# The Fixed Point Theory and the Existence of the Periodic Solution on a Nonlinear Differential Equation

Authors: Ni Hua
Journal: Journal of Applied Mathematics
 (2018)

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## Abstract

This paper deals with a nonlinear differential equation, by using the fixed point theory. The existence of the periodic solution of the nonlinear differential equation is obtained; these results are new.


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## Body

## 1. Introduction

The nonlinear Abel-type first-order differential equation(1)dxdt=atx3+btx2+ctx+dtplays an important role in many physical and technical applications; because of its importance, many scholars have studied it [1–7].Recently, A. Cima, A. Gasull, and F. Manosas [8] gave the maximum number of polynomial solutions of some integrable polynomial Abel differential equations; Jaume Giné Claudia and Valls [9] studied the center problem for Abel polynomial differential equations of second kind; Jianfeng Huang and Haihua Liang [10] were devoted to the investigation of Abel equation by means of Lagrange interpolation formula; they gave a criterion to estimate the number of limit cycles of the Abel’s equations; Berna Bülbül and Mehmet Sezer [11] introduced a numerical power series algorithm which is based on the improved Taylor matrix method for the approximate solution of Abel-type differential equations; Ni et al. [12] discussed the existence and stability of the periodic solutions of (1) and obtained the sufficient conditions which guaranteed the existence and stability of the periodic solutions for (1) from a particular one.In this paper, we consider the following more general nonlinear differential equation:(2)dxdt=ft,xx+gt,x;firstly, we give two results about the existence and uniqueness of the periodic solution of (2) by using the fixed point theory; then, we use Lyapunov function method and discuss the stability of the periodic solution; further, we discuss (1) and get the existence of the periodic solution of (1); some new results are obtained.


## 2. Preliminaries

Lemma 1 (see [12]).
Consider the following linear differential equation:(3)dxdt=atx+bt,where a(t),b(t) are ω-periodic continuous functions; if ∫0ωa(t)dt≠0, then for (3) there exists a unique ω-periodic continuous solution η(t), and η(t) can be written as follows:(4)ηt=∫-∞te∫staτdτbsds,∫0ωatdt<0-∫t+∞e∫staτdτbsds,∫0ωatdt>0
For the sake of convenience, suppose thatf(t) is an ω-periodic continuous function on R; we denote(5)fM=supt∈0,ωft,fL=inft∈0,ωft.The rest of the paper is arranged as follows. We will study the existence and stability property of the periodic solution of system (2) in the next section and discuss the existence of the periodic solution of system (1) on Section 4. We end this paper with two examples.


