Existence and Multiplicity Results for Nonlinear Differential … · 2019. 7. 31. · approximate method to study the following periodic boundary value problems: −x a t x λf t,x

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2012, Article ID 215617, 10 pagesdoi:10.1155/2012/215617

Research ArticleExistence and Multiplicity Results forNonlinear Differential Equations Depending ona Parameter in Semipositone Case

Hailong Zhu1 and Shengjun Li2

1 School of Statistics and Applied Mathematics, Anhui University of Finance and Economics,Bengbu 233030, China

2 College of Information Sciences and Technology, Hainan University, Haikou 570228, China

Correspondence should be addressed to Hailong Zhu, [email protected]

Received 27 September 2012; Accepted 13 October 2012

Academic Editor: Jifeng Chu

Copyright q 2012 H. Zhu and S. Li. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

The existence and multiplicity of solutions for second-order differential equations with aparameter are discussed in this paper. We are mainly concerned with the semipositone case. Theanalysis relies on the nonlinear alternative principle of Leray-Schauder and Krasnosel’skii’s fixedpoint theorem in cones.

1. Introduction

In this paper, we consider the problem of existence, multiplicity, and nonexistence of positivesolutions for the following boundary value problem (BVP):

−(a(t)x′)′ + b(t)x = λf(t, x), t ∈ I,

x(0) = x(2π), a(0)x′(0) = a(2π)x′(2π),(Eλ)

where I := [0, 2π], λ is a positive parameter, f(t, x) ∈ Car(I × R+,R), and a(t), b(t) are real-

valued measurable functions defined on [0, 2π] and satisfy the following condition:

a(t) > 0, b(t) ≥ 0, b(t)/≡ 0,∫2π

0

dt

a(t)< ∞,

∫2π

0b(t)dt < ∞. (H1)

2 Abstract and Applied Analysis

Here, the symbol Car(I×R+,R) denotes the set of functions satisfying the Carathedory

conditions on I × R+; that is,

(i) f(·, x) : I → R is Lebesgue integrable for each fixed x ∈ R+, and

(ii) f(t, ·) : R+ → R is continuous for a.e. t ∈ I.

Due to a wide range of applications in physics and engineering, second-order bound-ary value problems have been extensively investigated by numerous researchers in recentyears. For a small sample of such work, we refer the reader to [1–18] and the referencestherein. When a(t) = 1, b(t) = m2, λ = 1 of (Eλ), in [11, 18], by using Krasnosel’skii’sfixed point theorem, the existence and multiplicity of positive solutions are established tothe periodic boundary value problem:

−x′′ +m2x = f(t, x), t ∈ I,

x(0) = x(2π), x′(0) = x′(2π),(1.1)

where f(t, x) ∈ Car(I × R+,R+).

In [8], Graef et al. consider the second-order periodic boundary value problem:

−x′′ +m2x = λg(t)f(x), t ∈ I,

x(0) = x(2π), x′(0) = x′(2π),(1.2)

where g : I → R+ is continuous and f : R

+ → R+ is continuous and f(x) > 0 for x > 0. Under

different combinations of superlinearity and sublinearity of the function f , various existence,multiplicity, and nonexistence results for positive solutions are derived in terms of differentvalue of λ via Krasnosel’skii’s fixed point theorem.

Hao et al. [9] use the Global continuation theorem, fixed point index theory, andapproximate method to study the following periodic boundary value problems:

−x′′ + a(t)x = λf(t, x), t ∈ I,

x(0) = x(2π), x′(0) = x′(2π),(1.3)

where a ∈ L1(0, 2π) and f(t, x) ∈ Car(I × R+,R+).

In [10], by using the fixed point index theory, He et al. study the existence andmultiplicity of positive solutions to BVP (Eλ). Motivated by the above works, we establishthe results of existence, multiplicity, and nonexistence of positive solutions for BVP (Eλ)via Leray-Schauder alternative principle and Krasnosel’skii’s fixed point in the semipositonecase, that is, f(t, x) +M > 0 for some M > 0. Notice that we do not need f(t, x) > 0 for anyt ∈ [0, 2π] and x > 0, which is an essential condition of [9, 10].

