UNIQUENESS OF TRAVELING WAVE SOLUTIONS FOR NON-MONOTONE CELLULAR NEURAL NETWORKS WITH DISTRIBUTED DELAYS

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1 Electronic Journal of Differential Equations, Vol. 217 (217), No. 12, pp ISSN: URL: or UNIQUENESS OF TRAVELING WAVE SOLUTIONS FOR NON-MONOTONE CELLULAR NEURAL NETWORKS WITH DISTRIBUTED DELAYS HUI-LING ZHOU, ZHIXIAN YU Abstract. In this article, we study the uniqueness of traveling wave solutions for non-monotone cellular neural networks with distributed delays. First we establish a priori asymptotic behavior of the traveling wave solutions at infinity. Then, based on Ikehara s theorem, we prove the uniqueness of the solution ψ(n ct) with c c, where c < is the critical wave speed. 1. Introduction In this article, we study the uniqueness of traveling wave solution for the nonmonotone cellular neural networks with distributed delays x n(t) = x n (t) a i J i (y)f(x n i (t y))dy α J m1 (y)f(x n (t y))dy J m1j (y)f(x nj (t y))dy, (1.1) where the constants n Z, m, l N, τ, and the varible t R. We use the following assumptions: (H) (i) α >, a 1 >, a i (i = 2,..., m), β 1 > and (j = 2,..., l). a = m a i and β = l. (ii) J i : [, τ (, ) is the piecewise continuous function satisfying τ J i(y)dy = 1, where < τ <. b (H1) f C([, b, [, aαβ ), f() =, αf () 1 and there exists K > with K b such that (a α β)f(k) = K, f(u) f(v) f () u v for u, v [, b. (H2) (a α β)f(u) > u for u (, K) and (a α β)f(u) < u for u (K, b. (H3) There exist σ >, δ > and M > such that f(u) f ()u Mu 1σ for u [, δ. 21 Mathematics Subject Classification. 35C7, 92D25, 35B35. Key words and phrases. Cellular neural network; uniqueness; non-monotone; asymptotic behavior; distributed delays. c 217 Texas State University. Submitted March 1, 217. Published April 11,

2 2 H.-L. ZHOU, Z. YU EJDE-217/12 A traveling wave solution (1.1) with speed c is a nonnegative bounded solution of the form u n (t) = ψ(n ct) satisfying ψ() = and inf ξ ψ(ξ) >. Substituting u n (t) = ψ(n ct) in (1.1), we have the wave profile equation cψ (ξ) = ψ(ξ) a i J i (y)f(ψ(ξ i cy))dy α J m1 (y)f(ψ(ξ cy))dy J m1j (y)f(ψ(ξ j cy))dy. (1.2) When the output function f is monotone, the existence of traveling wave solutions for many versions of CNNs (1.1) with delays or without delays has been widely investigated. see for example [1, 11, 13, 14, 15, 16, 17, 18, 21, 23, 24, 28, 3, 26. The existence of entire solutions for (1.1) has been investigated by Wu and Hsu [23, 24. Letting J i = δ(y τ i ), i = 1,..., m l 1, (1.1) reduces to the multiple discrete delays equation w n(t) = w n (t) a i f(w n i (t τ i )) αf(w n (t τ m1 )) f(w nj (t τ m1j )). (1.3) Yu and Mei [28 investigated uniqueness and stability of traveling wave solutions for (1.3) with the monotone output function. In [28 the authors used the technique in [3 to study uniqueness of travelling wave soluitons for (1.3) with discrete delays. We will extend this method to (1.1) with distributed delays. For the non-monotone output function f, Yu et al. [27 only established the existence of non-critical traveling wave solutions. Yu and Zhao [31 further established the existence of the spreading speed, its coincidence with the minimal wave speed and the existence of critical waves for the non-monotone DCNNs (1.1). We summarize the existence of traveling wave solutions of (1.1) with the non-monotone output function in [27, 31 as follows. Proposition 1.1. Assume that (H)-(H3) hold. Then there exists c < (which is given in Lemma 2.1) such that for any c c, (1.1) admits a non-negative traveling wave solution ψ(n ct) with the wave speed c < and satisfying ψ() = and < inf ξ ψ(ξ) sup ψ(ξ) b. (1.4) ξ The uniqueness of monotone travelling wave solutions for various evolution systems has been established; see for example [1, 2, 4, 5, 19, 2 and the references therein. The proof of uniqueness strongly relies on the monotonicity of travelling waves. It seems very difficult to extend the techniques in those literatures to the non-monotone evolution systems because the wave profile may lose the monotonicity and the study of the corresponding uniqueness is very ited, see, e.g., [6, 8, 9. Recently, the authors in [25, 29 extend the technique in [3 to non-monotone lattice equations with discrete delays. In this article, we extend the technique in [3 to non-monotone CNNs with distributed delays.