## 3. Periodic Solution on a Nonlinear Differential Equation

Theorem 2.
Consider (2); f(t,x),g(t,x) are both ω-periodic continuous functions on t∈R and uniformly with respect to x∈R; suppose that the following conditions hold: (6)H1ft,x≤-α<0;H2ft,x-ft,y≤L1x-y,gt,x-gt,y≤L2x-y,L2<α;H3αL2+L1supt∈0,ω⁡gt,0-L22αα-L2<1; then (1)
for (2) there exists a unique ω-periodic continuous solution γ(t), and (7)γt≤1α-L2supt∈0,ωgt,0,(2)
γ(t) is uniformly asymptotic stable.Proof.
( 1 ) Let(8)B=φt∣φ∈CR,R,φt+ω=φt,φt≤1α-L2supt∈0,ωgt,0.Given any φ(t)∈B, the norm is defined as follows:(9)φ=supt∈0,ωφt;thus (B,·) is a Banach space. Consider the following equation: (10)dxdt=ft,φtx+gt,φt;by (H1), according to Lemma 1, (10) has a unique ω-periodic continuous solution as follows: (11)ηt=∫-∞te∫stfτ,φτdτgs,φsdsand (12)ηt≤∫-∞te-αt-sgs,φsds≤∫-∞te-αt-sL2φs+gs,0ds≤L2φM+supt∈0,ω⁡gt,0α=L21/α-L2supt∈0,ωgt,0+supt∈0,ωgt,0α=1α-L2supt∈0,ωgt,0 Define a map as follows: (13)Tφt=ηt=∫-∞te∫stfτ,φτdτgs,φsds;thus if given any φ(t)∈B, then (Tφ)(t)∈B; hence T:B→B; given any φ(t),ψ(t)∈B, we have (14)Tφt-Tψt=∫-∞te∫stfτ,φτdτgs,φsds-∫-∞te∫stfτ,ψτdτgs,ψsds=∫-∞te∫stfτ,φτdτgs,φs-gs,ψsds+∫-∞te∫stfτ,φτdτ-e∫stfτ,ψτdτgs,ψsds=∫-∞te∫stfτ,φτdτgs,φs-gs,ψsds+∫-∞teξ∫stfτ,φτdτ-∫stfτ,ψτdτgs,ψsds; here, ξ is between ∫stf[τ,φ(τ)]dτ and ∫stf[τ,ψ(τ)]dτ; thus ξ≤-α(t-s), so we have (15)Tφt-Tψt≤∫-∞te∫stfτ,φτdτgs,φs-gs,ψsds+∫-∞teξ∫stfτ,φτdτ-∫stfτ,ψτdτgs,ψsds≤∫-∞te∫stfτ,φτdτgs,φs-gs,ψsds+∫-∞teξ∫stfτ,φτ-fτ,ψτdτgs,ψsds≤L2∫-∞te-αt-sφs-ψsds+L1∫-∞te-αt-s∫stφτ-ψτdτgs,ψsds≤L2∫-∞te-αt-sφs-ψsds+L1∫-∞te-αt-s∫stφτ-ψτdτL2ψs+gs,0ds=L2∫-∞te-αt-sds+L1∫-∞te-αt-st-sL2ψs+gs,0dsφ-ψ≤L2α+L1L2ψM+supt∈0,ω⁡gt,0α2φ-ψ=L2α+L1L21/α-L2supt∈0,ωgt,0+supt∈0,ωgt,0α2φ-ψ=L2α+αL1supt∈0,ωgt,0/α-L2α2φ-ψ=αL2+L1supt∈0,ωgt,0-L22αα-L2φ-ψ By (H2),(H3), it follows that (16)0<αL2+L1supt∈0,ωgt,0-L22αα-L2<1;by (16), T is a compression mapping; according to the compression mapping principle, T has a fixed point on B, the fixed point is the unique ω-periodic continuous solution γ(t) of (2), and γt≤1/α-L2supt∈0,ωgt,0.
( 2 ) Define a Lyapunov function as follows: (17)Vt,xt=xt-γt2,where x(t) is the unique solution with the initial value x(t0)=x0 of (2); differentiating both sides of (17) along the solution of (2), we get (18)
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2 By (H2) and (H3), we have -α2+2αL2-L22+L1supt∈0,ωgt,0/α-L2<0; there is a positive number δ such that -α2+2αL2-L22+L1supt∈0,ωgt,0/α-L2≤-δ<0, and hence we have (19)dVt,xtdt≤-2δxt-γt2. Therefore, the periodic solution γ(t) is uniformly asymptotic stable.Theorem 3.
Consider (2); f(t,x),g(t,x) are both ω-periodic continuous functions on t∈R and uniformly, with respect to x∈R, suppose that the following conditions hold: (20)H1ft,x≥α>0;H2ft,x-ft,y≤L1x-y,gt,x-gt,y≤L2x-y,L2<α;H3αL2+L1supt∈0,ω⁡gt,0-L22αα-L2<1; then (1)
for (2) there exists a unique ω-periodic continuous solution γ(t), and (21)γt≤1α-L2supt∈0,ωgt,0,(2)
