BOUNDARY VALUE PROBLEMS WITH φ-LAPLACIAN AND THEIR APPLICATIONS

BOUNDARY VALUE PROBLEMS WITH φ-LAPLACIAN AND THEIR APPLICATIONS

RAVI P. AGARWAL, DONAL O’REGAN, AND SVATOSLAV STAN ˇEK Received 1 April 2005; Accepted 12 May 2005

The paper presents general existence principles which can be used for a large class of nonlocal boundary value problems of the form (φ(x))= f1(t,x,x) + f2(t,x,x)F1x+ f3(t,x,x)F2x,α(x)=0,β(x)=0, where f_jsatisfy local Carath´eodory conditions on some [0,T]×Ᏸj⊂R², fj are either regular or have singularities in their phase variables (j= 1, 2, 3), Fi:C¹[0,T]→C⁰[0,T] (i=1, 2), and α,β:C¹[0,T]→R are continuous. The proofs are based on the Leray-Schauder degree theory and use regularization and sequen- tial techniques. Applications of general existence principles to singular BVPs are given.

Copyright © 2006 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.

1. Introduction

LetT >0. As usual,C^j[0,T] and AC^j[0,T] (j=0, 1) denote the set of functions having thejth derivative continuous and absolutely continuous on [0,T], respectively.L1[0,T] is the set of Lebesgue integrable functions on [0,T]. In what follows,C⁰[0,T] andL1[0,T]

are equipped with the norms

x =maxx(t): 0≤t≤T, xL= _T

0

x(t)dt, (1.1)

respectively.

Assume thatG⊂R². Car([0,T]×G) stands for the set of functions f : [0,T]×G→ Rsatisfying local Carath´eodory conditions on [0,T]×G, that is: (j) for each (x,y)∈ G, the function f(·,x,y) : [0,T]→R is measurable; (jj) for a.e. t∈[0,T], the func- tion f(t,_·,·) :G_→Ris continuous; (jjj) for each compact setK_⊂G, sup_{|f(t,x,y)_|: (x,y)∈K} ∈L1[0,T]. For any measurableᏹ⊂R,μ(ᏹ) denotes the Lebesgue measure ofᏹ.

Hindawi Publishing Corporation Abstract and Applied Analysis

Volume 2006, Article ID 96826, Pages1–30 DOI10.1155/AAA/2006/96826

We will denote byᏴthe set of operatorsF:C¹[0,T]→C⁰[0,T] which are (a) continuous and

(b) bounded, that is, for anyr >0,

supFx:x∈C¹[0,T],x+x ≤r<∞. (1.2) Finally, letᏭdenote the set of functionalsα:C¹[0,T]→Rwhich are

(a) continuous and

(b) bounded, that is,α(Ω) is bounded (inR) for any boundedΩ⊂C¹[0,T].

We study singular nonlocal boundary value problems (BVPs) of the type φx(t)=f1

t,x(t),x(t)+f2

t,x(t),x(t)F1x(t) +f3

t,x(t),x(t)F2x(t) (1.3)

α(x)=0, β(x)=0, (1.4)

whereφ is an increasing homeomorphism fromRontoR, f_j∈Car([0,T]×Ᏸ^j), the setsᏰ^j=Ᏸ1^j×Ᏸ2^j⊂R² are not necessarily closed, f_j have singularities in their phase variables on the boundary∂Ᏸ^jofᏰ^j(j=1, 2, 3),Fi∈Ᏼ(i=1, 2) andα,β∈Ꮽ.

Letj∈ {1, 2, 3}. We say that fjhas a singularity on∂Ᏸ^jin its phase variablexi(i=1, 2) if there existsai,j∈∂Ᏸi^jsuch that

lim sup

xi→ai,j,xi∈Ᏸi^j

f_jt,x1,x2= ∞ (1.5)

for a.e.t∈[0,T] and allx3₋i∈D₃^j_−i.

A function x∈C¹[0,T] is said to be a solution of the BVP (1.3), (1.4) if φ(x)∈ AC[0,T], x satisfies the boundary conditions (1.4) and (1.3) holds for almost allt∈ [0,T].

