Based on a continuation theorem of Mawhin, a unique periodic solution is found for a second-order nonlinear differential equation with piecewise constant argument.

1. Introduction

Qualitative behaviors of first-order delay differential equations with piecewise constant arguments are the subject of many investigations (see, e.g., [1–19]), while those of higher-order equations are not.

However, there are reasons for studying higher-order equations with piecewise constant arguments. Indeed, as mentioned in [10], a potential application of these equations is in the stabilization of hybrid control systems with feedback delay, where a hybrid system is one with a continuous plant and with a discrete (sampled) controller. As an example, suppose that a moving particle with time variable mass r(t) is subjected to a restoring controller −ϕ(x[t]) which acts at sampled time [t]. Then Newton′s second law asserts that

(1.1)

Since this equation is “similar” to the harmonic oscillator equation

(1.2)

we expect that the well-known qualitative behavior of the later equation may also be found in the former equation, provided appropriate conditions on r(t) and ϕ(x) are imposed.

In this paper we study a slightly more general second-order delay differential equation with piecewise constant argument:

(1.3)

where f(t, x) is a real continuous function defined on R² with positive integer period ω for t; r(t) and p(t) are continuous function defined on R with period ω, r(t) > 0 for t ∈ R and

By a solution of (1.3) we mean a function x(t) which is defined on R and which satisfies the following conditions: (i) x^′(t) is continuous on R, (ii) r(t)x^′(t) is differentiable at each point t ∈ R, with the possible exception of the points [t] ∈ R where one-sided derivatives exist, and (iii) substitution of x(t) into (1.3) leads to an identity on each interval [n, n + 1) ⊂ R with integral endpoints.

In this note, existence and uniqueness criteria for periodic solutions of (1.3) will be established. For this purpose, we will make use of a continuation theorem of Mawhin. Let X and Y be two Banach spaces and L : Dom L ⊂ X → Y is a linear mapping and N : X → Y a continuous mapping. The mapping L will be called a Fredholm mapping of index zero if dim Ker L = codimImL < +∞, and Im L is closed in Y. If L is a Fredholm mapping of index zero, there exist continuous projectors P : X → X and Q : Y → Y such that Im P = Ker L and Im L = Ker Q = Im (I − Q). It follows that L_{|Dom L∩Ker P} : (I − P)X → Im L has an inverse which will be denoted by K_P. If Ω is an open and bounded subset of X, the mapping N will be called L-compact on if is bounded and is compact. Since Im Q is isomorphic to Ker L, there exists an isomorphism J : Im Q → Ker L.

Theorem 1 A (Mawhin′s continuation theorem [18]). Let L be a Fredholm mapping of index zero, and let N be L-compact on . Suppose that

(i)
for each λ ∈ (0,1), x ∈ ∂Ω, Lx ≠ λNx;
(ii)
for each x ∈ ∂Ω∩Ker L, QNx ≠ 0 and deg (JQN, Ω∩Ker , 0) ≠ 0.

Then the equation Lx = Nx has at least one solution in

2. Existence and Uniqueness Criteria

Our main results of this paper are as follows.

Theorem 2.1. Suppose that there exist constants D > 0 and δ⩾0 such that

(i)
f(t, x)sgn x > 0 for t ∈ R and |x| > D,
(ii)
lim _x→−∞max _0≤t≤ω(f(t, x)/x) ≤ δ (or lim _x→+∞max _0≤t≤ω(f(t, x)/x) ≤ δ).

If ω²δ(max _0≤t≤ω(1/r(t))) < 1, then (1.3) has an ω-periodic solution. Furthermore, the ω-periodic solution is unique if in addition one has the following.

(iii)
f(t, x) is strictly monotonous in x and there exists nonnegative constant b < (4/ω²)min _0≤t≤ωr(t) such that

(2.1)

Theorem 2.2. Suppose that there exist constants D > 0 and δ⩾0 such that

i^′ f(t, x)sgn x < 0 for t ∈ R and |x| > D,
ii^′ lim _x→−∞max _0≤t≤ω(f(t, x)/x)⩾−δ (or lim _x→+∞max _0≤t≤ω(f(t, x)/x)⩾−δ).

If ω²δ(max _0≤t≤ω(1/r(t))) < 1, then (1.3) has an ω-periodic solution. Furthermore, the ω-periodic solution is unique if in addition one has the following.

(iii)
f(t, x) is strictly monotonous in x and there exists nonnegative constant b < (4/ω²)min _0≤t≤ωr(t) such that (2.1) holds.

