# Positive Solutions for Nonlinear First-Orderm-Point Boundary Value Problem on Time Scales

Authors: İsmail Yaslan
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

By means of fixed-point theorems, we investigate the existence of positive solutions for nonlinear first-orderm-point boundary value problem xΔ(t)+a(t)x(σ(t))=f(t,x(σ(t))),t∈[t1,tm]⊂𝕋, x(t1)=∑k=2m-1αkx(tk)+α1x(σ(tm)), where 𝕋 is a time scale, 0≤t1<t2<⋯<tm-1<tm, α1,α2,…,αm-1≥0 are given constants.


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## Body

## 1.  Introduction

The theory of time scales, which has recently received a lot of attention, was introduced by Stefan Hilger in his Ph.D. thesis in 1988 (see [1]). The time scales calculus has a tremendous potential for applications in some mathematical models of real processes and phenomena studied in physics, chemical technology, population dynamics, biotechnology and economics, neural networks, and social sciences; see the monographs of Aulbach and Hilger [2], Bohner and Peterson [3, 4], and Lakshmikantham et al. [5] and the references therein.A time scale𝕋 is an arbitrary nonempty closed subset of real numbers ℝ. A book on the subject of time scales by Bohner and Peterson [3] also summarizes and organizes much of the time scale calculus. The closed interval in 𝕋 is defined as
(1.1)[a,b]={t∈𝕋:a≤t≤b},
where a,b∈𝕋 with a<ρ(b).In this study, we consider the nonlinear first-orderm-point boundary value problem
(1.2)xΔ(t)+a(t)x(σ(t))=f(t,x(σ(t))),t∈[t1,tm]⊂𝕋,x(t1)=∑k=2m-1‍αkx(tk)+α1x(σ(tm)),
where 𝕋 is a time scale, 0≤t1<t2<⋯<tm-1<tm,α1,α2,…,αm-1≥0 are given constants. a is regressive and rd-continuous, and f:[t1,σ(tm)]×ℝ+→ℝ+ is continuous.In [6], Cabada studied the following first-order periodic boundary value problem on time scales:
(1.3)uΔ(t)=f(t,u(t)),t∈[a,b]⊂𝕋,u(a)=u(σ(b)).
He developed the monotone method in the presence of lower and upper solutions to obtain the existence of extremal solutions. When α1=1,α2=⋯=αm-1=0, and a(t)≡0, BVP (1.2) is reduced to (1.3).In [7], Sun studied the first-order boundary value problem
(1.4)xΔ(t)=f(x(σ(t))),t∈[0,T]⊂𝕋,x(0)=βx(σ(T)),
where 0<β<1. Some existence results for at least two positive solutions were established, by using Avery-Henderson fixed-point theorem. When α2=⋯=αm-1=0 and a(t)≡0, BVP (1.2) is reduced to (1.5).In [8], Shu and Chunhua are concerned with the existence of three positive solutions for the following nonlinear first-order boundary value problem on time scale:
(1.5)xΔ(t)=f(x(σ(t))),t∈[0,T]⊂𝕋,x(0)=ηx(σ(T)),
where T>0 is fixed, 0,T∈𝕋, and f:[0,∞)→[0,∞) is continuous. When α2=⋯=αm-1=0 and a(t)≡0, BVP (1.2) is reduced to (1.5).Sun and Li [9] studied the following first-order periodic boundary value problem on time scales:
(1.6)xΔ(t)+p(t)x(σ(t))=g(t,x(σ(t))),t∈[0,T]⊂𝕋,x(0)=x(σ(T)).
Conditions for the existence of at least one solution were obtained by using novel inequalities and the Schaefer fixed-point theorem. When α1=1 and α2=⋯=αm-1=0, BVP (1.2) is reduced to (1.6).In [10], Tian and Ge studied the existence and uniqueness results for first-order three-point boundary value problem
(1.7)xΔ(t)+p(t)x(σ(t))=f(t,x(σ(t))),t∈[0,T]⊂𝕋,x(0)-αx(ξ)=βx(σ(T)),
by using several well-known fixed-point theorems. When α3=⋯=αm-1=0, BVP (1.2) is reduced to (1.7).Motivated by [6–10], we establish some new and more general results for the existence of positive solutions for the problem (1.2) by applying fixed-point theorems in cones.We have arranged the paper as follows. In Section2, we give some lemmas which are needed later. In Section 3, we apply the Krasnosel'skii fixed-point theorem, Avery-Henderson fixed-point theorem, and Leggett-Williams fixed-point theorem to prove the existence of at least one, two, and three positive solutions to BVP (1.2). In Section 4, as an application, the examples are included to illustrate our results.


