# Positive Solution of Singular Fractional Differential Equation in Banach Space

Authors: Jianxin Cao; Haibo Chen
Journal: Journal of Applied Mathematics
 (2011)

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## Abstract

We investigated a singular multipoint boundary value problem for
fractional differential equation in Banach space. The nonlinear termf(t,x,y) is
positive and singular at x=θ and (or) y=θ. Employing regularization, sequential techniques, and diagonalization methods, we obtained some new existence results of positive solution.


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## Body

## 1. Introduction

Recently, fractional differential equations have been investigated extensively. The motivation for those works rises from both the development of the theory of fractional calculus itself and the applications of such constructions in various sciences such as physics, chemistry, aerodynamics, and electrodynamics of the complex medium. For examples and details, see [1–5] and the references therein.Prompted by the application of multipoint boundary value problem (BVP for short) to applied mathematics and physics, these problems have provoked a great deal of attention by many authors. Here, for fractional differential equations, we refer the reader to [6–12]. Rehman and Khan [7] studied the problem(1.1)Dαy(t)+f(t,y(t),Dβy(t))=0,t∈(0,1),y(0)=0,Dβy(1)-∑i=1m-2biDβy(ξi)=y0,
where 1<α≤2, 0<β≤α-1, bi≥0, 0<ξi<1, (i=1,2,…,m-2) with γ=∑i=1m-2biξiα-β-1<1, and Dβ represents the standard Riemann-Liouville fractional derivative. The existence and uniqueness of solutions were obtained, by means of Schauder fixed-point theorem and Banach contraction principle. Importantly, they gave the Green function of the multipoint BVP (1.1). But they have not proved the positivity of Green function, so the existence of positive solution is unobtainable. However, only positive solutions are useful for many applications, as some physicists pointed out.The authors of [13–17] investigated singular problem for fractional differential equations with bounded domain. In particular, Agarwal et al. [13] considered the following Dirichlet problem:
(1.2)Dαy(t)+f(t,y(t),Dμy(t))=0,y(0)=y(1)=0,
where 1<α≤2, 0<μ≤α-1. f(t,x,y) satisfies the Carathéodory conditions and is singular at x=0. In order to overcome the singularity, they used regularization and sequential techniques for the existence of a positive solution.When the domain where the problem is considered is unbounded, there are few papers about BVP for fractional differential equations in literatures. This situation has changed recently. One can find some works, for example, see [18–22].In [21], the following BVP:(1.3)Dαy(t)+f(t,y(t))=0,t∈(0,+∞),α∈(1,2),y(0)=0,lim⁡t→+∞Dα-1y(t)=βy(ξ)
was studied. Using the equicontinuity on any compact intervals and the equiconvergence at infinity of a bounded set, the authors proved that the corresponding operator was completely continuous, then the existence of solutions was obtained by the Leray-Schauder nonlinear alternative theorem.Let(E,∥·∥) be a real Banach space. P is a cone in E which defines a partial ordering in E by x≤y if and only if y-x∈P. P is said to be normal if there exists a positive constant N such that θ≤x≤y implies ∥x∥≤N∥y∥, where θ denotes the zero element of E, and the smallest N is called the normal constant of P (it is clear that N≥1). If x≤y and x≠y, we write x<y. Let P+=P∖{θ}. So, x∈P+ if and only if x>θ. For details on cone theory, see [23].In this paper, we are concerned with the existence of positive solution of a BVP for fractional differential equation with bounded domain(1.4)Dαy(t)+f(t,y(t),Dβy(t))=θ,a.e.t∈[0,T],(1.5)y(0)=θ,Dβy(T)-∑i=1m-2aiy(ξi)-∑i=1m-2biDβy(ξi)=y0,
or with unbounded domain(1.6)Dαy(t)+f(t,y(t),Dβy(t))=θ,a.e.t∈[0,∞),(1.7)y(0)=θ,lim⁡t→+∞Dβy(t)-∑i=1m-2aiy(ξi)-∑i=1m-2biDβy(ξi)=y0.
Here, 1<α≤2, 0<β≤α-1, ξi>0, ai,bi≥0(i=1,2,…,m-2), y0≥θ are real numbers, and Dα is the standard Riemann-Liouville fractional derivative. And f:[0,+∞)×P+×P+→P+ is singular at x=θ and y=θ and satisfies other conditions which will be specified later. In addition, f(t,x,y) is the Carathéodory function.We say thatf satisfies the Carathéodory conditions on [0,+∞)×B, B=P+×P+(f∈Car([0,+∞)×B)) if(i)
f(·,x,y):[0,+∞)→E is measurable for all (x,y)∈B,(ii)
f(t,·,·):B→E is continuous for a.e.t∈[0,+∞),(iii)
for each compact setK⊂B, there is a function ϕK∈L1[0,+∞) such that
(1.8)‖f(t,x,y)‖≤ϕK(t),fora.e.t∈[0,+∞),∀(x,y)∈K.No contribution exists, as far as we know, concerning   the existence of positive solution of the problems (1.4)-(1.5) and (1.6)-(1.7). In the present paper, we consider, firstly, the case of bounded domain, that is, BVP (1.4)-(1.5), and give some existence results by means of regularization process combined with fixed-point theorem due to Krasnosel'skii. Then we investigate the BVP (1.6)-(1.7). As we know, [0,∞) is noncompact. In order to overcome these difficulties, based on the results of BVP (1.4)-(1.5), we use diagonalization process to establish the existence of positive solutions for BVP (1.6)-(1.7). Let us mention that this method was widely used for integer-order differential equations, see, for instance, [5, 22]. Using diagonalization process, Agarwal et al. [20] have considered a class of boundary value problems involving Riemann-Liouville fractional derivative on the half line. And Arara et al. [19] continued this study by considering a BVP with the Caputo fractional derivative.The remainder of this paper is organized as follows. In Section2, we introduced some notations, definitions, and preliminary facts about the fractional calculus, which are used in the next two sections. In Section 3, the case with bounded domain is considered. In Section 4, we discuss the existence of a positive solution for the BVP (1.6)-(1.7). We end this paper with giving an example to demonstrate the application of our results in Section 5.


