# Positive Solutions for Multipoint Boundary Value Problem of Fractional Differential Equations

Authors: Wenyong Zhong
Journal: Abstract and Applied Analysis
 (2010)

---

## Abstract

We study the existence and multiplicity of positive solutions for the fractionalm-point boundary value problem D0+αu(t)+f(t,u(t))=0, 0<t<1, u(0)=u′(0)=0, u′(1)=∑i=1m-2aiu′(ξi), where 2<α<3, D0+α is the standard Riemann-Liouville fractional derivative, and f:[0,1]×[0,∞)↦[0,∞) is continuous. Here, ai⩾0 for i=1,…,m-2, 0<ξ1<ξ2<⋯<ξm-2<1, and ρ=∑i=1m-2aiξiα-2 with ρ<1. In light of some fixed point theorems, some existence and multiplicity results of positive solutions are obtained.


---

## Body

## 1. Introduction

Fractional calculus is a generalization of ordinary differentiation and integration to arbitrary (noninteger) order. It has been applied to almost every field of science, engineering, and mathematics in the last three decades [1–5]. But the investigation of the theory of fractional differential equations has only been started quite recently.Among all the researches on the theory of the fractional differential equations, the study of the boundary value problems for fractional differential equations recently has attracted a great deal of attention from many researchers. And some results have been obtained on the existence of solutions (or positive solutions) of the boundary value problems for some specific fractional differential equations [6–11].More specifically, Bai [12] discussed the existence of positive solutions for the boundary value problem (BVP for short) (1.1)D0+αu(t)+f(t,u(t))=0,0<t<1,1<α⩽2,u(0)=0,u(1)=βu(η),
where 0<βηα-1<1,0<η<1,D0+α is the standard Riemann-Liouville fractional derivative. Some existence results of at least one positive solution for the above-mentioned   BVP are obtained by the use of fixed point index theory.In [13], Salem considered the existence of Pseudosolutions for the nonlinear m-point   BVP,(1.2)Dαx(t)+q(t)f(t,x(t))=0,0<t<1,α∈(n-1,n],n⩾2,x(0)=x′(0)=x′′(0)=⋯=x(n-2)=0,x(1)=∑i=1m-2ξix(ηi),
where x takes values in a reflexive Banach space E. Here, 0<η1<⋯<ηm-2<1,ξi>0 with ∑i=1m-2ξiηiα-1<1, and x(k) denotes the kth Pseudoderivative of x while Dα denotes the Pseudofractional differential operator of order α. In light of the fixed point theorem given by O'Regan, the criteria for the existence of at least one Pseudo solution for the m-point   BVP are established.Very recently, M. El-Shahed [14] considered the existence and nonexistence of positive solutions to the fractional differential equation (1.3)D0+αu(t)+λf(u(t))=0,0<t<1,2<α<3,
subject to the boundary conditions (1.4)u(0)=u′(0)=0,u′(1)=0.
Their analysis relies on Krasnoselskii's fixed point theorem.Goodrich [15] then considered the BVP for the higher-dimensional fractional differential equation as follows: (1.5)Dαx(t)+f(t,x(t))=0,0<t<1,α∈(n-1,n],n>3,xi(0)=0,0⩽i⩽n-2,Dνx(t)∣t=1=0,1⩽ν⩽n-2,
and a Harnack-like inequality associated with the Green's function related to the above problem is obtained improving the results in [16].Motivated by the aforementioned results and techniques in coping with those boundary value problems of the fractional differential equations, we then turn to investigate the existence and multiplicity of positive solutions for the following   BVP:(1.6)D0+αu(t)+f(t,u(t))=0,0<t<1,2<α<3,u(0)=u′(0)=0,u′(1)=∑i=1m-2aiu′(ξi),
where D0+α is the standard Riemann-Liouville fractional derivative of order α. Here, by a positive solution of   BVP (1.6), we mean a function which is positive on (0,1) and satisfies the equation and boundary conditions in (1.6).Throughout the paper, we will assume that the following conditions hold.(H1)
f:[0,1]×[0,∞)→[0,∞) is continuous.(H2)
ai⩾0 for i=1,…,m-2, 0<ξ1<ξ2<⋯<ξm-2<1, and ρ=∑i=1m-2aiξiα-2 with ρ<1.There is a vast literature concerning the multipoint BVPs for the integer-order differential equations. An important recent paper was given by Webb and Infante [17]; and they established a new unified method for the existence of multiple positive solutions to a large number of nonlinear nonlocal BVPs for the integer-order differential equations. While in the setting of the fractional-order derivatives, as far as we know, the existence of positive solutions for the multipoint   BVP (1.6) has not been discussed in the literature.The rest of the paper is organized as follows. Section2 preliminarily provides some definitions and lemmas which are crucial to the following discussion. In Section 3, we obtain the existence and multiplicity results of positive solutions for the   BVP (1.6) by means of some fixed point theorems. Finally, we give a concrete example to illustrate the possible application of our analytical results.


