# Existence Results for a Nonlinear Semipositone Telegraph

Authors: Fanglei Wang; Yunhai Wang; Yukun An
Journal: Mathematical Problems in Engineering
 (2011)

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## Abstract

Using the fixed point theorem of cone expansion/compression, we consider the existence results of positive solutions for a nonlinear semipositone telegraph system with repulsive weak singular forces.


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## Body

## 1. Introduction

In this paper, we are concerned with the existence of positive solutions for the nonlinear telegraph system:(1.1)utt-uxx+c1ut+a1(t,x)u=f(t,x,v),vtt-vxx+c2vt+a2(t,x)v=g(t,x,u),
with doubly periodic boundary conditions(1.2)u(t+2π,x)=u(t,x+2π)=u(t,x),(t,x)∈R2,v(t+2π,x)=v(t,x+2π)=v(t,x),(t,x)∈R2.
In particular, the function f(t,x,v) may be singular at v=0 or superlinear at v=+∞, and g(t,x,u) may be singular at u=0 or superlinear at u=+∞.In the latter years, the periodic problem for the semilinear singular equation(1.3)x′′+a(t)x=b(t)xλ+c(t),
with a, b, c∈L1[0,T] and λ>0, has received the attention of many specialists in differential equations. The main methods to study (1.3) are the following three common techniques:(i)
the obtainment of a priori bounds for the possible solutions and then the applications of topological degree arguments;(ii)
the theory of upper and lower solutions;(iii)
some fixed point theorems in a cone.We refer the readers to see [1–7] and the references therein.Equation (1.3) is related to the stationary version of the telegraph equation(1.4)utt-uxx+cut+λu=f(t,x,u),
where c>0 is a constant and λ∈R. Because of its important physical background, the existence of periodic solutions for a single telegraph equation or telegraph system has been studied by many authors; see [8–16]. Recently, Wang utilize a weak force condition to enable the achievement of new existence criteria for positive doubly periodic solutions of nonlinear telegraph system through a basic application of Schauder’s fixed point theorem in [17]. Inspired by these papers, here our interest is in studying the existence of positive doubly periodic solutions for a semipositone nonlinear telegraph system with repulsive weak singular forces by using the fixed point theorem of cone expansion/compression.Lemma 1.1 (see [18]).
LetE be a Banach space, and let K⊂E be a cone in E. Assume that Ω1, Ω2 are open subsets of E with 0∈Ω1, Ω¯1⊂Ω2, and let T:K∩(Ω¯2∖Ω1)→K be a completely continuous operator such that either(i)
∥Tu∥≤∥u∥,u∈K∩∂Ω1 and ∥Tu∥≥∥u∥,u∈K∩∂Ω2; or(ii)
∥Tu∥≥∥u∥,u∈K∩∂Ω1 and ∥Tu∥≤∥u∥,u∈K∩∂Ω2.Then, T has a fixed point in K∩(Ω¯2∖Ω1).This paper is organized as follows: in Section2, some preliminaries are given; in Section 3, we give the main results.


