R E S E A R C H
Open Access
Positive solutions for a system of
second-order discrete boundary value
problems
Ravi P. Agarwal
1and Rodica Luca
2**Correspondence: [email protected]
2Department of Mathematics, Gh.
Asachi Technical University, Iasi, Romania
Full list of author information is available at the end of the article
Abstract
We study the existence and multiplicity of positive solutions for a system of nonlinear second-order difference equations subject to multi-point boundary conditions, under some assumptions on the nonlinearities of the system which contains concave functions. In the proofs of our main results we use some theorems from the fixed point index theory.
MSC: 39A10; 39A12
Keywords: Difference equations; Multi-point boundary conditions; Positive solutions; Existence; Multiplicity
1 Introduction
In this paper, we consider the system of nonlinear second-order difference equations
⎧ ⎨ ⎩
2un–1+f(n,un,vn) = 0, n= 1,N– 1, 2v
n–1+g(n,un,vn) = 0, n= 1,N– 1,
(S)
subject to the multi-point boundary conditions
u0= p
i=1
aiuξi, uN=
q
i=1
biuηi, v0=
r
i=1
civζi, vN=
l
i=1
divρi, (BC)
whereN∈N,N≥2,p,q,r,l∈N,is the forward difference operator with stepsize 1, un=un+1–un,2un–1=un+1– 2un+un–1, andn=k,mmeans thatn=k,k+ 1, . . . ,m fork,m∈N,ξi∈Nfor alli= 1,p,ηi∈Nfor alli= 1,q,ζi∈Nfor alli= 1,r,ρi∈Nfor all i= 1,l, 1≤ξ1<· · ·<ξp≤N– 1, 1≤η1<· · ·<ηq≤N– 1, 1≤ζ1<· · ·<ζr≤N– 1, and 1≤ρ1<· · ·<ρl≤N– 1.
Under sufficient conditions on the nonnegative nonlinearities f andg which contain some concave functions, we investigate the existence and multiplicity of positive solu-tions of problem (S)–(BC) by using the fixed point index theory. By a positive solution of (S)–(BC), we mean a pair of sequences (u,v) = ((un)n=0,N, (vn)n=0,N) satisfying (S) and
(BC) withun≥0 andvn≥0 for alln= 0,N, andun> 0 for alln= 1,N orvn> 0 for all n= 1,N. The existence and nonexistence of nonnegative and nontrivial solutions (u,v) (un≥0,vn≥0 for alln= 0,N and (u,v)= (0, 0)) of problem (S)–(BC) with some posi-tive parameters in system (S) were studied in the papers [14] and [15] by using the Guo– Krasnosel’skii fixed point theorem. We also mention the paper [19], where the authors investigated the existence and multiplicity of positive solutions for problem (S)–(BC) un-der some assumptions on the functionsf andg which are different than those we use in this paper. The existence, nonexistence, and multiplicity of positive solutions for system (S) with parameters or without parameters, subject to the multi-point coupled boundary conditions
u0= 0, uN = p
i=1
aivξi, v0= 0, vN=
q
i=1
biuηi, (BC1)
were studied in the papers [16] and [18].
The mathematical modeling of many nonlinear problems from computer science, eco-nomics, mechanical engineering, control systems, biological neural networks, and others leads to the consideration of nonlinear difference equations (see [2,4,21,23]). In the last decades, many authors have investigated such problems by using various methods, such as fixed point theorems, the critical point theory, upper and lower solutions, the fixed point index theory, and the topological degree theory (see, for example, [1,6–12,17,20, 24–28]).
The paper is organized as follows. In Sect.2, we investigate a system of second-order lin-ear difference equations subject to the boundary conditions (BC), and we present the prop-erties of the corresponding Green functions. In Sect.3, we prove the main theorems for the existence and multiplicity of positive solutions of problem (S)–(BC) which are based on some theorems from the fixed point index theory, and we present two examples to support our results.
2 Preliminary results
We begin this section with a result from [14] related to the following system of second-order difference equations:
2un–1+yn= 0, n= 1,N– 1, (1)
subject to the multi-point boundary conditions
u0= p
i=1
aiuξi, uN=
q
i=1
biuηi, (2)
wherep,q∈N,ξi∈Nfor alli= 1,p,ηi∈Nfor alli= 1,q, 1≤ξ1<· · ·<ξp≤N– 1, 1≤η1< · · ·<ηq≤N– 1, andyn∈Rfor alln= 1,N– 1.