Nonlocal BVPs for second-order differential equations with nonlinear left-hand sides and having singularities in their phase variables were studied in [1,2,5,10–12]. In [1]

the differential equation (g(x))= f(t,x, (g(x))) is discussed together with the non- local boundary conditions x(0)=x(T), min{x(t) : 0≤t≤T} =0. In [2] the authors present conditions guaranteeing that the BVP (φ(x))=μ f(t,x,x), x(0)=0=x(T), max{x(t) : 0≤t≤T} =Ahas for anyA >0 a positive solution with a positive value of the parameterμ. The existence of a solution of (φ(x))= f(t,x,x) satisfyingx(0)=x(T), max{x(t) : 0≤t≤T} =Ais considered in [11], satisfyingx(0)=x(T)= −γmin{x(t) : 0≤t≤T}(γ∈(0,∞)) in [10] and satisfying min{x(t) : 0≤t≤T} =0,χ(x)=0 where χis a continuous functional in [12]. Existence results for functional-differential equations with nonlinear functional left-hand sides and nonlocal functional boundary conditions are also presented in [5].

The aim of this paper is to present general existence principles for solving regular and singular nonlocal BVPs for second-order functional-differential equations with φ- Laplacian and to give applications of these general principles. The general existence prin- ciple for regular nonlocal BVPs can be used either for solving regular BVPs or in the case

of singular BVPs for solving a sequence of auxiliary regular BVPs obtaining by regulariza- tion and sequential techniques. We note that our general existence principle for singular nonlocal BVPs is related to that given in [9] for singular BVPs

x⁽ⁿ⁾(t)=ht,x(t),. . .,x⁽ⁿ⁻¹⁾(t), x∈᏿, (1.6) wherehhas singularities in all its phase variables and᏿is a closed subset ofCⁿ⁻¹[0,T].

To obtain a solution of the BVP (1.3), (1.4), we use regularization and sequential tech- niques. To use these techniques we consider a sequence of regular functional-differential equations

φx(t)=f_1,nt,x(t),x(t)+ f_2,nt,x(t),x(t)F1x(t)

+f_3,nt,x(t),x(t)F2x(t), (1.7) where fj,n∈Car([0,T]×R²),n∈N,j=1, 2, 3. Ifxnis a solution of the BVP (1.7), (1.4), then a solution of the BVP (1.3), (1.4) is obtained as the limit (inC¹[0,T]) of a subse- quence of{x_n}. In limiting processes one usually uses the Lebesgue dominated conver- gence theorem. Note that in our case with general nonlocal boundary conditions (1.4), we often cannot find a Lebesgue integrable majorant function for the auxiliary sequence of regular functions connected to the BVP (1.3), (1.4). In such a case our limiting processes are based on Vitali’s convergence theorem, where the assumption about the existence of a Lebesgue integrable majorant function is replaced by a more general assumption about the uniform integrability.

A collectionᐂ⊂L1[0,T] is called uniformly integrable (UI) on [0,T] if to givenε >0 there existsδ >0 such that ifρ∈ᐂandᏹ⊂[0,T],μ(ᏹ)< δ, then

ᏹ

ρ(t)dt < ε. (1.8)

Theorem 1.1 (Vitali’s convergence theorem, [3]). Let{ρn}be a sequence inL1[0,T] which convergent toρfor a.e.t∈[0,T]. Then the following statements are equivalent:

(a)ρ∈L1[0,T] and limn→∞ρn−ρL=0, (b) the sequence{ρn}is UI on [0,T].

Remark 1.2. Assumption (b) inTheorem 1.1is equivalent to the following condition: to givenε >0 there existsδ >0 such that for any at most countable set{aj,bj}j∈Jof mutually disjoint intervals (aj,bj)⊂(0,T),_j∈J(bj−aj)< δ, we have

j∈J

_bj

aj

ρn(t)dt < ε, n∈N. (1.9)

The rest of the paper is organized as follows. InSection 2, we present a general ex- istence principle for regular nonlocal BVP and general existence principles for singular nonlocal BVP. Applications of both principles are given in Sections3and4.Section 3dis- cusses singular nonlocal BVPs where nonlinearities in the singular differential equations

are positive.Section 4 is devoted to the study of positive solutions to singular Dirich- let BVPs for functional-differential equations with right-hand sides changing their sign.

