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Limits of solutions to a nonlinear second-order ODE
Nonlinear Analysis 75 (2012) 5139–5144
Contents lists available at SciVerse ScienceDirect
Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
Limits of solutions to a nonlinear second-order ODE Cristian Vladimirescu Department of Mathematics, University of Craiova, 13 A.I. Cuza Str., Craiova RO 200585, Romania
article
info
Article history: Received 28 December 2011 Accepted 11 April 2012 Communicated by S. Ahmad This paper is dedicated to the memory of Professor Cezar Avramescu.
abstract In this paper, the existence of solutions to the equation x¨ + 2f (t ) x˙ + β(t )x + g (t , x) = 0, t ≥ 0, is discussed. Our approach allows us achieve extension to the case of the whole real line, for which the existence of homoclinic solutions having zero limit at ±∞ is deduced. The result is obtained through the method of the Lyapunov function and differential inequalities. © 2012 Elsevier Ltd. All rights reserved.
MSC: 34A40 34C37 Keywords: Second-order ODE Lyapunov function Solutions having zero limit at ∞ Homoclinic solution
1. Introduction Consider the nonlinear second-order ODE x¨ + 2f (t )˙x + β(t )x + g (t , x) = 0,
t ∈ R+ ,
(1.1)
where f , β : R+ → R, and g : R+ × R → R are three given functions; R+ = [0, +∞). We give sufficient conditions for which Eq. (1.1) admits at least one solution x : R+ → R fulfilling a condition of the type
w (t ) ≤ β0 x2 (t ) + (˙x(t ) + f (t )x(t ))2 ≤ v(t ),
t ∈ R+ ,
(1.2)
where v, w : R+ → are two functions depending on f and β , β0 > := (0, +∞). Next, we present sufficient conditions for which v(+∞) := limt →+∞ v(t ) = 0. This fact assures the existence of a solution x to Eq. (1.1), which is not identically zero, such that x(+∞) = x˙ (+∞) = 0. Eq. (1.1) is a basic mathematical model for the representation of damped nonlinear oscillatory phenomena. The properties which could be interesting are the stability, the boundedness and the vanishing of solutions at +∞, and these have been intensively studied in, for example, [1–14]. The asymptotic stability of the null solution to Eq. (1.1) is researched in [3,4] (in the case where β(t ) = 1 and g (t , x) = x), in [10] (in the case where β(t ) = 1) and in [11], by using ingenious transformations (introduced in [3,4]), differential inequalities, and fixed point theorems. Here, we reconsider Eq. (1.1) under more general assumptions and prove an existence result for solutions not identically zero and vanishing at +∞ (see Theorem 2.1 below). The left side of inequality (1.2) is used to prove that the solution found is not identically zero. To establish inequality (1.2), we will use the method of the Lyapunov function and differential inequalities.
R∗+
E-mail address: [email protected]. 0362-546X/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2012.04.029
0 is constant, and R∗+
5140
C. Vladimirescu / Nonlinear Analysis 75 (2012) 5139–5144
Less studied but nonetheless important is the asymptotic behavior of the solutions on the whole real axis R. Recently, in [15–17,11,18] the stability, the boundedness of solutions and the vanishing of solutions at ±∞ are studied. In the present paper, our hypotheses are more general than the ones from [11,18] and allow us to achieve extension of Theorem 2.1 to the whole real axis (see Theorem 4.1 below, for which the existence of homoclinic solutions to Eq. (4.1) vanishing at ±∞ is deduced). 2. The main result The following hypotheses will be required: (i) f ∈ C 1 (R+ ) and f (t ) ≥ 0 for all t ≥ 0. (ii)
+∞ 0
f (t )dt = +∞.
(iii) There exist two constants h, K ≥ 0 such that
|f ′ (t ) + f 2 (t )| ≤ Kf (t ), ∀t ∈ [h, +∞). +∞ (iv) β ∈ C (R+ ), 0 |β(t ) − β0 |dt < +∞, and K < 2/γ ,
√
where β0 > 0 is constant and γ = max{1; 1/ β0 }. (v) g ∈ C (R+ × R) and g is locally Lipschitzian in x. (vi) g satisfies the following estimate:
|g (t , x)| ≤ f (t )o(|x|),
∀t ∈ R+ ,
where ‘‘o’’ denotes the usual Landau symbol. The main result of this paper is the following theorem: Theorem 2.1. If the assumptions (i)–(vi) are fulfilled, then Eq. (1.1) admits a solution x not identically zero and having x(+∞) = x˙ (+∞) = 0. 3. Proof of Theorem 2.1 Define on R2 the Lyapunov function V (z ) = β0 x2 + y2 ,
z = (x, y)T ∈ R2 .
