Existence of periodic solutions for fully nonlinear first-order differential equations

Nonlinear Analysis 52 (2003) 1095 – 1109

www.elsevier.com/locate/na

Existence of periodic solutions for fully nonlinear "rst-order di$erential equations
Desheng Lia;∗ , Yantang Liangb a Department

of Mathematics and Information Science, Yantai University, Yantai 264005, Shandong, People’s Republic of China b Department of Mathematics, Lanzhou Teachers College, Lanzhou 730070, Gansu, People’s Republic of China Received 22 April 2000; accepted 22 February 2002

Abstract In this paper, we establish some existence results for periodic solutions of fully nonlinear "rst-order scalar di$erential equation: F(t; x; x
)=0. The approach here uses the viscosity solutions method developed in recent decades and the classical upper–lower solutions method. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Existence; Periodic solution; Fully nonlinear; Di$erential equation; Viscosity solutions method

1. Introduction In recent years, the existence of periodic solutions (closed solutions) for semilinear "rst-order di$erential equation x
= f(t; x) were extensively studied in the literature including [1,12–15,17–20,27–29], etc., in which, the authors established quite general and beautiful results. Many new methods, such as the upper–lower solutions method, monotone technics and stability method were developed. Unfortunately, we "nd that these fruitful methods can be hardly extended and applied directly to those equations, which, may not be solvable in the derivative “x
”. In this paper we consider the existence of periodic solutions for the following fully nonlinear "rst-order di$erential equation: F(t; x; x
) = 0


Supported by the National Natural Science Foundation of China (10071066).

∗

Corresponding author. E-mail address: [email protected] (D. Li).

0362-546X/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 0 2 ) 0 0 1 5 3 - 0

(1.1)

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D. Li, Y. Liang / Nonlinear Analysis 52 (2003) 1095 – 1109

in case F is continuous and is periodic in t. This consideration is also motivated by an increasing interest in such types of equations in recent years; see, for instance, [2,3,5 –8,10,11,16,21–27,30 –34], etc. We will establish some existence results for periodic solutions of Eq. (1.1) under appropriate conditions, making use of the theory of viscosity solutions (cf. [4,9]) combining with the lower and upper solutions method (cf. [13,19,20], etc.). We recall that a function x is said to be a solution (lower solution, upper solution) of (1.1) (in CarathGeodory sense) on an open interval J , if it is locally absolutely continuous on J and F(t; x; x
) = 0

(resp: 6 0; ¿ 0)

(1.2)

1

for a.e. t ∈ J . If x ∈ C (J ) and satis"es (1.2) for any t ∈ J , then we say that x is a classical solution (resp. lower solution, upper solution) of (1.1) on J . We will always assume that F ∈ C(R3 ) and is T -periodic in t (T ¿ 0). The main results are the following theorems. Theorem 1.1. Assume that for any R ¿ 0; lim inf F(t; x; p) ¿ 0 and lim sup F(t; x; p) ¡ 0 p→∞

p→−∞

(1.3)

uniformly with respect to (t; x) ∈ R1 × [ − R; R]; moreover; Eq. (1.1) possesses a T -periodic classical lower solution a and upper solution b on R1 with a 6 b. Then Eq. (1.1) has at least one T -periodic solution x with a 6 x 6 b. Theorem 1.2. Assume that for any R ¿ 0; lim sup F(t; x; p) ¡ 0 and lim inf F(t; x; p) ¿ 0 p→−∞

p→∞

(1.4)

uniformly with respect to (t; x) ∈ R1 × [ − R; R]; moreover; Eq. (1.1) possesses a T -periodic classical lower solution a and upper solution b on R1 with a 6 b. Then Eq. (1.1) has at least one T -periodic solution x with a 6 x 6 b. Since the equation under consideration is continuous (i.e., the function F is continuous), moreover, we assumed the existence of classical lower and upper solutions in the theorems, one may expect that under the hypothesis of Theorems 1.1 and 1.2, Eq. (1.1) has a classical T -periodic solution. Unfortunately, this is impossible in general, even if we further assume that F is C 1 and is monotone in p (not strictly). In Section 5, we give an easy example in which F ∈ C 1 (R3 ) and is monotone in p, moreover all the conditions needed in Theorem 1.1 are satis"ed; however, the equation has no classical T -periodic solutions. It is worthwhile to note that if lim inf F(t; x; 0) ¿ 0 x→∞

and

lim sup F(t; x; 0) ¡ 0 x→−∞

uniformly with respect to t ∈ [0; T ], then the constant functions a(t) ≡ −C and b(t)≡C on R1 with C ¿ 0 suHciently large are a classical T -periodic lower solution and upper solution of (1.1), respectively.

