BOUNDARY VALUE PROBLEMS WITH MIXED DERIVATIVES

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BOUNDARY VALUE PROBLEMS WITH MIXED DERIVATIVES

MARTIN BOHNER AND HUA LUO

Received 12 September 2005; Accepted 26 October 2005

We study a certain singular second-orderm-point boundary value problem on a time scale and establish the existence of a solution. The proof of our main result is based upon the Leray-Schauder continuation theorem.

Copyright © 2006 M. Bohner and H. Luo. 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.

1. Introduction

Singular nonlinear boundary value problems for diﬀerential equations and diﬀerence equations have been extensively studied in the literature; see [1,4,11, 12,16,18–22]

and the references therein. However, the research for singular boundary value problems on time scales is still in its beginning stages. In [8], the authors investigate the existence of a positive solution for the three-point dynamic boundary value problem

y^ΔΔ+f(x,y)=0, x∈(0, 1], y(0)=0, y(p)=yσ²(1), (1.1) whereTis a time scale, the interval (0, 1]∩Tis abbreviated by (0, 1],p∈(0, 1) is fixed, and f(x,y) is singular aty=0 and possibly atx=0,y= ∞.

Throughout we denote byTa time scale, that is, a nonempty closed subset of the real numbers. In this paper we study the singular second-orderm-point dynamic boundary value problem

x^Δ^∇=ft,x,x^Δ+e(t), t∈(a,b], x^Δ(a)=0, xσ(b)=^m

−2

i=1

a_ixξ_i, (1.2)

whereai∈R,ξi∈(a,σ(b)),i∈ {1, 2,...,m−2}, and f : (a,σ(b))×R²→Rsatisfies the Carath´eodory conditions, that is, for each (x,y)∈R², the function f(·,x,y) is measurable

Hindawi Publishing Corporation Advances in Diﬀerence Equations Volume 2006, Article ID 54989, Pages1–15 DOI10.1155/ADE/2006/54989

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on (a,σ(b)) and for (seeDefinition 2.1)∇-a.e.t∈(a,σ(b)), the function f(t,·,·) is continuous onR². Here we allow f andeto be singular att=σ(b).

In particular, when the nonlinearity f does not contain x^Δ, the problem (1.2) has been investigated for the nonsingular case by some authors, see He [10]; whenT=R, the problem (1.2) has been studied for the nonsingular case by Gupta et al. [9] and Ma [14]

to name a few. Recently, Ma and O’Regan [16] established the existence of a solution to the singular problem (1.2) in the special caseT=Rby making use of the ideas of [4,9].

The motivation for this paper is [16].

The paper is organized as follows. InSection 2, we state some preliminary definitions and results about Lebesgue delta and nabla integrals. We then give all spaces relevant to our work and present the main assumptions ensuring us to obtain the main results.

Section 3is devoted to the study of the properties of Green’s function. We also state and prove some lemmas which are required for discussing the problem (1.2). Then we establish the existence of one solution to the problem (1.2) inSection 4.

The time scale related notations adopted in this paper can be found, if not explained specifically, in almost all literature related to time scales. The readers who are unfamiliar with this area can consult for example [2,3,5–8,10,13,15] for details.

2. The Lebesgue delta and nabla integrals

The integrals mentioned in this paper refer to the Lebesgue integrals on the time scaleT. For the main notions and facts from Lebesgue measures and Lebesgue integrals theory, we refer the reader to [5] and [7, pages 157–163]. Here we give some definitions and lemmas for the convenience of the reader.

LetμΔandμ∇be the LebesgueΔ-measure and the Lebesgue∇-measure onT, respectively. IfA⊂TsatisfiesμΔ(A)=μ∇(A), then we callAmeasurable onTand denote by μ(A) this same value, named the Lebesgue measure ofA.

Definition 2.1. LetPdenote a proposition with respect tot∈T,A⊂T.

(1) If there existsE1⊂AwithμΔ(E1)=0 such thatP holds onA\E1, thenPis said to holdΔ-a.e. onA.

