Boundary Value Problems
Volume 2009 (2009), Article ID 103867, 34 pages
doi:10.1155/2009/103867
Research Article

An Approximation Approach to Eigenvalue Intervals for Singular Boundary Value Problems with Sign Changing and Superlinear Nonlinearities

Haishen Lü,1 Ravi P. Agarwal,2,3 and Donal O'Regan4

1Department of Applied Mathematics, Hohai University, Nanjing 210098, China
2Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901-6975, USA
3KFUPM Chair Professor, Mathematics and Statistics Department, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
4Department of Mathematics, National University of Ireland, Galway, Ireland

Received 25 June 2009; Accepted 5 October 2009

Academic Editor: Ivan T. Kiguradze

Copyright © 2009 Haishen Lü et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper studies the eigenvalue interval for the singular boundary value problem − 𝑢 = 𝑔 ( 𝑡 , 𝑢 ) + 𝜆 ℎ ( 𝑡 , 𝑢 ) , 𝑡 ∈ ( 0 , 1 ) , 𝑢 ( 0 ) = 0 = 𝑢 ( 1 ) , where 𝑔 + ℎ may be singular at 𝑢 = 0 ,   𝑡 = 0 , 1 , and may change sign and be superlinear at 𝑢 = + ∞ . The approach is based on an approximation method together with the theory of upper and lower solutions.

1. Introduction

The singular boundary value problems of the form

− 𝑢 = 𝑓 ( 𝑡 , 𝑢 ) , 𝑡 ∈ ( 0 , 1 ) , 𝑢 ( 0 ) = 0 = 𝑢 ( 1 ) ( 1 . 1 ) occurs in several problems in applied mathematics, see [1–6] and their references. In many papers, a critical condition is that

𝑓 ( 𝑡 , 𝑟 ) ≥ 0 f o r ( 𝑡 , 𝑟 ) ∈ ( 0 , 1 ) × ( 0 , ∞ ) ( 1 . 2 ) or there exists a constant 𝐿 > 0 such that for any compact set 𝐾 ⊂ ( 0 , 1 ) , there is 𝜀 = 𝜀 𝐾 > 0 such that

] , 𝑓 ( 𝑡 , 𝑟 ) ≥ 𝐿 ∀ 𝑡 ∈ 𝐾 , 𝑟 ∈ ( 0 , 𝜀 l i m 𝑟 → ∞ 𝑓 ( 𝑡 , 𝑟 ) 𝑟 = 0 ∀ 𝑡 ∈ ( 0 , 1 ) . ( 1 . 3 ) We refer the reader to [1–4]. In the case, when 𝑓 ( 𝑡 , 𝑟 ) may change sign in a neighborhood of 𝑟 = 0 and l i m s u p 𝑟 → + ∞ ( 𝑓 ( 𝑡 , 𝑟 ) / 𝑟 ) = + ∞ for 𝑡 ∈ ( 0 , 1 ) , very few existence results are available in literature [1].

In this paper we study positive solutions of the second boundary value problem

− 𝑢 = 𝑔 ( 𝑡 , 𝑢 ) + 𝜆 ℎ ( 𝑡 , 𝑢 ) , 𝑡 ∈ ( 0 , 1 ) , 𝑢 ( 0 ) = 0 = 𝑢 ( 1 ) ; ( 1 . 4 ) here 𝑔 ∶ ( 0 , 1 ) × ( 0 , ∞ ) → 𝑅 and ℎ ∶ ( 0 , 1 ) × [ 0 , ∞ ) → ( 0 , ∞ ) are continuous, so as a result, our nonlinearity may be singular at 𝑡 = 0 , 1 and 𝑢 = 0 . Also our nonlinearity may change sign and be superlinear at 𝑢 = + ∞ . Our main existence results (Theorems 1.1, 1.2 and 1.4) are new (see Remark 1.5, Examples 3.1 and 3.2).

A function 𝑢 is a solution of the boundary value problem (1.4) if 𝑢 ∶ [ 0 , 1 ] → 𝑅 , 𝑢 satisfies the differential equation (1.4) on ( 0 , 1 ) and the stated boundary data.

