AN EXISTENCE–UNIQUENESS THEOREM FOR A CLASS OF BOUNDARY VALUE PROBLEMS

M. R. MOKHTARZADEH, M. R. POURNAKI, AND A. RAZANI

Abstract. In this paper the solutions of a two–endpoint boundary value problem is studied and under suitable assumptions the existence and uniqueness of a solution is proved. As a consequence, a condition to guarantee the existence of at least one periodic solution for a class of Liénard equations is presented.

1. Introduction and the statement of the main result

It is well known that Liénard equations are considered in several problems in mechanics, engineering, and electrical circuits theory. There are some existence and multiplicity results for such equations with nonconstant forced terms; see for example [6, 7, 8, 10, 13, 14, 15, 16, 17, 21]. In the following we state and prove an existence–uniqueness type theorem for a class of two–endpoint boundary value problems associated with the second order forced Liénard equations.

Theorem 1.1. Let a_{1} < a_{2} and B > 0 be real numbers and put A = max{2|a_{1}|, 2|a_{2}|}. Suppose f and g are real functions on ℝ which are locally Lipschitz and at least one of the f or g is nonconstant on |x| ≤ A; and p is a continuous real function on [0, T], T > 0. Also suppose M_{0} is the maximum value of |p| on [0, T]; M_{1}, M_{2} are the maximum values of |f|, |g| on |x| ≤ A; and M′_{1}, M′_{2} are the Lipschitz constants of f, g on |x| ≤ A, respectively. Consider M = 2/(M′_{1}B +M′_{2} +M_{1}), N = 1/(M_{1}B +M_{2} +M_{0}), and 0 < T_{0} < min\{T, 2\sqrt{AN}, 2BN, 2\sqrt{M + 1} − 2\}. Then for each a_{1} ≤ b ≤ a_{2} the boundary value problem

(1)  \{ \begin{matrix}x′′ + f(x)x′ + g(x) = p(t) & : 0 ≤ t ≤ T_{0} \\ x(0) = x(T_{0}) = b \\ |x(t)| ≤ A, |x′(t)| ≤ B & : 0 ≤ t ≤ T_{0} \end{matrix}

has a unique solution.

Proof. Consider the equation x′′ = 0 with boundary condition x(0) = x(T_{0}) = b. The existence of Green’s function for a typical two–endpoint problem was suggested by a simple physical example in [1] and is as follows:

2000 Mathematics Subject Classification. Primary: 34B15; Secondary: 34C25.

Key words and phrases. Nonlinear boundary value problem, Liénard equation, Periodic solution, Banach space, Schauder’s fixed point theorem.

The research of M. R. Pournaki was in part supported by a grant from the Academy of Sciences for the Developing World (TWAS–UNESCO Associateship – Ref. FR3240126591). The research of A. Razani was in part supported by a grant from IPM (No. 89470126).

G(t, s) = \{ \begin{matrix}s(t − T_{0})/T_{0} & : 0 ≤ s ≤ t ≤ T_{0} \\ t(s − T_{0})/T_{0} & : 0 ≤ t ≤ s ≤ T_{0} \end{matrix}

If we now consider the integral equation

x(t) = b + ∫_{0}^{T_{0}} G(t, s)(f(x(s))x′(s) + g(x(s)) − p(s))ds,

then it is easy to see that the solutions x(t) of this integral equation which are satisfied in |x(t)| ≤ A and |x′(t)| ≤ B for each 0 ≤ t ≤ T_{0} are exactly the solutions of given boundary value problem. Hence, to prove the theorem, it is enough to show that the above integral equation has a unique solution x(t) satisfying |x(t)| ≤ A and |x′(t)| ≤ B for each 0 ≤ t ≤ T_{0}. In order to do so, suppose X = C^{1}([0, T_{0}], ℝ), and for ϕ ∈ X define ||ϕ|| = max_{0≤t≤T_{0}} |ϕ(t)|+max_{0≤t≤T_{0}} |ϕ′(t)|. It is clear that X is a Banach space. Now, consider

Ω = {ϕ ∈ X : |ϕ(t)| ≤ A and |ϕ′(t)| ≤ B hold for each 0 ≤ t ≤ T_{0}},

which is obviously a closed, bounded, and convex subspace of X. Define the operator S : Ω → X by mapping ϕ to S(ϕ), where S(ϕ) is defined by

S(ϕ)(t) = b + ∫_{0}^{T_{0}} G(t, s)(f(ϕ(s))ϕ′(s) + g(ϕ(s)) − p(s))ds.