We only give the proof of Theorem 2.1, as Theorem 2.2 can be proved similarly.

First we make the simple observation that x(t) is an ω-periodic solution of the following equation:

(2.2)

if, and only if, x(t) is an ω-periodic solution of (1.3). Next, let X_ω be the Banach space of all real ω-periodic continuously differentiable functions of the form x = x(t) which is defined on R and endowed with the usual linear structure as well as the norm

. Let Y_ω be the Banach space of all real continuous functions of the form y = αt + h(t) such that y(0) = 0, where α ∈ R and h(t) ∈ X_ω, and endowed with the usual linear structure as well as the norm ∥y∥₂ = |α| + ∥h∥₁. Let the zero element of X_ω and Y_ω be denoted by θ₁ and θ₂ respectively.

Define the mappings L : X_ω → Y_ω and N : X_ω → Y_ω, respectively, by

(2.3)

(2.4)

Let

(2.5)

Since

and

, N is a well-defined operator from X_ω to Y_ω. Let us define P : X_ω → X_ω and Q : Y_ω → Y_ω, respectively, by

(2.6)

for x = x(t) ∈ X_ω and

(2.7)

for y(t) = αt + h(t) ∈ Y_ω.

Lemma 2.3. Let the mapping L be defined by (2.3). Then

(2.8)

Proof. It suffices to show that if x(t) is a real ω-periodic continuously differentiable function which satisfies

(2.9)

then x(t) is a constant function. To see this, note that for such a function x = x(t),

(2.10)

Hence by integrating both sides of the above equality from 0 to t, we see that

(2.11)

Since r(t) is positive, continuous, and periodic,

(2.12)

Since x(t) is bounded, we may infer from (2.11) that x^′(0) = 0. But then (2.9) implies x^′(t) = 0 for t ∈ R. The proof is complete.

Lemma 2.4. Let the mapping L be defined by (2.3). Then

(2.13)

Proof. It suffices to show that for each y = y(t) ∈ X_ω that satisfies y(0) = 0, there is a x = x(t) ∈ X_ω such that

(2.14)

But this is relatively easy, since we may let

(2.15)

(2.16)

Then it may easily be checked that (2.14) holds. The proof is complete.

Lemma 2.5. The mapping L defined by (2.3) is a Fredholm mapping of index zero.

Indeed, from Lemmas 2.3 and 2.4 and the definition of Y_ω, dim Ker L = codim Im L = 1 < +∞. From (2.13), we see that Im L is closed in Y_ω. Hence L is a Fredholm mapping of index zero.

Lemma 2.6. Let the mapping L, P, and Q be defined by (2.3), (2.6), and (2.7), respectively. Then Im P = Ker L and Im L = Ker Q.

Indeed, from Lemmas 2.3 and 2.4 and defining conditions (2.6) and (2.7), it is easy to see that Im P = Ker L and Im L = Ker Q.

Lemma 2.7. Let L and N be defined by (2.3) and (2.4), respectively. Suppose that Ω is an open and bounded subset of X_ω. Then N is L-compact on .

Proof. It is easy to see that for any

(2.17)

so that

(2.18)

(2.19)

These lead us to

(2.20)

where α is defined by (2.15). By (2.18), we see that

is bounded. Noting that (2.7) holds and N is a completely continuous mapping, by means of the Arzela-Ascoli theorem we know that

is relatively compact. Thus N is L-compact on

. The proof is complete.

Lemma 2.8. Suppose that g(t) is a real, bounded and continuous function on [a, b) and exists. Then there is a point ξ ∈ (a, b) such that

(2.21)

The above result is only a slight extension of the integral mean value theorem and is easily proved.

Lemma 2.9. Suppose that condition (i) in Theorem 2.1 holds. Suppose further that x(t) ∈ X_ω satisfies

(2.22)

Then there is t₁ ∈ [0, ω] such that |x(t₁)| ≤ D.

Proof. From (2.22) and Lemma 2.8, we have ξ_i ∈ (i − 1, i) for i = 1, …, ω such that

(2.23)

In case ω = 1, from the condition (i) in Theorem 2.1 and (2.23), we know that |x(0)| ≤ D. Suppose ω ≥ 2. Our assertion is true if one of x(0), x(1), …, x(ω − 1) has absolute value less than or equal to D. Otherwise, there should be x(η₁) and x(η₂) among x(0), x(1), … and x(ω − 1) such that x(η₁) > D and x(η₂)<−D. Since x(t) is continuous, in view of the intermediate value theorem, there is x(η₃) such that −D ≤ x(η₃) ≤ D, (here η₁ > η₃ > η₂ or η₂ > η₃ > η₁). Since x(t) is periodic, there is t₁ ∈ [0, ω] such that |x(t₁)| = |x(η₃)| ≤ D. The proof is complete.

