Positive solutions of Neumann problems with singularities

J. Math. Anal. Appl. 337 (2008) 1267–1272 www.elsevier.com/locate/jmaa

Positive solutions of Neumann problems with singularities Jifeng Chu a,∗ , Yigang Sun a , Hao Chen b a Department of Applied Mathematics, College of Science, Hohai University, Nanjing 210098, China b The Second Department, Army Command College, Nanjing 210045, China

Received 13 September 2006 Available online 5 May 2007 Submitted by Steven G. Krantz

Abstract In this paper, we study the existence of positive solutions to second order singular equations with Neumann boundary conditions. The proof of the main result relies on a nonlinear alternative principle of Leray–Schauder, together with a truncation technique. © 2007 Elsevier Inc. All rights reserved. Keywords: Positive solutions; Neumann boundary conditions; Singular equations; Leray–Schauder alternative principle; Truncation technique

1. Introduction In this paper, we establish the existence of positive solutions to the second order equation x  + m2 x = f (t, x) + e(t),

(1.1)

with Neumann boundary conditions x  (0) = 0,

x  (1) = 0.

(1.2)

Here 0 < m < μ1 = π2 is a constant, e(t) ∈ C[0, 1] and the nonlinearity f (t, x) may be singular at x = 0. Note that μ1 is the first eigenvalue of the linear equation with boundary conditions (1.2). Recently, Neumann boundary value problems have deserved the attention of many researchers during the last two decades [1,3,4,10]. Here we mention the following two results. In [6], Jiang and Liu studied the existence of one positive solution of (1.1)–(1.2) with e(t) ≡ 0 under assumption that f (t, x) is either superlinear or sublinear. In [9], Sun and Li obtained some existence results for at least two positive solutions under the weaker conditions than those in [6]. The proof in the above two papers is based on Krasnosel’skii fixed point theorem on compression and expansion of cones [8]. Besides fixed point theorems in cones, the method of upper and lower solutions [5] is also used in the literature [1,2,10]. In this paper, we establish the existence of positive solutions to problem (1.1)–(1.2), by using a nonlinear alternative principle of Leray–Schauder, which was used in [7] to deal with periodic singular problems. Here we emphasize that * Corresponding author.

E-mail address: [email protected] (J. Chu). 0022-247X/$ – see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2007.04.070

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our results are applicable to the case of a strong singularity as well as the case of a weak singularity, and that in our results e may take negative values. The remaining part of the paper is organized as follows. In Section 2, some preliminary results will be given. In Section 3, we will state and prove the main results. Some illustrating examples will also be given. Let us fix some notation to be used. Given ϕ ∈ L1 [0, 1], we write ϕ  0 if ϕ  0 for a.e. t ∈ [0, 1] and it is positive in a set of positive measure. R+ denotes the set of positive real numbers. Let us denote by p ∗ and p∗ the essential supremum and infimum of a given function p ∈ L1 [0, 1], if they exist. 2. Preliminaries Lemma 2.1. (See [6].) Suppose h : [0, 1] → [0, ∞) is continuous, 0 < m < μ1 . Then the linear equation x  + m2 x = h(t),

(2.1)

with boundary conditions (1.2) has a unique solution x ∈ C2 [0, 1] with the representation 1 x(t) =

G(t, s)h(s) ds, 0

where

 cos m(1−t) cos ms

, m sin m cos m(1−s) cos mt , m sin m

G(t, s) =

0  s  t  1, 0  t  s  1.

It is easy to see that G(t, s) > 0 for all m ∈ (0, μ1 ). Let A = min0t1 G(t, s), B = max0t1 G(t, s), σ = A/B. 2 1 2 Then B > A > 0 and 0 < σ < 1. In fact, a direct calculation shows that A = mcossinmm , B = m sin m , σ = cos m. In order to prove the main result of this paper, we need the following nonlinear alternative of Leray–Schauder, which can be found in [8]. Theorem 2.1. Assume Ω is a relatively subset of a convex set K in a normed space X. Let T : 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 exists x ∈ ∂Ω and 0 < λ < 1 such that x = λT x. 3. Main results In this section, we state and prove the main results of this paper. Let us define the function 1 γ (t) =

G(t, s)e(s) ds, 0

which is just the unique solution of the linear problem (2.1)–(1.2) with h(t) = e(t). Theorem 3.1. Suppose that there exists a constant r > 0 such that (H1 ) there exist continuous, nonnegative functions g(x), h(x) and k(t), such that   0  f (t, x)  k(t) g(x) + h(x) for all (t, x) ∈ [0, 1] × (0, r], g(x) > 0 is nonincreasing and h(x)/g(x) is nondecreasing in x ∈ (0, r]; 1 r−γ ∗ ∗ (H2 ) h(r) > K , here K(t) = 0 G(t, s)k(s) ds; g(σ r){1+ g(r) }

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(H3 ) there exists a continuous function φr  0 such that f (t, x)  φr (t) for all (t, x) ∈ [0, 1] × (0, r]; (H4 ) φr (t) + e(t)  0 for all t ∈ [0, 1]. Then problem (1.1)–(1.2) has at least one positive solution x with 0 < x < r. Proof. Since (H2 ) holds, we can choose n0 ∈ {1, 2, . . .} such that   h(r) 1 ∗ K g(σ r) 1 + +γ∗ + < r. g(r) n0

1 n0

< r and

Let N0 = {n0 , n0 + 1, . . .}. Fix n ∈ N0 . Consider the family of equations

m2 , x  + m2 x = λfn t, x(t) + λe(t) + n with boundary conditions (1.2), where λ ∈ [0, 1] and  f (t, x) if x  n1 , fn (t, x) = f (t, n1 ) if x  n1 .