## 2. Preliminaries

Let⊤2 be the torus defined as(2.1)⊤2=(R2πZ)×(R2πZ).
Doubly 2π-periodic functions will be identified to be functions defined on ⊤2. We use the notations(2.2)Lp(⊤2),C(⊤2),Cα(⊤2),D(⊤2)=C∞(⊤2),…
to denote the spaces of doubly periodic functions with the indicated degree of regularity. The space D′(⊤2)denotes the space of distributions on ⊤2.By a doubly periodic solution of (1.1)-(1.2) we mean that a (u,v)∈L1(⊤2)×L1(⊤2) satisfies (1.1)-(1.2) in the distribution sense; that is,(2.3)∫⊤2u(φtt-φxx-c1φt+a1(t,x)φ)dtdx=∫⊤2f(t,x,v)φdtdx,∫⊤2v(φtt-φxx-c2φt+a2(t,x)φ)dtdx=∫⊤2g(t,x,u)φdtdx,∀φ∈D(⊤2).
First, we consider the linear equation(2.4)utt-uxx+ciut-λiu=hi(t,x),inD′(⊤2),
where ci>0, λi∈R, and hi(t,x)∈L1(⊤2), (i=1,2).Let£λi be the differential operator(2.5)£λi=utt-uxx+ciut-λiu,
acting on functions on ⊤2. Following the discussion in [14], we know that if λi<0, then £λi has the resolvent Rλi:(2.6)Rλi:L1(⊤2)⟶C(⊤2),hi⟼ui,
where ui is the unique solution of (2.4), and the restriction of Rλi on Lp(⊤2)(1<p<∞) or C(⊤2) is compact. In particular, Rλi:C(⊤2)→C(⊤2) is a completely continuous operator.Forλi=-ci2/4, the Green function Gi(t,x) of the differential operator £λi is explicitly expressed; see lemma  5.2 in [14]. From the definition of Gi(t,x), we have(2.7)Gi̲:=essinf⁡Gi(t,x)=e-3ciπ/2(1-e-ciπ)2,Gi¯:=esssup⁡Gi(t,x)=(1+e-ciπ)2(1-e-ciπ)2.
Let E denote the Banach space C(⊤2) with the norm ∥u∥=max(t,x)∈⊤2|u(t,x)|, then E is an ordered Banach space with cone(2.8)K0={u∈E∣u(t,x)≥0,∀(t,x)∈⊤2}.
For convenience, we assume that the following condition holds throughout this paper:(H1)
ai(t,x)∈C(⊤2,R+), 0<ai(t,x)≤ci2/4 for (t,x)∈⊤2, and ∫⊤2ai(t,x)dtdx>0.Next, we consider (2.4) when -λi is replaced by ai(t,x). In [10], Li has proved the following unique existence and positive estimate result.Lemma 2.1.
Lethi(t,x)∈L1(⊤2);E is the Banach space C(⊤2). Then; (2.4) has a unique solution ui=Pihi;Pi:L1(⊤2)→C(⊤2) is a linear bounded operator with the following properties;(i)
Pi:C(⊤2)→C(⊤2) is a completely continuous operator;(ii)
ifhi(t,x)>0,thena.e.(t,x)∈⊤2,Pi[hi(t,x)] has the positive estimate(2.9)Gi̲‖hi‖L1≤Pi[hi(t,x)]≤Gi¯Gi̲‖ai‖L1‖hi‖L1.


