Positive solutions for second-order differential equations with singularities and separated integral

Positive solutions for second-order differential equations with singularities and separated integral

boundary conditions

Dedicated to Professor Jeffrey R. L. Webb on the occasion of his 75th birthday

Yanlei Zhang

, Kenzu Abdella

and Wenying Feng

1Department of Mathematics and Statistics, Queen’s University, Kingston, ON, K7L 3N6, Canada

2Department of Mathematics, Trent University, Peterborough, ON, K9L 0G2, Canada

Received 5 August 2020, appeared 21 December 2020 Communicated by Gennaro Infante

Abstract. We study the existence of positive solutions for second-order differential equations with separated integral boundary conditions. The nonlinear part of the equation involves the derivative and may be singular for the second and third space variables. The result ensures existence of a positive solution when the parameters are in certain ranges. The proof depends on general properties of the associated Green’s function and the Krasnosel’skii–Guo fixed point theorem applied to a perturbed Ham- merstein integral operator. Both numerical and analytical examples are constructed to illustrate applications of the theorem to a group of equations. The result generalizes previous work.

Keywords: fixed point, Green’s function, Hammerstein integral operator, positive solu- tion, singular boundary value problem.

2020 Mathematics Subject Classification: 34B10, 34B16, 34B18.

1 Introduction

We are interested in the following singular Boundary Value Problem (BVP) for second-order differential equations with non-local boundary conditions involving integrals:











u⁰⁰(t) + f(t,u(t),u⁰(t)) =0, t ∈[0, 1], θu(₀)−αu⁰(₀) =R1

0 g1(_s)_u(_s)_ds, γu(1) +βu⁰(1) =R1

0 g₂(s)u(s)ds,

(1.1)

where the parametersθ,α,β>0,γ≥0. The nonlinear function f is continuous, non-negative on [_{0, 1}]×(_0,∞)×(_0,∞)and may be singular at zero on its space variables.

BCorresponding author. Email: wfeng@trentu.ca

When γ = 0, BVP (1.1) reduces to the problem studied in [17]. It also includes the anti- symmetric boundary conditions u(0) = u⁰(0),u(1) = −u⁰(1) [13]. The local singular BVP studied in [16] is a special case of the boundary conditions of (1.1) whenγ=0, and g₁= g₂= 0 as well. Similar boundary conditions have been studied for fractional differential equations in [2] which assumed thatθ =γandα= −β.

In the study of BVPs and their applications, nonlocal boundary conditions usually involve discrete multi-point boundary conditions. Previously, the following three-point BVPs have been extensively studied [3–5,10,12]:

u(0) =0, u(1) =αu(η), or

u⁰(0) =0, u(1) =αu(η),

where 0 < η < 1, α is a parameter. Later, the boundary conditions were further extended to involve integrals and functionals [7–9,13–15]. In particular, in [14], existence of multiple positive solutions for nonlocal BVPs involving various integral conditions were obtained for the case that the nonlinear function f does not involve the first-order derivative. On the other side, results on non-existence of positive solutions for different types of nonlocal BVPs were discussed in [11].

Our main result on the existence of positive solutions of BVP (1.1) is proved by using the similar techniques that were applied in [17] and originally developed by Webb and Infante [13]. The idea is to restrict the singular function f to a subset [0, 1]×[ρ₁,∞)×[ρ₂,∞) of [0, 1]×(0,∞)×(0,∞), whereρ₁,ρ₂ > 0 are properly selected such that problem (1.1) can be converted to the following perturbed Hammerstein integral operator of the form

Fu(t) =

Z ₁

0 G(t,s)f(s,u(s),u⁰(s))ds+r(t)η[u] +w(t)ξ[u], (1.2) where η[u] and ξ[u] are positive linear functionals on C[0, 1], r and w satisfy certain upper bound conditions. Then existence of a positive solution for problem (1.1) is equivalent to a fixed point problem for the operatorF.