(2) If there existsE2⊂Awithμ∇(E2)=0 such thatPholds onA\E2, thenPis said to hold∇-a.e. onA.

(3) If there existsE1⊂AwithμΔ(E1)=0 andE2⊂Awithμ∇(E2)=0 such thatP holds onA\(E1∪E2), thenPis said to holdΔ∇-a.e. onA(or∇Δ-a.e. onA).

(4) If there existsE⊂Awithμ(E)=0 such thatPholds onA\E, thenP is said to hold a.e. onA.

Clearly, ifPholds a.e. onA⊂T, thenPholdsΔ-a.e. onA,∇-a.e. onA, andΔ∇-a.e.

onAsimultaneously.

Remark 2.2. In the caseT=R, all concepts defined above coincide with that of a.e. on R. In this case we haveμΔ=μ∇=μ=m, wheremis the usual Lebesgue measure onR. In the caseT=Z, for any subsetE⊂Z, we know thatμΔ(E)=μ∇(E) coincides with the number of points of the setE. Soμ(E)=μΔ(E)=μ∇(E)=0 if and only ifE= ∅.

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Combining [7, Theorems 5.82 and 5.84], we have the following example as a further illustration ofDefinition 2.1.

Example 2.3. Let f be a bounded function defined on the finite closed interval [r,s].

Assume that f is regulated. Consider the conditions:

(1) f is RiemannΔ-integrable fromrtos;

(2) f is Riemann∇-integrable fromrtos.

We have

(a) if (1) holds, then f is rd-continuousΔ-a.e. on [r,s);

(b) if (2) holds, then f is ld-continuous∇-a.e. on (r,s];

(c) if both (1) and (2) hold, then f is continuousΔ∇-a.e. on (r,s). If, moreover,r= minTands=maxT, then f is continuousΔ∇-a.e. on [r,s]. Here the continuity of f atrandsis understood as continuous from the right and left, respectively.

Definition 2.4. For a setE⊂Tand a functionf :E→R, the Lebesgue integrals of f over Edenoted by

Ef(t)Δt,

Ef(t)∇t (2.1)

are called the LebesgueΔ-integral of f overE and the Lebesgue∇-integral of f over EonT, respectively. Furthermore, we call f LebesgueΔ-integrable onEand Lebesgue

∇-integrable onEif_Ef(t)Δtand_Ef(t)∇tare finite, respectively.

Letr,s∈T,r≤s. We will use the notations _s

r f(t)Δt=

[r,s)f(t)Δt, ^s

r f(t)∇t=

(r,s]f(t)∇t, (2.2) respectively. Both intervals [r,r) and (s,s] are understood as the empty set.

From [7, page 159], we have that all theorems of the general Lebesgue integration theory hold also for the Lebesgue delta and nabla integrals onT.

Lemma 2.5. If f is LebesgueΔ-integrable on [r,s), then the indefinite integral_r^tf(ᐉ)Δᐉis absolutely continuous on [r,s].

Lemma 2.6. If f is Lebesgue∇-integrable on (r,s], then the indefinite integral_r^tf(ᐉ)∇ᐉis absolutely continuous on [r,s].

Lemma 2.7. If f is LebesgueΔ-integrable on [r,s), thenFdefined by F(t)=

_t

r f(ᐉ)Δᐉ, t∈[r,s) satisfiesF^Δ= f Δ-a.e. on [r,s). (2.3) Lemma 2.8. If f is Lebesgue∇-integrable on (r,s], thenFdefined by

F(t)= _t

r f(ᐉ)∇ᐉ, t∈(r,s] satisfiesF^∇= f ∇-a.e. on (r,s]. (2.4)

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Lemma 2.9. If f is everywhere finite and absolutely continuous on [r,s], then f^Δ exists Δ-a.e. and is LebesgueΔ-integrable on [r,s) and satisfies

f(t)= _t

r f^Δ(ᐉ)Δᐉ+f(r), t∈[r,s]. (2.5) Lemma 2.10. If f is everywhere finite and absolutely continuous on [r,s], then f^∇exists

∇-a.e. and is Lebesgue∇-integrable on (r,s] and satisfies f(t)=

_t

r f^∇(ᐉ)∇ᐉ+f(r), t∈[r,s]. (2.6) Lemma 2.11. Let f be defined on [r,s].