## 3. Main Result

In this section, we establish the existence of positive solutions for the telegraph system(3.1)vtt-vxx+c1vt+a1(t,x)v=f(t,x,u),vtt-vxx+c2vt+a2(t,x)v=g(t,x,u).
where ai∈C(R2,R+) and f(t,x,v) may be singular at v=0. In particular, f(t,x,v) may be negative or superlinear at v=+∞. g(t,x,u) has the similar assumptions. Our interest is in working out what weak force conditions of f(t,x,v) at v=0, g(t,x,u) at u=0 and what superlinear growth conditions of f(t,x,v) at v=+∞, g(t,x,u) at u=+∞ are needed to obtain the existence of positive solutions for problem (1.1)-(1.2).We assume the following conditions throughout.(H2)
f,g:⊤2×(0,∞)→R is continuous, and there exists a constant M>0 such that
(3.2)f1(t,x,u)+M≥0,f2(t,x,u)+M≥0,∀(t,x)∈⊤2andu,v∈(0,∞).(H3)
F(t,x,v)=f(t,x,v)+M≤j1(v)+h1(v) for (t,x,v)∈⊤2×(0,∞) with j1>0 continuous and nonincreasing on (0,∞), h1≥0 continuous on (0,∞) and h1/j1 nondecreasing on (0,∞).G(t,x,u)=g(t,x,u)+M≤j2(u)+h2(u) for (t,x,u)∈⊤2×(0,∞) with j2>0 continuous and nonincreasing on (0,∞), h2≥0 continuous on (0,∞) and h2/j2 nondecreasing on (0,∞).(H4)
F(t,x,v)=f(t,x,v)+M≥j3(v)+h3(v) for all (t,x,v)∈⊤2×(0,∞) with j3>0 continuous and nonincreasing on (0,∞), h3≥0 continuous on (0,∞) with h3/j3 nondecreasing on (0,∞);G(t,x,u)=g(t,x,u)+M≥j4(u)+h4(u) for all (t,x,u)∈⊤2×(0,∞) with j4>0 continuous and nonincreasing on (0,∞), h4≥0 continuous on (0,∞) with h4/j4 nondecreasing on (0,∞).(H5)
There exists(3.3)r>M‖ω1‖δ1,
such that
(3.4)r≥4π2G1¯G1̲‖a1‖L1I1⋅I2,
here
(3.5)I1=j1(G2̲j4(r){1+h4(δ1r-M‖ω1‖)j4(δ1r-M‖ω1‖)}4π2-M‖ω2‖),I2=1+h1((4π2G2¯/G2̲‖a2‖L1)j2(δ1r-M‖ω1‖){1+h2(r)/j2(r)})j1((4π2G2¯/G2̲‖a2‖L1)j2(δ1r-M‖ω1‖){1+h2(r)/j2(r)}),
where δi=(Gi̲2∥ai∥L1/Gi¯)∈(0,1), and ωi(t,x) is the unique solution to problem:
(3.6)utt-uxx+ciut+ai(t,x)u=1,u(t+2π,x)=u(t,x+2π)=u(t,x),(t,x)∈R2.(H6)
There existsR>r, such that
(3.7)4π2G1̲I3⋅I4≥R,δ2j4(R){1+h4(δ1r-M‖ω1‖)j4(δ1r-M‖ω1‖)}>M,
where
(3.8)I3=G1̲j3(4π2G2¯G2̲‖a2‖L1j2(δ1R-M‖ω1‖){1+h2(R)j2(R)}),I4=1+h3(G2̲j4(R){1+h4(δ1R-M‖ω1‖)/j4(δ1R-M‖ω1‖)}4π2-M‖ω2‖)j3(G2̲j4(R){1+h4(δ1R-M‖ω1‖)/j4(δ1R-M‖ω1‖)}4π2-M‖ω2‖).Theorem 3.1.
Assume that (H1)–(H6) hold. Then, the problem (1.1)-(1.2) has a positive doubly periodic solution (u,v).Proof.
To show that (1.1)-(1.2) has a positive solution, we will proof that
(3.9)utt-uxx+c1ut+a1(t,x)u=F(t,x,v-Mω2),vtt-vxx+c2vt+a2(t,x)v=G(t,x,u-Mω1)
has a solution (ũ,ṽ)=(u+Mω1,v+Mω2) with ũ>Mω1, ṽ>Mω2 for (t,x)∈⊤2. In addition, by Lemma 2.1, it is clear to see that (u,v)∈C2(⊤2)×C2(⊤2) is a solution of (3.9) if and only if (u,v)∈C(⊤2)×C(⊤2) is a solution of the following system:
(3.10)u=P1(F(t,x,v-Mω2)),v=P2(G(t,x,u-Mω1)).
Evidently, (3.10) can be rewritten as the following equation:
(3.11)u=P1(F(t,x,P2(G(t,x,u-Mω1))-Mω2)).
Define a coneK⊂E as
(3.12)K={u∈E:u≥0,u≥δ1‖u‖}.
We define an operator T:E→K by
(3.13)(Tu)(t,x)=P1(F(t,x,P2(G(t,x,u-Mω1))-Mω2))
for u∈E and (t,x)∈⊤2. We have the conclusion that T:E→E is completely continuous and T(K)⊆K. The complete continuity is obvious by Lemma 2.1. Now, we show that T(K)⊆K.
For anyu∈K, we have
(3.14)Tu=P1(F(t,x,P2(G(t,x,u-Mω1))-Mω2)).