For convenience, we give the following definition of an order cone P in a Banach space and the well-known Krasnosel’skii–Guo fixed point theorem on a conePthat will be applied to prove the existence result in Section 3.

Definition 1.1. A conePin a Banach spaceXis a closed convex set such thatλx∈Pfor every x∈ Pand for allλ≥0, and satisfyingP∩(−P) ={0}.

For anyr>0, we denoteΩr ={x∈ X:kxk<r}and∂Ωr= {x∈ X:kxk=r}.

Theorem 1.1(Krasnosel’skii–Guo [6]). Let T : P → P be a compact map. Assume that there exist two positive constantsr,Rwithr 6= Rsuch that

kTuk ≤ kukfor everyu∈ Pwithkuk=r, and

kTuk ≥ kukfor everyu∈ Pwithkuk=R.

Then there existsu0 ∈ Psuch thatTu0=u0and min{r,R} ≤ ku0k ≤max{r,R}.

In Section 2, we first prove some properties of the Green’s function G in (1.2) that are essential in the construction of the cone for our proof.

2 Preliminaries

Let h₁(t) = γ(1−t) +β, h₂(t) = α+θt and m = θγ+θ β+αγ. The following assumption ensures that problem (1.1) is non-resonant [5]:

(H1)

m−

Z ₁

0 g₁(s)h₁(s)ds m−

Z ₁

0 g₂(s)h₂(s)ds

−

Z ₁

0 g₁(s)h₂(s)ds Z ₁

0 g₂(s)h₁(s)ds6=0.

This condition implies that BVP (2.1) has only the trivial solution:











u⁰⁰(t) =_0, t∈ [_{0, 1}]_, θu(0)−αu⁰(0) =R1

0 g₁(s)u(s)ds, γu(1) +βu⁰(1) =R1

0 g2(s)u(s)ds.

(2.1)

Under condition (H1), BVP (1.1) can be converted to a fixed point problem for the nonlinear operator Fin (1.2), whereGis the Green’s function of the problem











u⁰⁰(t) +y(t) =0, t ∈[0, 1], θu(0)−αu⁰(0) =0,

γu(1) +βu⁰(1) =0,

(2.2)

r andware the unique solutions of











u⁰⁰(t) =0, t ∈[0, 1], θu(0)−αu⁰(0) =1, γu(1) +βu⁰(1) =0,

(2.3)

and 









u⁰⁰(t) =0, t ∈[0, 1], θu(0)−αu⁰(0) =0, γu(1) +βu⁰(1) =1,

(2.4)

respectively. By calculation, we can find thatr(t) = ^h1_m^(t), w(t) = ^h2_m^(t), and

G(t,s) =











h₂(s)h₁(t)

m , 0≤ s≤t ≤1, h₂(t)h₁(s)

m , 0≤ t≤s ≤1.

(2.5)

Condition (H1) is equivalent to

1−

Z ₁

0 g₁(s)r(s)ds 1−

Z ₁

0 g₂(s)w(s)ds

−

Z ₁

0 g₁(s)w(s)ds Z ₁

0 g₂(s)r(s)ds6=0. (2.6) Lemma 2.1. LetΦ(s) =G(s,s), then

c₀Φ(s)≤ G(t,s)≤_Φ(s), for 0<t,s<1, where

c₀=











α

α+θ, γ=0 or

γ6=0, and ^β_γ− ^α_θ ≥1 ,

β

β+γ, γ6=0, _γ^β −^α

θ ≤ −1,

αβ

(α+θ)(γ+β), γ6=0, −1< ^β

γ− ^α

θ <1.

(2.7)

Proof: For both cases of 0 ≤ s ≤ t ≤ 1 and 0 ≤ t ≤ s ≤ 1, we can easily verify that G(t,s)≤ G(s,s)from the inequalities: h₁(t) ≤ h₁(s) for 0 ≤ s ≤ t ≤ 1 and h2(t)≤ h2(s)for 0≤t≤s ≤1. Now consider

c₀G(s,s) = ^c0(−θγs²+ (γθ+βθ−αγ)s+α(γ+β))

m .

