Positive Solutions of Singular Complementary Lidstone Boundary Value Problems

Volume 2010, Article ID 368169,15pages doi:10.1155/2010/368169

Research Article

Positive Solutions of Singular Complementary

Lidstone Boundary Value Problems

Ravi P. Agarwal,

_{Donal O’Regan,}

_{and Svatoslav Stan ˇek}

FL 32901-6975, USA

2_{Department of Mathematics, National University of Ireland, Galway, Ireland}

3_{Department of Mathematical Analysis, Faculty of Science, Palack´y University, Tˇr. 17. listopadu 12,}

771 46 Olomouc, Czech Republic

Correspondence should be addressed to Ravi P. Agarwal,[email protected]

Received 7 October 2010; Accepted 21 November 2010

Academic Editor: Irena Rach ˚unkov´a

Copyrightq2010 Ravi P. Agarwal et al. 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 investigate the existence of positive solutions of singular problem−1mx2m1 _{ft, x, . . . ,}

x2m_{,x0 0,x}2i−1_{0 x}2i−1_{T 0, 1≤i≤m. Here,m≥1 and the Carath´eodory function}

ft, x0, . . . , x2mmay be singular in all its space variablesx0, . . . , x2m. The results are proved by regularization and sequential techniques. In limit processes, the Vitali convergence theorem is used.

1. Introduction

LetTbe a positive constant,J 0, TandÊ− −∞,0,Ê 0,∞,Ê0 Ê\{0}. We consider

the singular complementary Lidstone boundary value problem

−1mx2m1t ft, xt, . . . , x2mt, m≥1, 1.1

x0 0, x2i−10 x2i−1T 0, 1≤i≤m, 1.2

wherefsatisfies the local Carath´eodory function onJ× Df ∈CarJ× Dwith

D

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ Ê

2

×Ê0×Ê− ×Ê0×Ê × · · · ×Ê×Ê0

4k−1

ifm2k−1,

Ê

2

×Ê0×Ê− ×Ê0×Ê × · · · ×Ê−×Ê0

4k1

The functionft, x0, . . . , x2mis positive and may be singular at the value zero of all its space

variablesx0, . . . , x2m.

Leti∈ {0,1, . . . ,2m}. We say thatf is singular at the value zero of its space variablexiif

for a.e.t∈Jand allxj, 0≤j≤2m,j /isuch thatx0, . . . , xi, . . . , x2m∈ D, the relation

lim

xi→0ft, x0, . . . , xi, . . . , x2m ∞ 1.4

holds.

A functionx ∈ AC2m_{J i.e.,xhas absolutely continuous 2mth derivative on Jis}

a positive solution of problem 1.1, 1.2if xt > 0 for t ∈ 0, T, x satisfies the boundary conditions1.2and1.1holds a.e. onJ.

The regular complementary Lidstone problem

−1mx2m1t ht, xt, . . . , xqt, m≥1, qfixed, 0≤q≤2m,

x0 α0, x2i−10 αi, x2i−11 βi, 1≤i≤m

1.5

was discussed in1. Here,h:0,1×Êq1 → Êis continuous at least in the interior of the

domain of interest. Existence and uniqueness criteria for problem 1.5 are proved by the

complementary Lidstone interpolating polynomial of degree 2m. No contributions exist, as far as we know, concerning the existence of positive solutions of singular complementary Lidstone problems.

We observe that differential equations in complementary Lidstone problems as well as derivatives in boundary conditions are odd orders, in contrast to the Lidstone problem

−1mx2mt pt, xt, . . . , xrt, m≥1, r fixed, 0≤r ≤2m−1,

x2i_{0 a}

i, x2i1 bi, 1≤i≤m−1,

1.6

where the differential equation and derivatives in the boundary conditions are even orders. Forai bi 0 1 ≤ i ≤ m−1, regular Lidstone problems were discussed in 2–9, while

singular ones in10–15.

The aim of this paper is to give the conditions on the functionf in1.1which gua-rantee that the singular problem1.1,1.2has a solution. The existence results are proved by regularization and sequential techniques, and in limit processes, the Vitali convergence theorem16,17is applied.

Throughout the paper,x∞ max{|xt| : t ∈ J}and xCn n_k0xk_∞,n ≥ 1

stands for the norm in C0_{J and C}n_{J, respectively. L}1_{J denotes the set of functions}

Lebesgueintegrable onJand measMthe Lebesgue measure ofM ⊂J.