## 2. Preliminaries

Now, we introduce the Riemann-Liouville fractional- (arbitrary)-order integral and derivative as follows.Definition 2.1.
The fractional- (arbitrary)-order integral of the functionv(t)∈L1([0,b],ℝ) of μ∈ℝ+ is defined by
(2.1)Iμv(t)=1Γ(μ)∫0t(t-s)μ-1v(s)ds,t>0.Definition 2.2.
The Riemann-Liouville fractional derivative of orderμ>0 for a function v(t) given in the interval [0,∞) is defined by
(2.2)Dμv(t)=1Γ(n-μ)(ddt)n∫0t(t-s)n-μ-1v(s)ds
provided that the right hand side is point wise defined. Here, n=[μ]+1 and [μ] means the integral part of the number μ, and Γ is the Euler gamma function.The following properties of the fractional calculus theory are well known, see, for example, [2, 4]:(i)
DβIβv(t)=v(t) for a.e. t∈[0,T], where v(t)∈L1[0,T], β>0,(ii)
Dβv(t)=0 if and only if v(t)=∑j=1ncjtβ-j, where cj(j=1,2,…,n) are arbitrary constants, n=[β]+1, β>0,(iii)
Iβ:C([0,T])→C([0,T]), Iβ:L1([0,T])→L1([0,T]), β>0,(iv)
DβIα=Iα-β and Dβtα=(Γ(β+1)/Γ(β-α+1))tβ-α for t∈[0,T], α-β>0.More details on fractional derivatives and their properties can be found in [2, 4].For the sake of convenience, we introduce the following assumptions:(H0)
(2.3)Δ=Γ(α)Tα-β-1Γ(α-β)-∑i=1m-2aiξiα-1-∑i=1m-2biΓ(α)ξiα-β-1Γ(α-β)>0,(H1)
f∈Car([0,+∞)×B), B=(0,+∞)×(0,+∞),
(2.4)lim⁡‖x‖→0‖f(t,x,y)‖=+∞,fora.e.t∈[0,+∞)andally∈P+,lim⁡‖y‖→0‖f(t,x,y)‖=+∞,fora.e.t∈[0,+∞)andallx∈P+,
and there exists a positive constant ϖ such that for all T0≥T,
(2.5)‖f(t,x,y)‖≥ϖ(1-tT0)2+β-αfora.e.t∈[0,T0]andall(x,y)∈B,(H2)
f fulfills the estimate,
(2.6)‖f(t,x,y)‖≤γ(t)(γ0(t)+q1(‖x‖)+p1(‖y‖)+q(‖x‖)+p(‖y‖))fora.e.t∈[0,+∞),andall(x,y)∈B,
where γ,γ0∈L1[0,+∞), q1,p1,q,p∈C((0,+∞),ℝ+), q1,p1 are nonincreasing, and, for any T0≥T,
(2.7)∫0T0γ(t)q1(K1(tα-1(T0-t)2)T0)dt<+∞,K1=ϖ2Γ(α),∫0T0γ(t)p1(K2(tα-β-1(T0-t)2)T0)dt<+∞,K2=ϖ2Γ(α-β),
while q,p are nondecreasing and
(2.8)lim⁡‖x‖→+∞q(‖x‖)+p(‖x‖)‖x‖=0,(H3)
for a.e.t∈[0,+∞), and for all D⊂P, f(t,D,D) is relatively compact.Remark 2.3.
It follows from (2.4) that under condition (H2), lim∥x∥→0q1(∥x∥)=+∞ and lim∥y∥→0p1(∥y∥)=+∞.In the sequel,L1([0,T],ℝ) denote the Banach space of functions y:[0,T]→ℝ which are Lebesgue integrable with the norm
