NONUNIQUENESS THEOREM FOR A SINGULAR CAUCHY-NICOLETTI PROBLEM

CAUCHY-NICOLETTI PROBLEM

JOSEF KALAS

Received 20 September 2002

The problem of nonuniqueness for a singular Cauchy-Nicoletti boundary value problem is studied. The general nonuniqueness theorem ensuring the existence of two different solutions is given such that the estimating expressions are nonlinear, in general, and de- pend on suitable Lyapunov functions. The applicability of results is illustrated by several examples.

1. Introduction

The nonuniqueness of a regular or singular Cauchy problem for ordinary differential equations is studied in several papers such as [3,4,5,13,14,15,16,17]. Most of these results can also be found in the monograph [1]. The uniqueness of solutions of Cauchy initial value problem for ordinary differential equations with singularity is investigated in [7,8,9,12]. The topological structure of solution sets to a large class of boundary value problems for ordinary differential equations is studied in [2]. First results on the nonuniqueness for a singular Cauchy-Nicoletti boundary value problem are given in [10, 11,12] by Kiguradze, where sufficient conditions for the nonuniqueness are written in the form of one-sided inequalities for the components in the right-hand side f(t,x1,. . .,x_n) of the corresponding equation. An expression for the estimation of the jth component

fj(t,x1,. . .,xn) off depends ontandxjand is linear in|xj|.

In [6], we studied the nonuniqueness for a singular Cauchy problem. Our criteria in- volve vector Lyapunov functions and the estimations need not be linear. The present pa- per deals with the nonuniqueness of the singular Cauchy-Nicoletti problem and extends the results of [6] to this more general problem.

Supposing−∞ ≤a < A≤ ∞,b >0, we will use the following notations throughout the paper:R^kandR⁺denotek-dimensional real Euclidean space and the interval [0,∞), respectively.| · |is used for the notation of H¨older’s 1-norm (the sum of the absolute val- ues of components).x=(x1,. . .,x_n) denotes a variable vector fromRⁿwith components x1,. . .,xn, while x0=(x01,. . .,x0n) stands for a fixed vector from Rⁿ with components x01,. . .,x0n.Nis equal to the set{1,. . .,n}.ldenotes a fixed number from the set{1,. . .,n}.

Copyright©2004 Hindawi Publishing Corporation Abstract and Applied Analysis 2004:7 (2004) 591–602 2000 Mathematics Subject Classification: 34B15 URL:http://dx.doi.org/10.1155/S1085337504306147

i1,i2,. . .,i_l are fixed integers such that 1≤i1< i2<···< i_l≤n.I is set to be equal to {i1,. . .,i_l}. Prxdenotes a projection ofxsuch that Prx=(x_i1,. . .,x_il), while Pr*xdenotes a complementary projection to Prx. Clearly, Pr*x=(xj1,. . .,xjn−l), where 1≤j1<···<

jn−l≤n,{i1,. . .,il} ∩ {j1,. . .,jn−l} = ∅.R^k_α,β;b(x0) and ˜R^k_a,Aare used for the notation of the set{(t,x)∈R^k+1:α < t < β,|x−x0| ≤b}and the set{(t,x)∈R^k+1:a < t < A,x∈R^k}, respectively. The symbol ˆRⁿ_a,Awill be used for the set{(t,x)∈Rⁿ⁺¹:a≤t≤A,x∈Rⁿ}.

∆(α,β) denotes the interval (min(α,β), max(α,β)).