3 EJDE-217/12 UNIQUENESS OF TRAVELING WAVE SOLUTIONS 3 The rest of this article is organized as follows. Section 2 is devoted to studying the asymptotic behavior of the traveling wave solutions. In Section 3, we prove the uniqueness of the solution. 2. Asymptotic behavior of traveling wave solutions In this section, we consider the asymptotic behavior at negative infinity of any traveling wave solutions of (1.1). The characteristic equation of (1.2) at is m (c, λ) = cλ 1 f () a i J i (y)e λ( icy) dy α J m1 (y)e λcy dy J m1j (y)e λ(jcy) dy. (2.1) Lemma 2.1 ([26, Lemma 2.1). Assume that (H) and αf () 1 hold. Then there exist a unique pair of c < and λ > such that (i) (c, λ ) =, (c,λ) λ c=c,λ=λ = ; (ii) For any c > c and λ [, ), (c, λ) < ; (iii) For any c < c, (c, λ) = has two positive roots λ 2 λ 1 >. Moreover, if c < c, (c, λ) > for any λ (λ 1, λ 2 ); if c = c, then λ 1 = λ 2 = λ. Now we give a different version of Ikehara s Theorem, which can be found in [3. Proposition 2.2. Let F (λ) := u(x)e λx dx, where u(x) is a positive decreasing function. Assume F (λ) can be written as F (λ) = h(λ) (λ µ) k1, where k > 1 and h(λ) is analytic in the strip µ Rλ <. Then x u(x) x k e µx = h( µ) Γ(µ 1). Remark 2.3. Changing the variable t = x, and modifying the proof for Ikehara s Theorem given in [7, we can show the following version of Proposition 2.2. Let F (λ) := u(t)e λt dt, where u(t) is a positive increasing function. Assume F (λ) can be written as F (λ) = h(λ) (µ λ) k1, where k > 1 and h(λ) is analytic in the strip µ ɛ < Rλ µ for some < ɛ < µ. Then x u(x) x k e µx = h(µ) Γ(µ 1). To apply Ikehara s Theorem, we need to assure that traveling wave solutions are positive. Lemma 2.4. Assume that (H) (H3) hold and let ψ(n ct) be a non-negative traveling wave of (1.1) with c c satisfying (1.4). Then ψ(ξ) > for ξ R.

4 4 H.-L. ZHOU, Z. YU EJDE-217/12 Proof. Assume that there exists ξ such that ψ(ξ ) =. Without loss of generality, we may assume ξ is the left-most point. According to ψ(ξ) for ξ R, we can easily see that ψ(ξ) attains the minimum at ξ and ψ (ξ ) =. According to (H) and (H1), it follows from (1.2) that J m1 (y)f(ψ(ξ cy))dy =, which implies that f(ψ(ξ cy)) = for any y [, τ. Thus, choosing some sufficiently small number y >, we can obtain ψ(ξ cy ) = according to the continuity of ψ(ξ) and c <. This contradicts to the choice of ξ, and completes the proof. Lemma 2.5. Assume that (H1) (H3) hold and let ψ(n ct) be any non-negative traveling wave of (1.1) with c c and satisfy (1.4). Then there exists a positive number ρ > such that ψ(ξ) = O(e ρξ ) as ξ. Proof. Since f ()(a α β) > 1, there exists ɛ > such that A := (1 ɛ )f ()(a α β) 1 >. For such ɛ >, there exist δ 1 > such that f(u) (1 ɛ )f ()u for any u [, δ 1. Since ψ() =, there exists M > and ξ M such that ψ(ξ) < δ 1. Integrating (1.2) from to ξ with ξ l M, it follows that c[ψ(ξ) ψ() = α α = A ψ(x)dx a i J m1 (y)f(ψ(x cy)) dy dx J i (y)f(ψ(x i cy)) dy dx J m1j (y)f(ψ(x j cy)) dy dx m ψ(x)dx f ()(1 ɛ ) a i J m1 (y)ψ(x cy) dy dx J m1j (y)ψ(x j cy) dy dx [ ψ(x)dx f ()(1 ɛ ) α a i J i (y)(ψ(x i cy) ψ(x)) dy dx J i (y)ψ(x i cy) dy dx J m1 (y)(ψ(x cy) ψ(x)) dy dx J m1j (y)(ψ(x j cy) ψ(x)) dy dx. (2.2)