γt is unstable.Proof.
( 1 ) Let (22)B=φt∣φ∈CR,R,φt+ω=φt,φt≤1α-L2supt∈0,ωgt,0.Given any φ(t)∈B, the norm is defined as follows:(23)φ=supt∈0,ωφt;thus (B,·) is a Banach space. Consider the following equation: (24)dxdt=ft,φtx+gt,φt;by (H1), according to Lemma 1, (24) has a unique ω-periodic continuous solution as follows: (25)ηt=-∫t+∞e∫stfτ,φτdτgs,φsdsand (26)ηt≤∫t+∞e-αs-tgs,φsds≤∫t+∞e-αs-tL2φs+gs,0ds≤L2φM+supt∈0,ω⁡gt,0α=1α-L2supt∈0,ωgt,0 Define a map as follows: (27)Tφt=ηt=-∫t+∞e∫stfτ,φτdτgs,φsds;thus if given any φ(t)∈B, then (Tφ)(t)∈B; hence T:B→B; given any φ(t),ψ(t)∈B, we have (28)Tφt-Tψt=-∫t+∞e∫stfτ,φτdτgs,φsds+∫t+∞e∫stfτ,ψτdτgs,ψsds=∫t+∞e∫stfτ,φτdτgs,φs-gs,ψsds+∫t+∞e∫stfτ,φτdτ-e∫stfτ,ψτdτgs,ψsds=∫t+∞e∫stfτ,φτdτgs,φs-gs,ψsds+∫t+∞eξ∫stfτ,φτdτ-∫stfτ,ψτdτgs,ψsds; here, ξ is between ∫stf[τ,φ(τ)]dτ and ∫stf[τ,ψ(τ)]dτ; thus ξ≤-α(s-t), so we have (29)Tφt-Tψt≤L2∫t+∞e-αt-sφs-ψsds+L1∫t+∞e-αt-s∫stφτ-ψτdτgs,ψsds≤L2∫t+∞e-αt-s]dτφs-ψsds+L1∫t+∞e-αt-s∫stφτ-ψτdτL2ψs+gs,0ds=L2∫t+∞e-αt-sds+L1∫t+∞e-αt-st-sL2ψs+gs,0dsφ-ψ≤L2α+L1L2ψM+supt∈0,ω⁡gt,0α2φ-ψ=L2α+L1L21/α-L2supt∈0,ωgt,0+supt∈0,ωgt,0α2φ-ψ=L2α+αL1supt∈0,ωgt,0/α-L2α2φ-ψ=αL2+L1supt∈0,ωgt,0-L22αα-L2φ-ψ By (H2),(H3), it follows that(30)0<αL2+L1supt∈0,ωgt,0-L22αα-L2<1by (30), T is a compression mapping; according to the compression mapping principle, T has a fixed point on B, the fixed point is the unique ω-periodic continuous solution γt of (2), and γt≤1/α-L2supt∈0,ωgt,0.
( 2 ) Define a Lyapunov function as follows: (31)Vt,xt=xt-γt2,where x(t) is the unique solution with the initial value x(t0)=x0 of (2); differentiating both sides of (31) along the solution of (2), we get (32)
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; By (H2) and (H3), we have α2-2αL2-L1supt∈0,ωgt,0+L22/α-L2>0; there is a positive number δ such that α2-2αL2-L1supt∈0,ωgt,0+L22/α-L2≥δ>0, and hence we have (33)dVt,xtdt≥2δxt-γt2 Therefore, the periodic solution γ(t) is unstable.
Consider the following nonlinear differential equation:(34)dxdt=ft,xx+gt;it is easy for us to draw the two corollaries about the existence and stability of the periodic solution on (34).Corollary 4.
Consider (34); f(t,x) is ω-periodic continuous functions on t∈R and uniformly with respect to x∈R, g(t) is an ω-periodic continuous function on t∈R; suppose that the following conditions hold: (35)H1ft,x≤-α<0;H2ft,x-ft,y≤Lx-y;H3LgM<α2; then (1)
for (34) there exists a unique ω-periodic continuous solution γ(t), and (36)γt≤gMα,(2)
γ(t) is uniformly asymptotic stable.Corollary 5.
Consider (34); f(t,x) is ω-periodic continuous functions on t∈R and uniformly with respect to x∈R, g(t) is an ω-periodic continuous function on t∈R, suppose that the following conditions hold: (37)H1ft,x≥α>0;H2ft,x-ft,y≤Lx-y;H3LgM<α2;then (1)
for (34) there exists a unique ω-periodic continuous solution γ(t), and (38)γt≤gMα,(2)
γ(t) is unstable.