We denote1= (1 –
q
i=1bi)
p
i=1aiξi+ (1 –
p
i=1ai)(N–
q
Lemma 2.1([14]) If1= 0,then the solution(un)n=0,N of problem(1)–(2)is given by un=
N–1
j=1 G1(n,j)yjfor all n= 0,N,where the Green function G1is defined by
G1(n,j) =g0(n,j) + 1 1
(N–n)
1 – q
k=1 bk +
q
i=1
bi(N–ηi)
p
i=1
aig0(ξi,j)
+ 1 1
n
1 – p
k=1 ak +
p
i=1 aiξi
q
i=1
big0(ηi,j), n= 0,N,j= 1,N– 1, (3)
and
g0(n,j) = 1 N
⎧ ⎨ ⎩
j(N–n), 1≤j≤n≤N,
n(N–j), 0≤n≤j≤N– 1. (4)
Next we will present some properties of the functiong0and the Green functionG1.
Lemma 2.2 The function g0given by(4)has the following properties:
(a) g0(n,j)≥0for alln= 0,N,j= 1,N– 1;
(b) g0(n,j)≤h(j)for alln= 0,N,j= 1,N– 1,whereh(j) =g0(j,j) =N1j(N–j)for all j= 1,N– 1;
(c) g0(n,j)≥k(n)h(j)for alln= 0,N,j= 1,N– 1,wherek(n) =N(N1–1)n(N–n)for all n= 0,N.
Proof For the proofs of (a) and (b), see [7]. For (c), if 1≤j≤n≤N, then we have
1
Nj(N–n)≥ 1
N(N– 1)n(N–n) 1
Nj(N–j) ⇔ N(N– 1)≥n(N–j),
which is satisfied for allj= 1,N– 1 andn= 0,N. If 0≤n≤j≤N– 1, then we obtain
1
Nn(N–j)≥ 1
N(N– 1)n(N–n) 1
Nj(N–j) ⇔ N(N– 1)≥j(N–n),
which is satisfied for allj= 1,N– 1 andn= 0,N.
Lemma 2.3 If ai≥0for all i= 1,p,
p
i=1ai< 1,bi≥0for all i= 1,q,
q
i=1bi< 1,then the Green function G1of problem(1)–(2)given by(3)satisfies the inequalities
(a) G1(n,j)≤Ah(j)for alln= 0,N,j= 1,N– 1,where
A= 1 + 1 1
N– q
i=1 biηi
p
i=1 ai +
1 1
N– p
i=1
ai(N–ξi) q
i=1
bi > 0.
Proof By the assumptions on the coefficientsai,i= 1,pandbj,j= 1,q, we can easily see
that is, we obtain inequalities (a) and (b).
Lemma 2.4 Assume that ai≥0for all i= 1,p,
Proof By using Lemmas2.1–2.3, we deduce
un=
We can also formulate similar results as Lemmas2.1–2.4for the discrete boundary value problem We denote by
G2(n,j) =g0(n,j) + 1 2
(N–n)
1 – l
k=1 dk +
l
i=1
di(N–ρi)
r
i=1
cig0(ζi,j)
+ 1 2
n
1 – r
k=1 ck +
r
i=1 ciζi
l
i=1
dig0(ρi,j), n= 0,N,j= 1,N– 1,
B= 1 + 1 2
N– l
i=1 diρi
r
i=1 ci +
1 2
N– r
i=1
ci(N–ζi) l
i=1
di > 0.
Then we deduce the inequalitiesG2(n,j)≤Bh(j) andG2(n,j)≥k(n)h(j) for alln= 0,N, j= 1,N– 1. In addition the solution (vn)n=0,N of problem (5)–(6) satisfies the inequality vn≥B1k(n)vmfor alln,m= 0,N.
We recall now some theorems concerning the fixed point index theory. LetEbe a real Banach space with the norm · ,P⊂Ebe a cone, “≤” be the partial ordering defined by P, and 0 be the zero element inE. For> 0, letB={u∈E,u<}be the open ball of
radiuscentered at 0, and its boundary∂B={u∈E,u=}. The proofs of our results
are based on the following fixed point index theorems (see [3,5,13,22]).
