Solvability of Second Order Delta-Nabla p-Laplacian m-point Eigenvalue Problem on Time Scales

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SOLVABILITY OF SECOND ORDER DELTA-NABLA p-LAPLACIAN

m-POINT EIGENVALUE PROBLEM ON TIME SCALES

SABBAVARAPU NAGESWARA RAO1, §

Abstract. In this paper, we are concerned with the following eigenvalue problem of m-point boundary value problem forp-Laplacian dynamic equation on time scales,

(

ϕp(u∆(t))

)∇

+λh(t)f(u(t))= 0, t∈[a, b]_T, u(a)−u∆(a) =

m∑−2

i=1

u∆(ξi), u∆(b) = 0, m≥3,

whereϕp(u) =|u|p−2u, p >1 andλ >0 is a real parameter. Under certain assumptions,

some new results on existence of one or two positive solutions and nonexistence are obtained for λ evaluated in diﬀerent intervals by using Guo-Krasnosel’skii ﬁxed point theorem.

Keywords: eigenvalue, time scale,p-Laplacian, positive solution, ﬁxed point, cone. AMS Subject Classiﬁcation: 34B18, 34A34

1. Introduction

The theory of dynamic equations on time scales was introduced ﬁrstly by Stefan Hilger in 1988. Since then, more and more scholars are interested in this area. The main reason is that the time scale theory can not only unite continuous and discrete dynamic equa-tions but also have important applicaequa-tions, for example, in the study of insect population models, neural networks, heat transfer, economic, stock market, and epidemic models. Throughout this work we assume a working knowledge of time scales and time scales natation, where any nonempty closed subset ofR can serve as a time scale T,see Hilger [18], Bohner and Peterson [8]

Very recently, there is an increasing attention paid to question of positive solution for second order boundary value problem on time scales [4, 5, 9, 10, 11, 22]. But very little work has been done to the existence of positive solutions forp-Laplacian boundary value problem on time scales [6, 24, 17]. In particular, we would like to mention some results of Agarwal, L¨u and O’Regan [3], Anderson, Avery and Henderson [6], Fan and Li [12], Guo and Sun [14], Goodrich [15, 16], Nageswararao [21], Prasad, Nageswararao and Murali [22], Sun and Li [23, 24], Sun, Tang and Wang [25], which are motivate us to consider our problem.

1 _{Department of Mathematics, Jazan University, Jazan, Kingdom of Saudi Arabia.}

e-mail: [email protected]; § Manuscript received: July 04, 2014.

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In this paper we consider the eigenvalue problem of m-point boundary value problem for the one-dimensionalp-Laplacian dynamic equation on time scales

(

ϕp(u∆(t))

)_∇

+λh(t)f(u(t)) = 0, t∈[a, b]_T (1)

u(a)−u∆(a) =

m_∑−2

i=1

u∆(ξi), u∆(b) = 0, m≥3. (2)

Under certain assumptions, results on existence of one or two positive solutions and nonex-istence are obtained forλevaluated in diﬀerent intervals.

For convenience, throughout this paper, we denoteϕp(u) as thep-Laplacian operator, i.e.,

ϕp(u) =|u|p−2u forp >1 with (ϕp)−1=ϕq,where 1_p +1_q = 1.

We make the following assumptions throughout:

(A1) f ∈ C([0,∞),[0,∞)) does not vanish identically on any closed subinterval of [a, b]_T.

(A2) λ >0 is a parameter, the function h: (a, b)→[0,∞) is left-dense continuous such that h(t0)>0 for at least one t0 ∈[a, b) and 0<

∫b

ah(τ)∇τ <∞.

We deﬁne the positive extended real numbersf0, f0, f∞ and f∞ by f0 = lim sup

u→0+

f(u) ϕp(u)

, f0= lim inf

u→0+

f(u) ϕp(u)

, f∞= lim sup

u→∞

f(u) ϕp(u)

and f_∞= lim inf

u→∞

f(u) ϕp(u)

assume that they will exist.

The rest of this paper is organized as follows. In Section 2, we shall provide some preliminaries. For convenience, we also state the Krasnosel’skii’s ﬁxed point theorem in a cone. In Section 3, we establish a criteria to determine eigenvalue intervals for which there exist at least one or two positive solutions. In the last Section, we will consider the conditions of the nonexistence of the positive solution. An example is also given to illustrate the main results.

