Existence of Positive Solutions for Semipositone Higher-Order BVPS on Time Scales

doi:10.1155/2010/235296

Research Article

Existence of Positive Solutions for Semipositone Higher-Order BVPS on Time Scales

Yuguo Lin and Minghe Pei

Department of Mathematics, Beihua University, JiLin City 132013, China

Correspondence should be addressed to Minghe Pei,[email protected] Received 4 December 2009; Revised 16 March 2010; Accepted 29 March 2010 Academic Editor: Alberto Cabada

Copyrightq2010 Y. Lin and M. Pei. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We offer conditions on semipositone functionft, u0, u1, . . . , u_n−2such that the boundary value problem,u^Δnt ft, uσⁿ⁻¹t, u^Δσⁿ⁻²t, . . . , u^Δn−2σt 0,t∈0,1∩T,n≥2,u^Δi0 0, i0,1, . . . , n−3,αu^Δn−20−βu^Δn−10 0,γu^Δn−2σ1 δu^Δn−1σ1 0, has at least one positive solution, whereTis a time scale andft, u0, u1, . . . , u_n−2 ∈ C0,1 ×R0,∞ⁿ⁻¹,R−∞,∞is continuous withft, u0, u1, . . . , u_n−2≥ −Mfor some positive constantM.

1. Introduction

Throughout this paper, letTbe a time scale, for anya, b∈R −∞,∞b > a, the interval a, b defined asa, b {t ∈ T | a ≤ t ≤ b}. Analogous notations for open and half-open intervals will also be used in the paper. We also use the notationRc, d to denote the real interval{t∈R |c≤ t≤ d}. To understand further knowledge about dynamic equations on time scales, the reader may refer to1–3 for an introduction to the subject.

In this paper, we present results governing the existence of positive solutions to the differential equation on time scales of the form

u^Δnt f t, u

σⁿ⁻¹t , u^Δ

σⁿ⁻²t

, . . . , u^Δn−2σt

0, t∈0,1, n≥2 1.1

subject to the two-point boundary conditions

u^Δi0 0, i0,1, . . . , n−3, αu^Δn−20−βu^Δn−10 0, γu^Δn−2σ1 δu^Δn−1σ1 0,

1.2

whereα, γ, β, δ≥0, d:αδγβαγσ1>0 and δγ

σ1−σ²1

≥0. 1.3

Throughout, we assume that ft, u0, u₁, . . . , u_n−2 ∈ C0,1 × R0,∞ⁿ⁻¹,R−∞,∞ is continuous withft, u0, u1, . . . , u_n−2≥ −Mfor some positive constantM.

LetC_rdⁿ0,1 denote the space of functions Cⁿ_rd0,1

u:u∈C0, σⁿ1 , . . . , u^Δn−1∈C0, σ1 , u^Δn ∈C_rd0,1

. 1.4

We say thatutis a positive solution of BVP1.1and1.2, ifut∈Cⁿ_rd0,1 is a solution of BVP1.1and1.2andu^Δit>0, t∈0, σⁿ⁻ⁱ1, i0,1, . . . , n−2.

Various cases of BVP1.1and1.2have attracted a lot of attention in the literature.

When n 2, BVP 1.1 and 1.2 has been studied by many specialists. For example, Agarwal et al. 4 have established the existence of positive solutions for continuous case of the semipositone Sturm-Liouville BVPs. Erbe and Peterson 5 andHao et al. 6 dealt with Sturm-Liouville BVPs on time scale of positone nonlinear term. In addition, Agarwal and O’Regan7 obtained positive solution of second-order right focal BVPs on time scale by using nonlinear alternative of Leray-Schauder type. In 2005, Chyan and Wong 8 obtained triple solutions of the same BVPs with7 . Recently,Sun and Li9,10 investigated semipositone Dirichlet BVPs on time scale. For higher-order BVPs, continuous case of BVP 1.1and1.2have been investigated by Agarwal and Wong11 , Wong and Agarwal12 and Wong13 . The discrete positone case of BVP1.1and1.2has been tackled by using a fixed point theorem for mappings that are decreasing with respect to a cone in14 . Especially, time-scale case of 1.1 with four-point boundary condition has been studied by Liu and Sang15 . Besides, BVP1.1and1.2of nonlinear positone termft, ut, ut, . . . , uⁿ⁻¹t which satisfied Nagumo-type conditions have been dealt with in 16 . Motivated by the works mentioned above, the purpose of this paper is to tackle semipositone BVP1.1and 1.2. In fact, BVPs appeared in7–14 can be looked at as special case of BVP1.1and1.2 in this paper. For other related works, we also refer to17–19 .