(1) Ifγ=0,

c0G(s,s) = ^c0(θs+α)

θ < ^c0(θ+α)

θ = ^α

θ ≤ (α+θt(ors))

θ = G(t,s).

(2) Ifγ6=0, leth(s):=−θγs²+ (γθ+βθ−αγ)s+α(γ+β). Thenhhas the critical point:

s₀ = ¹ 2 +¹

2 β

γ −^α θ

. Assume that _γ^β −^α

θ ≥_{1, max}{h(s)_,s∈ [_{0, 1}]}=h(₁)_, c₀G(s,s)≤ ^c0(α+θ)β

m ≤ ^c0(α+θ)(β+γ(1−t(or s))) m

≤ (α+θs(ort))(γ+β−γt(ors))

m = G(t,s).

On the other hand, if _γ^β −^α_θ ≤ −1, max{h(s),s∈[0, 1]}=h(0), c₀G(s,s)≤ ^c0α(γ+β)

m ≤ ^c0(α+θs(ort))(γ+β) m

≤ (α+θs(ort))(γ+β−γt(ors))

m =G(t,s).

In the case of−1< ^β_γ− ^α_θ <1, we have−αγ<γθ−βθ,

max{h(s)_,s∈[_{0, 1}]}=α(γ+β) +(γθ+βθ−αγ)²

4θγ < α(γ+β) +_θγ.

Therefore,

c₀h(s)< c₀(α(γ+β) +θγ) = ^α α+θ

αβ+ ^βθγ γ+β

<αβ< (α+θs(ort))(γ+β−γt(ors)),

and

c₀G(s,s)< (α+θs (or t))(γ+β−γt (or s))

m =G(t,s).

The following simple property of the constantc₀will be useful in the sequel.

Property 2.2. Letc0be defined as (2.7). Thenc0≤min _α

α+θ,_γ+^β_β .

Proof: If γ = 0, it is true since c0 = _α+^α_θ < 1. It is also clear for the case of γ 6= 0 and

−1 < _γ^β − ^α_θ < 1. Assume thatγ6=0 and _γ^β − ^α_θ ≥1. Then _γ^β > ^α_θ implies _β+^γ_γ < _α+^θ_θ. Hence c₀ = ^α

α+θ < ^β

β+γ. Similarly, it can be shown that c₀ = ^β

β+γ < ^α

α+θ with the assumption of

β γ− ^α

θ ≤ −1.

3 Main result

LetC¹[_{0, 1}]be the Banach space of continuously differentiable functions with the normkuk= max{kuk_∞,ku⁰k_∞} and kuk_∞ = max{|u(t)| : t ∈ [0, 1]}. Following similar approaches of [13,17], we consider the BVP for ef, the restriction of f on [0, 1]×[ρ₁,∞]×[ρ₂,∞], where ρ₁>_0, ρ₂>_0:











u⁰⁰(t) + ef(t,u(t),u⁰(t)) =0, t ∈[0, 1], θu(0)−αu⁰(0) =R1

0 g₁(s)u(s)ds, γu(1) +βu⁰(1) =R1

0 g₂(s)u(s)ds.