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This work is organized as follows. In Section 2 we recall some basic knowledge in the theory of viscosity solutions. In Section 3 we give an existence and uniqueness result for periodic viscosity solutions of (1.1) in case F(t; x; p) is strictly increasing in x. The proofs of Theorems 1.1 and 1.2 will be given in Section 4. Section 5 consists of an example mentioned above and some supplementary results. A multiplicity result on equation with parameter : F(t; x; x
) = h(; t) will also be given in this section. 2. Basic notions and lemmas in the theory of viscosity solutions We recall some basic notions and lemmas in the theory of viscosity solutions. Let J be a subset of R1 , x : J → R1 be given. x is said to be lower (upper) semicontinuous at t ∈ J , if       lim inf x(s) ¿ x(t) lim sup x(s) 6 x(t) :   s∈J; s=t s∈J; s=t s→t

s→t

We denote by LSC(J ) (USC(J )) the set of functions that are lower (upper) semicontinuous at any point on J . We will denote by x∗ , x∗ the upper and lower semicontinuous envelopes of x, respectively, x∗ (t) = lim sup{x(s): |s − t| 6 r}; r→0

x∗ (t) = lim inf {x(s): |s − t| 6 r}: r→0

Clearly x∗ 6 x 6 x∗ on J . Denition 2.1. Let J be an open interval of R1 . For t ∈ J ; the superjet @+ x(t) and subjet @− x(t) of x(t) at t are de"ned to be the subsets of R1 : @+ x(t) = {
(t):  ∈ C 1 (R1 ) such that t is a local maximum of x − }; @− x(t) = {
(t):  ∈ C 1 (R1 ) such that t is a local minimum of x − }: A function x ∈ USC(J ) is said to be a viscosity lower solution of Eq. (1.1) on J if F(t; x(t); p) 6 0;

∀t ∈ J; p ∈ @+ x(t):

A function x ∈ LSC(J ) is said to be a viscosity upper solution of Eq. (1.1) on J if F(t; x(t); p) ¿ 0;

∀t ∈ J; p ∈ @− x(t):

If x ∈ C(J ) is both a viscosity lower solution and upper solution of Eq. (1.1) on J , then we say that x is a viscosity solution of Eq. (1.1) on J .

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D. Li, Y. Liang / Nonlinear Analysis 52 (2003) 1095 – 1109

Note that if x is di$erentiable at t, then @+ x(t) = @− x(t) = {x
(t)}. Due to this basic fact, if a viscosity solution x is di$erentiable at t, then F(t; x; x
) = 0. One also easily sees that any classical lower, upper solutions are necessarily viscosity lower, upper solutions, respectively. This statement is in general not true for lower, upper solutions in CarathGeodory sense. We also observe that @± x(t) = −@∓ (−x)(t). Lemma 2.2. Let J be an open interval; x ∈ C(J ). Then (1) if p 6 0;

for ∀p ∈ @+ x(t); t ∈ J

(2.1)

p 6 0;

for ∀p ∈ @− x(t); t ∈ J;

(2.2)

or then x is nonincreasing on J ; (2) if p ¿ 0;

for ∀p ∈ @+ x(t); t ∈ J

(2.3)

p ¿ 0;

for ∀p ∈ @− x(t); t ∈ J;

(2.4)

or then x is nondecreasing on J . Proof. This is a basic knowledge in the theory of viscosity solutions. Here; we give a proof for the reader’s convenience. Recalling that @± x(t) = −@∓ (−x)(t); we see that the second conclusion (2) can be obtained by applying (1) directly to −x. Therefore; we only need to give a proof for (1). First, assume that (2.1) holds. If the conclusion fails to be true, then there exist t0 ; t2 ∈ J , t0 ¡ t2 such that x(t0 ) ¡ x(t2 ). Let t1 = sup{t ∈ [t0 ; t2 ): x(t) 6 x(t0 )}. Then by continuity, we have t1 ¡ t2 and x(t1 ) = x(t0 ). By the de"nition of t1 , x(t1 ) ¡ x(t);

∀t ∈ (t1 ; t2 ):

(2.5)

Now two cases may occur. Case 1. There exist s1 ; s2 ∈ (t1 ; t2 ), s1 ¡ s2 such that x(s1 ) ¿ x(s2 ). In this case we de"ne a function y as y(t) = x(t) − k(t − s2 ) − x(s2 );

k = (x(s2 ) − x(t1 ))=(s2 − t1 ):

(2.6)

By (2.5), k ¿ 0. Clearly y(t1 ) = y(s2 ) = 0. Since x(s1 ) ¿ x(s2 ), we see that y(s1 ) ¿ 0 and hence y attains its maximum on [t1 ; s2 ] at some point z ∈ (t1 ; s2 ). By the de"nition of superjet, k ∈ @+ x(z). This contradicts the assumption. Case 2. x is nondecreasing on (t1 ; t2 ). In this case x is a.e. di$erentiable. By continuity, we can take an s2 ∈ (t1 ; t2 ] such that x(t1 ) ¡ x(s2 ) and x is di$erentiable at s2 (hence @+ x(s2 ) = {x
(s2 )}). De"ne y and k as in (2.6). Then k ¿ 0. Note that y(t1 ) = y(s2 ) = 0. Now, if there exists s1 ∈ (t1 ; s2 ) such that y(s1 ) ¿ 0, then y attains its maximum on [t1 ; s2 ] at some point z ∈ (t1 ; s2 )

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and hence k ∈ @+ x(z). This contradicts the assumption. Thus we assume that y(t) 6 0 on (t1 ; s2 ). This is equivalent to saying that k 6 (x(t) − x(s2 ))=(t − s2 );

∀t ∈ (t1 ; s2 );

which implies that x
(s2 ) ¿ k ¿ 0. Again this contradicts the assumption. Now we assume (2.2) instead of (2.1). In this case the proof can be given in a quite similar manner as above. If the conclusion fails to be true, then there exist t0 ; t2 ∈ J , t0 ¡ t2 such that x(t0 ) ¡ x(t2 ). Let t1 = inf {t ∈ (t0 ; t2 ]: x(t) ¿ x(t2 )}. Then by de"nition of t1 and the continuity of x, we must have t0 ¡ t1 and x(t1 ) = x(t2 ). By the de"nition of t1 , we "nd x(t) ¡ x(t1 );

