# On the Nonhomogeneous Fourth-Orderp-Laplacian Generalized Sturm-Liouville Nonlocal Boundary Value Problems

Authors: Jian Liu; Zengqin Zhao
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## Abstract

We study the nonlinear nonhomogeneousn-point generalized Sturm-Liouville fourth-order p-Laplacian boundary value problem by using Leray-Schauder nonlinear alternative and Leggett-Williams fixed-point theorem.


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## Body

## 1. Introduction

In this paper, we prove the existence of one and multiple positive solutions of the following differential equations:(1.1)(ϕp(u′′(t)))′′-k2ϕp(u′′(t))=g(t)f(t,u(t)),t∈[0,1],u′′(0)=ϕq(a),u′′(1)=ϕq(b),αu(0)-βu′(0)=∑i=1n-2‍aiu(ξi),γu(1)+δu′(1)=∑i=1n-2‍biu(ξi),
where ϕp is p-Laplacian operator, that is, ϕp(u)=|u|p-2u, p>1, ϕp-1=ϕq, 1/p+1/q=1m, k≠0, α,β,γ,δ⩾0, ξi∈(0,1), a,b,ai,bi∈(0,∞)(i=1,2,…,n-2), f∈C([0,1]×[0,∞),[0,∞)), f(t,0)≢0, g(t)∈C([0,1],[0,∞)).Recently, much attention has been paid to the existence of positive solutions for nonlocal nonlinear boundary value problems (BVPs for short), see [1–4] and references therein. Such problems have potential applications in physics, biology, chemistry, and so forth. For example, a second-order three-point is used as a model for the membrane response of a spherical cap in nonlinear diffusion generated by nonlinear sources and in chemical reactor theory.At the same time, the boundary value problems withp-Laplacian operator have been discussed extensively, for example, see [1–3, 5–7].In [1], Feng et al. researched the boundary value problem
(1.2)(ϕp(u′))′(t)=q(t)f(t,u(t)),0⩽t⩽1,u(0)=∑i=1m‍aiu(ξi),u(1)=∑i=1m‍biu(ξi);
they obtained at least one or two positive solutions under some assumptions imposed on the nonlinearity of f by applying Krasnoselskii fixed-point theorem.Zhou and Ma studied the existence and iteration of positive solutions for the following third-order generalized right-focal boundary value problem withp-Laplacian operator in [3]:
(1.3)(ϕp(u′′))′(t)=q(t)f(t,u(t)),0⩽t⩽1,u(0)=∑i=1m‍αiu(ξi),u′(η)=0,u′′(1)=∑i=1n‍βiu′′(θi);
they established a corresponding iterative scheme for (1.4) by employing the monotone iterative technique.We would also like to mention the work of Zhang and Liu in [7], in which they considered the existence of positive solutions for
(1.4)(ϕp(u′′(t)))′′=f(t,u(t)),0<t<1,u(0)=∑i=1n-2‍aiu(ξi),u(1)=0,u′′(0)=∑i=1n-2‍biu′′(ξi),u′′(1)=0,
by virtue of monotone iterative techniques, and they established a necessary and sufficient condition of positive solutions for their problem.However, to the best of our knowledge, there are not many results concerning about the existence and multiple solutions of fourth-orderp-Laplacian generalized Sturm-Liouville n-point boundary value problems. In this paper, motivated by the study of [4, 8], we committed to consider the fourth-order p-Laplacian generalized Sturm-Liouville nonlocal boundary value problem without assuming any monotonicity condition on the nonlinearity f.The rest of the paper is arranged as follows. We state some definitions and several preliminary results in Section2 that we will use in the sequel. Then in Section 3 we present the existence of one positive solution of BVP (1.1) by Leray-Schauder nonlinear alternative. In Section 4 we get three solutions by Leggett-Williams fixed-point theorem.