The paper is outlined as follows. In Section2, we will present some notations and lemmas which will be used later. In Section3, by using Krasnoselskii’s fixed point theorem in a cone, we offer criteria for the existence of positive solution of BVP1.1and1.2.

2. Preliminary

In this section, we offer some notations and lemmas, which will be used in main results.

Throughout this paper, we always use the following notations:

C1Kt, s is the Green’s function of the differential equation −u^Δnt 0, t ∈ 0,1 subject to the boundary conditions1.2;

C2kt, s is the Green’s function of the differential equation−u^ΔΔt 0, t ∈ 0,1 subject to the boundary conditions

αu0−βu^Δ0 0, γuσ1 δu^Δσ1 0; 2.1

C3DefineTi:0,1 → R,i0,1, . . . , n−2 as

T₀t≡qt, T_it _t

0

T_i−1τΔτ, i1,2, . . . , n−2, 2.2

where

qt: min

t∈0,σ²1

αtβ

ασ²1 β,γσ1−t δ

γσ1 δ

. 2.3

Lemma 2.1. For the Green’s functionkt, sthe following hold:

0≤qtkσs, s≤kt, s≤kσs, s, t, s∈ 0, σ²1

×0,1 . 2.4

Proof. It is clear that

kt, s K^Δn−2t t, s

⎧⎪

⎪⎨

⎪⎪

⎩ 1 d

αtβ

γσ1−σs δ

, t≤s, 1

d

ασs β

γσ1−t δ

, σs≤t.

2.5

From the expression ofkt, s, we can easily obtain

0≤qtkσs, s≤kt, s≤kσs, s, t, s∈ 0, σ²1

×0,1 . 2.6

Lemma 2.2. Letwtbe the solution of BVP

−u^Δnt M, t∈0,1 , u^Δi0 0, i0,1, . . . , n−3,

αu^Δn−20−βu^Δn−10 0, γu^Δn−2σ1 δu^Δn−1σ1 0.

2.7

Then

0≤w^Δit≤cMT_n−2−it, t∈ 0, σⁿ⁻ⁱ1

, i0,1, . . . , n−2, 2.8

where

c:=

γσ1 δ

ασ²1 β

d σ1, 2.9

andM∈R0,∞is a positive constant.

Proof. Fort≤s,

kt, s 1 d

αtβ

γσ1−σs δ

≤ 1 d

αtβ

γσ1−t δ

αtβ

ασ²1 β·γσ1−t δ

γσ1 δ ·

γσ1 δ

α²σ1 β d

≤ cqt σ1.

2.10

Forσs≤t,

kt, s 1 d

ασs β

γσ1−t δ

≤ 1 d

αtβ

γσ1−t δ

γσ1−t δ

γσ1 δ · αtβ

ασ²1 β·

γσ1 δ

ασ²1 β d

≤ cqt σ1.

2.11

So

0≤kt, s≤ cqt

σ1, t, s∈ 0, σ²1

×0,1 . 2.12

By definingwtaswt _σ1

0 Kt, sM ds, t∈0, σⁿ1 , it is clear that

w^Δn−2t _σ1

0

kt, sM ds, t∈ 0, σ²1

. 2.13

Then

0≤w^Δn−2t≤ cqt σ1

_σ1

0

MΔscMqt, t∈ 0, σ²1

. 2.14

Further, sincew^Δit 0, i0,1, . . . , n−3, we get

0≤w^Δit≤cMT_n−2−it, t∈ 0, σⁿ⁻ⁱ1

, i0,1, . . . , n−2. 2.15

Lemma 2.3see20 . LetEbe a Banach space, and letC⊂Ebe a cone inE. Assume thatΩ1,Ω2

are open subsets ofEwith 0 ∈ Ω1 ⊂ Ω1 ⊂ Ω2, and letT : C∩Ω2\Ω1 → Cbe a completely continuous operator such that either

iTu ≤ u, u∈C∩∂Ω1;Tu ≥ u, u∈C∩∂Ω2or iiTu ≥ u, u∈C∩∂Ω1;Tu ≤ u, u∈C∩∂Ω2. Then,T has a fixed point inC∩Ω2\Ω1.

3. Main Results

In this section, by using Lemma2.3, we offer criteria for the existence of positive solution of BVP1.1and1.2.