Now, we consider that following equation:

(2.24)

where λ ∈ (0,1).

Lemma 2.10. Suppose that conditions (i) and (ii) of Theorem 2.1 hold. If ω²δ(max _0≤t≤ω(1/r(t))) < 1, then there are positive constants D₀ and D₁ such that for any ω-periodic solution x(t) of (2.24),

(2.25)

Proof. Let x(t) be a ω-periodic solution of (2.24). By (2.24) and our assumption that we have

(2.26)

By Lemma 2.9, there is t₁ ∈ [0, ω] such that

(2.27)

Since x(t) and x^′(t) are with period ω, thus for any t ∈ [t₁, t₁ + ω], we have

(2.28)

From (2.28), we see that for any t ∈ [t₁, t₁ + ω],

(2.29)

It is easy to see from (2.27) and (2.29) that for any t ∈ [0, ω]

(2.30)

In view of the condition ω²δ(max _0≤t≤ω(1/r(t))) < 1, we know that there is a positive number ε such that

(2.31)

From condition (ii), we see that there is a ρ > D such that for t ∈ R and x < −ρ,

(2.32)

Let

(2.33)

(2.34)

(2.35)

(2.36)

By (2.32) and (2.33), we have

(2.37)

From (2.34) and (2.36), we have

(2.38)

In view of condition (i), (2.26), (2.37), and (2.38), we get

(2.39)

It follows from (2.37), (2.38), and (2.39) that

(2.40)

Since x(0) = x(ω), thus there is a t₁ ∈ (0, ω) such that x^′(t₁) = 0. In view of (2.24) and the fact that x^′(t₁) = 0, we conclude that for any t ∈ [t₁, t₁ + ω],

(2.41)

From (2.40) and (2.41), we see that

(2.42)

It follows from (2.30), (2.31), and (2.42) that

(2.43)

where

(2.44)

Let D₀ = M₁/(1 − η₁), then from (2.43) we have

(2.45)

From (2.42) and (2.45), for any t ∈ [0, ω], we have

(2.46)

where

(2.47)

The proof is complete.

Lemma 2.11. Suppose that condition (iii) of Theorem 2.1 is satisfied. Then (1.3) has at most one ω-periodic solution.

Proof. Suppose that x₁(t) and x₂(t) are two ω-periodic solutions of (1.3). Set z(t) = x₁(t) − x₂(t). Then we have

(2.48)

Case 2 (i). For all t ∈ [0, ω], z(t) ≠ 0. Without loss of generality, we assume that z(t) > 0, that is, x₁(t) > x₂(t) for t ∈ [0, ω]. Integrating (2.48) from 0 to ω, we have

(2.49)

Combining condition (iii) and x₁(t) > x₂(t), either

(2.50)

(2.51)

holds. This is contrary to (2.49).

Case 2 (ii). There exist ξ ∈ [0, ω] such that z(ξ) = 0. As in the proof of (2.30) in Lemma 2.10, we have

(2.52)

On the other hand, since z(0) = z(ω), thus there is a t₁ ∈ (0, ω) such that z^′(t₁) = 0. In view of (2.48), we conclude that for any t ∈ [t₁, t₁ + ω],

(2.53)

By (2.53) and the fact that z^′(t₁) = 0, we have for any t ∈ [t₁, t₁ + ω],

(2.54)

It follows that for any t ∈ [t₁, t₁ + ω],

(2.55)

We know that for any t ∈ [0, ω],

(2.56)

From (2.56), we have

(2.57)

By (2.52), we get

(2.58)

It is easy to see from (2.57) and (2.58) that

(2.59)

By condition (iii) of Theorem 2.1, we see that (bω²/4)(max _0≤t≤ω(1/r(t))) < 1. Thus (2.58) leads us to max _0≤t≤ω | z(t)| = 0, which is contrary to x₁ ≠ x₂. So (1.3) has at most one ω-periodic solution. The proof is complete.