From (H1)–(H3) and Lemma 2.1, we have
(3.15)Tu=P1(F(t,x,P2(G(t,x,u-Mω1))-Mω2))≥G1̲‖F(t,x,P2(G(t,x,u-Mω1))-Mω2)‖L1,‖Tu‖=‖P(F(t,x,P2(G(t,x,u-Mω1))-Mω2))‖≤G1¯G1̲‖a1‖L1‖F(t,x,P2(G(t,x,u-Mω1))-Mω2)‖L1.
So, we get
(3.16)Tu≥G1̲2‖a1‖L1G1¯‖Tu‖≥δ1‖Tu‖,
namely, T(K)⊆K.
Let(3.17)Ωr={u∈E:‖u‖<r},ΩR={u∈E:‖u‖<R}.
Since r≤∥u∥≤R for any u∈K∩(ΩR¯∖Ωr), we have 0<δ1r-M∥ω∥≤u-Mω1≤R.
First, we show(3.18)‖Tu‖≤‖u‖, foru∈K∩∂Ωr.
In fact, if u∈K∩∂Ωr, then ∥u∥=r and u≥δ1r>M∥ω1∥ for(t,x)∈⊤2. By (H3) and (H4), we have
(3.19)P2(G(t,x,u-Mω1))≤G2¯G2̲‖a2‖L1‖G(t,x,u-Mω1)‖L1≤G2¯G2̲‖a2‖L1‖j2(u-Mω1)(1+h2(u-Mω1)j2(u-Mω1))‖L1≤G2¯G2̲‖a2‖L1j2(δ1r-M‖ω1‖){1+h2(r)j2(r)}4π2,(3.20)P2(G(t,x,u-Mω1))≥G2̲‖G(t,x,u-Mω1)‖L1≥G2̲‖j4(u-Mω1)(1+h4(u-Mω1)j4(u-Mω1))‖L1≥G2̲j4(r){1+h4(δ1r-M‖ω1‖)j4(δ1r-M‖ω1‖)}4π2.
In addition, we also have
(3.21)P2(G(t,x,u-Mω1))≥G2̲j4(r){1+h4(δ1r-M‖ω1‖)j4(δ1r-M‖ω1‖)}4π2≥G2̲j4(R){1+h4(δ1r-M‖ω1‖)j4(δ1r-M‖ω1‖)}4π2>G2¯G2̲‖a2‖L1M4π2≥Mω2,
by (H5), (H6), and (3.20).
So, we have(3.22)Tu=P1(F(t,x,v-Mω2))≤G1¯G1̲‖a1‖L1‖F(t,x,v-Mω2)‖L1≤G1¯G1̲‖a1‖L1‖j1(v-Mω2){1+h1(v-Mω2)j1(v-Mω2)}‖L1≤G1¯G1̲‖a1‖L1‖j1(P2(G(t,x,u-Mω1))-Mω2)×{1+h1(P2(G(t,x,u-Mω1))-Mω2)j1(P2(G(t,x,u-Mω1))-Mω2)}‖L1≤G1¯G1̲‖a1‖L1j1(G2̲j4(r){1+h4(δ1r-M‖ω1‖)j4(δ1r-M‖ω1‖)}4π2-M‖ω2‖)×{1+h1((G2¯/G2̲‖a2‖L1)j2(δ1r-M‖ω1‖){1+h2(r)/j2(r)}4π2)j1((G2¯/G2̲‖a2‖L1)j2(δ1r-M‖ω1‖){1+h2(r)/j2(r)}4π2)}4π2≤r=‖u‖
for (t,x)∈⊤2, since δ1r-M∥ω1∥≤u-Mω1≤r.
This implies that∥Tu∥≤∥u∥; that is, (3.18) holds.
Next, we show(3.23)‖Tu‖≥‖u‖,foru∈K∩∂ΩR.
If u∈K∩∂ΩR, then ∥u∥=R and u≥δR>M∥ω1∥ for (t,x)∈⊤2. From (H4) and (H6), we have(3.24)Tu=P1(F(t,x,v-Mω1))≥G1̲‖j3(v-Mω2){1+h3(v-Mω2)j3(v-Mω2)}‖L1≥G1̲‖j3(P2(G(t,x,u-Mω1))-Mω2)×{1+h3(P2(G(t,x,u-Mω1))-Mω2)j3(P2(G(t,x,u-Mω1))-Mω2)}‖L1≥G1̲‖j3(G2¯G2̲‖a2‖L1j2(δ1R-M‖ω1‖){1+h2(R)j2(R)}4π2)×{1+h3(G2̲j4(R){1+h4(δ1R-M‖ω1‖)/j4(δ1R-M‖ω1‖)}4π2-M‖ω2‖)j3(G2̲j4(R){1+h4(δ1R-M‖ω1‖)/j4(δ1R-M‖ω1‖)}4π2-M‖ω2‖)}‖L1≥R=‖u‖
for (t,x)∈⊤2, since δ1R-M∥ω1∥≤u-Mω1≤R.
This implies thatTu≥∥u∥; that is, (3.23) holds.
Finally, (3.18), (3.23), and Lemma 1.1 guarantee that T has a fixed point u∈K∩ΩR¯∖Ωr with r≤∥u∥≤R. Clearly, u>Mω1.
Since(3.25)P2(G(t,x,u-Mω1))≥G2̲‖G(t,x,Mω1)‖L1≥G2̲‖j4(u-Mω1)(1+h4(u-Mω1)j4(u-Mω1))‖L1≥G2̲j4(R){1+h4(δ1r-M‖ω1‖)j4(δ1r-M‖ω1‖)}4π2>G2¯G2̲‖a2‖L1M4π2≥Mω2,
then we have a doubly periodic solution (u,v) of (3.9) with u>Mω1, v>Mω2, namely, (u-Mω1,v-Mω2)>(0,0) is a positive solution of (1.1) with (1.2).Similarly, we also obtain the following result.Theorem 3.2.
Assume that (H1)–(H4) hold. In addition, we assume the following.(H7)
There exists(3.26)r>M‖ω2‖δ2,
such that
(3.27)r≥4π2G2¯G2̲‖a2‖L1I5⋅I6,
here
(3.28)I5=j2(4π2G1̲j3(r){1+h3(δ2r-M‖ω2‖)j3(δ2r-M‖ω2‖)}-M‖ω1‖),I6=1+h2((4π2G1¯/G1̲‖a1‖L1)j1(δ2r-M‖ω2‖){1+h1(r)/j1(r)})j2((4π2G1¯/G1̲‖a1‖L1)j1(δ2r-M‖ω2‖){1+h1(r)/j1(r)}).(H8)
There existsR>r, such that(3.29)4π2G2̲I7⋅I8≥R,δ1j3(R){1+h3(δ2r-M‖ω2‖)j3(δ2r-M‖ω2‖)}>M,
where
(3.30)I7=j4(4π2G1¯G1̲‖a1‖L1j1(δ2R-M‖ω2‖){1+h1(R)j1(R)}),I8=1+h4(4π2G1̲j3(R){1+h3(δ2R-M‖ω2‖)/j3(δ2R-M‖ω2‖)}-M‖ω1‖)j4(4π2G1̲j3(R){1+h3(δ2R-M‖ω2‖)/j3(δ2R-M‖ω2‖)}-M‖ω1‖).
Then, problem (1.1)-(1.2) has a positive periodic solution.