Results are demonstrated with examples throughout.

2. General existence principles

We first denote byᏸthe set of functionalsF:C¹[0,T]→L1[0,T] which are (a) continuous and

(b) for eachr >0,

sup(Fx)(t):x∈C¹[0,T],x+x ≤r∈L1[0,T]. (2.1) Notice that for each f ∈Car([0,T]×R²), the operator F:C¹[0,T]→L1[0,T] with (Fx)(t)=f(t,x(t),x(t)), belongs to the setᏸ.

Consider the regular functional-differential equation

φx(t)=(Fx)(t), (2.2)

whereF∈ᏸ. We give a general existence principle for the BVP (2.2), (1.4).

Theorem 2.1 (general existence principle for regular nonlocal BVP). Let φ be an increasing homeomorphism fromRontoR,F∈ᏸandα,β∈Ꮽ. Suppose there exist positive constantsS0andS1such that

x< S0, x< S1 (2.3)

for all solutionsxto the BVP

φx(t)=λ(Fx)(t)

α(x)=0, β(x)=0 (2.4)

and eachλ∈[0, 1]. Also assume there exists positive constantsΛ0andΛ1such that

|A|<Λ0, |B|<Λ1 (2.5) for all solutions (A,B)∈R²of the system

α(A+Bt)−μα(−A−Bt)=0

β(A+Bt)−μβ(−A−Bt)=0 (2.6)

and eachμ∈[0, 1]. Then the BVP (2.2), (1.4) has a solution.

Proof. Set Ω=

x:x∈C¹[0,T],x<maxS0,Λ0+Λ1T,x<maxS1,Λ1

. (2.7)

ThenΩis an open, bounded and symmetric with respect to 0∈C¹[0,T] subset of the Banach spaceC¹[0,T]. Define the operatorᏼ: [0, 1]×Ω→C¹[0,T] by the formula

ᏼ(λ,x)(t)=x(0) +α(x) + _t

0φ⁻¹

φx(0) +β(x)+λ _s

0(Fx)(v)dv

ds. (2.8)

A standard argument shows thatᏼis a continuous operator. We claim thatᏼ([0, 1]×Ω) is compact inC¹[0,T]. Indeed, sinceΩis bounded inC¹[0,T],

α(x)≤r, β(x)≤r, (Fx)(t)≤ρ(t) (2.9)

for a.e.t∈[0,T] andx∈Ω, whereris a positive constant andρ∈L1[0,T]. Then ᏼ(λ,x)(t)≤maxS0,Λ0+Λ1T+r+Tφ⁻¹φmaxS1,Λ1

+r+ρL

, ᏼ(λ,x)(t)≤φ⁻¹φmaxS1,Λ1

+r+ρL

, φᏼ(λ,x)t2

−φᏼ(λ,x)t1≤ _t2

t1

ρ(t)dt

(2.10)

fort,t1,t2∈[0,T] and (λ,x)∈[0, 1]×Ω. Henceᏼ([0, 1]×Ω) is bounded inC¹[0,T] and {φ[ᏼ(λ,x)(t)]}is equicontinuous on [0,T]. From

ᏼ(λ,x)t2

−ᏼ(λ,x)t1=φ⁻¹φᏼ(λ,x)t2

−φ⁻¹φᏼ(λ,x)t1 (2.11) and φ⁻¹ being an increasing homeomorphism from RontoR, we deduce that{ᏼ(λ, x)(t)}is also equicontinuous on [0,T]. Now the Arzel`a-Ascoli theorem shows thatᏼ([0, 1]×Ω) is compact inC¹[0,T]. Thusᏼis a compact operator.