## 2. Preliminaries and Some Lemmas

The basic space used in this paper isE=C[0,1]. It is well known that E is a real Banach space with the norm ∥u∥=maxt∈[0,1]|u(t)|.Denote(2.1)φ(t)=β+αt,ψ(t)=γ+δ-γt,t∈[0,1],ρ=αγ+βγ+αδ,Δ=|-∑i=1n-2‍aiφ(ξi)ρ-∑i=1n-2‍aiψ(ξi)ρ-∑i=1n-2‍biφ(ξi)-∑i=1n-2‍biψ(ξi)|.Definition 2.1.
A functionu is said to be a solution of the boundary value problem (1.1) if u∈C2[0,1] satisfies (1.1) and ϕp(u)∈C2[0,1]. In addition, u is said to be a positive solution if u(t)>0 for t∈(0,1), and u is a solution of BVP (1.1).Throughout the paper, we assume the following condition is satisfied:(H0)
ρ>0,ρ-∑i=1n-2aiψ(ξi)>0,ρ-∑i=1n-2biφ(ξi)>0,Δ<0,f(t,u(t))⩾k2(a+b)/mint∈[0,1]g(t).Let y(t)=-ϕp(u′′(t)), then BVP (1.1) is divided into the following two parts:
(2.2)-y′′+k2y=g(t)f(t,u(t)),t∈(0,1),y(0)=a,y(1)=b,(2.3)u′′+ϕq(y)=0,t∈(0,1),αu(0)-βu′(0)=∑i=1n-2‍aiu(ξi),γu(1)+δu′(1)=∑i=1n-2‍biu(ξi).It is not difficult that we can transform (2.2) into the following differential equations:
(2.4)-y′′+k2y=g(t)f(t,u(t))-k2a(1-t)-k2bt,t∈(0,1),y(0)=0,y(1)=0.By routine calculations we can get the following three Lemmas.Lemma 2.2.
The BVP (2.4) has a unique solution
(2.5)y(t)=∫01‍G1(t,s)[g(s)f(s,u(s))-k2a(1-s)-k2bs]ds,
where
(2.6)G1(t,s)=1ρ{sinhks·sinhk(1-t)ksinhk,0⩽s⩽t⩽1,sinhkt·sinhk(1-s)ksinhk0⩽t⩽s⩽1.Lemma 2.3.
The BVP (2.3) has a unique solution
(2.7)u(t)=∫01‍G2(t,s)ϕq(y(s))ds+Aφ(t)+Bψ(t),
where
(2.8)G2(t,s)=1ρ{ψ(t)φ(s),0⩽s⩽t⩽1,ψ(s)φ(t),0⩽t⩽s⩽1,A=1Δ|∑i=1n-2‍ai∫01‍G2(ξi,s)ϕq(y(s))dsρ-∑i=1n-2‍aiψ(ξi)∑i=1n-2‍bi∫01‍G2(ξi,s)ϕq(y(s))ds-∑i=1n-2‍aiψ(ξi)|,B=1Δ|-∑i=1n-2‍aiφ(ξi)∑i=1n-2‍ai∫01‍G2(ξi,s)ϕq(y(s))dsρ-∑i=1n-2‍aiφ(ξi)∑i=1n-2‍bi∫01‍G2(ξi,s)ϕq(y(s))ds|.The proof of Lemma2.3 is similar to that of Lemma 5.5.1 in [8], so we omit it here. From Lemmas 2.2 and 2.3 we can get that u(t) is a solution of BVP (1.1) if and only if
(2.9)u(t)=∫01‍G2(t,s)ϕq{∫01‍G1(s,τ)[g(τ)f(τ,u(τ))-k2a(1-τ)-k2bτ]}ds+A(ϕq(y))φ(t)+B(ϕq(y))ψ(t),
where
(2.10)A(ϕq(y))=1Δ|∑i=1n-2‍ai∫01‍G2(ξi,s)ϕq(∫01‍G1(s,τ)[g(τ)f(τ,u(τ))-k2a(1-τ)-k2bτ]dτ)dsρ-∑i=1n-2‍aiψ(ξi)∑i=1n-2‍bi∫01‍G2(ξi,s)ϕq(∫01‍G1(s,τ)[g(τ)f(τ,u(τ))-k2a(1-τ)-k2bτ]dτ)ds-∑i=1n-2‍aiψ(ξi)|,B(ϕq(y))=1Δ|-∑i=1n-2‍aiφ(ξi)∑i=1n-2‍ai∫01‍G2(ξi,s)ϕq(∫01‍G1(s,τ)[g(τ)f(τ,u(τ))-k2a(1-τ)-k2bτ]dτ)dsρ-∑i=1n-2‍aiφ(ξi)∑i=1n-2‍bi∫01‍G2(ξi,s)ϕq(∫01‍G1(s,τ)[g(τ)f(τ,u(τ))-k2a(1-τ)-k2bτ]dτ)ds|.Lemma 2.4.
Consider,(2.11)Gi(t,s)⩽Gi(s,s),t,s∈[0,1],i=1,2,G1(t,s)⩾Λ0G1(s,s),t∈[ϵ,1-ϵ],s∈[0,1],G2(t,s)⩾ΛG2(s,s),t∈[ϵ,1-ϵ],s∈[0,1],
where