If we use the transformation (as in [3]) y := x˙ + f (t )x then Eq. (1.1) becomes z˙ = F (t , z ),
(3.1)
where F (t , z ) =
y − f (t )x (f ′ (t ) + f 2 (t ) − β(t ))x − f (t )y − g (t , x)
and z = (x, y)T ∈ R2 . The derivative V˙ of V along system (3.1) (see [19], pp. 50, 99) is V˙ (z ) = (grad V , F ). Therefore, we obtain V˙ (z ) = −2f (t )(β0 x2 + y2 ) + 2xy[f ′ (t ) + f 2 (t ) + β0 − β(t )] − 2g (t , x)y.
(3.2)
Consider for z = (x, y) ∈ R the norm ∥z ∥ = β0 x2 + y2 . Since V : R2 → R+ is continuous, V (0) = 0, lim∥z ∥→+∞ V (z ) = +∞, and V (R2 ) is connected, it follows that for every r0 > 0, there exists z0 ∈ R, z0 ̸= 0, such that V (z0 ) = r0 . Therefore, ∀n ∈ N∗ := {1, 2, . . .} one can consider zn = (xn , yn )T the unique solution to system (3.1) for which T
V (zn (1/n)) = exp
2
1/n
0
v(s)ds ,
(3.3)
C. Vladimirescu / Nonlinear Analysis 75 (2012) 5139–5144
5141
where
v(t ) :=
(γ θ − 2)f (t ) + γ |f ′ (t ) + f 2 (t )| + γ |β(t ) − β0 |, [γ (K + θ ) − 2]f (t ) + γ |β(t ) − β0 |, ∀t ≥ h.
∀t ∈ [0, h),
Notice that even if |f ′ (h)+ f 2 (h)| ̸= Kf (h), v is still integrable on each compact of R+ , since, if ζ > h, then v is continuous on [0, h) (h, ζ ] and therefore is integrable on [0, ζ ]. Take some
θ ∈ (0, 2/γ − K ).
(3.4)
By hypothesis (vi), it follows that there exists a ρ > 0 such that if |x| < ρ , then
|g (t , x)| ≤ θ f (t )|x|. Now, we define the function g˜ : R+ × R → R by g (t , x), g (t , ρ), g (t , −ρ),
g˜ (t , x) :=
|x| < ρ, x ≥ ρ, x ≤ −ρ,
if if if
for all t ≥ 0. It is readily seen that for every (t , x) ∈ R+ × R,
|˜g (t , x)| ≤ θ f (t )|x|, g˜ is of class C (R+ × R), and is locally Lipschitzian in x. With this in mind, we will admit from now on that the original function g satisfies all the properties of g˜ . From (3.2) we get the estimate V˙ (zn (t )) ≤ v(t )V (zn (t )),
∀t ≥ 1/n, ∀n ∈ N∗ .
(3.5)
Since zn (1/n) ̸= 0, we infer that V (zn (t )) > 0, ∀t ≥ 1/n, ∀n ∈ N . By (3.5) we obtain then ∗
V˙ (zn (t )) V (zn (t ))
≤ v(t ),
∀t ≥ 1/n, ∀n ∈ N∗
and, by taking into account (3.3), we deduce V (zn (t )) ≤ α(t ),
∀t ≥ 1/n, ∀n ∈ N∗ ,
where α : R+ → R∗+ is defined by
α(t ) = exp
t
v(s)ds ,
∀t ≥ 0.
0
By hypotheses (ii)–(iv) and relation (3.4), it follows that lim α(t ) = lim exp
t →+∞
t
t →+∞
v(s)ds
0 h
v(s)ds +
= lim exp t →+∞
0
t
([γ (θ + K ) − 2]f (s) + γ |β(s) − β0 |)ds = 0.
h
The conclusion is that for every n ∈ N∗ , there exists a function zn such that z˙n (t ) = F (t , zn (t )),
∀t ∈ [1/n, n]
(3.6)
and V (zn (t )) ≤ α(t ),
∀t ∈ [1/n, n].
We extend zn to the whole semiline R+ by setting zn (1/n), zn (t ) := zn (t ), zn (n),
if if if
t ∈ [0, 1/n], t ∈ [1/n, n], t ≥ n.
Consider the function space Cc := {z : R+ → R2 , z continuous},
(3.7)
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C. Vladimirescu / Nonlinear Analysis 75 (2012) 5139–5144
endowed with the topology of the uniform convergence on compact subsets of R+ ; as is known, this topology can be defined through the following family of seminorms:
|z |n := sup{∥z (t )∥, t ∈ [1/n, n]},
n ∈ N∗ .