The notationC[Γ,Ω] is used for the notation of the class of all continuous mappings Γ→Ω.AC[[a,A],R^k] andAC[[a,A], R^k] denote the class of all absolutely continuous mappings [a,A]_→R^kand the class of all mappings fromC[[a,A],R^k] which are abso- lutely continuous on any interval [α,β], wherea < α < β < A, respectively. The class of all Lebesgue-integrable mappings [a,A]→R⁺is denoted byL[[a,A],R⁺].ᏸτ[ ˆRⁿ_a,A,R^+k] stands for the class of all functions V(t,x) : ˆRⁿ_a,A→R^+k with the following property:

V(t,·) is uniformly continuous, and if a < α < β < A, τ∈[α,β], then V(t,x(t)) is absolutely continuous on [α,β] for any absolutely continuous functionx: [α,β]→Rⁿ. Kσ1,...,σp[ ˆR^k_a,A,R^m] denotes the class of all mappings ˆR^k_a,A→R^mwhich satisfy Carath´eodory conditions on R^k_α,β;ρ(0) for any α,β,a≤α < β≤A,σ_j∈[α,β](j=1,. . .,p), ρ∈(0,∞), σ1,. . .,σp being numbers from [a,A].N0(a,A;τ1,. . .,τn) is used for the notation of the class{Λ=(λ_{i j}(t))ⁿ_i,j=1:λ_{i j}∈L[[a,A],R⁺]}such that the system of differential inequali- ties|x_i(t)| ≤_n

j=1λ_{i j}(t)|x_j(t)|,t∈[a,A],i∈N, possesses no nontrivial solutionx(t)= (x1(t),. . .,xn(t))∈AC[[a,A],Rⁿ] satisfyingxi(τi)=0 (i=1,. . .,n).

The fundamental role in the proof of our main theorem will be played by the following theorem by Kiguradze, which is adapted from [12] (see also [10]) in a simplified form.

Kiguradze Theorem . Leta≤τ_i≤A,xˆ_0i∈Rfori=1,. . .,n. Suppose thatf∈K_σ1,...,σp[ ˆRⁿ_a,A, Rⁿ]. Assume that the components fiof f satisfy

fi(t,x) sgnt−τi

xi−xˆ0i

≤ n j=1

λi j(t)xj+µi(t) (i=1,. . .,n) (1.1) for (t,x)=(t,x1,. . .,xn)∈R˜ⁿ_a,A, where xˆ0i=0 if τi∈ {σ1,. . .,σp}. Suppose that Λ(t)= (λi j(t))ⁿ_i,j=1∈N0(a,A;τ1,. . .,τn),µi∈L[[a,A],R⁺]. Then the Cauchy-Nicoletti problem

x= f(t,x), xi τi

=0 (i=1,. . .,n) (1.2)

has at least one solutionx(t)=(x1(t),. . .,xn(t))∈AC[[a,A],Rⁿ].

2. Results

Consider a Cauchy-Nicoletti boundary value problem

x=f(t,x), x_it_i=x_0i (i=1,. . .,n), (2.1) where f(t,x)=(f1(t,x1,. . .,x_n),. . .,f_n(t,x1,. . .,x_n)), f ∈K_σ1,...,σp[ ˆRⁿ_a,A,Rⁿ], x_0i∈R, and t_i∈[a,A] (i∈N).

Theorem2.1. Suppose that there are numbersc_i∈R(i∈N),B_i∈[a,A]\ {t_i,σ1,. . .,σ_p} (i∈I), a matrix functionΛ=(λ_{i j})ⁿ_i,j=1∈N0(a,A;τ1,. . .,τ_n)and functionsµ_i∈L[[a,A],R⁺] (i∈N)such thatci=x0ifori∈N\Iand

fi(t,x)sgnt−τi

xi−ci

≤ n j=1

λi j(t)xj+µi(t) (i∈N) (2.2) holds for(t,x)=(t,x1,. . .,xn)∈_R˜ⁿ_a,A, whereτi=tiorτi=Biwheneveri∈N\I ori∈I, respectively.