∀t ∈ (t0 ; t1 ):

(2.7)

Two cases may also occur. Case 1. There exist s1 ; s2 ∈ (t0 ; t1 ), s1 ¡ s2 such that x(s1 ) ¿ x(s2 ). Let k = (x(t1 ) − x(s1 ))=(t1 − s1 ). Then by (2.7), k ¿ 0. Consider the function y(t) = x(t) − k(t − s1 ) − x(s1 );

t ∈ [s1 ; t1 ]:

(2.8)

Note that y(s1 )=y(t1 )=0 and y(s2 ) ¡ 0. Let z ∈ (s1 ; t1 ) be a minimum of y on [s1 ; t1 ]. Then k ∈ @− x(z). This contradicts (2.2). Case 2. x is nondecreasing on (t0 ; t1 ). In this case we can take s1 ∈ [t0 ; t1 ) such that x(s1 ) ¡ x(t1 ) and x is di$erentiable at s1 (hence @− x(s1 ) = {x
(s1 )}). Let k = (x(t1 ) − x(s1 ))=(t1 − s1 ). Then k ¿ 0. Consider on [s1 ; t1 ] the function y de"ned by (2.8). By similar argument as in Case 2, one can also easily get a contradiction. As a consequence of Lemma 2.2, we have Lemma 2.3. Let J be an open interval of R1 ; x ∈ C(J ). If for some L ¿ 0; we have either (1) p 6 L for ∀p ∈ @+ x(t) and t ∈ J ; (2) p ¿ − L for ∀p ∈ @− x(t) and t ∈ J ; or (3) p 6 L for ∀p ∈ @− x(t) and t ∈ J ; (4) p ¿ − L for ∀p ∈ @+ x(t) and t ∈ J ; then x is Lipschits on J with |x|Lip 6 L; where |x|Lip is the Lipschits constant of x on J . Proof. Let s ∈ J . Consider the function y(t)=x(t)−x(s)−L(t −s). Clearly y(s)=0. We observe that @± y(t) = {p − L: p ∈ @± x(t)}. By Lemma 2.2; either (1) or (3) implies that y(t) is nonincreasing on J and hence y(t) 6 0 for t ∈ Js = J ∩ [s; ∞); which yields x(t) − x(s) 6 L(t − s);

t ∈ Js :

Similarly by considering the function y(t) = x(t) − x(s) + L(t − s); we can show that if (2) or (4) holds; then x(t) − x(s) ¿ − L(t − s);

t ∈ Js :

Combining the above conclusions together; one obtains the desired results immediately.

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Lemma 2.4 (Ishii [9], Proposition 2.4). Let J be an open interval of R1 . Let W be a family of viscosity lower solutions of Eq. (1.1) on J . De:ne a function x on J as: x(t) = sup{w(t): w ∈ W }. Assume that |x∗ (t)| ¡ ∞ for t ∈ J . Then x∗ is a viscosity lower solution of Eq. (1.1) on J . Lemma 2.5. Let J be an open interval of R1 ; x ∈ C(J ); t ∈ J and p ∈ @± x(t). Suppose that {x n } ⊂ C(J ) (n ∈ N ) converges to x on a neighborhood of t uniformly. Then there exists tn ∈ J ; pn ∈ @± x n (tn ) such that (tn ; x n (tn ); pn ) → (t; x(t); p). The proof of Lemma 2.5 is actually the same as that of [4, Proposition 4.3]. Also it is implied in the proof of [9, Proposition 2.4]. 3. Existence and uniqueness of periodic viscosity solutions In this section we establish a result on existence and uniqueness of periodic viscosity solutions for Eq. (1.1) in case F(t; x; p) is monotone in x. The main result is Theorem 3.1. Assume (1.3) or (1.4) holds; moreover; F(t; x; p) is strictly increasing in x. Suppose that Eq. (1.1) possesses a bounded T -periodic viscosity lower solution u and upper solution v. Then Eq. (1.1) possesses a unique T -periodic viscosity solution x with u 6 x 6 v. In order to prove Theorem 3.1, we "rst establish the following comparison result. Theorem 3.2 (Comparison Principle). Assume (1.3) or (1.4) holds; moreover; F(t; x; p) is strictly increasing in x. Let u; v be a T -periodic viscosity lower and upper solution of Eq. (1.1) respectively; moreover; u and v are bounded. Then u 6 v on R1 . Proof. We argue by contradiction and assume that u(t) ¿ v(t) for some t ∈ R1 . Then by the upper semicontinuity and the periodicity of u − v; there exists z ∈ [0; T ] such that u(z) − v(z) = maxR1 (u − v) = $ ¿ 0. Let  ¿ 0. De"ne w : R2 → R1 as follows:  w(s; t) = u(t) − v(s) − |t − s|2 2

(s; t) ∈ R2 :

Set M = supR2 w. Then M ¿ u(z) − v(z) = $:

(3.1)

Since u; v are bounded; we deduce that there exists a constant a ¿ 0 (a may depend on ) such that M = sup{w(s; t): (s; t) ∈ R2 ; |s − t| 6 a}: Therefore we can take sequences sn ; tn ∈ R1 with |sn − tn | 6 a such that M = limn→∞ w(sn ; tn ). Let kn ∈ Z be such that tn = kn T + tn
with |tn
| ¡ T . We also write sn as sn = kn T + sn
. Then |sn
− tn
| = |sn − tn | 6 a and hence |sn
| 6 a + |tn
| ¡ a + T .