Furthermore, we know, from the Ascoli–Arzelà lemma, that a family A ⊂ Cc is relatively compact if and only if A is equi-continuous and uniformly bounded on compact subsets of R+ (see, e.g., [19], p. 30). We want to show that the family { zn }n∈N∗ is relatively compact on compact subsets of R+ . To this end, let us consider k ∈ N∗ . Obviously, there exists n0 ∈ N∗ such that [1/k, k] ⊂ [1/n, n] for all n ∈ N∗ , n ≥ n0 , and therefore,
zn (t ) = zn (t ),
∀t ∈ [1/k, k], n ≥ n0 .
By (3.7) we obtain V (zn (t )) ≤ α(t ),
∀t ∈ [1/k, k],
and therefore
∥zn (t )∥ ≤ Mk ,
∀t ∈ [1/k, k], n ≥ n0 ,
where 1
Mk := (sup{α(t ) : t ∈ [1/k, k]}) 2 . Hence, the family { zn }n∈N∗ is uniformly bounded on [1/k, k]. Next, on setting Lk := sup{∥F (t , z )∥, t ∈ [1/k, k], ∥z ∥ ≤ Mk }, from (3.6) it follows that
∥ zn′ (t )∥ = ∥F (t , zn (t ))∥ ≤ Lk ,
∀t ∈ [1/k, k], n ≥ n0 .
Then the family { zn }n∈N∗ is equi-continuous on [1/k, k], since the derivatives { zn′ }n∈N∗ are uniformly bounded on [1/k, k]. Hence, by passing to subsequences, one may suppose that
zn → z ,
in Cc .
From zn (t ) = F (t , zn (t )), it follows that { zn′ (t )}n∈N∗ converges uniformly on [1/k, k] to z˙ . So, for all t ∈ [1/k, k],
′
z˙ (t ) = F (t , z (t )), V (z (t )) ≤ α(t ).
(3.8)
But, since each t ∈ R∗+ belongs to an interval [1/k, k], it follows that (3.8) holds for every t ∈ R∗+ . Defining z ∗ : R+ → R2 by z (t ) = ∗
z (t ), lim z (t ),
if if
t →0
t > 0, t = 0,
we get a solution to Eq. (1.1) on R+ having properties (3.8), like z . To end the proof of Theorem 2.1, it remains to show that z ∗ is not identically zero. From (3.2) we deduce another estimate, V˙ (zn (t )) ≥ w(t )V (zn (t )),
∀t ≥ 1/n,
where
w(t ) :=
(−γ θ − 2)f (t ) − γ |f ′ (t ) + f 2 (t )| − γ |β(t ) − β0 |, [−γ (K + θ ) − 2]f (t ) − γ |β(t ) − β0 |, ∀t ≥ h.
∀t ∈ [0, h),
Remark that w is integrable on each compact of R+ , using the same argument as for v . Hence V (zn (t )) ≥ V (zn (1/n)) exp
t 1/n
1/n
w(s)ds = exp 0
v(s)ds +
t 1/n
w(s)ds ,
∀t ≥ 1/n.
By setting t = t ∗ ∈ [1/n, h) in the previous inequality, since w ≤ 0 on [0, h], we obtain V (zn (t ∗ )) ≥ exp
1/n
0
v(s)ds +
h 1/n
w(s)ds .
(3.9)
C. Vladimirescu / Nonlinear Analysis 75 (2012) 5139–5144
5143
If z ∗ was identically zero, then limn→∞ zn (t ∗ ) = z ∗ (t ∗ ) would imply lim V ( zn (t ∗ )) = 0.