We work with the following conditions on the functionfin1.1.

H1f ∈CarJ× Dand there existsa∈0,∞such that

a≤ft, x0, . . . , x2m, 1.7

H2For a.e.t∈Jand for allx0, . . . , x2m∈ D, the inequality

ft, x0, . . . , x2m≤h ⎛ ⎝_t,2m

j0

_xj⎞⎠2m j0

ωjxj 1.8

is fulfilled, whereh∈ CarJ×0,∞is positive and nondecreasing in the second variable,ωj:Ê → Êis nonincreasing, 0≤j ≤2m,

lim sup

v→ ∞ 1 v

_T

0

ht, Kvdt <1, K

⎧ ⎪ ⎨ ⎪ ⎩

T2m1₋₁

T−1 if T /1,

2m1 if T 1,

₁

0 ω2j

s2_{ds <∞,}

₁

0

ω2j1sds <∞ if 0≤j≤m−1,

1

0

ω2msds <∞.

1.9

The paper is organized as follows. In Section2, we construct a sequence of auxiliary regular differential equations associated with 1.1. Section 3 is devoted to the study of auxiliary regular complementary Lidstone problems. We show that the solvability of these problems is reduced to the existence of a fixed point of an operatorH. The existence of a fixed

point ofHis proved by a fixed point theorem of cone compression type according to

Guo-Krasnosel’skii18,19. The properties of solutions to auxiliary problems are also investigated here. In Section4, applying the results of Section3, the existence of a positive solution of the singular problem1.1,1.2is proved.

2. Regularization

Let mbe from1.1. For n ∈ Æ, define χn, ϕn, τn,m ∈ C

0

Ê, Ên ⊂ Ê, andDn ⊂ Ê

2m1 _by

the formulas

χnu ⎧ ⎪ ⎨ ⎪ ⎩

u foru≥ 1

n, 1

n foru < 1 n,

ϕnu ⎧ ⎪ ⎨ ⎪ ⎩

−1_n foru >−1 n,

u foru≤ −1

n,

τn,m ⎧ ⎨ ⎩

χn ifm2k−1,

ϕn ifm2k,

Ên

−∞,−1

n

∪

1 n,∞

,

DnÊ

2_×

Ên×Ê×Ên×Ê× · · · ×Ê×Ên

2m1

.

Letf∈CarJ× D. Chosen∈Æand put

f∗

nt, x0, x1, x2, x3, x4, . . . , x2m−1, x2m

ft, χnx0, χnx1, x2, ϕnx3, x4, . . . , τn,mx2m−1, x2m

2.2

fort, x0, x1, x2, x3, x4, . . . , x2m−1, x2m∈J× Dn. Now, define an auxiliary functionfnby means

of the following recurrence formulas:

fn,0t, x0, x1, . . . , x2m fn∗t, x0, x1, . . . , x2m fort, x0, x1, . . . , x2m∈J× Dn,

fn,it, x0, x1, . . . , x2m

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

fn,i−1t, x0, x1, . . . , x2m if|x2i| ≥

1 n, n

2

fn,i−1

t, x0, . . . , x2i−1,_n1, x2i1, . . . , x2m

x2i1_n

−fn,i−1

t, x0, . . . , x2i−1,−1_n, x2i1, . . . , x2m

x2i− 1_n

if|x2i|< 1 n,

2.3

for 1≤i≤m, and

fnt, x0, x1, . . . , x2m fn,mt, x0, x1, . . . , x2m fort, x0, x1, . . . , x2m∈J×Ê

2m1_{. 2.4}

Then, under conditionH1,fn∈CarJ×Ê

2m1_and

a≤fnt, x0, x1, . . . , x2m for a.e. t∈J and allx0, x1, . . . , x2m∈Ê

2m1_{. 2.5}

ConditionH2gives

fnt, x0, x1, . . . , x2m≤h ⎛

⎝_t,2m12m j0

_xj⎞⎠2m j0

ωjxjωj1,

for a.e. t∈J and allx0, x1, . . . , x2m∈Ê

2m1

0 ,

2.6

fnt, x0, x1, . . . , x2m≤h ⎛

⎝_t,2m12m j0

_xj⎞⎠2m j0

ωj

1 n

,

for a.e. t∈J and allx0, x1, . . . , x2m∈Ê

2m1_.