5 EJDE-217/12 UNIQUENESS OF TRAVELING WAVE SOLUTIONS 5 Since ψ(x) is differentiable, we have Similarly, (ψ(x i cy) ψ(x))dx = = (ψ(x cy) ψ(x))dx = (ψ(x j cy) ψ(x))dx = Letting in (2.2), we obtain A ψ(x)dx icy icy cy m cψ(ξ) f ()(1 ɛ ) a i cy α J m1 (y)ψ(ξ s) ds dy jcy jcy ψ (x s)dsdx (ψ(ξ s) ψ( s))ds. (ψ(ξ s) ψ( s))ds, icy J m1j (y)ψ(ξ s) ds dy. (ψ(ξ s) ψ( s))ds. J i (y)ψ(ξ s) ds dy (2.3) From (2.3), we know that ψ(x)dx <. Letting Φ(ξ) = ψ(x)dx and integrating (2.3) from to ξ, we have A Φ(x)dx m cφ(ξ) f ()(1 ɛ ) a i cy α ϱφ(ξ κ) J m1 (y)φ(ξ s) ds dy jcy icy J m1j (y)φ(ξ s) ds dy J i (y)φ(ξ s) ds dy (2.4) for some κ > and ϱ > according to the monotonicity of Φ(ξ), Letting ϖ > such that ϱ < Aϖ, and for ξ l M, it follows that Φ(ξ ϖ) 1 ϖ ξ ϖ Φ(x)dx 1 ϖ Φ(x)dx ϱ Φ(ξ κ). (2.5) Aϖ Define h(ξ) = Φ(ξ)e ρξ, where ρ = 1 ρϖ ln Aϖ ϱ >. Hence, h(ξ ϖ) = Φ(ξ ϖ)e ρ(ξ ϖ) ϱ Aϖ eρ(κϖ) h(ξ κ) = h(ξ κ),

6 6 H.-L. ZHOU, Z. YU EJDE-217/12 which implies h is bounded. Therefore, Φ(ξ) = O(e ρξ ) when ξ. Integrating (1.2) from to ξ, it follows from (H2) that cψ(ξ) = a i α f () αf () f () a i J i (y)f(ψ(x i cy)) dx dy J m1 (y)f(ψ(x cy)) dx dy J m1j (y)f(ψ(x j cy)) dx dy Φ(ξ) = Φ(ξ) f () J i (y)ψ(x i cy) dx dy J m1 (y)ψ(x cy) dx dy J m1j (y)ψ(x j cy) dx dy Φ(ξ) a i J i (y)φ(ξ i cy)dy αf () J m1 (y)φ(ξ cy)dy f () J m1j (y)φ(ξ j cy)dy. (2.6) Thus, we have ψ(ξ) = O(e ρξ ) when ξ. With the help of Ikehara s theorem, we obtain the asymptotic behavior of traveling wave solutions at. Proposition 2.6. Assume that (H1) (H3) hold and let ψ(n ct) be any nonnegative traveling wave of (1.1) with the wave speed c c and satisfy (1.4). Then ψ(ξ) ψ(ξ) ξ e λ1ξ exists for c < c, ξ ξ e λ ξ exists for c = c. (2.7) Proof. According to Lemma 2.5, we define the two-sided Laplace transform of ψ for < Rλ < ρ, L(λ) ψ(x)e λx dx. We claim that L(λ) is analytic for < Rλ < λ 1 and has a singularity at λ = λ 1. Note that cψ (ξ) ψ(ξ) f () a i J i (y)ψ(ξ i cy)dy αf () J m1 (y)ψ(ξ cy)dy

