# Existence of Positive Solutions for Fourth-Order Boundary Value Problems with Sign-Changing Nonlinear Terms

Authors: Xingfang Feng; Hanying Feng
Journal: ISRN Mathematical Analysis
 (2013)

---

## Abstract

the existence of positive solutions for a fourth-order boundary value problem with a sign-changing nonlinear term is investigated. By using Krasnoselskii’s fixed point theorem, sufficient conditions that guarantee the existence of at least one positive solution are obtained. An example is presented to illustrate the application of our main results.


---

## Body

## 1. Introduction

In this paper, we consider the existence of positive solutions to the following fourth-order boundary value problem (BVP):(1)u(4)(t)-λ[f(t,u)+p(t)]=0,a.e.t∈(0,1),u(0)=u(1)=u′′(0)=u′′(1)=0,
where λ is a positive parameter, f:(0,1)×[0,∞)→[0,∞) is continuous and may be singular at t=0,1, and p:(0,1)→(-∞,∞) is Lebesgue integrable and has finitely many singularities in [0,1].Boundary value problems for ordinary differential equations play a very important role in both theory and applications. They are used to describe a large number of physical, biological, and chemical phenomena. The work of Timoshenko [1] on elasticity, the monograph by Soedel [2] on deformation of structures, and the work of Dulcska [3] on the effects of soil settlement are rich sources of such applications. There has been a great deal of research work on BVPs for second and higher order differential equations, and we cite as recent contributions the papers of Anderson and Davis [4], Baxley and Haywood [5], and Hao et al. [6]. For surveys of known results and additional references, we refer the readers to the monographs by Agarwal et al. [7, 8].Many authors have studied the existence of positive solutions for fourth-order boundary value problems where the nonlinearity takes nonnegative values, see [9–13]. However, for problems with sign-changing nonlinearities, only a few studies have been reported.Owing to the importance of high order differential equations in physics, the existence and multiplicity of the solutions to such problems have been studied by many authors, see [9, 12–17]. They obtained the existence of positive solutions provided f is superlinear or sublinear in u by employing the cone expansion-compression fixed point theorem.In [18], by using the strongly monotone operator principle and the critical point theory to discuss BVP
(2)u(4)(t)-f(t,u)=0,t∈[0,1],u(0)=u(1)=u′′(0)=u′′(1)=0,
the authors established some sufficient conditions for f to guarantee that the problem has a unique solution, at least one nonzero solution, or infinitely many solutions.In [10], Feng and Ge considered the fourth-order singular differential equation
(3)u(4)(t)-λg(t)f(u(t))=0,t∈(0,1),
subject to one of the following boundary conditions:
(4)u(0)=u(1)=u′′(0)=u′′(1)=0,u(0)=u′(1)=u′′(0)=u′′′(1)=0,
where λ>0. By using a fixed point index theorem in cones and the upper and lower solutions method, the authors discussed the existence of positive solutions for the above BVP.However, most papers only focus on attention to the case where the nonlinearity has no singularities or/and takes nonnegative values on[0,1] and p(t)=0. Inspired by the work of the above papers, our aim in the present paper is to investigated the existence of positive solutions to BVP (1) by employing the fixed point theorem of cone expansion and compression of norm type. Some well-known results in the literature are generalized and improved.By singularity we mean that the functionf(t,u),p(t) in BVP (1) are allowed to be unbounded at some point. In the paper, BVP (1) is allowed to have finitely many singularities in [0,1]. In BVP(1), p(t) are allowed to change sign and tend to negative infinity. An element u∈C[0,1] for a.e. t∈[0,1] is called a positive solution of BVP (1) if it satisfies BVP (1) and u(t)≥0 for any t∈[0,1].