Assume that

(i)there exist vector functionsgi=(gi1,. . .,giki)∈Ka,A,ti,Bi[ ˆR^k_a,Aⁱ ,R^ki] (i∈I)such that sgn(t−t_i)g_{i j}(t,u1,. . .,u_j−1,·,u_j,. . .,u_ki)is nondecreasing for j=1,. . .,k_iand there is a solutionϕ_i(t)=(ϕ_i1(t),. . .,ϕ_iki(t))of

u_i=gi

t,u1,. . .,uki

(2.3)

satisfying

ϕ_i(t)>0 fort∈∆t_i,B_i, lim

t→ti

ϕ_i(t)=0, lim inf

t→Bi

ϕ_i(t)>0 (2.4) fori∈I;

(ii)V_i(t,x)=(V_i1(t,x),. . .,V_iki(t,x))∈ᏸti[ ˆRⁿ_a,A,R^+ki] (i∈I)are such that there exists y0∈R^lwith the property

sup Vi j

Bi,y:y∈Rⁿ, Pry=y0

<lim inf

t→Bi

ϕi j(t) j=1,. . .,ki

(2.5) Vi(t,x)≥Ψixi−zi(t) fort∈∆ti,Bi

, (2.6)

whereΨi∈C[R⁺,R⁺],z_i∈C[(a,A),R]are such that Ψi(0)=0, Ψi(u)>0 foru >0, lim

t→tizi(t)=x0i (2.7) fori∈I;

(iii)there exist positive functionsεik∈C[(a,A),R⁺] (i∈I;k=1,. . .,ki)such that sgnB_i−t_iV_{i j}t,x(t)

≥sgnB_i−t_ig_{i j}t,ϕ_i1(t),. . .,ϕ_i,j−1(t),V_{i j}t,x(t),ϕ_i,j+1(t),. . .,ϕ_iki(t) (2.8) holds fori∈I, j=1,. . .,ki, and for any solutionx(t)of (2.1) a.e. on any interval (α_i1,α_i2)⊆∆(t_i,B_i)for which

Vik

t,x(t)< ϕik(t) +εik(t) onαi1,αi2 k=1,. . .,ki

,

V_{i j}t,x(t)> ϕ_{i j}(t) onα_i1,α_i2. (2.9) Then the Cauchy-Nicoletti boundary value problem (2.1) has at least two different solutions on[a,A], either of which satisfiesV_i(t,x(t))≤ϕ_i(t)fort∈∆(t_i,B_i)and i∈I.

Proof. Without loss of generality, it can be assumed thatI= {1,. . .,l}, Prx=

x1,. . .,x_l, Pr*x=

x_l+1,. . .,x_n. (2.10) For anyi∈Iandj∈ {1,. . .,k_i}, denote

L_{i j}=lim inf

t→Bi

ϕ_{i j}(t), S_{i j}=sup V_{i j}B_i,y:y∈Rⁿ, Pry=y0

. (2.11)

According to (2.5) and to the uniform continuity ofV_{i j}(B_i,·), we have a relation V_{i j}B_i,y^∗≤V_{i j}B_i,y+V_{i j}B_i,y^∗−V_{i j}B_i,y

≤1 2

Li j+Si j +1

4

Li j−Si j

=3 4Li j+1

4Si j< Li j

(2.12)

fory∈Rⁿ, Pry=y0, and for y^∗∈Rⁿsufficiently close toy. Hence it can be supposed without loss of generality thaty0=Prx0.

Further, the uniform continuity ofVi j(Bi,·) implies that the inequality sup Vi j

Bi,y:y∈Rⁿ, Pry=y0−λy0−Prx0

<lim inf

t→Bi

ϕi j(t) i∈I;j=1,. . .,ki

(2.13) holds provided thatλ >0 is sufficiently small. Therefore, we can choose ˜x1, ˜x2∈R^l, ˜x1=

˜

x2, such that maxk=1,2

sup Vi j

Bi,y:y∈Rⁿ, Pry=x˜k

<lim inf

t→Bi

ϕi j(t) i=1,. . .,l;j=1,. . .,ki . (2.14) Choose ˜ξ∈ {x˜1, ˜x2}arbitrary. Putξ=x0−( ˜ξ, Pr*x0),X=x−x0+ξ, and f^∗(t,X)= f(t,x0+X−ξ) for (t,X)=(t,X1,. . .,Xn)∈Rˆⁿ_a,A.