(3.10)
n→∞
But, passing to limit as n → ∞ in (3.9), since zn (t ) = zn (t ), for n big enough, we would get ∗
lim V ( zn (t ∗ )) ≥ exp
n→∞
h
∗
w(s)ds
0
h ((γ θ + 2)f (s) + γ |f ′ (s) + f 2 (s)| + γ |β(s) − β0 |)ds > 0, = exp − 0
which contradicts relation (3.10). Here are some examples of typical functions f , β, g which satisfy the assumptions (i)–(vi): 1 f (t ) = , ∀t ≥ h > 0, t sin t , ∀t ∈ R+ , β(t ) = 3 + 2 t +1 g (t , x) = f (t )xα , α > 1, ∀t ∈ R+ , ∀x ∈ R, where f is extended to a smooth nonnegative function defined on R+ . Remark 3.1. Under our hypothesis (iv), it follows that β does not necessarily have to be of class C 1 (R+ ) or decreasing, as assumed in [11,18]. Moreover, in the particular case where β(t ) = 1 + e−t (as√considered in [11]), it follows that β0 = 1, and hence hypothesis (iv) of Theorem 2.1 in [11] is fulfilled for K ∈ [0, β0 ) = [0, 1), while hypothesis (iv) here is fulfilled for K ∈ [0, 2/γ ) = [0, 2). 4. Extensions to R In this section we present some remarks concerning the possibility of extending Theorem 2.1 to the whole real line R. So, let us consider the equation x¨ + 2f (t )˙x + β(t )x + g (t , x) = 0,
t ∈ R,
(4.1)
where f , β : R → R and g : R × R → R are three given functions, satisfying the following hypotheses: ′
(i) f ∈ C 1 (R) and t · f (t ) > 0 for all t ∈ R, t ̸= 0. +∞ ′ 0 (ii) −∞ f (t )dt = −∞ and 0 f (t )dt = +∞. ′
(iii) There exist two constants h, K ≥ 0, such that |f ′ (t ) + f 2 (t )| ≤ K |f (t )|, ∀t with |t | ≥ h. +∞ 0 ′ (iv) β ∈ C (R), 0 |β(t ) − β0 |dt < +∞, −∞ |β(t ) − β0 |dt < +∞, and K < 2/γ , where β0 > 0 is constant and γ = max{1, 1/β0 }. ′ (v) g ∈ C (R × R) and g is locally Lipschitzian in x. ′ (vi) g satisfies the following estimate: |g (t , x)| ≤ |f (t )|o(|x|), ∀t ∈ R, where ‘‘o’’ denotes the usual Landau symbol. Example 4.1. Let f be defined by f (t ) =
1 t
if |t | ≥ h > 0 and extended on the interval (−h, h) in such a way that f ∈ C 1 (R)
and t · f (t ) ≥ 0, ∀t ∈ R (for instance, if h = 1, we can choose f (t ) = t (3 − 2|t |) if |t | < 1). Also, let β(t ) = 3 + α
t ∈ R, β0 = 3, and g (t , x) = f (t ) · x ,
sin t for t 2 +1 ′ ′
all
α > 1, for all (t , x) ∈ R . It is easily seen that these functions satisfy (i) –(vi) . 2
Notice that on making the changes s = −t ,
u(s) = x(−s),
t ≤ 0,
Eq. (4.1) for t ≤ 0 becomes d2 u
du + 2f ∗ (s) + β ∗ (s)u + g ∗ (s, u) = 0, s ∈ R+ , ds2 ds where f ∗ (s) = −f (−s), β ∗ (s) = β(−s), and g ∗ (s, u) = g (−s, u). Taking into account this remark as well as Theorem 2.1, we can state the following result. Theorem 4.1. If the assumptions (i)′ –(vi)′ are fulfilled, then Eq. (4.1) admits a homoclinic solution x having x(±∞) = x˙ (±∞) = 0.
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C. Vladimirescu / Nonlinear Analysis 75 (2012) 5139–5144
Acknowledgments The author wishes to express his sincere thanks to the reviewers for the suggestions and valuable comments that helped improve the paper. References [1] C. Avramescu, Limits of a solution of a nonlinear differential equation, Nonlinear Analysis Forum 7 (2002) 209–215. [2] C. Avramescu, C. Vladimirescu, Limits of solutions of a perturbed linear differential equation, Electronic Journal of Qualitative Theory of Differential Equations 3 (2002) 1–11. [3] T.A. Burton, T. Furumochi, A note on stability by Schauder’s theorem, Funkcialaj Ekvacioj 44 (2001) 73–82. [4] T.A. Burton, L. Hatvani, Asymptotic stability of second order ordinary functional and partial differential equations, Journal of Mathematical Analysis and Applications 176 (1993) 261–281. [5] L. Hatvani, On the stability of the zero solution of certain second order nonlinear differential equations, Acta Scientiarum Mathematicarum 32 (1971) 1–9. [6] L. Hatvani, On the asymptotic behavior of the solutions of (p(t )x′ )′ + q(t )f (x) = 0, Publicationes Mathematicae 19 (1972) 225–237. [7] L. Hatvani, A generalization of the Barbashin–Krasovskij theorems to the partial stability in nonautonomous systems, in: Colloquia Mathematica Societatis Janos Bolyai, in: Qualitative Theory of Differential Equations, vol. 30, Szeged, Hungary, 1979, pp. 381–409. [8] L. Hatvani, Integral conditions on the asymptotic stability for the damped linear oscillator with small damping, Proceedings of the American Mathematical Society 124 (1996) 