First, we show that S maps Ω into itself. In order to do this, note that for each x, x′, and t such that |x| ≤ A, |x′| ≤ B, and 0 ≤ t ≤ T_{0} we have

|f(x)x′ + g(x) − p(t)| ≤ M_{1}B + M_{2} + M_{0} = \frac{1}{N} .

Also for each 0 ≤ t ≤ T_{0} we have

\begin{matrix}∫_{0}^{T_{0}} |G(t, s)|ds = \frac{1}{2}t(T_{0} − t) ≤ \frac{T_{0}^{2}}{8} , \\ ∫_{0}^{T_{0}} |\frac{∂}{∂t}G(t, s)|ds = \frac{1}{T_{0}} t^{2} − t + \frac{1}{2}T_{0} ≤ \frac{T_{0}}{2} . \end{matrix}

Hence we conclude that for each ϕ ∈ Ω and 0 ≤ t ≤ T_{0},

\begin{matrix}|S(ϕ)(t)| ≤ |b| + \frac{1}{N} ∫_{0}^{T_{0}} |G(t, s)|ds ≤ |b| + \frac{T_{0}^{2}}{8N} ≤ \frac{A}{2} + \frac{A}{2} = A, \\ |S(ϕ)′(t)| ≤ \frac{1}{N} ∫_{0}^{T_{0}} |\frac{∂}{∂t}G(t, s)|ds ≤ \frac{T_{0}}{2N} ≤ B. \end{matrix}

These mean that for each ϕ ∈ Ω, S(ϕ) ∈ Ω and therefore S is an operator from Ω to Ω.

Next, we show that S is a compact operator on Ω. For this, it is enough to show that each bounded sequence {ϕ_{n}} on Ω has a subsequence {ϕ_{n_{i}}} for which {S(ϕ_{n_{i}})} is convergent on Ω. Therefore, let {ϕ_{n}} be a given sequence on Ω which is automatically bounded by definition of Ω. Suppose ϵ > 0 is given. Since G is a uniformly continuous function on [0, T_{0}] × [0, T_{0}], there exists δ, 0 < δ < ϵN, such that (t_{1}, s_{1}), (t_{2}, s_{2}) ∈ [0, T_{0}] × [0, T_{0}] and \sqrt{(t_{1} − t_{2})^{2} + (s_{1} − s_{2})^{2}} < δ imply that |G(t_{1}, s_{1}) − G(t_{2}, s_{2})| < ϵN/2T_{0}. We now conclude that for each n and for each t_{1}, t_{2} ∈ [0, T_{0}], if |t_{1} − t_{2}| < δ, then

\begin{matrix}|S(ϕ_{n})(t_{1}) − S(ϕ_{n})(t_{2})| ≤ \frac{1}{N} ∫_{0}^{T_{0}} |G(t_{1}, s) − G(t_{2}, s)|ds < ϵ, \\ |S(ϕ_{n})′(t_{1}) − S(ϕ_{n})′(t_{2})| ≤ \frac{1}{N} ∫_{0}^{T_{0}} |\frac{∂}{∂t}G(t_{1}, s) − \frac{∂}{∂t}G(t_{2}, s)|ds = \frac{1}{N} |t_{1} − t_{2}| < ϵ. \end{matrix}

Hence {S(ϕ_{n})(t)} and {S(ϕ_{n})′(t)} are equicontinuous family of functions on [0, T_{0}] and by classical Ascoli–Arzela theorem, there exists a subsequence {ϕ_{n_{i}}(t)} of {ϕ_{n}(t)} for which {S(ϕ_{n_{i}})(t)} and {S(ϕ_{n_{i}})′(t)} are uniformly convergent on [0, T_{0}]. This shows that {S(ϕ_{n_{i}})} is convergent on Ω and so S is a compact operator.

