Contractive Mapping Theorems in Partially Ordered ... - Springer Link

Order (2005) 22: 223–239 DOI: 10.1007/s11083-005-9018-5

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Springer 2006

Contractive Mapping Theorems in Partially Ordered Sets and Applications to Ordinary Differential Equationsj ´ PEZ JUAN J. NIETO* and ROSANA RODRI´GUEZ-LO Departamento de Ana´lisis Matema´tico, Facultad de Matema´ticas, Universidad de Santiago de Compostela, 15782 Santiago de Compostela, Spain. e-mail: {amnieto, amrosana}@usc.es (Received: 15 September 2004; in final form: 12 September 2005) Abstract. We prove the existence and uniqueness of solution for a first-order ordinary differential equation with periodic boundary conditions admitting only the existence of a lower solution. To this aim, we prove an appropriate fixed point theorem in partially ordered sets. Mathematics Subject Classifications: Primary: 47H10, 34B15. Key Words: fixed point, partially ordered set, first-order differential equation, lower and upper solutions.

1. Preliminaries Existence of fixed point in partially ordered sets has been considered recently in [5], where some applications to matrix equations are presented. In this paper we extend the theoretical results of fixed points in partially ordered sets and apply them to study a problem of ordinary differential equations. In the literature, we can find results on existence of solution for ordinary as well as functional differential equations in presence of both lower and upper solutions relative to the problem considered. In this paper, we assume the existence of just one of them for the periodic boundary value problem
0 u ðtÞ ¼ f ðt; uðtÞÞ; t 2 I ¼ ½0; T
; ð1Þ uð0Þ ¼ uðT Þ; where T > 0, and f : I  R ! R is a continuous function.

j

Partially supported by Ministerio de Ciencia y Tecnologı´a/ FEDER, project BFM2001-3884C02-01; and by Xunta de Galicia/ FEDER, project PGIDIT02PXIC20703PN. * Corresponding author.

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´ PEZ JUAN J. NIETO AND ROSANA RODRI´GUEZ-LO

DEFINITION 1. A solution to (1) is a function u 2 C1(I, R) satisfying conditions in (1). DEFINITION 2. A lower solution for (1) is a function
2 C1(I, R) such that
0 ðtÞ  f ðt;
ðtÞÞ; t 2 I;
ð0Þ 
ðT Þ: An upper solution for (1) satisfies the reversed inequalities. It is well known [4] that the existence of a lower solution
and an upper solution  with
e  implies the existence of a solution of (1) between
and . In Section 2, we present a new extension of Banach contractive mapping theorem to partially ordered sets that allows to consider discontinuous functions, showing some examples. Then, in Section 3, existence of a unique solution for problem (1) is obtained under suitable conditions.

2. Fixed Point Theorems DEFINITION 3. If (X, e) is a partially ordered set and f : X Y X, we say that f is monotone nondecreasing if x; y 2 X ; x  y ) f ð xÞ  f ð yÞ: This definition coincides with the notion of a nondecreasing function in the case where X ¼ R and e represents the usual total order in R. THEOREM 2.1. Let (X, e) be a partially ordered set and suppose that there exists a metric d in X such that (X, d ) is a complete metric space. Let f : X Y X be a continuous and nondecreasing mapping such that there exists k 2 ½ 0, 1Þ with d ð f ð xÞ; f ð yÞÞ  k d ðx; yÞ; 8x  y: If there exists x0 2 X with x0 e f (x0), then f has a fixed point. Proof. If f(x0) = x0, then the proof is finished. Suppose that f ðx0 Þ 6¼ x0 : Since x0 e f (x0) and f is nondecreasing, we obtain by induction that x0  f ðx0 Þ  f 2 ðx0 Þ ¼ f ð f ðx0 ÞÞ  f 3 ðx0 Þ  . . .  f n ðx0 Þ  f nþ1 ðx0 Þ  . . .