## 2. Preliminaries and Several Important Lemmas

LetE=C[0,1] be equipped with norm ∥u∥=maxt∈[0,1]|u(t)|, then (E,∥·∥) is a real Banach space.Definition 1.
We define a ordering inE by u≼v for u,v∈E if and only if
(5)u(t)≤v(t),t∈[0,1].
In the following, let us define a cone P in E by
(6)P={u∈E:u≥G(t,t)·∥u∥fort∈[0,1]},
where
(7)G(t,s)={s(1-t),0≤s≤t≤1,t(1-s),0≤t≤s≤1,
is the Green’s function of the following BVP
(8)-u′′(t)=0,t∈(0,1),u(0)=u(1)=0.
Obviously
(9)G(t,s)≤G(t,t),G(t,s)≤G(s,s),G(t,s)≤14,hhhhhhhhhhhhhsfor  anyt∈[0,1],s∈[0,1].
For convenience, we list the following assumptions:(H1)
f:(0,1)×[0,∞)→[0,∞) is continuous and there exist constants α,β,0<α<β<+∞, such that for any (t,u)∈(0,1)×[0,∞),
(10)cβf(t,u)≤f(t,cu)≤cαf(t,u)if0≤c≤1,(H2)p:(0,1)→(-∞,∞) is Lebesgue integrable such that
(11)0<∫01‍p-(s)ds=r<+∞,0<∫01‍G(s,s)[f(s,1)+p+(s)]ds=l<+∞,where p+(s)=max{p(s),0}, p-(s)=max{-p(s),0}.Remark 2.
The inequality (10) is equivalent to the following inequality:
(12)cαf(t,u)≤f(t,cu)≤cβf(t,u)ifc≥1.
For any x∈C[0,1], let us define a function [·]* by
(13)[x(t)]*={x(t),x(t)≥0,0,x(t)<0.
Letw(t)=∫01ds∫01λG(t,s)G(s,τ)p-(τ)dτ, t∈[0,1]. Obviously w(t) is continuous on [0,1]. By (H2), we obtain
(14)0<w(t)=∫01ds∫01λG(t,s)G(s,τ)p-(τ)dτ≤116λ∫01‍p-(τ)dτ<+∞,
so w(t) is well defined in E. By direct computation, we have
(15)w(4)(t)-λp-(t)=0,a.e.t∈(0,1),w(0)=w(1)=w′′(0)=w′′(1)=0,
which imply that w(t) is a positive solutions of the following BVP:
(16)u(4)(t)-λp-(t)=0,a.e.t∈(0,1),u(0)=u(1)=u′′(0)=u′′(1)=0.
Now, we consider the following BVP:(17)u(4)(t)-λ[f(t,[u(t)-w(t)]*)+p+(t)]=0,a.e.t∈(0,1),u(0)=u(1)=u′′(0)=u′′(1)=0.
It is well known that u∈C[0,1] for a.e. t∈(0,1) is a solution of BVP (17) if and only if u is a solution of the following nonlinear integral equation:
(18)u(t)=∫01ds∫01λG(t,s)G(s,τ)[f(τ,[u(τ)-w(τ)]*)hhhhhhhhhhhhhhhhhhh+p+(τ)]dτ,t∈[0,1].
Define an operatorT:E→E as
(19)Tu(t)=∫01ds∫01λG(t,s)G(s,τ)[f(τ,[u(τ)-w(τ)]*)hhhhhhhhhhhhhhhhhhhhh+p+(τ)]dτ,t∈[0,1].
Obviously, the existence of solutions of the BVP (17) is equivalent to the existence of fixed points of the operator T in the real Banach space E.Lemma 3.
Suppose that(H1) holds, then f(t,u) is nondecreasing in u in [0,+∞), for any fixed t∈(0,1).Proof.