Therefore, by Schauder’s fixed point theorem, there exists ϕ ∈ Ω such that S(ϕ) = ϕ. So for each 0 ≤ t ≤ T_{0}, we have S(ϕ)(t) = ϕ(t) which is to say

ϕ(t) = b + ∫_{0}^{T_{0}} G(t, s)(f(ϕ(s))ϕ′(s) + g(ϕ(s)) − p(s))ds.

This means that ϕ ∈ Ω is a solution of the mentioned integral equation with restrictions |ϕ(t)| ≤ A and |ϕ′(t)| ≤ B for each 0 ≤ t ≤ T_{0} and therefore is a solution of the given boundary value problem.

We now suppose that ψ is another solution of the given boundary value problem. This means that ψ ∈ Ω, ψ ≠ ϕ, and S(ψ) = ψ. By the locally Lipschitz condition for f and g, note that for each x, y, x′, and y′ such that |x| ≤ A, |y| ≤ A, |x′| ≤ B, and |y′| ≤ B we have

\begin{aligned}|(f(x)x′ + g(x)) − (f(y)y′ + g(y))| & = |(f(x) − f(y))x′ + f(y)(x′ − y′) + g(x) − g(y)| \\ & ≤ (M′_{1}B + M′_{2})|x − y| + M_{1}|x′ − y′|. \end{aligned}

Therefore by the above inequality, for each 0 ≤ t ≤ T_{0},

|S(ϕ)(t) − S(ψ)(t)| ≤ \frac{T_{0}^{2}}{8} (M′_{1}B + M′_{2} + M_{1})||ϕ − ψ||
= \frac{T_{0}^{2}}{8} \frac{2}{M} ||ϕ − ψ||
= \frac{T_{0}^{2}}{4M} ||ϕ − ψ||,

|S(ϕ)′(t) − S(ψ)′(t)| ≤ \frac{T_{0}}{2} (M′_{1}B + M′_{2} + M_{1})||ϕ − ψ||
= \frac{T_{0}}{2} \frac{2}{M} ||ϕ − ψ||
= \frac{T_{0}}{M} ||ϕ − ψ||.

Hence,

||ϕ − ψ|| = ||S(ϕ) − S(ψ)||

= max_{0≤t≤T_{0}}|S(ϕ)(t) − S(ψ)(t)| + max_{0≤t≤T_{0}}|S(ϕ)′(t) − S(ψ)′(t)|

≤ (\frac{T_{0}^{2}}{4M} + \frac{T_{0}}{M})||ϕ − ψ||.

Therefore, we obtain T_{0}^{2} + 4T_{0} ≥ 4M, or T_{0} ≥ 2\sqrt{M + 1} − 2 which is contradictory with the definition of T_{0}. So ϕ is the unique solution of the given boundary value problem. □

Remark 1.2. In the above proof, we obtained a more deeper property for the operator S, which is contractivity condition. Therefore, we can apply the Banach’s fixed point theorem directly to Eq. (1). We will verify this property in Section 3.

2. An application

The analysis of periodic Liénard equations have long been a topic of interest. In this direction, an important question, which has been studied extensively by a number of authors (see, for example [2, 3, 5, 9, 11, 12, 18, 19, 20]), is whether Liénard equations can support periodic solutions or not. In this section, as a consequence of Theorem 1.1, we investigate the existence of periodic solutions for a class of the second order forced Liénard equations

x′′ + f(x)x′ + g(x) = p(t),

where f and g are real functions on ℝ and p is a real function on [0, T], T > 0. These equations appear in a number of physical models and one important question is whether these equations can support periodic solutions. In the following we state and prove a result which yields a condition to guarantee the existence of at least one periodic solution for the above equation.