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225

Now,   d f nþ1 ðx0 Þ; f n ðx0 Þ  k n d ð f ðx0 Þ; x0 Þ; n 2 N:

ð2Þ

Indeed, for n = 1, using that f(x0) Q x0, we get   d f 2 ðx0 Þ; f ðx0 Þ  k d ð f ðx0 Þ; x0 Þ: Supposing that (2) is true for n and using that f n+1(x0) Q f n(x0), we obtain         d f nþ2 ðx0 Þ; f nþ1 ðx0 Þ ¼ d f f nþ1 ðx0 Þ ; f ð f n ðx0 ÞÞ  k d f nþ1 ðx0 Þ; f n ðx0 Þ  k k n d ð f ðx0 Þ; x0 Þ ¼ k nþ1 d ð f ðx0 Þ; x0 Þ: This implies that f f n ðx0 Þgn 2 N is a Cauchy sequence in X. Indeed, let m > n, then,     d ð f m ðx0 Þ; f n ðx0 ÞÞ  d f m ðx0 Þ; f m1 ðx0 Þ þ    þ d f nþ1 ðx0 Þ; f n ðx0 Þ    k m1 þ k m2 þ    þ k n d ð f ðx0 Þ; x0 Þ ¼

kn  km kn d ð f ðx0 Þ; x0 Þ  d ð f ðx0 Þ; x0 Þ: 1k 1k

Since X is a complete metric space, then there exists y 2 X such that lim f n ðx0 Þ ¼ y:

n!þ1

Finally, we prove that y 2 X is a fixed point of f, that is, f ( y) = y. Let  > 0. Using the continuity of f at the point y, given 2 > 0, there exists  > 0 such that d(z, y) <  implies that  d ð f ð yÞ; f ðzÞÞ < : 2   Now, by the convergence of { f n(x0)} to y, given  ¼ min 2 ;  > 0, there exists n0 2 N such that, for all n 2 N, n Q n0, we get d ð f n ðx0 Þ; yÞ < : Therefore, taking n 2 N, n Q n0,    d ð f ð yÞ; yÞ  d ð f ð yÞ; f ð f n ðx0 ÞÞÞ þ d f nþ1 ðx0 Þ; y < þ   : 2 This proves that d( f (y), y) = 0, and y is a fixed point of f.

Ì

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Note that this is essentially Theorem 2.1 in [5]. Remark 1. The hypothesis of Lipschitz type on f applies to every pair of elements of X that are related by the order relation e. Indeed, if x e y, then d ð f ð xÞ; f ð yÞÞ ¼ d ð f ð yÞ; f ð xÞÞ  k d ð y; xÞ ¼ k d ðx; yÞ: Last result is still valid for f not necessarily continuous, assuming an additional hypothesis on X. THEOREM 2.2. Let (X, e) be a partially ordered set and suppose that there exists a metric d in X such that (X, d) is a complete metric space. Assume that X satisfies if a nondecreasing sequence fxn g ! x in X ; then xn  x; 8n:

ð3Þ

Let f : X Y X be a monotone nondecreasing mapping such that there exists k 2 ½ 0, 1Þ with d ð f ð xÞ; f ð yÞÞ  k d ðx; yÞ; 8x Q y: If there exists x0 2 X with x0 e f(x0), then f has a fixed point. Proof. Following the proof of Theorem 2.1, we only have to check that f(y) = y. Let  > 0. Then, using that {f n(x0)} converges to y, given 2 > 0, there exists n1 2 N such that, for all n 2 N, n Q n1, we obtain d ð f n ðx0 Þ; yÞ
0; there exists y 2 Y with d ð y; sup Y Þ < : Proof. Indeed, let fxn gþ1 n¼1 " x, then x1  x2  x3  . . .  xn  . . . is a chain in X, hence there exists z = sup{xn} 2 X and xn e z, for all n 2 N. We prove that z = x and, thus, xn e x, for all n 2 N. Let  > 0. Given 2 > 0, by convergence, there exists N 2 N such that for n 2 N, n Q N, d(x, xn) < 2 and, by the property of the supremum, there exists xm with m Q N such that d(xm, z) < 2 and d ðx; zÞ  d ðx; xm Þ þ d ðxm ; zÞ < :