## 4. Periodic Solution on Abel’s Differential Equation

Theorem 6.
Consider (1); a(t),b(t),c(t),d(t) are all ω-periodic continuous functions; suppose that the following conditions hold: (39)H1at<0;H2b2t-3atct<0;H3-2aLdM2+αbMdM-α3<0;here, α=(b2-4ac/4a)L; then (1)
for (1) there exists a unique ω-periodic continuous solution γ(t), and (40)γt≤dMα,(2)
γ(t) is uniformly asymptotic stable.Proof.
( 1 ) Define a set (41)B=φt∣φ∈CR,R,φt+ω=φt,φt≤dMα;given any φ(t)∈B, the norm is defined as follows: (42)φ=supt∈0,ωφt;thus (B,·) is a Banach space. Consider the following equation: (43)dxdt=atφ2t+btφt+ctx+dtBy (H1) and (H2), we get that (44)b2t-4atct<0 and (45)atφ2t+btφt+ct≤4atct-b2t4at≤4ac-b24aM=-b2-4ac4aL=-α<0;by Lemma 1, (43) has a unique continuous ω-periodic solution as follows: (46)ηt=∫-∞te∫staτφ2τ+bτφτ+cτdτdsds;by (46), we get (47)ηt≤dMα.Define a map as follows: (48)Tφt=ηt=∫-∞te∫staτφ2τ+bτφτ+cτdτdsds;thus if given any φ(t)∈B, then (Tφ)(t)∈B; hence T:B→B; given any φ(t),ψ(t)∈B, we have (49)Tφt-Tψt=∫-∞te∫staτφ2τ+bτφτ+cτdτ-e∫staτψ2τ+bτψτ+cτdτdsds=∫-∞teξ∫staτφ2τ-ψ2τ+bτφτ-ψτdτdsds=∫-∞teξ∫staτφτ+ψτ+bτφτ-ψτdτdsds; here, ξ is between ∫st[a(τ)φ2(τ)+b(τ)φ(τ)+c(τ)]dτ and ∫st[a(τ)ψ2(τ)+b(τ)ψ(τ)+c(τ)]dτ; thus ξ≤-α(t-s), so we have (50)
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φ
t
-
T
ψ
t
≤
∫
-
∞
t
e
-
α
t
-
s
∫
s
t
a
τ
φ
τ
+
ψ
τ
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τ
d
τ
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d
s
φ
-
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t
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α
t
-
s
∫
s
t
2
a
M
φ
M
+
b
M
d
τ
d
M
d
s
φ
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ψ
=
∫
-
∞
t
e
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α
t
-
s
2
a
M
φ
M
+
b
M
t
-
s
d
M
d
s
φ
-
ψ
=
2
a
M
φ
M
+
b
M
d
M
α
2
φ
-
ψ
=
-
2
a
L
φ
M
+
b
M
d
M
α
2
φ
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ψ
=
-
2
a
L
d
M
/
α
+
b
M
d
M
α
2
φ
-
ψ
; therefore, (51)Tφ-Tψ≤-2aLdM+αbMdMα3φ-ψ.By (H3), it follows that(52)0<-2aLdM+αbMdMα3<1by (51) and (52), T is a compression mapping; according to the compression mapping principle, T has a fixed point on B, the fixed point is the unique ω-periodic continuous solution γ(t) of (1), and γt≤dM/α.
( 2 ) Define a Lyapunov function as follows: (53)Vt,xt=xt-γt2;where x(t) is the unique solution with the initial value x(t0)=x0 of (1); differentiating both sides of (53) along the solution of (1), we get (54)dVt,xtdt=2xt-γtdxtdt-dγtdt=2xt-γtatx3t-γ3t+btx2t-γ2t+ctxt-γt=2xt-γt2atx2t+xtγt+γ2t+btxt+γt+ct=2xt-γt2atx2t+atγt+btxt+atγ2t+btγt+ct≤2xt-γt24atatγ2t+btγt+ct-atγt+bt24at=2xt-γt23a2tγ2t+2atbtγt+4atct-b2t4at≤xt-γt22at12a2t4atct-b2t-4a2tb2t12a2t=23atct-b2t)3atxt-γt2 By (H2), there is a positive number δ such that 3a(t)c(t)-b2(t)≥δ>0, and hence we have (55)dVt,xtdt≤2δ3aLxt-γt2. Therefore, the periodic solution γ(t) of (1) is uniformly asymptotic stable.Theorem 7.
Consider (1); a(t),b(t),c(t),d(t) are all ω-periodic continuous functions; suppose that the following conditions hold: (56)H1at>0;H2b2t-3atct<0;H32aMdM2+αbMdM-α3<0,where α=4ac-b2/4aL; then (1)
for (1) there exists a unique ω-periodic continuous solution γ(t), and (57)γt≤dMα,(2)
γ(t) is unstable.Proof.
( 1 ) Define a set (58)B=φt∣φ∈CR,R,φt+ω=φt,φt≤dMα;
given anyφ(t)∈B, the norm is defined as follows:(59)φ=supt∈0,ωφt;thus (B,·) is a Banach space. Consider the following equation: (60)dxdt=atφ2t+btφt+ctx+dtBy (H1) and (H2), we get that (61)b2t-4atct<0 and (62)atφ2t+btφt+ct≥4atct-b2t4at≥4ac-b24aL=α>0.by Lemma 1, (60) has a unique ω-periodic continuous solution as follows: (63)ηt=-∫t+∞e∫staτφ2τ+bτφτ+cτdτdsdsand (64)ηt≤∫t+∞e-αs-tdsds≤∫t+∞e-αs-tdMds=dMα.Define a map as follows: (65)Tφt=ηt=-∫t+∞e∫staτφ2τ+bτφτ+cτdτdsds;thus if given any φ(t)∈B, then (Tφ)(t)∈B; hence T:B→B; given any φ(t),ψ(t)∈B, we have (66)Tφt-Tψt=∫t+∞e∫staτψ2τ+bτψτ+cτdτ-e∫staτφ2τ+bτφτ+cτdτdsds=∫t+∞eξ∫staτψ2τ-φ2τ+bτψτ-φτdτdsds=∫t+∞eξ∫staτφτ+ψτ+bτψτ-φτdτdsds, where ξ is between ∫st[a(τ)φ2(τ)+b(τ)φ(τ)+c(τ)]dτ and ∫st[a(τ)ψ2(τ)+b(τ)ψ(τ)+c(τ)]dτ; thus
ξ ≤ - α ( s - t ), so we have (67)Tφt-Tψt≤∫t+∞e-αs-t∫staτφτ+ψτ+bτdτdsdsφ-ψ≤∫t+∞e-αs-t∫st2aMφM+bMdτdMdsφ-ψ=∫t+∞e-αs-t2aMφM+bMt-sdMdsφ-ψ=2aMφM+bMdMα2φ-ψ=2aMdM/α+bMdMα2φ-ψ;therefore, (68)Tφ-Tψ≤2aMdM/α+bMdMα2φ-ψ.By (H3), it follows that (69)0<2aMdM+αbMdMα3<1by (68) and (69), T is a compression mapping; according to the compression mapping principle, T has a fixed point on B, the fixed point is the unique ω-periodic continuous solution γ(t) of (1), and γt≤dM/α.
(2) Define a Lyapunov function as follows:(70)Vt,xt=xt-γt2;where x(t) is the unique solution with the initial value x(t0)=x0 of (1); differentiating both sides of (70) along the solution of (1), we get (71)dVt,xtdt=2xt-γtdxtdt-dγtdt=2xt-γtatx3t-γ3t+btx2t-γ2t+ctxt-γt=2xt-γt2atx2t+xtγt+γ2t+btxt+γt+ct=2xt-γt2atx2t+atγt+btxt+atγ2t+btγt+ct≥2xt-γt24atatγ2t+btγt+ct-atγt+bt24at=2xt-γt23a2tγ2t+2atbtγt+4atct-b2t4at≥xt-γt22at12a2t4atct-b2t-4a2tb2t12a2t=23atct-b2t)3atxt-γt2By (H2), there is a positive number δ such that 3a(t)c(t)-b2(t)≥δ>0, and hence we have (72)dVt,xtdt≥2δ3aMxt-γt2.Therefore, the periodic solution γ(t) of (1) is unstable.