## 2. Preliminaries

Letℬ denote the Banach space C[t1,σ(tm)] with the norm ∥x∥=supt∈[t1,σ(tm)]|x(t)|. For h∈ℬ, we consider the following linear boundary value problem:(2.1)xΔ(t)+p(t)x(σ(t))=h(t),t∈[t1,tm]⊂𝕋,x(t1)=∑k=2m-1‍αkx(tk)+α1x(σ(tm)).Lemma 2.1.
Forh∈ℬ, BVP (2.1) has the unique solution
(2.2)x(t)=1ea(t,t1){Γ[α1∫t1σ(tm)ea(s,t1)h(s)Δsea(σ(tm),t1)+∑k=2m-1‍αk∫t1tkea(s,t1)h(s)Δsea(tk,t1)]+∫t1tea(s,t1)h(s)Δs[α1∫t1σ(tm)ea(s,t1)h(s)Δsea(σ(tm),t1)+∑k=2m-1‍αk∫t1tkea(s,t1)h(s)Δsea(tk,t1)]},t∈[t1,σ(tm)],
where Γ=[1-∑k=2m-1(αk/ea(tk,t1))-(α1/ea(σ(tm),t1))]-1.Proof.
FromxΔ(t)+a(t)x(σ(t))=h(t), we have
(2.3)x(t)=1ea(t,t1)[x(t1)+∫t1tea(s,t1)h(s)Δs].
By using the boundary condition, we get
(2.4)[1-α1ea(σ(tm),t1)-∑k=2m-1‍αkea(tk,t1)]x(t1)=α1∫t1σ(tm)ea(s,t1)h(s)Δsea(σ(tm),t1)+∑k=2m-1‍αk∫t1tkea(s,t1)h(s)Δsea(tk,t1).
Thus, x satisfies (2.2).LetG(t,s) be Green's function for the boundary value problem
(2.5)xΔ(t)+a(t)x(σ(t))=h(t),t∈[t1,tm]⊂𝕋,x(t1)=∑k=2m-1‍αkx(tk)+α1x(σ(tm)).
By Lemma 2.1, we obtain
(2.6)G(t,s)={G1(t,s),t1≤s≤σ(s)≤t2,G2(t,s),t2≤s≤σ(s)≤t3,⋮Gm-2(t,s),tm-2≤s≤σ(s)≤tm-1,Gm-1(t,s),tm-1≤s≤tm,
where
(2.7)Gj(t,s)={ea(s,t1)ea(t,t1){Γ[α1ea(σ(tm),t1)+∑k=j+1m-1‍αkea(tk,t1)]+1},σ(s)≤t,Γea(s,t1)ea(t,t1)[α1ea(σ(tm),t1)+∑k=j+1m-1‍αkea(tk,t1)],t≤s,
for all j=1,2,…,m-1.Lemma 2.2.
Green's functionG(t,s) in (2.6) has the following properties:(i)
G(t,s)≥0 for (t,s)∈[t1,σ(tm)]×[t1,tm].(ii)
m≤G(t,s)≤M, where m=Γα1/(ea(σ(tm),t1))2 and M=Γ∑k=1m-1αk+ea(tm-1,t1).(iii)
G(t,s)≥(m/M)sup(t,s)∈[t1,σ(tm)]×[t1,tm]G(t,s) for (t,s)∈[t1,σ(tm)]×[t1,tm].Letℬ denote the Banach space C[t1,σ(tm)] with the norm ∥x∥=maxt∈[t1,σ(tm)]|x(t)|. Define the cone P⊂ℬ by
(2.8)P={x∈ℬ:x(t)≥0,x(t)≥mM∥x∥on[t1,σ(tm)]}.
Equation (1.2) is equivalent to the nonlinear integral equation
(2.9)x(t)=∫t1σ(tm)G(t,s)f(s,x(σ(s)))Δs.We can define the operatorA:P→ℬ by
(2.10)Ax(t)=∫t1σ(tm)G(t,s)f(s,x(σ(s)))Δs.
Therefore solving (2.9) in P is equivalent to finding fixed-points of the operator A.From Lemma2.2, Ax(t)≥0 for t∈[t1,σ(tm)]. In addition, by using Lemma 2.2 we get
(2.11)Ax(t)=∫t1σ(tm)G(t,s)f(s,x(σ(s)))Δs≥mMsup(t,s)∈[t1,σ(tm)]×[t1,tm]G(t,s)∫t1σ(tm)f(s,x(σ(s)))Δs≥mMsupt∈[t1,σ(tm)]∫t1σ(tm)G(t,s)f(s,x(σ(s)))Δs=mM∥Ax∥.So, we haveA:P→P.