D. Li, Y. Liang / Nonlinear Analysis 52 (2003) 1095 – 1109

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By the periodicity; we have w(sn ; tn ) = w(sn
; tn
); thus M = lim w(sn
; tn
): n→∞

{sn
}; {tn
}

Since are bounded; we may assume that (sn
; tn
) → (s ; t ). By the upper semicontinuity of w; M = w(s ; t ). Note that t is a maximum of u(t) − v(s ) − (=2)|t − s |2 and s is a minimum of v(s) + (=2)|t − s|2 ; we see that p ∈ @+ u(t ) ∩ @− v(s ); where p = (t − s ). We claim that {p } ( ¿ 0) is bounded. Otherwise; there exists a sequence {n } such that pn → ∞ (− ∞) as n → ∞. Assume that F satis"es (1.3). Since u; v are bounded; we will have F(tn ; u(tn ); pn ) ¿ 0 (in case pn → ∞) or F(sn ; v(sn ); pn ) ¡ 0 (in case pn → − ∞) for n suHciently large. This is a contradiction; as u; v are a viscosity lower solution and upper solution of Eq. (1.1); respectively. In case F satis"es (1.4); we can get a contradiction by similar argument. We observe that  0 ¡ $ 6 M = u(t ) − v(s ) − |t − s |2 6 u(t ) − v(s ): 2 Now, assume that F(t; x; p) is strictly increasing in x. By periodicity of F(t; x; p) in t and the continuity, we deduce that for any R ¿ 0, there exists ' ¿ 0 such that F(t; x2 ; p) − F(t; x1 ; p) ¿ ' for ∀t ∈ R1 and ∀x1 ; x2 ; p ∈ [ − R; R] with x2 − x1 ¿ $. In particular, there exists '0 ¿ 0 such that for any  ¿ 0, '0 6 F(t ; u(t ); p ) − F(t ; v(s ); p ):

(3.2)

By de"nition of viscosity lower and upper solutions, one sees that F(t ; u(t ); p ) 6 0 6 F(s ; v(s ); p ): It follows by (3.2) that '0 6 F(s ; v(s ); p ) − F(t ; v(s ); p ):

(3.3)

Since F is continuous, for any R ¿ 0, F is uniformly continuous on R1 × [ − R; R] × [ − R; R]. Note that the boundedness of {p } implies |t − s | → 0 as  → ∞. We set  → ∞ in (3.3) and obtain directly that '0 6 0. A contradiction! Proof of Theorem 3.1 (Perron’s Method): The scheme of argument here is standard in the theory of viscosity solutions, see [4], Section 4. Let u; v be the T -periodic viscosity lower, upper solution of Eq. (1.1) given in the theorem, respectively. By Theorem 3.2, we have u 6 v. Set W = {w: w is a T -periodic viscosity lower solution of Eq: (1:1) with u6w6v}; then W = ∅, as u ∈ W . Let x(t) = sup w(t); w∈W

∀t ∈ R1 :

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Since u; v are bounded, one sees that −M 6 x∗ 6 x 6 x∗ 6 M for some constant M ¿ 0. Thanks to Lemma 2.4, x∗ is a T -periodic viscosity lower solution of Eq. (1.1). By Theorem 3.2, x∗ 6 v and hence x∗ ∈ W . It then follows from the definition of x that x = x∗ , i.e., x is the maximal T -periodic viscosity lower solution of Eq. (1.1) in W . We claim that x∗ is a T -periodic viscosity upper solution of Eq. (1.1). To see this we argue by contradiction and assume that x∗ fails to be a viscosity upper solution at some point z, i.e., F(z; x∗ (z); p) ¡ 0

for some p ∈ @− x∗ (z):

(3.4)

For the sake of simplicity we put z = 0 in the following argument. We "rst observe that at the point 0, x∗ (0) ¡ v(0):

(3.5) −

Indeed, since x∗ 6 v, if x∗ (0)=v(0), then by the de"nition of subjet, we have @ x∗ (0) ⊂ @− v(0); further, recalling that v is a viscosity upper solution of Eq. (1.1), we deduce that F(0; x∗ (0); q) = F(0; v(0); q) ¿ 0;

∀q ∈ @− x∗ (0);

which contradicts to (3.4). Let  ∈ C 1 (R1 ) be a function such that 0 is a local minimum of x∗ (t) − (t) with
 (0) = p. We may assume that for some '0 ¿ 0, x∗ (t) − (t) ¿ t 2 ; t ∈ (−'0 ; '0 ); (3.6) 2 ˜ otherwise we can replace (t) by (t)=x∗ (0)+(t)−(0)−t . Assume that '0 ¡ T=2. In view of (3.4) and(3.5), by continuity, there exists '1 ∈ (0; '0 =2) such that x∗ (0) = (0);

F(t; (t) + '12 ; 
(t)) 6 0;

t ∈ (−2'1 ; 2'1 )

(3.7)

and (also by lower semicontinuity of v) (t) + '12 6 v;

t ∈ (−2'1 ; 2'1 ):

(3.8)

