BOUNDARY VALUE PROBLEMS
GENNARO INFANTE AND J. R. L. WEBB Received 19 October 2002
We establish the existence of positive solutions of somem-point boundary value problems under weaker assumptions than previously employed. In particular, we do not require all the parameters occurring in the boundary conditions to be positive. Our results allow more general behaviour for the nonlinear term than being either sub- or superlinear.
1. Introduction
Recently, much attention has been paid to the study of certain nonlocal bound- ary value problems (BVPs), whose study has been motivated by the work of Bit- sadze and Samarskii [1] and Il’in and Moiseev [7].
In particular, existence of solutions for the so-calledm-point BVPs
u(t) +g(t)fu(t)=0 (0< t <1) (1.1) under one of the boundary conditions (BCs)
u(0)=0, u(1)=
m−2 i=1
αiuηi
, 0< η1< η2<···< ηm−2<1, (1.2a) u(0)=0, u(1)=
m−2 i=1
αiuηi, 0< η1< η2<···< ηm−2<1, (1.2b) has been thoroughly studied by Gupta et al., see, for example, [3,4,5].
The existence of positive solutions has been investigated by other authors. For example, Ma [11] has studied the second set of boundary conditions when all the αiare nonnegative andmi=−12αiηi<1 under the assumption that f is either sub- or superlinear.
Copyright©2003 Hindawi Publishing Corporation Abstract and Applied Analysis 2003:18 (2003) 1047–1060
2000 Mathematics Subject Classification: 34B10, 34B18, 47H10, 47H30 URL:http://dx.doi.org/10.1155/S1085337503301034
More general boundary conditions have been studied by Ma and Castaneda [12], again when f is either sub- or superlinear.
The special case of the 3-point BVP has been studied in greater detail, one reason being that them-point BVP can be reduced to a 3-point BVP when all the coefficients αi are positive [3,7]. The existence of a positive solution for the 3-point version of (1.2b) was established by Ma [10] under the condition 0< αη <1 for f either sub- or superlinear. Under weaker conditions on f, He and Ge [6] showed the existence of three (and multiple) nonnegative solutions for the 3-point version of (1.2b) when 0< αη <1 while Webb [14] studied the existence of multiple positive solutions when 0< α <1 for (1.2a) and 0< αη <1 for (1.2b).
The usual approach has been to write the BVP as an equivalent Hammerstein integral equation
u(t)= 1
0k(t, s)g(s)fu(s)ds=:Tu(t) (1.3) and find a solution as a fixed point of the operatorTby using the classical theory of fixed-point index in cones. A different method is employed by Palamides [13], which also allows f to depend on first-order derivatives and has a more general boundary condition at 0, namely,au(0)−bu(0)=0.
In the present paper, we want to show that requiring all theαi to be non- negative ismuch too restrictive, and that positive solutions exist more generally for both sets of boundary conditions. As in [14], our results allow more general behaviour on f than being either sub- or superlinear.
In order to keep the calculations at a reasonable level, we concentrate on the 4-point BVPs. We supposeη1,η2are given and we determine in each case nec- essary conditions on the parametersα1,α2 so that the kernelk(t, s)≥0 for all 0≤t, s≤1. This determines a region in the (α1, α2) plane which is unbounded and is much larger than the triangle
α1≥0, α2≥0, α1η1+α2η2<1, (1.4) which has been previously used for the BVP (1.2b).
We then show that if the parameters lie strictly inside these regions, then one or multiple positive solutions exist under suitable conditions on f. Our method utilises some known results of Lan [8] for the Hammerstein integral equation.
2. Positive solutions of some Hammerstein integral equations
We begin by recalling some results for the following Hammerstein integral equa- tion:
u(t)= 1
0k(t, s)g(s)fu(s)ds≡Tu(t). (2.1)
Although it is possible to give more general results (e.g., it is possible to re- placeg(s)f(u(s)) by f(s, u(s)) which satisfies Carath´eodory conditions, and we can treat some discontinuous kernels k), for simplicity in the sequel, we will make the following assumptions on f,g, and the kernelk:
(C1)k: [0,1]×[0,1]→[0,∞) is continuous;
(C2) f :R→[0,∞) is continuous;
(C3)g∈L1(0,1) andg(s)≥0 a.e.;
(C4) there exist a measurable functionΦ: [0,1]→[0,∞), a subinterval [a, b]
on whichabg(s)ds >0, andc∈(0,1] such that k(t, s)≤Φ(s) fort, s∈[0,1],
k(t, s)≥cΦ(s) fort∈[a, b], s∈[0,1]. (2.2) This allows us to use the following coneK, of a type due to Guo (see, e.g., [2]), which is a subset of the conePof positive functions:
K=
u∈C[0,1] :u≥0,minu(t) :a≤t≤b≥cu
. (2.3)
Lemma2.1 (see [8,9]). Under the above hypotheses, the mapT defined in (2.1) mapsKintoKand is compact.
Definition 2.2. We define the following numbers:
m= max
t∈[0,1]
1
0k(t, s)g(s)ds −1
, M= min
t∈[a,b]
b
ak(t, s)g(s)ds −1
, f0,ρ= max
u∈[0,ρ]
f(u)
ρ , f0=lim sup
u→0+
f(u)
u , f∞=lim sup
u→+∞
f(u) u , fcρ,ρ= min
u∈[cρ,ρ]
f(u)
ρ , f0=lim inf
u→0+
f(u)
u , f∞=lim inf
u→+∞
f(u) u .
(2.4)
This notation allows us to state the following theorem, a special case of some results from [8] proved by using the theory of fixed-point index.
Theorem2.3. If (C1), (C2), (C3), and (C4) hold, then (2.1) has a positive solution inKif one of the following conditions holds:
(h1) 0≤ f0< mandM < f∞≤ ∞; (h2)M < f0≤ ∞and0≤f∞< m.
Equation (2.1) has two positive solutions inK if there isρ >0such that either of the following conditions holds:
(S1) 0≤ f0< m, fcρ,ρ> cM, and0≤ f∞< m;
(S2)M < f0≤ ∞,f0,ρ< m, andM < f∞≤ ∞.
Under the hypothesis (S1), there are in fact 3 nonnegative solutions, but the third may be 0. This result is similar to the result of [6], but the constantmhere is better (larger) than the constant used in [6].
3. Positive solutions of the 4-point BVP (1.2a)
We now consider the two 4-point BVPs in detail. We first consider the BVP u(t) +g(t)fu(t)=0, a.e on [0,1], (3.1) with boundary conditions
u(0)=0, u(1)=α1uη1
+α2uη2
. (3.2)
If we writeγ1=1−α1−α2, then the solution ofu= −ysubject to the BCs (3.2) is
u(t)= 1 γ1
1
0(1−s)y(s)ds−α1
η1
0
η1−sy(s)ds−α2
η2
0
η2−sy(s)ds
− t
0(t−s)y(s)ds.
(3.3) Thus the kernel (Green’s function) is
k(t, s)= 1 γ1
(1−s)
−α1
γ1
η1−s, s≤η1, 0, s > η1, −
α2
γ1
η2−s, s≤η2, 0, s > η2,−
t−s, s≤t, 0, s > t.
(3.4)
For existence ofpositivesolutions of (2.1), the standard assumption made is thatk(t, s)≥0 for allt, s. If, for example,k(t0, s)<0 forsin some interval, then even the linear problem with a positive right-hand side can have a solution withu(t0)<0. Hence we will investigate whenk(t, s)≥0 for all t, s. This will determine a region in the (α1, α2) plane and we will show that if (α1, α2) lies in the interior of this region, then the hypothesis (C4) is also satisfied, and hence positive solutions for the nonlinear problem can be shown to exist.
The requirementk(t, s)≥0 for allt,sneedsγ1>0, that is,α1+α2<1 plus some other conditions which we explore now.
For a givens,t→k(t, s) is a decreasing function oft, so we investigate when k(1, s)≥0 for eachs.
Forη2< s <1,
k(1, s)=1−s
γ1 −(1−s)=
α1+α2
γ1 (1−s). (3.5)
Forη1< s≤η2,
k(1, s)= 1 γ1
α1+α2
(1−s)−α2
η2−s. (3.6)
α1+α2=0
α1+α2=1 α1(1−η1)+α2(1−η2)=0
α1 α2
Figure 3.1. Region where the kernel is positive.
For 0< s≤η1,
k(1, s)= 1 γ1
α1
1−η1
+α2
1−η2
. (3.7)
Thus we needα1+α2=1−γ1≥0 andd1:=α1(1−η1) +α2(1−η2)≥0. These also ensure (α1+α2)(1−s)−α2(η2−s)≥0 forη1≤s≤η2.
The region of the (α1, α2) plane for whichk(t, s)≥0 is therefore as shown in Figure 3.1.
Remark 3.1. We obtain the “region” for the 3-point BVP as the projection on the lineα1=0, which gives the known condition 0< α <1, see [14].
We now show that if (α1, α2) lies in the interior of the region ofFigure 3.1 then the kernel satisfies (C4), that is, suitableΦ, subinterval [a, b], andcexist.
Upper bounds. For eachs, the maximum ofk(t, s) occurs whent=0. So we may takeΦ(s)=k(0, s).
Hence we have
Φ(s)=
1
γ1(1−s), η2< s≤1, 1
γ1
1−s−α2
η2−s, η1< s≤η2, 1
γ1
d2−
1−α1−α2
s, 0< s≤η1,
(3.8)
whered2:=1−α1η1−α2η2 and d2>1−α1−α2>0 inside the region where k≥0.
Lower bounds on [a, b]. We take [a, b]=[0, η1]. The minimum of k(t, s) for s fixed is k(η1, s). Hence we want to determine c as large as possible so that k(η1, s)≥cΦ(s).
Forη2< s <1,
kη1, s= 1
γ1(1−s). (3.9)
Forη1< s≤η2,
kη1, s= 1 γ1
1−s−α2
η2−s. (3.10)
For 0< s≤η1, letting
d3:=1−η1−α2
η2−η1
=d1+1−η1
γ1>0, (3.11) we have
kη1, s= 1 γ1
1−η1−α2
η2−η1
= 1
γ1d3. (3.12) So,kmin≥cΦ(s) fort∈[a, b] if
1
γ1d3≥c1 γ1
d2−
1−α1−α2
s (3.13)
for 0< s≤η1. Thusc=d3/d2.
The constantsm,M fromDefinition 2.2can be calculated for an explicitly giveng. We give the results for the special caseg(s)≡1 as follows:
1
m=max
t∈[0,1]
1
0k(t, s)ds= 1
0k(0, s)ds= 1 2γ1
1−α1η21−α2η22, 1
M = min
t∈[0,η1]
η1
0 k(t, s)ds= η1
0 kη1, sds
= 1 γ1η1
1−η1−α2
η2−η1
= 1 γ1η1d3.
(3.14)
This gives the following result.
Theorem3.2. Letg(s)≡1,c=d3/d2, and letm,M be as given in (3.14). Then, for(α1, α2)in the interior of the region of Figure 3.1, the BVP (3.1), (3.2) has at least one positive solution if either (h1) or (h2) ofTheorem 2.3holds, and has two positive solutions if there isρ >0such that either (S1) or (S2) ofTheorem 2.3holds.
4. Positive solutions of the 4-point BVP (1.2b) We now study the BVP
u(t) +g(t)fu(t)=0 a.e on [0,1], (4.1) with boundary conditions
u(0)=0, u(1)=α1uη1
+α2uη2
. (4.2)
For the BCs (4.2), if we setγ=1−α1η1−α2η2>0, the kernel is k(t, s)= t
γ(1−s)
−α1t γ
η1−s, s≤η1
0, s > η1
−α2t γ
η2−s, s≤η2
0, s > η2
−
t−s, s≤t, 0, s > t.
(4.3)
We will show thatk(t, s)≥0 for allt, sif 0< γ≤1 and d1:=α1
1−η1
+α2
1−η2
≥0. (4.4)
Fors > η2andt < s,
k(t, s)=1
γt(1−s)≥0. (4.5)
Fors > η2andt≥s, k(t, s)=1
γ
t(1−s)−γ(t−s)=1 γ
t(1−γ−s) +γs (4.6)
andk(t, s)≥0 sincek(s, s) andk(1, s) are both positive.
Forη1< s≤η2andt < s, k(t, s)=1
γ
t(1−s)−α2tη2−s= t γ
1−s−α2
η2−s. (4.7) Now
1−s−α2
η2−s≥min1−η2,1−η1−α2
η2−η1
, (4.8)
where
d4:=1−η1−α2
η2−η1
=γ1−η1
+d1η1≥0. (4.9) Note thatd4=d3, but we have different hypotheses from the previous BC, so the positivity ofd4has to be shown. Then we have
k(t, s)≥ t
γmin1−η2, d4
≥0. (4.10)
Forη1< s≤η2andt≥s, the minimum occurs either whent=sor whent=1, so
k(t, s)≥min 1
γs1−s−α2
η2−s,1 γ
(1−γ)(1−s)−α2
η2−s. (4.11)
Here
s(1−s)−α2
η2−s≥minη2 1−η2
, η1d4
≥0, (1−γ)(1−s)−α2
η2−s≥min(1−γ)1−η2
, d1
≥0. (4.12)
For 0≤s≤η1andt < s,
k(t, s)=1
γtγ−s1−α1−α2
≥0 (4.13)
sinceγ−η1(1−α1−α2)=d4≥0.
For 0≤s≤η1andt≥s, the minimum occurs either whent=sor whent=1, and we have
k(1, s)=1 γ
γs−s1−α1−α2
=1 γsd1
≥0,
k(s, s)=1 γ
γs−s21−α1−α2
≥0
(4.14)
since it is equal to 0 whens=0, and whens=η1, kη1, η1
=η1
γ
γ−η1
1−α1−α2
=1
γd4>0. (4.15) The region of the (α1, α2) plane for whichk(t, s)≥0 is therefore as shown in Figure 4.1, which is clearly much larger than the triangle in the first quadrant which is essentially the region previously used by other authors.
Remark 4.1. Projecting onto the lineα1=0 gives the “region” for the 3-point BVP, 0< αη <1.
We now determineΦand show that we may take [a, b]=[η2,1].
Upper bounds. Sincek(0, s)=0 andt →k(t, s) is linear, with a jump in the gra- dient att=s, the maximum occurs either whent=sor whent=1.
Fors > η2andt < s,
k(t, s)=1
γt(1−s)≤1−s
γ . (4.16)
Fors > η2andt≥s, k(t, s)=1
γ
t(1−s)−γ(t−s)=1
γt(1−γ)−s+s. (4.17)
α1η1+α2η2=0 α1η1+α2η2=1
α1(1−η1)+α2(1−η2)=0
α1 α2
Figure 4.1. Region for positive kernel.
Herek(s, s)=(1/γ)s(1−s)≤(1/γ)(1−s) and k(1, s)=(1−γ)
γ (1−s)≤1−s
γ . (4.18)
Forη1< s≤η2,
k(s, s)= s γ
1−s−α2
η2−s,
k(1, s)=1 γ
(1−s)−α2
η2−s+s
=1 γ
(1−γ)(1−s)−α2
η2−s.
(4.19)
For 0≤s≤η1, the maximum is either k(s, s)= s
γ γ−
1−α1−α2
s (4.20)
or
k(1, s)=1 γsγ−
1−α1−α2
=1
γsd1. (4.21)
Hence we can takeΦ(s) as follows:
Φ(s)=
1
γ(1−s), η2< s≤1,
1 γ
1−s−α2
η2−s, η1< s≤η2, 1
γsγ−
1−α1−α2
s, 0≤s≤η1, α1+α2≤1, 1
γsα1
1−η1
+α2
1−η2
, 0≤s≤η1, α1+α2>1.
(4.22)
Lower bounds on[a, b]=[η2,1]. For the subinterval [a, b], we must havea >0, and, guided by our knowledge of the 3-point BVP, we choose [a, b]=[η2,1].
Fors > η2andη2≤t < s, k(t, s)=1
γt(1−s)≥1
γη2(1−s). (4.23)
Fors > η2andt≥s,
k(t, s)=1 γ
t(1−γ−s) +γs, (4.24)
and the minimum occurs either att=sor att=1 as follows:
k(s, s)=1 γ
s(1−γ−s) +γs
=1
γs(1−s)≥1
γη2(1−s), k(1, s)=1−γ
γ (1−s).
(4.25)
Hence,kmin≥cΦ(s) fors > η2if
c≤minη2,1−γ. (4.26)
Forη2≥s > η1andt∈[η2,1], k(t, s)=1 γ
t(1−s)−α2
η2−s−γ+s,
kmin=min 1
γη2
1−s−α2
η2−s−η2+s,1 γ
1−s−α2
η2−s−1 +s
. (4.27) We wantkmin≥cΦ(s), where
Φ(s)=1 γ
1−s−α2
η2−s forη1< s≤η2. (4.28)
This requires
η2Φ(s)−η2+s≥cΦ(s), (4.29)
Φ(s)−1 +s≥cΦ(s). (4.30)
Condition (4.29) needs η2−s≤
η2−cΦ(s) forη1< s≤η2. (4.31) This is satisfied ifc≤η2and
η2−η1≤
η2−cΦη1
=
η2−cd4
γ. (4.32)
Note that we must have
η2−
η2−η1
γ d4
>0 (4.33)
forcto exist. In fact, using (4.9), we have η2−
η2−η1 γ d4 > η2−
η2−η1 1−η1 =η1
1−η2
1−η1 >0. (4.34) Hence we want
c≤η2−
η2−η1
γ d4
. (4.35)
Condition (4.30) needs
1−s≤(1−c)Φ(s) forη1< s≤η2. (4.36) Whens=η2, (4.36) is
1−η2≤(1−c) 1−η2
γ , (4.37)
soc≤1−γsuffices.
Whens=η1, (4.36) is
1−η1≤(1−c)d4
γ . (4.38)
Sinceη1d1+ (1−η1)γ=d4from (4.9), this yields c≤1−
1−η1
γ d4 =η1d1
d4
. (4.39)
For 0≤s≤η1andt≥η2,
k(t, s)=1
γtα1+α2−1s+s. (4.40) Ifα1+α2>1, thenk(t, s) is increasing int, so the minimum is att=η2as follows:
kmin=1 γ
η2
α1+α2−1s+γs=1 γ
1−η2+α1
η2−η1
s. (4.41)
In order thatkmin≥cΦ(s), we need c≤
1−η2+α1
η2−η1
d1 . (4.42)
Note that
1−η2+α1η2−α1η1=γ+η2
α1+α2−1> γ >0 (4.43) since this is the caseα1+α2>1. Hence, for this case, we can take
c≤ γ
d1. (4.44)
Whenα1+α2≤1,k(t, s) is decreasing int, so kmin=1
γ
α1+α2−1s+γs= s
γd1. (4.45)
We want
s γd1≥c1
γsγ−
1−α1−α2
s, (4.46)
hence we wantc≤d1/d4. The total requirement is therefore c≤min
1−γ, η1d1
d4
, γ d1
, η2−
η2−η1
γ d4
. (4.47)
As all the requisite conditions have been now verified, we immediately have the following theorem.
Theorem 4.2. Let [a, b]=[η2,1] and suppose that η12g(s)ds >0. Letc satisfy (4.47) and letm,M be as inDefinition 2.2. Then, for(α1, α2)in the interior of the region ofFigure 4.1, the BVP (4.1), (4.2) has at least one positive solution if either (h1) or (h2) ofTheorem 2.3holds, and has two positive solutions if there is ρ >0such that either (S1) or (S2) ofTheorem 2.3holds.
In the special case wheng(s)≡1,m,Mare readily calculated. In fact, 1
m=max
t∈[0,1]
1 0k(t, s)ds
=max
t∈[0,1]
1 2γ
t1−α1η21−α2η22−γt2
= 1 8γ2
1−α1η12−α2η222, 1
M = min
t∈[η2,1]
1 η2
k(t, s)ds
= min
t∈[η2,1]
1 2γ
t1−η2
2
−γt−η2
2
=minη2,1−γ 1−η2
2
2γ .
(4.48)
Conclusion. It is possible to extend our methods to deal with 5,6, . . .-point BVPs, but we feel this would be only worthwhile if required by an explicit application.
Our aim is to show that the “obvious” extension of the condition of the 3-point BVP that requires positivity of the coefficientsαiis far from optimal.
Acknowledgment
This research was partially supported by a visiting professorship of GNAMPA.
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Gennaro Infante: Dipartimento di Matematica, Universit`a della Calabria, 87036 Arcava- cata di Rende, Cosenza, Italy
E-mail address:[email protected]
J. R. L. Webb: Department of Mathematics, University of Glasgow, Glasgow G12 8QW, UK
E-mail address:[email protected]
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Intelligent Computational Methods for Financial Engineering
Call for Papers
As a multidisciplinary field, financial engineering is becom- ing increasingly important in today’s economic and financial world, especially in areas such as portfolio management, as- set valuation and prediction, fraud detection, and credit risk management. For example, in a credit risk context, the re- cently approved Basel II guidelines advise financial institu- tions to build comprehensible credit risk models in order to optimize their capital allocation policy. Computational methods are being intensively studied and applied to im- prove the quality of the financial decisions that need to be made. Until now, computational methods and models are central to the analysis of economic and financial decisions.
However, more and more researchers have found that the financial environment is not ruled by mathematical distribu- tions or statistical models. In such situations, some attempts have also been made to develop financial engineering mod- els using intelligent computing approaches. For example, an artificial neural network (ANN) is a nonparametric estima- tion technique which does not make any distributional as- sumptions regarding the underlying asset. Instead, ANN ap- proach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting pa- rameters to obtain the desired results. The main aim of this special issue is not to merely illustrate the superior perfor- mance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making. In this sense, the submissions should es- pecially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelli- gent, easy-to-use, and/or comprehensible computational sys- tems (e.g., decision support systems, agent-based system, and web-based systems)
This special issue will include (but not be limited to) the following topics:
• Computational methods: artificial intelligence, neu- ral networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learn- ing, multiagent learning
• Application fields: asset valuation and prediction, as- set allocation and portfolio selection, bankruptcy pre- diction, fraud detection, credit risk management
• Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, imple- mentation
Authors should follow the Journal of Applied Mathemat- ics and Decision Sciences manuscript format described at the journal site http://www.hindawi.com/journals/jamds/.
Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Track- ing System athttp://mts.hindawi.com/, according to the fol- lowing timetable:
Manuscript Due December 1, 2008 First Round of Reviews March 1, 2009 Publication Date June 1, 2009
Guest Editors
Lean Yu,Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;
Department of Management Sciences, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong;
Shouyang Wang,Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China; [email protected]
K. K. Lai,Department of Management Sciences, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong; [email protected]
Hindawi Publishing Corporation http://www.hindawi.com