Suppose thatx0is a fixed point of the operatorᏼ(1,·). Then x0(t)=x0(0) +αx0

+ _t

0φ⁻¹

φx₀(0) +βx0

+ _s

0

Fx0

(v)dv

ds. (2.12)

Henceα(x0)=0,β(x0)=0 andx0is a solution of (2.2). Thereforex0is a solution of the BVP (2.2), (1.4) and to prove our theorem, it suffices to show that

DᏵ−ᏼ(1,·),Ω, 0 =0, (2.13)

where “D” stands for the Leray-Schauder degree and Ᏽ is the identity operator on C¹[0,T]. For this purpose let the compact operator᏷: [0, 1]×Ω→C¹[0,T] be given by

᏷(μ,x)(t)=x(0) +α(x)−μα(−x) +x(0) +β(x)−μβ(−x)t. (2.14)

Then᏷(1,·) is odd (i.e.,᏷(1,−x)= −᏷(1,x) forx∈Ω) and

᏷(0,·)=ᏼ(0,·). (2.15)

If᏷(μ0,x0)=x0for someμ0∈[0, 1] andx0∈Ω, then x0(t)=x0(0) +αx0

−μ0α−x0

+x0(0) +βx0

−μ0β−x0

t, t∈[0,T].

(2.16) Therefore x0(t)=A0+B0t where A0=x0(0) +α(x0)−μ0α(−x0), B0=x₀(0) +β(x0)− μ0β(−x0), and alsoα(x0)−μ0α(−x0)=0,β(x0)−μ0β(−x0)=0. Hence

αA0+B0t−μ0α−A0−B0t=0,

βA0+B0t−μ0β−A0−B0t=0. (2.17) Thus

A0<Λ0, B0<Λ1 (2.18) since (A0,B0) is a solution of (2.6) withμ=μ0. Thenx0<Λ0+Λ1T,x0<Λ1, which givesx0 ∈∂Ω. Now, by the Borsuk antipodal theorem and a homotopy property (see, e.g., [6]),

DᏵ−᏷(0,·),Ω, 0=DᏵ−᏷(1,·),Ω, 0 =0. (2.19) Finally assume thatᏼ(λ_∗,x_∗)=x_∗for someλ_∗∈[0, 1] andx_∗∈Ω. Thenx_∗is a solution of the BVP (2.4) withλ=λ_∗and, by our assumptions,x_∗< S0andx_∗< S1. Hence x_∗ ∈∂Ωand a homotopy property yields

DᏵ−ᏼ(0,·),Ω, 0=DᏵ−ᏼ(1,·),Ω, 0. (2.20) From the last equality, (2.15) and (2.19) it follows that (2.13) is true. Therefore the BVP

(2.2), (1.4) has a solution.

Remark 2.2. Let the functionalsα,β∈Ꮽbe linear. Then the system (2.6) has the form Aα(1) +Bα(t)=0,

Aβ(1) +Bβ(t)=0, (2.21)

and all of its solutions (A,B) are bounded if and only ifα(1)β(t)−α(t)β(1) =0 (and then (A,B)=(0, 0)).

Special cases ofTheorem 2.1are the general existence principles presented in [8] for the differential equation (φ(x))=q(t)f(t,x,x) with the Dirichlet and mixed boundary conditions.

Now consider the singular functional-differential equation (1.3) whereφis an increas- ing homeomorphism fromRontoR, fj∈Car([0,T]×Ᏸ^j), the setsᏰ^j=Ᏸ1^j×Ᏸ2^j⊂R²

are not necessarily closed, fj have singularities in their phase variables on the bound- ary∂Ᏸ^j of Ᏸ^j (j=1, 2, 3) andFi∈Ᏼ(i=1, 2). Also consider the sequence of regular differential equations (1.7) where f_j,n∈Car([0,T]×R²),n∈N,j=1, 2, 3.

Theorem 2.3 (general existence principle for singular nonlocal BVP). Letα,β∈Ꮽ. Suppose there exist a bounded setΩ⊂C¹[0,T] such that

(i) for eachn∈N, the regular BVP (1.7), (1.4) has a solutionxn∈Ω,

(ii) the sequences{f_j,n(t,x_n(t),x_n(t))}are uniformly integrable on [0,T] forj=1, 2, 3.

Then we have

(a) there existx∈Ωand a subsequence{xkn}of{xn}such that limn→∞xkn=xinC¹([0, T]),

(b)xis a solution of the BVP (1.3), (1.4) if

nlim→∞fj,kn

t,xkn(t),x_kn(t)= fj

t,x(t),x(t), j=1, 2, 3, (2.22) for almost allt∈[0,T].