2.7

We investigate the regular differential equation

−1mx2m1_{t f}

n

t, xt, . . . , x2m_{t. 2.8}

3. Auxiliary Regular Problems

Letj∈Æ and denote byGjt, sthe Green function of the problem

x2j_{t 0, x}2i_{0 x}2i_{T 0, 0≤i≤j−1. 3.1}

Then,

G1t, s

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

s

Tt−T for 0≤s≤t≤T, t

Ts−T for 0≤t≤s≤T.

3.2

By2,3,20, the Green functionGjcan be expressed as

Gjt, s _T

0

G1t, τGj−1τ, sdτ, j >1, 3.3

and it is known thatsee, e.g.,3,20

−1jGjt, s>0 fort, s∈0, T×0, T, j≥1. 3.4

Lemma 3.1see10, Lemmas 2.1 and 2.3. Fort, s∈J×Jandj∈Æ, the inequalities

−1jGjt, s≤T

2j−3

6j−1 sT−s, 3.5

−1jGjt, s≥ T

2j−5

30j−1tsT−tT−s 3.6

hold.

Letγ∈L1_{Jand letu∈AC}2m−1_{Jbe a solution of the differential equation}

−1mu2mt γt, 3.7

satisfying the Lidstone boundary conditions

u2i_{0 u}2i_{T 0, 0≤i≤m−1. 3.8}

It follows from the definition of the Green functionGjthat

−1ju2jt −1m−j

_T

0

It is easy to check thatx∈AC2m_{Jis a solution of problem2.8,1.2if and only ifx0 0,} and its derivativexis a solution of a problem involving the functional differential equation

−1mu2mt fn

t,

_t

0

usds, ut, . . . , u2m−1_t

3.10

and the Lidstone boundary conditions3.8. From3.9 forj0, we see thatu∈AC2m−1J is a solution of problem3.10,3.8exactly if it is a solution of the equation

ut −1m

_T

0

Gmt, sfn

s,

_s

0

uτdτ, us, . . . , u2m−1_{sds, 3.11}

in the setC2m−1_{J. Consequently,xis a solution of problem2.8,1.2if and only if it is a} solution of the equation

xt −1m

_t

0

T

0

Gms, τfn

τ, xτ, . . . , x2m_τdτ

ds, 3.12

in the setC2m_{J. It means thatxis a solution of problem2.8,1.2ifxis a fixed point of the}

operatorH:C2m_{J → C}2m_{Jdefined as}

Hxt −1m

_t

0

_T

0

Gms, τfn

τ, xτ, . . . , x2m_τdτ

ds. 3.13

We prove the existence of a fixed point of H by the following fixed point result of cone

compression type according to Guo-Krasnosel’skiisee, e.g.,18,19.

Lemma 3.2. LetX be a Banach space, and letP ⊂ X be a cone inX. LetΩ1,Ω2be bounded open balls ofX centered at the origin withΩ1 ⊂Ω2. Suppose thatF:P ∩Ω2\Ω1 → P is completely continuous operator such that

Fx ≥ x forx∈P∩∂Ω1, Fx ≤ x forx∈P∩∂Ω2 3.14

holds. Then,Fhas a fixed point inP∩Ω2\Ω1.

We are now in the position to prove that problem2.8,1.2has a solution.

Lemma 3.3. Let (H1) and (H2) hold. Then, problem2.8,1.2has a solution.

Proof. Let the operatorH:C2m_{J → C}2m_{Jbe given in3.13, and let}

P x∈C2m_{J:xt≥0 for t∈J. 3.15}

Then,P is a cone inC2m_{Jand since−1}m_G

mt, s>0 fort, s∈0, T×0, Tby3.4and

operator follows fromfn∈CarJ×Ê

2m1_{, from Lebesgue dominated convergence theorem,} and from the Arzel`a-Ascoli theorem.