(2.9)‖y‖L1=∫0T|y(t)|dt.We give now some auxiliary lemmas in scalar space, which will   take an important role throughout the paper.Lemma 2.4.
Suppose thath(t)∈L1([0,T]) and that (H0) holds, then the unique solution of linear BVP Dαy(t)+h(t)=0, a.e. t∈[0,T] with the boundary condition (1.5) is given by
(2.10)y(t)=∫0TG(t,s)h(s)ds+y0Δtα-1,
where
(2.11)G(t,s)=1Δ{(T-s)α-β-1tα-1Γ(α-β)-tα-1Γ(α)∑j=im-2aj(ξj-s)α-1-tα-1Γ(α-β)∑j=im-2bj(ξj-s)α-β-1-Δ(t-s)α-1Γ(α),t≥s,ξi-1<s≤ξi,i=1,2,…,m-1,(T-s)α-β-1tα-1Γ(α-β)-tα-1Γ(α)∑j=im-2aj(ξj-s)α-1-tα-1Γ(α-β)∑j=im-2bj(ξj-s)α-β-1,t≤s,ξi-1<s≤ξi,i=1,2,…,m-1.Proof.
The proof is similar to that of [7, Lemma 2.2], so we omit it.Lemma 2.5.
Suppose that(H0) holds, then G(t,s) defined as (2.11) has the following properties:(i)
G(t,s) is uniformly continuous about t in [0,T],(ii)
G(t,s)≥0 for all (t,s)∈[0,T]×[0,T] and G(t,s)≤E, where
(2.12)E=T2α-β-2ΔΓ(α-β),(iii)
∫0TG(t,s)R(s)ds≥tα-1(T-t)2/2TΓ(α) if R(s)≥(1-t/T)2+β-α.Proof.
From (2.3), it is easy to verify (i) and (ii). We now show that (iii) is true. Firstly, if t≥s, then (2.11) gives
(2.13)G(t,s)=1Δ{(T-s)α-β-1tα-1Γ(α-β)-tα-1Γ(α)∑j=im-2aj(ξj-s)α-1-tα-1Γ(α-β)∑j=im-2bj(ξj-s)α-β-1-Δ(t-s)α-1Γ(α),(ξi-1<s≤ξi,i=1,2,…,m-1).
Then,
(2.14)G(t,s)≥1Δ{(T-s)α-β-1tα-1Γ(α-β)-tα-1Γ(α)∑j=1m-2aj(ξj-s)α-1-tα-1Γ(α-β)∑j=1m-2bj(ξj-s)α-β-1-Δ(t-s)α-1Γ(α)}≥tα-1Δ{(T-s)α-β-1Γ(α-β)-1Γ(α)∑j=1m-2ajξjα-1(1-sξj)α-1-1Γ(α-β)∑j=1m-2bjξjα-β-1(1-sξj)α-β-1-Δ(1-s/t)α-1Γ(α)}≥tα-1(1-s/T)α-β-1Δ{Tα-β-1Γ(α-β)-1Γ(α-β)∑j=1m-2bjξjα-β-1-(1-s/T)βΓ(α)(∑j=1m-2ajξjα-1+Δ)}.
From (2.14) and (2.3), we deduce from the Lagrange mean value theorem that
(2.15)G(t,s)≥tα-1(1-s/T)α-β-1(∑j=1m-2ajξjα-1+Δ)ΔΓ(α)×(1-(1-sT)β)≥tα-1(1-s/T)α-β-1(∑j=1m-2ajξjα-1+Δ)ΔΓ(α)βξβ-1sT.
In view of (1-s/T)≤ξ≤1 and β<1, one can obtain for t≥s that
(2.16)G(t,s)≥tα-1(1-s/T)α-β-1(∑j=1m-2ajξjα-1+Δ)ΔΓ(α)Tβs.
Analogously, if t≤s, one has
(2.17)G(t,s)≥tα-1(1-s/T)α-β-1Γ(α).
It follows from (2.16) and (2.17) that
(2.18)∫0TG(t,s)R(s)ds=∫0tG(t,s)R(s)ds+∫tTG(t,s)R(s)ds≥∫0ttα-1(∑j=1m-2ajξjα-1+Δ)ΔΓ(α)Tβs(1-sT)ds+∫tTtα-1Γ(α)(1-sT)ds≥tα-1(T-t)22TΓ(α).
The proof is complete.