Proof. SinceΩis bounded inC¹[0,T] and{xn} ⊂Ω, there are positive constantsKandr such that

F1x_n(t)≤K, F2x_n(t)≤K, x_n≤r, x_n≤r, n∈N. (2.23) Now (ii) guarantees that for eachε >0 there existsδ >0 such that for eacht1,t2∈[0,T],

|t2−t1|< δ, andn∈N, φx_nt2

−φx_nt1≤ _t2

t1

f1,n

t,xn(t),x_n(t)dt

+K

3

j=2

_t2

t1

fj,n

t,xn(t),x_n(t)dt< ε.

(2.24)

Therefore{φ(x_n(t))}is equicontinuous on [0,T], and then using (2.23) and the fact that φis continuous and increasing onR, we see that{x_n(t)}is equicontinuous on [0,T] as well. The Arzel`a-Ascoli theorem guarantees the existence of a subsequence{xkn}of{xn} converging inC¹[0,T] tox∈Ω.

Suppose that lim_n→∞f_j,kn(t,x_kn(t),x_kn(t))= f_j(t,x(t),x(t)) for a.e.t∈[0,T] and for j=1, 2, 3. By (ii),{fj,kn(t,xkn(t),x_kn(t))}are uniformly integrable on [0,T]. Therefore, by Vitali’s convergence theorem, fj(t,x(t),x(t))∈L1[0,T] forj=1, 2, 3 and also letting n→ ∞in

φx_kn(t)=φx_kn(0)+ _t

0 f_1,kns,x_kn(s),x_kn(s)ds +

_t

0 f_2,kns,x_kn(s),x_kn(s)F1x_kn(s)ds +

_t

0 f_3,kns,x_kn(s),x_kn(s)F2x_kn(s)ds

(2.25)

fort∈[0,T] andn∈N, we have φx(t)=φx(0)+

_t

0 f1

s,x(s),x(s)ds

+ _t

0 f2

s,x(s),x(s)F1x(s)ds

+ _t

0 f3

s,x(s),x(s)F2x(s)ds, t_∈[0,T].

(2.26)

Consequently,φ(x)∈AC[0,T] andxis a solution of (1.3). In addition, since limn→∞xkn

=xinC¹[0,T] andαandβare continuous inC¹[0,T], it follows thatα(x)=0,β(x)=0.

Hencexis a solution of the BVP (1.3), (1.4).

Remark 2.4. Let f_jin (1.3) have singularities only at the value 0 of their phase variables and fj,nin (1.7) satisfy

f_j,n(t,x,y)= f_j(t,x,y), j=1, 2, 3 (2.27) for a.e.t∈[0,T] and all (x,y)∈Ᏸ^j,n∈N,|x| ≥1/nand|y| ≥1/n. Then the condition

nlim→∞fj,kn

t,xkn(t),x_kn(t)=fj

t,x(t),x(t), j=1, 2, 3 (2.28) for a.e.t∈[0,T] is satisfied ifxandxhave a finite number of zeros.

Remark 2.5. The absolute continuity of the Lebesgue integral yields that condition (ii) in Theorem 2.3is satisfied if there exists a functionϕ∈L1[0,T] such that

fj,n

t,xn(t),x_n(t)≤ϕ(t) (2.29) for a.e.t∈[0,T] and eachn∈N,j∈ {1, 2, 3}.

We now discuss the special case of (1.3) with f1=f and f2=f3=0, that is the differ- ential equation

φx(t)= ft,x(t),x(t), (2.30) wheref∈Car([0,T]×(0,∞)×(R\ {0})). Together with (2.30), we consider the sequence of regular differential equations

φx(t)=fn

t,x(t),x(t), (2.31)

where fn∈Car([0,T]×R²) and

f_n(t,x,y)=f(t,x,y) for a.e.t∈[0,T] and allx≥1/n,|y| ≥1/n. (2.32) For the solvability of the singular BVP (2.30), (1.4) the following result holds.

Theorem 2.6. Suppose that

(j) there existsγ∈L1[0,T] such that fn(t,x,y)≥γ(t)>0 for a.e.t∈[0,T] and each (x,y)∈R²,n∈N,

(jj) there exists a bounded setΩ⊂C¹[0,T] such that for eachn∈N, the BVP (2.31), (1.4) has a solutionxn∈Ω,xn(ξn)=x(ξn)=0 whereξn∈[0,T] and{x_n(t)}is equicontinuous on [0,T].