Clearly f^∗∈Kσ1,...,σp[ ˆRⁿ_a,A,Rⁿ]. By using (2.2), we obtain f_i^∗(t,X) sgnt−τi

Xi+ ˜ξi−ci

≤ l j=1

λi j(t)Xj+ ˜ξj+ n j=l+1

λi j(t)Xj+x0j+µi(t)

≤ n j=1

λ_{i j}(t)X_j+ ˜µ_i(t)

(2.15) for (t,X)∈R˜ⁿ_a,A,i=1,. . .,l, and

f_i^∗(t,X) sgnt−τi

Xi

≤ l j=1

λi j(t)Xj+ ˜ξj+ n j=l+1

λi j(t)Xj+x0j+µi(t)

≤ n j=1

λ_{i j}(t)X_j+ ˜µ_i(t)

(2.16)

for (t,X)∈R˜ⁿ_a,A,i=l+ 1,. . .,n, where

˜ µi(t)=

l j=1

λi j(t)ξ˜j+ n j=l+1

λi j(t)x0j+µi(t) (2.17) fori=1, 2,. . .,n. As ˜µ_i∈L[[a,A],R⁺] holds, Kiguradze theorem implies that the bound- ary value problem

X=f^∗(t,X), Xi τi

=0 (i=1,. . .,n) (2.18)

has at least one solutionX(t)∈AC[[a,A],Rⁿ]. Hencex(t)=X(t) +x0−ξis a solution of x=f(t,x), xi

τi

=ξ˜i (i=1,. . .,l), xi

τi

=x0i (i=l+ 1,. . .,n). (2.19) Now we will prove that limt→tixi(t)=x0ifori=1,. . .,l. Putmi(t)=Vi(t,x(t)),mi j(t)= Vi j(t,x(t)) fori=1,. . .,landj=1,. . .,ki. In view of (2.14), the inequality

mi(t)< ϕi(t) (2.20)

holds fort∈(a,A) sufficiently close toBi. Suppose for definiteness thatti< Bi, that is,

∆(t_i,B_i)=(t_i,B_i) for somei∈ {1,. . .,l}. We will show thatm_i(t)≤ϕ_i(t) fort∈(t_i,B_i).

Assume on the contrary that there is aτ∈(ti,Bi) such that mi(τ)≤ϕi(τ) is not true.

Sincex(t) is continuous and (2.20) holds fort∈(a,A) sufficiently close toBi, there exist j∈ {1,. . .,k_i}and an intervalJ_i=(τ_i1,τ_i2) such thatτ < τ_i1< τ_i2< B_i,

mi j

τi2

=ϕi j

τi2

,

ϕi j(s)< mi j(s)< ϕi j(s) +εi j(s) fors∈Ji, mik(s)< ϕik(s) +εik(s) fors∈Ji,k=1,. . .,ki.

(2.21)

Using (2.8), we get m_{i j}(s)≥gi j

s,ϕi1(s),. . .,ϕi,j−1(s),mi j(s),ϕi,j+1(s),. . .,ϕiki(s) (2.22) a.e. onJi. Asgi j(t,u1,. . .,uj−1,·,uj+1,. . .,un(s)) is nondecreasing, we have

m_{i j}(s)≥gi j(s,ϕi1(s),. . .,ϕiki(s))=ϕ_{i j}(s) (2.23) a.e. onJi. Therefore, the functionmi j(t)−ϕi j(t) is nondecreasing onJi, which is a con- tradiction tomi j(τi2)=ϕi j(τi2). Thus

0≤mi(t)≤ϕi(t) fort∈ ti,Bi

. (2.24)

Now the condition limt→ti+ϕi(t)=0 implies limt→ti+mi(t)=0. With respect to the con- tinuity ofx_i(t) on [a,A], we havex_i(t_i)=limt→tix_i(t)₌x_0i. The inequality (2.24) implies

V_i(t,x(t))≤ϕ_i(t) fort∈∆(t_i,B_i).