## 3. Existence Results for BVP (1.4)-(1.5)

In this section, we discuss the uniqueness, existence, and continuous dependence of positive solution for problem (1.4)-(1.5). To this end, we introduce some auxiliary technical lemmas.Let𝔼={x∈C([0,T],E):Dβx∈C([0,T],E)} equipped with the norm ∥x∥*=max{∥x∥,∥Dβx∥}, then 𝔼 is a real Banach space (see [24]).Since the nonlinear termf(t,x,y) is singular at x=θ and y=θ, we use the following regularization process. For each m∈N+, define fm by the formula(3.1)fm(t,x,y)={f(t,x,y)ifx≥cm,y≥cm,f(t,cm,y)if0≤x<cm,y≥cm,f(t,x,cm)ifx≥1m,0≤y<1m,f(t,cm,cm)if0≤x<cm,0≤y<cm,
where c>θ is a given element of 𝔼 and ∥c∥=1.Remark 3.1.
The functionfm defined by (3.1) satisfies fm∈Car([0,T]×B*), B*=P×P. And conditions (H1) and (H2) imply
(3.2)‖fm(t,x,y)‖≥ϖ(1-tT)1+β-α,fora.e.t∈[0,T]andall(x,y)∈B*,(3.3)‖fm(t,x,y)‖≤γ(t)(γ0(t)+q1(‖1m‖)+p1(‖1m‖)+q(1)+p(1)+q(‖x‖)+p(‖y‖)),fora.e.t∈[0,T]andall(x,y)∈B*,(3.4)‖fm(t,x,y)‖≤γ(t)(γ0(t)+q1(‖x‖)+p1(‖y‖)+q(1)+p(1)+q(‖x‖)+p(‖y‖)),fora.e.t∈[0,T]andall(x,y)∈B.Remark 3.2.
The functionfm defined by (3.1) satisfies limm→+∞fm=f.Define operatorQm:P+→P+ by the following: (3.5)(Qmy)(t)=∫0TG(t,s)fm(s,y(s),Dβy(s))ds+y0Δt(α-1).Lemma 3.3.
Suppose that(H0) holds, then
(3.6)(DβQmy)(t)=∫0TDβG(t,s)fm(s,y(s),Dβy(s))ds+y0Γ(α)ΔΓ(α-β)t(α-β-1),
where
(3.7)DβG(t,s)=Γ(α)ΔΓ(α-β){(T-s)α-β-1tα-β-1Γ(α-β)-tα-β-1Γ(α)∑j=im-2aj(ξj-s)α-1-tα-β-1Γ(α-β)∑j=im-2bj(ξj-s)α-β-1-Δ(t-s)α-β-1Γ(α),t≥s,ξi-1<s≤ξi,i=1,2,…,m-1,(T-s)α-β-1tα-β-1Γ(α-β)-tα-β-1Γ(α)∑j=im-2aj(ξj-s)α-1-tα-β-1Γ(α-β)∑j=im-2bj(ξj-s)α-β-1,t≤s,ξi-1<s≤ξi,i=1,2,…,m-1.Proof.
The proof is similar to that of [7, Lemma 2.2], and we omit it.Lemma 3.4.
Suppose that(H0) holds, then DβG(t,s) defined as (3.7) has the following properties:(i)
DβG(t,s) is uniformly continuous about t in [0,T],(ii)
DβG(t,s)≥0 for all (t,s)∈[0,T]×[0,T] and DβG(t,s)≤ED, where
(3.8)ED=Γ(α)T2α-2β-2Δ[Γ(α-β)]2,(iii)
∫0TDβG(t,s)R(s)ds≥tα-β-1(T-t)2/2TΓ(α-β) if R(s)≥(1-t/T)2+β-α.Proof.
The proof of this Lemma is similar to that of Lemma2.5. Hence it is omitted.Lemma 3.5.
Suppose that(H0) and following condition (H4) hold:(H4)
there exist positive constantsL,LD such that(3.9)‖f(s,x(s),Dβx(s))-f(s,y(s),Dβy(s))‖≤L‖x-y‖+LD‖Dβx-Dβy‖
and τ=max{E(L+LD),ED(L+LD)}<1, then Qm has a unique fixed point.Proof.
Obviously,fm defined by the formula (3.1) satisfy also the condition (H4). By (3.5) and (3.6), it is easy to show that ∥Qmx-Qmy∥*<τ∥x-y∥*, then Banach contraction principle implies that the operator Qm has a unique fixed point, which completes this proof.The following fixed-point result of cone compression type is due to Krasnosel'skii, which is fundamental to establish another auxiliary existence result (Lemma3.8).Lemma 3.6 (see, e.g., [23, 25]).
LetY be a Banach space, and let P⊂Y be a cone in Y. Let Ω1,Ω2 be bounded open balls of Y centered at the origin with Ω1¯⊂Ω2. Suppose that A:P∩(Ω2¯∖Ω1)→P is a completely continuous operator such that
(3.10)‖Ax‖≥‖x‖forx∈P∩∂Ω1,‖Ax‖≤‖x‖forx∈P∩∂Ω2
hold, then A has a fixed point in P∩(Ω2¯∖Ω1).Lemma 3.7.
Let(H0)–(H3) hold, then Qm:P→P and Qm is a completely continuous operator.Proof.
