doi:10.1155/2009/829735
Research Article
Positive Solutions for Boundary Value Problems
of Second-Order Functional Dynamic Equations on
Time Scales
Ilkay Yaslan Karaca
Department of Mathematics, Ege University, Bornova, 35100 Izmir, Turkey
Correspondence should be addressed to Ilkay Yaslan Karaca,ilkay.karaca@ege.edu.tr
Received 3 February 2009; Accepted 26 February 2009
Recommended by Alberto Cabada
Criteria are established for existence of least one or three positive solutions for boundary value problems of second-order functional dynamic equations on time scales. By this purpose, we use a fixed-point index theorem in cones and Leggett-Williams fixed-point theorem.
Copyrightq2009 Ilkay Yaslan Karaca. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
In a recent paper 1, by applying a fixed-point index theorem in cones, Jiang and Weng studied the existence of positive solutions for the boundary value problems described by second-order functional differential equations of the form
yx rxfyωx0, 0< x <1, αyx−βyx ξx, a≤x≤0, γyx δyx ηx, 1≤x≤b.
1.1
Aykut 2 applied a cone fixed-point index theorem in cones and obtained sufficient conditions for the existence of positive solutions of the boundary value problems of functional difference equations of the form
−Δ2yn−1 fn, yωn, n∈a, b, αyn−1−βΔyn−1 ξn, n∈τ1, a
, γyn δΔyn ηn, n∈b, τ2
.
In this article, we are interested in proving the existence and multiplicity of positive solutions for the boundary value problems of a second-order functional dynamic equation of the form
−yΔ∇t qtyt ft, yωt, t∈a, b, αyρt−βyΔρtξt, t∈τ1, a
, γyt δyΔt ηt, t∈b, τ2
.
1.3
Throughout this paper we letTbe any time scalenonempty closed subset ofRanda, b
be a subset ofTsuch thata, b {t∈T, a≤t≤b},and fort∈τ1, a, tis not right scattered
and left dense at the same time.
Some preliminary definitions and theorems on time scales can be found in books3,4 which are excellent references for calculus of time scales.
We will assume that the following conditions are satisfied.
H1qt∈ Ca, b, qt≥0.
H2f :a, b×R → Ris continuous with respect toyandft, y≥0 fory∈R, where
Rdenotes the set of nonnegative real numbers.
H3ωtdefined ona, bsatisfies
cinfωt:a≤t≤b< b,
dsupωt:a≤t≤b> a. 1.4
LetE1:{t∈E:a≤ωt≤ρb}be nonempty subset of
E:t∈a, b:a≤ωt≤b. 1.5
H4α, β, γ, δ≥0, αβ >0, γδ >0;
ifqt≡0 a≤t≤b,thenαγ >0;−γ/δ∈ R forδ >0, whereRdenotes the set of all positively regressive and rd-continuous functions.
H5ξt and ηt are defined on τ1, σa and b, στ2, respectively, where τ1 :
min{a, c}, τ2:max{b, d}; furthermore,ξa ηb 0;
ξσt≥0, fort∈τ1, a
asβ0;
a
t
ξse α/βs,0∇s≥0, fort∈τ1, a
asβ >0;
ηt≥0, for t∈b, τ2
asδ0;
e−γ/δt,0
t
b
ηse −γ/δσs,0Δs≥0, fort∈b, τ2
asδ >0.
There have been a number of works concerning of at least one and multiple positive solutions for boundary value problems recent years. Some authors have studied the problem for ordinary differential equations, while others have studied the problem for difference equations, while still others have considered the problem for dynamic equations on a time scale 5–10. However there are fewer research for the existence of positive solutions of the boundary value problems of functional differential, difference, and dynamic equations
1,2,11–13.
Our problem is a dynamic analog of the BVPs1.1and1.2. But it is more general than them. Moreover, conditions for the existence of at least one positive solution were studied for the BVPs 1.1 and 1.2. In this paper, we investigate the conditions for the existence of at least one or three positive solutions for the BVP1.3. The key tools in our approach are the following fixed-point index theorem14, and Leggett-Williams fixed-point theorem15.