## 2. Preliminaries

In this section, we preliminarily provide some definitions and lemmas which are useful in the following discussion.Definition 2.1 (see [3]).
The fractional integral of orderα>0 of a function y:(0,∞)↦ℝ is given by
(2.1)I0+αy(t)=1Γ(α)∫0t(t-s)α-1y(s)ds
provided the right side is pointwise defined on (0,∞).Definition 2.2 (see [3]).
The standard Riemann-Liouville fractional derivative of orderα>0 of a continuous function y:(0,∞)↦ℝ is given by
(2.2)D0+αy(t)=1Γ(n-α)(ddt)n∫0ty(s)(t-s)α-n+1ds,
where n=[α]+1 provided the right side is pointwise defined on (0,∞).Lemma 2.3 (see [6]).
Assume thatu∈C(0,1)∩L(0,1) with a fractional derivative of order α>0, then
(2.3)I0+αD0+αu(t)=u(t)+c1tα-1+c2tα-2+⋯+cNtα-N,
for some ci∈ℝ,i=0,1,…,N, where N is the smallest integer greater than or equal to α.By Lemma2.3, we next present an integral presentation of the solution for the   BVP of the linearized equation associated with the   BVP (1.6).Lemma 2.4.
Lety∈C[0,1], then the  BVP
(2.4)D0+αu(t)+y(t)=0,0<t<1,2<α<3,u(0)=u′(0)=0,u′(1)=∑i=1m-2aiu′(ξi)
has a unique solution
(2.5)u(t)=∫01G(t,s)y(s)ds,
where the Green function G is given by G(t,s)=G1(t,s)+G2(t,s), where
(2.6)G1(t,s)={(1-s)α-2tα-1-(t-s)α-1Γ(α),0⩽s⩽t⩽1,(1-s)α-2tα-1Γ(α),0⩽t⩽s⩽1,G2(t,s)=1(1-ρ)Γ(α)(ρ(1-s)α-2-∑i=1m-2ai(ξi-s)α-2χEi(s))tα-1.
Here, Ei=[0,ξi], and χEi denotes the characteristic function of the set Ei for i=1,2,…,m-2.Proof.
Lemma2.3 yields
(2.7)u(t)=-I0+αy(t)+c1tα-1+c2tα-2+c3tα-3.
This, with the condition that u(0)=0, gives c3=0. Furthermore, differentiating both sides of the expression of u(t) with respect to t, we obtain
(2.8)u′(t)=-I0+α-1y(t)+c1(α-1)tα-2+c2(α-2)tα-3.
Now, by the conditions that u′(0)=0 and u′(1)=∑i=1m-2aiu′(ξi), we get that c2=0 and
(2.9)c1=1(α-1)(1-ρ)(I0+α-1y(1)-∑i=1m-2aiI0+α-1y(ξi)).
Hence,
(2.10)u(t)=-1Γ(α)∫0t(t-s)α-1y(s)ds+tα-1Γ(α)∫01(1-s)α-2y(s)ds+tα-1(1-ρ)Γ(α)(ρ∫01(1-s)α-2y(s)ds-∑i=1m-2ai∫0ξi(ξi-s)α-2y(s)ds)=-1Γ(α)∫0t(t-s)α-1y(s)ds+tα-1Γ(α)∫01(1-s)α-2y(s)ds+tα-1(1-ρ)Γ(α)(ρ∫01(1-s)α-2y(s)ds-∫01∑i=1m-2ai(ξi-s)α-2χEi(s)y(s)ds)=∫01G(t,s)y(s)ds.
The proof is complete.The functionsGi(t,s)(i=1,2) have important properties as follows.Lemma 2.5.
Assume (H2) holds, thenGi(1,s)⩾Gi(t,s)⩾q(t)Gi(1,s)⩾0 for all t,s∈[0,1] and i=1,2, where q(t)=tα-1.Proof.
The asserted relation fori=1 is a direct result of Lemma 2.8 in [14]. In addition, the case for i=2 follows from a direct application of the definition of G2(t,s) and the fact that
(2.11)G2(t,s)=1(1-ρ)Γ(α)(∑i=1m-2aiξiα-2(1-s)α-2-∑i=1m-2aiξiα-2(1-sξi)α-2χEi(s))tα-1⩾1(1-ρ)Γ(α)(∑i=1m-2aiξiα-2(1-s)α-2-∑i=1m-2aiξiα-2(1-s)α-2)tα-1=0