7 EJDE-217/12 UNIQUENESS OF TRAVELING WAVE SOLUTIONS 7 f () = J i m 1 jψ(ξ j cy)dy a i J i (y)[f(ψ(ξ i cy)) f ()ψ(ξ i cy)dy α J m1 (y)[f(ψ(ξ cy)) f ()ψ(ξ i cy)dy J m1j [f(ψ(ξ j cy)) f ()ψ(ξ i cy)dy =: Q(ψ)(ξ). Multiplying the two sides of the above equality by e λξ and integrating ξ on R, we obtain (c, λ)l(λ) = e λx Q(ψ)(x)dx. (2.8) We know that the left-hand side of (2.8) is analytic for < Rλ < ρ. According to (H3), for any u >, there exists d > such that f(u) f ()u du σ1, for all u [, u, where d := max{d, γ (σ1) max u [γ,u {f ()u f(u)}}. Thus, m d a i J i (y)ψ σ1 (ξ i cy)dy α Q(ψ)(ξ). J m1j ψ σ1 (ξ j cy)dy J m1 (y)ψ σ1 (ξ cy)dy Choose υ > such that υ σ < ρ. Then for any Rλ (, ρ υ), we have d α e λx Q(ψ)(x)dx e λξ m a i J i (y)ψ σ1 (ξ i cy)dy J m1 (y)ψ σ1 (ξ cy)dy = d[ a i J m1j ψ σ1 (ξ j cy)dy dξ J i (y)e λ( icy) dy α J m1 (y)e λcy dy J m1j e λ(jcy) dy m d a i e λx ψ σ1 (x)dx J i (y)e λ( icy) dy α J m1 (y)e λcy dy (2.9)

8 8 H.-L. ZHOU, Z. YU EJDE-217/12 ( J m1j e λ(jcy) dy L(λ υ) sup ξ R ) σ ψ(ξ)e υξ σ <. We use properties of Laplace transform [22, p. 58. Since ψ > according to Lemma 2.4, there exists a real number D such that L(λ) is analytic for < Rλ < D and L(λ) has a singularity at λ = D. Thus, when c c, L(λ) is analytic for Rλ (, λ 1 ) and L(λ) has a singularity at λ = λ 1. According to (2.8), we have F (λ) := ψ(x)e λx dx = e λx Q(ψ)(x)dx ψ(x)e λx dx. (c, λ) Define H(λ) = F (λ)(λ 1 λ) k1, where k = if c < c and k = 1 if c = c. We claim that H(λ) is analytic in the strip S := {λ C < Rλ λ 1 }. Indeed, define G(λ) = e λx Q(ψ)(x)dx (c, λ)/(λ 1 λ) k1 = L(λ)(λ 1 λ) k1. It is easily seen that G(λ) is analytic in the strip {λ C < Rλ < λ 1 }. To prove that G(λ) is analytic for Rλ = λ 1, we only need to prove that (c, λ) = does not have any zero with Rλ = λ 1 other than λ = λ 1. Indeed, letting λ = λ 1 i λ, we have m = c λ 1 f () a k J k (y)e λ1( kcy) cos( k cy) λdy k=1 α J m1 (y)e λ1cy cos cy λdy J m1j (y)e λ1(jcy) cos(j cy) λdy (2.1) and m = c λ f () a k J k (y)e λ1( kcy) sin( k cy) λdy k=1 α J m1 (y)e λ1cy sin cy λdy J m1j (y)e λ1(jcy) sin(j cy) λdy. (2.11) It follows from (2.1) and (2.11) that λ =. According to the above argument, G(λ) is analytic in S, and H(λ) is also analytic in S. Moreover, we claim that H(λ 1 ) >. Indeed, notice that H(λ 1 ) = G(λ 1 ). On the other hand, e λ1x Q(ψ)(x)dx < according to (2.9) and λ λ (c, λ)/(λ 1 λ) k1 < according to Lemma Since ψ(ξ) may be non-monotone, Ikehara s Theorem could be directly used. Thus, we need to make a function transformation, i.e., ψ(ξ) = ψ(ξ)e pξ, where

