Existence theory for functional p-Laplacian equations with variable exponents

Nonlinear Analysis 52 (2003) 557 – 572

www.elsevier.com/locate/na

Existence theory for functional p-Laplacian equations with variable exponents
Alberto Cabada ∗ , Rodrigo L. Pouso Departamento de An alise Matem atica, Facultade de Matem aticas, Universidade de Santiago de Compostela, 15782, Santiago de Compostela, Spain Received 23 October 2001; accepted 12 November 2001

Abstract In this paper we consider the solvability of equations of the form d − ’(t; u; u(t); u
(t)) = f(t; u; u(t); u
(t)) for a:e: t ∈ I = [a; b] dt subject to a general type of functional-boundary conditions which cover Dirichlet and periodic boundary data as particular cases. Our approach is that of upper and lower solutions together with growth restrictions of Nagumo’s type. An example is provided where a p-Laplacian with variable p is shown to have a solution between given upper and lower solutions. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: p-Laplacian; Upper and lower solutions; Nonlinear equations; Boundary value problems; Functional equations

1. Introduction For a given real number p ¿ 1, the general form of the one-dimensional p-Laplacian equation is d − (|u
(t)|p−2 u
(t)) = f(t; u(t); u
(t)); t ∈ I = [a; b]: (1.1) dt The reader is referred to [6–8,12], and references therein, where existence results for (1.1) with Dirichlet boundary data are proven. Usually, the p-Laplacian operator is replaced by abstract and more general versions of it which lead to clearer expositions and a better understanding of the methods that


Partially supported by DGESIC, project PB97-0552-C01, and Xunta de Galicia, project XUGA 20701B98.

∗

Corresponding author. Fax: +34-981-597054. E-mail address: [email protected] (A. Cabada).

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 2 2 - 0

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are employed to derive the existence results. The most common generalization of (1.1) has the form d (1.2) − ’(u
(t)) = f(t; u(t); u
(t)); t ∈ I = [a; b]; dt where ’ is an increasing homeomorphism from R onto R, see [2–5,11,14,15,17,18] (in [9] the case ’ depending on t and u
is considered for comparison results). In this paper, we are concerned with the existence of solutions for equations of the type of (1.2) in presence of a pair of well-ordered upper and lower solutions. In this sense, we follow the line of [2– 6,14,15,17,18]. Moreover, in this framework of upper an lower solutions, we consider the case in which p is not a constant in Eq. (1.1). In particular, p will depend on t, u(t) and u (this last dependence is functional, i.e., it involves the global behavior of the solution). As far as we are aware, we know of no previous work in this direction, which we are convinced that it is interesting because there seems to be many other points of view to face this type of problems and, thus, a lot of work is left to be done. Following the previous ideas, it is advisable to replace the diGerential operator by an abstract version of it. The adequate formulation we propose here is the following one: d − ’(t; u; u(t); u
(t)) = f(t; u; u(t); u
(t)) for a:e: t ∈ I = [a; b]; dt where the conditions on ’ and f will be detailed in next sections. On the other hand, we shall study the solvability of the above equation together with a general type of functional-boundary conditions such as those considered in [1,4]. This setting includes the usual Dirichlet and periodic boundary conditions as particular cases and, moreover, covers a wide class of nonlinear and, even, functional conditions. Finally, note that there is also a functional dependence on the right-hand side of the equation. This has been taken into account in order to deal with a variety of generalized diGerential equations under a homogeneous notation (for instance, delay diGerential equations, diGerential equations with deviated arguments and integro-diGerential equations). This paper is organized as follows: In Section 2 we prove the existence of solutions for, roughly speaking, a problem with bounded nonlinear parts. In Section 3 we present the deHnitions of lower and upper solution and, using the existence result of Section 2, we prove our main result. As a previous step a new Nagumo-type condition has to be deHned. In Section 4, we give an example to illustrate the applicability of our result. 2. Existence in the small: bounded nonlinearities Let a; b ∈ R be such that a ¡ b and call I = [a; b]. We start considering the functional problem   − d ’(t; u; u(t); u
(t)) = f(t; u; u(t); u
(t)) for a:e: t ∈ I; dt (P)  u(a) = A(u); u(b) = B(u);

A. Cabada, R.L. Pouso / Nonlinear Analysis 52 (2003) 557 – 572

559

under the following set of assumptions: (H1 ) (a) ’ : I × C(I ) × R2 → R is a continuous function and, for each (t; ; x) ∈ I × C(I ) × R, the mapping ’t; ; x (·) ≡ ’(t; ; x; ·) is increasing and such that ’t; ; x (R) = R. (We shall employ the notation (t; ; x; z) ≡ ’−1 t; ; x (z) for all 2 (t; ; x; z) ∈ I × C(I ) × R .) (b) For every y ∈ R there exists R ≡ Ry ¿ 0 such that |’(t; ; x; y)| 6 R

for all (t; ; x) ∈ I × C(I ) × R:

(c) For every z ∈ R there exists S ≡ Sz ¿ 0 such that |(t; ; x; z)| 6 S

for all (t; ; x) ∈ I × C(I ) × R:

(H2 ) f : I × C(I ) × R2 → R is such that f(·; u; u(·); u
(·)) is measurable in I and lim f(t; un ; un (t); un
(t)) = f(t; u; u(t); u
(t))

n→∞

whenever un → u in C1 (I ). Moreover, there exists |f(t; ; x; y)| 6 (t)

for a:e: t ∈ I ∈ L1 (I ) such that

for a:e: t ∈ I and all ( ; x; y) ∈ C(I ) × R2 :

(H3 ) A and B are real-valued mappings which are deHned, bounded and continuous in C1 (I ). Some remarks about condition (H1 ) (i) When ’ does not depend on (t; ; x), i.e., ’(t; ; x; y) = ’(y), condition (H1 ) becomes “’(·) is an increasing homeomorphism from R onto R”, which is the usual condition when studying p-Laplacian-like operators (see [4,5,11,15,17]). (ii) Condition (H1 ) implies that function (t; ; x; z) = ’−1 t; ; x (z) is well deHned and that 2 it is continuous in I × C(I ) × R . Indeed, assume (tn ; n ; x n ; zn ) → (t0 ; 0 ; x0 ; z0 ) ∈ I × C(I ) × R2 . From the deHnition of  we have that yn = (tn ; n ; x n ; zn ) ⇔ zn = ’(tn ; n ; x n ; yn )

for all n ∈ N:

Claim 1. {yn }n is bounded. If it were not then we would have, up to a subsequence, that yn → +∞ (or −∞, what involves similar arguments to reach a contradiction). Let z ¿ 0 be such that zn ¡ z for all n ∈ N. By (H1 ), part (c), there exists S ≡ Sz such that (t; ; x; z) 6 S

for all (t; ; x) ∈ I × C(I ) × R;

and thus, applying ’(t; ; x; ·), we obtain zn ¡ z 6

inf

(t; ;x)∈I ×C(I )×R

’(t; ; x; S) 6 ’(tn ; n ; x n ; S)

for all n ∈ N:

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Since there exists n0 ∈ N such that yn0 ¿ S and ’(tn0 ; n0 ; x n0 ; ·) is strictly increasing, we have that ’(tn0 ; n0 ; x n0 ; yn0 ) ¿ ’(tn0 ; n0 ; x n0 ; S) ¿ zn0 ; attaining a contradiction. Claim 2. {yn }n converges to (t0 ; 0 ; x0 ; z0 ). By Claim 1 there is a subsequence {ynk }k that converges to some y0 . Since znk = ’(tnk ; nk ; x nk ; ynk )

for all k ∈ N

and ’ is continuous, we obtain, by passing to the limit, that z0 = ’(t0 ; 0 ; x0 ; y0 ), or equivalently, (t0 ; 0 ; x0 ; z0 ) = y0 . Since any convergent subsequence of {yn }n tends to (t0 ; 0 ; x0 ; z0 ) claim 2 holds. Remark 2.1. Assumption (H2 ) may look strange at Hrst sight; since it is only slightly more general than the well-known CarathMeodory conditions and these ones are usually quite more easier to check in practical situations. The usefulness of regarding (H2 ) instead of CarathMeodory conditions is technical and will be clear in next section; where we shall deal with a concrete problem for which the veriHcation of condition (H2 ) will be immediate. We say that u is a solution of (P) if u ∈ C1 (I ), ’(·; u; u(·); u
(·)) ∈ AC(I ) and u fulHlls both the diGerential equation (a.e. in I ) and the functional conditions (here AC(I ) denotes the space of absolutely continuous functions in I ). The following lemma is one of the main contributions of this section and it plays an essential role in the construction of a proper operator between function spaces whose Hxed points are the solutions of problem (P). Lemma 2.1. Assume conditions (H1 ); (H2 ) and (H3 ) hold. Then for every v ∈ C1 (I ) there exists a unique v ∈ R such that   r  b   r; v; v(r); v − f(s; v; v(s); v
(s)) ds dr = B(v) − A(v): a

a

(2.1)

Moreover, there exists a constant C ¿ 0, which depends only on A, B, ’ and such that |v | 6 C for all v ∈ C1 (I ). Proof. Let v ∈ C1 (I ) be Hxed and deHne the real function   b   r  r; v; v(r);  − f(s; v; v(s); v
(s)) ds dr Fv () = a

a

,

for all  ∈ R:

As an immediate consequence of our assumptions, Fv is increasing and continuous, thus it suNces to prove that lim Fv () = −∞ and

→−∞

lim Fv () = +∞

→+∞

to deduce the existence and uniqueness of the value v deHned in (2.1).