The main result of the present paper is summarized as follows.

Theorem 1.1. Assume that

f0 := limx→ 0+

maxt∈[0,2π]

f(t, x)x

< ∞, f∞ := limx→+∞

maxt∈[0,2π]

f(t, x)x

< ∞. (1.4)

Abstract and Applied Analysis 3

Then, there exist 0 < λ < λ such that (Eλ) has no positive solution for λ < λ and at least two positivesolutions for λ ≥ λ.

Remark 1.2. The main result above is a generalization of [9, Theorem 1.2] and [10, Theorem1.2] and some other known results, in which f0 and f∞ must be zero, besides f(t, x) ispositive.

The remaining part of the paper is organized as follows. Some preliminary results willbe given in Section 2. In Section 3, existence results are obtained using a nonlinear alternativeof Leray-Schauder and fixed point theorem in cones when λ is large enough; the proof ofTheorem 1.1 is also given.

2. Preliminaries and Lemmas

In this section, we present some preliminary results which will be needed in subsequentsections. Denote by u(x) and v(x) the solutions of the corresponding homogeneous equation:

−(a(t)x′)′ + b(t)x = 0, t ∈ I, (2.1)

under the initial conditions

u(0) = 1, a(0)u(0) = 0, v(0) = 0, a(0)v(0) = 1. (2.2)

Lemma 2.1 (see [2, Theorem 2.4], [10, Lemma 2.1]). Assume that (H1) holds and h ∈ C(I,R+).Then for the solution x(t) of the BVP

−(a(t)x′)′ + b(t)x = h(t), t ∈ I,

x(0) = x(2π), a(0)x′(0) = a(2π)x′(2π),(2.3)

the formula

x(t) = (Lh)(t) :=∫2π

0G(t, s)h(s)ds, t ∈ I (2.4)

holds, where

G(t, s) =v(2π)D

u(t)u(s) − a(2π)u′(2π)D

v(t)v(s)

+

⎧⎪⎪⎨

⎪⎪⎩

a(2π)v′(2π) − 1D

u(t)v(s) − u(2π) − 1D

u(s)v(t), 0 ≤ s ≤ t ≤ 2π,

a(2π)v′(2π) − 1D

u(s)v(t) − u(2π) − 1D

u(t)v(s), 0 ≤ t ≤ s ≤ 2π,

(2.5)

and D = u(2π) + a(2π)v′(2π) − 1 > 0.


Lemma 2.2 (see [2, Theorem 2.5], [10, Lemma 2.2]). Under condition (H1), the Green’s functionof the BVP (2.3) is positive, that is, G(t, s) > 0 for t, s ∈ I.

Remark 2.3. We denote

A = min0≤s,t≤2π

G(t, s), B = max0≤s,t≤2π

G(t, s), σ =A

B. (2.6)

Thus, B > A > 0 and 0 < σ < 1. In this paper, we use ω(t) to denote the unique periodicsolution of (2.3)with h(t) = 1, that is, ω(t) = (L1)(t). Obviously, A ≤ ‖ω‖∞/2π ≤ B.

Remark 2.4. If a(t) = 1, b(t) = m2 > 0, then the Green’s function G(t, s) of the boundary valueproblem (2.3) has the form

G(t, s) = G(|t, s|) =

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

exp(m(t − s)) + exp(m(2π − t + s))2m

(exp(2mπ) − 1

) , 0 ≤ s ≤ t ≤ 2π,

exp(m(s − t)) + exp(m(2π − s + t))2m

(exp(2mπ) − 1

) , 0 ≤ t ≤ s ≤ 2π.(2.7)

It is obvious that G(t, s) > 0 for 0 ≤ s, t ≤ 2π , and a direct calculation shows that

A =emπ

m(e2mπ − 1), B =

1 + e2mπ

2m(e2mπ − 1), σ =

2emπ

1 + e2mπ< 1. (2.8)

In the obtention of the second periodic solution of (Eλ), we need the following well-known fixed point theorem of compression and expansion of cones [19].

