An existence result for homoclinic solutions to a nonlinear second-order ODE through differential inequalities

Nonlinear Analysis 68 (2008) 3217–3223 www.elsevier.com/locate/na

An existence result for homoclinic solutions to a nonlinear second-order ODE through differential inequalities Cristian Vladimirescu Department of Mathematics, University of Craiova, 13 A.I. Cuza Str., Craiova RO 200585, Romania Received 8 February 2007; accepted 8 March 2007

Abstract In this paper an existence result of homoclinic solutions to a nonlinear second-order ODE is presented. To this end, a method based on differential inequalities is used. c 2007 Elsevier Ltd. All rights reserved.

MSC: 34C37; 34A40 Keywords: Homoclinic solutions; Differential inequalities

1. Introduction In this paper we are concerned with the existence of homoclinic solutions to the equation x 00 + 2 f (t)x 0 + β(t)x + g(t, x) = 0,

t ∈ R,

(1.1)

i.e. solutions x : R → R satisfying the boundary conditions x(±∞) = x 0 (±∞) = 0,

(1.2)

where f : R → R and g : R × R → R are two given functions, and x(±∞) := lim x(t), t→±∞

x 0 (±∞) := lim x 0 (t). t→±∞

Our method is mainly based on Lemma 2.2 below. Considering Eq. (1.1) on the positive real semi-line R+ , we deduce the equation x 00 + 2 f (t)x 0 + β(t)x + g(t, x) = 0,

t ∈ R+ ,

(1.3)

which is a basic mathematical model for the representation of damped nonlinear oscillatory phenomena. Extensive studies around the damped nonlinear oscillators have been published (see, e.g., [3,7,14–18], and the references therein).

E-mail address: [email protected]. c 2007 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2007.03.015

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For interesting studies on homoclinic solutions we refer the reader to [4,5,9–11]. Regarding the stability of the zero solution to different types of perturbations to Eq. (1.3), the reader can find in [1, 12,14–18] a rich bibliography. In [6] (which is the first general introduction to stability of ordinary and functional differential equations by means of fixed point techniques), Section 4.1, and in [7] the authors proved the stability of the zero solution to Eq. (1.3) by using Schauder’s fixed point theorem (in the case when β(t) = 1). In [15], by using relatively classical arguments, the stability for the zero solution to Eq. (1.3) is researched. 2. Preliminary results The following results will be used. Let t0 , b, p0 ∈ R, b > t0 , H : [t0 , b) × R → R, a function of class C([t0 , b) × R), and q : [t0 , b) → R, a function of class C 1 ([t0 , b)). Admit that the Cauchy problem  0 p = H (t, p), t ∈ [t0 , b), p(t0 ) = p0 , has a unique solution, say p(t), defined on [t0 , b). Lemma 2.1 ([8], Theorem 3.5, p. 49). If q satisfies  0 q (t) ≤ H (t, q(t)), ∀t ∈ [t0 , b), q(t0 ) ≤ p0 , then q(t) ≤ p(t),

∀t ∈ [t0 , b).

Another result which will be used is in fact a generalization of a known result due to Krasnoselskii (see [13]). Let s : R → R be a bounded and positive function and G : R × Rd → Rd a continuous function (d ≥ 1); denote by k · k an arbitrary norm in Rd . In addition, consider a sequence (tn )n∈N\{0} , with tn → +∞ and 0 < tn < tn+1 . Set In := [−tn , tn ]. Lemma 2.2 (Avramescu [2]). Suppose that for each n ∈ N \ {0}, there exists xn : In → Rd , such that the following hypotheses are satisfied: (i) xn satisfies on In the differential equation x 0 = G(t, x);

(2.1)

(ii) the following inequality holds: kxn (t)k ≤ s(t),

∀t ∈ In .

Then, Eq. (2.1) admits a solution x : R → Rd fulfilling the inequality kx(t)k ≤ s(t),

(2.2)

∀t ∈ R.

3. The main result The following hypotheses will be required: (i) f ∈ C 1 (R) and t f (t) > 0, for all t ∈ R, t 6= 0; R0 R +∞ (ii) −∞ f (t)dt = −∞ and 0 f (t)dt = +∞; (iii) there exist h ≥ 0 and K ∈ (0, 1) such that | f 0 (t) + f 2 (t)| ≤ K | f (t)|,

for all |t| ≥ h;

(3.1)

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(iv) β ∈ C 1 (R), β is decreasing, and β1 ≥ β(t) ≥ β0 > (γ K )2 ,

(3.2)

∀t ∈ R, √ where β0 , β1 are constants and γ := β1 /β0 ; (v) g ∈ C(R × R) and g is locally Lipschitzian in x; (vi) g satisfies the following estimate: |g(t, x)| ≤ M| f (t)||x|α ,

(3.3)

∀x ∈ R, ∀t ∈ R.