2. Preliminaries

Let the Banach space B=Cld[a, b] be endowed with the norm

∥u∥= sup

t∈[a,b]_T

|u(t)|,

and choose the coneP ⊂ B deﬁned by

P =

{

u∈ B :u(t)≥0, on [a, b]_T and u∆∇(t)≤0, u∆(t)≥0,

fort∈[a, b]_T, u(a)−u∆(a) =

m∑−2

i=1

u∆(ξi), m≥3

}

clearly, we can obtain∥u∥=u(b) for u∈ P.

A function u : T → R is said to be a solution of the BVP (1)-(2) provided that u is delta diﬀerential,u∆and(ϕp(u∆(t))

)_∇

are continuous on [a, b]_T, andu satisﬁes the BVP (1)-(2).

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Lemma 2.1. Assume that (A1) and (A2) are satisfied. Then u(t) is the solution of the BVP (1)-(2)if and only if

u(t) =ϕq

( ∫ b

a

λh(τ)f(u(τ))∇τ

)

+

m_∑−2

i=1 ϕq

( ∫ b

ξi

)

+

∫ t

a

ϕq

( ∫ b

s

)

∆s.

Proof. Firstly, we prove the necessity. Let u(t) be the solution of the BVP (1)-(2). ∇ -integrating on (1) from t to b, we haveϕp(u∆(b))−ϕp(u∆(t)) =−

∫b

t λh(τ)f(u(τ))∇τ.By

using second boundary condition, we get u∆(t) =ϕq

( ∫b

t λh(τ)f(u(τ))∇τ

)

∆-integrating

from a to t, we getu(t)−u(a) =∫_atϕq

( ∫b

s λh(τ)f(u(τ))

)

∆τ. Therefore,

u(t) =u(a) +

∫ t

a

ϕq

( ∫ b

s

λh(τ)f(u(τ))

)

∆τ.

We have u∆_{(a) =} _ϕ

q

( ∫b

aλh(τ)f(u(τ))∇τ

)

, u∆_(ξ

i) = ϕq

( ∫b

ξiλh(τ)f(u(τ))∇τ )

. Now we submittingu∆(a), u∆(ξi) into the ﬁrst boundary condition of the (2), we obtain

u(a) =ϕq

( ∫ b

a

)

+

m_∑−2

i=1 ϕq

( ∫ b

ξi

)

. Thus, we have

u(t) =ϕq

( ∫ b

a

)

+

m_∑−2

i=1 ϕq

( ∫ b

ξi

)

+

∫ t

a

ϕq

( ∫ b

s

)

∆s.

Secondly, we can prove the suﬃciency easily.

Deﬁne the operator T :P → B, foru∈ P,by (T u)(t) =ϕq

( ∫ b

a

)

+

m_∑−2

i=1 ϕq

( ∫ b

ξi

)

+

∫ t

a

ϕq

( ∫ b

s

)

∆s.

(3)

By using Lemma 2.1,u(t) is the solution of the BVP (1)-(2) if and only ifu(t) = (T u)(t).

Lemma 2.2. Assume that (A1) and (A2) hold. T :P → P is a completely continuous operator.

Proof. Firstly we proveT :P → P.For u∈ P,by Lemma 2.1, (A1),and (A2),we obtain T u(t)≥0, t∈[a, b]_T,

(T u)∆(t) =ϕq

( ∫ b

t

)

, (T u)∆∇(t)≤0,

(T u)(a)−(T u)∆(a) =

m_∑−2

i=1 ϕq

( ∫ b

ξi

)

=

m_∑−2

i=1

(4)

which impliesT u∈ P.

Secondly, T maps a bounded set into a bounded set. Assume c > 0 is a constant and u∈Pc={x∈P :∥x∥≤c}.Note that the continuity of f(u) is continuous, there exists a

C >0 such thatf(u)≤ϕp(C). Hence, fort∈[a, b]T, u∈Pc, we have

|(T u)∆(t)|=ϕq

( ∫ b

t

λh(τ)f(u(τ))∇τ)< Cϕq(λ)ϕq

( ∫ b

a

h(τ)∇τ

)

<∞,

|(T u)(t)|=ϕq

( ∫ b

a

)

+

m_∑−2

i=1 ϕq

( ∫ b

ξi

)

+

∫ t

a

ϕq

( ∫ b

s

)

∆s

≤C(m−1 +b)ϕq(λ)ϕq

( ∫ b

a

h(τ)∇τ

)

<∞. Consequently,T Pcis bounded.