(2.12)Λ0=sinhkϵsinhk,ϵ∈(0,12),Λ=min{φ(ϵ)φ(1),ψ(1-ϵ)ψ(0)},ϵ∈(0,12).Denote(2.13)Λ1=max{∥φ∥,∥ψ∥},Λ2=min{mint∈[ϵ,1-ϵ]φ(t),mint∈[ϵ,1-ϵ]ψ(t)},λ=min{Λ,Λ2Λ1}.Lemma 2.5.
Let(2.14)K={u∣u∈C[0,1],u(t)⩾0,∀t∈[0,1],mint∈[ϵ,1-ϵ]u(t)⩾λ∥u∥},Tu(t)=∫01‍G2(t,s)ϕq{∫01‍G1(s,τ)[g(τ)f(τ,u(τ))-k2a(1-τ)-k2bτ]}ds+A(ϕq(y))φ(t)+B(ϕq(y))ψ(t),
then
(2.15)T(K)⊆K.Proof.
Firstly, we prove thatTu(t)⩾0. For f∈C([0,1]×[0,∞),[0,∞), g(t)∈C([0,1],[0,∞)), then we can get ϕq(y)=ϕq(∫01‍G1(s,τ)[g(τ)f(τ,u(τ))-k2a(1-τ)-k2bτ]dτ)>0, for all u∈K. Furthermore, condition (H0) leads to A(ϕq(y))⩾0, and B(ϕq(y))⩾0, thus, we get Tu(t)⩾0.
Secondly, fort∈[ϵ,1-ϵ], we can get
(2.16)mint∈[ϵ,1-ϵ]Tu(t)=mint∈[ϵ,1-ϵ]{∫01‍G2(t,s)ϕq(∫01‍G1(s,τ)[g(τ)f(τ,u(τ))-k2a(1-τ)-k2bτ]dτ)ds=mint∈[ϵ,1-ϵ]+A(ϕq(y))φ(t)+B(ϕq(y))ψ(t)∫01‍}⩾Λ∫01‍G2(s,s)ϕq(∫01‍G1(s,τ)[g(τ)f(τ,u(τ))-k2a(1-τ)-k2bτ]dτ)ds+A(ϕq(y))φ(t)+B(ϕq(y))ψ(t)⩾Λ∫01‍G2(s,s)ϕq(∫01‍G1(s,τ)[g(τ)f(τ,u(τ))-k2a(1-τ)-k2bτ]dτ)ds+Λ2Λ1Λ1[A(ϕq(y))+B(ϕq(y)])⩾λ{∫01‍G2(s,s)ϕq(∫01‍G1(s,τ)[g(τ)f(τ,u(τ))-k2a(1-τ)-k2bτ]dτ)ds=mint∈[ϵ,1-ϵ]+Λ1[A(ϕq(y))+B(ϕq(y))]∫01‍}=λ∥Tu∥.
Thus we can get thatmint∈[ϵ,1-ϵ]Tu(t)⩾λ∥Tu∥, which means T(K)⊆K.We present here several definitions.Given a coneK in a real Banach space E, a map α is said to be a nonnegative continuous concave (resp., convex) functional on K provided that α:K→[0,+∞) is continuous and
(2.17)α(tx+(1-t)y)⩾tα(x)+(1-t)α(y),(resp.,α(tx+(1-t)y)⩽tα(x)+(1-t)α(y)),
for all, x,y∈K and t∈[0,1].Let0<a<b be given, and let α be a nonnegative continuous concave functional on K. Define the convex sets Pr and P(α,a,b) by
(2.18)Pr={x∈K∣∥x∥<r},P(α,a,b)={x∈K∣a⩽α(x),∥x∥⩽b}.For the convenience of the reader, we present here the Leggett-Williams fixed-point theorem and the Leray-Schauder nonlinear alternative theorem.Lemma 2.6 (see [9], Leggett-Williams fixed-point theorem).
LetA:Pc¯→Pc¯ be a completely continuous operator, and let α be a nonnegative continuous concave functional on K such that α(x)⩽∥x∥  for all x∈Pc¯. Suppose there exist 0<a<b<d⩽c such that(A1)
{x∈P(α,b,d):α(x)>b}≠∅, and α(Ax)>b for x∈P(α,b,d);(A2)
∥Ax∥<a for ∥x∥⩽a;(A3)
α(Ax)>b for x∈P(α,b,c) with ∥Ax∥>d.
ThenA has at least three fixed points x1, x2, and x3 and such that ∥x1∥<a, b<α(x2) and ∥x3∥>a, with α(x3)<b.Now we cite the Leray-Schauder nonlinear alternative.Lemma 2.7 (see [10]).
LetF be a Banach space and Ω a bounded open subset of F,0∈Ω. T:Ω¯→F be a completely continuous operator. Then, either there exists x∈∂Ω,λ>1 such that T(x)=λx, or there exists a fixed point x*∈Ω¯.