These assumptions are inspired by those in [7]. Remark 3.1. A prototype of functions f , g, and β is f (t) = 1/t for |t| ≥ h (h > 0) and extended on the interval (−h, h) in such a way that f ∈ C 1 (R) and t · f (t) > 0 for all t ∈ R, t 6= 0 (for instance we can choose f (t) = t (3/ h 2 −2|t|/ h 3 ), |t| < h), g(t, x) = f (t)x α , α > 1, and β(t) = 2− π2 arctan t. Then, hypotheses (i)–(vi) are fulfilled. Remark 3.2. Notice that (i) and (iii) imply that f is uniformly bounded (and so we are in the case of small damping (see [14], Remark 2.2)). Remark 3.3. If f satisfies (i)–(iii), then | f (t)| > 0, for all t ∈ R, |t| ≥ h (see Remark 3.1 in [15]). Our main result has the following statement. Theorem 3.1. Suppose that hypotheses (i)–(vi) are fulfilled. Then, Eq. (1.1) admits homoclinic solutions. 4. Proof of the main result As in [7], we write Eq. (1.1) as the following first-order system: z 0 = A(t)z + B(t)z + F(t, z),

(4.1)

where   x , y  F(t, z) =

z=

 − f (t) A(t) = −β(t)  0 . −g(t, x)

 1 , − f (t)

 B(t) =

0 0 f (t) + f 2 (t)

 0 , 0

It is readily seen that our question on the existence of homoclinic solutions to Eq. (1.1) reduces to the existence of homoclinic solutions to system (4.1). Denote by Z (t), t ∈ R, the fundamental matrix of the linear system z 0 = A(t)z, which is equal to the identity matrix for t = 0. If z := (x, y)> is a vector of R2 , we set kzk := (β0 x 2 + y 2 )1/2 . Then, classical estimates lead us to the following inequality:   Z t kZ (t)Z −1 (s)zk ≤ exp − f (τ )dτ γ kzk, t, s ∈ R, t ≥ s, z ∈ R2 .

(4.2)

s

Let us consider n ∈ N, n > h, arbitrarily fixed. The proof will be given in four steps. Step 1. We consider first the system (4.1) on [−n, −h] and let z(t) be a solution to (4.1). By our assumptions, we know that z(t) is defined on a maximal right interval, say [−n, ω1 ). This solution satisfies the integral equation Z t z(t) = Z (t)Z −1 (−n)z(−n) + Z (t)Z −1 (s)[B(s)z(s) + F(s, z(s))]ds, (4.3) −n

for all t ∈ [−n, ω1 ). In fact we can show that ω1 = −h if kz(−n)k is small enough. Suppose by contradiction that ω1 < −h.

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As in [7] we deduce from (3.1), (3.2), (4.2) and (4.3), Z t h p Rt Rt − −n f (s)ds e− s f (τ )dτ γ K / β0 kz(s)k kz(t)k ≤ e γ kz(−n)k + −n

i

+ M/( β0 )α kz(s)kα | f (s)|ds =: ρ(t), p

∀t ∈ [−n, ω1 ).

By (4.4) easy estimates lead us to ( p α p ρ 0 (t) ≤ −(γ K / β0 + 1) f (t)ρ(t) − γ M/ β0 f (t)ρ(t)α , ρ(−n) = γ kz(−n)k. In addition, consider the Cauchy problem (   p α p u¯ 0 = − γ K / β0 + 1 f (t)u¯ − γ M/ β0 f (t)u¯ α , u(−n) ¯ = u¯ 0 > 0,

∀t ∈ [−n, ω1 ),

∀t ∈ [−n, ω1 ),

(4.4)

(4.5)

(4.6)

the unique solution of which on [−n, ω1 ) is ( u(t) ¯ = e

Rt √ (α−1)(γ K / β0 +1) −n f (s)ds

u¯ 1−α 0

1 √
α ! √
α ) 1−α γ M/ β0 γ M/ β0 − + . √ √ γ K / β0 + 1 γ K / β0 + 1

We remark that if ( √
α γ M/ β0  (α−1)(γ K /√β0 +1) R −n −h e u¯ 0 < √ γ K / β0 + 1

f (τ )dτ



)