Theorem 1.1see14. LetEbe Banach space andK ⊂ Ebe a cone inE. Letr > 0, and define
Ωr {x∈K:x< r}. AssumeA:Ωr → Kis a completely continuous operator such thatAx /x forx∈∂Ωr.
iIfAu ≤ uforu∈∂Ωr, theniA,Ωr, K 1.
iiIfAu ≥ uforu∈∂Ωr, theniA,Ωr, K 0.
Theorem 1.2see15. LetPbe a cone in the real Banach spaceE. Set
Pr :
x∈ P:x< r,
Pψ, p, q:x∈ P:p≤ψx, x ≤q.
1.7
Suppose thatA:Pr → Pr is a completely continuous operator andψ is a nonnegative continuous concave functional onPwithψx≤ xfor allx∈ Pr. If there exists0 < p < q < s≤rsuch that the following conditions hold:
i{x∈ Pψ, q, s:ψx> q}/{}andψAx> qfor allx∈ Pψ, q, s;
iiAx< pforx ≤p;
iiiψAx> qforx∈ Pψ, q, rwithAx> s.
ThenAhas at least three fixed pointsx1, x2,andx3inPrsatisfying
2. Preliminaries
First, we give the following definitions of solution and positive solution of BVP1.3.
Definition 2.1. We say a functionytis a solution of BVP1.3if it satisfies the following.
1ytis nonnegative onρτ1, στ2.
2yt yτ1;tast∈ρτ1, a, whereyτ1;t:ρτ1, a → 0,∞is defined as
yτ1;t
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
eα/βt,0
1
β a
t
ξse α/βs,0∇se α/βa,0ya
, if β >0,
1
αξ
σt, if β0.
2.1
3yt yτ2;tast∈b, στ2, whereyτ2;t:b, στ2 → 0,∞is defined as
yτ2;t
⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩
e−γ/δt,0
1
δ t
b
ηse −γ/δσs,0Δse −γ/δb,0yb
, ifδ >0,
1
γηt, ifδ0.
2.2
4yisΔ-differentiable,yΔ : ρa, b → Ris∇-differentiable ona, band yΔ∇ :
a, b → Ris continuous.
5−yΔ∇t qtyt ft, yωt,fort∈a, b.
Furthermore, a solutionytof1.3is called a positive solution ifyt>0 fort∈a, b.
Denote byϕtandψtthe solutions of the corresponding homogeneous equation
−yΔ∇t qtyt 0, t∈a, b, 2.3
under the initial conditions
ϕρaβ, ϕΔρaα,
ψb δ, ψΔb −γ. 2.4
Set
DWtψ, ϕ ψtϕΔt−ψΔtϕt. 2.5
Since the Wronskian of any two solutions of 2.3 is independent of t, evaluating at t ρa, tband using the initial conditions2.4yield
Using the initial conditions2.4, we can deduce from2.3forϕtandψt, the following equations:
ϕt βαt−ρa t
ρa
τ
ρaqsϕs∇sΔτ, 2.7
ψt δγb−t b
t
b
τ
qsψs∇sΔτ. 2.8
See8.
Lemma 2.2see8. Under the conditions (H1) and the first part of (H4) the solutionsϕtand ψtpossess the following properties:
ϕt≥0, t∈ρa, σb, ψt≥0, t∈ρa, b, ϕt>0, t∈ρa, σb, ψt>0, t∈ρa, b, ϕΔt≥0, t∈ρa, b, ψΔt≤0, t∈ρa, b.
2.9
LetGt, sbe the Green function for the boundary value problem:
−yΔ∇t qtyt 0, t∈a, b, αyρa−βyΔρa0,
γyb δyΔb 0,
2.10
given by
Gt, s 1 D
ψtϕs, ifρa≤s≤t≤σb,
ψsϕt, ifρa≤t≤s≤σb, 2.11
whereϕtandψtare given in2.7and2.8, respectively. It is obvious from2.6,H1 andH4, thatD >0 holds.
Lemma 2.3. Assume the conditions (H1) and (H4) are satisfied. Then
i0≤Gt, s≤Gs, sfort, s∈ρa, b,
iiGt, s≥ΓGs, sfort∈a, ρbands∈ρa, b,
where
Γ minI1, I2
where definitionKandLemma 2.3that
Lemma 2.5. A:K → Kis completely continuous.
Lemma 2.6. If
lim v→0
ft, v
v ∞, vlim→∞
ft, v
v ∞, 2.27
for allt ∈a, b, then there exist0 < r0 < R0 < ∞such thatiA, Kr, K 0, for0 < r ≤r0and iA, KR, K 0, forR≥R0.
Proof. ChooseM >0 such that
Γ2M
E1
Gs, s∇s >1. 2.28
By using the first equality of2.27, we can chooser0>0 such that
ft, v≥Mv, 0≤v≤r0. 2.29
Ifu∈∂Kr0< r≤r0, then fort0∈a, ρb, we have
Aut0 b
ρaG
t0, s
fs, uωsu0
ωs∇s
≥Γ
b
ρaGs, sf
s, uωsu0
ωs∇s
≥Γ
E1
Gs, sfs, uωs∇s
≥ΓM
E1
Gs, suωs∇s
≥Γ2uM
E1
Gs, s∇s
>u.
2.30
Therefore we get
Au>u, ∀u∈∂Kr. 2.31
Thus, we have fromTheorem 1.1,iA, Kr, K 0, for 0 < r ≤ r0. On the other hand, the
second equality of2.27implies for everyM >0, there is anR0> r0, such that
Here we chooseM >0 satisfying2.28. Foru∈∂KR, R≥R0, we have definition ofKRthat
ut≥Γu ΓR, t∈a, ρb. 2.33
It follows from2.32that
Aut0 b
ρa
Gt0, s
fs, uωsu0
ωs∇s
≥Γ
b
ρaGs, sf
s, uωsu0
ωs∇s
≥Γ
E1
Gs, sfs, uωs∇s
≥ΓM
E1
Gs, suωs∇s
≥Γ2MR
E1
Gs, s∇s
> Ru.
2.34
This shows that
Au>u, ∀u∈KR. 2.35
Thus, byTheorem 1.1, we conclude thatiA, KR, K 0 forR ≥ R0. The proof is therefore
complete.
3. Existence of One Positive Solution
In this section, we investigate the conditions for the existence of at least one positive solution of the BVP1.3.
In the next theorem, we will also assume that the following condition onft, v.
H6:
lim inf v→0 tmin∈a,b
ft, v
v > kλ1, lim supv→∞ max t∈a,b
ft, v
v < qλ1, 3.1
wherek >0 is large enough such that
kΓ
E1
ϕ1s∇s≥ b
andq >0 is small enough such that
Theorem 3.1. If (H1)–(H6) are satisfied, then the BVP1.3has at least one positive solution.
Proof. Fix 0< m <1< m1and letf1u umum1foru≥0. Then,f1usatisfies2.27. Define
We now prove thatHt, u/ufor allt∈0,1andu∈∂Kr1. In fact, if there existst0 ∈0,1
and the boundary conditions
αu1
by parts in the left-hand side two times, we obtain
We also have
λ1 b
ρaϕ1su1s∇s≤λ1
u1b
ρaϕ1s∇s. 3.14
Equations3.13and3.14lead to
λ1≥λ1
k. 3.15
This is impossible. ThusHt, u/uforu ∈ ∂Kr1 andt ∈ 0,1. By3.7and the homotopy
invariance of the fixed-point indexsee11, we get that
iA, Kr1, K
iH0,·, Kr1, K
iH1,·, Kr1, K
i A, Kr1, K
0. 3.16
On the other hand, according to the second inequality ofH6, there exist >0 andR> R0
such that
ft, u≤qλ1−
u, u > R, t∈a, b. 3.17
We define
C: max
a≤t≤b,0≤u≤Rft, u−
qλ1−
u1, 3.18
then it follows that
ft, u≤qλ1−
uC, u≥0, t∈a, b. 3.19
DefineH1 :0,1×K → KbyH1t, u tAu, thenH1is a completely continuous operator.
We claim that there existsR1≥Rsuch that
In fact, ifH1t0, u1 u1for someu1∈Kand 0≤t0≤1, then
λ1 b
ρau1sϕ1s∇s≤
b
ρaf
s, u1
ωsu0
ωsϕ1s∇s
≤q
λ1−
q
u1u0 b
ρaϕ1s∇sC
b
ρaϕ1s∇s
≤q
λ1−
q
u1 b
ρaϕ1s∇sC1
b
ρaϕ1s∇s,
3.21
λ1 b
ρau1sϕ1s∇s≥λ1
ρb
ρau1sϕ1s∇s
≥λ1Γu1 ρb
ρa
ϕ1s∇s
≥λ1qu1 b
ρa
ϕ1s∇s,
3.22
whereC1qλ1−/qu0C.Combining3.21with3.22, we have
u1≤ C1
R1. 3.23
LetR1max{R,R1}1.Then we get
H1t, u/u, fort∈0,1, u∈K,u ≥R1. 3.24
Consequently, by the homotopy invariance of the fixed-point index, we have
iA, KR1, K
iH11,·, KR1, K
iH10,·, KR1, K
iΘ, KR1, K
1, 3.25
whereΘis zero operator. Use3.16and3.25to conclude that
iA, KR1\Kr1, K
iA, KR1, K
−iA, Kr1, K
1−01. 3.26
Hence,Ahas a fixed point inKR1\Kr1.
Letyt ut u0t. Sinceyt utfort ∈ a, band 0 < r1 ≤ u ua,b
ya,b≤R1.
H7
lim sup v→0 tmax∈a,b
ft, v v < qλ1,
lim inf v→∞ tmin∈a,b
ft, v
v > kλ1, ξt≡0, ηt≡0.
Theorem 3.2. If (H1)–(H5) and (H7) are satisfied, then the BVP 1.3 has at least one positive solution.
Proof. DefineH1 : 0,1×K → KbyH1t, u tAu, thenH1 is a completely continuous
operator. By the first inequality inH7, there exist >0 and 0< r1≤r0such that
ft, v≤qλ1−
v, ∀t∈a, b,0≤v≤r1. 3.28
We claim thatH1t, u/ufor 0 ≤ t ≤ 1 andu ∈ ∂Kr1. In fact, if there exist 0 ≤ t0 ≤ 1 and u1 ∈∂Kr1 such thatH1t0, u1 u1, thenu1tsatisfies the boundary condition3.11. Since ξt≡0, ηt≡0, we haveu0t≡0. Then we have
−uΔ1∇t qtu1t t0f
t, u1
ωt, a≤t≤b. 3.29
Multiplying the last equation byϕ1tintegrating fromatob, by3.28, we obtain
λ1 b
ρa
ϕ1su1s∇st0 b
ρa
fs, u1
ωsϕ1s∇s
≤
b
ρa
fs, u1
ωsϕ1s∇s
≤qλ1−u1
b
ρaϕ1s∇s,
3.30
then we have
λ1 b
ρau1sϕ1s∇s≥λ1
ρb
ρau1sϕ1s∇s
≥λ1Γu1 ρb
ρa
ϕ1s∇s
≥λ1qu1 b
ρa
ϕ1s∇s.
3.31
Equations3.30and3.31lead to
λ1q≤λq1−. 3.32
This is impossible. By homotopy invariance of the fixed-point index, we get that
iA, Kr1, K
iH11,·, Kr1, K
iH10,·, Kr1, K
iΘ, Kr1, K
DefineH:0,1×K → KbyHt, u 1−tAutAu , thenHis a completely continuous
then, it is obvious that
Consequently, by3.8and the homotopy invariance of the fixed-point index, we have
iA, KR1, K
iH0,·, KR1, K
iH1,·, KR1, K
i A, KR1, K
0. 3.40
In view of3.33and3.40, we obtain
iA, KR1\Kr1, K
iA, KR1, K
−iA, Kr1, K
0−1−1. 3.41
Therefore,Ahas a fixed point inKR1\Kr1. The proof is completed.
Corollary 3.3. Using the following (H8) or (H9) instead of (H6) or (H7), the conclusions of Theorems 3.1and3.2are true. Fort∈a, b,
H8
lim v→0
ft, v
v
∞, lim
v→∞
ft, v
v
0 sublinear; 3.42
H9
lim v→0
ft, v
v
0, lim v→∞
ft, v
v
∞ superlinear, ξt≡0, ηt≡0. 3.43
4. Existence of Three Positive Solutions
In this section, usingTheorem 1.2the Leggett-Williams fixed-point theoremwe prove the existence of at least three positive solutions to the BVP1.3.
Define the continuous concave functional ψ : K → 0,∞ to be ψu : mint∈a,ρbut, and the constants
M:
min t∈a,ρb
ρb
ρa
Gt, s∇s −1
, 4.1
N:
b
ρaGs, s∇s. 4.2
Theorem 4.1. Suppose there exists constants0< p < q < q/Γ≤rsuch that
D1ft, v< p/Nfort∈a, b, v∈0, p;
D2ft, v≥qMfort∈a, ρb, v∈q, q/Γ;
D3one of the following is satisfied:
alim supv→ ∞maxt∈a,bft, v/v<1/N,
where Γ,M, and R, are as defined in2.12,4.1, 4.2, respectively. Then the boundary value problem1.3has at least three positive solutionsu1, u2,andu3satisfying
u1< p, min
t∈a,ρbu2t> q, p <u3 witht∈mina,ρbu3t< q. 4.3
Proof. The technique here similar to that used in5Again the coneK, the operatorAis the same as in the previous sections. For allu ∈Kwe haveψu≤ u.Ifu∈Kr, thenu ≤r and the conditionaofD3imply that
lim sup v→ ∞ tmax∈a,b
ft, v v <
1
N. 4.4
Thus there exist aζ > 0 and < 1/Rsuch that ifv > ζ, then maxt∈a,bft, v/v < . For
λ:max{ft, v:v∈0, ζ, t∈a, b}, we haveft, v≤vλfor allv≥0, for allt∈a, b.
Pick any
r >max
q
Γ,
λ
1/N−
. 4.5
Thenu∈Kr implies that
Au max
t∈a,b
b
ρa
Gt, sfs, uωs∇s
≤uλN < rNr1−N r. 4.6
ThusA:Kr → Kr.
The condition b of D3 implies that there exists a positive number r such that
ft, v≤r/Nfort∈a, bandv∈0, r. Ifu∈Kr, then
Au max
t∈a,b
b
ρa
Gt, sfs, uωs∇s≤
r N
N ≤r. 4.7
ThusA: Kr → Kr.Consequently, the assumptionD3holds, then there exist a numberr such thatr > q/ΓandA:Kr → Kr.
The remaining conditions ofTheorem 1.2will now be shown to be satisfied.
ByD1and argument above, we can get thatA:Kp → Kp.Hence, conditioniiof Theorem 1.2is satisfied.
We now consider conditioniofTheorem 1.2. ChooseuKt≡sq/2 fort∈a, b, wheresq/Γ. ThenuKt∈Kψ, q, q/ΓandψuK ψsq/2> q,so that{Kψ, p, q/Γ:
ψu> q}/{}. Foru∈ Kψ, q, q/Γ, we haveq≤ ut ≤q/Γ,t∈a, ρb. Combining with
D2, we get
fort∈a, ρb. Thus, we have
ψAu min
t∈a,ρb
b
ρaGt, sf
s, uωs∇s
> min t∈a,ρb
ρb
ρaGt, sf
s, uωs∇s≥ qM
M q.
4.9
As a result,u∈Kψ, q, syieldsψAu> q.
Lastly, we considerTheorem 1.2iii. Recall that A : K → K. Ifu ∈ Kψ, q, rand
Au> q/Γ, then
ψAu min
t∈a,ρbAut≥ΓAu>Γ q
Γ q. 4.10
Thus, all conditions ofTheorem 1.2are satisfied. It implies that the TPBVP1.3has at least three positive solutionsu1, u2, u3with
u1< p, ψu2> q, p <u3 with ψu3< q. 4.11
5. Examples
Example 5.1. LetT{n/2:n∈N0}.Consider the BVP:
−yΔ∇t yt
y2t−1, t∈0,3⊂T, yρt−2yΔρtt2, t∈−1,0,
3yt 4yΔt t−3, t∈3,5.
5.1
Thena0, b3, τ1−1, τ25, α1, β2, γ 3, δ4,and
qt 1, ξt t2, ηt t−3, ft, v √v, ωt 2t−1. 5.2
Since limv→0ft, v/v ∞, limv→∞ft, v/v 0. It is clear thatH1–H5andH8 are satisfied. Thus, byCorollary 3.3, the BVP5.1has at least one positive solution.
Example 5.2. Let us introduce an example to illustrate the usage ofTheorem 4.1. Let
T
4n2
n29 :n∈N0
Consider the TPBVP:
−yΔ∇t 500y
2t21 y2t21300, t∈
2 5,2
,
y0 0, yΔt 0, t∈2,5.
5.4
Thena2/5, b2, τ12/5, τ25, α1, β0, δ1, γ 0,and
ωt t21, ξt
t−2
5
2
, ηt 0, ft, v 500v
2
v2300. 5.5
The Green function of the BVP5.4has the form
Gt, s ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
s, if 0≤s≤t≤ 64
25,
t, if 0≤t≤s≤ 64
25.
5.6
Clearly,fis continuous and increasing0,∞. We can also see thatαγ >0. By2.12,4.1, and4.2, we getΓ 1/5, M65/32, andN11496/4225.
Now we check that D1, D2, and b of D3 are satisfied. To verify D1, as
f1/10 0.01666611113, we takep1/10, then
fy< p
N 0.03675191371, y∈0, p 5.7
andD1holds. Note thatf3/2 3.722084367, when we setq3/2,
fy≥qM3.046875000, y∈q,5q 5.8
holds. It means thatD2is satisfied. Letr 1500, we have
fy≤500< r
N 551.2787056, y∈0, r, 5.9
from limy→ ∞fy 500, so thatbofD3is met. Summing up, there exist constantsp
1/10, q3/2,andr 1500 satisfying
0< p < q < qΓ < r. 5.10
Thus, byTheorem 4.1, the TPBVP5.4has at least three positive solutionsy1, y2, y3with
y1< 1
10, ψy2> 3 2,
1
10 <y3 withψy3< 3
References
1 D. Jiang and P. Weng, “Existence of positive solutions for boundary value problems of second-order functional-differential equations,”Electronic Journal of Qualitative Theory of Differential Equations, no. 6, pp. 1–13, 1998.
2 N. Aykut, “Existence of positive solutions for boundary value problems of second-order functional difference equations,”Computers & Mathematics with Applications, vol. 48, no. 3-4, pp. 517–527, 2004.
3 M. Bohner and A. Peterson,Dynamic Equations on Time Scales: An Introduction with Applications, Birkh¨auser, Boston, Mass, USA, 2001.
4 M. Bohner and A. Peterson, Eds.,Advances in Dynamic Equations on Time Scales, Birkh¨auser, Boston, Mass, USA, 2003.
5 D. R. Anderson, “Existence of solutions for nonlinear multi-point problems on time scales,”Dynamic Systems and Applications, vol. 15, no. 1, pp. 21–33, 2006.
6 D. R. Anderson, R. Avery, J. Davis, J. Henderson, and W. Yin, “Positive solutions of boundary value problems,” inAdvances in Dynamic Equations on Time Scales, pp. 189–249, Birkh¨auser, Boston, Mass, USA, 2003.
7 D. R. Anderson, R. Avery, and A. Peterson, “Three positive solutions to a discrete focal boundary value problem,”Journal of Computational and Applied Mathematics, vol. 88, no. 1, pp. 103–118, 1998.
8 F. M. Atici and G. Sh. Guseinov, “On Green’s functions and positive solutions for boundary value problems on time scales,”Journal of Computational and Applied Mathematics, vol. 141, no. 1-2, pp. 75–99, 2002.
9 C. J. Chyan, J. M. Davis, J. Henderson, and W. K. C. Yin, “Eigenvalue comparisons for differential equations on a measure chain,”Electronic Journal of Differential Equations, vol. 1998, no. 35, pp. 1–7, 1998.
10 J. M. Davis, P. W. Eloe, and J. Henderson, “Comparison of eigenvalues for discrete Lidstone boundary value problems,”Dynamic Systems and Applications, vol. 8, no. 3-4, pp. 381–388, 1999.
11 C.-Z. Bai and J.-X. Fang, “Existence of multiple positive solutions for functional differential equations,”Computers & Mathematics with Applications, vol. 45, no. 12, pp. 1797–1806, 2003.
12 J. M. Davis, K. R. Prasad, and W. K. C. Yin, “Nonlinear eigenvalue problems involving two classes of functional differential equations,”Houston Journal of Mathematics, vol. 26, no. 3, pp. 597–608, 2000.
13 E. R. Kaufmann and Y. N. Raffoul, “Positive solutions for a nonlinear functional dynamic equation on a time scale,”Nonlinear Analysis: Theory, Methods & Applications, vol. 62, no. 7, pp. 1267–1276, 2005.
14 D. J. Guo and V. Lakshmikantham,Nonlinear Problems in Abstract Cones, vol. 5 ofNotes and Reports in Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1988.