Lemma 2.5 (see Krasnosel’skii [19]). Let X be a Banach space and K(⊂ X) a cone. Assume thatΩ1, Ω2 are open subsets of X with 0 ∈ Ω1, Ω1 ⊂ Ω2, and let

T : K ∩(Ω2 \Ω1

)−→ K (2.9)

be a continuous and compact operator such that either

(i) ‖Tu‖ ≥ ‖u‖, u ∈ K ∩ ∂Ω1 and ‖Tu‖ ≤ ‖u‖, u ∈ K ∩ ∂Ω2, or

(ii) ‖Tu‖ ≤ ‖u‖, u ∈ K ∩ ∂Ω1 and ‖Tu‖ ≥ ‖u‖, u ∈ K ∩ ∂Ω2.

Then T has a fixed point in K ∩ (Ω2 \Ω1).

In the applications below, we take X = C[0, 2π] with the supremum norm ‖ · ‖ anddefine

K ={x ∈ X : x(t) ≥ 0 ∀t, min

0≤t≤2πx(t) ≥ σ‖x‖

}, (2.10)

where ‖x(t)‖ = max0≤t≤2π |x(t)|.


Onemay readily verify thatK is a cone inX. Finally, we define an operator T : X → Kby

(Tx)(t) =∫2π

0G(t, s)F(s, x(s))ds, (2.11)

for x ∈ X and t ∈ [0, 2π], where F : [0, 2π] × R → [0,∞) is continuous and G(t, s) is theGreen function defined above.

Lemma 2.6 (see [12, Lemmas 2.2, 2.3], [13, Lemma 2.4]). T is well defined and maps X into K.Moreover, T : X → K is continuous and completely continuous.

3. Proof of Theorem 1.1

In this section we establish the existence, multiplicity, and nonexistence of positive solutionsto the periodic boundary problem (Eλ). The first existence result is based on the followingnonlinear alternative of Leray-Schauder, which can be found in [15].

Lemma 3.1. Assume Ω is a relatively compact subset of a convex set K in a normed space X. LetT : Ω → K be a compact map with 0 ∈ Ω. Then one of the following two conclusions holds:

(I) T has at least one fixed point in Ω.

(II) There exist x ∈ Ω and 0 < λ < 1 such that x = λTx.

Since we are mainly interested in the semipositone case, without loss of generality, wemay assume that f(t, x) satisfies the following.

(F1) There is a constantM > 0 such that f(t, x)+M > 0 for all (t, x) ∈ [0, 2π]×(0,∞) andlet F(t, x) := λ(f(t, x) +M) > 0. Besides, we introduce the following assumption onf(t, x).

(F2) there exists a continuous, nonnegative function g(x) on (0,∞) such that

f(t, x) ≤ g(x), ∀(t, x) ∈ [0, 2π] × (0,∞), (3.1)

that is,

F(t, x) ≤ λ(g(x) +M

), ∀(t, x) ∈ [0, 2π] × (0,∞), (3.2)

and g(x) > 0 is nondecreasing in x ∈ (0,∞).

Theorem 3.2. Suppose f(t, x) satisfies (F1) and (F2). Suppose further that(F3) there exists r > M‖ω‖/σ such that

r

λ(g(r) +M

) > ‖ω‖, (3.3)

where σ and ω are as in Section 2.


Then (Eλ) has at least one positive periodic solution with 0 < ‖x +Mω‖ < r.