Theorem 2.1 Let A:B∩P→P be a completely continuous operator which has no fixed
points on∂B∩P.IfAu ≤ ufor all u∈∂B∩P,then i(A,B∩P,P) = 1.
Theorem 2.2 Let A:B∩P→P be a completely continuous operator.If there exists u0∈
P\ {0}such that u–Au=λu0for allλ≥0and u∈∂B∩P,then i(A,B∩P,P) = 0.
Theorem 2.3 LetΩ⊂E be a bounded open set with0∈Ω.Assume that A:Ω∩P→P is a completely continuous operator.
(a) Ifu≤Aufor allu∈∂Ω∩P,then the fixed point indexi(A,Ω∩P,P) = 1. (b) IfAu≤ufor allu∈∂Ω∩P,then the fixed point indexi(A,Ω∩P,P) = 0.
3 Existence and multiplicity of positive solutions
In this section we present sufficient conditions on the functionsfandgsuch that problem (S)–(BC) has positive solutions with respect to a cone.
We present the assumptions that we shall use in the sequel. (H1) ai≥0,ξi∈Nfor alli= 1,p,1≤ξ1<· · ·<ξp≤N– 1,
bi≥0,ηi∈Nfor alli= 1,q,1≤η1<· · ·<ηq≤N– 1, ci≥0,ζi∈Nfor alli= 1,r,1≤ζ1<· · ·<ζr≤N– 1, di≥0,ρi∈Nfor alli= 1,l,1≤ρ1<· · ·<ρl≤N– 1, and
p
i=1ai< 1,
q
i=1bi< 1,
r
i=1ci< 1,
l
i=1di< 1.
(H2) The functionsf,g:{1, . . . ,N– 1} ×R+×R+→R+are continuous, (R+= [0,∞)). (H3) There exist functionsa,b∈C(R+,R+)such that
(a) a(·)is concave and strictly increasing onR+witha(0) = 0;
(b)
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
f0i=lim infv→0+f(an(,uv),v)∈(0,∞],
uniformly with respect to(n,u)∈ {1, . . . ,N– 1} ×R+, and
g0i=lim infu→0+g(bn(,uu,)v)∈(0,∞],
(c) limu→0+a(Cbu(u))=∞exists for any constantC> 0. (H4) There existα1,α2> 0withα1α2≤1such that
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
fs
∞=lim supv→∞ f(n,u,v)
vα1 ∈[0,∞),
uniformly with respect to(n,u)∈ {1, . . . ,N– 1} ×R+, and
g∞s =limu→∞g(unα,u2,v)= 0
exists uniformly with respect to(n,v)∈ {1, . . . ,N– 1} ×R+.
(H5) There exist the functionsc,d∈C(R+,R+)such that (a) c(·)is concave and strictly increasing onR+;
(b)
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
fi
∞=lim infv→∞f(nc(,uv),v)∈(0,∞],
uniformly with respect to(n,u)∈ {1, . . . ,N– 1} ×R+, and
gi
∞=lim infu→∞g(dn(,uu),v) ∈(0,∞],
uniformly with respect to(n,v)∈ {1, . . . ,N– 1} ×R+;
(c) limu→∞c(Cdu(u))=∞exists for any constantC> 0. (H6) There existβ1,β2> 0withβ1β2≥1such that
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
f0s=lim supv→0+f(n,u,v)
vβ1 ∈[0,∞),
uniformly with respect to(n,u)∈ {1, . . . ,N– 1} ×R+, and g0s=limu→0+g(un,βu2,v)= 0
exists uniformly with respect to(n,v)∈ {1, . . . ,N– 1} ×R+.
(H7) The functionsf(n,u,v)andg(n,u,v)are nondecreasing with respect touandv, and there existsN0> 0such that
f(n,N0,N0) <
3N0
(N2– 1)max{A,B} and g(n,N0,N0) <
3N0 (N2– 1)max{A,B}
for alln∈ {1, . . . ,N– 1}.
By using the Green functions G1 andG2 from Sect.2, our problem (S)–(BC) can be written equivalently as the following system:
⎧ ⎨ ⎩
un=Ni=1–1G1(n,i)f(i,ui,vi), n= 0,N, vn=
N–1
i=1 G2(n,i)g(i,ui,vi), n= 0,N.
