Positive solutions to semi-positone second-order three ... - CiteSeerX
Applied Mathematics and Computation 215 (2010) 3713–3720
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Positive solutions to semi-positone second-order three-point problems on time scales Douglas R. Anderson a,b,*, Chengbo Zhai c a
Department of Mathematics and Computer Science, Concordia College, Moorhead, MN 56562, USA Visiting Fellow, School of Mathematics, The University of New South Wales, Sydney 2052, Australia c School of Mathematical Sciences, Shanxi University, Taiyuan, 030006 Shanxi, PR China b
a r t i c l e
i n f o
Keywords: Fixed-point theorems Time scales Dynamic equations Cone Semipositone Three-point problem
a b s t r a c t Using a fixed point theorem of generalized cone expansion and compression we establish the existence of at least two positive solutions for the nonlinear semi-positone three-point boundary value problem on time scales
uDr ðtÞ þ kf ðt; uðtÞÞ ¼ 0;
uðaÞ ¼ 0;
auðgÞ ¼ uðTÞ:
Here t 2 ½a; TT , where T is a time scale, a > 0; g 2 ða; qðTÞÞT ; aðg aÞ < T a, and the parameter k > 0 belongs to a certain interval. These results are new for difference equations as well as for general time scales. An example is provided for differential, difference, and q-difference equations. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction to the boundary value problem We will be concerned with proving the existence of positive solutions to the semi-positone second-order three-point nonlinear boundary value problem on a time scale T given by
uDr ðtÞ þ kf ðt; uðtÞÞ ¼ 0; t 2 ða; TÞT ; uðaÞ ¼ 0; auðgÞ ¼ uðTÞ;
ð1:1Þ ð1:2Þ
where D is the delta derivative and r is the nabla derivative. Throughout the paper we assume g 2 ða; qðTÞÞT for a 2 Tj ; T 2 Tj ; a > 0, and aðg aÞ < T a. We likewise assume that f : ½a; TT ½0; 1Þ ! R is continuous, and f ðt; Þ does not vanish identically on any subset of ½a; TT . By a positive solution of (1.1) and (1.2) we understand a function u which is positive on ða; TÞT and satisfies dynamic equation (1.1) and boundary conditions (1.2). For more on time scales and the time-scale calculus, please see the book by Bohner and Peterson [7]. Eq. (1.1) is a dynamic equation on time scales, dynamic in the sense that the specific choice of time scale determines the interpretation of the delta and nabla derivatives. For example, we have the following versions of Eq. (1.1):
T ¼ R ðdifferential equationsÞ :
u00 ðtÞ þ kf ðt; uðtÞÞ ¼ 0;
T ¼ Z ðdifference equationsÞ :
2
T ¼ qZ ðquantum equationsÞ :
t 2 ða; TÞR ;
D uðt 1Þ þ kf ðt; uðtÞÞ ¼ 0; t 2 ða; TÞZ ; Dq Dq u ðtÞ þ kf ðt; uðtÞÞ ¼ 0; t 2 ða; TÞq ;
* Corresponding author. Address: Department of Mathematics and Computer Science, Concordia College, Moorhead, MN 56562, USA. E-mail addresses: [email protected] (D.R. Anderson), [email protected], [email protected] (C. Zhai). URL: http://www.cord.edu/faculty/andersod/ (D.R. Anderson). 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.11.010
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where DxðtÞ ¼ xðt þ 1Þ xðtÞ with D2 xðtÞ ¼ DðDxðtÞÞ, while for q > 1 we have
xðtÞ xðt=qÞ t t=q
Dq xðtÞ ¼
and Dq xðtÞ ¼
xðqtÞ xðtÞ : tðq 1Þ
Boundary value problem (1.1) and (1.2) was studied in the differential equations case recently [15]; the aim of this sequel is to extend these results to difference equations, quantum equations, and arbitrary dynamic equations on time scales. Other recent work on second-order boundary value problems on time scales include [2–5,8–11]. Earlier papers on second-order three-point problems in the continuous case include [12–14]. The nabla derivative was introduced in [6]. We now present a fixed point theorem of generalized cone expansion and compression which will be used in the later proofs. Let E be a real Banach space and P be a cone in E, and let h denote the null element. The map c : P ! R1 is said to be a convex functional on P provided that cðtx þ ð1 tÞyÞ 6 tcðxÞ þ ð1 tÞcðyÞ for all x; y 2 P and t 2 ½0; 1. See [16] for further information. Theorem 1.1 (See [16]). Let X1 ; X2 be two open bounded subsets in E with h 2 X1 and X1 X2 . Suppose that A : P \ ðX2 n X1 Þ ! P is completely continuous and c : P ! ½0; 1Þ is a uniformly continuous convex functional with cðhÞ ¼ 0 and cðxÞ > 0 for x – h. If one of the two conditions (i) cðAxÞ 6 cðxÞ for all x 2 P \ @ X1 and inf x2P\@ X2 cðxÞ > 0; cðAxÞ P cðxÞ for all x 2 P \ @ X2 , and (ii) inf x2P\@ X1 cðxÞ > 0; cðAxÞ P cðxÞ for all x 2 P \ @ X1 and cðAxÞ 6 cðxÞ for all x 2 P \ @ X2 is satisfied, then A has at least one fixed point in P \ ðX2 n X1 Þ. The paper is organized as follows. In Section 2, we give some preliminary results that will be used in the proof of the main result. In Section 3, we prove the existence of at least two positive solutions for problem (1.1) and (1.2). In the end, we illustrate a simple use of the main result. 2. Foundational lemmas To prove the main existence result we will employ several straightforward lemmas. These lemmas are based on the boundary value problem
uDr ðtÞ þ yðtÞ ¼ 0;
t 2 ða; TÞT ;
ð2:1Þ
auðgÞ ¼ uðTÞ:
uðaÞ ¼ 0;
ð2:2Þ
Lemma 2.1. If aðg aÞ – T a, then for y 2 C ld ½a; TT the boundary value problem (2.1) and (2.2) has the unique solution
uðtÞ ¼
Z
t
ðt sÞyðsÞrs
aðt aÞ d
a
Z
g
ðg sÞyðsÞrs þ a
ta d
Z
T
ðT sÞyðsÞrs;
ð2:3Þ
a
where d :¼ T a aðg aÞ – 0. Proof. Let u be as in (2.3). Routine calculations verify that u satisfies the boundary conditions in (2.2). By [7, Theorem 8.50 (iii)],
Z
t
D Z t f ðt; sÞrs ¼ f ðrðtÞ; rðtÞÞ þ f D ðt; sÞrs
a
if f ; f
D
a
are continuous. Using this theorem to take the delta derivative of (2.3) we have
uD ðtÞ ¼
Z
t
yðsÞrs
a
a d
Z
g
ðg sÞyðsÞrs þ
a
1 d
Z
T
ðT sÞyðsÞrs:
a
Taking the nabla derivative of this expression yields uDr ðtÞ ¼ yðtÞ, so that u given in (2.3) is a solution of (2.1) and (2.2). The rest of the proof is similar to [2, Lemma 2]. h Lemma 2.2. If uðaÞ ¼ 0 and uDr 6 0, then
uðTÞ Ta
6 uðtÞ for all t 2 ða; TT . ta Dr
. Then hðaÞ ¼ hðTÞ ¼ 0 and h Proof. Let hðtÞ :¼ uðtÞ ðtaÞuðTÞ Ta
6 0 so that hðtÞ P 0 on ½a; TT . h
Lemma 2.3. Let 0 < a < Ta ga. If y 2 C ld ½a; TT and y P 0, the unique solution u of (2.1) and (2.2) satisfies
uðtÞ P 0;
t 2 ½a; TT :
Proof. From the fact that uDr ðtÞ ¼ yðtÞ 6 0, we know that the graph of u is concave down on ða; TÞT . If uðTÞ P 0, then the concavity of u and the boundary condition uðaÞ ¼ 0 imply that uðtÞ P 0 for t 2 ½a; TT . If uðTÞ < 0, then we have uðgÞ < 0 and
D.R. Anderson, C. Zhai / Applied Mathematics and Computation 215 (2010) 3713–3720
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uðTÞ auðgÞ uðgÞ ¼ > ; T a T a ga a contradiction of Lemma 2.2. h Lemma 2.4. Let 0 < a < Ta ga. If y 2 C ld ½a; TT and y P 0, then the unique solution u as in (2.3) of (2.1) and (2.2) satisfies
inf uðtÞ P rkuk;
ð2:4Þ
t2½g;TT
where
r :¼ min
aðT gÞ aðg aÞ g a ;
d
T a
;
T a
> 0;
ð2:5Þ
where d :¼ ðT aÞ aðg aÞ > 0. Proof. First consider the case where 0 < a < 1. By the second boundary condition we know that uðgÞ P uðTÞ. Pick t 0 2 ða; TÞT such that uðt0 Þ ¼ kuk. If t 0 6 g < T, then
min uðtÞ ¼ uðTÞ
t2½g;TT
and
uðt0 Þ 6 uðTÞ þ
uðTÞ uðgÞ duðTÞ ða TÞ ¼ : T g aðT gÞ
Therefore
aðT gÞ
min uðtÞ P
d
t2½g;TT
kuk:
gÞ uðt 0 Þ If g 6 t0 < T, again we have uðTÞ ¼ mint2½g;TT uðtÞ. As in Lemma 2.2, uð ga P t0 a. Using the boundary condition auðgÞ ¼ uðTÞ, we find that
uðTÞ >
aðg aÞ T a
uðt 0 Þ;
so that
aðg aÞ
min uðtÞ >
T a
t2½g;TT
kuk:
Now consider the case 1 6 a < Ta ga. The boundary condition this time implies uðgÞ 6 uðTÞ. Set uðt 0 Þ ¼ kuk. Note that by the gÞ uðt 0 Þ concavity of u we have t 0 2 ½g; TT and mint2½g;TT uðtÞ ¼ uðgÞ. Once again by Lemma 2.2 it follows that uð ga P t0 a, so that
min uðtÞ P
t2½g;TT
ga T a
kuk:
The proof is complete. h Lemma 2.5. Let 0 < a < Ta ga. If y 2 C ld ½a; TT and y P 0, then the unique solution u as in (2.3) of (2.1) and (2.2) satisfies
uðtÞ P
rðt aÞ kuk; ga
t 2 ½a; gT ;
ð2:6Þ
where r is given by (2.5). Proof. The result follows from Lemmas 2.2 and 2.4. h ^2 , respectively, via Lemma 2.6. If we define the delta and nabla polynomials h2 and h
h2 ðt; aÞ ¼
Z
t
ðs aÞDs;
a
^ ðt; aÞ ¼ h 2
Z
t
ðs aÞrs;
ðt; aÞ 2 T T;
ð2:7Þ
a
then the equation
h2 ðt; aÞ ¼
Z
t
ðt sÞrs
a
holds for ðt; aÞ 2 T T.
ð2:8Þ
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D.R. Anderson, C. Zhai / Applied Mathematics and Computation 215 (2010) 3713–3720
Proof. Following Bohner and Peterson [7, Section 1.6] and Anderson [1, Section 2], define the delta and nabla polynomials h2 ^2 , respectively, as above in (2.7). Thus, by [1, Theorem 9], and h
^2 ða; tÞ ¼ h2 ðt; aÞ ¼ h
Z
a
ðs tÞrs ¼
Z
t
The proof is complete.
t
ðt sÞrs: a
h
As examples of Lemma 2.6, we have
1 ðt aÞ2 ; 2 1 T ¼ hZ ðdifference equationsÞ : h2 ðt; aÞ ¼ ðt aÞðt a hÞ; h > 0; 2 1 Z T ¼ q ðquantum equationsÞ : h2 ðt; aÞ ¼ ðt aÞðt qaÞ; q > 1; 1þq
pffiffiffiffiffi pffiffiffi pffiffi 1 pffiffi pffiffiffi pffiffi pffiffiffi T ¼ N2 ðsquared differencesÞ : h2 ðt; aÞ ¼ ð t aÞð t a 1Þ 3t þ 3a þ 6 at þ a t 1 : 6
T ¼ R ðdifferential equationsÞ :
h2 ðt; aÞ ¼
Lemma 2.7. If the function w is given by
wðtÞ ¼ h2 ðt; aÞ þ
ta ½h2 ðT; aÞ ah2 ðg; aÞ; d
ð2:9Þ
where d ¼ ðT aÞ aðg aÞ > 0 and h2 ðt; aÞ is from (2.8) , then we have the following conclusions: (i) wðaÞ ¼ 0; awðgÞ ¼ wðTÞ; wDr ðtÞ 1; ½h2 ðT; aÞ ah2 ðg; aÞ 6 Ta ½h2 ðT; aÞ ah2 ðg; aÞ for all t 2 ½a; TT . (ii) wðtÞ 6 ta d d
Proof. From Lemmas 2.1 and 2.3 we have that w P 0 on ½a; TT and
wðtÞ ¼
Z
t
ðt sÞrs
a
aðt aÞ
Z
d
g
ðg sÞrs þ
a
ta d
Z
T
ðT sÞrs; a
which is (2.9). Part (ii) follows from the fact that h2 ðt; aÞ 6 0 for t 2 ½a; TT . h 3. Existence of two positive solutions We will employ Theorem 1.1 to establish the existence of at least two positive solutions for the second-order three-point boundary value problem (1.1) and (1.2). We will assume the following conditions. (C1) There exists a constant M > 0 such that f ðt; uÞ P M for ðt; uÞ 2 ½a; TT ½0; 1Þ. (C2) There exist two real constants b; c 2 ð0; 1Þ such that
0 < f ðt; uÞ 6 b for ðt; uÞ 2 ½a; TT ½0; c: (C3) or L :¼ minf2r ; cg and M 1 :¼ maxff ðt; uÞ þ M : ðt; uÞ 2 ½a; TT ½0; 2g, the parameter k satisfies
0 < k 6 s :¼
rd r 2 L ; ; min ; ðT aÞ½h2 ðT; aÞ ah2 ðg; aÞ M M1 M2
where r is given in (2.5), and M 2 :¼ maxff ðt; uÞ : ðt; uÞ 2 ½a; TT ½0; Lg. (C4) There exists R > 2 such that f ðt; uÞ þ M P Nu for t 2 ½g; TT and u P 12 Rr 2 , where
NP
2d krðg aÞh2 ðT; gÞ
for fixed k 2 ð0; s:
Remark 3.1. It follows from ðC 2 Þ and the continuity of f that
lim
u!0þ
f ðt; uÞ ¼ 1 uniformly on ½a; TT : u
Theorem 3.2. Suppose (C1)–(C4) hold. Then problem (1.1) and (1.2) has at least two positive solutions u1 and u2 , where ku1 k P r and ku2 k 6 L 6 r=2.
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D.R. Anderson, C. Zhai / Applied Mathematics and Computation 215 (2010) 3713–3720
Proof. Define the cone
P¼
u : u 2 C½a; TT ; uðtÞ P 0; t 2 ½a; TT ; min uðtÞ P rkuk t2½g;TT
and let z ¼ kMw, where r and w are given via (2.5) and (2.9), respectively. It is straightforward to see that (1.1) and (1.2) has a ~ ¼ u þ z is a solution of positive solution u if and only if u
~ zÞ ¼ 0; ~ Dr ðtÞ þ kgðt; u u ~ ðTÞ ~ ðaÞ ¼ 0; au ~ðgÞ ¼ u u
t 2 ða; TÞT ;
ð3:1Þ ð3:2Þ
~ > z on ða; TÞT , where g : ½a; TT R ! Rþ is defined by with u
gðt; uÞ ¼
f ðt; uÞ þ M : ðt; uÞ 2 ½a; TT ½0; 1Þ; f ðt; 0Þ þ M : ðt; uÞ 2 ½a; TT ð1; 0Þ:
For u 2 P, denote by Au the unique solution of (3.1) and (3.2), so that by Lemma 2.1 we have
AuðtÞ ¼
Z
t
ðt sÞkgðs; uðsÞ zðsÞÞrs
aðt aÞ
a
Z
d
g
ðg sÞkgðs; uðsÞ zðsÞÞrs þ a
ta d
Z
T
ðT sÞkgðs; uðsÞ zðsÞÞrs:
a
By Lemmas 2.3 and 2.4 we see that AðPÞ P; moreover, A is completely continuous by an application of the Arzela–Ascoli theorem. Let the uniformly continuous convex functional w : P ! ½0; 1Þ be defined by
wðuÞ ¼ max uðtÞ; t2½g;TT
u 2 P:
ð3:3Þ
Then wð0Þ ¼ 0 and wðuÞ > 0 for u – 0. Define the sets
X1 ¼ fu 2 C½a; TT : wðuÞ < 2rg and X2 ¼ fu 2 C½a; TT : wðuÞ < Rrg: Clearly X1 and X2 are bounded open sets in C½a; TT with 0 2 X1 and X1 X2 . If u 2 P \ X1 , then
kuk 6
1 1 1 min uðtÞ 6 max uðtÞ ¼ wðuÞ < 2; r t2½g;TT r t2½g;TT r
which implies that P \ X1 is bounded; similarly, P \ X2 is also bounded. If u 2 P \ @ X1 , then wðuÞ ¼ 2r, and thus kuk 6 2. Then we have
Z t Z aðt aÞ g ðt sÞkgðs; uðsÞ zðsÞÞrs ðg sÞkgðs; uðsÞ zðsÞÞrs t2½a;TT d a a Z T ta þ ðT sÞkgðs; uðsÞ zðsÞÞrs d a Z t Z Z aðt aÞ g ta T 6 max ðt sÞrs ðg sÞrs þ ðT sÞrs kM 1 ¼ kM 1 max wðtÞ t2½a;TT t2½a;TT d d a a a ðC 3 Þ T a 6 kM 1 ½h2 ðT; aÞ ah2 ðg; aÞ 6 2r ¼ wðuÞ: d
wðAuÞ 6 kAuk ¼ max
This shows that wðAuÞ 6 wðuÞ for all u 2 P \ @ X1 . If u 2 P \ X2 , then wðuÞ ¼ rR, so that rR 6 kuk 6 R. Consequently, we can see that inf u2P\@ X2 wðuÞ > 0, and for u 2 P \ @ X2 we have
zðsÞ ¼ kMwðsÞ 6 kM
ðC 3 Þ T a uðsÞ 1 ½h2 ðT; aÞ ah2 ðg; aÞ 6 r2 6 r 6 uðsÞ d kuk R
for s 2 ½g; TT . Therefore
1 uðsÞ; uðsÞ zðsÞ P 1 R
s 2 ½g; TT :
ð3:4Þ
Considering (3.4) and Lemma 2.4, we conclude that
uðsÞ zðsÞ P
1 1 1 uðsÞ P rkuk P Rr2 ; 2 2 2
s 2 ½g; TT :
This together with (C4) implies that
gðs; u zÞ ¼ f ðs; u zÞ þ M P Nðu zÞ P
1 2 Rr N; 2
s 2 ½g; TT :
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D.R. Anderson, C. Zhai / Applied Mathematics and Computation 215 (2010) 3713–3720
Moreover, we also have from (C4) that
wðAuÞ ¼ max AuðtÞ P AuðgÞ t2½g;TT
¼
Z
g
ðg sÞkgðs; uðsÞ zðsÞÞrs
aðg aÞ
Z
d
a
g
ðg sÞkgðs; uðsÞ zðsÞÞrs þ
a
ga d
Z
T
ðT sÞkgðs; uðsÞ a
zðsÞÞrs Z g Z ga T ðg sÞkgðs; uðsÞ zðsÞÞrs þ ðT sÞkgðs; uðsÞ zðsÞÞrs d a a Z T Z Z g T g ga ga T 1 ðs aÞkgðs; uðsÞ zðsÞÞrs þ ðT sÞkgðs; uðsÞ zðsÞÞrs P ðT sÞk Rr2 Nrs ¼ d d 2 d a g g T a ¼ d
¼
ga 2d
ðC 4 Þ
kRr2 Nh2 ðT; gÞ P Rr ¼ wðuÞ:
Thus, wðAuÞ P wðuÞ for all u 2 P \ X2 . It then follows from the first part of Theorem 1.1 that A has a fixed point ~ 2 P \ ðX2 n X1 Þ such that u
~Þ 6 rR; 2r 6 wðu
~ k 6 R: and thus 2r 6 ku
ð3:5Þ
Moreover, by combining (3.5) with (C3), and using Lemmas 2.4 and 2.7, we have that
~ ðtÞ P rku ~ k P 2r 2 P u
2kM ðT aÞ½h2 ðT; aÞ ah2 ðg; aÞ P 2kMwðtÞ; d
t 2 ½g; TT :
ð3:6Þ
In addition, for t 2 ½a; gT , by Lemma 2.5 and (C3) we have
~ ðtÞ P u
rðt aÞ 2r 2 ðt aÞ 2kMðt aÞ ~k P ku P ðT aÞ½h2 ðT; aÞ ah2 ðg; aÞ P 2kMwðtÞ: dðg aÞ ga ga
ð3:7Þ
As a consequence of (3.6) and (3.7) we see that
~ ðtÞ P 2kMwðtÞ ¼ 2zðtÞ; u
t 2 ½a; TT :
ð3:8Þ
~ z is a positive solution of (1.1) and (1.2). In addition, from (3.5) and (3.8) it follows that Hence, u1 ¼ u
ku1 k P
1 ~ k P r: ku 2
ð3:9Þ
To find the second positive solution of (1.1) and (1.2), we begin by setting
f ðt; uÞ ¼
f ðt; uÞ : ðt; uÞ 2 ½a; TT ½0; c;
ð3:10Þ
f ðt; cÞ : ðt; uÞ 2 ½a; TT ½c; 1Þ:
By (C2) we then have that 0 < f ðt; uÞ 6 b for ðt; uÞ 2 ½a; TT ½0; 1Þ. If we consider the auxiliary dynamic equation
uDr ðtÞ þ kf ðt; uðtÞÞ ¼ 0;
t 2 ða; TÞT
ð3:11Þ
with familiar boundary conditions
auðgÞ ¼ uðTÞ;
uðaÞ ¼ 0;
ð3:12Þ
we know that solutions to the problem (3.11) and (3.12) are equivalent to those of the operator equation u ¼ Fu, where
FuðtÞ ¼
Z
t
ðt sÞkf ðs; uðsÞÞrs
aðt aÞ
a
d
Z
g
ðg sÞkf ðs; uðsÞÞrs þ
a
ta d
Z
T
ðT sÞkf ðs; uðsÞÞrs: a
It is straightforward that F : P ! P is completely continuous and f ðPÞ P. Using Remark 3.1 we have that
limþ
u!0
f ðt; uÞ ¼ 1 uniformly on ½a; TT : u
Thus there exists a constant 0 < ‘ < L for the L is (C3) such that f ðt; uÞ P bu for ðt; uÞ 2 ½a; TT ½0; ‘, where b > 0 is chosen so that
bkr
aðg aÞh2 ðT; gÞ d
P 1:
ð3:13Þ
Choose
X3 ¼ fu 2 C½a; TT : wðuÞ < Lrg;
X4 ¼ fu 2 C½a; TT : wðuÞ < ‘rg:
Then P \ X3 and P \ X4 are bounded open sets in C½a; TT with inf u2P\@ X4 wðuÞ > 0.
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D.R. Anderson, C. Zhai / Applied Mathematics and Computation 215 (2010) 3713–3720
If u 2 P \ @ X3 , then from Lemmas 2.3 and 2.7 we have that
wðFuÞ ¼ max FuðtÞ 6 max FuðtÞ t2½g;TT
t2½a;TT
Z t Z Z aðt aÞ g ta T ðg sÞkf ðs; uðsÞÞrs þ ðT sÞkf ðs; uðsÞÞrs ¼ max ðt sÞkf ðs; uðsÞÞrs t2½a;TT d d a a a Z t Z g Z T aðt aÞ ta 6 max ðt sÞrs ðg sÞrs þ ðT sÞrs kM 2 ¼ kM 2 max wðtÞ t2½a;TT t2½a;TT d d a a a 6 kM 2
ðC 3 Þ T a ½h2 ðT; aÞ ah2 ðg; aÞ 6 Lr ¼ wðuÞ: d
This shows that wðFuÞ 6 wðuÞ for all u 2 P \ @ X3 . If u 2 P \ @ X4 , then r‘ 6 kuk 6 ‘, and we have
wðFuÞ ¼ max FuðtÞ P FuðTÞ t2½g;TT
¼
Z
T
ðT sÞkf ðs; uðsÞÞrs
a
aðg aÞ
¼
d
aðT gÞ
¼
d
P
Z
aðT aÞ
Z
d
T
ðT sÞkf ðs; uðsÞÞrs g
ðs aÞkf ðs; uðsÞÞrs þ
ðg sÞkf ðs; uðsÞÞrs þ
a
aðT aÞ d
a
Z
g
aðg aÞ
a
d
Z
T a d
Z
T
ðT sÞkf ðs; uðsÞÞrs
a
g
ðg sÞkf ðs; uðsÞÞrs a
Z
T
ðT sÞkf ðs; uðsÞÞrs P
aðg aÞ
g
d
Z
T
ðT sÞkf ðs; uðsÞÞrs
g
ð3:13Þ kbaðg aÞrkuk h2 ðT; gÞ P r‘ ¼ wðuÞ: d
Thus, wðFuÞ P wðuÞ for all u 2 P \ X4 . It then follows from the second part of Theorem 1.1 that the problem given in (3.11) and (3.12) has a positive solution u2 satisfying r‘ 6 wðu2 Þ 6 Lr. Consequently, ku2 k 6 L 6 r=2. In light of (C3) and (3.10) we conclude that u2 is also a solution of the original problem 1.1 and 1.2. From (C3) and (3.9) we then have that the problem (1.1) and (1.2) has two distinct positive solutions u1 and u2 . h Corollary 3.3. Suppose ðC 1 Þ and ðC 4 Þ hold. If
0 2 such that f ðt; uÞ þ M P Nu for all t 2 ½4; 128T and Since limu!1 e1 ðt;1Þu u 2 9 R, where u P 12 Rr ¼ 32258
N¼
30734 9kh2 ð128; 1Þ
for fixed k 2 ð0; s. As all of the conditions of Theorem 3.2 are satisfied, problem (4.1), (4.2) has at least two positive solutions u1 and u2 with 3 3 and ku2 k 6 254 . Note that 1; 4; 128 2 T for the following three key time scales: ku1 k P 127
1 ðt 1Þ2 ; e1 ðt; 1Þ ¼ et1 ; 2 1 T ¼ Z ðdifference equationsÞ : h2 ðt; 1Þ ¼ ðt 1Þðt 2Þ; e1 ðt; 1Þ ¼ 2t1 ; 2 1þlog Y 2t 1 T ¼ 2Z ðquantum equationsÞ : h2 ðt; 1Þ ¼ ðt 1Þðt 2Þ; e1 ðt; 1Þ ¼ ð1 þ 2k Þ 3 k¼0 T ¼ R ðdifferential equationsÞ : h2 ðt; 1Þ ¼
with the convention that
Q1
k¼0 ð1
þ 2k Þ ¼ 1; this exponential for qZ is also known as a q-Pochhammer function. h
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
D.R. Anderson, Taylor polynomials for nabla dynamic equations on time scales, PanAmerican Math. J. 12 (4) (2002) 17–27. D.R. Anderson, Solutions to second-order three-point problems on time scales, J. Difference Equations Appl. 8 (8) (2002) 673–688. D.R. Anderson, Existence of solutions for nonlinear multi-point problems on time scales, Dynam. Syst. Appl. 15 (2006) 21–34. D.R. Anderson, R.Y. Ma, Second-order n-point eigenvalue problems on time scales, Adv. Difference Equ. (2006) 1–17. Article ID 5957. D.R. Anderson, P.J.Y. Wong, Positive solutions for second-order semipositone problems on time scales, Computers Math. Appl. 58 (2009) 281–291. F.M. Atici, G.Sh. Guseinov, On Green’s functions and positive solutions for boundary value problems on time scales, J. Comput. Appl. Math. 141 (2002) 75–99. M. Bohner, A. Peterson, Dynamic Equations on Time Scales, An Introduction with Applications, Birkhäuser, Boston, 2001. J.J. DaCunha, J.M. Davis, P.K. Singh, Existence results for singular three-point boundary value problems on time scales, J. Math. Anal. Appl. 295 (2004) 378–391. L.H. Erbe, A.C. Peterson, Positive solutions for a nonlinear differential equation on a measure chain, Math. Computer Modelling 32 (2000) 571–585. L.H. Erbe, A.C. Peterson, Green’s functions and comparison theorems for differential equations on measure chains, Dynam. Continuous, Discrete Impulsive Systems 6 (1999) 121–137. H. Luo, Positive solutions to singular multi-point dynamic eigenvalue problems with mixed derivatives, Nonlinear Anal. 70 (2009) 1679–1691. R.Y. Ma, Positive solutions of a nonlinear three-point boundary-value problem, Electronic J. Differential Eqs. 1999 (34) (1999) 1–8. R.Y. Ma, Multiplicity of positive solutions for second-order three-point boundary-value problems, Computers Math. Appl. 40 (2000) 193–204. R.Y. Ma, Positive solutions for second-order three-point boundary-value problems, Appl. Math. Letters 14 (2001) 1–5. C.B. Zhai, Positive solutions for semi-positone three-point boundary value problems, J. Comput. Appl. Math. 228 (2009) 279–286. G.W. Zhang, J.X. Sun, A generalization of the cone expansion and compression fixed point theorem and applications, Nonlinear Anal. 67 (2007) 579– 586.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Positive solutions to semi-positone second-order three-point problems on time scales Douglas R. Anderson a,b,*, Chengbo Zhai c a
Department of Mathematics and Computer Science, Concordia College, Moorhead, MN 56562, USA Visiting Fellow, School of Mathematics, The University of New South Wales, Sydney 2052, Australia c School of Mathematical Sciences, Shanxi University, Taiyuan, 030006 Shanxi, PR China b
a r t i c l e
i n f o
Keywords: Fixed-point theorems Time scales Dynamic equations Cone Semipositone Three-point problem
a b s t r a c t Using a fixed point theorem of generalized cone expansion and compression we establish the existence of at least two positive solutions for the nonlinear semi-positone three-point boundary value problem on time scales
uDr ðtÞ þ kf ðt; uðtÞÞ ¼ 0;
uðaÞ ¼ 0;
auðgÞ ¼ uðTÞ:
Here t 2 ½a; TT , where T is a time scale, a > 0; g 2 ða; qðTÞÞT ; aðg aÞ < T a, and the parameter k > 0 belongs to a certain interval. These results are new for difference equations as well as for general time scales. An example is provided for differential, difference, and q-difference equations. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction to the boundary value problem We will be concerned with proving the existence of positive solutions to the semi-positone second-order three-point nonlinear boundary value problem on a time scale T given by
uDr ðtÞ þ kf ðt; uðtÞÞ ¼ 0; t 2 ða; TÞT ; uðaÞ ¼ 0; auðgÞ ¼ uðTÞ;
ð1:1Þ ð1:2Þ
where D is the delta derivative and r is the nabla derivative. Throughout the paper we assume g 2 ða; qðTÞÞT for a 2 Tj ; T 2 Tj ; a > 0, and aðg aÞ < T a. We likewise assume that f : ½a; TT ½0; 1Þ ! R is continuous, and f ðt; Þ does not vanish identically on any subset of ½a; TT . By a positive solution of (1.1) and (1.2) we understand a function u which is positive on ða; TÞT and satisfies dynamic equation (1.1) and boundary conditions (1.2). For more on time scales and the time-scale calculus, please see the book by Bohner and Peterson [7]. Eq. (1.1) is a dynamic equation on time scales, dynamic in the sense that the specific choice of time scale determines the interpretation of the delta and nabla derivatives. For example, we have the following versions of Eq. (1.1):
T ¼ R ðdifferential equationsÞ :
u00 ðtÞ þ kf ðt; uðtÞÞ ¼ 0;
T ¼ Z ðdifference equationsÞ :
2
T ¼ qZ ðquantum equationsÞ :
t 2 ða; TÞR ;
D uðt 1Þ þ kf ðt; uðtÞÞ ¼ 0; t 2 ða; TÞZ ; Dq Dq u ðtÞ þ kf ðt; uðtÞÞ ¼ 0; t 2 ða; TÞq ;
* Corresponding author. Address: Department of Mathematics and Computer Science, Concordia College, Moorhead, MN 56562, USA. E-mail addresses: [email protected] (D.R. Anderson), [email protected], [email protected] (C. Zhai). URL: http://www.cord.edu/faculty/andersod/ (D.R. Anderson). 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.11.010
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D.R. Anderson, C. Zhai / Applied Mathematics and Computation 215 (2010) 3713–3720
where DxðtÞ ¼ xðt þ 1Þ xðtÞ with D2 xðtÞ ¼ DðDxðtÞÞ, while for q > 1 we have
xðtÞ xðt=qÞ t t=q
Dq xðtÞ ¼
and Dq xðtÞ ¼
xðqtÞ xðtÞ : tðq 1Þ
Boundary value problem (1.1) and (1.2) was studied in the differential equations case recently [15]; the aim of this sequel is to extend these results to difference equations, quantum equations, and arbitrary dynamic equations on time scales. Other recent work on second-order boundary value problems on time scales include [2–5,8–11]. Earlier papers on second-order three-point problems in the continuous case include [12–14]. The nabla derivative was introduced in [6]. We now present a fixed point theorem of generalized cone expansion and compression which will be used in the later proofs. Let E be a real Banach space and P be a cone in E, and let h denote the null element. The map c : P ! R1 is said to be a convex functional on P provided that cðtx þ ð1 tÞyÞ 6 tcðxÞ þ ð1 tÞcðyÞ for all x; y 2 P and t 2 ½0; 1. See [16] for further information. Theorem 1.1 (See [16]). Let X1 ; X2 be two open bounded subsets in E with h 2 X1 and X1 X2 . Suppose that A : P \ ðX2 n X1 Þ ! P is completely continuous and c : P ! ½0; 1Þ is a uniformly continuous convex functional with cðhÞ ¼ 0 and cðxÞ > 0 for x – h. If one of the two conditions (i) cðAxÞ 6 cðxÞ for all x 2 P \ @ X1 and inf x2P\@ X2 cðxÞ > 0; cðAxÞ P cðxÞ for all x 2 P \ @ X2 , and (ii) inf x2P\@ X1 cðxÞ > 0; cðAxÞ P cðxÞ for all x 2 P \ @ X1 and cðAxÞ 6 cðxÞ for all x 2 P \ @ X2 is satisfied, then A has at least one fixed point in P \ ðX2 n X1 Þ. The paper is organized as follows. In Section 2, we give some preliminary results that will be used in the proof of the main result. In Section 3, we prove the existence of at least two positive solutions for problem (1.1) and (1.2). In the end, we illustrate a simple use of the main result. 2. Foundational lemmas To prove the main existence result we will employ several straightforward lemmas. These lemmas are based on the boundary value problem
uDr ðtÞ þ yðtÞ ¼ 0;
t 2 ða; TÞT ;
ð2:1Þ
auðgÞ ¼ uðTÞ:
uðaÞ ¼ 0;
ð2:2Þ
Lemma 2.1. If aðg aÞ – T a, then for y 2 C ld ½a; TT the boundary value problem (2.1) and (2.2) has the unique solution
uðtÞ ¼
Z
t
ðt sÞyðsÞrs
aðt aÞ d
a
Z
g
ðg sÞyðsÞrs þ a
ta d
Z
T
ðT sÞyðsÞrs;
ð2:3Þ
a
where d :¼ T a aðg aÞ – 0. Proof. Let u be as in (2.3). Routine calculations verify that u satisfies the boundary conditions in (2.2). By [7, Theorem 8.50 (iii)],
Z
t
D Z t f ðt; sÞrs ¼ f ðrðtÞ; rðtÞÞ þ f D ðt; sÞrs
a
if f ; f
D
a
are continuous. Using this theorem to take the delta derivative of (2.3) we have
uD ðtÞ ¼
Z
t
yðsÞrs
a
a d
Z
g
ðg sÞyðsÞrs þ
a
1 d
Z
T
ðT sÞyðsÞrs:
a
Taking the nabla derivative of this expression yields uDr ðtÞ ¼ yðtÞ, so that u given in (2.3) is a solution of (2.1) and (2.2). The rest of the proof is similar to [2, Lemma 2]. h Lemma 2.2. If uðaÞ ¼ 0 and uDr 6 0, then
uðTÞ Ta
6 uðtÞ for all t 2 ða; TT . ta Dr
. Then hðaÞ ¼ hðTÞ ¼ 0 and h Proof. Let hðtÞ :¼ uðtÞ ðtaÞuðTÞ Ta
6 0 so that hðtÞ P 0 on ½a; TT . h
Lemma 2.3. Let 0 < a < Ta ga. If y 2 C ld ½a; TT and y P 0, the unique solution u of (2.1) and (2.2) satisfies
uðtÞ P 0;
t 2 ½a; TT :
Proof. From the fact that uDr ðtÞ ¼ yðtÞ 6 0, we know that the graph of u is concave down on ða; TÞT . If uðTÞ P 0, then the concavity of u and the boundary condition uðaÞ ¼ 0 imply that uðtÞ P 0 for t 2 ½a; TT . If uðTÞ < 0, then we have uðgÞ < 0 and
D.R. Anderson, C. Zhai / Applied Mathematics and Computation 215 (2010) 3713–3720
3715
uðTÞ auðgÞ uðgÞ ¼ > ; T a T a ga a contradiction of Lemma 2.2. h Lemma 2.4. Let 0 < a < Ta ga. If y 2 C ld ½a; TT and y P 0, then the unique solution u as in (2.3) of (2.1) and (2.2) satisfies
inf uðtÞ P rkuk;
ð2:4Þ
t2½g;TT
where
r :¼ min
aðT gÞ aðg aÞ g a ;
d
T a
;
T a
> 0;
ð2:5Þ
where d :¼ ðT aÞ aðg aÞ > 0. Proof. First consider the case where 0 < a < 1. By the second boundary condition we know that uðgÞ P uðTÞ. Pick t 0 2 ða; TÞT such that uðt0 Þ ¼ kuk. If t 0 6 g < T, then
min uðtÞ ¼ uðTÞ
t2½g;TT
and
uðt0 Þ 6 uðTÞ þ
uðTÞ uðgÞ duðTÞ ða TÞ ¼ : T g aðT gÞ
Therefore
aðT gÞ
min uðtÞ P
d
t2½g;TT
kuk:
gÞ uðt 0 Þ If g 6 t0 < T, again we have uðTÞ ¼ mint2½g;TT uðtÞ. As in Lemma 2.2, uð ga P t0 a. Using the boundary condition auðgÞ ¼ uðTÞ, we find that
uðTÞ >
aðg aÞ T a
uðt 0 Þ;
so that
aðg aÞ
min uðtÞ >
T a
t2½g;TT
kuk:
Now consider the case 1 6 a < Ta ga. The boundary condition this time implies uðgÞ 6 uðTÞ. Set uðt 0 Þ ¼ kuk. Note that by the gÞ uðt 0 Þ concavity of u we have t 0 2 ½g; TT and mint2½g;TT uðtÞ ¼ uðgÞ. Once again by Lemma 2.2 it follows that uð ga P t0 a, so that
min uðtÞ P
t2½g;TT
ga T a
kuk:
The proof is complete. h Lemma 2.5. Let 0 < a < Ta ga. If y 2 C ld ½a; TT and y P 0, then the unique solution u as in (2.3) of (2.1) and (2.2) satisfies
uðtÞ P
rðt aÞ kuk; ga
t 2 ½a; gT ;
ð2:6Þ
where r is given by (2.5). Proof. The result follows from Lemmas 2.2 and 2.4. h ^2 , respectively, via Lemma 2.6. If we define the delta and nabla polynomials h2 and h
h2 ðt; aÞ ¼
Z
t
ðs aÞDs;
a
^ ðt; aÞ ¼ h 2
Z
t
ðs aÞrs;
ðt; aÞ 2 T T;
ð2:7Þ
a
then the equation
h2 ðt; aÞ ¼
Z
t
ðt sÞrs
a
holds for ðt; aÞ 2 T T.
ð2:8Þ
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D.R. Anderson, C. Zhai / Applied Mathematics and Computation 215 (2010) 3713–3720
Proof. Following Bohner and Peterson [7, Section 1.6] and Anderson [1, Section 2], define the delta and nabla polynomials h2 ^2 , respectively, as above in (2.7). Thus, by [1, Theorem 9], and h
^2 ða; tÞ ¼ h2 ðt; aÞ ¼ h
Z
a
ðs tÞrs ¼
Z
t
The proof is complete.
t
ðt sÞrs: a
h
As examples of Lemma 2.6, we have
1 ðt aÞ2 ; 2 1 T ¼ hZ ðdifference equationsÞ : h2 ðt; aÞ ¼ ðt aÞðt a hÞ; h > 0; 2 1 Z T ¼ q ðquantum equationsÞ : h2 ðt; aÞ ¼ ðt aÞðt qaÞ; q > 1; 1þq
pffiffiffiffiffi pffiffiffi pffiffi 1 pffiffi pffiffiffi pffiffi pffiffiffi T ¼ N2 ðsquared differencesÞ : h2 ðt; aÞ ¼ ð t aÞð t a 1Þ 3t þ 3a þ 6 at þ a t 1 : 6
T ¼ R ðdifferential equationsÞ :
h2 ðt; aÞ ¼
Lemma 2.7. If the function w is given by
wðtÞ ¼ h2 ðt; aÞ þ
ta ½h2 ðT; aÞ ah2 ðg; aÞ; d
ð2:9Þ
where d ¼ ðT aÞ aðg aÞ > 0 and h2 ðt; aÞ is from (2.8) , then we have the following conclusions: (i) wðaÞ ¼ 0; awðgÞ ¼ wðTÞ; wDr ðtÞ 1; ½h2 ðT; aÞ ah2 ðg; aÞ 6 Ta ½h2 ðT; aÞ ah2 ðg; aÞ for all t 2 ½a; TT . (ii) wðtÞ 6 ta d d
Proof. From Lemmas 2.1 and 2.3 we have that w P 0 on ½a; TT and
wðtÞ ¼
Z
t
ðt sÞrs
a
aðt aÞ
Z
d
g
ðg sÞrs þ
a
ta d
Z
T
ðT sÞrs; a
which is (2.9). Part (ii) follows from the fact that h2 ðt; aÞ 6 0 for t 2 ½a; TT . h 3. Existence of two positive solutions We will employ Theorem 1.1 to establish the existence of at least two positive solutions for the second-order three-point boundary value problem (1.1) and (1.2). We will assume the following conditions. (C1) There exists a constant M > 0 such that f ðt; uÞ P M for ðt; uÞ 2 ½a; TT ½0; 1Þ. (C2) There exist two real constants b; c 2 ð0; 1Þ such that
0 < f ðt; uÞ 6 b for ðt; uÞ 2 ½a; TT ½0; c: (C3) or L :¼ minf2r ; cg and M 1 :¼ maxff ðt; uÞ þ M : ðt; uÞ 2 ½a; TT ½0; 2g, the parameter k satisfies
0 < k 6 s :¼
rd r 2 L ; ; min ; ðT aÞ½h2 ðT; aÞ ah2 ðg; aÞ M M1 M2
where r is given in (2.5), and M 2 :¼ maxff ðt; uÞ : ðt; uÞ 2 ½a; TT ½0; Lg. (C4) There exists R > 2 such that f ðt; uÞ þ M P Nu for t 2 ½g; TT and u P 12 Rr 2 , where
NP
2d krðg aÞh2 ðT; gÞ
for fixed k 2 ð0; s:
Remark 3.1. It follows from ðC 2 Þ and the continuity of f that
lim
u!0þ
f ðt; uÞ ¼ 1 uniformly on ½a; TT : u
Theorem 3.2. Suppose (C1)–(C4) hold. Then problem (1.1) and (1.2) has at least two positive solutions u1 and u2 , where ku1 k P r and ku2 k 6 L 6 r=2.
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D.R. Anderson, C. Zhai / Applied Mathematics and Computation 215 (2010) 3713–3720
Proof. Define the cone
P¼
u : u 2 C½a; TT ; uðtÞ P 0; t 2 ½a; TT ; min uðtÞ P rkuk t2½g;TT
and let z ¼ kMw, where r and w are given via (2.5) and (2.9), respectively. It is straightforward to see that (1.1) and (1.2) has a ~ ¼ u þ z is a solution of positive solution u if and only if u
~ zÞ ¼ 0; ~ Dr ðtÞ þ kgðt; u u ~ ðTÞ ~ ðaÞ ¼ 0; au ~ðgÞ ¼ u u
t 2 ða; TÞT ;
ð3:1Þ ð3:2Þ
~ > z on ða; TÞT , where g : ½a; TT R ! Rþ is defined by with u
gðt; uÞ ¼
f ðt; uÞ þ M : ðt; uÞ 2 ½a; TT ½0; 1Þ; f ðt; 0Þ þ M : ðt; uÞ 2 ½a; TT ð1; 0Þ:
For u 2 P, denote by Au the unique solution of (3.1) and (3.2), so that by Lemma 2.1 we have
AuðtÞ ¼
Z
t
ðt sÞkgðs; uðsÞ zðsÞÞrs
aðt aÞ
a
Z
d
g
ðg sÞkgðs; uðsÞ zðsÞÞrs þ a
ta d
Z
T
ðT sÞkgðs; uðsÞ zðsÞÞrs:
a
By Lemmas 2.3 and 2.4 we see that AðPÞ P; moreover, A is completely continuous by an application of the Arzela–Ascoli theorem. Let the uniformly continuous convex functional w : P ! ½0; 1Þ be defined by
wðuÞ ¼ max uðtÞ; t2½g;TT
u 2 P:
ð3:3Þ
Then wð0Þ ¼ 0 and wðuÞ > 0 for u – 0. Define the sets
X1 ¼ fu 2 C½a; TT : wðuÞ < 2rg and X2 ¼ fu 2 C½a; TT : wðuÞ < Rrg: Clearly X1 and X2 are bounded open sets in C½a; TT with 0 2 X1 and X1 X2 . If u 2 P \ X1 , then
kuk 6
1 1 1 min uðtÞ 6 max uðtÞ ¼ wðuÞ < 2; r t2½g;TT r t2½g;TT r
which implies that P \ X1 is bounded; similarly, P \ X2 is also bounded. If u 2 P \ @ X1 , then wðuÞ ¼ 2r, and thus kuk 6 2. Then we have
Z t Z aðt aÞ g ðt sÞkgðs; uðsÞ zðsÞÞrs ðg sÞkgðs; uðsÞ zðsÞÞrs t2½a;TT d a a Z T ta þ ðT sÞkgðs; uðsÞ zðsÞÞrs d a Z t Z Z aðt aÞ g ta T 6 max ðt sÞrs ðg sÞrs þ ðT sÞrs kM 1 ¼ kM 1 max wðtÞ t2½a;TT t2½a;TT d d a a a ðC 3 Þ T a 6 kM 1 ½h2 ðT; aÞ ah2 ðg; aÞ 6 2r ¼ wðuÞ: d
wðAuÞ 6 kAuk ¼ max
This shows that wðAuÞ 6 wðuÞ for all u 2 P \ @ X1 . If u 2 P \ X2 , then wðuÞ ¼ rR, so that rR 6 kuk 6 R. Consequently, we can see that inf u2P\@ X2 wðuÞ > 0, and for u 2 P \ @ X2 we have
zðsÞ ¼ kMwðsÞ 6 kM
ðC 3 Þ T a uðsÞ 1 ½h2 ðT; aÞ ah2 ðg; aÞ 6 r2 6 r 6 uðsÞ d kuk R
for s 2 ½g; TT . Therefore
1 uðsÞ; uðsÞ zðsÞ P 1 R
s 2 ½g; TT :
ð3:4Þ
Considering (3.4) and Lemma 2.4, we conclude that
uðsÞ zðsÞ P
1 1 1 uðsÞ P rkuk P Rr2 ; 2 2 2
s 2 ½g; TT :
This together with (C4) implies that
gðs; u zÞ ¼ f ðs; u zÞ þ M P Nðu zÞ P
1 2 Rr N; 2
s 2 ½g; TT :
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D.R. Anderson, C. Zhai / Applied Mathematics and Computation 215 (2010) 3713–3720
Moreover, we also have from (C4) that
wðAuÞ ¼ max AuðtÞ P AuðgÞ t2½g;TT
¼
Z
g
ðg sÞkgðs; uðsÞ zðsÞÞrs
aðg aÞ
Z
d
a
g
ðg sÞkgðs; uðsÞ zðsÞÞrs þ
a
ga d
Z
T
ðT sÞkgðs; uðsÞ a
zðsÞÞrs Z g Z ga T ðg sÞkgðs; uðsÞ zðsÞÞrs þ ðT sÞkgðs; uðsÞ zðsÞÞrs d a a Z T Z Z g T g ga ga T 1 ðs aÞkgðs; uðsÞ zðsÞÞrs þ ðT sÞkgðs; uðsÞ zðsÞÞrs P ðT sÞk Rr2 Nrs ¼ d d 2 d a g g T a ¼ d
¼
ga 2d
ðC 4 Þ
kRr2 Nh2 ðT; gÞ P Rr ¼ wðuÞ:
Thus, wðAuÞ P wðuÞ for all u 2 P \ X2 . It then follows from the first part of Theorem 1.1 that A has a fixed point ~ 2 P \ ðX2 n X1 Þ such that u
~Þ 6 rR; 2r 6 wðu
~ k 6 R: and thus 2r 6 ku
ð3:5Þ
Moreover, by combining (3.5) with (C3), and using Lemmas 2.4 and 2.7, we have that
~ ðtÞ P rku ~ k P 2r 2 P u
2kM ðT aÞ½h2 ðT; aÞ ah2 ðg; aÞ P 2kMwðtÞ; d
t 2 ½g; TT :
ð3:6Þ
In addition, for t 2 ½a; gT , by Lemma 2.5 and (C3) we have
~ ðtÞ P u
rðt aÞ 2r 2 ðt aÞ 2kMðt aÞ ~k P ku P ðT aÞ½h2 ðT; aÞ ah2 ðg; aÞ P 2kMwðtÞ: dðg aÞ ga ga
ð3:7Þ
As a consequence of (3.6) and (3.7) we see that
~ ðtÞ P 2kMwðtÞ ¼ 2zðtÞ; u
t 2 ½a; TT :
ð3:8Þ
~ z is a positive solution of (1.1) and (1.2). In addition, from (3.5) and (3.8) it follows that Hence, u1 ¼ u
ku1 k P
1 ~ k P r: ku 2
ð3:9Þ
To find the second positive solution of (1.1) and (1.2), we begin by setting
f ðt; uÞ ¼
f ðt; uÞ : ðt; uÞ 2 ½a; TT ½0; c;
ð3:10Þ
f ðt; cÞ : ðt; uÞ 2 ½a; TT ½c; 1Þ:
By (C2) we then have that 0 < f ðt; uÞ 6 b for ðt; uÞ 2 ½a; TT ½0; 1Þ. If we consider the auxiliary dynamic equation
uDr ðtÞ þ kf ðt; uðtÞÞ ¼ 0;
t 2 ða; TÞT
ð3:11Þ
with familiar boundary conditions
auðgÞ ¼ uðTÞ;
uðaÞ ¼ 0;
ð3:12Þ
we know that solutions to the problem (3.11) and (3.12) are equivalent to those of the operator equation u ¼ Fu, where
FuðtÞ ¼
Z
t
ðt sÞkf ðs; uðsÞÞrs
aðt aÞ
a
d
Z
g
ðg sÞkf ðs; uðsÞÞrs þ
a
ta d
Z
T
ðT sÞkf ðs; uðsÞÞrs: a
It is straightforward that F : P ! P is completely continuous and f ðPÞ P. Using Remark 3.1 we have that
limþ
u!0
f ðt; uÞ ¼ 1 uniformly on ½a; TT : u
Thus there exists a constant 0 < ‘ < L for the L is (C3) such that f ðt; uÞ P bu for ðt; uÞ 2 ½a; TT ½0; ‘, where b > 0 is chosen so that
bkr
aðg aÞh2 ðT; gÞ d
P 1:
ð3:13Þ
Choose
X3 ¼ fu 2 C½a; TT : wðuÞ < Lrg;
X4 ¼ fu 2 C½a; TT : wðuÞ < ‘rg:
Then P \ X3 and P \ X4 are bounded open sets in C½a; TT with inf u2P\@ X4 wðuÞ > 0.
3719
D.R. Anderson, C. Zhai / Applied Mathematics and Computation 215 (2010) 3713–3720
If u 2 P \ @ X3 , then from Lemmas 2.3 and 2.7 we have that
wðFuÞ ¼ max FuðtÞ 6 max FuðtÞ t2½g;TT
t2½a;TT
Z t Z Z aðt aÞ g ta T ðg sÞkf ðs; uðsÞÞrs þ ðT sÞkf ðs; uðsÞÞrs ¼ max ðt sÞkf ðs; uðsÞÞrs t2½a;TT d d a a a Z t Z g Z T aðt aÞ ta 6 max ðt sÞrs ðg sÞrs þ ðT sÞrs kM 2 ¼ kM 2 max wðtÞ t2½a;TT t2½a;TT d d a a a 6 kM 2
ðC 3 Þ T a ½h2 ðT; aÞ ah2 ðg; aÞ 6 Lr ¼ wðuÞ: d
This shows that wðFuÞ 6 wðuÞ for all u 2 P \ @ X3 . If u 2 P \ @ X4 , then r‘ 6 kuk 6 ‘, and we have
wðFuÞ ¼ max FuðtÞ P FuðTÞ t2½g;TT
¼
Z
T
ðT sÞkf ðs; uðsÞÞrs
a
aðg aÞ
¼
d
aðT gÞ
¼
d
P
Z
aðT aÞ
Z
d
T
ðT sÞkf ðs; uðsÞÞrs g
ðs aÞkf ðs; uðsÞÞrs þ
ðg sÞkf ðs; uðsÞÞrs þ
a
aðT aÞ d
a
Z
g
aðg aÞ
a
d
Z
T a d
Z
T
ðT sÞkf ðs; uðsÞÞrs
a
g
ðg sÞkf ðs; uðsÞÞrs a
Z
T
ðT sÞkf ðs; uðsÞÞrs P
aðg aÞ
g
d
Z
T
ðT sÞkf ðs; uðsÞÞrs
g
ð3:13Þ kbaðg aÞrkuk h2 ðT; gÞ P r‘ ¼ wðuÞ: d
Thus, wðFuÞ P wðuÞ for all u 2 P \ X4 . It then follows from the second part of Theorem 1.1 that the problem given in (3.11) and (3.12) has a positive solution u2 satisfying r‘ 6 wðu2 Þ 6 Lr. Consequently, ku2 k 6 L 6 r=2. In light of (C3) and (3.10) we conclude that u2 is also a solution of the original problem 1.1 and 1.2. From (C3) and (3.9) we then have that the problem (1.1) and (1.2) has two distinct positive solutions u1 and u2 . h Corollary 3.3. Suppose ðC 1 Þ and ðC 4 Þ hold. If
0 2 such that f ðt; uÞ þ M P Nu for all t 2 ½4; 128T and Since limu!1 e1 ðt;1Þu u 2 9 R, where u P 12 Rr ¼ 32258
N¼
30734 9kh2 ð128; 1Þ
for fixed k 2 ð0; s. As all of the conditions of Theorem 3.2 are satisfied, problem (4.1), (4.2) has at least two positive solutions u1 and u2 with 3 3 and ku2 k 6 254 . Note that 1; 4; 128 2 T for the following three key time scales: ku1 k P 127
1 ðt 1Þ2 ; e1 ðt; 1Þ ¼ et1 ; 2 1 T ¼ Z ðdifference equationsÞ : h2 ðt; 1Þ ¼ ðt 1Þðt 2Þ; e1 ðt; 1Þ ¼ 2t1 ; 2 1þlog Y 2t 1 T ¼ 2Z ðquantum equationsÞ : h2 ðt; 1Þ ¼ ðt 1Þðt 2Þ; e1 ðt; 1Þ ¼ ð1 þ 2k Þ 3 k¼0 T ¼ R ðdifferential equationsÞ : h2 ðt; 1Þ ¼
with the convention that
Q1
k¼0 ð1
þ 2k Þ ¼ 1; this exponential for qZ is also known as a q-Pochhammer function. h
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