## 4. An Example

Consider the following system:(4.1)utt-uxx+2ut+sin⁡2(t+x)u=μ(v-α+vβ+k1(t,x)),vtt-vxx+2vt+cos⁡2(t+x)v=λ(u-τ+uσ+k2(t,x)),u(t+2π,x)=u(t,x+2π)=u(t,x),(t,x)∈R2,v(t+2π,x)=v(t,x+2π)=v(t,x),(t,x)∈R2,
where c1=c2=2, μ,λ>0, α,τ>0,β,σ>1, a1(t,x)=sin2(t+x), a2(t,x)=cos2(t+x)∈C(⊤2,R+), ki:⊤2→R is continuous. When μ is chosen such that(4.2)μ<sup⁡u∈((M‖ω1‖)/δ1,∞)G̲‖a1‖L1G¯4π2I1I2,
here we denote
(4.3)I1=u(G̲λu-τ{1+(δ1u-M‖ω1‖)σ+τ}4π2-M‖ω2‖)α,I2=1+(G¯G̲‖a2‖L1λ(δ1u-M‖ω1‖)-τ(1+uσ+τ+2Huτ)4π2)β+α+2H(G¯G̲‖a2‖L1λ(δ1u-M‖ω1‖)-τ(1+uσ+τ+2Huτ)4π2),
where H=max{∥k1∥,∥k2∥} and the Green function G1=G2=G. Then, problem (4.1) has a positive solution.To verify the result, we will apply Theorem3.1 with M=max{μH,λH} and(4.4)j1(v)=j3(v)=μv-α,h1(v)=μ(vβ+2H),h3(v)=μvβ,j2(u)=j4(u)=λu-τ,h2(u)=μ(uσ+2H),h4(u)=μuσ.
Clearly, (H1)–(H4) are satisfied.Set(4.5)T(u)=G̲‖a1‖L1G¯4π2I1I2,u∈((M‖ω1‖)δ1,+∞).
Obviously, T((M∥ω1∥)/δ1)=0, T(∞)=0, then there exists r∈((M∥ω1∥)/δ1,+∞) such that(4.6)T(r)=sup⁡u∈((M‖ω1‖)/δ1,∞)G̲‖a1‖L1G¯4π2I1I2.
This implies that there exists(4.7)r∈((M‖ω1‖)δ1,+∞),
such that(4.8)μ<sup⁡u∈((M‖ω1‖)/δ1,∞)G̲‖a1‖L1G¯4π2I1I2.
So, (H5) is satisfied.Finally, since(4.9)R((G¯/G̲‖a2‖L1)λ(δ1R-M‖ω1‖)-τ(1+Rσ+τ+2HRτ)4π2)αμG̲[1+(G̲λR-τ{1+(δ1R-M‖ω1‖)σ+τ}4π2-M‖ω2‖)α+β]⟶0asR⟶∞,
this implies that there exists R. In addition, for fixed r,R, choosing λ sufficiently large, we have(4.10)δ2λR-τ{1+(δ1r-M‖ω1‖)σ+τ}>M.
Thus, (H6) is satisfied. So, all the conditions of Theorem 3.1 are satisfied.



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