(7)
Then (u,v) = ((un)n=0,N, (vn)n=0,N) is a solution of problem (S)–(BC) if and only if (u,v) is a solution of system (7).
We consider the Banach spaceX=RN+1={u= (u
(u,v)Y=u+v. We define the cones
P1=
u∈X,u= (un)n=0,N,un≥ 1
Ak(n)u,∀n= 0,N
⊂X,
P2=
v∈X,v= (vn)n=0,N,vn≥ 1
Bk(n)v,∀n= 0,N
⊂X,
andP=P1×P2⊂Y.
We introduce the operatorsQ1,Q2:Y→XandQ:Y→Ydefined by
Q1(u,v) =Q1n(u,v)
n=0,N, Q2(u,v) =
Q2n(u,v)
n=0,N,
Q1n(u,v) = N–1
i=1
G1(n,i)f(i,ui,vi), n= 0,N,
Q2n(u,v) = N–1
i=1
G2(n,i)g(i,ui,vi), n= 0,N,
Q(u,v) =Q1(u,v),Q2(u,v)
, (u,v) =(un)n=0,N, (vn)n=0,N
∈Y.
The pair (u,v) is a solution of problem (S)–(BC) if and only if (u,v) is a fixed point of operatorQin the spaceY. So, we will investigate the existence of fixed points of opera-torQ. Under assumptions (H1) and (H2) and by using Lemma2.4, we can easily prove thatQ(P)⊂Pand the operatorQ:P→Pis completely continuous.
Theorem 3.1 Assume that(H1), (H2), (H3),and(H4)hold.Then problem(S)–(BC)has at least one positive solution.
Proof By (H3), there existC1> 0,C2> 0, and a sufficiently smallr1> 0 such that
f(n,u,v)≥C1a(v), ∀(n,u)∈ {1, . . . ,N– 1} ×R+,v∈[0,r1],
g(n,u,v)≥C2b(u), ∀(n,v)∈ {1, . . . ,N– 1} ×R+,u∈[0,r1],
(8)
and
aC3b(u)≥72C3max{A,B}N 3u
C1C2(N2– 1)2 , ∀u∈[0,r1], (9)
whereC3=max{(N–1)C2
N h(j),j= 1,N– 1}.
We will show that (Q1(u,v),Q2(u,v))≤(u,v) for all (u,v)∈∂Br1 ∩P. We suppose that
there exists (u,v)∈∂Br1 ∩P, that is,(u,v)Y =r1, such that (Q1(u,v),Q2(u,v))≤(u,v).
Thenu≥Q1(u,v) andv≥Q2(u,v). By using the monotonicity and concavity ofa(·), the Jensen inequality, Lemma2.3, relations (8) and (9), we obtain
un≥Q1n(u,v) = N–1
i=1
G1(n,i)f(i,ui,vi)≥C1 N–1
i=1
h(i)k(n)a(vi)
≥C1k(1) N–1
i=1
h(i)a(vi) = C1 N
N–1
i=1 h(i)a
N–1
j=1
≥C1
In a similar manner, we deduce
≥ C2 NC3
N–1
j=1
h(j)72C3max{A,B}N 3 C1C2(N2– 1)2 uj
=72N
2max{A,B} (N2– 1)2C1
N–1
j=1 h(j)
N–1
θ=1
G1(j,θ)f(θ,uθ,vθ)
≥72N2max{A,B} (N2– 1)2C1
N–1
j=1 h(j)
N–1
θ=1
h(θ)k(j)C1a(vθ)
≥72Nmax{A,B} (N2– 1)2
N–1
j=1 h(j)
N–1
θ=1
h(θ)a(vθ)
=12Nmax{A,B} N2– 1
N–1
θ=1
h(θ)a(vθ)
≥12Nmax{A,B} N2– 1
N–1
θ=1 h(θ)a
1 Bk(θ)v
≥12Nmax{A,B} N2– 1
N–1
θ=1 h(θ) a
1 BNv
≥2Nmax{A,B} 1 BNa
v
≥2av, ∀i= 1,N– 1.