415–422. [9] V. Lakshmikantham, S. Leela, Differential and Integral Inequalities. Theory and Applications, in: Ordinary Differential Equations, vol. I, Academic Press, New York, London, 1969. [10] Gh. Moroşanu, C. Vladimirescu, Stability for a nonlinear second order ODE, Funkcialaj Ekvacioj 48 (2005) 49–56. [11] Gh. Moroşanu, C. Vladimirescu, Stability for a damped nonlinear oscillator, Nonlinear Analysis 60 (2005) 303–310. [12] P. Pucci, J. Serrin, Precise damping conditions for global asymptotic stability for nonlinear second order systems, Acta Mathematica 170 (1993) 275–307. [13] P. Pucci, J. Serrin, Precise damping conditions for global asymptotic stability for non-linear second order systems, II, Journal of Differential Equations 113 (1994) 505–534. [14] P. Pucci, J. Serrin, Asymptotic stability for intermittently controlled nonlinear oscillators, SIAM Journal on Mathematical Analysis 25 (1994) 815–835. [15] C. Avramescu, Existence problems for homoclinic solutions, Abstract and Applied Analysis 7 (1) (2002) 1–29. [16] C. Avramescu, C. Vladimirescu, Homoclinic solutions for linear and linearizable ordinary differential equations, Abstract and Applied Analysis 5 (2) (2000) 65–85. [17] C. Avramescu, C. Vladimirescu, g-bounded solutions for ordinary differential equations, Annals of the University of Craiova - Mathematics and Computer Science Series XXIX (2002) 72–90. [18] C. Vladimirescu, An existence result for homoclinic solutions to a nonlinear second order ODE through differential inequalities, Nonlinear Analysis: Theory, Methods & Applications 68 (10) (2008) 3217–3223. [19] C. Corduneanu, Principles of Differential and Integral Equations, Allyn and Bacon, Boston, 1971.
Contents lists available at SciVerse ScienceDirect
Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
Limits of solutions to a nonlinear second-order ODE Cristian Vladimirescu Department of Mathematics, University of Craiova, 13 A.I. Cuza Str., Craiova RO 200585, Romania
article
info
Article history: Received 28 December 2011 Accepted 11 April 2012 Communicated by S. Ahmad This paper is dedicated to the memory of Professor Cezar Avramescu.
abstract In this paper, the existence of solutions to the equation x¨ + 2f (t ) x˙ + β(t )x + g (t , x) = 0, t ≥ 0, is discussed. Our approach allows us achieve extension to the case of the whole real line, for which the existence of homoclinic solutions having zero limit at ±∞ is deduced. The result is obtained through the method of the Lyapunov function and differential inequalities. © 2012 Elsevier Ltd. All rights reserved.
MSC: 34A40 34C37 Keywords: Second-order ODE Lyapunov function Solutions having zero limit at ∞ Homoclinic solution
1. Introduction Consider the nonlinear second-order ODE x¨ + 2f (t )˙x + β(t )x + g (t , x) = 0,
t ∈ R+ ,
(1.1)
where f , β : R+ → R, and g : R+ × R → R are three given functions; R+ = [0, +∞). We give sufficient conditions for which Eq. (1.1) admits at least one solution x : R+ → R fulfilling a condition of the type
w (t ) ≤ β0 x2 (t ) + (˙x(t ) + f (t )x(t ))2 ≤ v(t ),
t ∈ R+ ,
(1.2)
where v, w : R+ → are two functions depending on f and β , β0 > := (0, +∞). Next, we present sufficient conditions for which v(+∞) := limt →+∞ v(t ) = 0. This fact assures the existence of a solution x to Eq. (1.1), which is not identically zero, such that x(+∞) = x˙ (+∞) = 0. Eq. (1.1) is a basic mathematical model for the representation of damped nonlinear oscillatory phenomena. The properties which could be interesting are the stability, the boundedness and the vanishing of solutions at +∞, and these have been intensively studied in, for example, [1–14]. The asymptotic stability of the null solution to Eq. (1.1) is researched in [3,4] (in the case where β(t ) = 1 and g (t , x) = x), in [10] (in the case where β(t ) = 1) and in [11], by using ingenious transformations (introduced in [3,4]), differential inequalities, and fixed point theorems. Here, we reconsider Eq. (1.1) under more general assumptions and prove an existence result for solutions not identically zero and vanishing at +∞ (see Theorem 2.1 below). The left side of inequality (1.2) is used to prove that the solution found is not identically zero. To establish inequality (1.2), we will use the method of the Lyapunov function and differential inequalities.