9 EJDE-217/12 UNIQUENESS OF TRAVELING WAVE SOLUTIONS 9 p = 1 c >. It follows from (1.2) that ψ (ξ) = 1 c [ m a i J i (y)f(ψ(ξ i cy))dy α J m1j (y)f(ψ(ξ j cy))dy e pξ >. J m1 (y)f(ψ(ξ cy))dy Therefore, ψ(ξ) is increasing and ψ(ξ) >. Now we apply the Ikehara s Theorem to ψ(ξ). Let Then F (λ) := F (λ) = ψ(x)e λx dx = F (λ p). Ĥ(λ) ((λ 1 p) λ) k1, where Ĥ(λ) = H(λ p) is analytic for p < Rλ λ 1 p and Ĥ(λ 1 p) = H(λ 1 ) >. According to Remark 2.3, the its exists and i.e., ψ(ξ) ξ This completes the proof. ξ ψ(ξ) ξ k e (λ1p)ξ = ξ e λ1ξ exists for c < c, ξ ψ(ξ) ξ k e λ1ξ, ψ(ξ) ξ e λ ξ exists for c = c. 3. Uniqueness of traveling wave solutions In this section, we show the following unique result of traveling wave solutions of (1.1). Theorem 3.1. Assume that (H1) (H3) hold. Let ψ(n ct) be a traveling wave of (1.1) with the wave speed c c, which is given in Proposition 1.1. If φ(n ct) is any non-negative traveling wave of (1.1) with the same wave speed c satisfying (1.4), then φ is a translation of ψ; more precisely, there exists ξ R such that φ(n ct) = ψ(n ct ξ). Proof. From Proposition 2.6, there exist two positive numbers ϑ 1 and ϑ 2 such that ξ φ(ξ) ξ k e λ1ξ = ϑ 1, ξ ψ(ξ) ξ k e λ1ξ = ϑ 2 where k = for c < c, and k = 1 for c = c. For ɛ >, define ω(ξ) := φ(ξ) ψ(ξ ξ) φ(ξ) ψ(ξ ξ) e λ1ξ for c < c, ω ɛ (ξ) := (ɛ ξ 1)e λ ξ for c = c, (3.1) where ξ = 1 λ 1 ln ϑ1 ϑ 2. Then ω(± ) = and ω ɛ (± ) =. First, we consider c < c. Since ω(± ) =, sup ξ R {ω(ξ)} and inf ξ R {ω(ξ)} are finite. Without loss of generality, we assume sup ξ R {ω(ξ)} inf ξ R {ω(ξ)}. If ω(ξ), there exists ξ such that ω(ξ ) = max ξ R ω(ξ) = sup ω(ξ) >, ω (ξ ) =. ξ R

10 1 H.-L. ZHOU, Z. YU EJDE-217/12 We claim that for all i, j Z, we have ω(ξ i cy) = ω(ξ cy) = ω(ξ j cy) = ω(ξ ) for y [, τ. Suppose on the contrary that one of three inequalities ω(ξ icy) < ω(ξ ), ω(ξ cy) < ω(ξ ) and ω(ξ j cy) < ω(ξ ) for some i, j must hold. According to (1.2), (3.1) and (H2), we obtain = cω (ξ ) = cλ 1 ω(ξ ) ω(ξ ) e λ1ξ e λ1ξ α e λ1ξ m a i J i (y)[f(φ(ξ i cy)) f(ψ(ξ ξ i cy))dy J m1 (y)[f(φ(ξ cy)) f(ψ(ξ ξ cy))dy J m1j (y)[f(φ(ξ j cy)) f(ψ(ξ ξ j cy))dy m > cλ 1 ω(ξ ) ω(ξ ) f ()ω(ξ ) a i J i (y)e λ1( icy) dy α J m1 (y)e λ1cy dy = ω(ξ ) (c, λ 1 ) =, J m1j (y)e λ1(cyj) dy which is a contradiction. Thus, ω(ξ cy ) = ω(ξ ) also holds for y (, τ). Again by bootstrapping, ω(ξ kcy ) = ω(ξ ) for all k Z and ω( ) =. Therefore, we have φ(ξ) ψ(ξ ξ) for ξ R, which contradicts to ω(ξ). Next, we consider c = c. Assume sup ξ R {ω(ξ)} inf ξ R {ω(ξ)}. If ω ɛ (ξ), there exists ξ ɛ such that ω ɛ (ξ) ɛ = max {ω ɛ(ξ)} = sup{ω ɛ (ξ)} >, ω ɛ(ξ ) ɛ =. ξ R ξ R Now we divide this part into three cases: Case 1: Suppose that ξ ɛ as ɛ. It follows from (3.1) and (1.2) that c [φ (ξ) ɛ ψ (ξ ɛ ξ) = φ(ξ) ɛ a i J i (y)[f(φ(ξ ɛ i c y)) f(ψ(ξ ɛ ξ i c y))dy α J m1 (y)[f(φ(ξ ɛ c y)) f(ψ(ξ ɛ ξ c y))dy ψ(ξ ɛ ξ) J m1j (y)f(φ(ξ ɛ j c y)) f(ψ(ξ ɛ ξ j c y))dy. We claim that for all i, j Z, ω ɛ (ξ ɛ i c y) = ω ɛ (ξ ɛ c y) = ω ɛ (ξ ɛ j c y) = ω ɛ (ξ ɛ ) for y [, τ. Suppose for the contrary that one of three inequalities ω ɛ (ξ ɛ i c y) < ω ɛ (ξ ɛ ), ω ɛ (ξ ɛ c y) < ω ɛ (ξ ɛ ) and ω ɛ (ξ ɛ j c y) < ω ɛ (ξ ɛ ) for some i and