For any fixedt∈(0,1) and for any u1,u2∈[0,+∞), without the loss of the generality, let 0≤u1≤u2. If u2=0, obviously the equation f(t,u1)=f(t,u2)=f(t,0) holds. If u2≠0, let c0=u1/u2, then we obtain 0≤c0≤1. It follows from (10) that
(20)f(t,u1)=f(t,c0u2)≤c0αf(t,u2)≤f(t,u2),
that is, f(t,u) is nondecreasing in u in [0,+∞).Lemma 4.
Ifv with w≤v is a positive solution of the BVP (17), then v-w is a positive solution of BVP (1).Proof.
Assume thatv is a positive solution of BVP (17) such that w≤v, then from (17) and the definition of [·]*, we have
(21)v(4)(t)-λ[f(t,[v(t)-w(t)]*)+p+(t)]=0,a.e.t∈(0,1),v(0)=v(1)=v′′(0)=v′′(1)=0.
Let x(t)=v(t)-w(t), then x(4)(t)=v(4)(t)-w(4)(t) for a.e. t∈(0,1), which imply that
(22)v(4)(t)=x(4)(t)+λp-(t)fora.e.t∈(0,1).
Thus, (21) becomes
(23)x(4)(t)-λ[f(t,x(t))+p+(t)-p-(t)]=0,oooooooooooooooooooooa.e.t∈(0,1),x(0)=x(1)=x′′(0)=x′′(1)=0.
Notice that p(t)=p+(t)-p-(t) and (23), we know that x(t) is a positive solution of BVP (1), that is, v-w is a positive solution of BVP (1).Lemma 5.
Assume that(H1) and (H2) hold. Then, T:P→P is well defined and T:P→P is a completely continuous operator.Proof.
For anyu∈P, choose 0<σ<1 such that σ∥u∥<1, then we obtain σ[u(t)-w(t)]*≤σu(t)≤σ∥u∥<1. Thus, by (10), (12), and Lemma 3, we have
(24)f(t,[u(t)-w(t)]*)=f(t,1σ·σ[u(t)-w(t)]*)≤(1σ)βf(t,σ[u(t)-w(t)]*)≤σ-βf(t,σ∥u∥)≤σα-β∥u∥αf(t,1).
Hence, for any t∈[0,1], we get
(25)Tu(t)=∫01ds∫01λG(t,s)G(s,τ)[f(τ,[u(τ)-w(τ)]*)hhhhhhhhhhhhhhhhhhhhh+p+(τ)]dτ≤∫01ds∫01λG(t,t)G(τ,τ)[f(τ,[u(τ)-w(τ)]*)hhhhhhhhhhhhhhhhhhhhh+p+(τ)]dτ≤14λ∫01‍G(τ,τ)[f(τ,[u(τ)-w(τ)]*)+p+(τ)]dτ≤K1∫01‍G(τ,τ)[f(τ,1)+p+(τ)]dτ<+∞,
where K1=(1/4)λ·max{σα-β∥u∥α,1}. Thus,  T:P→E is well defined.
Next, for anyu∈P, let v(t)=Tu(t). Then, there exists t0∈[0,1] such that ∥v∥=v(t0). Since
(26)G(t,s)G(t0,s)={s(1-t)s(1-t0)=1-t1-t0≥1-t≥t(1-t),s≤t0,t,s(1-t)t0(1-s)≥s(1-t)s(1-s)=1-t1-s≥1-t≥t(1-t),t0≤s≤t,t(1-s)s(1-t0)≥t(1-t0)s(1-t0)=ts≥t≥t(1-t),t≤s≤t0,t(1-s)t0(1-s)=tt0≥t≥t(1-t),t,t0≤s,
we obtain
(27)G(t,s)≥G(t,t)G(t0,s).
Hence, we have
(28)v(t)=∫01ds∫01λG(t,s)G(s,τ)[f(τ,[u(τ)-w(τ)]*)hhhhhhhhhhhhhhhhhhhhh+p+(τ)]dτ≥∫01ds∫01λG(t0,s)G(t,t)G(s,τ)hhhhhhhhh×[f(τ,[u(τ)-w(τ)]*)+p+(τ)]dτ=G(t,t)v(t0)=G(t,t)∥v∥,t∈[0,1].