(3.1)

Problem (3.1)–(1.2) is equivalent to the following fixed point problem 1 x(t) = λ 0



G(t, s)fn s, x(s) ds + λ

1 G(t, s)e(s) ds +

1 1 = λ(Tn x)(t) + λγ (t) + . n n

(3.2)

0

We claim that any fixed point x of (3.2) for any λ ∈ [0, 1] must satisfy x = r. Otherwise, assume that x is a solution of (3.2) for some λ ∈ [0, 1] such that x = r. Note that fn (t, x) + e(t)  φr (t) + e(t)  0 for 0 < x  r, it is easy to see that    1 1  . x(t) −  σ x −  n n Hence, for all t ∈ [0, 1], we have x(t)  n1 and  



 1  + 1  σ x − 1 + 1 = σ r − 1 + 1  σ r. x − x(t)  σ   n n n n n n Thus we have from condition (H1 ), for all t ∈ [0, 1], 1 x(t) = λ



 1 G(t, s) fn s, x(s) + e(s) ds + = λ n

0

1



 1 G(t, s) f s, x(s) + e(s) ds + n

0

  1

h(r) 1 1 G(t, s)k(s) ds + γ (t) +  G(t, s)f s, x(s) ds + γ (t) +  g(σ r) 1 + n g(r) n 0 0   1 h(r) K∗ + γ ∗ + .  g(σ r) 1 + g(r) n0 1

(3.3)

Therefore,

  1 h(r) K∗ + γ ∗ + . r = x  g(σ r) 1 + g(r) n0

This is a contradiction to the choice of n0 and the claim is proved. From this claim, Theorem 2.1 guarantees that equation

m2 , x  + m2 x = fn t, x(t) + e(t) + n

(3.4)

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with boundary conditions (1.2) has a solution xn with xn < r. Since xn (t)  n1 > 0 for all t ∈ [0, 1] and xn is actually a positive solution. Next we claim that these solutions xn have a uniform positive lower bound, i.e., there exists a constant δ > 0, independent of n ∈ N0 , such that min xn (t)  δ

(3.5)

t∈[0,1]

for all n ∈ N0 . To see this, let xr (t) be the unique solution to the problem (2.1)–(1.2) with h = φr (t). Since (H3 )–(H4 ) holds, we have 1 xn (t) =

G(t, s)fn s, xn (s) ds +

1

0

0

1 

1 G(t, s)e(s) ds + = n

G(t, s)φr (s) ds + γ (t) +

1



1 G(t, s)f s, xn (s) ds + γ (t) + n

0

1  Φ∗ + γ∗ = δ > 0, n

0

1

here Φ(t) = 0 G(t, s)φr (s) ds. Next we prove the fact   x   H n

(3.6)

for some constant H > 0 and for all n  n0 . To this end, integrating (3.4) from 0 to 1, we obtain 1 

1 xn (t) dt =

2

m

0

Then



m2 fn t, xn (t) + e(t) + dt. n

0

 t  t        2     

m     x  = max x (t) = max  x  (s) ds  = max  fn s, xn (s) + e(s) + − m2 xn (s) ds  n n n  0t1  n 0t1 0t1 0

1  

0



m2 ds + m2 fn s, xn (s) + e(s) + n

0

1

1 xn (s) ds = 2m

xn (s) ds < 2m2 r = H.

2

0

0

The fact xn < r and (3.6) show that {xn }n∈N0 is a bounded and equicontinuous family on [0, 1]. Now the Arzela– Ascoli Theorem guarantees that {xn }n∈N0 has a subsequence, {xnk }k∈N , converging uniformly on [0, 1] to a function x ∈ C[0, 1]. From the fact xn < r and (3.5), x satisfies δ  x(t)  r for all t ∈ [0, 1]. Moreover, xnk satisfies the integral equation 1 xnk (t) =



G(t, s)f s, xnk (s) ds +

0

1 G(t, s)e(s) ds +

1 . nk

0

Let k → ∞ and we arrive at 1 1

x(t) = G(t, s)f s, x(s) ds + G(t, s)e(s) ds, 0

0

where the uniform continuity of f (t, x) on [0, 1] × [δ, r] is used. Therefore, x is a positive solution of problem (1.1)–(1.2). Finally it is not difficult to show that x < r. 2 Remark 3.1. In [7], the same technique was used to deal with the singular periodic problems. However, we emphasize that our conditions are weaker than those in [7], because in our results, e(t) may take negative values, and it is not assumed that f (t, x) + e(t)  0,

for all (t, x) ∈ [0, 1] × (0, ∞).