(3.1)

If u₀ is a positive solution of the regular BVP (3.1), thenu₀(t)≥ ρ₁ > 0 andu₀⁰(t)≥ ρ₂, so u₀ is a positive solution of (1.1). In addition to (H1), we introduce more assumptions on function

ef and the coefficientsθ, α, γandβthat appear in (3.1). Let l₁=

Z ₁

0 g₁(s)ds, l₂ =

Z ₁

0 g₂(s)ds,

and m = θ(γ+β) +αγ as defined in Section 2. Assume there exist 0 < r < R andK,k > 0 such that:

(H2) c0min

1,^α_θ r ≥ρ₁ andcmin

1,^α_θ r ≥ρ₂; (H3) ^ef(t,u,v_R ⁾≤K≤^{2(m−βl1−αl2−θl2)}

θγ+2(θ+α)β for(t,u,v)∈[0, 1]×Rc₀min

1,^α_θ ,R

×[Rcmin

1,^α_θ ,R]; (H4) ^ef(t,u,v_r ⁾ ≥k ≥ ^{2(m−c0βmin}{^1,α_θ}^l1−c₀(α+θ)min{^1,α_θ}^l2)

(2α+θ)β for(t,u,v)∈[0, 1]×[rc₀min

1,^α_θ ,r]× [rcmin

1,^α_θ ,r]; (H5) (θl₂−γl₁)min

1,^α_θ r−RKγ(θ+α)>0.

Of conditions (H2)–(H4), the constantcis defined as

c:= −RKγ(θ+α) + (θl2−γl₁)min 1,^α_θ r (α+θ)(γ+β)RK+ [(γ+β)l₁+ (α+θ)l₂]R. Since

θl₂−γl₁ <(γ+β)l₁+ (α+θ)l₂, minn 1,α

θ o

r <R, We have

(θl₂−γl₁)minn 1,α

θ o

r−RKγ(θ+α)<(α+θ)(γ+β)RK+ [(γ+β)l₁+ (α+θ)l₂]R.

Condition (H5) implies that 0≤c<1.

Theorem 3.1. Under the assumptions (H1)–(H5), the regular BVP (3.1) has a positive solution usatisfying

c0minn 1,α

θ o

r ≤u(t)≤R, and

cminn 1,α

θ o

r≤u⁰(t)≤R.

Proof: Similar as (1.2), we consider (Fue )(t) =

Z ₁

0 G(t,s)ef(s,u(s),u⁰(s))ds+r(t)

Z ₁

0 g₁(s)u(s)ds+w(t)

Z ₁

0 g2(s)u(s)ds (3.2) and its derivative

(Fue )⁰(t) =

Z _t

0

−γ(α+θs)

m ef(s,u(s),u⁰(s))ds+

Z ₁

t

θ(γ+β−γs)

m ef(s,u(s),u⁰(s))ds

− ^γ m

Z ₁

0 g₁(s)u(s)ds+ ^θ m

Z ₁

0 g₂(s)u(s)ds.

Define the conePof C¹[0, 1]as P=ⁿu∈C¹[0, 1]:u(0)≥ ^α

θku⁰k_∞, u⁰(t)≥ckuk_∞, u(t)≥c₀kuk_∞, t∈ [0, 1]^o. (3.3) Notice that the constantcin Pinvolves the upper bound RKand the lower boundrk of feon the closed subsets. Ifu∈ P, then

u(t)≥ c₀kuk_∞≥ c₀u(0)≥c₀α θku⁰k_∞. Hence

u(t)≥maxn

c₀kuk_∞, c₀α

θku⁰k_∞^o

≥c0minn 1,α

θ o

max

kuk_∞,ku⁰k_∞ =c0minn 1,α

θ okuk. Also

u⁰(t)≥ ckuk_∞ ≥cα θku⁰k_∞. Therefore

u⁰(t)≥cmax

nkuk_∞, ^α θ

ku⁰k_∞^o≥cmin n

1,α θ

okuk_.