R∗+
E-mail address: [email protected]. 0362-546X/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2012.04.029
0 is constant, and R∗+
5140
C. Vladimirescu / Nonlinear Analysis 75 (2012) 5139–5144
Less studied but nonetheless important is the asymptotic behavior of the solutions on the whole real axis R. Recently, in [15–17,11,18] the stability, the boundedness of solutions and the vanishing of solutions at ±∞ are studied. In the present paper, our hypotheses are more general than the ones from [11,18] and allow us to achieve extension of Theorem 2.1 to the whole real axis (see Theorem 4.1 below, for which the existence of homoclinic solutions to Eq. (4.1) vanishing at ±∞ is deduced). 2. The main result The following hypotheses will be required: (i) f ∈ C 1 (R+ ) and f (t ) ≥ 0 for all t ≥ 0. (ii)
+∞ 0
f (t )dt = +∞.
(iii) There exist two constants h, K ≥ 0 such that
|f ′ (t ) + f 2 (t )| ≤ Kf (t ), ∀t ∈ [h, +∞). +∞ (iv) β ∈ C (R+ ), 0 |β(t ) − β0 |dt < +∞, and K < 2/γ ,
√
where β0 > 0 is constant and γ = max{1; 1/ β0 }. (v) g ∈ C (R+ × R) and g is locally Lipschitzian in x. (vi) g satisfies the following estimate:
|g (t , x)| ≤ f (t )o(|x|),
∀t ∈ R+ ,
where ‘‘o’’ denotes the usual Landau symbol. The main result of this paper is the following theorem: Theorem 2.1. If the assumptions (i)–(vi) are fulfilled, then Eq. (1.1) admits a solution x not identically zero and having x(+∞) = x˙ (+∞) = 0. 3. Proof of Theorem 2.1 Define on R2 the Lyapunov function V (z ) = β0 x2 + y2 ,
z = (x, y)T ∈ R2 .
If we use the transformation (as in [3]) y := x˙ + f (t )x then Eq. (1.1) becomes z˙ = F (t , z ),
(3.1)
where F (t , z ) =
y − f (t )x (f ′ (t ) + f 2 (t ) − β(t ))x − f (t )y − g (t , x)
and z = (x, y)T ∈ R2 . The derivative V˙ of V along system (3.1) (see [19], pp. 50, 99) is V˙ (z ) = (grad V , F ). Therefore, we obtain V˙ (z ) = −2f (t )(β0 x2 + y2 ) + 2xy[f ′ (t ) + f 2 (t ) + β0 − β(t )] − 2g (t , x)y.
(3.2)
Consider for z = (x, y) ∈ R the norm ∥z ∥ = β0 x2 + y2 . Since V : R2 → R+ is continuous, V (0) = 0, lim∥z ∥→+∞ V (z ) = +∞, and V (R2 ) is connected, it follows that for every r0 > 0, there exists z0 ∈ R, z0 ̸= 0, such that V (z0 ) = r0 . Therefore, ∀n ∈ N∗ := {1, 2, . . .} one can consider zn = (xn , yn )T the unique solution to system (3.1) for which T
V (zn (1/n)) = exp
2
1/n
0
v(s)ds ,
(3.3)
C. Vladimirescu / Nonlinear Analysis 75 (2012) 5139–5144
5141
where
v(t ) :=
(γ θ − 2)f (t ) + γ |f ′ (t ) + f 2 (t )| + γ |β(t ) − β0 |, [γ (K + θ ) − 2]f (t ) + γ |β(t ) − β0 |, ∀t ≥ h.
∀t ∈ [0, h),
Notice that even if |f ′ (h)+ f 2 (h)| ̸= Kf (h), v is still integrable on each compact of R+ , since, if ζ > h, then v is continuous on [0, h) (h, ζ ] and therefore is integrable on [0, ζ ]. Take some
θ ∈ (0, 2/γ − K ).
(3.4)
By hypothesis (vi), it follows that there exists a ρ > 0 such that if |x| < ρ , then
|g (t , x)| ≤ θ f (t )|x|. Now, we define the function g˜ : R+ × R → R by g (t , x), g (t , ρ), g (t , −ρ),
g˜ (t , x) :=
|x| < ρ, x ≥ ρ, x ≤ −ρ,
if if if
for all t ≥ 0. It is readily seen that for every (t , x) ∈ R+ × R,
|˜g (t , x)| ≤ θ f (t )|x|, g˜ is of class C (R+ × R), and is locally Lipschitzian in x. With this in mind, we will admit from now on that the original function g satisfies all the properties of g˜ . From (3.2) we get the estimate V˙ (zn (t )) ≤ v(t )V (zn (t )),
∀t ≥ 1/n, ∀n ∈ N∗ .