'12

(3.7) shows that (t) + is a classical lower solution of Eq. (1.1) on (−2'1 ; 2'1 ). Set  max{(t − kT ) + '12 ; x(t − kT )} when t ∈ (kT − 2'1 ; kT + 2'1 ); y(t) = x(t) otherwise; where k ∈ Z. Clearly y is T -periodic. We show that y is a viscosity lower solution of Eq. (1.1). First, for each k ∈ Z, it is easy to check that y is upper semicontinuous on the open interval Ik = (kT − 2'1 ; kT + 2'1 ) (recall that x ∈ USC(R1 )). By Lemma 2.4, y (=y∗ on Ik ) is a viscosity lower solution of (1.1) on Ik . Now, by (3.6) (note that 2'1 ¡ '0 ¡ T=2), we have x(s) ¿ x∗ (s) ¿ (s) + '12 ;

for ∀s ∈ R1 with '1 ¡ |s| ¡ 2'1 :

By (3.9) and the de"nition of y, one easily sees that y = x on  [kT − '1 ; kT + '1 ]: * := R1 { k∈Z

(3.9)

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Since x is a viscosity lower solution of Eq. (1.1) on R1 and * is open, we conclude immediately that y is a viscosity lower solution of (1.1) on

  Ik = R 1 : * k∈Z

Hence the result. Thanks to (3.8), we "nd that y ∈ W . On the other hand, by the de"nition of semicontinuous envelopes, there exists tn → 0 (as n → ∞) such that x(tn ) → x∗ (0). Since x∗ (0) = (0) (see (3.6)), we "nd that (tn ) + '12 ¿ x(tn ) for n suHciently large, hence supR1 (y − x) ¿ 0. This contradicts to the de"nition of x and thus proves our claim. Now, by Theorem 3.2, we have x∗ 6 x∗ , thus x = x∗ = x∗ ∈ C(R1 ) and is a T -periodic viscosity solution of Eq. (1.1). The uniqueness is a direct consequence of Theorem 3.2.

4. Proofs of Theorems 1.1 and 1.2 Proof of Theorem 1.1. We "rst prove the existence of periodic solutions in case (1.3) is replaced by a slightly stronger assumption: (1:3
) for any R ¿ 0, limp→± ∞ F(t; x; p) = ± ∞ uniformly with respect to (t; x) ∈ R1 × [ − R; R]; Let Q = {(t; x): (t; x) ∈ R2 ; a(t) 6 x 6 b(t)}. De"ne c ∈ C(R2 ) as    b(t); x ¿ b(t); a(t) 6 x 6 b(t); c(t; x) = x;   a(t); x ¡ a(t): We consider the modi"ed equation: F(t; c(t; x); x
) + x − c(t; x) = 0

(4.1)

that obviously reduces to Eq. (1.1) on Q. We claim that any T -periodic viscosity solution x of (4.1) satis"es: a 6 x 6 b on R1 . Thus if a T -periodic viscosity solution x of (4.1) is locally absolutely continuous, then it is a periodic solution of Eq. (1.1). Indeed, if a periodic viscosity solution x does not satisfy a 6 x 6 b, say for instance, there exists t ∈ R1 such that x(t) ¿ b(t), then x − b has a maximum s, at which x(s) ¿ b(s). By the de"nition of superjet, we have b
(s) ∈ @+ x(s). On the other hand, we have F(s; c(s; x(s)); b
(s)) + x(s) − c(s; x(s)) = F(s; b(s); b
(s)) + x(s) − b(s) ¿ 0; which contradicts the fact that x is a viscosity lower solution of (4.1). In the following we show that Eq. (4.1) has at least one T -periodic viscosity solution x. We also show that x is Lipschits and hence it is a periodic solution of (1.1) in CarathGeodory sense.

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D. Li, Y. Liang / Nonlinear Analysis 52 (2003) 1095 – 1109

Consider the approximate equation F(t; c(t; x); x
) + (1 + ')x − c(t; x) = 0;

(4.2)

where ' ∈ (0; 1). We "rst prove that (4.2) has a T -periodic viscosity solution x' . Since c(t; x) is bounded, we see that there exists M ¿ 0 independent of ' such that F(t; c(t; x); 0) + (1 + ')M − c(t; M ) ¿ 0; F(t; c(t; x); 0) − (1 + ')M − c(t; −M ) 6 0;

∀t; x ∈ R1 ; ∀t; x ∈ R1 :

(4.3) (4.4)

Let CT0 be the set of T -periodic functions that are continuous on R1 . Then CT0 is a Banach space with the norm  ·  de"ned by x = maxR1 |x(t)|. We de"ne K : CT0 → CT0 as ∀y ∈ CT0 , Ky is the T -periodic viscosity solution of equation F(t; c(t; y); x
) + (1 + ')x − c(t; x) = 0:

(4.5)

Note that (1 + ')x − c(t; x) is strictly increasing in x, by Theorem 3.1 and (4.3) and (4.4), Eq. (4.5) has a unique T -periodic viscosity solution x with x 6 M;

(4.6)

hence K is well de"ned. For any y ∈ CT0 , in view of (4.6) and the boundedness of the function c, by (1:3
) and the de"nition of viscosity (lower, upper) solutions, we deduce that there is an L ¿ 0 that is independent of y and ' such that p 6 L; p ¿ −L;

∀p ∈ @+ Ky(t); ∀t ∈ R1 ; ∀p ∈ @− Ky(t); ∀t ∈ R1 :