LetEdenote the space of functions

E

u:u∈C0, σⁿ1 , . . . , u^Δn−3 ∈C 0, σ³1

, u^Δn−2∈C 0, σ²1

. 3.1

Let B {u ∈ E : u^Δi0 0, i 0,1, . . . , n−3} be a Banach space with the norm u sup_t∈0,σ21 |u^Δn−2t|, and let

C

u∈ B:u^Δn−2t≥qtu, t∈ 0, σ²1

. 3.2

It is obvious thatCis a cone inB. Fromu^Δi0 0, i 0,1, . . . , n−3, it follows that for all u∈C,

T_n−2−itu ≤u^Δit≤σ₀u, t∈ 0, σⁿ⁻ⁱ1

, i0,1, . . . , n−2, 3.3

where

σ₀: σⁿ1 ⁿ⁻². 3.4

Throughout the rest of the section, we assume that the set0, σ1 is such that

ξmin

τ ∈T:τ≥ σ1 4

, ζmin

τ∈T:τ ≤ 3σ1 4

3.5

exist and satisfy

σ1

4 ≤ξ < ζ≤ 3σ1

4 . 3.6

In addition, we denote that

η_i min

t∈^{ξ, σn−i−1ζ} T_n−2−it, i0,1, . . . , n−2. 3.7

In order to obtain positive solutions of BVP1.1and1.2, we need to consider the following boundary value problem:

u^Δnt f^∗ t, v

σⁿ⁻¹t , v^Δ

σⁿ⁻²t

, . . . , v^Δn−2σt

0, t∈0,1, u^Δi0 0, i0,1, . . . , n−3,

αu^Δn−20−βu^Δn−10 0, γu^Δn−2σ1 δu^Δn−1σ1 0,

3.8

where

vt ut−wt, wtis as in Lemma 2.2, f^∗t, u0, u₁, . . . , u_n−2 f

t, ρ₀, ρ₁, . . . , ρ_n−2

M 3.9

and for alli0,1, . . . , n−2,

ρ_i

⎧⎨

⎩

u_i, u_i≥0,

0, u_i<0. 3.10

Let the operatorS:C → Bbe defined by

Sut _σ1

0

Kt, sf^∗ s, v

σⁿ⁻¹s , v^Δ

σⁿ⁻²s

, . . . , v^Δn−2σs

Δs, t∈0, σⁿ1 ,

Su^Δn−2t _σ1

0

kt, sf^∗ s, v

σⁿ⁻¹s , v^Δ

σⁿ⁻²s

, . . . , v^Δn−2σs

Δs, t∈ 0, σ²1 . 3.11

Lemma 3.1. The operatorSmapsCintoC.

Proof. From Lemma2.1, we know that fort∈0, σ²1 ,

Su^Δn−2t _σ1

0

kt, sf^∗ s, v

σⁿ⁻¹s , v^Δ

σⁿ⁻²s

, . . . , v^Δn−2σs Δs

≤ _σ1

0

kσs, sf^∗ s, v

σⁿ⁻¹s , v^Δ

σⁿ⁻²s

, . . . , v^Δn−2σs Δs.

3.12

So fort∈0, σ²1 ,

Su ≤ _σ1

0

kσs, sf^∗ s, v

σⁿ⁻¹s , v^Δ

σⁿ⁻²s

, . . . , v^Δn−2σs

Δs. 3.13

From Lemma2.1again, it follows that fort∈0, σ²1 ,

Su^Δn−2t _σ1

0

kt, sf^∗ s, v

σⁿ⁻¹s , v^Δ

σⁿ⁻²s

, . . . , v^Δn−2σs Δs

≥ _σ1

0

qtkσs, sf^∗

s, v

σⁿ⁻¹s , v^Δ

σⁿ⁻²s

, . . . , v^Δn−2σs Δs

≥qtSu.

3.14

Hence,SmapsCintoC.

Lemma 3.2. The operatorS:C → Cis completely continuous.

Proof. First we shall prove that the operator S is continuous. Let u_m,u ∈ C be such that limm→ ∞um−u0. Fromu^Δi0 0, i0,1, . . . , n−3, we have

sup

t∈0,σⁿ⁻ⁱ1

u^Δ_mⁱt−u^Δit−→0, i0,1, . . . , n−2. 3.15

Then, it is easy to see that asm → ∞

ρ_m: sup

s∈0,1

f^∗ s, u_m

σⁿ⁻¹s

−w

σⁿ⁻¹s

, . . . , u^Δ_mⁿ⁻²σs−w^Δn−2σs

−f^∗ s, u

σⁿ⁻¹s

−w

σⁿ⁻¹s

, . . . , u^Δn−2σs−w^Δn−2σs−→0.