Then

(a) there existx∈Ωand a subsequence{xkn}of{xn}such that limn→∞xkn=xinC¹[0, T],x(ξ)=x(ξ)=0 with aξ∈[0,T] andx >0 on [0,T]\ {ξ}.

(b)xis a solution of the BVP (2.30), (1.4) if to givenε >0, there existδε∈L1[0,T] and nε∈Nsuch that

f_knt,x_kn(t),x_k

n(t)≤δ_ε(t) (2.33)

for a.e.t∈[0,T]\[ξ−ε,ξ+ε] and eachn≥nε.

Proof. Since (φ(x_n(t)))=fn(t,xn(t),x_n(t))≥γ(t)>0 for a.e.t∈[0,T] and eachn∈N, x_nis increasing on [0,T] andx_n(ξ_n)=x_n(ξ_n)=0 implies

x_n(t)≤ −φ _ξn

t γ(s)ds

, x_n(t)≥ _ξn

t φ _ξn

s γ(v)dv

ds, t∈ 0,ξ_n, x_n(t)≥φ

_t

ξn

γ(s)ds

, x_n(t)≥ _t

ξn

φ _s

ξn

γ(v)dv

ds, t∈ ξ_n,T.

(2.34)

We know, by the assumptions, that{xn}is bounded inC¹[0,T] and{x_n(t)}is equicon- tinuous on [0,T]. Hence there exist a convergent subsequence{x_kn}converging tox in C¹[0,T] and a convergent subsequence{ξ_kn}converging toξ inR. Thenx∈C¹[0,T], ξ∈[0,T], and taking the limit asn→ ∞in (2.34) withkninstead ofn, we get

x(t)≤ −φ ξ

t γ(s)ds

, x(t)≥ _ξ

t φ ξ

s γ(v)dv

ds, t∈[0,ξ], x(t)≥φ

t ξγ(s)ds

, x(t)≥ _t

ξφ s

ξγ(v)dv

ds, t∈[ξ,T].

(2.35)

Thereforex(ξ)=x(ξ)=0,x >0,|x|>0 on [0,T]\ {ξ}, and consequently (see (2.32))

nlim→∞f_knt,x_kn(t),x_kn(t)=ft,x(t),x(t) (2.36) for almost allt∈[0,T].

From{φ(x_kn(0))}and{φ(x_kn(T))}being bounded, φx_kn(T)−φx_kn(0)=

_T

0 f_knt,x_kn(t),x_kn(t)dt, n∈N, (2.37)

and fkn(t,xkn(t),x_kn(t))≥γ(t)>0, it follows that f(t,x(t),x(t))∈L1[0,T], by Fatou’s theorem. If the condition in (b) is satisfied, then lettingn→ ∞in

φx_kn(t)=φx_kn(0)+ _t

0fkn

s,xkn(s),x_kn(s)ds,

φx_kn(t)=φx_kn(T)− _t

T f_kns,x_kn(s),x_kn(s)ds,

(2.38)

we obtain

φx_n(t)=φx(0)+ _t

0 fs,x(s),x(s)ds, t∈[0,ξ), φx(t)=φx(T)−

_t

T fs,x(s),x(s)ds, t∈(ξ,T]

(2.39)

by the Lebesgue dominated convergence theorem. Now using f(t,x(t),x(t))∈L1[0,T], we haveφ(x)∈AC[0,T] andxis a solution of (2.30). Also fromα(x_kn)=0,β(x_kn)=0 and the continuity ofα and βon C¹[0,T] it follows α(x)=0,β(x)=0. Hencex is a

solution of the BVP (2.30), (1.4).

3. Application to singular nonlocal BVPs with positive nonlinearities

3.1. Introduction. Denote byᏮthe set of all functionalsα:C¹[0,T]→Rwhich are (i) continuous,

(ii) sign preserving with respect to the derivative of functions in the following sense:

x∈C¹[0,T],εx>0 on [0,T] whereε∈ {−1, 1} ⇒εα(x)>0, (iii) bounded that isΩ⊂C¹[0,T] bounded⇒α(Ω) bounded.