Thanks to Lemma 2.3, we have |Ky|Lip 6 L;

∀y ∈ CT0 ;

(4.7)

where |Ky|Lip is the Lipschitz constant of Ky. We now show that K is continuous and hence K is a completely continuous operator. For this purpose, it suHces to show that, if {yn } ⊂ CT0 , yn → y0 in CT0 , then {yn } has a subsequence {ynk } such that Kynk → Ky0 in CT0 . Indeed, since {Kyn } is precompact in CT0 , it possesses a subsequence {Kynk } that converges in CT0 to a function x0 . For simplicity we write Kynk as xk . For any t ∈ R1 and p ∈ @+ x0 (t), by Lemma 2.5, there exists tk ∈ R1 , pk ∈ @+ xk (tk ) such that (tk ; xk (tk ); pk ) → (t; x0 (t); p) as k → ∞. By de"nition of viscosity (lower) solution, we have F(tk ; c(tk ; ynk (tk )); pk ) + (1 + ')xk (tk ) − c(tk ; xk (tk )) 6 0: We pass to the limit in the above equation to obtain that F(t; c(t; y0 (t)); p) + (1 + ')x0 (t) − c(t; x0 (t)) 6 0;

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1105

which shows that x0 is a viscosity lower solution of (4.5) with y=y0 . Similar argument applies to prove that x0 is a viscosity upper solution of (4.5) with y = y0 . By de"nition of K, we have Ky0 = x0 . This completes the proof of the continuity of K. (4.6) and (4.7) show that KB(M ) is a precompact subset of B(M ), where B(M ) = {x ∈ CT0 : x 6 M }. Since K is continuous, by the well-known Schauder "xed-point theorem, K has at least one "xed point x' on B(M ). x' is a T -periodic viscosity solution of Eq. (4.2); moreover, by (4.6) and (4.7), we have x'  6 M;

|x' |Lip 6 L:

Now, since M and L are both '-independent, we can take a sequence {'
} ⊂ (0; 1) that converges to 0 such that x'
converges in CT0 to a function x ∈ CT0 . Necessarily x 6 M;

|x|Lip 6 L:

(4.8)

Using a similar argument as in showing the continuity of the operator K, we can prove that x is a viscosity solution of (4.1). We now assume (1.3) instead of (1:3
). Let Q = {(t; x): (t; x) ∈ R2 ; a(t) 6 x 6 b(t)}. Then there exists p0 ¿ 0 such that F(t; x; p) ¿ 0;

∀(t; x) ∈ Q; p ¿ p0

F(t; x; p) ¡ 0;

∀(t; x) ∈ Q; p ¡ − p0 :

(4.9)

and


∗

(4.10)


∗

1

Take p ¿ p0 suHciently large so that |a (t)|; |b (t)| 6 p for any t ∈ R . De"ne  ∗ x ¿ p∗ ;  p ; −p∗ 6 p 6 p∗ ; d(p) = p;   −p∗ ; x ¡ − p∗ : Consider the modi"ed equation: x
− d(x
) + F(t; x; x
) = 0:

(4.11)

Clearly a; b are classical lower and upper solutions of (4.11), respectively. By the result we have just obtained above, Eq. (4.11) has at least one T -periodic solution x with a 6 x 6 b. By the de"nition of the function d and (4.9) and (4.10), we "nd that |x
(t)| 6 p0 ;

∀t ∈ D;

1

where D = {t ∈ R : x is di$erentiable at t}. Thus x is a solution of Eq. (1.1). The proof of the theorem is complete. Proof of Theorem 1.2. The proof of Theorem 1.2 is an analog of that of Theorem 1.1; we omit it. 5. Some remarks and a multiplicity result Remark 5.1. To derive a periodic solution x of Eq. (1.1); we assume in Theorems 1.1 and 1.2 that the equation possesses a T -periodic classical lower solution a and upper

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D. Li, Y. Liang / Nonlinear Analysis 52 (2003) 1095 – 1109

solution b with a 6 b. We remark that if F(t; x; p) is monotone in p; then the word “classical” in the above assumption can be removed. Thus we have Theorem 5.2. Assume that F(t; x; p) is nondecreasing in p and satis:es (1.3). Then if Eq. (1.1) possesses a T -periodic lower solution a and upper solution b on R1 with a 6 b; Eq. (1.1) has at least one T -periodic solution x with a 6 x 6 b. Theorem 5.3. Assume that F(t; x; p) is nonincreasing in p and satis:es (1.4). Then if Eq. (1.1) possesses a T -periodic lower solution a and upper solution b on R1 with a 6 b; Eq. (1.1) has at least one T -periodic solution x with a 6 x 6 b. The proof of Theorem 5.2 is just the same as that of Theorem 1.1 except that, instead of showing T -periodic viscosity solutions of the modi"ed equation (4.1) are those of Eq. (1.1), we show that any T -periodic solution of (4.1) is one of Eq. (1.1) in the proof of Theorem 5.2 (This can be easily done by making use of the monotonicity assumption on F(t; x; p) in p and the basic fact that if a function x is absolutely continuous on an open interval J , then at any extremum point z ∈ J , one can "nd in any neighborhood of z a subset A as well as a subset B with positive measure such that x is di$erentiable at any point t ∈ A ∪ B with x
(t) 6 0 for t ∈ A and x
(t) ¿ 0 for t ∈ B), we omit the argument. The same remark holds true for Theorem 5.3. In the following we give an easy example which shows that even if F ∈ C 1 and is monotone in p (not strictly), Eq. (1.1) may have no classical T -periodic solutions. (Thus even if Theorems 5.2 and 5.3 are also meaningful.) Example 5.4. Let H ∈ C 1 (R1 ) be a function such that H (p) is nondecreasing in p with H (p) = 0;