(2.2)

A. Cabada, R.L. Pouso / Nonlinear Analysis 52 (2003) 557 – 572

561

From condition (H2 ) we have that  b (r; v; v(r);  − 1 ) dr 6 Fv () a



6

b

a

(r; v; v(r); v + 1 ) dr;

∀ ∈ R:

On the other hand,  b (r; v; v(r);  − 1 ) dr ¿ (b − a) a

and



b

a

(r; v; v(r);  + 1 ) dr 6 (b − a)

(2.3)

inf

(t; ; x;  − 1 )

sup

(t; ; x;  + 1 );

(t; ;x)∈I ×C(I )×R

(t; ;x)∈I ×C(I )×R

hence, to conclude that (2.2) holds, it suNces to prove that   inf (t; ; x;  − 1 ) = +∞ lim →+∞

and

(t; ;x)∈I ×C(I )×R

 lim

→−∞

(2.4)

 sup

(t; ;x)∈I ×C(I )×R

(t; ; x;  + 1 )

= −∞:

(2.5)

We shall only prove (2.4) (as (2.5) can be treated in an analogous way) reasoning by contradiction: assume there exist a constant M ¿ 0 and a sequence of real numbers {n }n such that n → +∞ and inf

(t; ;x)∈I ×C(I )×R

(t; ; x; n − 1 ) 6 M

for all n ∈ N:

By deHnition of inHmum, for each n ∈ N there exists (tn ; n ; x n ) ∈ I × C(I ) × R such that (tn ; n ; x n ; n − 1 ) ¡ M + 1; hence, applying ’(tn ; n ; x n ; ·), we have n − 1 ¡ ’(tn ; n ; x n ; M + 1)

for all n ∈ N;

which is impossible since, by (H1 ) (b), ’(·; ·; ·; M + 1) is bounded in I × C(I ) × R. Now we are going to prove that the values v are uniformly bounded in C1 (I ). As a byproduct of the previous computations we also have that the functions F− () = (b − a)

inf

(t; ; x;  − 1 )

for all  ∈ R

sup

(t; ; x;  + 1 )

for all  ∈ R

(t; ;x)∈I ×C(I )×R

and F+ () = (b − a)

(t; ;x)∈I ×C(I )×R

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are non-decreasing and such that lim→± F± () = ±∞. Thus since the operator B − A is bounded in C1 (I ), there exist values ± (that only depend on A, B, ’ and ) such that F− (− ) ¿ B(v) − A(v) = Fv (v ) ¿ F+ (+ )

for all v ∈ C1 (I ):

From these last inequalities and (2.3), we conclude that + 6 v 6 − for all v ∈ C1 (I ). Remark 2.2. Note that the continuity of A and B is not needed to prove Lemma 2.1. Indeed; only the fact that the operator B − A is bounded is used in the proof. In the conditions of Lemma 2.1 we can deHne for each v ∈ C1 (I ) the function Tv, where, for each t ∈ I ,   r  t   r; v; v(r); v − f(s; v; v(s); v
(s)) ds dr (2.6) (Tv)(t) = A(v) + a

a

and v is the value associated to v by relation (2.1). Theorem 2.2. If conditions (H1 ); (H2 ) and (H3 ) hold; problem (P) has at least one solution. Proof. It is easy to prove that u is a solution of (P) if and only if u is a Hxed point of the operator T : C1 (I ) → C1 (I ) deHned in (2.6). Claim 1. Operator T is continuous in C1 (I ). Suppose un → u in C 1 (I ), let us prove that Tun → Tu in C1 (I ). By Lemma 2.1, for each n ∈ N there is a unique n ≡ un which veriHes (2.1) for v = un and there is a unique  ≡ u satisfying the analogous relation for v = u. O for some O ∈ R. By Also by Lemma 2.1 we have that, up to a subsequence, n → , assumption (H2 ) and Lebesgue’s Dominated Convergence Theorem we have that    t  t f(s; un ; un (s); un
(s)) ds = O − f(s; u; u(s); u
(s)) ds lim n − n→∞

a

a

uniformly in t ∈ I , hence, by the uniform continuity of  over compact subsets of I × C(I ) × R2 ,    t f(s; un ; un (s); un
(s)) ds lim  t; un ; un (t); n − n→∞





=  t; u; u(t); O −

a

a

t




f(s; u; u(s); u (s)) ds

 uniformly in t ∈ I:

(2.7)

Thus, by continuity of the operators A and B, passing to the limit when n tends to ∞ in (2.1) with v = un , we obtain   b   r
 r; u; u(r); O − f(s; u; u(s); u (s)) ds dr = B(u) − A(u); a

a

A. Cabada, R.L. Pouso / Nonlinear Analysis 52 (2003) 557 – 572

563

which implies that O = . Moreover, using this last fact and (2.7), we conclude that (Tun )
→ (Tu)
uniformly in I and, by integration, we obtain Tun → Tu in C1 (I ). Claim 2. T is completely continuous; i.e.; T maps bounded sets into relatively compact sets. Let R ¿ 0 be Hxed and call SR = {Tu: u ∈ C1 (I ); u C1 6 R}. Since    t (Tu)
(t) =  t; u; u(t); u − f(s; u; u(s); u
(s)) ds for t ∈ I; a

we have, by condition (H2 ), Lemma 2.1 and the properties of , that the set SR
= {(Tu)
: u ∈ C1 (I ); u C1 6 R} is bounded in C(I ). Hence, by integration and taking into account that A is bounded, we conclude that SR is bounded in C1 (I ) (which, in particular, means that SR is uniformly equicontinuous in I ). Finally, we have to prove that SR
is uniformly equicontinuous in I . Indeed, since BC1 [0; R] = {u ∈ C1 (I ): u C1 6 R} is a uniformly equicontinuous set in I , for each " ¿ 0 there is "O1 ¿ 0 such that [t; s ∈ I; |t − s| 6 "O1 ] ⇒ |u(t) − u(s)| 6 "

for all u ∈ BC1 [0; R]:

Moreover, by the properties of f, there is "O2 ¿ 0 such that  t
O [t; s ∈ I; |t − s| 6 "2 ] ⇒ f(r; u; u(r); u (r)) dr 6 " for all u ∈ C1 (I ); s

and, on the other hand, there is K ¿ 0 such that  t
u − f(s; u; u(s); u (s)) ds 6 K for all t ∈ I and all u ∈ C1 (I ): a

On the other hand, the set B = BC1 [0; R], where the closure is computed with respect to the topology of C(I ), is compact in C(I ). Now, since  is uniformly continuous in H = I × B × [ − R; R] × [ − K; K], for each j ¿ 0 there is " ¿ 0 such that O x; O x; [(t; ; x; y); (s; ; O y) O ∈ H; (t; ; x; y) − (s; ; O y) O ∞ 6 "] O x; ⇒ |(t; ; x; y) − (s; ; O y)| O 6 j: By the previous considerations, for the " ¿ 0 given above it suNces to take % ∈ (0; min{"O1 ; "O2 ; "}) so as to have [t; s ∈ I; |t − s| 6 %] ⇒ |(Tu)
(t) − (Tu)
(s)| 6 j

for all u ∈ BC1 [0; R];

and the claim follows. By Schauder Hxed point theorem (see [10]), the operator T has at least one Hxed point, which concludes the proof.

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3. Existence in the large: upper and lower solutions and Nagumo conditions We are going to prove an existence result whose main interest with respect to Theorem 2.2 is that boundedness conditions are relaxed in assumptions (H1 ) and (H2 ). This new result’s proof is based on a priori bounds on some of the solutions of the problem, which are given in terms of a lower and an upper solution and a Nagumo condition. The problem we shall study in this part is  d

   − dt ’(t; u; u(t); u (t)) = f(t; u; u(t); u (t)) for a:e: t ∈ I = [a; b]; ∗ (P ) L1 (u(a); u(b); u
(a); u
(b); u) = 0;    L2 (u(a); u(b)) = 0; where ’, f and Li , i = 1; 2, satisfy the following conditions: (H1∗ ) (a) ’ : I × R2 → R veriHes part (a) of condition (H1 ) and, for every ' ∈ R the mappings ’(·; ·; ·; ') and (·; ·; ·; ') are bounded on bounded subsets of I × C(I ) × R. (b) ’(t; ; x; 0) = 0 for all (t; ; x) ∈ I × C(I ) × R. (H2∗ ) f is a CarathMeodory function, that is: f(t; ·; ·; ·) is continuous in C(I ) × R2 for a.e. t ∈ I ; f(·; ; x; y) is measurable for all ( ; x; y) ∈ C(I ) × R2 ; and for every R ¿ 0 there exists ≡ R ∈ L1 (I ) such that for a.e. t ∈ I |f(t; ; x; y)| 6 (t) ∀( ; x; y) ∈ C(I ) × R2 with ( ; x; y) ∞ 6 R: (H3∗ ) L1 ∈ C(R4 × C(I ); R) is non-decreasing in the third variable, non-increasing in the fourth and non-decreasing in the Hfth one. On the other hand, L2 : R2 → R is a continuous function and it is non-increasing with respect to its Hrst variable. Remark 3.1. Condition (H1∗ )(b) is not a restriction provided that ’(·; ; ·; 0) ∈ C1 (I ×R) for every ∈ C(I ). Indeed, if ’(t; ; x; 0) = 0 for some (t; ; x) ∈ I × C(I ) × R we deHne ’(t; ˜ ; x; y) = ’(t; ; x; y) − ’(t; ; x; 0)

for all (t; ; x; y) ∈ I × C(I ) × R2 :

Obviously, ’(t; ˜ ; x; 0) = 0 for all (t; ; x) ∈ I × C(I ) × R and it is easy to prove that u is a solution of the equation d − ’(t; u; u(t); u
(t)) = f(t; u; u(t); u
(t)) for a:e: t ∈ I dt if and only if u solves d − ’(t; ˜ u; u(t); u
(t)) = g(t; u; u(t); u
(t)) for a:e: t ∈ I; dt where g(t; ; x; y) = f(t; ; x; y) + D1 ’(t; ; x; 0) + D3 ’(t; ; x; 0)y for a.e. t ∈ I . Note that g clearly veriHes (H2∗ ) if so does f.