(i) If f is continuous on [r,s), then_r^sf(ρ(t))∇t=_s

r f(t)Δt;

(ii) if f is continuous on (r,s], then_r^sf(σ(t))Δt=_s

r f(t)∇t.

Proof. We only show (i) as the proof of (ii) is similar to the proof of (i). Since f is continuous on [r,s), there existsF: [r,s]→Rsuch thatF^Δ=f holds on [r,s). Then

F^∇(t)=F^Δρ(t)= fρ(t), ∀t∈(r,s] (2.7) by [6, Theorem 8.49]. So

_s

r f(t)Δt= _s

rF^Δ(t)Δt=F(s)−F(r), _s

r fρ(t)∇t= _s

rF^Δρ(t)∇t= _s

rF^∇(t)∇t=F(s)−F(r).

(2.8)

This implies that (i) holds.

Now we define the Banach spacesC[a,σ(b)],C^Δ[a,σ(b)], andL^∇(a,σ(b)] to be the sets of all continuous functions on [a,σ(b)] with the sup norm·∞, allΔ-diﬀerentiable functions with continuousΔ-derivative on [a,σ(b)] with the normx =max{x∞, x^Δ∞}, and all Lebesgue ∇-integrable functions on (a,σ(b)] with the norm x = _σ(b)

a |x(t)|∇t, respectively. Let L^∇loc

a,σ(b)=

x: x|(a,d]∈L^∇(a,d] for every interval (a,d]⊆

a,σ(b). (2.9) We denote byAC[a,σ(b)] the space of all absolutely continuous functions on [a,σ(b)]

and set

ACloc a,σ(b)=

x:x|[a,d]∈AC[a,d] for every interval [a,d]⊆ a,σ(b). (2.10) LetEbe the Banach space

E=

x∈L^∇_loca,σ(b): σ(b)−ρx∈L^∇a,σ(b), (2.11)

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equipped with the norm

x_E= _σ(b)

a σ(b)−ρ(t)x(t)∇t, (2.12)

and letXbe the Banach space X=

u∈C^Δ a,σ(b):u∈C a,σ(b), lim

t→σ(b) σ(b)−tu^Δ(t) exists, (2.13) equipped with the norm

uX=maxu∞, σ(b)−τu^Δ_∞, whereτ(t) :=t,∀t∈T. (2.14) A functionx: [a,σ(b)]→Ris said to be a solution of the problem (1.2) providedxisΔ- diﬀerentiableΔ-a.e. on [a,σ(b)),x^Δis∇-diﬀerentiableΔ∇-a.e. on (a,b],x^Δ^∇: (a,b]→R satisfies the dynamic equation in (1.2), andxfulfills the boundary conditions in (1.2).

We make the following assumptions throughout this paper.

(A0)σ(b)=maxT,ξ_i∈(a,σ(b)) fori∈ {1, 2,...,m−2},a < ξ1< ξ2<···< ξ_m−2<

σ(b),ai∈Rfori∈ {1, 2,...,m−2},m≥3, and

m−2 i=1

ai=1. We defineA:=1 + _m₋2

i=1 ai 1−_m₋2

i=1 ai. (2.15) (A1) There existp,q,r∈Esuch that for (u,v)∈R²we have

|f(t,u,v)| ≤p(t)|u|+ [σ(b)−t]q(t)|v|+r(t),∇-a.e. on (a,σ(b)]. (2.16) (A2)e∈E, that is,e∈L^∇_loc(a,σ(b)) and_a^σ⁽^b⁾[σ(b)−ρ(t)]|e(t)|∇t <∞.