−1

1 1−α

=: δ1 ,

then u(t) ¯ is defined on [−n, −h]. By (4.5) and (4.6), and applying Lemma 2.1, we deduce that (kz(−n)k ≤ u¯ 0 /γ ) H⇒ (ρ(t) ≤ u(t), ¯ ∀t ∈ [−n, ω1 )) . Since ρ(t) is bounded on [−n, ω1 ), we get by (4.4) that z(t), z 0 (t) are both bounded on [−n, ω1 ), and therefore z(t) can be extended to the right of ω1 . This fact contradicts the maximality of ω1 . In conclusion, z(t) does exist on [−n, −h]. Notice that the Cauchy problem (   p α p β0 f (t)u α , u 0 = − γ K / β0 + 1 f (t)u − γ M/ (4.7) u(−h) = u 0 > 0, has the unique solution ( u(t) = e

Rt √ (α−1)(γ K / β0 +1) −h f (s)ds

u 1−α 0

1 √
α ! √
α ) 1−α γ M/ β0 γ M/ β0 + − √ √ γ K / β0 + 1 γ K / β0 + 1

(4.8)

and u(t) does exist on (−∞, −h], ∀u 0 > 0, having u(−∞) = 0. We set u¯ 0 = u(−n). Then u¯ 0 < δ1 and, from the uniqueness of the solution to problem (4.6), we get u(t) ¯ = u(t), ∀t ∈ [−n, −h]. Hence, if kz(−n)k ≤ u(−n)/γ , then kz(t)k ≤ u(t), ∀t ∈ [−n, −h]. The conclusion is that there exists a function u : (−∞, −h] → (0, +∞) having u(−∞) = 0 such that for all n > h there exists z n : [−n, −h] → R2 , a solution to (4.1) with the property that if kz n (−n)k ≤ u(−n)/γ , then kz n (t)k ≤ u(t), ∀t ∈ [−n, −h]. Step 2. Consider now the interval [−h, h] and let z(t) be a solution to (4.1). By our assumptions, we know that z(t) is defined on a maximal right interval, say [−h, ω2 ). This solution satisfies the integral equation

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z(t) = Z (t)Z −1 (−h)z(−h) +

Z

t

Z (t)Z −1 (s)[B(s)z(s) + F(s, z(s))]ds,

−h

for all t ∈ [−h, ω2 ). Let us prove that ω2 = h for kz(−h)k small enough. Suppose by contradiction that ω2 < h. Set n o L := sup | f 0 (t) + f 2 (t)| t∈[−h,h]

and we obtain as in Step 1, −

kz(t)k ≤ e

Rt −h

f (s)ds

γ kz(−h)k +

Z

t

e−

Rt s

f (τ )dτ

h γ Lkz(s)k

−h

i p α β0 kz(s)kα | f (s)|ds =: µ(t), + M/

∀t ∈ [−h, ω2 ).

Straightforward computations lead us to ( p α µ0 (t) ≤ (γ L − f (t))µ(t) + γ M/ β0 | f (t)|µ(t)α , µ(−h) = γ kz(−h)k. But the Cauchy problem ( p α v 0 = (γ L − f (t))v + γ M/ β0 | f (t)|v α , v(−h) = v0 > 0,

(4.9)

∀t ∈ [−h, ω2 ),

(4.10)

∀t ∈ [−h, ω2 ),

(4.11)

admits the unique solution v(t) = e

Rt

−h [γ L− f (s)]ds



v01−α

Z p α + γ M/ β0 (1 − α)

t

| f (s)|e

(α−1)

Rs

−h [γ L− f (τ )]dτ

1  1−α

ds

,

(4.12)

−h

for all t ∈ [−h, ω2 ). Let  Z p α δ2 := γ M/ β0 (α − 1)

h

| f (s)|e

(α−1)

Rs

−h [γ L− f (τ )]dτ

1  1−α

ds

.