Fort1, t2 ∈[a, b]T, u∈Pc,we get

|(T u)(t1)−(T u)(t2)| ≤

∫ t2

t1

ϕq

( ∫ b

s

)

∆s

≤ ∫ t2

t1

ϕq

( ∫ b

s

λh(τ)∇τ

)

∆s

≤ |t1−t2|Cϕq(λ)ϕq

( ∫ b

a

h(τ)∇τ

)

→0, t1 →t2.

So, by applying Arzela-Ascoli theorem on time scales [2], we obtain thatT Pcis relatively

compact. In view of Lebesgue’s dominated convergence theorem on time scales [7], it is easy to prove thatT is continuous. Hence, T is completely continuous.

Theorem 2.1. [Krasnosel’skii] [13, 20] Let B be a Banach space, and let P ⊂ B be a cone inB. Assume that Ω1 and Ω2 are open subsets of B with0∈Ω1 ⊂Ω1 ⊂Ω2, and let

T :P ∩(Ω2\Ω1)→ P

be a completely continuous operator such that either

(i) ∥T u∥≤∥u∥, u∈ P ∩∂Ω1, and ∥T u∥≥∥u∥, u∈ P ∩∂Ω2, or (ii) ∥T u∥geq ∥u∥, u∈ P ∩∂Ω1, and ∥T u∥≤∥u∥, u∈ P ∩∂Ω2. Then,T has a fixed point in P ∩(Ω2\Ω1).

3. Eigenvalue intervals

In this section, we shall apply Theorem 2.1 to derive explicit eigenvalue intervals, and also give suﬃcient conditions that the BVP (1)-(2) has at least one or two positive solu-tions.

To begin, we shall deﬁne some important constants

L1 = (m+b−a−1)ϕq

( ∫ b

a

h(τ)∇τ

)

L2 =

(_ξ

m−2−a b−a

)

(m+ξm−2−a−1)ϕq

( ∫ b

ξm−2

h(τ)∇τ

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Theorem 3.1. Suppose (A1)−(A2) hold. Then for each

(f_∞)−1ϕq

(_(ξ

m−2−a)L2 b−a

)

< λ <(f0)−1ϕq(L1), (4)

the BVP (1)-(2) has at least one positive solution. Here we impose

(f_∞)−1ϕq

(

(ξm−2−a)L2

b−a

)

= 0 if f_∞=∞ and (f0)−1ϕq(L1) =∞ if f0= 0.

Proof. Letλsatisfy (4) and ϵ >0 be such that (f_∞−ϵ)−1ϕq

(_(ξ_m₋₂₋_a)L2

b−a

)

≤λ≤(f0+ϵ)−1ϕq(L1). (5)

By the deﬁnition off0,we see that there existsr1 >0 such that

f(u)≤(f0+ϵ)ϕp(u), 0< u < r1. (6)

So, ifu∈∂Pr1,then by (5) and (6) we have ∥T u∥= (T u)(b)≤ϕq

( ∫ b

a

)

+ (m−2)ϕq

( ∫ b

a

)

+ (b−a)ϕq

( ∫ b

a

)

≤(m+b−a−1)u ϕq(λ(f0+ϵ))ϕq

( ∫ b

a

h(τ)∇τ

)

≤ϕq(λ(f0+ϵ))L1∥u∥≤∥u∥.