By (A1) and (A2), we allow f(·,u,v) ande(·) to be singular att=σ(b). Whenσ(b)=b, their singularities are clear. Whenσ(b)> b, their singularities are reflected on that both

f(·,u,v) ande(·) may not be defined att=σ(b). If we put f(σ(b),u,v)= ∞, then _σ(b)

a f(t,u,v)∇t= _b

a f(t,u,v)∇t+fσ(b),u,v σ(b)−b= ∞. (2.17) Now (2.16) means that∞ = ∞providedp(σ(b))=q(σ(b))=r(σ(b))= ∞.

3. Green’s function and preliminary lemmas

LetGbe Green’s function of the second-order boundary value problem

−x^Δ^∇=0, on (a,b], x^Δ(a)=0, xσ(b)=0, (3.1)

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which can be explicitly given by G(t,s)=

⎧⎨

⎩

σ(b)−s ifa≤t≤s≤σ(b),

σ(b)−t ifa≤s≤t≤σ(b). (3.2) From this explicit representation, the following lemma is clear.

Lemma 3.1. We have

0≤G(t,s)≤G(s,s), ∀s,t∈ a,σ(b). (3.3) For eachy∈E, we define

u(t)= _σ(b)

a G(t,s)y(s)∇s, fort∈ a,σ(b). (3.4) Since

_σ(b)

a G(t,s)y(s)∇s≤ _σ(b)

a G(s,s)y(s)∇s

= _σ(b)

a σ(b)−sy(s)∇s

≤ _σ(b)

a σ(b)−ρ(s)y(s)∇s

= yE<∞,

(3.5)

we know thatu: [a,σ(b)]→Ris well defined.

Lemma 3.2. Lety∈E. Then _σ(b)

a G(·,s)y(s)∇s∈ACloc a,σ(b). (3.6) Proof. We have

_σ(b)

a G(t,s)y(s)∇s= _t

a σ(b)−ty(s)∇s+ _σ(b)

t σ(b)−sy(s)∇s. (3.7) Sincey∈E, we have y∈L^∇loc(a,σ(b)) and [σ(b)−τ]y∈L^∇(a,σ(b)]. Thus (3.6) follows

fromLemma 2.6.

Lemma 3.3. Lety∈E. Then

y∈L^∇a,σ(b), wherey(t) : = _t

ay(s)∇s,∀t∈T. (3.8) Proof. Set

Φ(t,s)=

⎧⎨

⎩

y(s) ifa≤s≤t≤σ(b),

0 ifa≤t < s≤σ(b). (3.9)

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Since _σ₍_b₎

a

_σ₍_b₎

a

Φ(t,s)∇t∇s= _σ₍_b₎

a

_σ₍_b₎

ρ(s)

y(s)∇t∇s

= _σ(b)

a σ(b)−ρ(s)y(s)∇s= y_E<∞,

(3.10)

we get by the Fubini theorem [2] that _t

a

y(s)∇s= _σ(b)

a

Φ(t,s)∇s∈L^∇a,σ(b). (3.11) Furthermore,

_σ(b) a

_t

ay(s)∇s∇t= _σ(b)

a

_σ(b)

a Φ(t,s)∇s∇t

≤ _σ₍_b₎

a

_σ₍_b₎

a

Φ(t,s)∇s∇t

= _σ(b)

a

_σ(b)

a

Φ(t,s)∇t∇s <∞.

(3.12)

Thus (3.8) holds.

t→limσ(b)

_σ(b)

a G(t,s)y(s)∇s=0. (3.13)

Proof. We have

t→limσ(b)

_σ₍_b₎

a G(t,s)y(s)∇s= lim

t→σ(b)

_t

a σ(b)−ty(s)∇s+ _σ₍_b₎

t σ(b)−sy(s)∇s

. (3.14) Sincey∈E, we have [σ(b)−τ]y∈L^∇(a,σ(b)]. So

t→limσ(b)

_σ(b)

t σ(b)−sy(s)∇s=0. (3.15)

Now we verify that

t→limσ(b)

_t

a σ(b)−ty(s)∇s=0 (3.16)

holds, which completes the proof. We have _σ(b)

t

_r

a y(s)∇s∇r= r

_r

ay(s)∇s

σ(b)

t −

_σ(b)

t ρ(r)y(r)∇r

= σ(b)−t^t

ay(s)∇s+ _σ(b)

t σ(b)−ρ(s)y(s)∇s.