Firstly, lety∈P, because fm∈Car([0,T]×B*) is positive. It follows from Lemma 2.5 (i) and (ii) that Qmy∈C([0,T],𝔼) and (Qmy)(t)≥θ for t∈[0,T]. Similarly, from Lemma 3.4 (i) and (ii) we can get that DβQmy∈C([0,T],𝔼) and (DβQmy)(t)≥θ for t∈[0,T]. To summarize, Qm:P→P.
Secondly, we prove thatQm is a continuous operator. Let {xk}⊂P be a convergent sequence and limk→+∞∥xk-x∥*=0, then x⊂P and ∥xk∥*≤S, where S is a positive constant. In view of fm∈Car([0,T]×B*), we have limk→+∞fm(t,xk(t),Dβxk(t))=fm(t,x(t),Dβx(t)). Since by (3.2), (3.3),
(3.11)0<‖fm(t,xk(t),Dβxk(t))‖≤γ(t)(γ0(t)+q1(1m)+p1(1m)+q(1)+p(1)+q(S)+p(S)),
the Lebesgue dominated convergence theorem gives
(3.12)lim⁡k→+∞∫0T‖fm(t,xk(t),Dβxk(t))-fm(t,x(t),Dβx(t))‖dt=0.
Now, from (3.12), Lemma 2.5(ii), Lemma 3.4(ii) and from the inequalities (cf. (3.5), (3.6))
(3.13)‖(Qmxk)(t)-(Qmx)(t)‖≤E∫0T‖fm(t,xk(t),Dβxk(t))-fm(t,x(t),Dβx(t))‖dt,‖(DβQmxk)(t)-(DβQmx)(t)‖≤ED∫0T‖fm(t,xk(t),Dβxk(t))-fm(t,x(t),Dβx(t))‖dt,
we have that limk→+∞∥Qmxk-Qmx∥*=0, which proves that Qm is a continuous operator.
Thirdly, letΩ⊂P be bounded in 𝔼 and let ∥x∥*≤L for all x∈Ω, where L is a positive constant. We are in position to prove that Qm(Ω) is bounded. Keeping in mind fm∈Car([0,T]×B*), there exists ϕ∈L1([0,T]) such that
(3.14)0<‖fm(t,xk(t),Dβxk(t))‖≤ϕ(t)fora.e.t∈[0,T]andallx∈Ω,
then (cf. (3.5))
(3.15)‖(Qmx)(t)‖≤E∫0T‖fm(s,x(s),Dβx(s))‖ds+y0Δt(α-1)≤E‖ϕ‖L1+‖y0‖ΔT(α-1),
and (cf. (3.6))
(3.16)‖(DβQmx)(t)‖≤ED∫0T‖fm(s,x(s),Dβx(s))‖ds+‖y0Γ(α)ΔΓ(α-β)t(α-β-1)‖≤ED‖ϕ‖L1+‖y0‖Γ(α)ΔΓ(α-β)T(α-β-1),
for t∈[0,T] and all x∈Ω. Therefore, Qm(Ω) is bounded in 𝔼.
Fourthly, by(H3) and (3.5), it is easy to show that Qm(Ω)(t) is relatively compact.
Finally, let0≤t1<t2≤T. From Lemma 2.5(i) and the functions tα-1,tα-β-1 being uniformly continuous on [0,T], for any arbitrary ϵ>0, there exists a positive number δ(ϵ), such that when |t1-t2|<δ(ϵ), one has |GT(t1,s)-GT(t2,s)|<ϵ and |t1α-1-t2α-1|<ϵ, then (cf. (3.14)) the inequality
(3.17)‖(Qmx)(t1)-(Qmx)(t2)‖<ϵ‖ϕ‖L1+‖y0‖Δϵ
holds. Hence the set of functions Qm(Ω) is equicontinuous on [0,T].
Therefore, by the Arzelá-Ascoli theorem,Qm(Ω) is relatively compact in 𝔼. We have proved that Qm is a completely continuous operator.Lemma 3.8.
Suppose that(H0)–(H3) hold, then the operator Qm has at least a fixed point.Proof.
By Lemma3.7, Qm:P→P is completely continuous. In order to apply Lemma 3.6, we construct two bounded open balls Ω1,Ω2 and prove that the conditions (3.10) are satisfied with respect to Qm.
Firstly, letΩ1={y∈𝔼:∥y∥*<r}, where r=supt∈[0,T]K1(tα-1(T-t)2)/T and K1 is defined as in (H2). It follows from Lemma 2.5, (H1), Remark 3.1 and from the definition of Qm that ∥(Qmy)(t)∥≥K1(tα-1(T-t)2/T). Then ∥Qmy∥≥r. Immediately:
(3.18)‖Qmy‖*≥‖y‖*,fory∈P∩∂Ω1.
Secondly, (3.3) and Lemma 2.5(ii) imply that, for x∈P,
(3.19)‖(Qmy)(t)‖≤E∫0T‖fm(s,y(s),Dβy(s))‖ds+‖y0‖ΔT(α-1)≤E∫0Tγ(t)[γ0(t)+q1(1m)+p1(1m)+q(1)+p(1)+q(y(s))+p(Dβy(s))(1m)]ds+‖y0‖ΔT(α-1)≤E∫0Tγ(t)[γ0(t)+q1(1m)+p1(1m)+q(1)+p(1)+q(‖y‖)+p(‖Dβy‖)(1m)]ds+‖y0‖ΔT(α-1)≤E{‖γγ0‖L1+[q1(1m)+p1(1m)+q(1)+p(1)+q(‖y‖)+p(‖Dβy‖)(1m)]‖γ‖L1(1m)}+‖y0‖ΔT(α-1),