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565

In the sequel we shall use the following notation: given functions *; + ∈ C(I ) such that *(t) 6 +(t) for all t ∈ I , we shall write [*; +] = {v ∈ C(I ): *(t) 6 v(t) 6 +(t)}: De%nition 3.1. Two functions *; + : I → R such that * 6 + are said to be a couple of a lower and an upper solution of problem (P∗ ) if the following conditions are satisHed: (i) * ∈ W 1; ∞ (I ) and D− *(t) 6 D+ *(t) for all t ∈ (a; b). Moreover; if t0 ∈ (a; b) is such that D− *(t0 ) = D+ *(t0 ) then there exists j ¿ 0 such that * ∈ C1 [t0 − j; t0 + j] and for every ∈ [*; +] ’(·; ; *(·); *
(·)) ∈ AC[t0 ; t0 + j] and d − ’(t; ; *(t); *
(t)) 6 f(t; ; *(t); *
(t)) for a:e: ∈ [t0 ; t0 + j]: dt (ii) D+ *(a); D− *(b) ∈ R and L1 (*(a); *(b); D+ *(a); D− *(b); *) ¿ 0; L2 (*(a); *(b)) = 0 and L2 (*(a); ·) is injective. On the other hand; (i
) + ∈ W 1; ∞ (I ) and D− +(t) ¿ D+ +(t) for all t ∈ (a; b). Moreover; if t0 ∈ (a; b) is such that D− +(t0 )=D+ +(t0 ) then there exists j ¿ 0 such that + ∈ C1 [t0 −j; t0 +j] and for every ∈ [*; +] ’(·; ; +(·); +
(·)) ∈ AC[t0 ; t0 + j] and d − ’(t; ; +(t); +
(t)) ¿ f(t; ; +(t); +
(t)) for a:e: t ∈ [t0 ; t0 + j]: dt (ii
) D+ +(a); D− +(b) ∈ R and L1 (+(a); +(b); D+ +(a); D− +(b); +) 6 0; L2 (+(a); +(b)) = 0 and L2 (+(a); ·) is injective. Remark 3.2. Note that; unlike the usual situation; we have to Hnd at the same time a lower solution and an upper one; since the deHnitions given above depend on each other. However; the reader should also take into account that this dependence disappears in the non-functional case; i.e.; when ’ and f simply depend on (t; u(t); u
(t)). These deHnitions of upper and lower solution follow the spirit of [5], allowing corners in the graphs of * and + and they can even be weakened by, for instance, following the ideas of [4]. To obtain a priori bounds on the derivatives of the solutions of (P∗ ) between a lower and an upper solution, we need a Nagumo-type condition. The reader must note that the Nagumo condition we are going to introduce in next deHnition does not cover the usual Nagumo condition for -Laplacian equations (see [3,4,17]): this is due to the fact that ’ is a function of some other variables besides u
, which makes it impossible to use the theorem of change of variable for the Lebesgue integral when trying to follow the standard arguments with the usual Nagumo condition (see [3–5,17]). De%nition 3.2. Let ’ be a function that veriHes (H1∗ ) and let f be in the conditions (H2∗ ). Let *; + ∈ C(I ) be such that *(t) 6 +(t) for all t ∈ I . We say that f satisHes a Nagumo condition with respect to ’ between * and + if there exist functions k ∈ Lq (I ); 1 6 q 6 ∞, and . : [0; ∞) → (0; ∞) continuous, such

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that for a.e. t ∈ I , |f(t; ; x; y)| 6 k(t).(|’(t; ; x; y)|) ∀ ∈ [*; +];

∀x ∈ [*(t); +(t)];

∀y ∈ R;

and also that there exists K ¿ 0 for which the three following relations hold: max{|*(a) − +(b)|; |*(b) − +(a)|} (i) K ¿ / = ; b−a inf I ×[*; +]×J*; + ’(t; ;x;K)



(ii) sup

I ×[*;+]×J*;+

’(t; ;x;/)

z (q−1)=q d z ¿ 21(q−1)=q k q ; .(z)

where J*; + = [mint∈I *(t); maxt∈I +(t)],  21 = and

K



K 1

+

sup

I ×[*;+]×J*;+

’(t; ; x; /) (b − a);

∈ L1 (I ) is the function given in (H2∗ ) such that for a.e. t ∈ I

|f(t; ; x; y)| 6

K (t)

∀ ∈ [*; +];

inf I ×[*; +]×J*; + ’(t; ;x;−/)



(iii) sup

I ×[*;+]×J*;+

where



 22 =

’(t; ;x;−K)

K 1 −

inf

I ×[*;+]×J*;+

∀x ∈ [*(t); +(t)] and |y| 6 K:

|z|(q−1)=q d z ¿ 22(q−1)=q k q ; .(|z|)

 ’(t; ; x; −/) (b − a):

Any such K ¿ 0 will be called Nagumo constant. Remark 3.3. Note that both sides in inequalities (ii) and (iii) depend on the Nagumo constant K ¿ 0 if q = 1. Thus; the version of the Nagumo condition which seems to be more easy to verify is the one corresponding to q = 1. Now we can prove the main result of this section. Theorem 3.1. Assume conditions (H1∗ ); (H2∗ ) and (H3∗ ) are veri9ed. Assume that *, + ∈ C(I ), with * 6 +, are a couple of a lower and an upper solution for problem (P ∗ ). If f veri9es a Nagumo condition with respect to ’ between * and + and there exists a Nagumo constant K ¿ max{ *
∞ ; +
∞ }, then problem (P∗ ) has at least one solution u ∈ [*; +] such that |u
(t)| 6 K for all t ∈ I . Proof. We deHne p(t; x) = max{*(t); min{x; +(t)}} for all (t; x) ∈ I × R: Next lemma’s proof is essentially given in [16]:

A. Cabada, R.L. Pouso / Nonlinear Analysis 52 (2003) 557 – 572

567

Lemma 3.2. Given v; vn ∈ C 1 (I ) such that vn → v in C 1 (I ); then (i)

d p(t; v(t)) exists for a:e: t ∈ I ; dt

(ii)

d d p(t; vn (t)) → p(t; v(t)) for a:e: t ∈ I: dt dt

Now call "K (y) = max{−K; min{y; K}} for all y ∈ R and deHne the following modiHed problem, which will be denoted by problem (P∗K ):     d   − d ’(t; O u; u(t); u
(t)) = f t; p(·; u(·)); p(t; u(t)); "K p(t; u(t)) ; dt dt   u(a) = A(u); u(b) = B(u); where ’(t; O ; x; y) = ’(t; p(·; (·)); p(t; x); y) for all (t; ; x; y) ∈ I × C(I ) × R2 , A(v) = p(a; v(a) + L1 (v(a); v(b); v
(a); v
(b); v)) and B(v) = p(b; v(b) − L2 (v(a); v(b))) for all v ∈ C1 (I ). By deHnition of p and (H1∗ )(a) it is easy to verify that ’O fulHlls condition (H1 ). On the other hand, for each u ∈ C1 (I ), we have that pu (·) ≡ p(·; u(·)) ∈ C(I ) and, by condition (H2∗ ), the mapping (t; x; y) ∈ I × R2 → f(t; pu ; x; y); is a CarathMeodory function in the usual sense. Hence, the composition    d t ∈ I → f t; pu ; p(t; u(t)); "K p(t; u(t)) dt is measurable in I . Furthermore, using Lemma 3.2 and condition (H2∗ ), one can easily check that condition (H2 ) is satisHed. Finally, condition (H3 ) also holds, so the problem (P∗K ) has at least one solution by virtue of Theorem 2.2. Now it suNces to prove that every solution of (P∗K ) is in fact a solution of (P∗ ). Step 1: Every solution of (P∗K ) belongs to [*; +]. Let u be a solution of (P∗K ). We shall only prove *(t) ≤ u(t) for all t ∈ I , since the inequality u(t) 6 +(t) for all t ∈ I can be proven in a symmetric way. By deHnition of the boundary conditions, we have that *(a) 6 u(a) 6 +(a) and *(b) 6 u(b) 6 +(b). Assume there exists t0 ∈ (a; b) such that (* − u)(t0 ) = max (* − u)(t) ¿ 0 and (* − u)(t0 ) ¿ (* − u)(t) t∈I

for all t ∈ (t0 ; b]: A necessary condition for a maximum, in terms of Dini derivatives, is that D+ *(t0 ) − u
(t0 ) 6 D− *(t0 ) − u
(t0 );

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A. Cabada, R.L. Pouso / Nonlinear Analysis 52 (2003) 557 – 572

so, by deHnition of lower solution, there exists 7 ¿ 0 such that * ∈ C1 [t0 − 7; t0 + 7] (in particular, *
(t0 ) = u
(t0 )), ’(·; pu ; *(·); *
(·)) ∈ AC[t0 − 7; t0 + 7] and d − ’(t; pu ; *(t); *
(t)) 6 f(t; pu ; *(t); *
(t)) a:e: in [t0 ; t0 + 7]; dt thus, integrating between t0 and t ∈ [t0 ; t0 + 7], we obtain −’(t; pu ; *(t); *
(t)) + ’(t0 ; pu ; *(t0 ); *
(t0 ))  t 6 f(s; pu ; *(s); *
(s)) ds for all t ∈ [t0 ; t0 + 7]: t0

(3.1)

Moreover, by continuity of * − u, we lose no generality by assuming that *(t) ¿ u(t) for all t ∈ [t0 ; t0 + 7], thus taking into account the deHnition of (P∗K ), we have d − ’(t; pu ; *(t); u
(t)) = f(t; pu ; *(t); *
(t)) a:e: in [t0 ; t0 + 7]; dt and hence − ’(t; pu ; *(t); u
(t)) + ’(t0 ; pu ; *(t0 ); u
(t0 ))  t = f(s; p(·; u(·)); *(s); *
(s)) ds for all t ∈ [t0 ; t0 + 7]: t0

(3.2)

From (3.1), (3.2), and using that u
(t0 ) = *
(t0 ), we obtain ’(t; pu ; *(t); u
(t)) 6 ’(t; pu ; *(t); *
(t))

for all t ∈ [t0 ; t0 + 7];

and hence, (* − u)
(t) ¿ 0 for all t ∈ [t0 ; t0 + 7], in contradiction with the choice of t0 . Step 2: Every solution of (P∗K ) ful9lls the boundary conditions. If u(b) − L2 (u(a); u(b)) ¡ *(b) the deHnition of B gives us that u(b)=*(b). Now using (H3∗ ) and the conclusion of Step 1, we arrive at a contradiction: *(b) ¿ *(b) − L2 (u(a); *(b)) ¿ *(b) − L2 (*(a); *(b)) = *(b): Analogously, if u(b)−L2 (u(a); u(b)) ¿ +(b) we have u(b)=+(b) and a contradiction is reached similarly. Now *(b) 6 u(b) − L2 (u(a); u(b)) 6 +(b) and hence L2 (u(a); u(b)) = 0. To prove that L1 (u(a); u(b); u
(a); u
(b); u) = 0 it is enough to show that *(a) 6 u(a) + L1 (u(a); u(b); u
(a); u
(b); u) 6 +(a): If u(a) + L1 (u(a); u(b); u
(a); u
(b); u) ¡ *(a) then u(a) = *(a) and then 0 = L2 (u(a); u(b)) = L2 (*(a); *(b)): Now, since L2 (*(a); ·) is injective and by the deHnition of lower solution, we have that u(b) = *(b). As a consequence, u − * is non-negative on I and attains its minimum in a and b, thus u
(b) 6 D− *(b) and u
(a) ¿ D+ *(a). Using the deHnition of lower solution and the properties of L1 we obtain a contradiction: *(a) ¿ *(a) + L1 (*(a); *(b); u
(a); u
(b); u) ¿ *(a) + L1 (*(a); *(b); D+ *(a); D− *(b); *) ¿ *(a):