(3.17)

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Sincey∈E, we have [σ(b)−ρ]y∈L^∇(a,σ(b)], so

t→limσ(b)

_σ(b)

t σ(b)−ρ(s)y(s)∇s=0. (3.18) On the other hand, we know fromLemma 3.3thaty∈L^∇(a,σ(b)], so

t→limσ(b)

_σ(b) t

_r

a y(s)∇s∇r= lim

t→σ(b)

_σ(b)

t y(r)∇r=0. (3.19) Therefore the limit in (3.16) exists and is equal to zero, that is, (3.16) holds.

For eachy∈E, we define (T y)(t)=

_σ(b)

a G(t,s)y(s)∇s+ 1 1−_m₋2

i=1 ai m−2

i=1

a_i _σ(b)

a Gξ_i,sy(s)∇s. (3.20) Now since (usingLemma 3.1and the notation introduced in (A0))

(T y)(t)≤ _σ₍_b₎

a G(s,s)y(s)∇s+ 1 1−_m₋₂

i=1 a_i

m−2 i=1

ai_σ₍_b₎

a G(s,s)y(s)∇s

=A _σ(b)

a σ(b)−sy(s)∇s≤AyE<∞,

(3.21) we know from (A0) thatT y: [a,σ(b)]→Ris well defined.

Lemma 3.5. Lety∈E. ThenT y∈Xand

(T y)^Δ^∇+y=0, Δ∇-a.e. on (a,b]. (3.22) Proof. By usingLemma 3.2,T y∈ACloc[a,σ(b)) fory∈E. Together withLemma 2.9, we have thatT yisΔ-diﬀerentiableΔ-a.e. on [a,σ(b)). Then

(T y)^Δ(t)= − _t

ay(s)∇s, (3.23)

so (T y)^Δ∈ACloc[a,σ(b)) sincey∈L^∇loc(a,σ(b)). Next,

(T y)^Δ^∇(t)= −y(t), Δ∇-a.e. on (a,b]. (3.24) (Note thatμ∇({b})=b−ρ(b)>0 whenρ(b)< b. Ifσ(b)=bholds at the same time, that is,bis an lsrd point, then this equality just holds forΔ∇-a.e.t∈(a,b). Further, by means of the definition ofΔ∇-a.e. and the fact ofμΔ({b})=0 forσ(b)=b, we get (3.24).) Next, since

(T y)(t)= _t

a(T y)^Δ(s)Δs+ (T y)(a) (3.25)

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and (T y)^Δ∈L^∇(a,σ(b)] fromLemma 3.3, we haveT y∈AC[a,σ(b)] by means ofLemma 2.6. Now we need to verify that lim_t_→_σ(b)[σ(b)−t](T y)^Δ(t) exists. Indeed, according to

t→limσ(b) σ(b)−t(T y)^Δ(t)= − lim

t→σ(b)

_t

a σ(b)−ty(s)∇s, (3.26) we obtain the existence of the above limit from the proof ofLemma 3.4. Therefore,T y∈

Xwheny∈E. The proof is complete.

(T y)^Δ(a)=0, (T y)(σ(b))=^m

−2

i=1

ai(T y)(ξi). (3.27)

Proof. The fact that (T y)^Δ∈C[a,σ(b)) andy∈L^∇_loc(a,σ(b)) imply that (T y)^Δ(a)=lim

t→a(T y)^Δ(t)= −lim

t→a

_t

ay(s)∇s=0. (3.28) FromT y∈C[a,σ(b)] andLemma 3.4, we have

(T y)σ(b)= lim

t→σ(b)(T y)(t)

= lim

t→σ(b)

_σ(b)

a G(t,s)y(s)∇s+ 1 1−_m₋2

i=1 a_i

m−2 i=1

ai

_σ(b)

a Gξi,sy(s)∇s

= 1 1−_m₋2

i=1 ai m−2

i=1

a_i _σ(b)

a Gξ_i,sy(s)∇s.