Corollary2.2. Letc_i∈R(i∈N),B_i∈[a,A]\ {t_i,σ1,. . .,σ_p}(i∈I), a matrix function Λ=(λ_{i j})ⁿ_i,j=1∈N0(a,A;τ1,. . .,τ_n), and functions µ_i∈L[[a,A],R⁺] (i∈N)be such that ci=x0ifori∈N\I and condition (2.2) is fulfilled, where τi=tiorτi=Biwheneveri∈ N\Iori∈I, respectively.

Assume that

(i)there exist functions gi∈Ka,A,ti,Bi[ ˆR¹_a,A,R] (i∈I)such that sgn(t−ti)gi(t,·) are nondecreasing and there are solutionsϕi(t)of

u_i=g_it,u_i (2.25)

satisfying (2.4);

(ii)there arezi∈AC[[a,A],R]andε=(εi1,. . .,εil)∈C[(a,A),R^+l]such thatzi(ti)= x_0i(i∈I)and the estimation

sgn(Bi−ti) sgnxi−zi(t)fi(t,x)−z_i(t)≥sgnBi−ti gi

t,xi−zi(t)(i∈I) (2.26) is fulfilled onΩˆ = {(t,x) :ϕ_i(t)<|x_i−z_i(t)|< ϕ_i(t) +ε_i(t),t∈∆(t_i,B_i)}for almost allt∈∆(ti,Bi). Then the Cauchy-Nicoletti boundary value problem (2.1) has at least two different solutions on[a,A], either of which satisfies |xi(t)−zi(t)| ≤ϕi(t)for t∈∆(t_i,B_i)andi∈I.

Proof. Without loss of generality, it can be supposed thatI= {1,. . .,l}and Prx=(x1,. . ., x_l). Putk_i=1 andV_i(t,x(t))=V_i1(t,x)= |x_i−z_i(t)|fori=1,. . .,l. Then

sgnBi−ti

V_i1t,x(t)≥sgnBi−ti

fi

t,x(t)−z_i(t)sgnxi(t)−zi(t)

≥sgnBi−ti gi

t,xi(t)−zi(t)

=sgnB_i−t_ig_it,V_i1t,x(t)

(2.27)

holds for any solution x(t) of (2.1) a. e. on any interval (αi1,αi2)⊆∆(ti,Bi) for which ϕi(t)< Vi(t,x(t))< ϕi(t) +εi(t) on (αi1,αi2). The assumptions ofTheorem 2.1are satisfied.

Example 2.3. Let f1,. . .,f_n∈K0[ ˆRⁿ_0,1,R] be such that

f1

t,x1,. . .,xn

sgnx1≥δ(t)x1^γ,

−fj

t,x1,. . .,xn

sgnxj≤ j k=1

λjk(t)xk+µj(t) (j=2,. . .,n) (2.28) for (t,x1,. . .,xn)∈R˜ⁿ0,1, whereγ∈(0, 1) and δ,λjk,µj∈L[[0, 1],R⁺],δ being a positive function. Consider the boundary value problem

x₁=f1

t,x1,. . .,xn

, x1(0)=0, x₂=f2

t,x1,. . .,xn

, x2(1)=0, ...

x_n=fn

t,x1,. . .,xn

, xn(1)=0.

(2.29)

Putt1=0,t2=t3= ··· =t_n=1, g1(t,u)=





δ(t)u^γ foru≥0,

0 foru <0, (2.30)

λ1k(t)≡0 (k=1,. . .,n),λjk(t)≡0 (j=2,. . .,n;k=j+ 1,. . .,n), andµ1(t)≡0. LetB1=1.