Let

Ω1= {u∈ C¹[0, 1]:kuk<r} and Ω2= {u∈C¹[0, 1]:kuk< R}. We show thatFe: P∩(_Ω2\_Ω1)→P. Ifu∈ P∩(_Ω2\_Ω1), thenFue ∈C¹[0, 1], and

c₀kFue k_∞ ≤c₀ Z ₁

0 G(s,s)fe(s,u(s),u⁰(s))ds+c₀β+γ m

Z ₁

0 g₁(s)u(s)ds +c0

α+θ m

Z ₁

0 g2(s)u(s)ds

≤

Z ₁

0 G(t,s)ef(s,u(s),u⁰(s))ds+ ^β+γ(1−t) m

Z ₁

0 g₁(s)u(s)ds + ^α+θt

m Z ₁

0 g₂(s)u(s)ds

= Fue (t)_{. (3.4)}

Next, conditions (H3) and (H4) imply thatf(t,u,v)≤RKfor(t,u,v)∈[0, 1]×[Rc₀min{1,^α_θ},R]×

Rcmin{_1,^α

θ}_,R

andf(t,u,v)≥rkfor(t,u,v)∈[_{0, 1}]×rc₀min

1,^α_θ ,r

×rcmin

1,^α_θ ,r .

For u∈P∩(_Ω2\_Ω1), we obtain that (Fue )⁰(t) =

Z _t

0

−γ(α+θs)

m ef(s,u(s)_,u⁰(s))ds+

Z ₁

t

θ(γ+β−γs)

m ef(s,u(s)_,u⁰(s))ds

− ^γ m

Z ₁

0 g₁(s)u(s)ds+ ^θ m

Z ₁

0 g2(s)u(s)ds

≥ RK−γ(α+θ)

m +rk

Z ₁

t

θ(γ+β−γs)

m ds+

Z ₁

0

−γg₁(s) +θg₂(s) m u(s)ds

≥ RK−γ(α+θ)

m +rkθ(γ+β)

m (1−t)−rkθγ

2m(1−t²) + ^θl2−γl₁

m rminn 1,α

θ o

= ^rkθγ

2m t²−^rkθ(γ+β)

m t−RK−γ(α+θ)

m + ^rkθ(γ+β)

m −rkθγ 2m + ^θl2−γl₁

m rmin n

1,α θ

o

= H(t)_{. (3.5)}

Ifγ=_{0, then}

H(t) =−^{rkθ β}

m t+ ^{rkθ β} m + ^θl2

mrminn 1,α

θ o

.

H is decreasing for t ∈ [0, 1]. The minimum occurs at t = 1. When γ 6= 0, H is a quadratic function with the critical point ^γ+_γ^β > 1. For t ∈ [0, 1], the minimum also occurs at t = 1.

Hence

(Fue )⁰(t)≥ H(₁) = −RKγ(α+θ) + (θl₂−γl₁)r min{1,^α_θ}

m . (3.6)

On the other hand, ckFue k_∞ ≤c

Z ₁

0

(α+θ)(γ+β)

m ef(s,u(s),u⁰(s))ds+cγ+β m

Z ₁

0 g₁(s)u(s)ds +cα+θ

m Z ₁

0

g₂(s)u(s)ds

≤c

RK(α+θ)(γ+β)

m + (γ+β)R

m l₁+ (α+θ)R m l₂

= −RKγ(θ+α) + (θl₂−γl₁)min 1,^α_θ r

m . (3.7)

From (3.6) and (3.7), we have

(Fue )⁰(t)≥ ckFue k_∞ ≥0, t ∈[0, 1]. (3.8) The non-negative property of (Fue )⁰ ensures that

α

θk(Fue )⁰k_∞= ^α θ max

t∈[0,1](Fue )⁰(t)

≤

Z ₁

0

α(γ+β−γs)

m ef(s,u(s),u⁰(s))ds− ^αγ θm

Z ₁

0 g₁(s)u(s)ds+ ^α m

Z ₁

0 g₂(s)u(s)ds

≤

Z ₁

0

α(γ+β−γs)

m ef(s,u(s),u⁰(s))ds+ ^γ+β m

Z ₁

0 g₁(s)u(s)ds+ ^α m

Z ₁

0 g₂(s)u(s)ds

= Fue (0). (3.9)

Combining (3.4), (3.8) and (3.9), we obtain that Femaps P∩(_Ω2\_Ω1)_to P.