3.16

Hence, we get from Lemma2.1that fort∈0, σ²1 , Sum^Δn−2t−Su^Δn−2t

_σ1

0

kt, s f^∗ s, u_m

σⁿ⁻¹s

−w

σⁿ⁻¹s

, . . . , u^Δ_mⁿ⁻²σs−w^Δn−2σs

−f^∗ s, u

σⁿ⁻¹s

−w

σⁿ⁻¹s

, . . . , u^Δn−2σs−w^Δn−2σs Δs

≤ρm

_σ1

0

kt, sΔs≤ρm

_σ1

0

kσs, sΔs−→0, as m−→ ∞.

3.17

This shows thatS:C → Cis continuous.

Next, to show complete continuity, we will apply Arzela-Ascoli theorem. LetΩbe a bounded subset ofC. Then there existsL >0 such that for allu∈Ω,

supu^Δn−2≤L, supu^Δi≤σ₀L, i0,1, . . . , n−3, 3.18

whereσ0is given in3.4. Let

M sup

s,ρ0,ρ1,...,ρn−2∈0,1 ×R0,σ0L ⁿ⁻²×R0,L

f

s, ρ₀, ρ₁, . . . , ρ_n−2

M. 3.19

Clearly, we have fort∈0, σ²1 , Su^Δn−2t≤M

_σ1

0

kt, sΔs≤M _σ1

0

kσs, sΔs 3.20

and fort, t∈0, σ²1 ,

Su^Δn−2t−Su^Δn−2

t ≤M _σ1

0

kt, s−k

t, sΔs. 3.21

The Arzela-Ascoli theorem guarantees that SΩ is relatively compact, so S : C → C is completely continuous.

Theorem 3.3. Assume that the following conditions hold:

ithere exist r ∈ RcM,∞ such that for any u0, u₁, . . . , u_n−2 ∈ Γr :R0, σ0r ⁿ⁻² × R0, r ,

Au0, u1, . . . , u_n−2: _σ1

0

kσs, s

fs, u0, u1, . . . , u_n−2 M

Δs≤r, 3.22

iithere existR ∈ RcM,∞ with R /r such that for any u0, u₁, . . . , u_n−2 ∈ ΓR : Rη0R, σ0R ×Rη1R, σ0R × · · · ×Rηn−2R, R ,

Bu0, u1, . . . , u_n−2:η_{n−2 ζ}

ξ

kσs, s

fs, u0, u1, . . . , u_n−2 M

Δs≥R, 3.23

whereσ0is given in3.4,ξ, ζare given in3.5,ηi, i0,1, . . . , n−2 are given in3.7, and

1−cM

R . 3.24

Then BVP1.1and1.2has a positive solution.

Proof. Without loss of generality, we assume thatr < R. Now we seek positive solutions of BVP3.8. Let

Ω1{u∈B:u ≤r}. 3.25

Foru∈∂Ω1∩C, it follows from3.3that

0≤u^Δit≤σ₀r, t∈ 0, σⁿ⁻ⁱ1

, i0,1, . . . , n−3. 3.26

Fromi, we obtain that foru∈∂Ω1∩C,

Su^Δn−2t _σ1

0

kt, sf^∗ s, v

σⁿ⁻¹s , v^Δ

σⁿ⁻²s

, . . . , v^Δn−2σs Δs

≤ _σ1

0

kσs, sf^∗ s, v

σⁿ⁻¹s , v^Δ

σⁿ⁻²s

, . . . , v^Δn−2σs Δs

≤r.

3.27

So

Su ≤ u, u∈∂Ω1∩C. 3.28

Let

Ω2{u∈B:u ≤R}. 3.29

Foru∈∂Ω2∩C, it follows from Lemma2.2and3.3that fors∈0, σ1 ,

v^Δi

σⁿ⁻ⁱ⁻¹s u^Δi

σⁿ⁻ⁱ⁻¹s

−w^Δi

σⁿ⁻ⁱ⁻¹s

≥u^Δi

σⁿ⁻ⁱ⁻¹s

−cMT_n−2−i

σⁿ⁻ⁱ⁻¹s u^Δi

σⁿ⁻ⁱ⁻¹s

−cMT_n−2−i

σⁿ⁻ⁱ⁻¹s u R

≥u^Δi

σⁿ⁻ⁱ⁻¹s

−cMu^Δi

σⁿ⁻ⁱ⁻¹s R

u^Δi

σⁿ⁻ⁱ⁻¹s

≥RT_n−2−i

σⁿ⁻ⁱ⁻¹s

, i0,1, . . . , n−2.