Choosex∈Pand putyt Hxtfort∈J. Then,cf.2.5

−1my2m1t fn

t, xt, . . . , x2m_{t≥a >0 for a.e. t∈J. 3.16}

Sincey0 0 andy2i−1_{0 y}2i−1_{T 0 for 1≤i≤m, the equalityy}j_ξ

j 0 holds with

someξj∈Jfor 0≤j ≤2m. We now use the equalityy2mξ2m 0 and have

y2m_t

_t

ξ2m

y2m1_sds

≥a|t−ξ2m| fort∈J. 3.17

Hence,y2m

∞≥aT/2, and so

Hx_C2m > aT

2 . 3.18

Next, we deduce from the relation

y2m_t

_t

ξ2m

fn

s, xs, . . . , x2m_sds

≤ _T

0 fn

s, xs, . . . , x2m_{sds 3.19}

and from2.7that

y2m_t≤T 0

hs,2m1x_C2mdsT

2m

j0 ωj

1 n

fort∈J. 3.20

Therefore,

y2m

∞≤

_T

0

hs,2m1x_C2mdsV, 3.21

whereV T2_jm₀ωj1/n. Sinceyjξj 0 for 0≤j≤2m, we have

yj ∞≤T

2m−j_y2m

∞, 0≤j≤2m. 3.22

The last inequality together with3.21gives

_{yC2m ≤Ky}2m

∞≤K

T

0

hs,2m1x_C2mdsV

, 3.23

x∈P, inequalities3.18and

Hx_C2m ≤K _T

0

hs,2m1x_C2mdsV

3.24

hold. ByH2, there existsC >0 such that

1 v

_T

0

hs,2m1KvdsV

≤1 ∀v≥ C

K, 3.25

and therefore,

K

_T

0

hs,2m1vdsV

≤v ∀v≥C. 3.26

Let

Ω1

x∈C2m_J:x

C2m < aT

2 , Ω2

x∈C2m_J:x

C2m < C . 3.27

Then, it follows from3.18,3.24, and3.26that

Hx_C2m ≥ x_C2m forx∈P∩∂Ω1, Hx_C2m ≤ x_C2m forx∈P∩∂Ω2. 3.28

The conclusion now follows from Lemma3.2forXC2m_JandFH.

The properties of solutions to problem2.8,1.2are collected in the following lemma.

Lemma 3.4. Let (H1) and (H2) be satisfied. Letxnbe a solution of problem2.8,1.2. Then, for all

n∈Æ, the following assertions hold:

i −1jx_n2j1t>0fort∈0, T,0≤j≤m−1, and−1mx_n2m1t≥afor a.e.t∈J,

iix_nis increasing onJ, and for0≤j ≤m−1,−1jx_n2j2is decreasing onJ, and there is a uniqueξ_j,n∈0, Tsuch thatx_n2j2ξ_j,n 0,

iiithere exists a positive constantAsuch that

x2m

n t≥A|t−ξm−1,n|,

x2j2

n t≥At−ξj,n2 if 0≤j≤m−2,

x2j1

n t≥AtT−t if0≤j≤m−1,

xnt≥At2,

3.29

fort∈J,

Proof. Let us choose an arbitraryn∈Æ. By2.5,

−1mx_n2m1t f_nt, x_nt, . . . , x_n2mt≥a for a.e. t∈J, 3.30

and it follows from the definition of the Green functionGjthat the equality

−1jxn2j1t −1m−j _T

0

Gm−jt, sfn

s, xns, . . . , xn2ms

ds 3.31

holds fort ∈ J and 0 ≤ j ≤ m−1. Now, using 1.2,3.4,3.30, and 3.31, we see that assertioniis true. Hence,−1jxn2j2is decreasing onJfor 0≤j ≤m−1 andxnis increasing

on this interval. Due toxn2i−10 xn2i−1T 0 for 1≤i≤m, there exists a uniqueξj,n∈0, T

such thatun2j2ξj,n 0 for 0≤j≤m−1. Consequently, assertioniiholds.

Next, in view of2.5,3.6, and3.31,

x2j1

n t≥ T

2m−j−5_a

30m−j−1 tT−t

_T

0

sT−sds

T2m−j−2a

6·30m−j−1tT−t fort∈J, 0≤j ≤m−1.

3.32

Since

x2j2

n t _t

ξj,n

x2j3

n sds 3.33

and, by13, Lemma 6.2,

_t

ξj,n

sT−sds≥ T

6

t−ξj,n2, 3.34

we have

x2j2

n t≥ T

2m−j−3_a

36·30m−j−2

t−ξj,n2 fort∈J, 0≤j≤m−2. 3.35

Furthermore,

x2m

n t

_t

ξm−1,n

fn

s, xns, . . . , xn2ms

andcf.3.32forj0

xnt _t

0 x

nsds≥ T

2m−2_a

6·30m−1

_t

0

sT−sds

T2m−2a

36·30m−1t

2_3T−2t≥ T2m−1a 36·30m−1t

2 _fort∈J,

3.37

sincex_n>0 on0, Tby assertionii. Let

Aa·min

!

1, A1, A2, T2m−1 36·30m−1

"

, 3.38

where

A1min

!