because q,p are nondecreasing as stated in (H2). Analogously, by (3.3) and Lemma 3.4(ii), one can get that for x∈P(3.20)‖(DβQmy)(t)‖≤ED{‖γγ0‖L1+[q1(1m)+p1(1m)+q(1)+p(1)+q(‖y‖)+p(‖Dβy‖)(1m)]×‖γ‖L1[q1(1m)+p1(1m)+q(1)+p(1)+q(‖y‖)+p(‖Dβy‖)(1m)]}+‖y0‖Γ(α)ΔΓ(α-β)T(α-β-1).
Let W1=max{E,ED} and W2=max{(∥y0∥/Δ)T(α-1),(∥y0∥Γ(α)/(ΔΓ(α-β)))T(α-β-1)}. Hence for x∈P, we have the following inequality
(3.21)‖Qmy‖*≤W1{‖γγ0‖L1+[q1(1m)+p1(1m)+q(1)+p(1)+q(‖y‖*)+p(‖y‖*)]×‖γ‖L1[q1(1m)+p1(1m)+q(1)+p(1)+q(‖y‖*)+p(‖y‖*)]}+W2.
Since lim∥x∥→∞(q(∥x∥)+p(∥x∥))/∥x∥=0 by (H2), there exists a sufficiently large number R>r such that
(3.22)W1{‖γγ0‖L1+[q1(1m)+p1(1m)+q(1)+p(1)+q(R)+p(R)]‖γ‖L1}+W2≤R.
Let Ω2={y∈𝔼:∥y∥*<R}, then (cf. (3.21) and (3.22))
(3.23)‖Qmy‖*≤‖y‖*,fory∈P∩∂Ω2.
Applying Lemma 3.6, we conclude from (3.18) and (3.23) that Qm has a fixed point in P∩(Ω2¯∖Ω1).Lemma 3.9.
Suppose that(H0)–(H3) hold, then the sequences {ym}m=1∞ and {Dβym}m=1∞ are relatively compact in C([0,T]), where ym be a fixed point of operator Qm defined by (3.5).Proof.
Letym be a fixed point of operator Qm, that is,
(3.24)ym(t)=∫0TG(t,s)fm(s,ym(s),Dβym(s))ds+y0Δt(α-1),t∈[0,T],m∈N.
And consider (cf. (3.6))
(3.25)Dβym(t)=∫0TDβG(t,s)fm(s,ym(s),Dβym(s))ds+y0Γ(α)ΔΓ(α-β)t(α-β-1),t∈[0,T],m∈N.
By Lemma 2.5(iii), Lemma 3.4(iii), and Remark 3.1, we have also
(3.26)‖ym(t)‖≥K1tα-1(T-t)2T+y0Γ(α)ΔΓ(α-β)t(α-β-1)≥K1tα-1(T-t)2T,t∈[0,T],m∈N,(3.27)‖Dβym(t)‖≥K2tα-β-1(T-t)2T,t∈[0,T],m∈N.
Hence (cf. (3.4)),
(3.28)‖fm(t,ym(t),Dβym(t))‖≤γ(t){γ0(t)+q1(K1tα-1(T-t)2T)+p1(K2tα-β-1(T-t)2T)+q(1)+p(1)+q(ym(t))+p(Dβym(t))(K1tα-1(T-t)2T)},
for a.e. t∈[0,T], and all m∈N. Therefore, by (3.26), (3.27), Lemma 2.5(ii), Lemma 3.4(ii), and Remark 3.1,
(3.29)‖ym(t)‖≤E{‖γγ0‖L1+U1+U2+[q(1)+p(1)+q(‖ym‖)+p(‖Dβym‖)]‖γ‖L1}+y0ΔT(α-1),(3.30)‖Dβym(t)‖≤ED{‖γγ0‖L1+U1+U2+[q(1)+p(1)+q(‖ym‖)+p(‖Dβym‖)]‖γ‖L1}+y0Γ(α)ΔΓ(α-β)T(α-β-1),
for t∈[0,T], m∈N, where
(3.31)U1=∫0Tγ(t)q1(K1tα-1(T-t)2T)dt<+∞,(3.32)U2=∫0Tγ(t)p1(K2tα-β-1(T-t)2T)dt<+∞.
In particular,
(3.33)‖ym‖*≤W1{‖γγ0‖L1+U1+U2+[q(1)+p(1)+q(‖ym‖*)+p(‖ym‖*)]‖γ‖L1}+W2,∀m∈N,
where W1,W2 are defined in the proof of Lemma 3.8. Since limx→+∞(q(x)+p(x))/x=0 by (H2), there exists a constant W>0 such that for each x>W,
(3.34)W1{‖γγ0‖L1+U1+U2+[q(1)+p(1)+q(x)+p(x)]‖γ‖L1}+W2<x.
Immediately, (cf. (3.33))
(3.35)‖ym‖*≤W,∀m∈N.
Hence, the sequences {ym}m=1∞ and {Dβym}m=1∞ are uniformly bounded.
We will take   similar discussions as in Lemma3.7 to show that {ym}m=1∞ and {Dβym}m=1∞ are equicontinuous on [0,T]. Let 0≤t1<t2≤T, then we have
(3.36)‖(ym)(t1)-(ym)(t2)‖≤∫0T|GT(t1,s)-GT(t2,s)|‖fm(s,x(s),Dβx(s))‖ds+‖y0‖Δ|t1α-1-t2α-1|,‖(Dβym)(t1)-(Dβym)(t2)‖≤∫0T|DβGT(t1,s)-DβGT(t2,s)|‖fm(s,x(s),Dβx(s))‖ds+‖y0‖Γ(α)ΔΓ(α-β)|t1α-β-1-t2α-β-1|.
Using (3.28), (3.35), one can get
(3.37)0<‖fm(t,ym(t),Dβym(t))‖≤γ(t){γ0(t)+q1(K1tα-1(T-t)2T)+p1(K2tα-β-1(T-t)2T)+q(1)+p(1)+q(W)+p(W)(K1tα-1(T-t)2T)}.
From Lemma 2.5 (i), Lemma 3.4 (i), and the functions tα-1,tα-β-1 being uniformly continuous on [0,T], choosing an arbitrary ϵ>0, there exists a positive number δ(ϵ). When |t1-t2|<δ(ϵ), we can get |GT(t1,s)-GT(t2,s)|<ϵ, |DβGT(t1,s)-DβGT(t2,s)|<ϵ, |t1α-1-t2α-1|<ϵ, and |t1α-β-1-t2α-β-1|<ϵ. Therefore (cf. (3.36) and (3.37)) the inequalities