−h

If v0 < δ2 , then v(t) is defined on [−h, −h]. By (4.10) and (4.11), and applying Lemma 2.1, we deduce that (kz(−h)k ≤ v0 /γ ) H⇒ (µ(t) ≤ v(t), ∀t ∈ [−h, ω2 )) . Since µ(t) is bounded on [−h, ω2 ), we get by (4.9) that z(t), z 0 (t) are both bounded on [−h, ω2 ), and therefore z(t) can be extended to the right of ω2 . This fact contradicts the maximality of ω2 . Then it follows that z(t) does exist on [−h, h]. The conclusion is that if v0 < δ2 and kz(−h)k ≤ v0 /γ , then kz(t)k ≤ v(t), t ∈ [−h, h]. Step 3. Consider now the interval [h, n] and let z(t) be a solution to (4.1). Let [h, ω3 ) be the maximal right interval of z(t). Then Z t z(t) = Z (t)Z −1 (h)z(h) + Z (t)Z −1 (s)[B(s)z(s) + F(s, z(s))]ds, h

for all t ∈ [h, ω3 ). If kz(h)k is small enough, we show that ω3 = n. Suppose by contradiction that ω3 < n. Then as in Step 1, Z t R h p p α i Rt t − h f (s)ds kz(t)k ≤ e γ kz(h)k + e− s f (τ )dτ γ K / β0 kz(s)k + M/ β0 kz(s)kα h

× f (s)ds =: ν(t),

∀t ∈ [h, ω3 ).

(4.13)

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But, (

 p α  p β0 f (t)ν(t)α , ν 0 (t) ≤ γ K / β0 − 1 f (t)ν(t) + γ M/ ν(h) = γ kz(h)k.

Let us consider the Cauchy problem (  p α  p w 0 = γ K / β0 − 1 f (t)w + γ M/ β0 f (t)w α , w(h) = w0 > 0,

∀t ∈ [h, ω3 ),

∀t ∈ [h, ω3 ),

(4.14)

(4.15)

the unique solution of which on [h, ω3 ) is (

Rt √ (α−1)(1−γ K / β0 ) h f (s)ds

w(t) = e

w01−α

1 √
α ! √
α ) 1−α β0 γ M/ β0 − + , √ √ 1 − γ K / β0 1 − γ K / β0

γ M/

(4.16)

for all t ∈ [h, ω3 ). Let δ3 :=

1 √
α ! 1−α β0 . √ 1 − γ K / β0

γ M/

If w0 < δ3 , then w(t) is defined on [h, n]. By (4.14) and (4.15), and applying Lemma 2.1, we deduce that (kz(h)k ≤ w0 /γ ) H⇒ (ν(t) ≤ w(t), ∀t ∈ [h, ω3 )). Since ν(t) is bounded on [h, ω3 ), we get by (4.13) that z(t), z 0 (t) are both bounded on [h, ω3 ), and therefore z(t) can be extended to the right of ω3 . This fact contradicts the maximality of ω3 . Then it follows that z(t) does exist on [h, n]. The conclusion is that if w0 < δ3 and kz(h)k ≤ w0 /γ , then kz(t)k ≤ w(t), ∀t ∈ [h, n]. Remark 4.1. Notice that in fact, if w0 < δ3 , then w(t) given by (4.16) does exist on the whole semi-line [h, +∞) and w(+∞) = 0. Step 4. Let us synthesize our conclusions from Steps 1–3. We have determined a function u : (−∞, −h] → (0, +∞) (given by (4.8)), a solution to problem (4.7) that we denote by u(t; u 0 ), defined for all u 0 > 0; we have determined a function w : [h, +∞) → (0, +∞) (given by (4.16)), a solution to problem (4.15) that we denote by w (t; w0 ), defined for w0 ∈ (0, δ3 ). In addition, u (−∞; u 0 ) = w (+∞; w0 ) = 0, for all u 0 > 0 and w0 ∈ (0, δ3 ). Furthermore, we have built a function v : [−h, h] → (0, +∞) (given by (4.12)), a solution to problem (4.11) that we denote by v(t; v0 ), defined for all v0 ∈ (0, δ2 ) =: I0 . Notice that for all t ∈ [−h, h], the mapping v0 7−→ v(t; v0 ) is continuous from I0 to Cc ([−h, h], R) (see, e.g., [8]). Therefore, the range of the interval I0 through the mapping v0 7−→ v(h; v0 ) is an interval, say (0, η) , η > 0. In addition, by (4.12), we easily deduce that limv0 →0 v(t; v0 ) = 0, uniformly for t ∈ [−h, h]. In particular, lim v (h; v0 ) = 0.

v0 →0

(4.17)

Let us consider ε := min {η, δ3 }. From (4.17) there exists δ > 0 such that for every v0 ∈ (0, δ), one has v(h; v0 ) < ε. We set v¯0 ∈ (0, min {δ, δ2 }), w¯ 0 := v (h; v¯0 ), and u¯ 0 := v (−h; v¯0 ). We define the function s : R → R through  u(t; u¯ 0 ), if t ≤ −h, s(t) := v(t; v¯0 ), if t ∈ [−h, h],  w(t; w¯ 0 ), if t ≥ h.