holds for t,s∈[0,1]. The proof is complete.Remark 2.6.
The definition ofG(t,s) and Lemma 2.5 yield
(2.12)G(1,s)⩾G(t,s)⩾q(t)G(1,s)⩾0,0⩽t,s⩽1.Remark 2.7.
It is necessary to mention that Bai and Lü [6] showed that their Green's function did not satisfy a classical Harnack-like inequality (HLI) for the homogenous two-point   BVP of fractional differential equation with order α in (1,2]. They proved that G(t,s)>γ(s)G(s,s), where γ(s)→0+ as s→0+, which is a challenge for our seeking positive solutions.
On the other hand, Goodrich [15], for the homogenous   BVP of fractional differential equation with order α in (n-1,n](n>3), established a HLI: mint∈[1/2,1]G(t,s)>γG(1,s), where γ is a positive constant. Using this inequality, the author obtained sufficient conditions on the existence of positive solutions. In Remark 2.6, we get a generalized HLI contrasting to the result in [15]. In fact, for a constant θ∈(0,1], it follows from Remark 2.6 that mint∈[θ,1]G(t,s)>θα-1G(1,s).
From the above discussion, we may infer that it is closely related to both the order of fractional differential equations and boundary conditions that whether or not the Green's function satisfies a traditional HLI, which needs more detailed and rigorous investigations.Next, we introduce some fixed point theorems which will be adopted to prove the main results in the following section.Lemma 2.8 (see [18]).
Let𝔅 be a Banach space, 𝒫⊆𝔅 a cone, and Ω1,Ω2 two bounded open balls of 𝔅 centered at the origin with Ω¯1⊂Ω2. Suppose that 𝒜:𝒫∩(Ω¯2∖Ω1)→𝒫 is a completely continuous operator such that either(B1)
∥𝒜x∥⩽∥x∥,x∈𝒫∩∂Ω1 and ∥𝒜x∥⩾∥x∥,x∈𝒫∩∂Ω2 or(B2)
∥𝒜x∥⩾∥x∥,x∈𝒫∩∂Ω1 and ∥𝒜x∥⩽∥x∥,x∈𝒫∩∂Ω2Holds, then 𝒜 has a fixed point in 𝒫∩(Ω¯2∖Ω1).Definition 2.9.
The mapθ is said to be a nonnegative continuous concave functional on a cone 𝒫 of a real Banach space 𝔅 provided that θ:𝒫→[0,∞) is continuous and θ(tx+(1-t)y)⩾tθ(x)+(1-t)θ(y) for all x,y∈𝒫 and 0⩽t⩽1.Lemma 2.10 (see [19]).
Let𝒫 be a cone in a real Banach space 𝔅,𝒫c={x∈𝒫∣∥x∥<c},θ a nonnegative continuous concave functional on 𝒫 such that θ(x)⩽∥x∥, for all x∈𝒫¯c, and 𝒫(θ,b,d)={x∈𝒫∣b⩽θ(x),∥x∥⩽d}. Suppose that 𝒜:𝒫¯c→𝒫¯c is completely continuous and there exist constants 0<a<b<d⩽c such that(C1)
{x∈𝒫(θ,b,d)∣θ(x)>b}≠∅ and θ(𝒜x)>b for x∈𝒫(θ,b,d),(C2)
∥𝒜x∥<a for x∈𝒫¯a,(C3)
θ(𝒜x)>b for x∈𝒫(θ,b,c) with ∥𝒜x∥>d,then 𝒜 has at least three fixed points x1, x2, and x3 such that
(2.13)‖x1‖<a,b<θ(x2),a<‖x3‖withθ(x3)<b.Remark 2.11.
If there holdsd=c, then condition (C1) of Lemma 2.10 implies condition (C3) of Lemma 2.10.