We now turn to the proof of Theorem 2.1. Suppose ω²δ(max _0≤t≤ω(1/r(t))) < 1. Let L, N, P, and Q be defined by (2.3), (2.4), (2.6), and (2.7), respectively. By Lemma 2.10, there are positive constants D₀ and D₁ such that for any ω-periodic solution x(t) of (2.24) such that (2.25) holds. Set

(2.60)

where

is a fixed number which satisfies

It is easy to see that Ω is an open and bounded subset of X_ω. Furthermore, in view of Lemmas 2.5 and 2.7, L is a Fredholm mapping of index zero and N is L-compact on

. Noting that

by Lemma 2.10, for each λ ∈ (0,1) and x ∈ ∂Ω, Lx ≠ λNx. Next note that a function x ∈ ∂Ω ∩ Ker L must be constant:

Hence by (i) and (2.17),

Hence by conditions (i), (iii) and (2.17),

(2.61)

so QNx ≠ θ₂. The isomorphism J : Im Q → Ker L is defined by J(tα) = α for α ∈ R and t ∈ R. Then

(2.62)

In particular, we see that if

then

(2.63)

and if

then

(2.64)

Consider the mapping

(2.65)

From (2.63) and (2.65), for each μ ∈ [0,1] and

we have

(2.66)

Similarly, from (2.64) and (2.65), for each μ ∈ [0,1] and

we have

(2.67)

By (2.66) and (2.67), H(x, μ) is a homotopy. This shows that

(2.68)

By Theorem A, we see that equation Lx = Nx has at least one solution in

In other words, (1.3) has an ω-periodic solution x(t). Furthermore, if (iii) is satisfied, from Lemma 2.11, we know that (1.3) has an ω-periodic solution only. The proof is complete.

3. Example

Consider the equation

(3.1)

and we can show that it has a nontrivial 5-periodic solution. Indeed, take

(3.2)

We see that min _0≤t≤5r(t) = e. Let D > 0 and δ = b = 1/25. Then condition (i) of Theorem 2.1 is satisfied:

(3.3)

Let D > 0 and δ = b = 1/25. Then conditions (i), (ii) and (iii), of Theorem 2.1 are satisfied. Note further that 5²δ(max _0≤t≤ω(1/r(t))) = e⁻¹ < 1. Therefore (3.1) has exactly one 5-periodic solution. Furthermore, it is easy to see that any solution of (3.1) must be nontrivial. We have thus shown the existence of a unique nontrivial 5-periodic solution of (3.1).

Acknowledgment

The first author is supported by the Natural Science Foundation of Guangdong Province of China under Grant no. 9151008002000012.