So, we conclude that TP⊂P.
LetD⊂P be any bounded set, then there exists a constant L>0 such that ∥u∥≤L for any u∈D,t∈[0,1], we have
(29)[u(t)-w(t)]*≤u(t)≤∥u∥≤L≤L+1.
By (12), (29) and Lemma 3, for any u∈D,t∈[0,1], we have
(30)f(t,[u(t)-w(t)]*)≤f(t,L+1)≤(L+1)βf(t,1).
From (9), (30), (H2) and Lemma 3, for any u∈D, we have
(31)Tu(t)=∫01ds∫01λG(t,s)G(s,τ)[f(τ,[u(τ)-w(τ)]*)hhhhhhhhhhhhhhhhhhhhhh+p+(τ)]dτ≤∫01ds∫01λG(t,t)G(τ,τ)[f(τ,[u(τ)-w(τ)]*)hhhhhhhhhhhhhhhhhhhhh+p+(τ)]dτ≤14λ∫01‍G(τ,τ)[(L+1)βf(τ,1)+p+(τ)]dτ≤K2∫01‍G(τ,τ)[f(τ,1)+p+(τ)]dτ<+∞,
where K2=(1/4)λ·max{(L+1)β,1}. Therefore, T(D) is uniformly bounded.
SinceG(t,s) is continuous in [0,1]×[0,1], G(t,s) is uniformly continuous. Hence, for any ε>0, there exists δ>0 such that |t1-t2|<δ, for any s∈[0,1] we have
(32)|G(t1,s)-G(t2,s)|≤ε(λmax{(L+1)β,1}l)-1,
from (30), (32) and (H2), for any u∈D, we obtain
(33)|Tu(t1)-Tu(t2)|=|∫01‍λ(G(t1,s)-G(t2,s))ds×∫01‍G(s,τ)[f(τ,[u(τ)-w(τ)]*)+p+(τ)]dτ|≤∫01‍λ|G(t1,s)-G(t2,s)|ds×∫01‍G(τ,τ)[f(τ,[u(τ)-w(τ)]*)+p+(τ)]dτ≤λ∫01‍|G(t1,s)-G(t2,s)|ds·max{(L+1)β,1}×∫01‍G(τ,τ)[f(τ,1)+p+(τ)]dτ<ε.
Therefore, T(D) is equicontinuous on [0,1]. According to the Ascoli-Arzela Theorem, T(D) is a relatively compact set.
At the end, Letun(n=1,2,3,…),u0∈P,un→u0(n→+∞). Then, un is bounded, let L2=sup{∥un∥,n=0,1,2,…}, for any t∈[0,1], we have
(34)[un(t)-w(t)]*≤un(t)≤∥un∥≤L2≤L2+1.
By (12), (34) and Lemma 3, we get
(35)f(t,[un(t)-w(t)]*)≤f(t,L2+1)≤(L2+1)βf(t,1).
From (35), the continuity of f and Lebesgue dominated convergence theorem, we have
(36)|Tun(t)-Tu0(t)|=λ∫01‍G(t,s)ds∫01‍G(s,τ)|f(τ,[un(τ)-w(τ)]*)hhhhhhhhhhhhhhhihhhhhhh-f(τ,[u0(τ)-w(τ)]*)|dτ≤14λ∫01‍G(τ,τ)|f(τ,[un(τ)-w(τ)]*)hhhhhhhhhhhhhh-f(τ,[u0(τ)-w(τ)]*)|dτ⟶0.
Therefore, T:P→P is continuous. So  T:P→P is a completely continuous operator.The proof of our main result is based upon an application of the following fixed point theorem in a cone.Theorem 6 (see [11]).
LetE be a Banach space, and K⊂E be a cone. Assume Ω1,Ω2 are open bounded subsets of E with 0∈Ω1,Ω¯1⊂Ω2, and let T:K∩(Ω¯2∖Ω1)→K be a completely continuous operator such that(i)
∥Tu∥≤∥u∥,u∈K∩∂Ω1, and ∥Tu∥≥∥u∥, u∈K∩∂Ω2; or(ii)
∥Tu∥≥∥u∥,u∈K∩∂Ω1, and ∥Tu∥≤∥u∥, u∈K∩∂Ω2.Then, T has at least one fixed point in K∩(Ω¯2∖Ω1).