(3.29) By (3.20), we have

m−2 i=1

ai(T y)ξi

=^m

−2

i=1

ai

_σ₍_b₎

a Gξi,sy(s)∇s+ 1 1−_m₋2

i=1 ai m−2

i=1

ai

_σ(b)

a Gξi,sy(s)∇s

= 1 1−_m₋₂

i=1 ai m−2

i=1

ai

_σ₍_b₎

a Gξi,sy(s)∇s=(T y)σ(b).

(3.30)

This completes the proof.

Forx∈X, we define a nonlinear operatorNby

(Nx)(t)= −ft,x(t),x^Δ(t)−e(t), fort∈

a,σ(b). (3.31)

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From (A1) and (A2), we conclude thatN:X→Eis well defined. In fact, ford < σ(b), _d

a

(Nx)(t)∇t≤ _d

a

ft,x(t),x^Δ(t)∇t+ _d

a

e(t)∇t

≤ _d

a p(t)x(t)∇t+

_d

a[σ(b)−t]q(t)|x^Δ(t)|∇t+

_d

ar(t)∇t+

_d

a

e(t)∇t

≤ xX

_d

a p(t)∇t+ _d

a q(t)∇t

+ _d

a r(t)∇t+ _d

a

e(t)∇t <∞. (3.32) SoNx∈L^∇_loc(a,σ(b)). Moreover,

_σ(b)

a σ(b)−ρ(t)(Nx)(t)∇t

≤ _σ(b)

a σ(b)−ρ(t)ft,x(t),x^Δ(t)+e(t)∇t

≤ _σ(b)

a σ(b)−ρ(t)p(t)x(t)∇t +

_σ(b)

a σ(b)−ρ(t) σ(b)−tq(t)x^Δ(t)∇t +

_σ(b)

a σ(b)−ρ(t)r(t)∇t+ _σ(b)

a σ(b)−ρ(t)e(t)∇t

≤ p_Ex∞+q_E σ(b)−τx^Δ_∞+r_E+e_E

≤ xX

pE+qE

+rE+eE<∞.

(3.33)

Thus [σ(b)−ρ](Nx)∈L^∇(a,σ(b)].

Lemma 3.7. TN:X→Xis completely continuous.

Proof. By the definitions ofTandN, we get that (TN)x(t)= −

_σ(b)

a G(t,s)fs,x(s),x^Δ(s)∇s− _σ(b)

a G(t,s)e(s)∇s

− 1 1−_m₋2

i=1 a_i

m−2 i=1

a_i _σ(b)

a Gξ_i,sfs,x(s),x^Δ(s)∇s

− 1 1−_m₋2

i=1 ai m−2

i=1

a_i _σ(b)

a Gξ_i,se(s)∇s.

(3.34)

For eachx1,x2∈X, (TN)x1−(TN)x2

X

=max(TN)x1−(TN)x2

∞, σ(b)−τ (TN)x1)^Δ− (TN)x2

_Δ

∞

. (3.35)

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Since f satisfies Carath´eodory’s conditions, together with (use (A0) andLemma 3.1) (TN)x1−(TN)x2

≤ _σ(b)

a G(t,s)fs,x2(s),x^Δ2(s)−fs,x1(s),x1^Δ(s)∇s

+ 1

1−_m₋2 i=1 ai^m⁻

2 i=1

a_i _σ(b)

a Gξ_i,sfs,x2(s),x^Δ2(s)−fs,x1(s),x^Δ1(s)∇s

≤A _σ(b)

a G(s,s)fs,x2(s),x^Δ2(s)−fs,x1(s),x^Δ1(s)∇s, σ(b)−t (TN)x1

_Δ (t)−

(TN)x2

_Δ (t)

≤ σ(b)−a^t

a

fs,x1(s),x^Δ1(s)−fs,x2(s),x^Δ2(s)∇s,

(3.36) it is easy to show thatTN:X→Xis continuous.