C. Vladimirescu / Nonlinear Analysis 68 (2008) 3217–3223

3223

Then, s(t) > 0,

∀t ∈ R and s(±∞) = 0.

(4.18)

Consider n ∈ N, n > h arbitrarily fixed. Let z 1n be the solution to (4.1) on [−n, −h] with the initial condition kz 1n (−n)k = s (−n) /γ ; z 2n the solution to (4.1) on [−h, h] with the initial condition z 2n (−h) = z 1n (−h); z 3n the solution to (4.1) on [h, n] with the initial condition z 3n (h) = z 2n (h). Then the function z n : [−n, n] → R defined by  z 1n (t), if t ∈ [−n, −h], z n (t) := z 2n (t), if t ∈ [−h, h],  z 3n (t), if t ∈ [h, n], fulfills (4.1) on [−n, n] and kz n (t)k ≤ s(t),

∀t ∈ [−n, n] .

Then, by applying Lemma 2.2 with d := 2, xn := z n , G (t, z) := A (t) z + B(t)z + F (t, z) , tn := n, we deduce that there exists z : R → R2 solution to (4.1) fulfilling the inequality kz(t)k ≤ s(t),

∀t ∈ R.

(4.19)

So, from relations (4.18) and (4.19) we obtain that z is a homoclinic solution to (4.1). The proof of Theorem 3.1 ends.  References [1] Z. Artstein, E.F. Infante, On the asymptotic stability of oscillators with unbounded damping, Quart. Appl. Mech. 34 (1976) 195–199. [2] C. Avramescu, Limits of solutions of a nonlinear differential equation, Nonlinear Anal. Forum 7 (2002) 209–215. [3] C. Avramescu, C. Vladimirescu, Limits of solutions of a perturbed linear differential equation, Electron. J. Qual. Theory Differ. Equ. 3 (2003) 1–11. [4] C. Avramescu, C. Vladimirescu, Homoclinic solutions for linear and linearizable ordinary differential equations, Abstr. Appl. Anal. 5 (2) (2000) 65–85. [5] C. Avramescu, C. Vladimirescu, An existence result for homoclinic solutions for a linear ordinary differential equation of second order, Ann. Univ. Craiova Math. Comput. Sci. Ser. 30 (2) (2003) 14–19. [6] T.A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications, Inc., Mineola, New York, 2006. [7] T.A. Burton, T. Furumochi, A note on stability by Schauder’s theorem, Funkcial. Ekvac. 44 (2001) 73–82. [8] C. Corduneanu, Principles of Differential and Integral Equations, Allyn and Bacon, Boston, 1971. [9] V. Coti Zelati, P.H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc. 4 (1991) 693–727. [10] V. Coti Zelati, M. Macr`ı, Existence of homoclinic solutions to periodic orbits in a center manifold, J. Differential Equations 202 (1) (2004) 158–182. [11] J. Gruendler, Homoclinic solutions and chaos in ordinary differential equations with singular perturbations, Trans. Amer. Math. Soc. 350 (9) (1998) 3797–3814. [12] L. Hatvani, Integral conditions on the asymptotic stability for the damped linear oscillator with small damping, Proc. Amer. Math. Soc. 124 (1996) 415–422. [13] M.A. Krasnoselskii, The Operator of Translation Along the Trajectories of Differential Equations (Nauka, Moscow, 1966), in: Translations of Mathematical Monographs, vol. 19, Amer. Math. Soc., Providence, R.I., 1968 (in Russian). English translation. [14] Gh. Moros¸anu, C. Vladimirescu, Stability for a nonlinear second order ODE, Funkcial. Ekvac. 48 (1) (2005) 49–56. [15] Gh. Moros¸anu, C. Vladimirescu, Stability for a damped nonlinear oscillator, Nonlinear Anal. TMA 60 (2) (2005) 303–310. [16] P. Pucci, J. Serrin, Precise damping conditions for global asymptotic stability for non-linear second order systems, Acta Math. 170 (1993) 275–307. [17] P. Pucci, J. Serrin, Precise damping conditions for global asymptotic stability for non-linear second order systems, II, J. Differential Equations 113 (1994) 505–534. [18] P. Pucci, J. Serrin, Asymptotic stability for intermittently controlled nonlinear oscillators, SIAM J. Math. Anal. 25 (1994) 815–835.