Of course,Ꮾ⊂ᏭwhereᏭis defined inSection 1.

Remark 3.1. Ifα∈Ꮾthenα(A)=0 for each A∈R. Indeed, let α∈Ꮾ,A∈Rand set xn(t)=A+t/n,yn(t)=A−t/nfort∈[0,T] andn∈N. Thenα(yn)<0< α(xn) by (ii), and{xn},{yn}are convergent inC¹[0,T] toA. Thus 0≥limn→∞α(yn)=α(A) and 0≤ limn→∞α(x_n)=α(A) yieldα(A)=0.

Example 3.2. Letγ:C¹[0,T]→[0,T] be a continuous functional,q,r∈C⁰(R),qbe pos- itive,ur(u)>0 foru =0 and 0≤t1< t2<···< tn≤T. Then the functionals

α1(x)=xγ(x), α2(x)= _T

0 qx(t)rx(t)dt, α3(x)=

n

j=1

qxtj

rxtj (3.1) and all their linear combination with positive coefficients belong to the setᏮ.

We discuss the singular nonlocal boundary value problems

φx(t)= ft,x(t),x(t), (3.2) minx(t) : 0≤t≤T=0, α(x)=0, (3.3) whereφ∈C⁰(R),α∈Ꮾ, f ∈Car([0,T]×(0,∞)×(R\ {0})) is positive and f may be

singular at the value 0 of both its phase variables in the following sense limx→0⁺f(t,x,y)=

∞for a.e.t∈[0,T] and eachy∈R\ {0}, limy→0f(t,x,y)= ∞for a.e.t∈[0,T] and each x∈(0,∞). Equation (3.2) is the special case of (1.3) withF1=F2=0.

A function x∈C¹[0,T] is said to be a solution of the BVP (3.2), (3.3) if φ(x)∈ AC[0,T],xsatisfies (3.3) and for a.e.t∈[0,T] fulfills (3.2).

We give conditions on the functionsφand f in (3.2) which guarantee the solvability of the BVP (3.2), (3.3) for eachα∈Ꮾin (3.3). Existence results are based on regulariza- tion and sequential techniques and use our general existence principles (Theorems2.1 and2.6). Notice that any solution of the BVP (3.2), (3.3) and its derivative “go through”

singularities of f somewhere inside of [0,T].

Throughout this section we will use the following assumptions.

(H1)φ∈C⁰(R) is odd and increasing onR, limx→∞φ(x)= ∞;

(H2) f ∈Car([0,T]×(0,∞)×(R\ {0})) and there existsδ∈L1[0,T], such that for a.e.t∈[0,T] and each (x,y)∈(0,∞)×(R\ {0}),

0< δ(t)≤f(t,x,y); (3.4)

(H3) For a.e.t∈[0,T] and each (x,y)∈(0,∞)×(R\ {0}), f(t,x,y)≤

h1(x) +h2(x)ω1φ(y)+ω2φ(y), (3.5) withh1,ω1∈C⁰[0,∞) positive and nondecreasing,h2,ω2∈C⁰(0,∞) positive and nonincreasing,h1+h2andω1+ω2nonincreasing on a neighbourhood of 0,

₁

0h2(s)ds <∞, (3.6)

lim inf

x→∞

G(x)

H(Tx)>1, (3.7)

where G(x)=

_φ(x)

0

φ⁻¹(s)

ω1(s) +ω2(s)ds, H(x)= _x

0

h1(s) +h2(s)ds (3.8) forx∈[0,∞).

3.2. Auxiliary regular BVPs. Let assumption (H2) be satisfied. For eachn∈N, define the function fn∈Car([0,T]×R²) by the formula

fn(t,x,y)=

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩

f(t,x,y) fort∈[0,T], x≥ 1

n,|y| ≥ 1 n f

t,1

n,y

fort∈[0,T], x <1

n,|y| ≥1 n n

2

f_n

t,x,1 n

y+1

n

−fn

t,x,−1

n

y−1 n

fort∈[0,T], x∈R,|y|<1 n.

(3.9)