Ì

CONTRACTIVE MAPPING THEOREMS

229

Note that it is enough for the hypotheses to be fulfilled for nondecreasing chains. Remark 4. Conditions in Lemma 1 do not imply the validity of (4), since it may occur for X to have non comparable elements which are never included in a chain and upper bounds may not exist. Condition (3) can be more general, since for every chain to have a supremum, we need some kind of upper boundness of the space X, what is not required for (3) to hold. If X ¼ R, then (3) is true (for a nondecreasing sequence convergent to x, if a term xm is greater than the limit, then xn Q xm for all n > m, contradicting the convergence), but there exist chains in R without supremum. This suggests that, in some cases, our results are applicable while existence of fixed points can not be obtained using neither Tarski’s Theorem [2, 6], where it is assumed that X is a complete lattice, nor Theorems of Knaster-Tarski [7] or Amann [1, 7], which can be applied in case of existence of supremum for every chain. This remark also points out the relation between our results and some fixed point theorems established in [3]. LEMMA 2. If X is a totally ordered set and d ða; cÞ  d ðb; cÞ;

for

a  b  c;

then (3) and (4) hold. Proof. Given x, y 2 X, then x e y or y e x, and there exists an upper and a lower bound of x and y. Let {xn}  X be a monotone nondecreasing sequence converging to x in X, and suppose that xm e x is not true for some m 2 N. Since X is totally ordered, we obtain xm > x (xm Q x, xm m x). The monotonicity implies that xn  xm > x; 8n  m; and d ðxn ; xÞ  d ðxm ; xÞ > 0; 8n  m; which, passing to the limit as n Y +1, leads to the contradiction 0  d ðxm ; xÞ > 0:

Ì

Next, we show an example where (4) is valid, but (3) fails. EXAMPLE 4. Let X3 = {(x, x): x 2 ½j1, 1
}  R2 , (X3, d2) complete metric space and consider the following order ½ðx; xÞ  ð y; yÞ () x  y
; if x 6¼ 0; y 6¼ 0; ð1; 1Þ  ð0; 0Þ; ð0; 0Þ  ð0; 0Þ; ð0; 0Þ  ð1; 1Þ:

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This defines an order relation in X3, since every element is comparable with itself, if ðx; xÞ  ð y; yÞ and ð y; yÞ  ðx; xÞ; then

– for x m 0 and y m 0, we obtain x e y e x and the elements are equal, – for x = 0 or y = 0, the unique possibility is ðx; xÞ ¼ ð0; 0Þ ¼ ð y; yÞ; and, for ðx; xÞ  ð y; yÞ and ð y; yÞ  ðz; zÞ; we get, distinguishing cases, that necessarily (x, x) e (z, z). Condition (4) is valid since, given two elements in X3, there exist an upper bound (1, 1) and a lower bound (j1, j1). However, (3) fails, since the monotone nondecreasing sequence
  1 1 fxn g ¼  ; n n n2N converges to (0, 0) in X3 but only the first term in the sequence is comparable to the limit. EXAMPLE 5. Finally, for X4 = {(x, x): x 2 ½j1, 0
}  R2 equipped with the distance d2 and the order relation ½ðx; xÞ  ð y; yÞ () x  y
; if x 6¼ 0; y 6¼ 0; ð0; 0Þ  ð0; 0Þ; there is neither upper nor lower bound for the elements (j1, j1) and (0, 0), so that (4) is not valid, but neither is (3), taking the sequence in the previous example. Uniqueness of the fixed point and consequent global convergence of the method of successive approximation can be obtained under hypotheses weaker than condition (4), in fact, it is sufficient to consider that X is such that every pair of elements has a lower bound or an upper bound: This condition is equivalent to for every x; y 2 X ; there exists z 2 X which is comparable to x and y:

ð5Þ

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CONTRACTIVE MAPPING THEOREMS

Indeed, if (5) holds, given x, y 2 X, the upper or lower bound of x and y is comparable both to x and y. Conversely, if x, y 2 X and z 2 X is comparable with them, we have the following possibilities:

– If x e z, y e z or z e x, z e y, then (5) holds. – If x e z e y or y e z e x, then x and y are comparable and (5) holds. THEOREM 2.3. Adding condition (5) to the hypotheses of Theorem 2.1 (resp. 2.2), we obtain uniqueness of the fixed point of f. Proof. If x 2 X is another fixed point of f, we prove that d(x, y) = 0, where y ¼ lim f n ðx0 Þ: n!þ1

We distinguish two cases: Case 1. If x is comparable to y, then f n(x) = x is comparable to f n( y) = y for every n = 0, 1, 2, . . . , and d ðx; yÞ ¼ d ð f n ð xÞ; f n ð yÞÞ  k n d ðx; yÞ; which implies d(x, y) = 0. Case 2. If x is not comparable to y, then there exists either an upper or a lower bound of x and y, that is, there exists z 2 X comparable to x and y. Monotonicity implies that f n(z) is comparable to f n(x) = x and f n(y) = y, for all n = 0, 1, . . . and n!þ1

d ðx; yÞ  d ð f n ð xÞ; f n ð zÞÞ þ d ð f n ðzÞ; f n ð yÞÞ  k n d ðx; zÞ þ k n d ðz; yÞ! 0;