## 3. Results of One Nontrivial Solution

In this section, we study the existence of one nontrivial solution of BVP (1.1) by Leray-Schauder nonlinear alternative.Denote(3.1)H1=ϕq(∫01‍G1(τ,τ)g(τ)p(τ)dτ),H2=ϕq(∫01‍G1(τ,τ)g(τ)r(τ)dτ),A^=1Δ|∑i=1n-2‍aiρ-∑i=1n-2‍aiψ(ξi)∑i=1n-2‍bi-∑i=1n-2‍aiψ(ξi)|,B^=1Δ|-∑i=1n-2‍aiφ(ξi)∑i=1n-2‍aiρ-∑i=1n-2‍aiφ(ξi)∑i=1n-2‍bi|,N=M·2q-1(1+A^Λ1+B^Λ1),l=NH21-NH1,Ω={u∈C[0,1],∥u∥<l}.Theorem 3.1.
AssumeNH1<1,f(t,0)≢0, and there exist nonnegative functions p,r∈L1[0,1] such that |f(t,u)|⩽p(t)|u|p-1+r(t),a.e.(t,u)∈[0,1]×[0,+∞), then BVP (1.1) has a nontrivial solution u*∈Ω¯.Proof.
If there exist two nonnegative functionsp,r∈L1[0,1] such that |f(t,u)|⩽p(t)|u|p-1+r(t),a.e.(t,u)∈[0,1]×[0,+∞), we can get that
(3.2)ϕq(∫01‍G1(s,τ)[g(τ)f(τ,u(τ))-k2a(1-τ)-k2bτ]dτ)⩽ϕq(∫01‍G1(s,τ)g(τ)f(τ,u(τ))dτ)⩽ϕq[∫01‍G1(τ,τ)g(τ)(p(τ)|u|p-1+r(τ))dτ]=2q-1[∥u∥ϕq(∫01‍G1(τ,τ)g(τ)p(τ)dτ+ϕq(∫01‍G1(τ,τ)a(τ)r(τ))dτ)]=2q-1(∥u∥H1+H2),
thus, we get
(3.3)∫01‍G2(ξi,s)ϕq(∫01‍G1(s,τ)[g(τ)f(τ,u(τ))-k2a(1-τ)-k2bτ]dτ)ds⩽∫01‍G2(ξi,s)ϕq(∫01‍G1(s,τ)g(τ)f(τ,u(τ))dτ)ds⩽M·2q-1(∥u∥H1+H2).
In the same way, we obtain
(3.4)A(ϕq(y))⩽M·2q-1(∥u∥H1+H2)A^,B(ϕq(y))⩽M·2q-1(∥u∥H1+H2)B^.
Thus we have
(3.5)∥Tu∥=max0⩽t⩽1|Tu(t)|=max0⩽t⩽1|∫01‍G2(t,s)ϕq(∫01‍G1(s,τ)[g(τ)f(τ,u(τ))-k2a(1-τ)-k2bτ])ds=max0⩽t⩽1+A(ϕq(y))φ(t)+B(ϕq(y))ψ(t)∫01‍|⩽max0⩽t⩽1|∫01‍G2(t,s)ϕq(∫01‍G1(s,τ)g(τ)f(τ,u(τ))dτ)ds=max0⩽t⩽1+A(ϕq(y))φ(t)+B(ϕq(y))ψ(t)∫01‍|⩽M·2q-1(∥u∥H1+H2)+M·2q-1(∥u∥H1+H2)(A^Λ1+B^Λ1)=M·2q-1(∥u∥H1+H2)(1+A^Λ1+B^Λ1)=N(∥u∥H1+H2).
Suppose that there existsμ>1 such that
(3.6)Tu=μu,u∈∂Ω.
Therefore,
(3.7)μl=μ∥u∥=∥Tu∥⩽l(NH1+NH2l),
which leads to μ⩽NH1+NH2/l=1, and this contradicts μ>1, then by Lemma 2.7, T has a fixed point u*∈Ω¯; since f(t,0)≢0, the BVP (1.1) has a nontrivial solution u*∈Ω¯. This completes the proof of Theorem 3.1.