3.30

So

v^Δi

σⁿ⁻ⁱ⁻¹s

≥η_iR, s∈ξ, ζ , i0,1, . . . , n−2, 3.31

whereη_iis given in3.7andis given in3.24.

Combining Lemma2.1,3.3, andiiwith3.31, we obtain that foru∈∂Ω2∩C,

Su^Δn−2t _σ1

0

kt, sf^∗ s, v

σⁿ⁻¹s , v^Δ

σⁿ⁻²s

, . . . , v^Δn−2σs Δs

≥ _ζ

ξ

kt, sf^∗ s, v

σⁿ⁻¹s , v^Δ

σⁿ⁻²s

, . . . , v^Δn−2σs Δs

≥η_{n−2 ζ}

ξ

kσs, sf^∗ s, v

σⁿ⁻¹s , v^Δ

σⁿ⁻²s

, . . . , v^Δn−2σs Δs

≥R.

3.32

So

Su ≥ u, u∈∂Ω2∩C. 3.33

Therefore, it follows from Lemma2.3that BVP3.8has a solutionu₁∈Csuch thatr≤ u1 ≤ R.

Finally, we will prove thatu1t−wtis a positive solution of BVP1.1and1.2. Let ut u1t−wt, then we have from Lemma2.2and3.3that fori0,1, . . . , n−2,

u^Δit u^Δ₁ⁱt−w^Δit≥u^Δ₁ⁱt−cMT_n−2−it

≥u^Δ₁ⁱt−cMT_n−2−itu1 r

≥u^Δ₁ⁱt−cMu^Δ₁ⁱt r

1−cM

r

u^Δ₁ⁱt≥r−cMT_n−2−it>0, t∈

0, σⁿ⁻ⁱ1 .

3.34

In addition,

u^Δnt u^Δ₁ⁿt M −f^∗

t, u1

σⁿ⁻¹t

−w

σⁿ⁻¹t

, . . . , u^Δ₁ⁿ⁻²σt−w^Δn−2σt M −f

t, u₁

σⁿ⁻¹t

−w

σⁿ⁻¹t

, . . . , u^Δ₁ⁿ⁻²σt−w^Δn−2σt −f

t, u

σⁿ⁻¹t

, . . . , u^Δn−2σt .

3.35

So,ut u1t−wtis a positive solution of BVP1.1and1.2. This completes the proof.

Corollary 3.4. Assume that

afor anyt, u0, u₁, . . . , u_n−2∈0,1 ×R0,∞ⁿ⁻¹,

ft, u0, u₁, . . . , u_n−2 M≤μtgu0, u₁, . . . , u_n−2, 3.36

whereg : R0,∞ⁿ⁻¹ → R0,∞is a continuous function which is nondecreasing inuj

for each fixedu0, . . . , u_j−1, u_j1, . . . , u_n−2andμtis a continuous nonnegative function on0,1 ,

bfor anyt, u0, u1, . . . , u_n−2∈ξ, ζ ×R0,∞ⁿ⁻¹,

ft, u0, u₁, . . . , u_n−2 M≥νthu0, u₁, . . . , u_n−2, 3.37

whereh:R0,∞ⁿ⁻¹ → R0,∞is a continuous function which is nondecreasing inu_j for each fixedu0, . . . , u_j−1, u_j1, . . . , u_n−2andνtis a continuous nonnegative function on0,1 ,

cthere existsr ∈RcM,∞such that

gσ0r, . . . , σ₀r, r _σ1

0

kσs, sμsΔs≤r, 3.38

dthere existsR∈RcM,∞withR /rsuch that

h

η0R, η1R, . . . , η_n−2R η_n−2

_ζ

ξ

kσs, sνsΔs≥R. 3.39

Then BVP1.1and1.2has a positive solution.

Proof. Fromaandc, we obtain that foru0, u₁, . . . , u_n−2∈Γr,

Au0, u1, . . . , u_n−2

_σ1

0

kσs, s

fs, u0, u1, . . . , u_n−2 M Δs

≤ _σ1

0

kσs, sμsgu0, u1, . . . , u_n−2Δs

≤gσ0r, . . . , σ₀r, r _σ1

0

kσs, sμsΔs≤r.