Next, foru∈ P∩_∂Ω2,kuk= R, Rc₀minn

1,α θ

o≤u(t)≤ R and Rcminn 1,α

θ

o≤u⁰(t)≤R.

kFue (t)k_∞= Fue (1)

=

Z ₁

0

G(_1,s)fe(s,u(s)_,u⁰(s))ds+ ^β m

Z ₁

0

g₁(s)u(s)ds

+^α+θ m

Z ₁

0 g2(s)u(s)ds

≤ KR Z ₁

0 G(1,s)ds+ ^βR

m l₁+(α+θ)R m l₂

= KR αβ

m + ^{θ β} 2m

+ ^βRl1+ (α+θ)Rl2

m ,

and

k(Fue )⁰(t)k_∞ ≤

Z ₁

0

θ(γ+β−γs)

m ef(s,u(s),u⁰(s))ds− ^γ m

Z ₁

0 g₁(s)u(s)ds + ^θ

m Z ₁

0 g₂(s)u(s)ds

≤ KR Z ₁

0

α(γ+β−γs)

m ds+^θl2 mR

= ^KR(^θγ₂ +θ β) +θl2

m .

Thus, (H3) implies

k(Fue )(t)k= maxn

kFuk_∞,k(Fue )⁰k_∞^o

≤ ^K(^θγ₂ +θ β+αβ) +βl₁+ (α+θ)l2

m R

≤ R=kuk. Foru∈ P∩_∂Ω1,kuk=r,

rc₀min n

1,α θ

o

≤u(t)≤r and rcmin n

1,α θ

o

≤u⁰(t)≤r.

From (H4), we obtain kFue k ≥ kFue k_∞

≥

Z ₁

0 G(1,s)f^e(s,u(s),u⁰(s))ds+ ^β m

Z ₁

0 g₁(s)u(s)ds+^α+θ m

Z ₁

0 g2(s)u(s)ds

≥kr Z ₁

0 G(1,s)ds+ ^βrc0min

1,^α_θ l₁

m + (α+θ)rc₀min 1,^α_θ l₂ m

≥kr αβ

m + ^{θ β} 2m

+ ^βrc0min

1,^α_θ l₁+ (α+θ)rc0min 1,^α_θ l2

m

= ^k(αβ+^{θ β}₂) + (βmin

1,^α_θ l₁+ (α+θ)min

1,^α_θ l₂)c₀

m r

≥r =kuk.

It can be shown thatFeis compact on P∩(_Ω2\_Ω1)following the standard arguments. Theo- rem1.1ensures that Fehas a fixed point inP∩(_Ω2\_Ω1)_.

4 Examples

We construct two examples to illustrate applications of Theorem3.1. Example4.1represents a group of BVPs satisfying the conditions of Theorem3.1 but results of [17] can not be applied.

Example4.3shows that it is possible for BVPs satisfying all conditions of Theorem3.1to have multiple solutions including negative solutions.

Example 4.1. Consider the boundary value problem











u⁰⁰(t) + (0.5t+1)^0.01

u(t)+ ^0.0001

u⁰(t)

=0, t∈[0, 1], u(0)−u⁰(0) =R1

0 g₁(s)u(s)ds, 0.01u(₁) +4u⁰(₁) =R1

0 g₂(s)u(s)ds,

(4.1)

where the parametersα=θ =_1, β=_4, γ=_{0.01. Let g1, g2} be selected such thatl1 = ₁₆¹ _and l₂ = 1. We can find that c₀ = 0.5, m = 4.02. For example, for g₁(s) = ₈^s, g₂(s) = 2s, we can verify that (H1) is true. LetR=2 andr=0.1. Condition (H3) is satisfied if

f(t,u,v)

2 ≤K<0.22< ^m−βl₁−(α+θ)l₂

θγ

2 +θ β+αβ

<0.23,

for(t,u,v)∈ [0, 1]×[1, 2]×[2c, 2], wherec= ⁻_16.04K^0.04K₊⁺_4.50125^0.09375. Sincecis decreasing with respect to K, by calculation, we havec≥0.011. From

1

2(0.5t+1) 0.01

u + ^0.0001 v

<0.01, for(t,u,v)∈[0, 1]×[1, 2]×[2c, 2], we know that (H3) and (H5) are valid forK∈ [0.01, 0.22].