## 4. Results of Multiple Positive Solutions

In the following parts, we will study the existence of multiple positive solutions of BVP (1.1) by using Leggett-Williams fixed-point theorem.Denote(4.1)Pc={u∈K∣∥u∥<c}.
Define the nonnegative continuous concave functional on K by
(4.2)α(u)=minϵ⩽t⩽1-ϵu(t).
It is obvious that for each u∈K,α(u)⩽∥u∥.LetM=max0⩽t⩽1∫01‍G2(t,s)ds, m=∫ϵ1-ϵ‍G2(s,s)ds, h=ϕq(∫ϵ1-ϵ‍k2(a+b)G1(τ,τ)dτ), and A^,B^,Λ,Λ0,Λ1 be defined in Sections 2 and 3.We list the following three hypotheses:(H1)
f(t,u)<ϕp(c/M(1+Λ1A^+Λ1B^))/∫01‍G1(τ,τ)g(τ)dτ, for all t∈[0,1], 0⩽u⩽c;(H2)
f(t,u)<ϕp(a/M(1+Λ1A^+Λ1B^))/∫01‍G1(τ,τ)g(τ)dτ, for all t∈[0,1], 0⩽u⩽a;(H3)
f(t,u)>ϕp(2q-1(b/mΛ(1+Λ1A^+Λ1B^)+h))/Λ0∫ϵ1-ϵ‍G1(τ,τ)g(τ)dτ, for all t∈[0,1], b⩽u⩽b/λ.Theorem 4.1.
Assume(H1)–(H3) hold, then BVP (1.1) has at least three positive solutions u1, u2, and u3, such that ∥u1∥<a,b<min[ϵ,1-ϵ]u2(t), and ∥u3∥>a, with min[ϵ,1-ϵ]u3(t)<b.Proof.
Firstly, we prove thatT:Pc¯→Pc¯. The operator T is completely continuous.
From condition(H1), we can get
(4.3)∫01‍G2(ξi,s)ϕq{∫01‍G1(s,τ)[g(τ)f(τ,u(τ))-k2a(1-τ)-k2bτ]dτ}ds⩽Mϕq(∫01‍G1(τ,τ)g(τ)f(τ,u(τ))dτ)⩽c1+Λ1A^+Λ1B^.
Hence,(4.4)∥Tu∥=max0⩽t⩽1|Tu(t)|=max0⩽t⩽1|∫01‍G2(t,s)ϕq{∫01‍G1(s,τ)[g(τ)f(τ,u(τ))-k2a(1-τ)-k2bτ]dτ}ds=max0⩽t⩽1+A(ϕq(y))φ(t)+B(ϕq(y))ψ(t)∫01‍|⩽max0⩽t⩽1|∫01‍G2(t,s)ϕq{∫01‍G1(s,τ)[g(τ)f(τ,u(τ)]dτ]}ds=mint∈[ϵ,1-ϵ]+A(ϕq(y))φ(t)+B(ϕq(y))ψ(t)|⩽c1+Λ1A^+Λ1B^+c1+Λ1A^+Λ1B^Λ1A^+c1+Λ1A^+Λ1B^Λ1B^=c.