Ì

so that d(x, y) = 0. Remark 5. In fact, it can be proved that lim f n ð xÞ ¼ lim f n ðx0 Þ ¼ y; for every x 2 X :

n!þ1

n!þ1

Indeed, let x 2 X. If x is comparable to y, then f n(x) is comparable to f n( y) = y for every n = 0, 1, 2, . . . , and thus d ð f n ð xÞ; yÞ  k n d ðx; yÞ; and if x is not comparable to y, let z 2 X be comparable both to x and y and f n(z) is comparable to f n(x) and f n(y) = y, for all n = 0, 1, . . . so that d ð f n ð xÞ; yÞ  d ð f n ð xÞ; f n ð zÞÞ þ d ð f n ðzÞ; f n ð yÞÞ  k n d ðx; zÞ þ k n d ðz; yÞ:

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As we have shown in Example 1, if condition (5) fails, there may exist more than one fixed point. However, the monotonicity of f as well as the contractivity hypothesis over comparable elements also play an important role in uniqueness. EXAMPLE 6. Let us present an example where the lack of monotonicity yields the existence of more than one fixed point. Consider the continuous function f : R2 ! R2 given by f (x, y) = (x,jx), where the distance d in R2 is defined by d ððx; yÞ; ðz; tÞÞ ¼ maxfjx  zj; j y  tjg: Consider the order relation in R2 given by ðx; yÞ  ðz; tÞ () x  z; y  t; and jx  zj 

1 j y  tj: 2

Let us prove that e is an order relation:

– Every element (x, y) 2 R2 satisfies that (x, y) e (x, y), since jx  xj ¼ 0 ¼

1 jy  yj: 2

– If (x, y) e (z, t) and (z, t) e (x, y), then x = z and y = t. – If (x, y) e (z, t) and (z, t) e (u, v), then x  z  u; y  t  v and jx  uj ¼ u  x ¼ ðu  zÞ þ ðz  xÞ 

1 1 1 ðv  tÞ þ ðt  yÞ ¼ jv  yj: 2 2 2

Besides, if (x, y) Q (z, t), then d ð f ðx; yÞ; f ðz; tÞÞ ¼ d ððx; xÞ; ðz; zÞÞ ¼ jz  xj  

1 jt  yj 2

1 1 maxfjz  xj; jt  yjg ¼ d ððx; yÞ; ðz; tÞÞ: 2 2

However, function f is not monotone nondecreasing, since for ðx; yÞ  ðz; tÞ; condition f(x, y) = (x,jx) e (z,jz) = f (z, t) is equivalent to x = z. Note that hypotheses (4) and (5) are true: if (x, y), (z, t) 2 R2 and x e z, then ðz; maxft; y þ 2jx  zjgÞ and ðx; minfy; t  2jx  zjgÞ

CONTRACTIVE MAPPING THEOREMS

233

are, respectively, upper and lower bounds of (x, y) and (z, t). Function f possesses a line of fixed points, the set fðx; xÞ : x 2 Rg: EXAMPLE 7. If X5 = ½0,1
 R and e is the usual order, then (4) and (5) are valid. Considering the metric derived from jj, the identity map is continuous and nondecreasing with an infinite number of fixed points. The condition that fails is contractivity. These examples show that our Theorems 2.1 and 2.2 joint to Theorem 2.3 are not comparable with Theorem 2.1 in [5]. If we restrict our attention to nondecreasing functions f, assuming the existence of x0 2 X with x0 e f(x0), our results improve Theorem 2.1 in [5], since our hypothesis for uniqueness is sharper and in Theorem 2.2 we eliminate the continuity assumption on f. EXAMPLE 8. Let X6 = ½0,+1Þ be a complete metric space with the distance d ðx; yÞ ¼ jx  yj; x; y 2 X6 ; and consider the order relation 8 x; y 2 ½0; 1
; and x  y < x  y () or : x; y 2 ðn; n þ 1
; for some n ¼ 1; 2; . . . and x  y: Let f be defined by f ð0Þ ¼ 0; f ð xÞ ¼

n 1 þ x; x 2 ðn; n þ 1
; n ¼ 0; 1; 2; . . . 2 2

which is not continuous at t = n + 1, for n = 0, 1, 2, . . . , since f ðn þ 1 Þ ¼ n þ