∀p ∈ [ − 1; 1]

and lim H (p) = ± ∞:

p→± ∞

Let (t) = |sin t|3 ;

∀t ∈ R1 :

It is trivial to check that  ∈ C 1 (R1 ). De"ne F ∈ C 1 (R3 ) as F(t; x; p) = H (p) + x3 − (t);

∀(t; x; p) ∈ R3 :

Then F(t; x; p) is 2-periodic in t. Consider the equation: F(t; x; x
) = 0:

(5.1)

One easily checks that all the conditions needed in Theorem 1.1 are satis"ed for Eq. (5.1), therefore it has at least a 2-periodic solution. Indeed, x=|sin t| is a 2-periodic solution of (5.1). We claim that it is the unique 2-periodic solution of (5.1).

D. Li, Y. Liang / Nonlinear Analysis 52 (2003) 1095 – 1109

1107

To see this, we assume the contrary and thus (5.1) has another 2-periodic solution y with x = y. We "rst assume that there exists a point t such that y(t) ¿ x(t). Let t0 be a maximum of y − x. Then y(t0 ) − x(t0 ) ¿ 0. Take a $ ¿ 0 suHciently small such that y(t) − x(t) ¿ 0

for t ∈ (t0 − $; t0 + $):

Since t0 is a maximum of y − x, we conclude that there exists a set D ⊂ (t0 − $; t0 + $) with positive measure such that y
(t) − x
(t) ¿ 0;

for t ∈ D:

Now, for t ∈ D, we have F(t; y(t); y
(t)) = H (y
(t)) + y3 (t) − (t) ¿ H (x
(t)) + x3 (t) − (t) = 0: This is a contradiction. Similarly we can get a contradiction in case there exists a point t such that y(t) ¡ x(t). As an easy application of our existence results, we consider the equation with parameter  ∈ R1 : F(t; x; x
) = h(; t)

(5:2 )

and derive a multiplicity result for periodic solutions, where h is also continuous and is T -periodic in t. We have Theorem 5.5. Assume that F(t; x; p) is nondecreasing in p and satis:es (1.3); h(; t) is nondecreasing in ; moreover; h(; t) ≡ h(
; t) for t ∈ R1 when  = 
. Assume   lim F(t; x; 0) = ∞ lim F(t; x; 0) = −∞ (5.3) |x|→∞

|x|→∞

then there exists a 0 with −∞ 6 0 6 ∞; such that (1) if  ¡ 0 (resp. ¿ 0 ); Eq. (5:2) has no T -periodic solution; (2) if  ¿ 0 (resp.  ¡ 0 ); Eq. (5:2) has at least two distinct T -periodic solutions. Proof of Theorem 5.5. We only consider the case lim F(t; x; 0) = ∞;

(5.4)

|x|→∞

the argument for the other case is similar. First, we show that if (5:2)
has a T -periodic classical solution x
for some 
∈ R1 , then for any  ¿ 
, (5:2) has at least two distinct T -periodic solutions. Indeed, if  ¿ 
, then F(t; x
; x
) 6 h(; t);

∀t ∈ R1 ;

(5.5)

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D. Li, Y. Liang / Nonlinear Analysis 52 (2003) 1095 – 1109

as h(; t) is nondecreasing in . By (5.4), we can take a M ¿ 0 suHciently large such that − M 6 x
(t) 6 M;

F(t; ±M; 0) ¿ |h(; t)|;

∀t ∈ R1 :

(5.6)