(3.5)
Since zn (1/n) ̸= 0, we infer that V (zn (t )) > 0, ∀t ≥ 1/n, ∀n ∈ N . By (3.5) we obtain then ∗
V˙ (zn (t )) V (zn (t ))
≤ v(t ),
∀t ≥ 1/n, ∀n ∈ N∗
and, by taking into account (3.3), we deduce V (zn (t )) ≤ α(t ),
∀t ≥ 1/n, ∀n ∈ N∗ ,
where α : R+ → R∗+ is defined by
α(t ) = exp
t
v(s)ds ,
∀t ≥ 0.
0
By hypotheses (ii)–(iv) and relation (3.4), it follows that lim α(t ) = lim exp
t →+∞
t
t →+∞
v(s)ds
0 h
v(s)ds +
= lim exp t →+∞
0
t
([γ (θ + K ) − 2]f (s) + γ |β(s) − β0 |)ds = 0.
h
The conclusion is that for every n ∈ N∗ , there exists a function zn such that z˙n (t ) = F (t , zn (t )),
∀t ∈ [1/n, n]
(3.6)
and V (zn (t )) ≤ α(t ),
∀t ∈ [1/n, n].
We extend zn to the whole semiline R+ by setting zn (1/n), zn (t ) := zn (t ), zn (n),
if if if
t ∈ [0, 1/n], t ∈ [1/n, n], t ≥ n.
Consider the function space Cc := {z : R+ → R2 , z continuous},
(3.7)
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endowed with the topology of the uniform convergence on compact subsets of R+ ; as is known, this topology can be defined through the following family of seminorms:
|z |n := sup{∥z (t )∥, t ∈ [1/n, n]},
n ∈ N∗ .
Furthermore, we know, from the Ascoli–Arzelà lemma, that a family A ⊂ Cc is relatively compact if and only if A is equi-continuous and uniformly bounded on compact subsets of R+ (see, e.g., [19], p. 30). We want to show that the family { zn }n∈N∗ is relatively compact on compact subsets of R+ . To this end, let us consider k ∈ N∗ . Obviously, there exists n0 ∈ N∗ such that [1/k, k] ⊂ [1/n, n] for all n ∈ N∗ , n ≥ n0 , and therefore,
zn (t ) = zn (t ),
∀t ∈ [1/k, k], n ≥ n0 .
By (3.7) we obtain V (zn (t )) ≤ α(t ),
∀t ∈ [1/k, k],
and therefore
∥zn (t )∥ ≤ Mk ,
∀t ∈ [1/k, k], n ≥ n0 ,
where 1
Mk := (sup{α(t ) : t ∈ [1/k, k]}) 2 . Hence, the family { zn }n∈N∗ is uniformly bounded on [1/k, k]. Next, on setting Lk := sup{∥F (t , z )∥, t ∈ [1/k, k], ∥z ∥ ≤ Mk }, from (3.6) it follows that
∥ zn′ (t )∥ = ∥F (t , zn (t ))∥ ≤ Lk ,
∀t ∈ [1/k, k], n ≥ n0 .
Then the family { zn }n∈N∗ is equi-continuous on [1/k, k], since the derivatives { zn′ }n∈N∗ are uniformly bounded on [1/k, k]. Hence, by passing to subsequences, one may suppose that
zn → z ,
in Cc .
From zn (t ) = F (t , zn (t )), it follows that { zn′ (t )}n∈N∗ converges uniformly on [1/k, k] to z˙ . So, for all t ∈ [1/k, k],
′
z˙ (t ) = F (t , z (t )), V (z (t )) ≤ α(t ).
(3.8)
But, since each t ∈ R∗+ belongs to an interval [1/k, k], it follows that (3.8) holds for every t ∈ R∗+ . Defining z ∗ : R+ → R2 by z (t ) = ∗
z (t ), lim z (t ),
if if
t →0
t > 0, t = 0,
we get a solution to Eq. (1.1) on R+ having properties (3.8), like z . To end the proof of Theorem 2.1, it remains to show that z ∗ is not identically zero. From (3.2) we deduce another estimate, V˙ (zn (t )) ≥ w(t )V (zn (t )),
∀t ≥ 1/n,
where
w(t ) :=
(−γ θ − 2)f (t ) − γ |f ′ (t ) + f 2 (t )| − γ |β(t ) − β0 |, [−γ (K + θ ) − 2]f (t ) − γ |β(t ) − β0 |, ∀t ≥ h.