1 6¼ n þ 1 ¼ f ðn þ 1þ Þ: 2

Taking into account the order relation and the definition of f, we obtain that f is monotone nondecreasing (since the slope is 12 in each interval (n, n + 1)). Moreover, if x Q y, then x, y 2 ½0, 1
or x, y 2 (n, n + 1], for some n = 1, 2, . . . and   n 1 n 1 1 d ð f ð xÞ; f ð yÞÞ ¼ d þ x; þ y ¼ d ðx; yÞ: 2 2 2 2 2 Condition (3) is trivially satisfied, since any nondecreasing sequence such that {xn}  ½ 0, +1Þ, {xn} Y x in ½ 0, +1Þ, is necessarily contained in ½0; 1
or ðn; n þ 1
for some n ¼ 1; 2; . . . ;

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then the limit belongs to the same interval and, hence, xn e x, for all n 2 N. Condition (5) is not valid since, given 1, 2 2 ½ 0, +1Þ, there is neither upper nor lower bound of 1 and 2. Nevertheless, there exists a unique fixed point of f in ½ 0, +1Þ. We give an explanation for this behavior of f. If we consider the restriction f1 ¼ fj½0;1
: ½0; 1
! ½0; 1
; condition (5) is satisfied, so that uniqueness of the fixed point x = 0 is guaranteed. If we take f2 ¼ fj½1;þ1Þ : ½1; þ1Þ ! ½1; þ1Þ considering the induced order relation in ½ 1, +1Þ, there are no fixed points for f2, and all the hypotheses in Theorem 2.2 are valid, excepting the existence of x0 2 ½ 1, +1Þ such that x0 e f2(x0) = f (x0). Indeed, f ð1Þ ¼ 12 and, for x 2 (1, +1), we obtain that x 2 ð n, n + 1
for some n = 1, 2, . . . and n 1 x  f ð xÞ ¼ þ x () x  n; 2 2 which is absurd. EXAMPLE 9. Take the same metric space of the previous example (X6, d ) and consider the following order relation in X6 = ½ 0, +1Þ x  y () x; y 2 ½n; n þ 1Þ for some n ¼ 0; 1; . . . and x  y: Let f : X6 Y X6 given by (1 x 2 ½0; 1Þ 2 x; f ð xÞ ¼ nþ1 1 2 þ 2 x; x 2 ½n; n þ 1Þ; n ¼ 1; 2; . . . f is monotone nondecreasing but not continuous and the contractivity condition over comparable elements holds. Condition (5) is not true, since there is neither upper nor lower bound for two different numbers in Zþ , and (3) is not valid for any interval ½ n, n + 1
, n = 0, 1, . . . However, f|[0,1] has a unique fixed point while fj½1;þ1Þ has no fixed points despite x0 ¼ 1  f ðx0 Þ ¼ 32 . THEOREM 2.4. Let (X, e) be a partially ordered set and suppose that there exists a metric d in X such that (X, d ) is a complete metric space. Let f : X Y X be a monotone nondecreasing mapping such that there exists k 2 ½0,1Þ with d ð f ð xÞ; f ð yÞÞ  k d ðx; yÞ; 8x  y: Assume that either f is continuous or X is such that if a nonincreasing sequence f xn g ! x in X ; then x  xn ; 8n: If there exists x0 2 X with x0 Q f(x0), then f has a fixed point.

ð6Þ

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235

Proof. The scheme of the proof is similar to the procedure followed in the proof of the previous theorems. If f (x0) = x0, the proof is finished, thus we assume that f (x0) m x0. Using that x0 Q f (x0) and that f is nondecreasing, we get in this case that the sequence { f n(x0)} is nonincreasing, x0  f ðx0 Þ  f 2 ðx0 Þ ¼ f ð f ðx0 ÞÞ  f 3 ðx0 Þ  . . .  f n ðx0 Þ  f nþ1 ðx0 Þ  . . . An analogous procedure to the proof of Theorem 2.1 leads to   d f n ðx0 Þ; f nþ1 ðx0 Þ  k n d ð f ðx0 Þ; x0 Þ; n 2 N;