11 EJDE-217/12 UNIQUENESS OF TRAVELING WAVE SOLUTIONS 11 j must hold. Choose ɛ > sufficiently small such that ξ ɛ m c τ >. Thus, c ω ɛ (ξ ɛ )ɛ c λ ω ɛ (ξ ɛ )(ɛξ ɛ 1) ω ɛ (ξ)(ɛ ξ ɛ ɛ 1) f () a i J i (y)e λ ( ic y) [ɛ ξ ɛ i c y 1ω ɛ (ξ ɛ i c y)dy αf () f () J m1 (y)e λ c y [ɛ ξ ɛ c y 1ω ɛ (ξ ɛ c y)dy J m1j (y)e λ (jc y) [ɛ ξ ɛ j c y 1ω ɛ (ξ ɛ j c y)dy < ω ɛ (ξ ɛ )(ɛ ξ ɛ 1) f () a i J i (y)e λ ( ic y) [ɛ ξ ɛ i c y 1ω ɛ (ξ)dy ɛ αf () J m1 (y)e λ c y [ɛ ξ ɛ c y 1ω ɛ (ξ)dy ɛ f () J m1j (y)e λ (jc y) [ɛ ξ ɛ j c y 1ω ɛ (ξ)dy, ɛ it follows that c ω ɛ (ξ)ɛ ɛ ω ɛ (ξ)(ɛξ ɛ ɛ 1) (c ; λ ) < f () a i J i (y)e λ ( ic y) ɛ( i c y)ω ɛ (ξ)dy ɛ αf () f () J m1 (y)e λ (c y) ɛc yω ɛ (ξ ɛ )dy J m1j (y)e λ (jc y) [ɛ(j c y)ω ɛ (ξ)dy. ɛ (3.2) This contradicts (c,λ) λ c=c,λ=λ =. Repeating the arguments, we have ω ɛ (ξ ɛ ) = ω ɛ (ξ ɛ kc y ) for all k Z and some y (, τ). It follows from that ω ɛ ( ) =, we can obtain φ(ξ) ψ(ξ ξ) for ξ R, which contradicts ω ɛ (ξ). Similar to the process in [3, φ(ξ) ψ(ξ ξ) for ξ R still holds for Case 2: Suppose ξ ɛ as ɛ and Case 3: Suppose ξ ɛ is bounded. This completes the proof. Acknowledgements. Zhi-Xian Yu was supported by the National Natural Science Foundation of China, by the Shanghai Leading Academic Discipline Project(No. XTKX212), by the Innovation Program of Shanghai Municipal Education Commission (No.14YZ96), and by the Hujiang Foundation of China (B145). References [1 M. Aguerrea, S. Trofimchuk, G. Valenzuela; Uniqueness of fast travelling fronts in reactiondiffusion equations with delay, Proc. Roy. Soc. A, 464 (28), [2 H. Berestyski, L. Nirenberg; Traveling fronts in cylinders, Ann. Inst. H. Poincar Anal Non. Lineaire, 9 (1992),