(3.38)‖(ym)(t1)-(ym)(t2)‖<ϵ{‖γγ0‖L1+U1+U2+[q(1)+p(1)+q(W)+p(W)]‖γ‖L1}+‖y0‖Δϵ,‖(Dβym)(t1)-(Dβym)(t2)‖<ϵ{‖γγ0‖L1+U1+U2+[q(1)+p(1)+q(W)+p(W)]‖γ‖L1}+‖y0‖Γ(α)ΔΓ(α-β)ϵ,
hold, where U1, U2 are defined as (3.31) and (3.32), respectively. As a result, {ym}m=1∞ and {Dβym}m=1∞ are equicontinuous on [0,T].
Finally, we prove that{ym(t)}m=1∞ and {Dβym(t)}m=1∞ are relatively compact. Because E is a Banach space, we need only to show that {ym(t)}m=1∞ and {Dβym(t)}m=1∞ are completely bounded. For all ε>0, by the Remark 3.2, there exists a sufficiently large positive integer N, such that if m>N,
(3.39)‖fm(t,y(t),Dβy(t))-f(t,y(t),Dβy(t))‖<εẼ,a.e.t∈[0,T],
where Ẽ=max{E,ED}.
Hence, by (3.5) and (3.6), we have ∥(Qmy)(t)-Q0∥<ε and ∥Dβ(Qmy)(t)-DβQ0∥<ε, for m>N, where
(3.40)Q0=∫0TG(t,s)f(s,y(s),Dβy(s))ds+y0Δt(α-1),DβQ0=∫0TDβG(t,s)f(s,y(s),Dβy(s))ds+y0Γ(α)ΔΓ(α-β)t(α-β-1).
This implies that {ym(t)}m=1∞ and {Dβym(t)}m=1∞ have an ε-net constituted by finite elements ({y1(t),y2(t),yN(t),Q0} and {Dβy1(t),Dβy2(t),DβyN(t),DβQ0}, resp.) of E, that is, completely bounded.
Therefore,{ym}m=1∞ and {Dβym}m=1∞ are relatively compact in C([0,T]) by the Arzelá-Ascoli theorem.
Using above results, we now give the existence of positive solution of singular problem (1.4)-(1.5).Theorem 3.10.
Suppose that(H0)–(H3) hold, then problem (1.4)-(1.5) has a positive solution y  and
(3.41)‖y(t)‖≥K1tα-1(T-t)2T,‖Dβy(t)‖≥K2tα-β-1(T-t)2T,t∈[0,T].
Moreover, y is continuous and ∥y∥*≤W, where W is a constant as in (3.35).Proof.
From Lemmas3.8 and 3.9, the operator Qm has a fixed point ym satisfying (3.26), (3.27), (3.35). And {ym}m=1∞ and {Dβym}m=1∞ are relatively compact in C([0,T]). Hence, {ym}m=1∞ is relatively compact in 𝔼. And therefore, there exist y∈𝔼 and a subsequence ymk of {ym}m=1∞ such that limk→∞ymk=y in 𝔼. Consequently, y is positive and continuous. Moreover y satisfies (3.47), ∥y∥*≤W. And
(3.42)lim⁡k→∞fmk(t,ymk(t),Dβymk(t))=f(t,y(t),Dβy(t)),fora.e.t∈[0,T].
Keeping in mind (3.35) holding, where W is a positive constant, it follows from inequalities (3.4) and (3.26) and from Lemma 2.5(ii) that
(3.43)0≤‖G(t,s)fm(s,ym(s),Dβym(s))‖≤Eγ(s){γ0(s)+q1(K1sα-1(T-s)2T)+p1(K2sα-β-1(T-s)2T)+q(1)+p(1)+q(W)+p(W)(K1sα-1(T-s)2T)},
for a.e. s∈[0,T] and all t∈[0,T], m∈N. Hence, by the Lebesgue dominated convergence theorem, we have
(3.44)lim⁡k→+∞∫0TG(t,s)fm(s,ym(s),Dβym(s))ds=∫0TG(t,s)f(s,y(s),Dβy(s))ds,
for t∈[0,T]. Now, passing to the limit as k→+∞ in
(3.45)ymk(t)=∫0TG(t,s)fmk(s,ymk(s),Dβymk(s))ds+y0Δt(α-1),
we have
(3.46)y(t)=∫0TG(t,s)f(s,y(s),Dβy(s))ds+y0Δt(α-1),fort∈[0,T].
Consequently, y is a positive solution of BVP (1.4)-(1.5) by Lemma 2.4.By Lemmas3.5 and 3.9, and Theorem 3.10, we give the following unique result without proof.Theorem 3.11.
Suppose that(H0)–(H4) hold, then problem (1.4)-(1.5) has a unique positive solution y and
(3.47)‖y(t)‖≥K1tα-1(T-t)2T,‖Dβy(t)‖≥K2tα-β-1(T-t)2T,t∈[0,T].
Moreover, y is continuous and ∥y∥*≤W, where W is a constant as in (3.35).