(5.5) and (5.6) show that x
and the constant M are a T -periodic lower solution and upper solution of (5:2) , respectively, with x
6 M , also x
and −M are a T -periodic upper solution and lower solution of equation: −F(t; x; x
) = −h(; t) with −M 6 x
. Thanks to Theorems 5.2 and 5.3, we deduce immediately that Eq. (5:2) has at least a T -periodic solution x with x
6 x 6 M and T -periodic solution y with −M 6 y 6 x
. Since h(; t) ≡ h(
; t) for t ∈ R1 , we "nd that x (t); y (t) ≡ x
(t) for t ∈ R1 , hence x (t) ≡ y (t) on R1 . Now, let 0 be the in"mum of those  which are such that (5:2) has at least one T -periodic solution. Then by the result obtained above, we conclude immediately that for any  ¿ 0 , Eq. (5:2) has at least two distinct T -periodic solutions and for  ¡ 0 , the equation has no T -periodic solution. References [1] K.M. Andersen, A. Sandqvist, Existence of closed solutions to an equation x˙ = f(t; x), where fx
(t; x) is weakly convex or concave in x, J. Math. Anal. Appl. 229 (1999) 480–500. [2] S. Carl, S. HeikkilUa, On discontinuous "rst order implicit boundary value problem, J. Di$erential Equations 148 (1998) 100–121. [3] S. Carl, S. HeikkilUa, Nonlinear Di$erential Equation in Ordered Spaces, Chapman & Hall=CRC, London, 2000. [4] M.G. Crandall, H. Ishii, P.L. Lions, User’s guide to viscosity solutions of second order PDE’s, Bull. Am. Math. Soc. (new series) 27 (1992) 1–67. [5] M. Frigon, T. Kaczynski, Boundary value problems for systems of implicit di$erential equations, J. Math. Anal. Appl. 179 (1993) 317–326. [6] S. HeikkilUa, M. Kumpulainen, S. Seikkala, Uniqueness and comparison results for implicit di$erential equations, Dyn. Systems Appl. 7 (1998) 237–244. [7] V.M. Hokkanen, Continuous dependence for an implicit nonlinear equation, J. Di$erential Equations 110 (1994) 67–85. [8] V.M. Hokkanen, Existence of a periodic solution for implicit nonlinear equations, Di$erential and Integral Equations 9 (1996) 745–760. [9] H. Ishii, Perron’s method for Hamilton-Jacobi equations, Duke Math. J. 55 (1987) 369–384. [10] P.A. Kravtscov, The Cauchy problem and implicit equations (Russian), Partial di$erential equations, Leningrad Gos. Ped. Inst. Leningrad, 1990. [11] P.A. Kravtsov, Existence and uniqueness of the solution of the Cauchy problem for an implicit system of di$erential equations (Russian), Di$erential equations (qualitative theory), Ryazan Gos. Ped. Inst., Ryazan, 1990. [12] V. Lakshmikantham, Periodic boundary value problems of "rst and second order di$erential equations, J. Appl. Math. Simulation 2 (1989) 131–138. [13] V. Lakshmikantham, S. Leela, Existence and monotone method for periodic solutions of "rst-order di$erential equations, J. Math. Anal. Appl. 91 (1983) 237–243. [14] V. Lakshmikantham, S. Leela, Remarks on "rst and second order periodic boundary value problems, Nonlinear Anal. TMA 8 (1984) 281–287. [15] S. Leela, Monotone technique for periodic solutions of di$erential equations, J. Math. Phys. Sci. 18 (1984) 73–82.

D. Li, Y. Liang / Nonlinear Analysis 52 (2003) 1095 – 1109

1109

[16] D.S. Li, Peano’s theorem on implicit di$erential equations, J. Math. Anal. Appl. 258 (2001) 591–616. [17] J.L. Massera, The existence of periodic solutions of systems of di$erential equations, Duke Math. J. 17 (1950) 457–475. [18] J. Mawhin, Points Fixes, Points Critiques et Probleme aux limites, Semi. Math. Sup. 92, Press Univ. de Montreal, 1985. [19] J. Mawhin, First order ordinary di$erential equations with several periodic solutions, Z. Angew. Math. Physics 38 (1987) 257–265. [20] M.N. Nkashama, A generalized upper and lower solutions methods and multiplicity results for nonlinear "rst-order ordinary di$erential equations, J. Math. Anal. Appl. 140 (1989) 381–395. [21] L.G. Prosenyuk, On the existence and asymptotic properties of a real ayatem of di$erential equations that are not solved with respect to the derivative, Ukr. Math. J. 45 (1993) 1644–1648. [22] P.J. Rabier, Implicit di$erential equations near a singular point, J. Math. Anal. Appl. (1989) 425 – 449. [23] P.J. Rabier, W.C. Rheinboldt, A geometric treatment of implicit algebraic equations, J. Di$erential Equations 109 (1994) 110–146. [24] B. Ricceri, Solutions Lipschitziennes d’Gequations di$Gerentielles sous forme implicite, C. R. Acad. Sci. Paris SGer. I Math. 295 (1982) 245–248. [25] B. Ricceri, Lipschitz solutions of the implicit Cauchy problem g(x
) = f(t; x); x(0) = 0, with f discontinuous in x, Rend. Circ. Mat. Palermo 34 (1985) 127–135. [26] B. Ricceri, On the Cauchy problem for the di$erential equation f(t; x; x
; : : : ; x(k) ) = 0, Glasgow Math. J. 33 (1991) 343–348. [27] A. Sandqvist, K.M. Andersen, On the number of closed solutions to an equation x˙ = f(t; x), where fxn (t; x) ¿ 0 (n = 1; 2 or 3), J. Math. Anal. Appl. 159 (1991) 127–146. [28] G. Sansone, R. Conti, Nonlinear Di$erential Equations, Pergamon Press, New York, 1964. [29] S. Seikkala, S. HeikkilUa, Uniqueness, comparison and existence results for discontinuous implicit di$erential equations, Nonlinear Anal., TMA 30 (1997) 1771–1780. [30] S. Vakhidov, Existence and uniqueness of the solution of a di$erential equation, which cannot be solved with respect to the derivative, in a normed space (Russian), Dokl. Akad. Nauk. Uzb. SSR 11 (1987) 13–15. [31] J.R.L. Webb, S.C. Welsh, Existence and uniqueness of initial value problems for a class of second-order di$erential equations, J. Di$erential Equations 82 (1989) 314–321. [32] A.E. Zernov, Solutions of a singular Cauchy Problem of implicit form, Transl. Ukr. Math. J. 43 (1991) 705–709. [33] A.E. Zernov, On the solvability and asymptotic properties of solutions of a singular Cauchy Problem, Transl. Di$erential Equations 28 (1992) 601–604. [34] A.E. Zernov, Asymptotic behavior of the solutions of Cauchy Problem that is not solved with respect to the derivative, Transl. Di$erential Equations 31 (1995) 31–37.