T2m−j−2

6·30m−j−1 : 0≤j≤m−1

"

,

A2min

!

T2m−j−3

36·30m−j−2 : 0≤j ≤m−2

"

.

3.39

Then estimate3.29follows from relations3.32–3.37.

It remains to prove the boundedness of the sequence{xn}inC2m_{J. We use estimate}

3.29, the properties ofωjgiven inH2, and the inequality

tT−t≥ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

T

2t for 0< t≤

T 2, T

2T−t for T

2 < t < T

3.40

and have

_T

0

ω2mxn2ms

ds≤ _T

0

ω2mA|s−ξm−1,n|ds

_A1

_Aξm−1,n

0

ω2msds

_{AT−ξm−1,n}

0

ω2msds

< _A2

_AT

0

ω2msds, _T

0

ω2j2xn2j2s

ds≤ _T

0 ω2j2

As−ξj,n2

ds

√1

A

√

AT−ξj,n

−√Aξj,n

ω2j2

< √2

A

√

AT

0

ω2j2

s2_{ds for 0≤j≤m−2,}

_T

0

ω2j1xn2j1s

ds≤ _T

0

ω2j1AsT−sds

<

_T/2

0 ω2j1

_ATs 2 ds _T T/2 ω2j1

_ATT−s

2

ds

< _AT4

_AT2_/4

0

ω2j1sds for 0≤j ≤m−1,

_T

0

ω0|xns|ds≤ _T

0

ω0As2_{ds √}1

A

√

AT

0

ω0s2_ds.

3.41

In particular,

_T

0

ω2mxn2ms

ds < _A2

_AT

0

ω2msds,

_T

0

ω2j2xn2j2s

ds < √2

A

√

AT

0

ω2j2

s2_{ds for 0≤j≤m−2,}

_T

0

ω2j1xn2j1s

ds < _AT4

_AT2_/4

0

ω2j1sds for 0≤j≤m−1,

_T

0

ω0|xns|ds≤ √1

A

√

AT

0

ω0s2_ds,

3.42

for alln∈Æ. Now, from the above estimates, from2.6and fromx

2m

n ξm−1,n 0 for some

ξm−1,n∈0, T, which is proved inii, we get

x2m

n t

_t

ξm−1,n

fn

s, xns, . . . , xn2ms ds ≤ _T 0 fn

s, xns, . . . , xn2ms ds ≤ _T 0 h ⎛

⎝_s,2m12m j0

xnjs

⎞

⎠_ds2m j0

_T

0

ωjxnjs

ωj1 ds < _T 0 h ⎛

⎝_s,2m12m j0

xnj_∞

⎞ ⎠_{ds Λ,}

where

Λ _A2

_AT

0

ω2msds√2

A

m−2

j0

√

AT

0

ω2j2

s2_ds

_AT4 m−1

j0

_AT2_/4

0

ω2j1sds 1

√

A

√

AT

0

ω0s2_dsT2m

j0 ωj1.

3.44

Notice thatΛ<∞byH2. Consequently,

x2m

n _∞< _T

0 h

⎛

⎝_s,2m12m j0

xnj_∞

⎞

⎠_{ds Λ forn∈}Æ. 3.45

Sincex_nj_∞ ≤T2m−j_x2m

n ∞for 0≤j ≤2m, which follows from the fact thatxnjvanishes

inJby1.2and assertionii, inequality3.45yields

x2m

n _∞< _T

0

hs,2m1Kxn2m_∞

ds Λ forn∈Æ, 3.46

whereKis fromH2. Due to the condition

lim sup

v→ ∞ 1 v

_T

0

ht, Kvdv <1 3.47

inH2, there exists a positive constantSsuch that for allv≥Sthe inequality

_T

0

ht,2m1Kvdt Λ≤v 3.48

is fulfilled. The last inequality together with estimate3.46givesx_n2m_∞ < Sforn ∈ Æ.

Consequently,x_nj_∞< T2m−j_{Sfor 0≤j≤2n,n∈}

Æ, and assertionivfollows.

The following result gives the important property of {fnt, xnt, . . . , xn2mt} for

applying the Vitali convergent theorem in the proof of Theorem4.1.