Thenτ1=τ2= ··· =τ_n=1, f1

t,x1,. . .,xn

sgnt−B1

x1

≤0, fj

t,x1,. . .,xn

sgn(t−1)xj

≤ n k=1

λjk(t)xk+µj(t) (j=2,. . .,n), (2.31) and the equationu1=g1(t,u) has a positive solution

ϕ1(t)=

(1−γ) _t

0δ(s)ds 1/(1−γ)

(2.32) in (0, 1] such that limt→0ϕ1(t)=0. The assumptions ofCorollary 2.2are fulfilled with I= {1},c1=0, andz(t)=z1(t)≡0. Therefore, the considered boundary value problem has at least two different solutions on [a,A]. Moreover, the first componentx1(t) of these solutions satisfies|x1(t)| ≤ϕ1(t) fort∈(0, 1].

Corollary2.4. Suppose that−∞< a < A <∞,c∈R,λ∈L[[a,A],R⁺], andµ∈L[[a,A], R⁺]. LetB∈[a,A]\ {tn,σ1,. . .,σp}be such that

f˜t,x1,. . .,xn

sgn(t−B)xn−c≤λ(t)xn+µ(t) (2.33) for(t,x)∈R˜ⁿ_a,A. Assume that

(i)there exists a functionq∈Ka,A,tn,B[ ˆR¹_a,A,R]such thatsgn(t−tn)q(t,·)is nondecreas- ing and there is a solutionϕ(t)of

u=q(t,u) (2.34)

satisfying

ϕ(t)>0 fort∈∆t_n,B, lim

t→tn

ϕ(t)=0, lim inf

t→B ϕ(t)>0; (2.35) (ii)there arez∈AC[[a,A], R]andε∈C[(a,A),R⁺]such thatz(tn)=x0nand

sgnB−tn

sgnxn−z(t)f˜t,x1,. . .,xn

−z(t)≥sgnB−tn

qt,xn−z(t) (2.36)

holds onΩˆ = {(t,x1,. . .,x_n) :ϕ(t)<|x_n−z(t)|< ϕ(t) +ε(t),t∈∆(t_n,B)}for almost allt∈∆(t_n,B). Then the boundary value problem

v⁽ⁿ⁾= f˜t,v,v,. . .,v⁽ⁿ⁻¹⁾, vt1

=x01, vt2

=x02,. . ., v⁽ⁿ⁻¹⁾t_n=x_0n (2.37) has at least two different solutions on[a,A].

Proof. PutI= {n},k1=1, Prx=x_n,c_n=c,g_n(t,u)=q(t,u), ϕ_n(t)=ϕ(t),c_i=x_0i for i=1,. . .,n−1,µi(t)=0 fori=1,. . .,n−1,µn(t)=µ(t),Bn=B, and

λi j(t)=











1 for 1≤i=j−1≤n−1, λ(t) fori=j=n,

0 otherwise.

(2.38)

Considering the system x₁=x2, x₂=x3,

... x_n−1=xn,

x_n=f˜t,x1,x2,. . .,xn

,

x1 t1

=x01, x2

t2

=x02, ... xn−1

tn−1

=x0n−1, xn

tn

=x0n,

(2.39)

and applyingCorollary 2.2, we get

f_nt,x1,. . .,x_nsgnt−B_nx_n−c_n≤ n j=1

λ_{n j}(t)x_j+µ_n(t), fi

t,x1,. . .,xn

sgnt−ti

xi−ci

≤xi+1≤λi,i+1xi+1

= n j=1

λ_{i j}(t)x_j+µ_i(t)

(2.40)

fori=1,. . .,n−1. The result follows fromCorollary 2.2.