Hence if we let Ω1={u∈ B:∥u∥< r1},then

∥T u∥≤∥y∥, foru∈ P ∩∂Ω1. (7)

Letr3 >0 be such that

f(u)≥(f_∞−ϵ)ϕp(u), u≥r3. (8)

Ifu∈ B with∥u∥=r2= max

{

2r1, (b−a)r3

ξm−2−a }

.Then in view of (8)we have

∥T u∥ ≥(T u)(ξm−2)≥

(_ξ

m−2−a b−a

)

(T u)(b)

≥(ξm−2−a b−a

)[

ϕq

( ∫ b

a

)

+

m∑−2

i=1 ϕq

( ∫ b

ξi

)

+

∫ b

a

ϕq

( ∫ b

s

)

∆s

]

≥(ξm−2−a b−a

)[

ϕq

( ∫ b

ξm−2

)

+ (m−2)ϕq

( ∫ b

ξm−2

)

+

∫ ξm−2

a

ϕq

( ∫ b

ξm−2

)

∆s

]

≥(ξm−2−a b−a

)

(m+ξm−2−a−1)ϕq

( ∫ b

ξm−2

λh(τ)(f_∞−ϵ)ϕp(u(τ))∇τ

)

≥L2ϕq

(

λ(f_∞−ϵ))(ξm−2−a b−a

)

(6)

Thus if we set Ω2 ={u∈ B :∥u∥< r2},then

∥T u∥≤∥u∥, foru∈ P ∩∂Ω2. (9) Now, (7),(9) and Theorem 2.1 guarantee that T has a ﬁxed point P ∩(Ω2\Ω1) with r1 ≤∥u∥≤r2,and clearlyu is a positive solution of (1)-(2).

Theorem 3.2. Suppose (A1)−(A2) hold. Then for each

(f0)−1ϕq

(_(ξ

m−2−a)L2 b−a

)

< λ <(f∞)−1ϕq(L1), (10)

the BVP (1)-(2) has at least one positive solution.

Proof. Letλsatisfy (10) and ϵ >0 be such that (f0−ϵ)−1ϕq

(_(ξ

m−2−a)L2 b−a

)

≤λ≤(f∞+ϵ)−1ϕq(L1). (11) By the deﬁnition off0,we see that there existsr1 >0 such that

f(u)≤(f0−ϵ)ϕp(u), 0< u≤r1. Further, ifu∈ P with ∥u∥=r1,then u(t)≥

(

ξm−2−a

b−a

)

∥u∥, t∈[ξm−2, b]T,and similar to the second part of Theorem 3.1, we can obtain that ∥ T u ∥≥∥ u ∥ . Thus, if we let Ω1 ={u∈ B :∥u∥< r1},then

∥T u∥≥∥u∥, foru∈ P ∩∂Ω1. (12) Next, we may chooseR2 >0 such that

f(u)≤(f∞+ϵ)ϕp(u), u≥R2. (13)

Here there are two cases to consider, namely, wheref is bounded andf is unbounded.

Case (i). Supposef is bounded, then there exists someM >0,such thatf(u)≤M, u∈ (0,∞).We deﬁne r3 = max

{

2r1, ϕq(λM)L1,

}

and u∈ P be such that∥u∥=r3,then

∥T u∥= (T u)(b)

≤ϕq

( ∫ b

a

)

+ (m−2)ϕq

( ∫ b

a

)

+ (b−a)ϕq

( ∫ b

a

)

≤ϕq(λM)L1≤r3 =∥u∥.

Hence,

∥T u∥≤∥u∥, foru∈∂Pr3. (14)

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Letu∈ P be such that ∥u∥=r4,then by (11), we have

∥T u∥= (T u)(b)

≤ϕq

( ∫ b

a

)

+ (m−2)ϕq

( ∫ b

a

)

+ (b−a)ϕq

( ∫ b

a

)

≤ϕq(λ(f∞+ϵ))L1∥u∥≤∥u∥.

Thus, (14) is also true. In both Case (i) and Case (ii), if we set

Ω2 =

{

u∈ B:∥u∥< r2 = max{r3, r4}

}

,

then (13) hold foru∈ P ∩∂Ω2.Now that we have obtained (12) and (14), it follows from Theorem 2.1 thatT has a ﬁxed pointu∈ P ∩(Ω2\Ω1),andr1 ≤∥u∥≤r2.It is clear that

uis a positive solution of (1)-(2).

In the rest of this section, we consider the existence of two positive solutions of (1)-(2). First, we give lemma.

Lemma 3.1. Assume that(A1)and(A2)hold. In addition, assume there existr2> r1 > 0,such that

max 0≤u≤r1

f(u)≤ ϕp(

r1

L1)

λ (16)

min

(_ξm₋2−a)r2 b−a

≥f(u)≤r2

ϕp(_Lr2₂)

λ . (17)

Then,(1)-(2) has a solutionu∈ P with r1 ≤∥u∥≤r2.