Lemma 3.5. Let (H1) and (H2) hold. Letxnbe a solution of problem2.8,1.2. Then, the sequence

fn

t, xnt, . . . , xn2mt

⊂L1_{J 3.49}

is uniformly integrable on J, that is, for each ε > 0, there exists δ > 0 such that ifM ⊂ J and measM< δ, then

Mfn

t, xnt, . . . , xn2mt

Proof. By Lemma3.4iv, there existsS > 0 such that forn∈ Æ, the inequalityxn C2m < S

holds. Now, we conclude from2.5and2.6, from the properties ofhandωjgiven inH2,

and finally from3.29that forj∈Jandn∈Æ, the estimate

a≤fn

t, xnt, . . . , xn2mt

≤ht,2m1S ω0At2

m−1

j0

ω2j1AtT−t

m−2

j0 ω2j2

At−ξj,n2

ω2mA|t−ξm−1,n|

m

j0 ωj1

3.51

is fulfilled, whereAis a positive constant. Since the functionsht,2m1S,ω0At2_{, and} ω2j1AtT −t 0 ≤ j ≤ m−1 belong to the setL1J by assumption H2, in order to prove that{fnt, xnt, . . . , xn2mt}is uniformly integrable onJ, it suffices to show that the

sequences

{ω2mA|t−ξm−1,n|},

ω2j2

At−ξj,n2

, 0≤j≤m−2 3.52

are uniformly integrable onJ. Due to#₀1ω2msds <∞and#₀1ω2js2ds <∞for 1≤j≤m−1

byH2, this fact follows from13, Criterion 11.10withbAandr1,2.

4. The Main Result

The following theorem is the existence result for the singular problem1.1,1.2.

Theorem 4.1. Let (H1) and (H2) hold. Then, problem 1.1, 1.2 has a positive solution x∈AC2m_Jand

xt>0 fort∈0, T, −1jx2j1t>0 fort∈0, T, 0≤j≤m−1. 4.1

Proof. Lemma 3.3guarantees that problem 2.8, 1.2 has a solution xn. Consider the

se-quence{xn}. By Lemma3.4,{xn}is bounded inC2m_J,

−1jx_n2j1t>0 fort∈0, T, 0≤j≤m−1, 4.2

andxnfulfils estimate3.29, whereAis a positive constant andξj,n ∈ 0, T. Furthermore,

the sequence {fnt, xnt, . . . , xn2mt} is uniformly integrable on J by Lemma 3.5, and

therefore, we deduce from the equality−1mx_n2m1t fnt, xnt, . . . , xn2mtfor a.e.t∈J

that{x_n2m} is equicontinuous onJ. Now, by the Arzel`a-Ascoli theorem and the Bolzano-Weierstrass theorem, we may assume without loss of generality that{xn}is convergent in C2m_Jand{ξ

0≤j≤m−1. Thenx∈C2m_{Jandxsatisfies the boundary conditions1.2. Lettingn → ∞} in3.29and4.2, we getfort∈J

x2m_t≥A|t−ξ

m−1|, x2j2t≥A

t−ξj2 if 0≤j≤m−2

−1jx2j1t≥AtT−t if 0≤j≤m−1, xt≥At2.

4.3

Keeping in mind the definition offn, we conclude from4.3that

lim

n→ ∞fn

t, xnt, . . . , xn2mt

ft, xt, . . . , x2m_{t for a.e. t∈J. 4.4}

Then, by the Vitali theorem,ft, xt, . . . , x2mt∈L1Jand

lim

n→ ∞

_t

0 fn

s, xns, . . . , xn2ms

ds

_t

0

fs, xs, . . . , x2m_{sds fort∈J. 4.5}

Lettingn → ∞in the equality

x2m

n t xn2m0 _t

0 fn

s, xns, . . . , xn2ms

ds, 4.6

we get

x2m_{t x}2m₀ t

0

fs, xs, . . . , x2m_{sds fort∈J. 4.7}

As a result,x∈AC2m_{Jandxis a solution of1.1. Consequently,xis a positive solution of} problem1.1,1.2and inequality4.1follows from4.3.

Example 4.2. Consider problem1.1,1.2with

ft, x0, . . . , x2m pt

2m

k0

akt|xk|αk bkt |xk|βk

4.8

onJ×D, wherep, ak∈L1J,bk∈L∞J that is,bkis essentially bounded and measurable on

Jare nonnegative,pt≥a >0 for a.e.t∈J. Ifα_k∈0,1for 0≤k ≤2mandβ2_k ∈0,1/2, β2m, β2k1∈0,1for 0≤k≤m−1, then, by Theorem4.1, the problem has a positive solution x∈AC2m_{Jsatisfying inequality4.1.}

Acknowledgment

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