A. Cabada, R.L. Pouso / Nonlinear Analysis 52 (2003) 557 – 572

569

It can be analogously proven that u(a) + L1 (u(a); u(b); u
(a); u
(b); u) 6 +(a) and so the result is established. Step 3: Every solution u of (P∗K ) is such that |u
(t)| 6 K for all t ∈ I . Let u ∈ C 1 (I ) be a solution of (P∗K ). By the mean value theorem, there exists t0 ∈ (a; b) such that u
(t0 ) =

u(b) − u(a) ; b−a

and, since u ∈ [*; +], we have −K ¡ − / 6 u
(t0 ) 6 / ¡ K: Let us call /0 = |u
(t0 )|. Suppose that there exists a point in the interval I for which u
¿ K or u
6 − K. By the continuity of u
we can choose t1 ∈ I verifying one of the following situations: (a) (b) (c) (d)

u
(t0 ) = /0 , u
(t1 ) = K and /0 6 u
(t) ¡ K for all t ∈ (t0 ; t1 ), u
(t1 ) = K, u
(t0 ) = /0 and /0 6 u
(t) ¡ K for all t ∈ (t1 ; t0 ), u
(t0 ) = −/0 , u
(t1 ) = −K and −K ¡ u
(t) 6 − /0 for all t ∈ (t0 ; t1 ), u
(t1 ) = −K, u
(t0 ) = −/0 and −K ¡ u
(t) 6 − /0 for all t ∈ (t1 ; t0 ).

Assume that situation (a) holds. Since −K ¡ /0 6 u
(t) ¡ K for all t ∈ (t0 ; t1 ) and u ∈ [*; +], we have −

d ’(t; u; u(t); u
(t)) = f(t; u; u(t); u
(t)) dt

for a:e: t ∈ (t0 ; t1 );

so, by the Nagumo condition, d ’(t; u; u(t); u
(t)) = |f(t; u; u(t); u
(t))| dt 6 k(t).(|’(t; u; u(t); u
(t))|)

(3.3)

for a.e. t ∈ (t0 ; t1 ). As 0 6 /0 6 /, by (H1∗ )(b), we have that ’(t; ; x; /) ¿ ’(t; ; x; /0 ) ¿ 0 hence



inf I ×[*;+]×J*;+ ’(t; ;x;K)

supI ×[*;+]×J

*;+

 6

’(t; ;x;/)

*;+

 6

z (q−1)=q dz .(z)

inf I ×[*;+]×J*;+ ’(t; ;x;K)

supI ×[*;+]×J

’(t; ;x;/0 )

’(t1 ;u;u(t1 );K)

’(t0 ;u;u(t0 );/0 )

for all (t; ; x) ∈ I × C(I ) × R;

z (q−1)=q dz .(z)

z (q−1)=q d z: .(z)

(3.4)

570

A. Cabada, R.L. Pouso / Nonlinear Analysis 52 (2003) 557 – 572

By the theorem of change of variable, see [13], and (3.3) we have  ’(t1 ;u;u(t1 );K) (q−1)=q z dz .(z) ’(t0 ;u;u(t0 );/0 )  =

t1

t0

 6

t1

t0

(’(t; u; u(t); u
(t)))(q−1)=q (’(t; u; u(t); u
(t)))
dt .(’(t; u; u(t); u
(t))) (’(t; u; u(t); u
(t)))(q−1)=q k(t) dt;

and, by HQolder’s inequality,   t1 (q−1)=q  ’(t1 ;u;u(t1 );K) (q−1)=q z d z 6 k q ’(t; u; u(t); u
(t)) dt : .(z) ’(t0 ;u;u(t0 );/0 ) t0

(3.5)

Integrating in the diGerential equation we obtain  t1 ’(t; u; u(t); u
(t)) dt t0

 =

t1



 ’(t0 ; u; u(t0 ); /0 ) −

t0

6 (’(t0 ; u; u(t0 ); /0 ) +

t

t0




f(s; u; u(s); u (s)) ds

K 1 )(b

 dt

− a) 6 21 :

(3.6)