Proof. The proof of Lemma 3.1 is similar to that of Theorem 3.2, we omit it here.

For the remainder of the paper, we will need the following condition: (A3)

sup

r>0

min

u∈((ξm−_b₋2_a−a)r, r)

f(u)>0.

λ1 = sup

r>0

ϕp(_Lr₁)

max0_≤u≤rf(u)

, (18)

λ2= inf

r>0

ϕp(_Lr₂)

min((_ξm₋₂−a)r b−a

)

≤u≤rf(u)

, (19)

In view of (A1) and (A3),we can easily obtain that 0< λ1 ≤ ∞and 0≤λ2 <∞.

Theorem 3.3. Suppose (A1)−(A3)hold, iff0 =∞ and f_∞=∞, then the BVP (1)-(2)

has at least two positive solutions for each λ∈(0, λ1).

Proof. Deﬁne

a(r) = ϕp(

r L1)

max0≤u≤rf(u)

, r >0,

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a(r0) = supr>0a(r) =λ1,then for λ∈(0, λ1),there exist constants c1, c2 (0< c1 < r0 < c2 <∞) witha(c1) =a(c2) =λ.Thus

f(u)≤ ϕp(

c1

L1)

λ foru∈[0, c1], (20)

f(u)≤ ϕp(

c2

L1)

λ foru∈[0, c2]. (21)

On the other hand, applying the condition f0 = ∞ and f∞ = ∞, there exist constants d1, d2 (0< d1 < c1 < r0< c2< d2 <∞),with

f(u) ϕp(u) ≥

λ−1ϕq

(_ξ

m−2−a)L2 b−a

)

foru∈(0, d1)∪

(_(ξ

m−2−a)d2 b−a ,∞

)

(22)

then

min

(_ξm₋₂−a)d₁ b−a ≤u≤d1

f(u)≥λ−1ϕp

(_d

1 L2

)

(23)

min

(_ξm₋₂−a)d₂ b−a ≤u≤d2

f(u)≥λ−1ϕp

(_d2

L2

)

. (24)

By (20) and (23), (21) and (24), Lemma 3.1, we can complete the proof.

Theorem 3.4. Suppose(A1)−(A3)hold, if f0= 0 andf_∞= 0,then for eachλ∈(λ2,∞)

the BVP (1)-(2) has at least two positive solutions.

Proof. Deﬁne

b(r) = ϕp(

r L2)

min(_ξm₋2−a)r b−a ≤u≤r

f(u), r∈(0,∞).

Byf0 = 0 and f∞= 0 we easily see thatb(r) : (0,∞)→(0,∞) is continuous and lim

r→0b(r) = limr→∞b(r) =∞.

Thus there exists r0 ∈(0,∞), such that b(r0) = infr>0b(r) =λ2.For λ∈(λ2,∞), there exist constantsd1, d2(0< d1 < r0 < d2 <∞) withb(d1) =b(d2) =λ.Therefore

f(u)≥ ϕp(

d1

L1)

λ foru∈

[_(ξ

m−2−a)d1 b−a , d1

]

,

f(u)≥ ϕp(

d2

L2)

λ foru∈

[_(ξ_m₋₂₋_a)d2

b−a , d2

]

,

On the other hand, usingf0 = 0,we know that there is a constantc1(0< c1 < d1) with f(u)

ϕp(u) ≤

λ−1ϕq(L1) for u∈(0, c1)

max 0≤u≤c1

f(u)≤ ϕp

(

c1

L1 )

λ . (25)

In view off_∞= 0,there exits a constantc2 ∈(d2,∞) such that f(u)

ϕp(u) ≤

ϕq(L1λ) for u∈(c2,∞).

LetM = sup_u_∈_[0_,c₂_]f(u) and c2 ≥L1ϕq(λM).It is easily seen that

max 0≤u≤c2

f(u)≤ ϕp(

c2

L1)

(9)

By (25), (26) and Lemma 3.1, we can complete the proof.

4. Nonexistence

In this section, we give some suﬃcient conditions for the nonexistence of positive solution to the BVP (1)-(2).