ð7Þ

and f f n ðx0 Þgn2N is a Cauchy sequence in X, hence there exists z 2 X such that lim f n ðx0 Þ ¼ z:

n!þ1

To prove that z 2 X is a fixed point of f, if f is continuous, the same reasoning of Theorem 2.1 is valid. In case (6) holds and using that { f n(x0)} converges to z, for 2 > 0, there exists n2 2 N such that, for all n 2 N, n Q n2, we get  d ð f n ðx0 Þ; zÞ < : 2 Taking n 2 N, n Q n2, and using that f n(x0) Q z, for all n 2 N, we check that   d ðz; f ð zÞÞ  d z; f nþ1 ðx0 Þ þ d ð f ð f n ðx0 ÞÞ; f ðzÞÞ    d z; f nþ1 ðx0 Þ þ k d ð f n ðx0 Þ; zÞ    d z; f nþ1 ðx0 Þ þ d ð f n ðx0 Þ; zÞ < : Since  > 0 is arbitrary, the result follows.

Ì

THEOREM 2.5. Condition (5) provides uniqueness of the fixed point of f in the hypotheses of Theorem 2.4. 3. Existence and Uniqueness of Solution to (1) Now, we prove the existence of solution for the first-order periodic problem (1) in presence of a lower solution. THEOREM 3.1. Consider problem (1) with f continuous and suppose that there exist  > 0, m > 0, with m <  such that for all x, y 2 R, y Q x, 0  f ðt; yÞ þ y  ½ f ðt; xÞ þ x
 ð y  xÞ: Then the existence of a lower solution for (1) provides the existence of a unique solution of (1).

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Proof. Problem (1) is written as ( 0 u ðtÞ þ uðtÞ ¼ f ðt; uðtÞÞ þ uðtÞ; t 2 I ¼ ½0; T
; : uð0Þ ¼ uðT Þ: This problem is equivalent to the integral equation Z T Gðt; sÞ½ f ðs; uðsÞÞ þ uðsÞ
ds; uðtÞ ¼ 0

where

Gðt; sÞ ¼

8 ðTþstÞ < e T ; 0  s < t  T; e 1 : eðstÞ ; eT 1

0  t < s  T:

Define A : C ðI; RÞ ! C ðI; RÞ by Z T Gðt; sÞ½ f ðs; uðsÞÞ þ uðsÞ
ds; t 2 I: ½Au
ðtÞ ¼

ð8Þ

0

Note that if u 2 C ðI; RÞ is a fixed point of A, then u 2 C 1 ðI; RÞ is a solution of (1). We check that hypotheses in Theorems 2.2 and 2.3 are satisfied. Indeed, X ¼ C ðI; RÞ is a partially ordered set if we define the following order relation in X : x; y 2 C ðI; RÞ; x  y if and only if xðtÞ  yðtÞ; 8t 2 I: Also (X, d ) is a complete metric space, if we choose d ðx; yÞ ¼ supjxðtÞ  yðtÞj; x; y 2 C ðI; RÞ: t2I

Take a monotone nondecreasing sequence {xn}  C ðI; RÞ converging to x in C ðI; RÞ. Then, for every t 2 I, we get x1 ðtÞ  x2 ðtÞ  x3 ðtÞ  . . .  xn ðtÞ  . . . ; and the convergence of this sequence of real numbers to x(t) implies xn ðtÞ  xðtÞ; for all t 2 I; n 2 N; and, therefore, xn e x for all n 2 N and the limit is an upper bound for all the terms xn in the sequence. Moreover, the mapping A is nondecreasing since, by hypotheses, for u Q v, f ðt; uÞ þ u  f ðt; vÞ þ v;