References

1 Cooke K. L. and Wiener J., Retarded differential equations with piecewise constant delays, Journal of Mathematical Analysis and Applications. (1984) 99, no. 1, 265–297, MR732717, https://doi.org/10.1016/0022-247X(84)90248-8, ZBL0557.34059.
10.1016/0022-247X(84)90248-8
Web of Science® Google Scholar
2 Shah S. M. and Wiener J., Advanced differential equations with piecewise constant argument deviations, International Journal of Mathematics and Mathematical Sciences. (1983) 6, no. 4, 671–703, MR729389, https://doi.org/10.1155/S0161171283000599, ZBL0534.34067.
10.1155/S0161171283000599
Google Scholar
3 Partheniadis E. C., Stability and oscillation of neutral delay differential equations with piecewise constant argument, Differential and Integral Equations. (1988) 1, no. 4, 459–472, MR945821, ZBL0723.34059.
10.57262/die/1372451948
Google Scholar
4 Aftabizadeh A. R., Wiener J., and Xu J.-M., Oscillatory and periodic solutions of delay differential equations with piecewise constant argument, Proceedings of the American Mathematical Society. (1987) 99, no. 4, 673–679, MR877038, https://doi.org/10.2307/2046474, ZBL0631.34078.
10.1090/S0002-9939-1987-0877038-7
Web of Science® Google Scholar
5 Aftabizadeh A. R. and Wiener J., Oscillatory and periodic solutions of an equation alternately of retarded and advanced type, Applicable Analysis. (1986) 23, no. 3, 219–231, https://doi.org/10.1080/00036818608839642, MR870489, ZBL0598.34059.
10.1080/00036818608839642
Google Scholar
6 Wiener J. and Aftabizadeh A. R., Differential equations alternately of retarded and advanced type, Journal of Mathematical Analysis and Applications. (1988) 129, no. 1, 243–255, MR921390, https://doi.org/10.1016/0022-247X(88)90246-6, ZBL0671.34063.
10.1016/0022-247X(88)90246-6
Web of Science® Google Scholar
7 Busenberg S. and Cooke L., V. Lakshmikantham, Models of vertically transmitted diseases with sequential-continuous dynamics, Nonlinear Phenomena in Mathematical Sciences, 1982, Academic Press, New York, NY, USA, 179–187, ZBL0512.92018.
10.1016/B978-0-12-434170-8.50028-5
Google Scholar
8 Aftabizadeh A. R. and Wiener J., Oscillatory and periodic solutions for systems of two first order linear differential equations with piecewise constant argument, Applicable Analysis. (1988) 26, no. 4, 327–333, https://doi.org/10.1080/00036818808839717, MR922978, ZBL0634.34051.
10.1080/00036818808839717
Google Scholar
9 Cooke K. L. and Wiener J., An equation alternately of retarded and advanced type, Proceedings of the American Mathematical Society. (1987) 99, no. 4, 726–732, MR877047, https://doi.org/10.2307/2046483, ZBL0628.34074.
10.1090/S0002-9939-1987-0877047-8
Web of Science® Google Scholar
10 Cooke K. L. and Wiener J., A survey of differential equations with piecewise continuous arguments, Delay Differential Equations and Dynamical Systems, 1991, 1475, Springer, Berlin, Germany, 1–15, Lecture Notes in Mathematics, MR1132014, https://doi.org/10.1007/BFb0083475, ZBL0737.34045.
10.1007/BFb0083475
Google Scholar
11 Gopalsamy K., Kulenović M. R. S., and Ladas G., On a logistic equation with piecewise constant arguments, Differential and Integral Equations. (1991) 4, no. 1, 215–223, MR1079622, ZBL0727.34061.
10.57262/die/1371569646
Google Scholar
12 Lin L. C. and Wang G. Q., Oscillatory and asymptotic behaviour of first order nonlinear differential equations with retarded argument [t], Chinese Science Bulletin. (1991) 36, no. 11, 889–891, MR1138294, ZBL0741.34046.
Google Scholar
13 Huang Y. K., Oscillations and asymptotic stability of solutions of first order neutral differential equations with piecewise constant argument, Journal of Mathematical Analysis and Applications. (1990) 149, no. 1, 70–85, MR1054794, https://doi.org/10.1016/0022-247X(90)90286-O, ZBL0704.34078.
10.1016/0022-247X(90)90286-O
Web of Science® Google Scholar
14 Papaschinopoulos G. and Schinas J., Existence stability and oscillation of the solutions of first order neutral delay differential equations with piecewise constant argument, Applicable Analysis. (1992) 44, no. 1-2, 99–111, https://doi.org/10.1080/00036819208840070, MR1284772, ZBL0739.34061.
10.1080/00036819208840070
Google Scholar
15 Shen J. H. and Stavroulakis I. P., Oscillatory and nonoscillatory delay equations with piecewise constant argument, Journal of Mathematical Analysis and Applications. (2000) 248, no. 2, 385–401, MR1775858, https://doi.org/10.1006/jmaa.2000.6908, ZBL0966.34063.
10.1006/jmaa.2000.6908
Web of Science® Google Scholar
16 Wiener J. and Heller W., Oscillatory and periodic solutions to a diffusion equation of neutral type, International Journal of Mathematics and Mathematical Sciences. (1999) 22, no. 2, 313–348, MR1695261, https://doi.org/10.1155/S0161171299223137, ZBL0935.35006.
10.1155/S0161171299223137
Google Scholar
17 Carvalho L. A. V. and Wiener J., A nonlinear equation with piecewise continuous argument, Differential and Integral Equations. (1988) 1, no. 3, 359–367, MR929923, ZBL0723.34061.
10.57262/die/1371669564
Google Scholar
18 Gaines R. E. and Mawhin J. L., Coincidence Degree, and Nonlinear Differential Equations, 1977, 586, Springer, Berlin, Germany, Lecture Notes in Mathematics, MR0637067.
10.1007/BFb0089537
Google Scholar
19 Wang G. Q. and Cheng S. S., Note on the set of periodic solutions of a delay differential equation with piecewise constant argument, International Journal of Pure and Applied Mathematics. (2003) 9, no. 2, 139–143, MR2019990, ZBL1057.34524.
Google Scholar

Citing Literature

All articles

Existence and Uniqueness of Periodic Solutions for a Second-Order Nonlinear Differential Equation with Piecewise Constant Argument

Abstract

1. Introduction

2. Existence and Uniqueness Criteria

3. Example

Acknowledgment

References

Citing Literature

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

Existence and Uniqueness of Periodic Solutions for a Second-Order Nonlinear Differential Equation with Piecewise Constant Argument

Abstract

1. Introduction

2. Existence and Uniqueness Criteria

3. Example

Acknowledgment

References

Citing Literature

References

Related

Information