## 3. Main Results

In order to apply the fixed point theorems to the   BVP (1.6), we first import some notations and operator.Let𝔅=C[0,1] be the classical Banach space with the norm ∥u∥=supt∈[0,1]|u(t)|. Furthermore, define the cone 𝒫 by (3.1)P={u∈B∣u(t)⩾q(t)‖u‖fort∈[0,1]}.
Notice that ∥u∥=u(1) for each u∈𝒫. For a positive number r, define the function space Ωr by (3.2)Ωr={u∈C[0,1]:‖u‖<r}.
Also define the operator 𝒜:𝒫→𝔅 by (3.3)[Au](t)=∫01G(t,s)f(s,u(s))ds.Next, we show some properties of the operator𝒜.Lemma 3.1.
If (H1)-(H2) hold, then𝒜(𝒫)⊂𝒫.Proof.
From the definition of the operator𝒜 and Remark 2.6, it follows that, for u∈𝒫,
(3.4)[Au](t)⩾q(t)∫01G(1,s)f(s,u(s))ds⩾q(t)max⁡t∈[0,1]{∫01G(t,s)f(s,u(s))ds}=q(t)‖[Au]‖,
Therefore, 𝒜(𝒫)⊂𝒫. This completes the proof.Lemma 3.2.
Assume that (H1)-(H2) hold, then the operator𝒜:Ω¯r∩𝒫→𝒫 is completely continuous.Proof.
Lemma3.1 implies that 𝒜(Ω¯r∩𝒫)⊂𝒫. Moreover, the uniform continuity of the function f(t,u) on the compact set [0,1]×[0,r] yields that the operator 𝒜:Ω¯r∩𝒫→𝒫 is continuous.
We now show that𝒜(Ω¯r∩𝒫) is bounded. To this end, let l=max{f(t,u):0⩽t⩽1,0⩽u⩽r}, then, for u∈Ω¯r∩𝒫, it follows from Remark 2.6 that
(3.5)0⩽[Au](t)=∫01G(t,s)f(s,u(s))ds⩽l∫01G(1,s)ds,
which implies that 𝒜(Ω¯r∩𝒫) is bounded.
In addition, for eachu∈Ω¯r∩𝒫,t1,t2∈[0,1],t1<t2, writing F(t1,t2)=|[𝒜u](t1)-[𝒜u](t2)|, we have the following estimation:
(3.6)F(t1,t2)⩽l∫01|G1(t2,s)-G1(t1,s)|ds+l∫01|G2(t2,s)-G2(t1,s)|ds⩽l∫0t1|G1(t2,s)-G1(t1,s)|ds+l∫t1t2|G1(t2,s)-G1(t1,s)|ds+l∫t21|G1(t2,s)-G1(t1,s)|ds+ρl(α-1)(1-ρ)Γ(α)(t2α-1-t1α-1)⩽lΓ(α)∫0t1((t2α-1-t1α-1)(1-s)α-2+(t2-s)α-1-(t1-s)α-1)ds+lΓ(α)∫t1t2((t2α-1-t1α-1)(1-s)α-2+(t2-s)α-1)ds+lΓ(α)∫t21(t2α-1-t1α-1)(1-s)α-2ds+ρl(t2α-1-t1α-1)(α-1)(1-ρ)Γ(α)⩽l(α-1)(1-ρ)Γ(α)(t2α-1-t1α-1)+lαΓ(α)(t2α-t1α).
Now, using the fact that the functions tα-1 and tα are uniformly continuous on [0,1], we conclude that 𝒜(Ω¯r∩𝒫) is an equicontinuous set on [0,1]. It follows from the Arzelà-Ascoli Theorem that 𝒜(Ω¯r∩𝒫) is a relatively compact set. As a consequence, we complete the whole proof.Lemma 3.3.
Assume that (H1)-(H2) hold, thenu∈C[0,1] is a solution of the  BVP (1.6) if and only if it is a fixed point of 𝒜 in 𝒫.Proof.
Ifu∈𝒫 and 𝒜u=u, then
(3.7)u(t)=-I0+αf(t,u(t))+1(α-1)(1-ρ)(I0+α-1f(1,u(1))-∑i=1m-2aiI0+α-1f(ξi,u(ξi)))tα-1.
Thus,
(3.8)D0+αu(t)+f(t,u(t))=0,
and u(0)=0. Furthermore, differentiation of (3.7) with respect to t produces
(3.9)u′(t)=-I0+α-1f(t,u(t))+1(1-ρ)(I0+α-1f(1,u(1))-∑i=1m-2aiI0+α-1f(ξi,u(ξi)))tα-2.
This yields u′(0)=0 and