237

CONTRACTIVE MAPPING THEOREMS

which implies, using that G(t, s) > 0, for (t, s) 2 I  I, that ½Au
ðtÞ ¼

Z

T

Gðt; sÞ½ f ðs; uðsÞÞ þ uðsÞ
ds 0



Z

T

Gðt; sÞ½ f ðs; vðsÞÞ þ vðsÞ
ds ¼ ½Av
ðtÞ; t 2 I; 0

and A is nondecreasing. Besides, for u Q v, d ðAu; AvÞ ¼ supj½Au
ðtÞ  ½Av
ðtÞj t2I Z T  sup Gðt; sÞj f ðs; uðsÞÞ þ uðsÞ  f ðs; vðsÞÞ  vðsÞj ds t2I 0 Z T  sup Gðt; sÞðuðsÞ  vðsÞÞ ds t2I 0 Z T   d ðu; vÞsup Gðt; sÞ ds t2I 0   1 1 ðTþstÞ it 1 ðsjtÞ iT þ e e ¼  d ðu; vÞsup T 0  t 1  t2I e  T   1 ¼  d ðu; vÞ e  1 ¼ d ðu; vÞ; ðeT  1Þ  this proves that A satisfies the corresponding hypothesis in Theorem 2.2 with k ¼  < 1. Finally, let us prove that
2 X is such that
e Að
Þ, and in consequence, A has a fixed point in C ðI; RÞ. Indeed,
0 ðtÞ þ 
ðtÞ  f ðt;
ðtÞÞ þ 
ðtÞ; t 2 I: Multiplying by et , we obtain that  
ðtÞet 0  ½ f ðt;
ðtÞÞ þ 
ðtÞ
et ; t 2 I; and t


ðtÞe 
ð0Þ þ

Z

t

½ f ðs;
ðsÞÞ þ 
ðsÞ
es ds; t 2 I; 0

which implies that
ð0ÞeT 
ðT ÞeT 
ð0Þ þ

Z 0

T

½ f ðs;
ðsÞÞ þ 
ðsÞ
es ds;

ð9Þ

238

´ PEZ JUAN J. NIETO AND ROSANA RODRI´GUEZ-LO

so that
ð0Þ 

Z

T 0

es ½ f ðs;
ðsÞÞ þ 
ðsÞ
ds: eT  1

From this inequality and (9), we obtain


ðtÞet 

Z

t

eðT þsÞ ½ f ðs;
ðsÞÞ þ 
ðsÞ
ds T  1 0 e Z T es þ ½ f ðs;
ðsÞÞ þ 
ðsÞ
ds; t 2 I; T  1 t e

and hence
ðtÞ 

Z

T

Gðt; sÞ½ f ðs;
ðsÞÞ þ 
ðsÞ
ds ¼ ½A

ðtÞ; t 2 I:

0

The uniqueness of solution comes from the application of Theorem 2.3.

Ì

Remark 6. In the proof of Theorem 3.1, the unique solution of (1) can be obtained as lim An ð xÞ; for every x 2 C ðI; RÞ:

n!þ1

If we choose x =
, then fAn ð
Þg is a monotone nondecreasing sequence uniformly convergent to the unique solution of (1). THEOREM 3.2. Theorem 3.1 still holds if we replace the existence of a lower solution to (1) by the existence of an upper solution to the same problem (1). Proof. We follow the previous proof and check that hypotheses in Theorem 2.4 are valid. If  is an upper solution for (1), with an analogous procedure as the exposed for the case of the lower solution
, we check that  ðt Þ 

Z

T

Gðt; sÞ½ f ðs;  ðsÞÞ þ  ðsÞ
ds ¼ ½A
ðtÞ; t 2 I:

0

Theorem 2.4 provides the existence of a fixed point for A, which is unique by Ì Theorem 2.5, that is, there exists a unique solution of (1).

CONTRACTIVE MAPPING THEOREMS

239

References 1. 2. 3. 4. 5. 6. 7.

Amann, H.: Order Structures and Fixed Points. Mimeographed Lecture Notes, RuhrUniversita¨t, Bochum, 1977. Cousot, P. and Cousot, R.: Constructive versions of Tarski’s fixed point theorems, Pacific J. Math. 82 (1979), 43–57. Heikkila¨, S. and Lakshmikantham, V.: Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations, Marcel Dekker, Inc., New York, 1994. Ladde, G.S., Lakshmikantham, V. and Vatsala, A.S.: Monotone Iterative Techniques for Nonlinear Differential Equations, Pitman, Boston, 1985. Ran, A.C.M. and Reurings, M.C.B.: A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Am. Math. Soc. 132 (2004), 1435–1443. Tarski, A.: A lattice-theoretical fixpoint theorem and its applications, Pacific J. Math. 5 (1955), 285–309. Zeidler, E.: Nonlinear Functional Analysis and Its Applications, Vol. I: Fixed-Point Theorems, Springer, New York, 1986.