ‖⋅‖_𝑋𝜆 in 𝑋_𝜆.

Now we start the proofs of Theorems 4 and 6.

Proof of Theorem 4. According to Lemma 19 and Schauder’s fixed point theorem, there exists 𝜙∈Γ such that 𝜙=𝑇(𝜙). Since 0≤𝑤⁻(𝜉)≤𝜙(𝜉)≤𝑤⁺(𝜉)≤𝑏 for 𝜉∈ℝ and 𝜆₂<0, we can easily obtain that, for 𝑐>𝑐^∗₂, lim_𝜉→+∞𝜙(𝜉)=0,lim_𝜉→+∞𝜙(𝜉)𝑒^−𝜆2𝜉=1,lim_𝜉→+∞𝜙^′(𝜉)𝑒^−𝜆2𝜉=𝜆₂. (74) By the same arguments as those of the proof of Theorem 3, we obtain the assertions of the theorem. This completes the proof.

Next, we consider the case 𝑎=0 and (𝛼+𝛽)𝑓^′(0)>1. For any 𝑐>0, let us redefine the continuous functions 𝑤^±(𝜉) by 𝑤⁺(𝜉)∶=min𝑏,𝑒^𝜆4𝜉,𝑤⁻(𝜉)∶=min𝑢^∗₋,𝑒^𝜆4𝜉,𝜉∈ℝ. (75) Then, similar to that of Theorem 5, we obtain the conclusion of Theorem 6.

5. Conclusion

In this paper, the motivation of our work is to consider the existence of traveling wave solutions for DCNN model with nonmonotonic output functions. We have developed a technique to construct two appropriate nondecreasing functions to squeeze the nonmonotonic output functions. By adopting Schauder’s fixed point theorem, the existence of traveling wave solutions has been derived.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

Zhi-Xian Yu is partially supported by the National Natural Science Foundation of China, Shanghai Leading Academic Discipline Project (no. XTKX2012), Innovation Program of Shanghai Municipal Education Commission (no. 14YZ096), and Hujiang Foundation of China (B14005). Rong Yuan and Ming-Shu Peng are partially supported by the National Natural Science Foundation of China and RFDP. Cheng-Hsiung Hsu is partially supported by the National Science Council and NCTS of Taiwan.