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Corollary 3.1. Assume that there exist continuous functions b, bˆ  0 and λ > 0 such that 0

(F)

ˆ b(t) b(t)  f (t, x)  λ , λ x x

for all x > 0 and t ∈ [0, 1].

Then problem (1.1)–(1.2) has at least one positive solution if one of the following two conditions holds: (i) e∗  0; ∗ λ (ii) e∗ < 0, bˆ∗ + ( σβ λ ) λ+1 e∗ > 0. Proof. We will apply Theorem 3.1. (H1 ) and (H3 ) are satisfied if we take ˆ b(t) 1 , k(t) = b(t), g(x) = λ , λ r x The existence condition (H2 ) and (H4 ) become φr (t) =

h(x) ≡ 0.

σ λ r λ+1 − σ λ r λ γ ∗ > β ∗

(3.7)

and bˆ∗ + e∗ > 0 (3.8) rλ for some r > 0. If e∗  0, then (3.8) always holds. Since λ > 0 and e∗  0, we can choose r > 0 large enough such that (3.7) is satisfied. ∗ 1 If e∗ < 0, then γ ∗ < 0. Now (3.7) is satisfied if we take r = ( σβ λ ) λ+1 . In this case, (3.8) becomes the inequality bˆ∗ +



β∗ σλ

λ λ+1

e∗ > 0.

By employing Theorem 3.1, we obtain the desired results (i) and (ii).

2

Finally in this section, we select the following example to illustrate our results. We only consider the case e∗  0. Example 3.1. Let the nonlinearity in (1.1) be

f (t, x) = μb(t) x −α + x β ,

(3.9)

where α > 0, β  0, b(t) ∈ C[0, 1] and b(t) > 0 for all t ∈ [0, 1], μ > 0 is a positive parameter. For each e(t) with e∗  0, (i) if β < 1, then (1.1)–(1.2) has at least one positive solution for each μ > 0. (ii) if β  1, then (1.1)–(1.2) has at least one positive solution for each 0 < μ < μ∗ , where μ∗ is some positive constant. Proof. We will apply Theorem 3.1. To this end, the assumption (H3 ) is fulfilled by φr (t) = μb(t)r −α . If we take g(x) = x −α ,

h(x) = x β ,

k(t) = μb(t),

then (H2 ) is satisfied. Sine e∗  0, (H4 ) is clearly satisfied. Let β(t) = tion (H3 ) becomes μ<

σ α r α+1 − γ ∗ σ α r α β ∗ (1 + r α+β )

for some r > 0. So (1.1)–(1.2) has at least one positive solution for

1 0

G(t, s)b(s) ds. Now the existence condi-

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σ α r α+1 − γ ∗ σ α r α . β ∗ (1 + r α+β ) r>0

0 < μ < μ∗ = sup

Note that μ∗ = ∞ if β < 1 and μ∗ < ∞ if β  1. We have the desired results.

2

Remark 3.2. Our main results remain valid if we consider the general equation x  + a(t)x = f (t, x) with boundary conditions (1.2), assuming that a(t) satisfies the following condition: (A) The associated Green function G(t, s) of the linear equation x  + a(t)x = h(t) with boundary conditions (1.2), is positive for all (t, s) ∈ [0, 1] × [0, 1]. Here we note that some results obtained recently in [4] may help to make the assumption (A) clear. References [1] A. Cabada, L. Sanchez, A positive operator approach to the Neumann problem for a second order ordinary differential equation, J. Math. Anal. Appl. 204 (1996) 774–785. [2] A. Cabada, R.R.L. Pouse, Existence result for the problem (φ(u )) = f (t, u, u ) with periodic and Neumann boundary conditions, Nonlinear Anal. 30 (1997) 1733–1742. [3] A. Cabada, P. Habets, S. Lois, Monotone method for the Neumann problem with lower and upper solutions in the reverse order, Appl. Math. Comput. 117 (2001) 1–14. [4] A. Canada, J.A. Montero, S. Villegas, Liapunov-type inequalities and Neumann boundary value problems at resonance, Math. Inequal. Appl. 3 (2005) 459–475. [5] C. De Coster, P. Habets, Upper and lower solutions in the theory of ODE boundary value problems: Classical and recent results, in: F. Zanolin (Ed.), Nonlinear Analysis and Boundary Value Problems for Ordinary Differential Equations, in: CISM–ICMS, vol. 371, Springer-Verlag, New York, 1996, pp. 1–78. [6] D. Jiang, H. Liu, Existence of positive solutions to second order Neumann boundary value problem, J. Math. Res. Exposition 20 (2000) 360–364. [7] D. Jiang, J. Chu, D. O’Regan, R.P. Agarwal, Multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces, J. Math. Anal. Appl. 286 (2003) 563–576. [8] M.A. Krasnosel’skii, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964. [9] J. Sun, W. Li, Multiple positive solutions to second order Neumann boundary value problems, Appl. Math. Comput. 146 (2003) 187–194. [10] N. Yazidi, Monotone method for singular Neumann problem, Nonlinear Anal. 49 (2002) 589–602.