Proof. The existence is proved using the Leray-Schauder alternative principle. Consider thefollowing equation:

−(a(t)x′)′ + b(t)x = μF(t, x(t) −Mω(t)), t ∈ I,

x(0) = x(2π), a(0)x′(0) = a(2π)x′(2π),(3.4)

where μ ∈ [0, 1]. Problem (3.4) is equivalent to the following fixed point problem in C[0, 2π]:

x = μTx, (3.5)

where T denotes the operator defined by (2.11), with F(t, x) replaced by F(t, x −Mω).We claim that any fixed point x of (3.5) for any μ ∈ [0, 1] must satisfy ‖x‖/= r.Then we have from condition (F2), for all t ∈ I,

x(t) = μTx(t)

= μ

∫2π

0G(t, s)F(s, x(s) −Mω(s))ds

≤∫2π


≤∫2π

0G(t, s)

(λ(g(x −Mω) +M

))ds

≤ λ(g(r) +M

)‖ω‖.

(3.6)

Therefore,

r = ‖x‖ ≤ λ(g(r) +M

)‖ω‖. (3.7)

This is a contradiction to the condition (F3). From this claim, the nonlinear alternative ofLeray-Schauder guarantees that (3.5) (with μ = 1) has a fixed point, denoted by x1(t), that is,

−(a(t)x′1

)′ + b(t)x1 = λ(f(t, x1(t) −Mω(t)) +M

), t ∈ I,

x1(0) = x1(2π), a(0)x′1(0) = a(2π)x′

1(2π).(3.8)

Using Lemma 2.5 and condition (F3), for all t ∈ I, we have

x1(t) ≥ σ‖x1‖ = σr > σ · M‖ω‖σ

= M‖ω‖ > 0, (3.9)


that is,

x1(t) −M‖ω‖ > 0. (3.10)

Let

x∗1(t) = x1(t) −Mω. (3.11)

It is easy to see that x∗1(t) is a solution of (Eλ) which satisfies 0 < ‖x∗

1 + Mω‖ < r. Thus, theproof of Theorem 3.2 is completed.

Theorem 3.3. Suppose that conditions (F1)–(F3) hold. In addition, it is assumed that the followingtwo conditions are satisfied.

(F4) There exists a continuous, nonnegative function h(x) on (0,∞) such that

f(t, x) +M ≥ h(x), ∀(t, x) ∈ [0, 2π] × (0,∞), (3.12)

that is,

F(t, x) ≥ λh(x), ∀(t, x) ∈ [0, 2π] × (0,∞), (3.13)

and h(x) > 0 is nondecreasing in x ∈ (0,∞).(F5) There exists a positive number R > r such that

R

λh(σR −M‖ω‖) ≤ ‖ω‖. (3.14)

Then, besides the periodic solution x constructed in Theorem 3.2, (Eλ) has another positiveperiodic solution x with r < ‖x +Mω‖ < R.

Proof. As in the proof of Theorem 3.2, we only need to show that (3.8) has a periodic solutionwith x2 ∈ C[0, 2π] with x2 > Mω and r < ‖x2‖ < R.

Let X = C[0, 2π] and K the cone in X in Section 2. Let Ω1 = Br and Ω2 = BR be ballsin X. The operator T : K ∩ (Ω2 \ Ω1) → K is defined by (2.11), with F(t, x) replaced byF(t, x − Mω). Note that any x ∈ K ∩ (Ω2 \ Ω1) satisfies 0 < σr ≤ x(t) ≤ R, thus T is welldefined.

First we have ‖Tx‖ ≤ ‖x‖ for x ∈ K ∩ ∂Ω1. In fact, if x ∈ K ∩ ∂Ω1, then ‖x‖ = r. Nowthe estimate ‖Tx‖ ≤ r can be obtained almost following the same ideas in proving (3.7). Weomit the details here.


Next we show that ‖Tx‖ ≥ ‖x‖ for x ∈ K ∩ ∂Ω2. To see this, let x ∈ K ∩ ∂Ω2, then‖x‖ = R and x ≥ σR; it follows from conditions (F4) and (F5) that, for 0 ≥ t ≥ 2π ,

Tx(t) =∫2π


≥∫2π

0G(t, s)(λ(h(x −Mω)))ds

≥ λh(σR −M‖ω‖)‖ω‖ ≥ R = ‖x‖.