## 3. Main Results

To prove the existence of at least one positive solution for the BVP (1.2), we will need the following (Krasnosel'skii) fixed-point theorem.Theorem 3.1 (Krasnosel'skii fixed-point theorem [11]).
LetE be a Banach space, and let K⊂E be a cone. Assume Ω1 and Ω2 are open bounded subsets of E with 0∈Ω1, Ω1¯⊂Ω2, and let
(3.1)A:K∩(Ω2¯∖Ω1)→K
be a completely continuous operator such that either(i)
∥Au∥≤∥u∥ for u∈K∩∂Ω1,∥Au∥≥∥u∥ for u∈K∩∂Ω2, or(ii)
∥Au∥≥∥u∥ for u∈K∩∂Ω1,∥Au∥≤∥u∥ for u∈K∩∂Ω2 hold. Then A has a fixed-point in K∩(Ω2¯∖Ω1).Theorem 3.2.
Let there exist numbersr,  R satisfying 0<r<R<∞ such that for t∈[t1,σ(tm)](3.2)f(t,x)<xMσ(tm)forx∈[0,r],f(t,x)≥Mxm2σ(tm)forx∈[R,∞).
Then BVP (1.2) has at least one positive solution x satisfying r≤x(t)≤RM/m, t∈[t1,σ(tm)].Proof.
It is easy to check by the Arzela-Ascoli theorem that the operatorA:P→P is completely continuous. Let us now define two bounded open sets as follows:
(3.3)Ω1={x∈ℬ:∥x∥<r},Ω2={x∈ℬ:∥x∥<RMm}.
Then Ω1¯⊂Ω2. For x∈P∩∂Ω1, we obtain
(3.4)Ax(t)=∫t1σ(tm)G(t,s)f(s,x(σ(s)))Δs≤M∫t1σ(tm)f(s,x(σ(s)))Δs≤M∫t1σ(tm)x(σ(s))ΔsMσ(tm)≤r=∥x∥.
Hence ∥Ax∥≤∥x∥ for x∈P∩∂Ω1.
Ifx∈P∩∂Ω2, then ∥x∥=RM/m and x(t)≥(m/M)∥x∥=R for t∈[t1,σ(tm)]. We have
(3.5)Ax(t)=∫t1σ(tm)G(t,s)f(s,x(σ(s)))Δs≥m∫t1σ(tm)f(s,x(σ(s)))Δs≥mM∫t1σ(tm)x(σ(s))Δsm2σ(tm)≥RMm=∥x∥.
Thus∥Ax∥≥∥x∥ for x∈P∩∂Ω2. By the first part of Theorem 3.1, A has a fixed-point in P∩(Ω2¯∖Ω1). Therefore, the BVP (1.2) has at least one positive solution satisfying r≤x(t)≤RM/m, t∈[t1,σ(tm)].Now, we will apply the following (Avery-Henderson) fixed-point theorem to prove the existence of at least two positive solutions to BVP (1.2).Theorem 3.3 (see [12]).
LetP be a cone in a real Banach space E. Set
(3.6)P(ϕ,r)={u∈P:ϕ(u)<r}.
If η and ϕ are increasing, nonnegative continuous functionals on P, let θ be a nonnegative continuous functional on P with θ(0)=0 such that, for some positive constants r and M,
(3.7)ϕ(u)≤θ(u)≤η(u),∥u∥≤Mϕ(u),
for all u∈P(ϕ,r)¯. Suppose that there exist positive numbers p<q<r such that
(3.8)θ(λu)≤λθ(u),∀0≤λ≤1,u∈∂P(θ,q).
If A:P(ϕ,r)¯→P is a completely continuous operator satisfying(i)
ϕ(Au)>r for all u∈∂P(ϕ,r),(ii)
θ(Au)<q for all u∈∂P(θ,q),(iii)
P(η,p)≠∅ and η(Au)>p for all u∈∂P(η,p),
thenA has at least two fixed-points u1 and u2 such that
(3.9)p<η(u1)withθ(u1)<q,q<θ(u2)withϕ(u2)<r.Theorem 3.4.
Suppose there exist numbersp, q, and r satisfying 0<p<q<r such that the function f satisfies the following conditions:(i)
f(t,x)>r/m for t∈[tm-1,σ(tm)] and x∈[r,rM/m(σ(tm)-tm-1)];(ii)
f(t,x)<q/Mσ(tm) for t∈[t1,σ(tm)] and x∈[0,qM/m];(iii)
f(t,x)>p/m(σ(tm)-tm-1) for t∈[tm-1,σ(tm)] and x∈[pm/M,p].
Then the BVP (1.2) has at least two positive solutions x1 and x2 such that
(3.10)p<supt∈[t1,σ(tm)]x1(t)withsupt∈[tm-1,σ(tm)]x1(t)<q,q<supt∈[tm-1,σ(tm)]x2(t)withinft∈[tm-1,σ(tm)]x2(t)<r.Proof .
Let the nonnegative increasing continuous functionalsϕ, θ, and η be defined on the cone P by
(3.11)ϕ(x)=inft∈[tm-1,σ(tm)]x(t),θ(x)=supt∈[tm-1,σ(tm)]x(t),η(x)=supt∈[t1,σ(tm)]x(t).
For each x∈P, we have ϕ(x)≤θ(x)≤η(x) and