Now letB⊂X be a bounded set. We need to show that (TN)(B)⊂Xis a relatively compact subset. Let{x_n}^∞_n₌1⊂Band denote

wn(t)=

(TN)xn

(t), zn(t)= σ(b)−t(TN)xnΔ

(t). (3.37)

We only need to show that there exists a subsequence with

wn−→w^∗, inC a,σ(b), (3.38)

zn−→z^∗, inC a,σ(b), (3.39)

wherez^∗(t)=[σ(b)−t](w^∗)^Δ(t) fort∈[a,σ(b)). We prove (3.38) and (3.39) by the following three steps.

Step 1. We prove that (TN)(B) is bounded. LetM=sup{xX:x∈B}. ThenM is a finite number. For eacht∈(a,σ(b)), we have

Nx_n(t)≤ft,x_n(t),x^Δ_n(t)+e(t)

≤p(t)xn

∞+ σ(b)−tq(t)x^Δ_n(t)+r(t) +e(t)

≤p(t)M+q(t)M+r(t) +e(t):=χ(t).

(3.40)

Clearly, (A1) and (A2) imply thatχ∈E. Thus Nxn

E≤ _σ₍_b₎

a σ(b)−ρ(t)χ(t)∇t:=K <∞. (3.41) It follows thatT((Nxn)(t)) is bounded. So (TN)(B) is bounded.

(12)

Step 2. We prove that{wn}^∞_n₌1is equicontinuous on [a,σ(b)]. For everyt1,t2∈[a,σ(b)]

witht1< t2, according toLemma 2.11(i), (3.23), and (3.40), we get w_nt1

−w_nt2= _t₁

t2

(TN)x_n^Δ(t)Δt≤ _t₂

t1

(TN)x_n^Δ(t)Δt

= _t₂

t1

_t

aNx_n(s)∇sΔt≤ _t₂

t1

_t

a

Nx_n(s)∇sΔt

≤ _t₂

t1

_t

aχ(s)∇sΔt= _t₂

t1

_ρ(t)

a χ(s)∇s∇t.

(3.42)

By the proof of Lemma 3.3, χ◦ρ∈L^∇(a,σ(b)]. Thus (3.42) shows that {w_n}^∞_n₌1 is equicontinuous on [a,σ(b)]. Therefore by the Arzel`a-Ascoli theorem, after taking a subsequence if necessary, (3.38) holds.

Step 3. We prove that{z_n}^∞_n₌1is equicontinuous on [a,σ(b)]. For everyt1,t2∈[a,σ(b)]

witht1< t2, we have byLemma 3.5and (3.40) that zn

t1

−zn t2≤

_t₂

t1

z_n^∇(t)∇t

= _t₂

t1

− (TN)xn

_Δ

(t) + σ(b)−ρ(t)(TN)xn

_Δ_∇ (t)∇t

≤ _t₂

t1

_t

a

Nxn(s)∇s∇t+ _t₂

t1

σ(b)−ρ(t)Nxn(t)∇t

≤ _t₂

t1

_t

aχ(s)∇s∇t+ _t₂

t1

σ(b)−ρ(t)χ(t)∇t

= _t₂

t1

_t

aχ(s)∇s+ σ(b)−ρ(t)χ(t)

∇t.

(3.43)

Sinceχ∈Eand because ofLemma 3.3, we know that

χ+ σ(b)−ρχ∈L^∇a,σ(b). (3.44) Thus (3.43) shows that{zn}^∞_n₌1is equicontinuous on [a,σ(b)]. Therefore by the Arzel`a- Ascoli theorem, after taking a subsequence if necessary, (3.39) holds.

This completes the proof.

4. Main result

Theorem 4.1. Let f : (a,σ(b))×R²→Rsatisfy Carath´eodory’s conditions. Assume that (A0), (A1), and (A2) hold. Then the problem (1.2) has at least one solution inXprovided

ApE+qE<1. (4.1)

Proof. FromLemmas 3.5and3.6, we know thatx∈Xis a solution of (1.2) if and only if

x=TNx. (4.2)