Example 2.5. Letγ∈(0, 1). Consider the boundary value problem v=p1(t,v)v^γsgnv+p2

t,v,v, v(0)=0, v(1)=0, (2.41) wherep1∈K1[ ˆR¹_0,1,R] andp2∈K1[ ˆR²_0,1,R] are such that

x2p2

t,x1,x2

≤0 fort,x1,x2

∈(0, 1)×R², p1

t,x1

≤ −δ(t) fort,x1

∈(0, 1)×R, (2.42)

δ∈L[[0, 1],R] being a positive function. Since

−p1

t,x1x2^γ−p2

t,x1,x2

sgnx2≥δ(t)x2^γ, (2.43)

the assumptions ofCorollary 2.4are fulfilled withn=2,a=0,A=1,t1=0,t2=1,c=0, B=0,z(t)≡0,λ(t)≡0,µ(t)≡0, and

q(t,u)=



−δ(t)u^γ foru≥0,

0 for u <0, ϕ(t)=

(1−γ)

1 t δ(s)ds

1/1−γ

. (2.44) Therefore, problem (2.41) has at least two different solutions on [0, 1].

Corollary2.6. Let the assumptions ofCorollary 2.2be fulfilled with the exception that the conditions (i), (ii) are replaced by (i), (ii):

(i)there exist functionsh_i,q_i∈K_a,A,ti,Bi[ ˆR¹_a,A,R] (i∈I) such that functions sgn(t− ti)hi(t,·)andsgn(t−ti)qi(t,·)are nondecreasing fori∈I and there are solutions ϕi(t),ψi(t)ofu_i=hi(t,ui)andv_i=qi(t,vi), respectively, satisfying

ϕ_i(t)>0 fort∈∆t_i,B_i, lim

t→ti

ϕ(t)=0, lim inf

t→Bi

ϕ(t)>0, ψ_i(t)>0 fort∈∆t_i,B_i, lim

t→ti

ψ(t)=0, lim inf

t→Bi

ψ(t)>0 (2.45) fort∈I;

(ii)there arezi∈AC[[a,A],R]andε=(εi1,. . .,εil)∈C[(a,A),R^+l]such thatzi(ti)= x_0iand the inequalities

sgnBj−tj

fj(t,x)−z_j(t)−hj

t,xj−zj(t)₊≥0 (j∈I) sgnB_j−t_j−

f_j(t,x)−z_j(t)−q_jt,x_j−z_j(t)₋≥0 (j∈I) (2.46) are fulfilled onΩˆ = {(t,x) :ϕj(t)< xj−zj(t)< ϕj(t) +εj(t),t∈∆(tj,Bj)}andΩˆˆ = {(t,x) :ψj(t)< zj(t)−xj< ψj(t) +εj(t),t∈∆(tj,Bj)}, respectively, for almost all t∈∆(t_j,B_j). Then the Cauchy-Nicoletti boundary value problem (2.1) has at least two different solutions on[a,A].

Proof. Without loss of generality, it can again be assumed thatI= {1,. . .,l}and Prx= (x1,. . .,xl). Putki=2,gi1(t,u)=hi(t,u), gi2(t,v)=qi(t,v),ϕi1(t)=ϕi(t), ϕi2(t)=ψi(t), V_i1(t,x)=(x_i−z_i(t))+,V_i2(t,x)=(x_i−z_i(t))₋, andV_i(t,x)=(V_i1(t,x),V_i2(t,x)) fori∈ I. Then we have

sgnBi−ti

V_i1t,x(t)≥sgnBi−ti fi

t,x(t)−z_i(t)

≥sgnBi−ti gi1

t,Vi1

t,x(t), sgnBi−ti

V_i2t,x(t)≥ −sgnBi−ti fi

t,x(t)−z_i(t)

≥sgnBi−ti

gi2

t,Vi2

t,x(t)

(2.47)

for any solutionx=x(t) of (2.1) a.e. on any interval (α_i1,α_i2)⊆∆(t_i,B_i) for which Vi1

t,x(t)< ϕi(t) +εi(t), Vi2

t,x(t)< ψi(t) +εi(t) (2.48)