Theorem 2.1. Suppose f and g are real functions on ℝ which are locally Lipschitz and p is a nonconstant, continuous, real function on [0, T], T > 0. Also suppose all solutions of the initial value problems associated with x′′ + f(x)x′ + g(x) = p(t) can be extended to [0, T]. If there exist real numbers a_{1} and a_{2} for which g(a_{1}) ≤ p(t) ≤ g(a_{2}) holds for each 0 ≤ t ≤ T, then there exists T_{0}, 0 < T_{0} < T, such that if p is T_{0}– periodic, x′′ + f(x)x′ + g(x) = p(t) has at least one T_{0}–periodic solution.

Proof. By the assumption we conclude that a_{1} ≠ a_{2} and so without loss of generality we can suppose that a_{1} < a_{2}. Define the functions g̃ and ĝ as follows which are obviously locally Lipschitz:

g̃(x) = \{ \begin{matrix}g(x) & : x ≤ a_{1} \\ g(a_{1}) + a_{1} − x & : x > a_{1} \end{matrix}

and

ĝ(x) = \{ \begin{matrix}g(x) & : x ≥ a_{2} \\ g(a_{2}) + a_{2} − x & : x < a_{2} \end{matrix}

Consider A = max{2|a_{1}|, 2|a_{2}|} and suppose B = 1. Let M_{0} be the maximum value of |p| on [0, T]; M_{1}, M_{2}, M̃_{2}, M̂_{2} be the maximum values of |f|, |g|, |g̃|, |ĝ| on |x| ≤ A; and M′_{1}, M′_{2}, M̃′_{2}, M̂′_{2} be the Lipschitz constants of f, g, g̃, ĝ on |x| ≤ A, respectively. Consider M = 2/(M′_{1} + M′_{2} + M_{1}), N = 1/(M_{1} + M_{2} + M_{0}), M̃ = 2/(M′_{1} + M̃′_{2} + M_{1}), Ñ = 1/(M_{1} + M̃_{2} + M_{0}), M̂ = 2/(M′_{1} + M̂′_{2} + M_{1}), N̂ = 1/(M_{1} + M̂_{2} + M_{0}), and 0 < T_{0} < min{L, L̃, L̂}, where

L = min \{T, 2\sqrt{AN}, 2N, 2\sqrt{M + 1} − 2\},
L̃ = min \{T, 2\sqrt{AÑ}, 2Ñ, 2\sqrt{M̃ + 1} − 2\},
L̂ = min \{T, 2\sqrt{AN̂}, 2N̂, 2\sqrt{M̂ + 1} − 2\}.

Theorem 1.1 now implies that for each a_{1} ≤ b ≤ a_{2}, the boundary value problem

\{ \begin{matrix}x′′ + f(x)x′ + g(x) = p(t) & : 0 ≤ t ≤ T_{0} \\ x(0) = x(T_{0}) = b \\ |x(t)| ≤ A, |x′(t)| ≤ 1 & : 0 ≤ t ≤ T_{0} \end{matrix}

has a unique solution, say x(t, b).

Lemma 2.2. For each 0 ≤ t ≤ T_{0}, we have x(t, a_{1}) ≤ a_{1} < a_{2} ≤ x(t, a_{2}).

Proof. We prove that x(t, a_{1}) ≤ a_{1} holds for each 0 ≤ t ≤ T_{0}. By Theorem 1.1, the boundary value problem