Thus we get ∥Tu∥⩽c; therefore, T:Pc¯→Pc¯. The operator T is completely continuous by an application of Ascoli-Arzela theorem.
In the same way, condition(H2) implies that condition (A2) of Lemma 2.6 is satisfied. In the following, we show that condition (A1) of Lemma 2.6 is satisfied.
Let(4.5)u0(t)=bλ,t∈[0,1],
then
(4.6)u0∈P(α,b,bλ),α(u0)=bλ>b,
thus, {u∈P(α,b,b/λ)∣α(u)>b}≠∅.
Ifu∈P(α,b,b/λ), then b⩽u(s)⩽b/λ, s∈[ϵ,1-ϵ].
By condition(H3), we obtain
(4.7)∫01‍G2(ξi,s)ϕq{∫01‍G1(s,τ)[g(τ)f(τ,u(τ)-k2a(1-τ)-k2bτ)]dτ}ds⩾Λ∫ϵ1-ϵ‍G2(s,s)[12q-1ϕq(∫ϵ1-ϵ‍G1(s,τ)g(τ)f(τ,u(τ)))]dτds-ϕq(∫ϵ1-ϵ‍G1(s,τ)(k2a(1-τ)+k2bτ)dτ]ds⩾Λ∫ϵ1-ϵ‍G2(s,s)[12q-1ϕq(∫ϵ1-ϵ‍G1(s,τ)g(τ)f(τ,u(τ)))]dτds-ϕq(∫ϵ1-ϵ‍G1(τ,τ)(k2a(1-τ)+k2bτ)dτ]ds⩾Λ∫ϵ1-ϵ‍G2(s,s)[12q-1ϕq(∫ϵ1-ϵ‍G1(s,τ)g(τ)f(τ,u(τ)))]dτ-ϕq(∫ϵ1-ϵ‍G1(τ,τ)(k2a+k2b)dτ]ds⩾b1+Λ1A^+Λ1B^.
Thus we get(4.8)α(Tu(t))=mint∈[ϵ,1-ϵ]Tu(t)⩾mint∈[ϵ,1-ϵ](∫ϵ1-ϵ‍G2(s,s)ϕq{∫01‍G1(s,τ)[g(τ)f(τ,u(τ)-k2a(1-τ)-k2bτ])dτ})ds+A(ϕq(y))φ(t)+B(ϕq(y))ψ((t))⩾b1+Λ2A^+Λ2B^+b1+Λ2A^+Λ2B^Λ2A^+b1+Λ2A^+Λ2B^Λ2B^=b.
Therefore, condition(A1) of Lemma 2.6 is satisfied.
Finally, we show that condition(A3) of Lemma 2.6 is satisfied.
Ifu∈P(α,b,c), and ∥Tu∥>b/λ, then α(Tu(t))=minϵ⩽t⩽1-ϵTu(t)⩾λ∥Tu∥>b.
Therefore, condition(A3) of Lemma 2.6 is also satisfied. By Lemma 2.6, there exist three positive solutions u1, u2, and u3 such that ∥u1∥<a,b<mint∈[ϵ,1-ϵ]u2(t), and ∥u3∥>a, with mint∈[ϵ,1-ϵ]u3(t)<b. Thus we completed the proof.



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