(3.12)x(t)=∫t1σ(tm)G(t,s)f(s,x(σ(s)))Δs≤Mmm∫t1σ(tm)f(s,x(σ(s)))Δs≤Mminf(t,s)∈[t1,σ(tm)]×[t1,tm]G(t,s)∫t1σ(tm)f(s,x(σ(s)))Δs≤Mminft∈[t1,σ(tm)]∫t1σ(tm)G(t,s)f(s,x(σ(s)))Δs=Mmϕ(x).
Then ∥x∥≤(M/m)ϕ(x). In addition, θ(0)=0 and for all x∈P, λ∈[0,1] we obtain θ(λx)=λθ(x).
Now we will verify the remaining conditions of Theorem3.3.
Claim
1. If x∈∂P(ϕ,r), then ϕ(Ax)>r. Since x∈∂P(ϕ,r), we have r=inft∈[tm-1,σ(tm)]x(t)≤∥x∥≤rM/m for t∈[tm-1,σ(tm)]. Then, we get
(3.13)ϕ(Ax)=∫t1σ(tm)mint∈[tm-1,σ(tm)]G(t,s)f(s,x(σ(s)))Δs≥m∫tm-1σ(tm)f(s,x(σ(s)))Δs>r
by hypothesis (i).
Claim
2. If x∈∂P(θ,q), then θ(Ax)<q. Since x∈∂P(θ,q), 0≤x(t)≤∥x∥≤(M/m)ϕ(x)≤(M/m)θ(x)=qM/m for t∈[t1,σ(tm)]. Thus, by hypothesis (ii) we have
(3.14)θ(Ax)=∫t1σ(tm)maxt∈[tm-1,σ(tm)]G(t,s)f(s,x(σ(s)))Δs≤M∫t1σ(tm)f(s,x(σ(s)))Δs<q.
Claim
3.
P
(
η
,
p
)
≠
∅ and η(Ax)>p for all x∈∂P(η,p). Since 0∈P and p>0, P(η,p)≠∅. If x∈∂P(η,p), we get (m/M)p≤ϕ(x)≤x(t)≤∥x∥=p for t∈[tm-1,σ(tm)]. Hence, we obtain
(3.15)η(Ax)≥∫t1σ(tm)G(t,s)f(s,x(σ(s)))Δs≥m∫tm-1σ(tm)f(s,x(σ(s)))Δs>p
by hypothesis (iii). This completes the proof.To prove the existence of at least three positive solutions for the BVP (1.2), we will apply the following (Leggett-Williams) fixed-point theorem.Theorem 3.5 (see [13]).
Let P be a cone in the real Banach spaceE. Set
(3.16)Pr:={x∈P:∥x∥<r},P(ψ,a,b):={x∈P:a≤ψ(x),∥x∥≤b}.
Suppose A:Pr¯→Pr¯ is a completely continuous operator and ψ is a nonnegative continuous concave functional on P with ψ(u)≤∥u∥ for all u∈Pr¯. If there exists 0<p<q<l≤r such that the following condition hold:(i)
{u∈P(ψ,q,l):ψ(u)>q}≠∅ and ψ(Au)>q for all u∈P(ψ,q,l);(ii)
∥Au∥<p for ∥u∥≤p;(iii)
ψ(Au)>q for u∈P(ψ,q,r) with ∥Au∥>l,
thenA has at least three fixed-points u1,u2, and u3 in Pr¯ satisfying
(3.17)∥u1∥<p,ψ(u2)>q,p<∥u3∥withψ(u3)<q.Theorem 3.6.
Suppose that there exist numbersp, q, and r satisfying 0<p<q<qM/m≤r such that for t∈[t1,σ(tm)] the function f satisfies the following conditions:(i)
f(t,x)≤r/Mσ(tm), x∈[0,r],(ii)
f(t,x)>q/mσ(tm), x∈[q,qM/m],(iii)
f(t,x)<p/Mσ(tm), x∈[0,p]. Then (1.2) has at least three positive solutions x1,x2, and x3 satisfying
(3.18)supt∈[t1,σ(tm)]x1(t)<p,q<inft∈[t1,σ(tm)]x2(t),p<supt∈[t1,σ(tm)]x3(t)withinft∈[t1,σ(tm)]x3(t)<q.Proof.
Define the nonnegative continuous concave functionalψ:P→[0,∞) to be ψ(x):=inft∈[t1,σ(tm)]x(t) and the cone P as in (2.8). For all x∈P, we have ψ(x)≤∥x∥. If x∈Pr¯, then 0≤x≤r and f(t,x)≤r/Mσ(tm) from the hypothesis (i). Then we get
(3.19)∥Ax∥=supt∈[t1,σ(tm)]∫t1σ(tm)G(t,s)f(s,x(σ(s)))Δs≤M∫t1σ(tm)f(s,x(σ(s)))Δs≤r
by Lemma 2.2. This proves that A:Pr¯→Pr¯. Similarly, by the hypothesis (iii), the condition (ii) of Theorem 3.5 is satisfied.
SinceqM/m∈P(ψ,q,qM/m) and ψ(qM/m)>q, {y∈P(ψ,q,qM/m):ψ(x)>q}≠∅. For all x∈P(ψ,q,qM/m), we have q≤inft∈[t1,σ(tm)]x(t)≤∥x∥≤qM/m for t∈[t1,σ(tm)]. Using the hypothesis (ii) and Lemma 2.2, we find
(3.20)ψ(Ax)=∫t1σ(tm)inft∈[t1,σ(tm)]G(t,s)f(s,x(σ(s)))Δs≥m∫t1σ(tm)f(s,x(σ(s)))Δs>q.
Hence, the condition (i) of Theorem 3.5 holds.
For the condition(iii) of Theorem 3.5, we suppose that x∈P(ψ,q,r) with ∥Ax∥>qM/m. Then, from Lemma 2.2 we obtain
(3.21)ψ(Ax)=inft∈[t1,σ(tm)]Ax(t)≥mM∥Ax∥>q.
This completes the proof.