(3.10)u′(1)=1(1-ρ)(ρI0+α-1f(1,u(1))-∑i=1m-2aiI0+α-1f(ξi,u(ξi)))=∑i=1m-2aiu′(ξi).
Therefore, u is a positive solution of the   BVP (1.6).
On the other hand, ifu is a positive solution of the   BVP (1.6), then Lemma 2.4 implies 𝒜u=u. Moreover, in view of the proof of Lemma 3.1, we also get u(t)⩾q(t)∥u∥ for t∈[0,1]. Hence, u is a fixed point of 𝒜 in 𝒫. We consequently complete the proof.In the following, fixη in (0,1) and set (3.11)L=(∫01G1(1,s)ds+ρ(α-1)(1-ρ)Γ(α))-1,M=(ηα-1∫η1G1(1,s)ds)-1.
We now present two main results on the existence of positive solutions for the BVP (1.6).Theorem 3.4.
Assume that (H1)-(H2) hold. In addition, and suppose that one of the following two conditions holds:(H3)
limu→0mint∈[0,1](f(t,u)/u)=∞,limu→∞maxt∈[0,1](f(t,u)/u)=0,(H4)
limu→0mint∈[0,1](f(t,u)/u)=0,limu→∞maxt∈[0,1](f(t,u)/u)=∞,then the  BVP (1.6) has at least one positive solution.Proof.
Notice that Lemma3.2 guarantees that the operator 𝒜:Ω¯r∩𝒫→𝒫 is completely continuous.
Now, assume that condition (H3) holds. Sincelimu→0mint∈[0,1](f(t,u)/u)=∞, there exists an r1>0 such that
(3.12)f(t,u)⩾ɛ1ufort∈[0,1],0⩽u⩽r1,
where the constant ɛ1>0 is chosen, so that
(3.13)ɛ1∫01G(1,s)q(s)ds>1.
Thus,
(3.14)f(t,u(t))⩾ɛ1u(t)foru∈P∩∂Ωr1,t∈[0,1].
This, together with the definitions of 𝒜 and Remark 2.6, implies that for any u∈𝒫∩∂Ωr1,
(3.15)‖Au‖=∫01G(1,s)f(s,u(s))ds⩾ɛ1∫01G(1,s)u(s)ds⩾ɛ1∫01G(1,s)q(s)ds‖u‖⩾‖u‖.
That is, for u∈𝒫∩∂Ωr1,∥𝒜u∥⩾∥u∥.
On the other hand, fromlimu→∞maxt∈[0,1](f(t,u)/u)=0, it follows that there exists a l1>0 such that
(3.16)f(t,u)⩽ɛ2ufort∈[0,1],u⩾l1,
where the constant ɛ2>0 satisfies
(3.17)ɛ2∫01G(1,s)ds<12.
Now, write l2=max{f(t,u):t∈[0,1],u∈[0,l1]}. Then the relation (3.16) yields
(3.18)f(t,u)⩽ɛ2u+l2fort∈[0,1],u⩾0.
Set r2=max{2r1,2l2∫01G(1,s)ds} and let u∈𝒫∩∂Ωr2, then Remark 2.6 and (3.18) imply
(3.19)‖Au‖=[Au](1)=∫01G(1,s)f(s,u(s))ds⩽∫01G(1,s)(ɛ2u(s)+l2)ds⩽ɛ2∫01G(1,s)ds‖u‖+l2∫01G(1,s)ds⩽‖u‖.
Thus, the operator 𝒜 satisfies condition (B2) of Lemma 2.8. Consequently, the operator 𝒜 has at least one fixed point u∈𝒫∩(Ω¯2∖Ω1), which is one positive solution of the   BVP (1.6).
Next, we suppose that (H4) holds. The proof is similar to that of the case in which assumption (H3) holds and will only be sketched here. Select two positive constantsɛ3 and ɛ4 with ɛ3∫01G(1,s)ds<1 and ɛ4ηα-1∫η1G(1,s)ds>1, respectively. Then, there exist two positive numbers r3 and l3, such that
(3.20)f(t,u)⩽ɛ3ufort∈[0,1],0⩽u⩽r3,(3.21)f(t,u)⩾ɛ4ufort∈[0,1],u⩾l3.
It follows from Remark 2.6 and (3.20) that for u∈𝒫∩∂Ωr3,
(3.22)‖Au‖=∫01G(1,s)f(s,u(s))ds⩽ɛ3∫01G(1,s)u(s)ds⩽‖u‖.