∀t ∈ [0, h),
Remark that w is integrable on each compact of R+ , using the same argument as for v . Hence V (zn (t )) ≥ V (zn (1/n)) exp
t 1/n
1/n
w(s)ds = exp 0
v(s)ds +
t 1/n
w(s)ds ,
∀t ≥ 1/n.
By setting t = t ∗ ∈ [1/n, h) in the previous inequality, since w ≤ 0 on [0, h], we obtain V (zn (t ∗ )) ≥ exp
1/n
0
v(s)ds +
h 1/n
w(s)ds .
(3.9)
C. Vladimirescu / Nonlinear Analysis 75 (2012) 5139–5144
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If z ∗ was identically zero, then limn→∞ zn (t ∗ ) = z ∗ (t ∗ ) would imply lim V ( zn (t ∗ )) = 0.
(3.10)
n→∞
But, passing to limit as n → ∞ in (3.9), since zn (t ) = zn (t ), for n big enough, we would get ∗
lim V ( zn (t ∗ )) ≥ exp
n→∞
h
∗
w(s)ds
0
h ((γ θ + 2)f (s) + γ |f ′ (s) + f 2 (s)| + γ |β(s) − β0 |)ds > 0, = exp − 0
which contradicts relation (3.10). Here are some examples of typical functions f , β, g which satisfy the assumptions (i)–(vi): 1 f (t ) = , ∀t ≥ h > 0, t sin t , ∀t ∈ R+ , β(t ) = 3 + 2 t +1 g (t , x) = f (t )xα , α > 1, ∀t ∈ R+ , ∀x ∈ R, where f is extended to a smooth nonnegative function defined on R+ . Remark 3.1. Under our hypothesis (iv), it follows that β does not necessarily have to be of class C 1 (R+ ) or decreasing, as assumed in [11,18]. Moreover, in the particular case where β(t ) = 1 + e−t (as√considered in [11]), it follows that β0 = 1, and hence hypothesis (iv) of Theorem 2.1 in [11] is fulfilled for K ∈ [0, β0 ) = [0, 1), while hypothesis (iv) here is fulfilled for K ∈ [0, 2/γ ) = [0, 2). 4. Extensions to R In this section we present some remarks concerning the possibility of extending Theorem 2.1 to the whole real line R. So, let us consider the equation x¨ + 2f (t )˙x + β(t )x + g (t , x) = 0,
t ∈ R,
(4.1)
where f , β : R → R and g : R × R → R are three given functions, satisfying the following hypotheses: ′
(i) f ∈ C 1 (R) and t · f (t ) > 0 for all t ∈ R, t ̸= 0. +∞ ′ 0 (ii) −∞ f (t )dt = −∞ and 0 f (t )dt = +∞. ′
(iii) There exist two constants h, K ≥ 0, such that |f ′ (t ) + f 2 (t )| ≤ K |f (t )|, ∀t with |t | ≥ h. +∞ 0 ′ (iv) β ∈ C (R), 0 |β(t ) − β0 |dt < +∞, −∞ |β(t ) − β0 |dt < +∞, and K < 2/γ , where β0 > 0 is constant and γ = max{1, 1/β0 }. ′ (v) g ∈ C (R × R) and g is locally Lipschitzian in x. ′ (vi) g satisfies the following estimate: |g (t , x)| ≤ |f (t )|o(|x|), ∀t ∈ R, where ‘‘o’’ denotes the usual Landau symbol. Example 4.1. Let f be defined by f (t ) =
1 t
if |t | ≥ h > 0 and extended on the interval (−h, h) in such a way that f ∈ C 1 (R)
and t · f (t ) ≥ 0, ∀t ∈ R (for instance, if h = 1, we can choose f (t ) = t (3 − 2|t |) if |t | < 1). Also, let β(t ) = 3 + α
t ∈ R, β0 = 3, and g (t , x) = f (t ) · x ,
sin t for t 2 +1 ′ ′
all
α > 1, for all (t , x) ∈ R . It is easily seen that these functions satisfy (i) –(vi) . 2
Notice that on making the changes s = −t ,
u(s) = x(−s),
t ≤ 0,
Eq. (4.1) for t ≤ 0 becomes d2 u
du + 2f ∗ (s) + β ∗ (s)u + g ∗ (s, u) = 0, s ∈ R+ , ds2 ds where f ∗ (s) = −f (−s), β ∗ (s) = β(−s), and g ∗ (s, u) = g (−s, u). Taking into account this remark as well as Theorem 2.1, we can state the following result. Theorem 4.1. If the assumptions (i)′ –(vi)′ are fulfilled, then Eq. (4.1) admits a homoclinic solution x having x(±∞) = x˙ (±∞) = 0.
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