## 4.  Examples

Example 4.1.
Let𝕋=ℤ. We consider the first-order four-point BVP as follows:
(4.1)xΔ(t)+x(σ(t))=x+5x4+1,t∈[0,5]⊂𝕋,x(0)=x(1)+x(2)+x(6).
Taking a(t)≡1,t1=0,t2=1,t3=2,t4=5,andα1=α2=α3=1, we have Γ=64/15,m=1/960, and M=384/5. If we take p=0.001,q=0.01, and r=0.02; then all the assumptions in Theorem 3.4 are satisfied. Finally, BVP (4.1) has at least two positive solutions x1 and x2 such that
(4.2)0.001<supt∈[0,6]x1(t)withsupt∈[2,6]x1(t)<0.01,0.01<supt∈[2,6]x2(t)withinft∈[2,6]x2(t)<0.02.Example 4.2.
Let𝕋=ℕ02. We consider the first-order four-point BVP as follows:
(4.3)xΔ(t)+x(σ(t))=f(t,x(σ(t))),t∈[0,9]⊂𝕋,x(0)=x(1)+x(4)+x(16),
where a(t)≡1,t1=0,t2=1,t3=4,t4=9,α1=α2=α3=1, and
(4.4)f(t,x)={x400,(t,x)∈[0,16]×[0,1],(6870-1400)x+2400-6870,(t,x)∈[0,16]×[1,2],15x882434+6870-15441217,(t,x)∈[0,16]×[2,∞).
Hence, we obtain Γ=384/143,m=1/54912, and M=2296/143. If we take p=1,q=2, and r=1764870; then all the assumptions in Theorem 3.6 are satisfied. Finally, BVP (4.3) has at least three positive solutions x1,x2, and x3 such that
(4.5)supt∈[0,16]x1(t)<1,2<inft∈[0,16]x2(t),1<supt∈[0,16]x3(t)withinft∈[0,16]x3(t)<2.



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