## 4. Existence Results for BVP (1.6)-(1.7)

We now give the existence of positive solution of BVP (1.6)-(1.7) by using diagonalization process.Theorem 4.1.
Suppose that(H0)–(H3) hold, then BVP (1.6)-(1.7) has a positive solution y, and Dβy is also positive.Proof.
Firstly, choose(4.1){Tn}n=1∞∈N*tobeasubsequenceofnumbers,suchthatT≤T1<T2<⋯<Tn<⋯↑∞,
then consider the BVP,
(4.2)Dαy(t)+f(t,y(t),Dβy(t))=θ,a.e.t∈[0,Tn],
subject to
(4.3)y(0)=θ,Dβy(Tn)-∑i=1m-2aiy(ξi)-∑i=1m-2biDβy(ξi)=y0.
Theorem3.10 guarantees that BVP (4.2)-(4.3) has a positive continuous solution yn. And for any n∈N,
(4.4)‖yn‖*≤Wn,fort∈[0,Tn],
where Wn is a constant defined similarly to W.
Secondly, we apply the following diagonalization process. Forn∈N, let
(4.5)un(t)={yn(t),t∈[0,Tn],yn(Tn),t∈[Tn,+∞).
Here, {Tn}n=1∞ is defined in (4.1). Notice that un(t)∈C[0,+∞) with
(4.6)0≤‖un(t)‖≤W1,0≤‖Dβun(t)‖≤W1,fort∈[0,T1].
Also for n∈N and t∈[0,T1], we get
(4.7)un(t)=∫0T1GT1(t,s)f(s,un(t),Dβun(t))ds+y0Δtα-1,
where GTn(t,s) are similarly defined as in (2.11), but all of T should be replaced by Tn. Then for t1,t2∈[0,T1], we have
(4.8)‖un(t1)-un(t2)‖≤∫0T1|GT1(t1,s)-GT1(t2,s)|‖f(s,unk(t),Dβunk(t))‖ds+y0Δ|t1α-1-t2α-1|,‖Dβun(t1)-Dβun(t2)‖≤∫0T1|DβGT1(t1,s)-DβGT1(t2,s)|×‖f(s,unk(t),Dβunk(t))‖ds+‖y0‖Γ(α)ΔΓ(α-β)|t1α-β-1-t2α-β-1|.
Thus, when |t1-t2|<δ(ϵ,1), similarly to (3.38),
(4.9)‖un(t1)-un(t2)‖<ϵ{‖γγ0‖L1+U11+U21+[q(1)+p(1)+q(W1)+p(W1)]‖γ‖L1}+‖y0‖Δϵ,‖Dβun(t1)-Dβun(t2)‖<ϵ{‖γγ0‖L1+U11+U21+[q(1)+p(1)+q(W1)+p(W1)]‖γ‖L1}+‖y0‖Γ(α)ΔΓ(α-β)ϵ
hold for an arbitrary ϵ>0, where δ(ϵ,1) is a suitable positive number and U11,U21 are defined similarly to U1,U2 as in (3.31) and (3.32), respectively. By using (H3), we know that, for a.e. t∈[0,+∞), f(t,DW1,DW1) is relatively compact, where DW1={u∈C([0,T1]:∥u∥*≤W1}∩P. Therefore, {un(t)}n=1∞ and {Dβun(t)}n=1∞ are relatively compact. The Arzelá-Ascoli theorem guarantees that there is a subsequence N1* of N and a function z1∈C([0,T1],E) with unk→z1 in C([0,T1],E) as k→+∞ through N1*. Obviously, z1 is positive. Let N1=N1*∖{1}, noticing that
(4.10)0≤‖un(t)‖≤W2,0≤‖Dβun(t)‖≤W2,fort∈[0,T2].
Similarly to above argumentation, we have that there is a subsequence N2* of N1 and a function z2∈C([0,T2],E) with unk→z2 in C([0,T2],E) as k→+∞ through N2*. Obviously, z2 is positive. Note that z1=z2 on [0,T1] since N2*⊂N1. Let N2=N2*∖{2}. Proceed inductively to obtain for m={2,3,…} a subsequence Nm* of Nm-1 and a function zm∈C([0,Tm],E) with unk→zm in C([0,Tm],E) as k→+∞ through Nm*. Also, zm is positive. Let Nm=Nm*∖{m}.
Define a functiony as follows. Fix t∈(0,+∞), and Let m∈N with s≤Tm, then define y(t)=zm(t). Hence y∈C([0,+∞),ℝ).
Again fixt∈[0,+∞) and Let m∈N with s≤Tm. Then for n∈Nm we get
(4.11)unk(t)=∫0TmGTm(t,s)f(s,unk(s),Dβunk(s))ds+y0Δt(α-1).
Let nk→+∞ through Nm to obtain
(4.12)zm(t)=∫0TmGTm(t,s)f(s,zm(s),Dβzm(s))ds+y0Δt(α-1),
that is,
(4.13)y(t)=∫0TmGTm(t,s)f(s,y(s),Dβy(s))ds+y0Δt(α-1).
We can use this method for each s∈[0,Tm] and for each m∈N. Hence,
(4.14)Dαy(t)+f(t,y(t),Dβy(t))=θ,a.e.t∈[0,Tm],
for each m∈N. Consequently, the constructed function y is a solution of (1.6)-(1.7). This completes the proof of the theorem.Remark 4.2.
In [21], the authors considered the BVP (1.3). Under some suitable conditions, they obtained the existence result of unbounded solution. In nature, BVP (1.3) is a special form of BVP (1.6)-(1.7). In that scalar situation, α-β=1, bi=0(i=1,2,…,m-2), b1>0, bi=0(i=2,3,…,m-2), y0=0, and f=f(t,y(t)) are not singular, then our Theorem 4.1 includes the result from [21]. But our approach is different from those of [21].Proceeding the similar arguments above, we list the following unique result for BVP (1.6)-(1.7), and the proof will be omitted.Theorem 4.3.
Suppose that(H0)–(H4) hold, then BVP (1.6)-(1.7) has a unique positive solution y, and Dβy is also positive.


## 5. Application

We end this paper with giving an example to demonstrate the application of our existence result.Example 5.1.
Consider the following BVP in scalar space:(5.1)Dαy(t)+(2-sin⁡13t-1)(e-t+1yη1+1(Dβy)κ1+yη+(Dβy)κ)=0,y(0)=0,lim⁡t→+∞Dβy(t)-14y(4)-2y(5)-Dβy(6)=1,
where 1<α≤2, 0<β<α-1. We will apply Theorem 4.1 with ϖ=1, γ(t)=2-sin(1/(3t-1)), γ0(t)=e-t, q1(x)=1/xη1, p1(x)=1/(x)κ1, q(x)=xη, p(t)=(x)κ. Clearly (H1) holds because f(t,x,y)≥1 for t∈[0,+∞)∖{1/3}, (x,y)∈(0,+∞)×(0,+∞). And also (H2) holds for η,κ∈(0,1), η1∈(0,1/2), and κ1∈(0,1/2). Hence, Theorem 4.1 guarantees the existence of positive solution of (5.1).



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*Source: 352341-2011-11-21.xml*