3.40

So, conditioniof Theorem3.3is satisfied. Frombandd, we obtain that foru0, u1, . . . , u_n−2∈ΓR,

Bu0, u₁, . . . , u_n−2 η_{n−2 ζ}

ξ

kσs, s

fs, u0, u₁, . . . , u_n−2 M Δs

≥η_{n−2 ζ}

ξ

kσs, sνshu0, u₁, . . . , u_n−2Δs

≥h

η₀R, η₁R, . . . , η_n−2R η_n−2

_ζ

ξ

kσs, sνsΔs≥R.

3.41

So, conditioniiof Theorem3.3is satisfied.

Therefore, from Theorem3.3, BVP1.1and1.2has a positive solution.

Corollary 3.5. Assume that conditionsaandcof Corollary3.4and the following condition hold:

u0u1...ulimn−2→ ∞min

t∈ξ,ζ

ft, u0, u₁, . . . , u_n−2 M u₀u₁· · ·u_n−2 ∈R

D_{1 n−2}

i0 η_i,∞

, 3.42

whereD₁ ηn−2_ζ

ξkσs, sΔs ⁻¹. Then BVP1.1and1.2has one positive solution.

Proof. We only need to prove that3.42implies conditioniiof Theorem3.3. From3.42, we know that there existsRRmay be chosen arbitrary largesuch that foru0, u₁, . . . , u_n−2∈ Rη0R,∞× · · · ×Rηn−2R,∞,

t∈ξ,ζ min

ft, u0, u₁, . . . , u_n−2 M

u₀u₁· · ·u_n−2 ≥ D1

_n−2

i0 η_i. 3.43

Hence, fort, u0, u1, . . . , u_n−2∈ξ, ζ ×ΓR,

ft, u0, u₁, . . . , u_n−2 M≥ D_{1 n−2}

i0 η_i n−2

i0

u_i≥ D_{1 n−2}

i0 η_i ·

n−2

i0

η_iRD₁R. 3.44

Thus, it follows that

Bu0, u₁, . . . , u_n−2 η_{n−2 ζ}

ξ

kσs, s

fs, u0, u₁, . . . , u_n−2 M Δs

≥η_{n−2 ζ}

ξ

kσs, sD1RΔsR.

3.45

So, conditioniiof Theorem3.3is satisfied.

Finally we present an example to illustrate our result.

Example 3.6. Consider the following boundary value problem:

u^Δ3t sin

u^Δσt

exp u

σ²t

u^Δσt /5

55uσ²t 0, t∈0,1∩T, u0 0, u^Δ0−u^Δ20 0, u^Δ2σ1 0,

3.46

where T 0 ∪ {t/8 : t ∈ N}, ft, u0, u₁ sinu₁ expu0 u₁/5/55 u₀,M 1, αβδ1,and γ0. Obviously,

dαδγβαγσ1 1, δγ

σ1−σ²1

1≥0, ξ 3

8, ζ 6

8. 3.47

Let μt νt 1, gu0, u1 2 expu0 u1/5/55 u0, and hu0, u1

expu0 u₁/5/55 u₀. So conditions a and b in Corollary 3.4 are satisfied. By direct calculation, we obtain that c 81/32, σ₀ 11/8, T₀t qt 4/9t1, t ∈ 0,10/8 ,and T1t _t

0qτΔτ

τ∈0,tστ−τ qτ. SinceTit, i0,1 are nondecreasing, η0 min_t∈3/8,1 T1t T13/8 3/16,η1 min_t∈3/8,7/8 qt q3/8 11/18. In addition, _σ1

0 kσs, sΔs117/64, _ζ

ξ kσs, sΔs39/64. Taker5, R63. So

gσ0r, r _σ1

0

kσs, sμsΔs

2expσ01 r/5 55σ0r

σ1 0

kσs, sΔs≈3.99<5r,

h

η0R, η1R η1

_ζ

ξ

kσs, sνsΔs

exp

R−cM

η₀η₁ /5 5

5 R−cMη0

η1

_ζ

ξ

kσs, sΔs≈71.25>60R.

3.48

Hence, conditions c and d in Corollary 3.4are satisfied. Therefore from Corollary 3.4, 3.46has at least one positive solution.

Remark 3.7. In Example3.6, because nonlinear termfmay attain negative value, the result in 15 is not applicable.

Acknowledgment

The authors thank the referee for valuable suggestions which led to improvement of the original manuscript.

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