In addition, letr4=max{2r3,l3η1-α}. If u∈𝒫∩∂Ωr4, then
(3.23)u(t)⩾q(t)‖u‖⩾ηα-1r4⩾l3fort∈[η,1],
and by (3.21),
(3.24)f(t,u(t))⩾ɛ4u(t)foru∈P∩∂Ωr4,t∈[η,1].
This yields that, for u∈∂Ωr4∩𝒫,
(3.25)‖Au‖=[Au](1)=∫01G(1,s)f(s,u(s))ds⩾ɛ4∫η1G(1,s)u(s)ds⩾ɛ4∫η1G(1,s)q(s)‖u‖ds⩾ɛ4ηα-1∫η1G(1,s)ds‖u‖⩾‖u‖.
Thus, ∥𝒜u∥⩾∥u∥ for u∈∂Ωr4∩𝒫. Hence, the operator 𝒜 satisfies condition (B1) in Lemma 2.8. As a consequence, the operator 𝒜 has at least one fixed point u∈𝒫∩(Ω¯4∖Ω3). This means that the   BVP (1.6) has at least one positive solution u. We complete the whole proof.Theorem 3.5.
Assume (H1)-(H2) hold. In addition, suppose that there exist constants0<a<b<c with Mb<Lc, such that the following assumptions hold:(H5)
f(t,u)<La, for (t,u)∈[0,1]×[0,a],(H6)
f(t,u)>Mb, for (t,u)∈[η,1]×[b,c],(H7)
f(t,u)⩽Lc, for (t,u)∈[0,1]×[0,c],then the  BVP (1.6) has at least three positive solutions.Proof.
Define the nonnegative continuous concave functionalθ on the cone 𝒫 by
(3.26)θ(u)=min⁡⁡η⩽t⩽1u(t).
Next, we intend to verify that all the conditions in Lemma 2.10 hold with respect to the operator 𝒜. Lemma 2.10 involves parameters a,b,c, and d with 0<a<b<d⩽c. Now, let d=c, then by Remark 2.11, it is sufficient to verify that the conditions (C1) and (C2) in Lemma 2.10 hold. To this end, let u∈𝒫¯c, then ∥u∥⩽c. This together with assumption (H7) implies that for 0⩽t⩽1,f(t,u(t))⩽Lc. This relation and Remark 2.6 yield
(3.27)‖Au‖⩽∫01G(1,s)f(s,u(s))ds⩽∫01G1(1,s)f(s,u(s))ds+∫01ρ(1-ρ)Γ(α)(1-s)α-2f(s,u(s))ds⩽Lc[∫01G1(1,s)ds+ρ(α-1)(1-ρ)Γ(α)]=c.
This, with Lemma 3.2, clearly manifests that the operator 𝒜:𝒫¯c→𝒫¯c is completely continuous. In a similar argument, if u∈𝒫¯a, then assumption (H5) yields f(t,u(t))<La,0⩽t⩽1. Therefore, condition (C2) of Lemma 2.10 is satisfied.
Moreover, the set{x∈𝒫(θ,b,c)∣θ(x)>b} is not empty, since the constant function u(t)≡(b+c)/2 is contained in the set {x∈𝒫(θ,b,c)∣θ(x)>b}. Now, let u∈𝒫(θ,b,c), then 0⩽u(t)⩽c for t∈[0,1] and b⩽minη⩽t⩽1u(t). Thus, we obtain
(3.28)b⩽u(t)⩽c,η⩽t⩽1.
Assumption (H6) and (3.28) imply
(3.29)f(t,u(t))>Mb,η⩽t⩽1.
Hence, it follows from (3.29) and Lemma 2.5 that
(3.30)θ(Au)=min⁡⁡η⩽t⩽1([Au](t))⩾ηα-1∫η1G1(1,s)f(s,u(s))ds>Mbηα-1∫η1G1(1,s)ds=b.
Accordingly, the validity of condition (C1) in Lemma 2.10 is verified. Consequently, by virtue of Lemma 2.10 and Remark 2.11, the operator 𝒜 has at least three fixed points u1,u2, and u3 satisfying
(3.31)max⁡0⩽t⩽1|u1(t)|<a,b<min⁡η⩽1⩽1|u2(t)|,a<max⁡⁡0⩽t⩽1|u3(t)|,min⁡⁡η⩽1⩽1|u3(t)|<b.
These fixed points are positive solutions for the   BVP (1.6). The proof is complete.