12 12 H.-L. ZHOU, Z. YU EJDE-217/12 [3 J. Carr, A. Chmaj; Uniqueness of traveling waves for nonlocal monostable equations, Proc.Amer.Math.Soc., 132 (24), [4 X. Chen; Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), [5 X. Chen, S. C. Fu, J. S. Guo; Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices, SIAM J. Math. Anal., 38 (26), [6 O. Diekmann, H. G. Kaper; On the bounded solutions of a nonlinear convolution equation, Nonlinear Anal. TMA, 2 (1978), [7 W. Ellison, F. Ellison; Prime Numbers, A Wiley-Intersience Publication/John Wiley and Sons, Inc./Hermann, New York, Paris, [8 J. Fang, X. Zhao; Existence and uniqueness of traveling waves for non-monotone integral equations with applications, J. Differential Equations, 248, (21), [9 J. Fang, J. Wei, X. Zhao; Uniquness of traveling waves for nonlocal lattice equations, Proc. Amer. Math. Soc., 25 (21), [1 C. Hsu, C. Li, S. Yang; Diversity of traveling wave solutions in delayed cellular neural networks, Internat. J. Bifur. Chaos, 18 (28), [11 C. Hsu, S. Lin, W. Shen; Traveling waves in cellular neural networks, Internat. J. Bifur. Chaos, 9 (1999), [12 C. Hsu, S. Lin; Existence and multiplicity of traveling waves in a lattice dynamical systems, J. Differential Equations, 164 (2), [13 C. Hsu, S. Yang; Structure of a class of traveling waves in delayed cellular neural networks, Discrete Contin. Dynam. Systems, 13 (25), [14 C. Hsu, S. Yang; Traveling wave solutions in cellular neural networks with multiple time delays, Discrete Contin. Dynam. Systems Suppl., (25), [15 C. H. Hsu, T. S. Yang, Existence, uniqueness, monotonicity and asymptotic behavior of traveling waves for a epidemic model, Nonlinearity, 26(213), Corrigendum: 26(213), [16 K. Li, J. H. Huang, X. Li; Asymptotic behavior and uniqueness of traveling wave fronts in a delayed nonlocal dispersal competitive system, Commu. Pure Appl. Anal., 16 (217), [17 K. Li, X. Li; Traveling wave solutions in a delayed diffusive competition system, Nonlinear Anal. TMA., 75 (212), [18 X. Liu, P.X. Weng, Z.T. Xu; Existence of traveling wave solutions in nonlinear delayed cellular neural networks, Nonlinear Anal. RWA, 1 (29), [19 S. Ma, J. Wu; Existence, uniqueness and asymptotic stability of traveling wavefronts in a non-local delayed diffusion equation, J.Dynam. Differential Equations, 19 (27) [2 S. Ma, X. Zou; Existence, uniqueness and stability of traveling waves in a discrete reactiondiffusion monostable equations with delay, J. Differential Equations, 217 (25), [21 P. Weng, J. Wu; Deformaton of traveling waves in delayed cellular neural networks, Internat. J. Bifur. Chaos, 13 (23), [22 D. Widder; The Laplace Transform, Princeton University Press, Princeton, [23 S. Wu, C. Hsu; Entire solutions of nonlinear cellular neural networks with distributed time delays, Nonlinearity, 25 (212), [24 S. Wu, C. Hsu; Entire solutions of non-quasi-monotone delayed reaction-diffusion equations with applications, Proc. Royal Soc. Edinb., 144A (214), [25 Z. X. Yu; Uniqueness of critical traveling waves for nonlocal lattice equations with delays, Proc. Amer. Math. Soc., 14 (212), [26 Z. X. Yu, R. Yuan, C.-H. Hsu, Q. Jiang; Traveling waves for nonlinear cellular neural networks with distributed delays, J. Differential Equations, 251 (211), [27 Z. X. Yu, R. Yuan, C.-H. Hsu, M. Peng; Traveling waves for delayed cellular neural networks with nonmonotonic output functions, Abstract and Applied Analysis, 214. [28 Z. X. Yu, M. Mei; Uniqueness and stability of traveling waves for cellular neural networks with multiple delays, J. Differential Equations, 26 (216), [29 Z. X. Yu, M. Mei; Asymptotics and uniqueness of travelling waves for non-monotone delayed systems on 2D lattices, Canad. Math. Bull., 56 (213), [3 Z. X. Yu, R. Yuan; Existence,Asymptotics and Uniqueness of traveling waves for nonlocal diffusion systems with delayed nonlocal response, Taiwanese J. Math., 17 (213),

13 EJDE-217/12 UNIQUENESS OF TRAVELING WAVE SOLUTIONS 13 [31 Z.X. Yu, X. Q. Zhao; Propagation phenomena for CNNs with asymmetric templates and distributed delays, preprinted. Hui-Ling Zhou College of Science, University of Shanghai for Science and Technology, Shanghai 293, China address: Zhixian Yu (corresponding author) College of Science, University of Shanghai for Science and Technology, Shanghai 293, China address: