# Positive Solutions of a Nonlinear Fourth-Order Dynamic Eigenvalue Problem on Time Scales

Authors: Hua Luo; Chenghua Gao
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

Let𝕋 be a time scale and a,b∈𝕋, a<ρ2(b). We study the nonlinear fourth-order eigenvalue problem on 𝕋, uΔ4(t)=λh(t)f(u(t),uΔ2(t)), t∈[a,ρ2(b)]𝕋, u(a)=uΔ(σ(b))=uΔ2(a)=uΔ3(ρ(b))=0 and obtain the existence and nonexistence of positive solutions when 0<λ≤λ* and λ>λ*, respectively, for some λ*. The main tools to prove the existence results are the Schauder fixed point theorem and the upper and lower solution method.


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## Body

## 1. Introduction

The deformation of an elastic beam with one end fixed and the other end free can be described by the nonlinear fourth-order dynamic eigenvalue problem on𝕋(1.1)uΔ4(t)=λh(t)f(u(t),uΔ2(t)),t∈[a,ρ2(b)]T,u(a)=uΔ(σ(b))=uΔ2(a)=uΔ3(ρ(b))=0,
where 𝕋 is a time scale, λ>0 is a parameter, a,b∈𝕋, and a<ρ2(b).Nonlinear dynamic eigenvalue problems of the above type have been studied by some authors, but most of them study only second-order dynamic equations. In 2000, Chyan and Henderson [1] obtained the existence of at least one positive solution for some λ to second-order case of the dynamic equation in problem (1.1) under conjugate boundary value condition and right focal boundary value condition, respectively. Anderson [2] discussed the same second-order dynamic equation under the Sturm-Liouville boundary value condition and directly generalized the result of [1]. Erbe et al. [3] then studied the general second-order Sturm-Liouville dynamic boundary value problem and obtained the existence, nonexistence, and multiplicity results of positive solutions. In 2005, Li and Liu [4] further studied the dependence of positive solutions on the parameter λ>0 for the second-order dynamic equation under the right focal boundary value condition. Luo and Ma [5] in 2006 were concerned with the existence and multiplicity of nodal solutions and obtained eigenvalue intervals of the nonlinear second-order dynamic eigenvalue problem under conjugate boundary value condition by using bifurcation methods. In 2007, Sun et al. [6] obtained some sufficient conditions for the nonexistence and existence of at least one or two positive solutions for the p-Laplacian three-point dynamic eigenvalue problem with mixed derivatives by using the Krasnosel’skii’s fixed point theorem in a cone. In 2009, Luo [7] derived the eigenvalue intervals in which there exist positive solutions of a singular second-order multipoint dynamic eigenvalue problem with mixed derivatives by making use of the fixed point index theory.As for the nonlinear higher-order dynamic eigenvalue problems, few papers can be found in the literature to the best of our knowledge. L. Kong and Q. Kong [8], and Boey and Wong [9] discussed the even-order dynamic eigenvalue problem and the right focal eigenvalue problem, respectively, but their problems do not contain (1.1). Particularly for fourth-order problems and special case λ≡1, Wang and Sun [10] studied the existence of positive solutions for dynamic equations under nonhomogeneous boundary-value conditions describing an elastic beam that is simply supported at its two ends. And both Karaca [11] and Pang and Bai [12] obtained the existence of a solution for two classes of fourth-order four-point problems on time scales by the Leray-Schauder fixed point theorem and the upper and lower solution method, respectively, but the problems they studied are different to (1.1).This paper studies the relationship between the existence and nonexistence of positive solutions and the value of parameterλ>0. We find the existence of a λ* such that problem (1.1) has positive solutions for 0<λ≤λ* and no positive solutions for λ>λ*.The rest of this paper is organized as follows: in Section2, we firstly introduce the time scales concepts and notations and present some basic properties on time scales which are needed later. Next, Section 3 gives some preliminary results relevant to our discussion, and Section 4 is devoted to establish our main theorems.


## 2. Introduction to Time Scales

The calculus theory on time scales, which unifies continuous and discrete analysis, is now still an active area of research. We refer the reader to [13–16] and the references therein for introduction on this theory. For the convenience of readers, we present some necessary definitions and results here.A time scale𝕋 is a nonempty closed subset of ℝ, assuming that 𝕋 has the topology that it inherits from the standard topology on ℝ. Define the forward and backward jump operators σ,ρ:𝕋→𝕋 by
(2.1)σ(t)=inf⁡{τ>t∣τ∈T},ρ(t)=sup⁡{τ<t∣τ∈T}.
Here we put inf∅=sup𝕋,sup∅=inf𝕋. Let μ(t)=σ(t)-t,t∈𝕋 be the graininess function. And 𝕋k which are derived from the time scale 𝕋 is
(2.2)Tk:={t∈T:tisnonmaximalorρ(t)=t},
and 𝕋kn:=(𝕋kn-1)k,n>1,n∈ℕ. Define interval I on 𝕋 by I𝕋=I∩𝕋.Definition 2.1.
Ifu:𝕋→ℝ is a function and t∈𝕋k, then the Δ-derivative of u at the point t is defined to be the number uΔ(t) (provided it exists) with the property that for each ɛ>0, there is a neighborhood U of t such that
(2.3)|u(σ(t))-u(s)-uΔ(t)(σ(t)-s)|⩽ɛ|σ(t)-s|
for all s∈U. The function u is called Δ-differentiable on 𝕋 if uΔ(t) exists for all t∈𝕋k.
The secondΔ-derivative ofuatt∈𝕋k2, if it exists, is defined to be uΔ2(t)=uΔΔ(t):=(uΔ)Δ(t). Similarly,
(2.4)uΔi(t):=(uΔi-1)Δ(t),i>2,i∈N
is called theith Δ-derivative ofuatt∈𝕋ki. We also define the function uσ:=u∘σ.Definition 2.2.
IfUΔ=u holds on 𝕋k, we define the CauchyΔ-integral by
(2.5)∫stu(τ)Δτ=U(t)-U(s),s,t∈Tk.Lemma 2.3.
Ifa,b∈𝕋, a<ρ(b), h:[a,b]𝕋→[0,+∞) is continuous and h(t)≢0 on [a,b)𝕋, then
(2.6)∫abh(t)Δt>0.Proof.
From [15, Theorems  1.28 (Viii) and  1.29], it is clear.Lemma 2.4 (See [14, Theorem  1.16]).
If theΔ-derivative of u exists at t∈𝕋k, then
(2.7)u(σ(t))=u(t)+μ(t)uΔ(t).Define the Banach spaceC(𝕋) to be the set of continuous functions u:𝕋→ℝ with the norm
(2.8)‖u‖∞=max⁡{|u(t)|∣t∈T}.
For i∈ℕ, we define the Banach space CΔi(𝕋) to be the set of the ith Δ-differential functions u:𝕋→ℝ for which uΔi∈C(𝕋ki) with the norm
(2.9)‖u‖i=max⁡{‖u‖∞,‖uΔ‖∞,…,‖uΔi‖∞},
where
(2.10)‖uΔj‖∞=max⁡{|uΔj(t)|∣t∈Tkj},j=0,1,…,i.


## 3. Preliminaries

Throughout this paper, we assume that both(3.1)ξ=min⁡{t∈T∣t≥σ2(b)+3a4},ω=max⁡{t∈T∣t≤3σ2(b)+a4}
exist and a<ξ<ρ(ω)≤ω≤ρ(b). So there exists a number m>0 such that
(3.2)ω-ξ≥m.We also make the following assumptions:(H1)
h:[a,σ(b)]𝕋→[0,+∞) is continuous and h(t)≢0 on [ξ,ω)𝕋;(H2)
f:[0,+∞)×(-∞,0]→[0,+∞) is continuous. f(u,w) is nondecreasing in u, nonincreasing in w and f(0,0)>0.Setv(t)=uΔ2(t),t∈[a,b]𝕋. Then problem (1.1) is equivalent to the system(3.3)uΔ2(t)=v(t),t∈[a,b]T,vΔ2(t)=λh(t)f(u(t),v(t)),t∈[a,ρ2(b)]T,u(a)=uΔ(σ(b))=0,v(a)=vΔ(ρ(b))=0.
According to [14, Corollary  4.84 and Theorem  4.70], the Green’s function of problems
(3.4)uΔ2(t)=0,t∈[a,b]T,u(a)=uΔ(σ(b))=0,vΔ2(t)=0,t∈[a,ρ2(b)]T,v(a)=vΔ(ρ(b))=0,
is of the same form
(3.5)G(t,s)={t-a,t≤s,σ(s)-a,σ(s)≤t,
and the solution of system (3.3) is
(3.6)u(t)=-∫aσ(b)G(t,s)v(s)Δs,t∈[a,σ2(b)]T,v(t)=-λ∫aρ(b)G(t,s)h(s)f(u(s),v(s))Δs,t∈[a,b]T.
Therefore, the solution of problem (1.1) is
(3.7)u(t)=λ∫aσ(b)[∫aρ(b)G(t,s)G(s,j)h(j)f(u(j),uΔ2(j))Δj]Δs,t∈[a,σ2(b)]T.Lemma 3.1.
Green’s function (3.5) is of the following properties:
(3.8)0≤G(t,s)≤min⁡{t-a,σ(s)-a},(t,s)∈[a,σ2(b)]T×[a,σ(b)]T,(3.9)G(t,s)≥14(σ(s)-a),(t,s)∈[ξ,ω]T×[a,σ(b)]T,(3.10)G(t,s)≥t-aσ2(b)-aG(x,s),(t,s)∈[a,σ2(b)]T×[a,σ(b)]T,x∈[a,σ2(b)]T,(3.11)G(t,s)≥14G(x,s),(t,s)∈[ξ,ω]T×[a,σ(b)]T,x∈[a,σ2(b)]T,(3.12)G(t,s)≥σ2(b)-a4,(t,s)∈[ξ,ω]T×[ξ,ω]T.Proof.
We here only give the proof of (3.10), and the others can be obtained easily. We divide the proof into the following four cases.
Case 1 (t≤s,x≤s).
We have(3.13)G(t,s)=t-a≥x-aσ2(b)-a(t-a)=t-aσ2(b)-a(x-a)=t-aσ2(b)-aG(x,s).
Case 2 (t≤s,σ(s)≤x).
We have(3.14)G(t,s)=t-a≥σ(s)-aσ2(b)-a(t-a)=t-aσ2(b)-a(σ(s)-a)=t-aσ2(b)-aG(x,s).
Case 3 (x≤s,σ(s)≤t).
We have(3.15)G(t,s)=σ(s)-a≥x-a≥t-aσ2(b)-a(x-a)=t-aσ2(b)-aG(x,s).
Case 4 (σ(s)≤t,σ(s)≤x).
We have(3.16)G(t,s)=σ(s)-a≥t-aσ2(b)-a(σ(s)-a)=t-aσ2(b)-aG(x,s).Define(3.17)E={u∈CΔ2([a,σ2(b)])∣u(a)=uΔ(σ(b))=uΔ2(a)=uΔ3(ρ(b))=0}.Lemma 3.2.
Foru∈E, one has
(3.18)‖u‖∞≤2[σ2(b)-a]‖uΔ‖∞,(3.19)‖uΔ‖∞≤2[σ2(b)-a]‖uΔ2‖∞,
where ∥u∥∞=max{|u(t)||t∈[a,σ2(b)]𝕋}, ∥uΔ∥∞=max{|uΔ(t)||t∈[a,σ(b)]𝕋}, ∥uΔ2∥∞=max{|uΔ2(t)||t∈[a,b]𝕋}.Proof.
Firstly, we show that (3.18) holds.
Forallt∈[a,σ(b)], we have from u(a)=0 that
(3.20)|u(t)|=|u(a)+∫atuΔ(s)Δs|≤∫at|uΔ(s)|Δs≤‖uΔ‖∞(t-a)≤[σ2(b)-a]‖uΔ‖∞.
Combining this with
(3.21)|u(σ2(b))|=|u(σ(b))+μ(σ(b))uΔ(σ(b))|≤[σ2(b)-a]‖uΔ‖∞+[σ2(b)-a]‖uΔ‖∞=2[σ2(b)-a]‖uΔ‖∞,
we have ∥u∥∞≤2[σ2(b)-a]∥uΔ∥∞.
Secondly, we show that (3.19) holds.
Forallt∈[a,b], we have from uΔ(σ(b))=0 that
(3.22)|uΔ(t)|=|-∫tbuΔ2(s)Δs+uΔ(b)|=|-∫tbuΔ2(s)Δs+uΔ(σ(b))-μ(b)uΔ2(b)|≤‖uΔ2‖∞(b-t)+‖uΔ2‖∞μ(b)≤2[σ2(b)-a]‖uΔ2‖∞.
Combining this with
(3.23)|uΔ(σ(b))|=0≤2[σ2(b)-a]‖uΔ2‖∞,
we have ∥uΔ∥∞≤2[σ2(b)-a]∥uΔ2∥∞.Define(3.24)‖u‖E=max⁡{‖u‖∞,2[σ2(b)-a]‖uΔ‖∞,4[σ2(b)-a]2‖uΔ2‖∞}=4[σ2(b)-a]2‖uΔ2‖∞.
Then E is a Banach space under the norm ∥·∥E. Set
(3.25)(Aλu)(t)=λ∫aσ(b)[∫aρ(b)G(t,s)G(s,j)h(j)f(u(j),uΔ2(j))Δj]Δs,t∈[a,σ2(b)]T.
Then Aλ:E→E. Since Aλ:CΔ2([a,σ2(b)])→CΔ4([a,σ2(b)])↪CΔ2([a,σ2(b)]), we have that Aλ is completely continuous.Lemma 3.3.
Suppose(H1) and (H2) hold, and u(t) is a solution of problem (1.1), then
(3.26)u(t)≥0,t∈[a,σ2(b)]T,uΔ2(t)≤0,t∈[a,b]T,(3.27)u(t)>0,uΔ2(t)<0,t∈[ξ,ω]T,(3.28)min⁡t∈[ξ,ω]T(-uΔ2(t))≥14‖uΔ2‖∞=116[σ2(b)-a]2‖u‖E,(3.29)min⁡t∈[ξ,ω]Tu(t)≥14‖u‖∞,(3.30)min⁡t∈[ξ,ω]Tu(t)≥m64[σ2(b)-a]‖u‖E.Proof.
Byλ>0, (3.7), (3.8) and the fact that h, f are nonnegative functions, (3.26) holds. For t∈[ξ,ω]𝕋, from (3.7), (3.26), (3.2) and (3.12), we have
(3.31)u(t)=λ∫aσ(b)[∫aρ(b)G(t,s)G(s,j)h(j)f(u(j),uΔ2(j))Δj]Δs≥λf(0,0)∫ξω[∫ξωG(t,s)G(s,j)h(j)Δj]Δs≥λm[σ2(b)-a]216f(0,0)∫ξωh(j)Δj>0,-uΔ2(t)=λ∫aρ(b)G(t,s)h(s)f(u(s),uΔ2(s))Δs≥λf(0,0)∫ξωG(t,s)h(s)Δs≥λσ2(b)-a4f(0,0)∫ξωh(s)Δs>0
Therefor (3.27) holds.
Forallt∈[ξ,ω]𝕋,forallx∈[a,σ2(b)]𝕋, from (3.7), (3.11), and (3.12), we have
(3.32)-uΔ2(t)=λ∫aρ(b)G(t,s)h(s)f(u(s),uΔ2(s))Δs≥14λ∫aρ(b)G(x,s)h(s)f(u(s),uΔ2(s))Δs=14[-uΔ2(x)],u(t)=λ∫aσ(b)[∫aρ(b)G(t,s)G(s,j)h(j)f(u(j),uΔ2(j))Δj]Δs≥14λ∫aσ(b)[∫aρ(b)G(x,s)G(s,j)h(j)f(u(j),uΔ2(j))Δj]Δs=14u(x),u(t)=λ∫aσ(b)[∫aρ(b)G(t,s)G(s,j)h(j)f(u(j),uΔ2(j))Δj]Δs≥σ2(b)-a4λ∫ξω[∫aρ(b)G(s,j)h(j)f(u(j),uΔ2(j))Δj]Δs≥σ2(b)-a16λ∫ξω[∫aρ(b)G(x,j)h(j)f(u(j),uΔ2(j))Δj]Δs≥m[σ2(b)-a]16[-uΔ2(x)]=m64[σ2(b)-a]4[σ2(b)-a]2[-uΔ2(x)].
Thus (3.28), (3.29), and (3.30) hold.At the end of this section, we state a lemma of the upper and lower solution method, which is needed for some proofs in next section.Lemma 3.4 (See [17, Theorem  3.3.8]).
LetP be a cone with nonempty interior in Banach space E, and A:P→P a completely continuous and increasing operator. Suppose the following conditions hold:
(i)
there existx0,y0∈P, such that x0≤Ax0,Ay0≤y0,x0≰y0;(ii)
there existϕ∈P and a constant α>0, such that Ax≥α∥Ax∥ϕ,forallx∈P;(iii)
y0 is in the interior of P, and there exists β>0, such that ϕ≥βx0.
Then A has at least one fixed point x≠0 in P.


## 4. The Main Result

Our main result is the following existence theorem.Theorem 4.1.
Suppose(H1) and (H2) hold, and either (H3) or (H4) holds. Here(H3)
lim(u,w)→(+∞,-∞)(f(u,w)/u)=+∞;(H4)
lim(u,w)→(+∞,-∞)(f(u,w)/(-w))=+∞.Then there exists λ*>0, such that problem (1.1) has at least one positive solution for λ∈(0,λ*], and has no positive solution for λ∈(λ*,+∞).To prove Theorem4.1, first, we show that problem (1.1) has positive solutions for some λ small enough.Theorem 4.2.
Suppose(H1) and (H2) hold. Then there exists λ*>0, such that problem (1.1) has at least one positive solution for λ∈(0,λ*].Proof.
The fixed point ofAλ defined in (3.25) is the solution of problem (1.1), so it will be enough to find the fixed point of Aλ.
Set(4.1)M̃=max⁡t∈[a,b]T∫aρ(b)G(t,s)h(s)Δs>0.
Let
(4.2)λ*=1M̃f(4[σ2(b)-a]2,-1).
For λ∈(0,λ*], we have
(4.3)f(4[σ2(b)-a]2,-1)≤1M̃λ.
Set
(4.4)D1={u∈E∣‖u‖E≤4[σ2(b)-a]2}.
If u∈D1, then
(4.5)‖u‖∞≤‖u‖E≤4[σ2(b)-a]2,‖uΔ2‖∞=‖u‖E4[σ2(b)-a]2≤1.
Consequently, for t∈[a,b]𝕋,
(4.6)|(Aλu)Δ2(t)|=|-λ∫aρ(b)G(t,s)h(s)f(u(s),uΔ2(s))Δs|≤M̃λf(‖u‖∞,-‖uΔ2‖∞)≤M̃λf(4[σ2(b)-a]2,-1)≤1.
Then
(4.7)‖Aλu‖E=4[σ2(b)-a]2‖(Aλu)Δ2‖∞≤4[σ2(b)-a]2,
that is, Aλ:D1→D1. By the Schauder fixed point theorem, Aλ has at least one fixed point uλ in E satisfying ∥uλ∥E≤4[σ2(b)-a]2. From Lemma 3.3, uλ is a positive solution of problem (1.1).Next, we show that there exist no positive solution for someλ large enough.Theorem 4.3.
Suppose that(H1) and (H2), hold, and either (H3) or (H4) holds. Then problem (1.1) has no positive solution for λ≫1.Proof.
Supposeuλ∈E is a solution to problem (1.1) for some λ>0. We divide our discussions into two cases.
Case 1 ((H1), (H2) and (H3) hold).
By(H3), for a fixed M1>0, there is H>0 such that
(4.8)f(u,w)≥M1u,u≥mH64[σ2(b)-a],w≤-H16[σ2(b)-a]2.
If u∈E with ∥uλ∥E≥H, then from (3.30) and (3.28), we have
(4.9)min⁡t∈[ξ,ω]Tuλ(t)≥m64[σ2(b)-a]‖un‖E≥mH64[σ2(b)-a],min⁡t∈[ξ,ω]T(-uλΔ2(t))≥14‖uλΔ2‖∞=‖uλ‖E16[σ2(b)-a]2≥H16[σ2(b)-a]2.
Further by (4.8),
(4.10)f(uλ(t),uλΔ2(t))≥M1uλ(t),t∈[ξ,ω]T.
If u∈E with ∥uλ∥E<H, then from (3.27), there is
(4.11)f(uλ(t),uλΔ2(t))≥M2uλ(t),t∈[ξ,ω]T
with
(4.12)M2=min⁡t∈[ξ,ω]Tf(uλ(t),uλΔ2(t))uλ(t)>0.
Set M̂=min{M1,M2}, we have from (4.10) and (4.11) that
(4.13)f(uλ(t),uλΔ2(t))≥M̂uλ(t),t∈[ξ,ω]T.
Combining this with (3.7), (4.13), (3.29), (3.12), and (3.2), we have for t∈[ξ,ω]𝕋,
(4.14)uλ(t)=λ∫aσ(b)[∫aρ(b)G(t,s)G(s,j)h(j)f(uλ(j),uλΔ2(j))Δj]Δs≥λM̂∫aσ(b)[∫ξωG(t,s)G(s,j)h(j)uλ(j)Δj]Δs≥λM̂‖uλ‖∞4∫ξω[∫ξωG(t,s)G(s,j)h(j)Δj]Δs≥λM̂m[σ2(b)-a]264‖uλ‖∞∫ξωh(j)Δj.
If we choose λ such that
(4.15)λM̂m[σ2(b)-a]264∫ξωh(j)Δj>1,
then
(4.16)uλ(t)>‖uλ‖∞,t∈[ξ,ω]T,
which is a contradiction.
Case 2 ((H1), (H2), and (H4) hold).
Similar to Case1, by (H4), there is also an M̂, such that
(4.17)f(uλ(t),uλΔ2(t))≥M̂(-uλΔ2(t)),t∈[ξ,ω]T.
Thus for t∈[ξ,ω]𝕋, we have from (3.7), (4.17), (3.28), and (3.12) that
(4.18)-uλΔ2(t)=λ∫aρ(b)G(t,s)h(s)f(uλ(s),uλΔ2(s))Δs≥λM̂∫ξωG(t,s)h(s)(-uλΔ2(s))Δs≥λM̂[σ2(b)-a]16‖uλΔ2‖∞∫ξωh(s)Δs.
If we take λ such that
(4.19)λM̂[σ2(b)-a]16∫ξωh(s)Δs>1,
then
(4.20)-uλΔ2(t)>‖uλΔ2‖∞,t∈[ξ,ω]T,
which is a contradiction.
Therefore, problem (1.1) has no positive solution for λ≫1.Define the setB={λ>0:problem(1.1)hasatleastonepositivesolutioninE}.Theorem 4.4.
One has(4.21)λ¯∈B⟹λ∈B,∀λ∈(0,λ¯].Proof.
Letλ* be defined as Theorem 4.2. For λ¯≤λ*, the result holds from Theorem 4.2. So, we discuss the case that λ¯>λ*.
Forλ∈(0,λ¯], three cases will be discussed.(1)
λ∈(0,λ*];(2)
λ∈(λ*,λ¯);(3)
λ=λ¯.Cases  (1) and  (3) are clear from Theorem 4.2 and the assumption λ¯∈B, respectively. Now, we deal with Case  (2).
Define(4.22)D2={u∈E∣u(t)≥0,t∈[a,σ2(b)],uΔ2(t)≤0,t∈[a,b]}.
Then D2 is a cone with nonempty interior in E. For λ∈(λ*,λ¯), let Aλ be defined as (3.25). Then Aλ:D2→D2 and Aλ is an increasing operator from (H2). Set u* and u¯ as positive solutions of problem (1.1) at λ* and λ¯, respectively. Then
(4.23)(Aλu*)(t)>(Aλ*u*)(t)=u*(t),(Aλu¯)(t)<(Aλ¯u¯)(t)=u¯(t).
So u¯ is an upper solution of the operator Aλ and u* is a lower solution. If u*≤u¯, then there exists a positive solution u satisfying u*(t)≤u(t)≤u¯(t) for t∈[a,σ2(b)]𝕋  by the upper and lower solution method. If u*≰u¯, we verify the conditions of Lemma 3.4.
Clearly, the condition (i) in Lemma3.4 holds for u*=x0,u¯=y0.
Forallu∈D2,forallt,z∈[a,σ2(b)]𝕋,x∈[ξ,ω]𝕋, we have from (3.10), (3.12), (3.11), and (3.2) that
(4.24)(Aλu)(t)=λ∫aσ(b)[∫aρ(b)G(t,s)G(s,j)h(j)f(u(j),uΔ2(j))Δj]Δs≥λt-aσ2(b)-a∫aσ(b)[∫aρ(b)G(x,s)G(s,j)h(j)f(u(j),uΔ2(j))Δj]Δs≥λt-a4∫ξω[∫aρ(b)G(s,j)h(j)f(u(j),uΔ2(j))Δj]Δs≥λt-a4∫ξω[∫aρ(b)14G(z,j)h(j)f(u(j),uΔ2(j))Δj]Δs≥m16(t-a)[-(Aλu)Δ2(z)].
Then
(4.25)Aλu≥m16‖(Aλu)Δ2‖∞(t-a)=m64[σ2(b)-a]2‖Aλu‖E(t-a),
and the condition (ii) of Lemma 3.4 is satisfied for ϕ=t-a,α=m/64[σ2(b)-a]2>0.
From (3.26) and (3.27), we have that u¯ is in the interior of D2. From (3.8), there is
(4.26)u*(t)=λ*∫aσ(b)[∫aρ(b)G(t,s)G(s,j)h(j)f(u*(j),u*Δ2(j))Δj]Δs≤λ*(t-a)∫aσ(b)[∫aρ(b)G(s,j)h(j)f(u*(j),u*Δ2(j))Δj]Δs≤λ*(t-a)[∫aσ(b)(s-a)Δs][∫aρ(b)h(j)f(u*(j),u*Δ2(j))Δj].
This implies ϕ(t)=t-a≥βu*(t),t∈[a,σ2(b)]𝕋 for
(4.27)β={λ*[∫aσ(b)(s-a)Δs][∫aρ(b)h(j)f(u*(j),u*Δ2(j))Δj]}-1>0,
and the condition (iii) of Lemma 3.4 is satisfied.
By Lemmas3.4 and 3.3, we get a positive solution uλ in D2∈E. That is, λ∈B.Now, we give the proof of Theorem4.1.Proof of Theorem4.1.
From Theorems4.2 and 4.3, B≠∅ is bounded. Thus, we can define λ*=supB. Firstly, we show that λ*∈B.
Choose a sequence{λn}n=1∞, λn∈B(n=1,2,…) which belongs to a compact subinterval in (0,∞), and λn→λ*(n→∞). Then there exists N1>0, for n≥N1,
(4.28)λn>λ*2.
Let un∈E satisfy
(4.29)Aλnun=un.
If {un}n=1∞ is uniformly bounded, then there exists u* such that un→u*(n→∞) and
(4.30)un=Aλnun⟶Aλ*u*(n⟶∞).
Consequently,
(4.31)u*=Aλ*u*.
By Lemma 3.3, u* is a positive solution of problem (1.1), and λ*∈B.
Next, we will prove that{un}n=1∞ is uniformly bounded and the discussions will be divided into two cases.
Case 1 ((H1), (H2), and (H3) hold).
From(H3), there exists H1>0,
(4.32)f(u,w)≥M1u,foru≥mH164[σ2(b)-a],-w≥H116[σ2(b)-a]2,
where M1>0 satisfies
(4.33)M1[σ2(b)-a]2mλ*128∫ξωh(j)Δj>1.
Suppose on the contrary that {un}n=1∞ is unbounded. Then
(4.34)‖un‖E⟶∞,n⟶∞,
which implies that there exists N2>0, ∥un∥E≥H1 for n≥N2. From (3.30) and (3.28), we have
(4.35)min⁡t∈[ξ,ω]Tun(t)≥m64[σ2(b)-a]‖un‖E≥mH164[σ2(b)-a],min⁡t∈[ξ,ω]T(-unΔ2(t))≥116[σ2(b)-a]2‖un‖E≥H116[σ2(b)-a]2.
Subsequently, for t∈[ξ,ω]𝕋, n≥max{N1,N2},
(4.36)un(t)=λn∫aσ(b)[∫aρ(b)G(t,s)G(s,j)h(j)f(un(j),unΔ2(j))Δj]Δs≥λnM1∫aσ(b)[∫ξωG(t,s)G(s,j)h(j)un(j)Δj]Δs≥λnM14‖un‖∞∫ξω[∫ξωG(t,s)G(s,j)h(j)Δj]Δs≥M1[σ2(b)-a]2mλ*128‖un‖∞∫ξωh(j)Δj>‖un‖∞
by (4.32), (3.29), (3.12), (4.28), (3.2), and (4.33). This is a contradiction.
Case 2 ((H1), (H2), and (H4) hold).
From(H4), there exists H2>0,
(4.37)f(u,w)≥M2(-w),foru≥mH264[σ2(b)-a],-w≥H216[σ2(b)-a]2,
where M2>0 satisfies
(4.38)M2[σ2(b)-a]λ*32∫ξωh(s)Δs>1.
Suppose on the contrary that {un}n=1∞ is unbounded. Then similar to Case 1, there exists N3>0 such that ∥un∥E≥H2 for n≥N3. Thus for t∈[ξ,ω]𝕋, n≥max{N1,N3}, we have
(4.39)-unΔ2(t)=λn∫aρ(b)G(t,s)h(s)f(un(s),unΔ2(s))Δs≥λnM2∫ξωG(t,s)h(s)(-unΔ2(s))Δs>M24λ*2‖unΔ2‖∞σ2(b)-a4∫ξωh(s)Δs>‖unΔ2‖∞
by (4.37), (3.28), (3.12), (4.28) and (4.38). This is also a contradiction.
According to the definition ofλ* and B, and Theorem 4.4, we complete the proof.



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*Source: 798796-2012-05-28.xml*