\{ \begin{matrix}x′′ + f(x)x′ + g̃(x) = p(t) & : 0 ≤ t ≤ T_{0} \\ x(0) = x(T_{0}) = a_{1} \\ |x(t)| ≤ A, |x′(t)| ≤ 1 & : 0 ≤ t ≤ T_{0} \end{matrix}

has a unique solution x(t). We claim that x(t) ≤ a_{1} holds for each 0 ≤ t ≤ T_{0}. Suppose for the purpose of a contradiction, there exists a point 0 ≤ t̃≤ T_{0} such that x(t̃) > a_{1}. Therefore the function x(t) − a_{1} has a positive maximum on the interval (0, T_{0}), say at t_{1}. Hence (x(t) − a_{1})′|_{t=t_{1}} = 0, or x′(t_{1}) = 0. Therefore we established

\begin{matrix}(x(t) − a_{1})′′ = & x′′(t_{1}) \\ = & −f(x(t_{1}))x′(t_{1}) − g̃(x(t_{1})) + p(t_{1}) \\ = & −g̃(x(t_{1})) + p(t_{1}) \\ = & −g(a_{1}) − a_{1} + x(t_{1}) + p(t_{1}) \\ = & (p(t_{1}) − g(a_{1})) + (x(t_{1}) − a_{1}) \\ > & 0, \end{matrix}

which is a contradiction since x(t) − a_{1} has a maximum at t_{1}. Therefore for each 0 ≤ t ≤ T_{0}, x(t) ≤ a_{1} and so by the definition of g̃, g̃(x(t)) = g(x(t)) holds for each 0 ≤ t ≤ T_{0}. This means that x(t) is a solution of

\{ \begin{matrix}x′′ + f(x)x′ + g(x) = p(t) & : 0 ≤ t ≤ T_{0} \\ x(0) = x(T_{0}) = a_{1} \\ |x(t)| ≤ A, |x′(t)| ≤ 1 & : 0 ≤ t ≤ T_{0} \end{matrix}

The uniqueness property now implies that for each 0 ≤ t ≤ T_{0}, x(t) = x(t, a_{1}) and so x(t, a_{1}) ≤ a_{1} holds for each 0 ≤ t ≤ T_{0}.

A similar argument applying to the function ĝ gives us the other inequality. □

Lemma 2.3. There exists b̂, a_{1} ≤ b̂ ≤ a_{2}, such that x′(0, b̂) = x′(T_{0}, b̂).

Proof. Define the function θ on [a_{1}, a_{2}] by

θ(b) = x′(0, b) − x′(T_{0}, b).

Using the Ascoli–Arzela theorem, one may easily verify that both x(t, b) and x′(t, b) are continuous on [0, T_{0}] × [a_{1}, a_{2}]. This implies that θ is continuous also. On the other hand, note that for i ∈ {1, 2},

x′(0, a_{i}) = lim_{t→0^{+}} \frac{x(t, a_{i}) − a_{i}}{t} , x′(T_{0}, a_{i}) = lim_{t→0^{+}} \frac{a_{i} − x(T_{0} − t, a_{i})}{t},

and therefore,

\begin{matrix}θ(a_{i}) = & x′(0, a_{i}) − x′(T_{0}, a_{i}) \\ = & lim_{t→0^{+}} \frac{x(t, a_{i}) + x(T_{0} − t, a_{i}) − 2a_{i}}{t} . \end{matrix}

So by Lemma 2.2, we obtain θ(a_{1}) ≤ 0 and θ(a_{2}) ≥ 0. Hence there exists b̂, a_{1} ≤ b̂ ≤ a_{2}, such that θ(b̂) = 0, or x′(0, b̂) = x′(T_{0}, b̂). □

Lemma 2.3 now implies that x(t, b̂) is a solution of the periodic boundary value problem

\{ \begin{matrix}x′′ + f(x)x′ + g(x) = p(t) & : 0 ≤ t ≤ T_{0} \\ x(0, b̂) = x(T_{0}, b̂) \\ x′(0, b̂) = x′(T_{0}, b̂) \end{matrix}

and therefore, by a similar method as the one used in [4], we can extend x(t, b̂) periodically with period T_{0} to obtain a periodic solution of the equation x′′ + f(x)x′ + g(x) = p(t). Note that this periodic solution is nontrivial, since p is a nonconstant forced function. □

3. An illustrative example

In this section, we give a concrete example satisfying the assumptions of the main result. In order to do this, consider the initial value problem

(2)  \{ \begin{matrix}x′′ = p(t, x, x′) = \frac{x}{10}x′ − (16x + \frac{x^{3}}{10}) − \frac{(1119999+480 cos(924t)+cos(48t)) sin(24t)}{20000} \\ x(0) = 0 \\ x′(0) = \frac{24}{10} \end{matrix}

There are several numerical methods to solve Eq. (2) in the standard texts of numerical analysis and numerical solutions of ordinary differential equations. For example using Runge–Kutta method, the error of approximation is about 10^{−8}, knowing the exact solution of Eq. (2) which is x(t) = 0.1 sin(24t).