(3.15)

Now Lemma 2.5 guarantees that T has a fixed point x2 ∈ K ∩ (Ω2 \ Ω1), thus r ≤‖x2(t)‖ ≤ R.

Finally, x∗2(t) = x2(t) − Mω will be the another desired positive periodic solution of

(Eλ). We omit the details because they are much similar to that in the proof of Theorem 3.2.

Now we are in a position to present the proof of Theorem 1.1.

Proof of Theorem 1.1. Consider v(x) > 0 be an eigenfunction satisfying

−(a(t)v′)′ + b(t)v = λ1v, t ∈ I,

v(0) = v(2π), a(0)v′(0) = a(2π)v′(2π),(3.16)

corresponding to the principal eigenvalue λ1. Let x be a positive solution of (Eλ). Multiplying(3.16) by x and (Eλ) by v, and subtracting we obtain

∫2π

0

(λf(t, x) − λ1x

)v dx = 0. (3.17)

Since f0 < ∞ and f∞ < ∞, there exist positive numbers η1, η2, ε1, and ε2 such thatε1 < ε2 and

∣∣f(t, x)∣∣ ≤ η1x for x ∈ [0, ε1],

∣∣f(t, x)∣∣ ≤ η2x for x ∈ [ε2,∞),

(3.18)

with t ∈ I. Let the positive number η3 be defined by

η3 = max{η1, η2, max

ε1<x<ε2

{∣∣∣∣f(t, x)

x

∣∣∣∣

}}. (3.19)

Then

∣∣f(t, x)∣∣ ≤ η3x for x ∈ [0,∞). (3.20)


Thus, there exists a λ > 0, for 0 < λ < λ satisfying |λ1/λ| > η3. (3.17) cannot hold, andhence (Eλ) has no positive solution for λ < λ.

Note that the sublinearity of f(t, x) near x = ∞, we can construct a suitable g(x) in(F2) which satisfies limr→∞g(r)/r < ∞. This means that there exists λ > λ1 satisfying (3.3)with r being large enough. There also exists λ > λ2 = R/‖ω‖(h(σR−M‖ω‖)) satisfying (3.14).Let λ = max(λ1, λ2). Thus, with the help of Theorems 3.2 and 3.3, (Eλ) has at least two positivesolution for λ > λ. This completes the proof of the theorem.

Example 3.4. Let the nonlinearity in (Eλ) be

f(t, x) = α(t)g(x) exp(−xγ), (3.21)

with γ > 0, α(t) is a continuous function for all t ∈ I and g(x) is a real coefficient polynomialfunction which has zero constant term. Then Theorem 1.1 is valid.

Proof. In this case, with the function f(t, x) = α(t)g(x) exp(−xγ), it is easy to verify

f0 := limx→ 0+

maxt∈[0,2π]

f(t, x)x

= limx→ 0+

maxt∈[0,2π]

α(t)g(x) exp(−xγ)x

< ∞,

f∞ := limx→+∞

maxt∈[0,2π]

f(t, x)x

= limx→+∞

maxt∈[0,2π]

α(t)g(x) exp(−xγ)x

= 0 < ∞.

(3.22)

Then the conclusion follows from Theorem 1.1 that there exists 0 < λ < λ such that(Eλ) has no positive solution for λ < λ and at least two positive solutions for λ ≥ λ.

Acknowledgments

H. Zhu was supported by the NSF of the Educational Bureau of Anhui province (nos.KJ2012B002, KJ2012B004, and 1208085QA11). S. Li was supported by the NSF of China (no.11161017).

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Existence and Multiplicity Results for Nonlinear Differential … · 2019. 7. 31. · approximate method to study the following periodic boundary value problems: −x a t x λf t,x