## 4. Illustrative Example

Consider the BVP(4.1)D0+5/2u(t)+f(t,u(t))=0,0<t<1,u(0)=u′(0)=0,u′(1)=12u′(164)+14u′(916),
where (4.2)f(t,u)={1t+100+52u3,0⩽t⩽1,u⩽1,ut+100u+u+51,0⩽t⩽1,1<u.Letη=3/4. Clearly, the parameters ρ=(1/2)(1/64)1/2+(1/4)(9/16)1/2=1/4<1,L=135π/88≈4.871,M=480π/173≈51.196. Choosing a=1/10,b=1,c=20, then Mb<Lc. Now, we can verify the validity of conditions (H5)–(H7) in Theorem 3.5. Indeed, indirect computations yield (4.3)f(t,u)=ut+100u+52u3⩽0.01+52a3<La,for(t,u)∈[0,1]×[0,a],f(t,u)=ut+100u+u+51⩾b+51⩾Mb,for(t,u)∈[34,1]×[b,c],f(t,u)⩽ut+100u+u+51⩽0.01+c+51<Lc,for(t,u)∈[0,1]×[0,c].
Hence, conditions (H5)–(H7) in Theorem 3.5 are satisfied for the above-specified functions and parameters. Therefore, in light of Theorem 3.5, we conclude that the above   BVP has at least three positive solutions u1, u2, and u3 defined on [0,1] satisfying max0⩽t⩽1|u1(t)|<a,b<minη⩽1⩽1|u2(t)|, and a<max0⩽t⩽1|u3(t)| with minη⩽1⩽1|u3(t)|<b.



---

*Source: 601492-2011-01-03.xml*