## 5. Examples

The following examples show the feasibility of our main results.Example 1.
Consider the following equation:(73)dxdt=-10+2sin⁡x+sin⁡tx+4+2cos⁡x+cos⁡tIt is easy to calculate that α=7,L1=2,L2=2,supt∈[0,2π]g(t,0)=7, and (74)ft,x-ft,y≤2x-y,gt,x-gt,y≤2x-y,L2=2<α;αL2+L1supt∈0,ω⁡gt,0-L22αα-L2=2435<1.Clearly, conditions (H1),(H2),(H3) of Theorem 2 are satisfied. It follows from Theorem 2 that system (73) has a unique 2π-periodic solution γ(t), which is uniformly asymptotic stable. Clearly, by the graph of solution curve, given any initial value x(0)=x0(e.g., x(0)=0), the solution curve of (73) tends to the curve of the periodic solution γ(t) (see Figure 1).Figure 1
The solution curve of (73) with initial value x(0)=0.Example 2.
Consider the following equation:(75)dxdt=-2+sin⁡tx3+1+cos⁡tx2-8+cos⁡tx+2+sin⁡tIt is easy to calculate that b2(t)-3a(t)c(t)<0,α=(b2-4ac)/4aL=7,aL=-3,dM=3,bM=2, and (76)-2aLdM2+αbMdM-α3=-247<0.Clearly, conditions (H1),(H2),(H3) of Theorem 6 are satisfied. It follows from Theorem 6 that (75) has a unique 2π-periodic solution γ(t), which is uniformly asymptotic stable.Clearly, by the graph of solution curve, given any initial valuex(0)=x0( e.g., x(0)=0), the solution curve of (75) tends to the curve of the periodic solution γ(t) (see Figure 2).Figure 2
The solution curve of (75) with initial value x(0)=0.


## 6. Conclusion

Abel’s differential equation plays an important role in many physical and technical applications; because of this, the paper also deals with Abel’s differential equation; we first give two results about the existence and uniqueness of the periodic solution on a nonlinear differential equation by using the fixed point theory; then, we get the existence of the periodic solution of Abel’s differential equation, the results having great application value.



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