Let 𝐶 [ 0 , 1 ] denote the class of maps 𝑢 continuous on [ 0 , 1 ] , with norm | 𝑢 | ∞ = m a x 𝑡 ∈ [ 0 , 1 ] | 𝑢 ( 𝑡 ) | . We put ⋀ 𝑏 m i n { 𝑎 , 𝑏 } = 𝑎 ; m a x { 𝑎 , 𝑏 } = 𝑎 ∨ 𝑏 . Given 𝛼 , 𝛽 ∈ 𝐶 [ 0 , 1 ] , 𝛼 ≤ 𝛽 , let

𝐷 𝛽 𝛼 [ ] = { 𝑣 ∣ 𝑣 ∈ 𝐶 0 , 1 , 𝛼 ≤ 𝑣 ≤ 𝛽 } . ( 1 . 5 ) Let

𝑀 = ℎ ∈ 𝐶 ( 0 , 1 ) ∶ 1 0 | | | | ℎ ( 𝑠 ) 𝑑 𝑠 < ∞ w i t h l i m 𝑡 → 0 + 𝑡 | | | | ℎ ( 𝑡 ) < ∞ , l i m 𝑡 → 1 − | | | | . ( 1 − 𝑡 ) ℎ ( 𝑡 ) < ∞ ( 1 . 6 )

In this paper, we suppose the following conditions hold:

( 𝐺 1 ) suppose there exist 𝑔 𝑖 ∶ ( 0 , 1 ) × ( 0 , ∞ ) → ( 0 , ∞ ) ( 𝑖 = 1 , 2 ) continuous functions such that

𝑔 𝑖 𝑔 ( 𝑡 , ⋅ ) i s s t r i c t l y d e c r e a s i n g f o r 𝑡 ∈ ( 0 , 1 ) , 1 ⋅ , 𝑟 𝜙 1 ( ⋅ ) , 𝑔 2 ( ⋅ , 𝑟 ) ∈ 𝑀 ∀ 𝑟 > 0 , − 𝑔 1 ( 𝑡 , 𝑟 ) ≤ 𝑔 ( 𝑡 , 𝑟 ) ≤ 𝑔 2 ( 𝑡 , 𝑟 ) f o r ( 𝑡 , 𝑟 ) ∈ ( 0 , 1 ) × ( 0 , ∞ ) , ( 1 . 7 ) where 𝜙 1 is defined in Lemma 2.1;

( 𝐻 1 ) there exist ℎ 𝑖 ∶ ( 0 , 1 ) × [ 0 , ∞ ) → [ 0 , ∞ ) ( 𝑖 = 1 , 2 ) continuous functions such that

ℎ 𝑖 ℎ ( 𝑡 , ⋅ ) i s i n c r e a s i n g f o r 𝑡 ∈ ( 0 , 1 ) , 1 ( ⋅ , 𝑟 ) , ℎ 2 ℎ ( ⋅ , 𝑟 ) ∈ 𝑀 f o r 𝑟 > 0 , 1 ( 𝑡 , 𝑟 ) ≤ ℎ ( 𝑡 , 𝑟 ) ≤ ℎ 2 [ ( 𝑡 , 𝑟 ) f o r ( 𝑡 , 𝑟 ) ∈ ( 0 , 1 ) × 0 , ∞ ) ; ( 1 . 8 )

( 𝐻 2 ) there exists 𝑟 > 0 such that ℎ 1 ( 𝑡 , 𝑟 ) > 0 for 𝑡 ∈ ( 0 , 1 ) .

The main results of the paper are the following.

Theorem 1.1. Suppose ( 𝐺 1 ) , ( 𝐻 1 ) , ( 𝐻 2 ) and the following conditions hold: ( 𝐺 2 ) for all 𝑟 2 > 𝑟 1 > 0 , there exists 𝛾 ( ⋅ ) ∈ 𝑀 such that 𝑔 2 ( ⋅ , 𝑟 ) + 𝛾 ( ⋅ ) 𝑟 is increasing in ( 𝑟 1 , 𝑟 2 ) : ( 𝐻 3 ) l i m 𝑟 → ∞ ℎ 1 ( 𝑡 , 𝑟 ) 𝑟 = 0 ∀ 𝑡 ∈ ( 0 , 1 ) ; ( 1 . 9 ) ( 𝐻 4 ) there exists a sequence { 𝑅 𝑗 } ∞ 𝑗 = 1 such that l i m 𝑗 → ∞ 𝑅 𝑗 = ∞ and l i m 𝑗 → ∞ ℎ 2 𝑠 , 𝑅 𝑗 + 𝑎 1 𝑅 𝑗 = 0 , ( 1 . 1 0 ) where 𝑎 1 ∫ = 1 + 1 0 𝑔 2 ( 𝑠 , 1 ) 𝑑 𝑠 .