Joining relations (3.4) – (3.6) we conclude that  inf I ×[*;+]×J ’(t; ;x;K) (q−1)=q *;+ z d z 6 k q 21(q−1)=q ; .(z) supI ×[*;+]×J ’(t; ;x;/) *;+

in contradiction with part (ii) of the Nagumo condition (see DeHnition 3.2). The remaining possibilities can be treated in the very same way (with the only diGerence that in cases (c) and (d) one reaches a contradiction with part (iii) of the Nagumo condition). Taking into account the results of the previous steps and the deHnition of (P∗K ) we conclude that problem (P∗ ) has at least one solution between * and +. Remark 3.4. Note that condition (H1∗ )(b) is only used in Step 3; to deal with the Nagumo condition. Thus; when the right-hand side of the diGerential equation does not depend on u
(t) and; hence; a Nagumo condition is not necessary; the result of Theorem 3.1 is valid replacing (H1∗ ) by (H1∗ )(a). 4. An example Let si ∈ [0; 1], i = 1; : : : ; 5 be Hxed (s4 6 s5 ). Consider the problem u2 (t) u3 (1 − t)  d 

maxs∈I1 (t) |u(s)| u (t)|u (t)| − = u4 (r) dr + u
(t); + + dt 2 4 I2 (t) t ∈ I = [0; 1];

A. Cabada, R.L. Pouso / Nonlinear Analysis 52 (2003) 557 – 572

 u(0) = 1=2 + min u(t) + t∈[s3 ;1]

s5

s4

u3 (r) dr;

571

u(1) = 0;

where I1 (t) = [t − s1 ; t] ∩ I , I2 (t) = [t; t + s2 ] ∩ I for each t ∈ [0; 1]. Note that we are considering here a kind of “integro-diGerential equation with maximum”. First of all, note that the above diGerential equation can be rewritten as follows: u2 (t) u3 (1 − t)  d  u(t) + u
(t)|u
(t)|maxs∈I1 (t) |u(s)| = u4 (r) dr; t ∈ I; − + + dt 2 4 I2 (t) hence, to Ht our previous notation, we deHne ’(t; ; x; y) = x + y|y|maxs∈I1 (t) | (s)| f(t; ; x; y) =

3 (1 − t) x2 + + 2 4

for (t; ; x; y) ∈ I × C(I ) × R2 ;

 I2 (t)

4 (r) dr 

L1 (x; y; z; w; ) = 1=2 − x + min (t) + t∈[s3 ;1]

s5

s4

for (t; ; x; y) ∈ I × C(I ) × R2 ;

3 (r) dr;

for (x; y; z; w; ) ∈ R4 × C(I ) and L2 (x; y) = y

for (x; y) ∈ R2 :

Since *(t) = 0 and +(t) = 1 − t, t ∈ I , are, respectively, a lower solution and an upper one for this problem, with * 6 + in I , Theorem 3.1 (see Remark 3.4) ensures the existence of at least one solution for it. References [1] A. Adje, Sur et sous-solutions gMenMeralisMees et problSemes aux limites du second ordre, Bull. Soc. Math. Bel. SMer. B 42 (1990) 347–368. [2] A. Cabada, P. Habets, R.L. Pouso, Optimal existence conditions for -Laplacian equations with upper and lower solutions in the reversed order, J. DiGerential Equations 166 (2) (2000) 385–401. [3] A. Cabada, R.L. Pouso, Existence result for the problem ((u
))
= f(t; u; u
) with nonlinear boundary conditions, Nonlinear Anal. 35 (1999) 221–231. [4] A. Cabada, R.L. Pouso, Extremal solutions of strongly nonlinear discontinuous second order equations with nonlinear functional boundary conditions, Nonlinear Anal. 42 (8) (2000) 1377–1396. [5] M. Cherpion, C. De Coster, P. Habets, Monotone Iterative methods for boundary value problems, DiGerential Integral Equations 12 (3) (1999) 309–338. [6] C. De Coster, Pairs of positive solutions for the one-dimensional p-Laplacian, Nonlinear Anal. 23 (1994) 669–681. [7] M. Del Pino, M. Elgueta, R. ManMasevich, A homotopic deformation along p of a Leray–Schauder degree result and existence for (|u
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[10] N.G. Lloyd, Degree Theory, Cambridge University Press, Cambridge, 1978. [11] R. ManMasevich, J. Mawhin, Periodic solutions for nonlinear systems with p-Laplacian-like operators, J. DiGerential Equations 145 (2) (1998) 367–393. [12] R. ManMasevich, F. Zanolin, Time-mappings and multiplicity of solutions for the one-dimensional p-Laplacian, Nonlinear Anal. 21 (1993) 269–291. [13] E.J. McShane, Integration, Princeton University Press, Princeton, 1967. [14] D. O’Regan, Some general principles and results for ((y
))
= qf(t; y; y
), 0 ¡ t ¡ 1, SIAM J. Math. Anal. 24 (1993) 648–668. [15] D. O’Regan, Existence theory for ((y
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= qf(t; y; y
), 0 ¡ t ¡ 1, Comm. Appl. Anal. 1 (1997) 33–52. [16] M.X. Wang, A. Cabada, J.J. Nieto, Monotone method for nonlinear second order periodic boundary value problems with CarathMeodory functions, Ann. Polon. Math. 58 (1993) 221–235. [17] J. Wang, W. Gao, Existence of solutions to boundary value problems for a nonlinear second order equation with weak CarathMeodory functions, DiGerential Equations Dynamics Systems 5 (2) (1997) 175–185. [18] J. Wang, W. Gao, Z. Lin, Boundary value problems for general second order equations and similarity solutions to the Rayleigh problem, Tˆohoku Math. J. 47 (1995) 327–344.