Theorem 4.1. Suppose (A1)−(A3)hold. If f0 <∞, f∞<∞,then there exists a λ0 >0

such that for all 0< λ < λ0,(1)-(2) has no positive solution. Proof. Since f0 _<_∞ _and_f∞_<_∞_,_{there exist positive numbers}_l

1, l2, r1 and r2 such that r1 < r2 and

f(u)≤l1ϕp(u) f or u∈[0, r1],

f(u)≤l2ϕp(u)f or u∈[r1,∞],

LetL= max

{

l1, l2,maxr1≤u≤r2{f(u)ϕq(u)} }

,then we have

f(u)≤Lϕp(u) for u∈[0,∞)

Assume thatv(t) is a positive solution of BVP (1)-(2). We will show that this leads to a contradiction for 0< λ < λ0 =L−1ϕq(L1).SinceT v(t) =v(t) for t∈[a, b]T,then

∥T v∥= (T v)(b)≤ϕq

( ∫ b

a

λh(τ)f(v(τ))∇τ

)

+ (m−2)ϕq

( ∫ b

a

)

+ (b−a)ϕq

( ∫ b

a

)

≤ϕq(λL)∥v∥L1 <∥v∥.

which is a contradiction. Therefore, (1)-(2) has no positive solutions.

Theorem 4.2. Suppose (A1)−(A3) hold, if f0 >0, f_∞ > 0, then there exist a λ0 > 0

such that for all λ > λ0,(1)-(2) has no positive solution.

Proof. By f0 >0, f∞ >0, we know that there existm1, m2, r1 and r2 such that r1 < r2 and

f(u)≥m1ϕp(u) foru∈[0, r1], f(u)≥m2ϕp(u) for u∈[r2,∞).

Letm3 = min

{

m1, m2,minr1≤u≤r2{f(u)ϕq(u)} }

>0,then we get

(10)

Assumev(t) is a positive solution of (1)-(2). We will show that this leads to a contradiction forλ > λ0= (m3)−1ϕp

(

(ξm−2−a)L2

b−a

)

.Since T v(t) =v(t) fort∈[a, b]_T,then

∥v∥ ≥v(ξm−2) = (T v)(ξm−2)≥

(_ξ

m−2−a b−a

)

(T v)(b)

≥(ξm−2−a b−a

)[

ϕq

( ∫ b

a

)

+

m_∑−2

i=1 ϕq

( ∫ b

ξi

)

+

∫ b

a

ϕq

( ∫ b

s

)

∆s

]

≥(ξm−2−a b−a

)[

ϕq

( ∫ b

ξm−2

)

+ (m−2)ϕq

( ∫ b

ξm−2

)

+

∫ ξm−2

a

ϕq

( ∫ b

ξm−2

)

∆s

]

=

(_ξ

m−2−a b−a

)

(m+ξm−2−a−1)ϕq

( ∫ b

ξm−2

)

≥ϕq

(

λm3)(ξm−2−a b−a

)

∥v∥L2 =∥v∥

which is a contradiction. Thus, (1)-(2) has no positive solution.

Example 4.1. Let T ={1−(₂1)N0} ∪ {₁}_, _where N

0 denotes the set of all nonnegative

integers. Takinga= 0, b= 1, p= 2, m= 3, ξ1 = 1₂,if we let h(s) = 1, thenL1 = 2, L2= ₂3.Suppose f(u) = 1+49₁₊_uu

(

3

2 + sinu

)

, u≥0,and

f0 =f0 = 3₂, f∞= 235, f_∞= 45.

By direct calculation, it is easy to get that

(f_∞)−1ϕq

(_(ξ1₋_a)L2

b−a

)

= 0.029, (f0)−1ϕq(L1) = 0.334,

L−1ϕq(L1) = 0.0122 and (m3)−1ϕp

(_(ξ

1−a)L2 b−a

)

= 4.

Thus the boundary value problem

u∆∇(t) +λf(u(t)) = 0, t∈[0,1]_T, u(0)−u∆(0) =u∆

(₁

2

)

, u∆(1) = 0

has at least one positive solutions for0.029< λ <0.334,has no solution on

P for0< λ <0.0122 orλ >4by Theorem 3.1, Theorem 4.1 and Theorem 4.2 respectively. Acknowledgement: The author thanks the referees for their valuable suggestions.

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