Then we conclude thata(v) =a(supi=0,Nvi)≥a(v1)≥2a(v), and hencea(v) = 0. By (H3)(a), we obtain
v= 0. (11)
Therefore, by (10) and (11), we deduce that(u,v)Y = 0, which is a contradiction. Hence (Q1(u,v),Q2(u,v))≤(u,v) for all (u,v)∈∂Br1∩P. By Theorem2.3(b), we conclude that the
fixed point index
i(Q,Br1∩P,P) = 0. (12)
On the other hand, by (H4) we deduce that there existC4> 0,C5> 0, andC6> 0 such that
f(n,u,v)≤C4vα1+C5, ∀(n,u,v)∈ {1, . . . ,N– 1} ×R+×R+,
g(n,u,v)≤ε1uα2+C6, ∀(n,u,v)∈ {1, . . . ,N– 1} ×R+×R+,
(13)
with
ε1=min
6 B(N2– 1)
3 4AC4(N2– 1)
α2
, 6
α2+1
8B(AC4)α2(N2– 1)α2+1
Then, by (13), we have
we conclude that there existsR1>r1such that
≤AC4N at least one positive solution.
Proof By (H5) there existCi> 0,i= 7, . . . , 11, such that
f(n,u,v)≥C7c(v) –C8, g(n,u,v)≥C9d(u) –C10,
and
=12Nmax{A,B} N2– 1
N–1
j=1
h(j)c(vj) –C18
≥12Nmax{A,B} N2– 1
N–1
j=1 h(j) 1
BNc
v–C18
≥2cv–C18, ∀n= 1,N– 1,
whereC17=C9C11(N2–1)
6NC12 +c(C14),C18=
12C13N2max{A,B}
C7(N2–1) +C17.
Thenc(v)≥c(v1)≥2c(v) –C18, and soc(v)≤C18. By (H5)(a) and (c), we deduce thatlimv→∞c(v) =∞. Thus there existsC19> 0 such that
v ≤C19. (26)
By (25) and (26), we conclude that(u,v)Y≤C16+C19for all (u,v)∈U. That is the setU is bounded. Then there exists a sufficiently largeR2> 0 such that (u,v)=Q(u,v) +λ(ϕ1,ϕ2) for all (u,v)∈∂BR2∩Pandλ≥0. By Theorem2.2we deduce that
i(Q,BR2∩P,P) = 0. (27)
On the other hand, by (H6) there existC20> 0 and a sufficiently smallr2> 0, (r2<R2, r2≤1) such that
f(n,u,v)≤C20vβ1, ∀(n,u)∈ {1, . . . ,N– 1} ×R+,v∈[0,r2],
g(n,u,v)≤ε2uβ2, ∀(n,v)∈ {1, . . . ,N– 1} ×R+,u∈[0,r2],
(28)
whereε2= (2ABβ1C20(N
2–1
6 )
β1+1)–1/β1> 0.
We will show that (u,v)≤Q(u,v) for all (u,v)∈∂Br2 ∩P. We suppose that there exists
(u,v)∈∂Br2 ∩P, that is,(u,v)Y =r2≤1, such that (u,v)≤(Q1(u,v),Q2(u,v)), or u≤
Q1(u,v) andv≤Q2(u,v). Then by (28) we obtain
un≤Q1n(u,v) = N–1
i=1
G1(n,i)f(i,ui,vi)≤AC20 N–1
i=1 h(i)vβ1i
≤AC20 N–1
i=1 h(i)
N–1
j=1
G2(i,j)g(j,uj,vj)
β1
≤AC20 N–1
i=1 h(i)
B N–1
j=1 h(j)ε2u
β2
j
β1
≤ABβ1C20(N2– 1)ε
β1
2 6
N–1
j=1 h(j)
β1
uβ1β2
=ABβ1C20
N2– 1 6
β1+1
εβ12 uβ1β2
≤ABβ1C
20ε
β1
2
N2– 1 6
β1+1
u=1
Thereforeu ≤12u, so
u= 0. (29)
In addition
vn≤Q2n(u,v) = N–1
i=1
G2(n,i)g(i,ui,vi)
≤B N–1
i=1 h(i)ε2u
β2
i ≤
Bε2(N2– 1)
6 u
β2, ∀n= 0,N. (30)
By (29) and (30) we deduce thatv= 0, and then(u,v)Y= 0, which is a contradiction because(u,v)Y=r2> 0. Then (u,v)≤Q(u,v) for all (u,v)∈∂Br2∩P. By Theorem2.3(a),
we conclude that
i(Q,Br2∩P,P) = 1. (31)
BecauseQhas no fixed points on∂Br2∪∂BR2, by (27) and (31), we deduce that
iQ, (BR2\Br2)∩P,P
=i(Q,BR2∩P,P) –i(Q,Br2∩P,P) = –1.