To findk satisfying condition (H4), we calculate that 0.5 > ^m−c₀ βmin

1,^α_θ l₁+ (α+θ)min

1,^α_θ l₂

θ β 2 +αβ

>0.49.

As ^f(t,u,v_r ⁾ ≥1.01 for(t,u,v)∈[0, 1]×[0.05, 0.1]×[0.1c, 0.1], (H4) is satisfied fork∈[0.50, 1.01]. By Theorem (3.1), BVP (3.8) has a positive solutionu∈ C¹[0, 1]such that 0.05≤ρ₁≤u(t)≤2 and 0.0011 ≤ρ₂ ≤u⁰(t)≤2.

Remark 4.2. More generally, for allα= θ,β= 4,γ = 0.01,l₂ = 1, the calculation of Example 4.1works as long asl₁is small enough. The extreme case isg₁(s) =0. Then the first boundary condition is reduced to u(0)−u⁰(0) = 0. We can verify that c0 = 0.5,m = 4.02α. Select the same values of R and r as that of Example 4.1, we can find the intervals for K ∈ [0.01, 0.25] and k ∈ [0.51, 1.01]. The solution and its derivative are still in the same range as obtained in Example4.1.

Example 4.3. The following problem is in the form of BVP (1.1):











u⁰⁰(t) + ^ln(t+2)

10⁴(t+2)²u(t)+ ¹

10⁴(t+2)³u⁰(t)+^0.98u_t+2^0(t) =0, u(₀)−u⁰(₀) =ξ₁R1

0 su(s)ds, 10⁻⁴u(1) +u⁰(1) =ξ₂R1

0 su(s)ds,

(4.2)

where θ = α = 1, γ = 10⁻⁴, β = 1, g₁(s) = ξ₁s and g₂(s) = ξ₂s. Selectξ₁ = _{3−ln 729}^{4 ln 2}₊⁻_{ln 256}² , ξ₂ = ^{3 ln 3+104}

7500(3−ln 729+ln 256), then 0.1979 < l₁ < 0.1980, 0.3413 < l₂ < 0.3414. It is easy to find that c₀ = ¹₂, and m = 1.0002. Let r = 0.02, R = 1, we can verify that all conditions (H1)- (H5) are satisfied. In fact, equation (4.2) is exact, we can find that u₁(t) = 0.1 ln(t+2) and u₂(t) =−0.1 ln(t+2)are two solutions of problem (4.2). This shows the existence of multiple and negative solutions.

Different from Example4.3, Example4.1cannot be solved analytically. In order to validate this result of Example4.1, we use the sinc-collocation numerical method based on the deriva- tive interpolation to obtain a numerical solution of BVP (4.1). The sinc-collocation method is a highly efficient numerical technique with exponential rate of convergence. The details of the approach can be found in [1]. The numerical algorithm is coded in Python. The graphs of the obtained solutions u andu⁰ for both cases of g₁(s) = 0 and g₁(s) = ₈^s are depicted in Figures 4.1 and 4.2 respectively. Clearly they all satisfy the bounds obtained from Example 4.1.

Figure 4.1: Numerical solution of BVP (4.1) (g₁=_0,g₂=2s)

Figure 4.2: Numerical solution of BVP (4.1) (g₁= ^s₈,g2=2s)

Acknowledgements

The authors thank the anonymous reviewer for valuable suggestions and comments. The re- search was supported by a grant from the Natural Sciences and Engineering Research Council of Canada (NSERC).

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