We now present a rather nonstandard symbolic–numeric scheme for generating approximate solution for this example. This method is based on transformation of the second order initial value problem to a system of the first order equations and then use Picard’s iteration method, with controlling the number of terms at each step. More precisely, at each step we ignore all the terms with an upper bound less than 10^{−12}. Using this method, we show that the corresponding Picard’s iteration converges, and also we give a crude approximation to the contraction factor of the Picard’s method.

Let C^{1}([0, T_{0}], ℝ) be the Banach space equipped with the norm

∥x∥_{∞} = max { max_{0≤t≤T_{0}} |x(t)|, max_{0≤t≤T_{0}} |x′(t)|}.

Assuming x = u_{1}, x′ = u_{2} and

u = [u_{1}, u_{2}]^{T} , u′ = [u′_{1}, u′_{2}]^{T} , F = [u_{2}, p(t, u_{1}, u_{2})]^{T} ,

Eq. (2) is equivalent to the system

\{ \begin{matrix}u′ = F (t, u) & : 0 < t ≤ T_{0} \\ u(0) = [0, 2.4]^{T} \end{matrix}

with corresponding Picard’s iteration formula given by

(3)  \{ \begin{matrix}u_{n}(t) = u(0) + ∫_{0}^{t} F (s, u_{n−1}(s))ds \\ u_{0}(t) = u(0) \end{matrix}

For n = 1, …, 7, we generate the sequence of the functions u_{1}, …, u_{7} and x_{1}, …, x_{7} and the approximation to Eq. (1) with exact solution x(t) = 0.1 sin(24t). Numerical values for expressions ∥x_{n}(t)−x_{n−1}(t)∥_{∞} and ∥x_{n}(t)−x(t)∥_{∞} are given in the following table.

\begin{matrix}n & 3 & 4 & 5 & 6 & 7 \\ ∥x_{n}(t) − x_{n−1}(t)∥_{∞} & 0.0692 & 0.0025 & 0.0000486 & 5.88 × 10^{−7} & 2.6306 × 10^{−9} \\ ∥x_{n}(t) − x(t)∥_{∞} & 2.11 & 0.0717 & 0.00255 & 0.0000492 & 5.91 × 10^{−7} \end{matrix}

Note that error decrease at most with the factor 0.1. This verifies numerically that the operator corresponding to the Picard’s iteration given in Eq. (3) is contraction mapping with contraction factor 0.1.

Numerical example given by Eq. (2) is generated by trail and error method designed in Mathematica version 5, and implemented in a cluster environment at Laboratory of Scientific Computation in Institute for Studies in Theoretical Physics and Mathematics (see http://www.scc.ipm.ac.ir/ganglia/).

Acknowledgments

The second author would like to thank the Academy of Sciences for the Developing World (TWAS) and the Delhi Center of the Indian Statistical Institute (ISID) for sponsoring his visits to New Delhi in July–August 2007 and January 2010. Especially he would like to express his thanks to Professor Rajendra Bhatia for the hospitality enjoyed at ISID.

References

[1] G. Birkhoff, G. G. Rota, Ordinary Differential Equations, Third ed., John Wiley and Sons, 1978.

[2] A. Capietto, Z. Wang, Periodic solutions of Liénard equations with asymmetric nonlinearities at resonance, J. London Math. Soc. (2) 68 (2003), no. 1, 119–132.

[3] A. Capietto, Z. Wang, Periodic solutions of Liénard equations at resonance, Differential Integral Equations 16 (2003), no. 5, 605–624.