## 3. The Main Results and Proofs

Theorem 7.
Suppose that(H1) and (H2) hold. Assume that there exist constants 0<ξ<η<1 and Γ such that
(37)mint∈[ξ,η]f(t,u)u≥Γ,ifu≥rξ(1-η),
where
(38)Γ=2[λξ(1-η)(η-ξ)∫01‍G(ξ,s)A(s)ds]-1,A(s)=min{s(1-η),ξ(1-s)}.
Then, for λ sufficiently small, BVP (1) has at least one positive solution u∈C[0,1] for a.e.  t∈[0,1] in P.Proof.
Set(39)Ωd={u∈P:∥u∥<d},∂Ωd={u∈P:∥u∥=d}.
Let
(40)ρ>r,λ0=min{1,4ρ·{[(max{ρ,1})β+1]l}-1},
where r,l are defined in (H2). For any u∈∂Ωρ, we have ∥u∥=ρ, since
(41)w(t)=∫01ds∫01λG(t,s)G(s,τ)p-(τ)dτ≤14λG(t,t)∫01‍p-(τ)dτ≤λrG(t,t),t∈[0,1],andu(t)≥G(t,t)∥u∥≥rG(t,t),
for any λ∈(0,λ0] and u∈∂Ωρ, we have
(42)u(t)-w(t)≥rG(t,t)-λrG(t,t)≥0,t∈[0,1].
Noting that
(43)0≤u(t)-w(t)≤u(t)≤∥u∥=ρ≤max{ρ,1},hhhhhhht∈[0,1].
From Lemma 3, we get
(44)f(t,[u(t)-w(t)]*)≤f(t,max{ρ,1})≤(max{ρ,1})βf(t,1).
Then, for any λ∈(0,λ0] and u∈∂Ωρ, we have
(45)Tu(t)=∫01ds∫01λG(t,s)G(s,τ)[f(τ,[u(τ)-w(τ)]*)hhhhhhhhhhhhhhhhhhhhh+p+(τ)]dτ≤∫01ds∫01λG(t,t)G(τ,τ)[f(τ,[u(τ)-w(τ)]*)hhhhhhhhhhhhhhhhhhhhh+p+(τ)]dτ≤14λ0[(max{ρ,1})β+1]∫01‍G(τ,τ)[f(τ,1)+p+(τ)]dτ=14λ0[(max{ρ,1})β+1]l≤ρ=∥u∥.
Thus, we obtain that
(46)∥Tu∥≤∥u∥,∀u∈∂Ωρ.
LetR>max{ρ,2r}. For t∈[ξ,η],u∈∂ΩR and λ∈(0,λ0], we have
(47)u(t)-w(t)≥RG(t,t)-λrG(t,t)≥RG(t,t)-λR2G(t,t)≥RG(t,t)-R2G(t,t)=R2G(t,t)≥R2ξ(1-η)>rξ(1-η)>0.
Hence, by (37) and (47), for any u∈∂ΩR and λ∈(0,λ0], we have
(48)Tu(ξ)=∫01ds∫01λG(ξ,s)G(s,τ)[f(τ,[u(τ)-w(τ)]*)hhhhhhhhhhhhhhhhhhhhhh+p+(τ)]dτ≥∫01ds‍∫ξη‍λG(ξ,s)G(s,τ)[f(τ,[u(τ)-w(τ)]*)hhhhhhhhhhhhhhhhhhhhhh+p+(τ)]dτ≥∫01‍λG(ξ,s)A(s)ds∫ξη‍f(τ,[u(τ)-w(τ)]*)dτ≥∫01‍λG(ξ,s)A(s)ds·Γ·∫ξη‍[u(τ)-w(τ)]dτ≥∫01‍λG(ξ,s)A(s)ds·Γ·R2ξ(1-η)(η-ξ)=R=∥u∥.
Thus, we get
(49)∥Tu∥≥Tu(ξ)≥∥u∥,∀u∈∂ΩR.
By Theorem6, we know that T has at least a fixed point u*∈Ω¯R∖Ωρ with ∥u*∥≥ρ.
Thus, for anyλ∈(0,λ0], we have
(50)u*(t)-w(t)≥G(t,t)∥u*∥-λG(t,t)∫01‍p-(τ)dτ≥ρG(t,t)-λrG(t,t)≥ρG(t,t)-λrG(t,t)≥0,t∈[0,1].
It follows from Lemma 4 that u*-w is a positive solution of BVP (1).Corollary 8.
Suppose that(H1) and (H2) hold. Assume that there exist constants 0<ξ1<η1<1 and Γ such that
(51)limu→+∞mint∈[ξ1,η1]f(t,u)u=+∞,
Then, for λ adequately small, BVP (1) has at least one positive solution u∈C[0,1] for a.e.  t∈[0,1] in P.Proof.
Obviously, (51) implies that (37) is satisfied. Thus, by Theorem 7, we know that Corollary 8 holds.


## 4. An Example

Now, we present an example to illustrate the main result.Example 1.
Consider the following BVP(52)u(4)(t)-λ[t(1-t)u3/2+12(11-t-1t)]=0,hhhhhhhhhhhhhhhhhhhhhhhhhha.e.t∈(0,1),u(0)=u(1)=u′′(0)=u′′(1)=0,
where λ is positive parameter, clearly
(53)f(t,u)=t(1-t)u3/2,p(t)=12(11-t-1t).
Let
(54)α=12,β=2,
then (H1) holds. By calculating, it is easy to obtain that
(55)r=∫01‍p-(s)ds=2-1,l=∫01‍G(s,s)[f(s,1)+p+(s)]ds=760(2-1),
thus (H2) holds. Obviously, for any fixed θ∈(0,1/2), we have
(56)limu→+∞mint∈[θ,1-θ]f(t,u)u=+∞.
Let
(57)ρ=2>r,
then
(58)λ0=min{1,4ρ·{[(max{ρ,1})β+1]l}-1}=1,
for any λ∈(0,λ0]. By Corollary 8, we know that BVP (52) has at least one positive solution u∈C[0,1] for a.e.  t∈[0,1] in P.



---

*Source: 349624-2013-11-19.xml*