So the operatorQhas at least one fixed point (u2,v2)∈(BR2\Br2)∩P, withr2<(u
2,v2) Y< R2, which is a positive solution for our problem (S)–(BC).
Theorem 3.3 Assume that assumptions(H1), (H2), (H3), (H5),and(H7)hold.Then prob-lem(S)–(BC)has at least two positive solutions.
Proof By using (H7), for any (u,v)∈∂BN0∩P, we obtain
Q1n(u,v)≤A N–1
i=1
h(i)f(i,N0,N0) < 3AN0 (N2– 1)max{A,B}
N–1
i=1
h(i)≤N0
2 , ∀n= 0,N,
Q2n(u,v)≤B N–1
i=1
h(i)g(i,N0,N0) <
3BN0 (N2– 1)max{A,B}
N–1
i=1
h(i)≤N0
2 , ∀n= 0,N.
Then we deduce
Q(u,v)Y =Q1(u,v)+Q2(u,v)<N0=(u,v)Y, ∀(u,v)∈∂BN0∩P.
BecauseQhas no fixed points on∂BN0, by Theorem2.1we conclude that
i(Q,BN0∩P,P) = 1. (32)
On the other hand, from (H3) and (H5), and the proofs of Theorems3.1and3.2, we know that there exist a sufficientlyr1> 0 (r1<N0) and a sufficiently largeR2>N0 such that
BecauseQhas no fixed points on∂Br1∪∂BR2∪∂BN0, by relations (32) and (33), we obtain
iQ, (BR2\ ¯BN0)∩P,P
=i(Q,BR2∩P,P) –i(Q,BN0∩P,P) = –1,
iQ, (BN0\ ¯Br1)∩P,P
=i(Q,BN0∩P,P) –i(Q,Br1∩P,P) = 1.
ThenQhas at least one fixed point (u1,v1)∈(BR2\ ¯BN0)∩Pand has at least one fixed
point (u2,v2)∈(B
N0\ ¯Br1)∩P. Therefore, problem (S)–(BC) has two distinct positive
so-lutions (u1,v1), (u2,v2).
Remark3.1 In (H3), ifa(v) =vpwithp≤1 andb(u) =uqwithq> 0, the condition from (H3)(c) is satisfied ifpq< 1. In (H5), ifc(v) =vp withp≤1, andd(u) =uqwithq> 0, the condition from (H5)(c) is satisfied ifpq> 1.
Examples
(1) We considerf(n,u,v) =nn+1(1 +e–(u+v))andg(n,u,v) = (1 +e–n)uθfor
(n,u,v)∈ {1, . . . ,N– 1} ×R+×R+. Fora(v) =vpwithp≤1, andb(u) =uqforq> 0
andpq< 1, then assumptions(H3)and(H4)are satisfied ifq>θandα2>θ. For
example, ifθ=54,p=13,q=43,α1=13, andα2= 3, we can apply Theorem3.1, and we deduce that problem (S)–(BC) has at least one positive solution.
(2) We considerf(n,u,v) = (1 +e–u)vθ1andg(n,u,v) = (1 +e–v)uθ2for
(n,u,v)∈ {1, . . . ,N– 1} ×R+×R+. Forc(v) =vpwithp≤1, andd(u) =uqforq> 0
andpq> 1, then assumptions(H5)and(H6)are satisfied ifp<θ1,q<θ2,β1<θ1,
andβ2<θ2. For example, ifθ1= 4,θ2= 2,p=35,q=95,β1= 3, andβ2=13, we can
apply Theorem3.2, and we conclude that problem (S)–(BC) has at least one positive solution.
Acknowledgements
The authors thank the referee for his/her valuable comments and suggestions.
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Author details
1Department of Mathematics, Texas A&M University, Kingsville, USA.2Department of Mathematics, Gh. Asachi Technical
University, Iasi, Romania.
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