[4] L. Cesari, Asymptotic Behavior and Stability Problems in Ordinary Differential Equations, Second ed., Academic Press, New York, 1963.

[5] S. K. Chang, H. S. Chang, Existence of periodic solutions of nonlinear systems of a generalized Liénard type, Kyungpook Math. J. 39 (1999), no. 2, 351–365.

[6] H. B. Chen, K. T. Li, D. S. Li, Existence of exactly one and two periodic solutions of the Liénard equation, Acta Math. Sinica 47 (2004), no. 3, 417–424.

[7] H. B. Chen, Y. Li, Exact multiplicity for periodic solutions of a first-order differential equation, J. Math. Anal. Appl. 292 (2004), no. 2, 415–422.

[8] H. B. Chen, Y. Li, X. Hou, Exact multiplicity for periodic solutions of duffing type, Nonlinear Anal. 55 (2003), no. 1-2, 115–124.

[9] E. Esmailzadeh, B. Mehri, G. Nakhaie-Jazar, Periodic solution of a second order, autonomous, nonlinear system, Nonlinear Dynam. 10 (1996), no. 4, 307–316.

[10] C. Fabry, J. Mawhin, M. N. Nkashama, A multiplicity result for periodic solutions of forced nonlinear second order ordinary differential equations, Bull. London Math. Soc. 18 (1986), no. 2, 173–180.

[11] J. F. Jiang, On the qualitative behavior of solutions of the equation ẍ+f_{1}(x)ẋ+f_{2}(x)ẋ^{2}+g(x) = 0, J. Math. Anal. Appl. 194 (1995), no. 3, 597–611.

[12] J. F. Jiang, The global stability of a class of second order differential equations, Nonlinear Anal. 28 (1997), no. 5, 855–870.

[13] G. Katriel, Uniqueness of periodic solutions for asymptotically linear duffing equations with strong forcing, Topol. Methods Nonlinear Anal. 12 (1998), no. 2, 263–274.

[14] A. C. Lazer, P. J. McKenna, On the existence of stable periodic solutions of differential equations of duffing type, Proc. Amer. Math. Soc. 110 (1990), no. 1, 125–133.

[15] J. Mawhin, Topological Degree and Boundary Value Problems for Nonlinear Differential Equations, Lecture Notes in Mathematics, vol. 1537, Springer-Verlag, Berlin, 1993.

[16] R. Ortega, Stability and index of periodic solutions of an equation of duffing type, Boll. Un. Mat. Ital. B (7) 3 (1989), no. 3, 533–546.

[17] G. Tarantello, On the number of solutions for the forced pendulum equation, J. Differential Equations 80 (1989), no. 1, 79–93.

[18] G. Villari, On the existence of periodic solutions for Liénard’s equation, Nonlinear Anal. 7 (1983), no. 1, 71–78.

[19] Z. H. Zheng, Periodic solutions of generalized Liénard equations, J. Math. Anal. Appl. 148 (1990), no. 1, 1–10.

[20] J. Zhou, On the existence and uniqueness of periodic solutions for Liénard-type equations, Nonlinear Anal. 27 (1996), no. 12, 1463–1470.

[21] A. Zitan, R. Ortega, Existence of asymptotically stable periodic solutions of a forced equation of Liénard type, Nonlinear Anal. 22 (1994), no. 8, 993–1003.

M. R. Mokhtarzadeh, School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran.

  E-mail address: mrmokhtarzadeh@ipm.ir

M. R. Pournaki, Department of Mathematical Sciences, Sharif University of Technology, P.O. Box 11155-9415, Tehran, Iran.

  E-mail address: pournaki@ipm.ir
  URL: http://math.ipm.ac.ir/pournaki/

A. Razani, Department of Mathematics, Faculty of Science, Imam Khomeini International University, P.O. Box 34194-288, Qazvin, Iran, and School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran.

  E-mail address: razani@ikiu.ac.ir
  URL: http://math.ipm.ac.ir/razani/