Then there exists 𝜆 ∗ 1 > 0 such that for every 𝜆 ≥ 𝜆 ∗ 1 , (1.4) has at least one positive solution 𝑢 ∈ 𝐶 [ 0 , 1 ] ∩ 𝐶 1 ( 0 , 1 ) and 𝑢 > 0 for 𝑡 ∈ ( 0 , 1 ) .

Theorem 1.2. Suppose ( 𝐺 1 ) , ( 𝐻 1 ) , ( 𝐻 2 ) and the following conditions hold: ( 𝐺 3 ) for all 𝑟 2 > 𝑟 1 > 0 there exists 𝛾 ( ⋅ ) ∈ 𝑀 such that 𝑔 ( 𝑡 , 𝑟 ) + 𝛾 ( 𝑡 ) 𝑟 is increasing in ( 𝑟 1 , 𝑟 2 ) ; ( 𝐺 4 ) there exists 𝑐 1 > 0 such that 0 ≤ 𝑔 ( 𝑡 , 𝑟 ) , 𝑡 ∈ ( 0 , 1 ) , 0 < 𝑟 < 𝑐 1 ; ( 1 . 1 1 ) ( 𝐺 5 ) there exists 𝑐 2 ∈ ( 0 , 𝑐 1 ) , 0 < 𝛽 < 1 such that for all 𝑟 ∈ ( 0 , 𝑐 2 ) 1 0 𝑡 ( 1 − 𝑡 ) 𝑔 1 ( 𝑡 , 𝑟 𝑙 ( 𝑡 ) ) 𝑑 𝑡 ≥ 𝑟 𝜋 , ( 1 . 1 2 ) where 𝑔 𝑚 𝑚 ( 𝑡 , 𝑟 ) = m i n 𝑔 ( 𝑡 , 𝑟 ) , 𝑟 𝛽 f o r 𝑚 ≥ 1 , ( 1 . 1 3 ) and 𝑙 ( 𝑡 ) = m i n { 𝑡 , 1 − 𝑡 } for 𝑡 ∈ [ 0 , 1 ] .

Then there exists 𝜆 ∗ 2 > 0 such that

(i)if 0 < 𝜆 < 𝜆 ∗ 2 , (1.4) has at least one solution 𝑢 ∈ 𝐶 [ 0 , 1 ] ∩ 𝐶 1 ( 0 , 1 ) and 𝑢 > 0 for 𝑡 ∈ ( 0 , 1 ) ;(ii)if 𝜆 > 𝜆 ∗ 2 , (1.4) has no solutions.

Remark 1.3. Notice that 𝑔 𝑚 ( 𝑡 , 𝑟 ) satisfies ( 𝐺 1 ) , ( 𝐺 3 ) , ( 𝐺 4 ) and for fixed 𝑚 ≥ 1 , 1 0 𝑡 ( 1 − 𝑡 ) 𝑔 𝑚 ( 𝑡 , 𝑟 𝑙 ( 𝑡 ) ) 𝑑 𝑡 ≥ 𝑟 𝜋 f o r 𝑟 ∈ 0 , 𝑐 2 , 𝑔 ( 𝑡 , 𝑟 ) ≥ 𝑔 𝑚 ( 𝑡 , 𝑟 ) ≥ 𝑔 1 ( 𝑡 , 𝑟 ) f o r 𝑡 ∈ ( 0 , 1 ) , 𝑟 ∈ ( 0 , ∞ ) . ( 1 . 1 4 )

Theorem 1.4. Suppose ( 𝐺 1 ) , ( 𝐻 1 ) , ( 𝐻 2 ) and the following conditions hold: ( 𝐺 6 ) there exists 𝜏 ≥ 𝜏 1 such that l i m 𝑟 → 0 + 𝜏 𝑟 + 𝑔 − ( 𝑡 , 𝑟 ) ℎ ( 𝑡 , 𝑟 ) = 0 , ( 1 . 1 5 ) where 𝜏 1 is defined in Lemma 2.1 and 𝑔 + ( 𝑡 , 𝑟 ) = m a x { 0 , 𝑔 ( 𝑡 , 𝑟 ) } , 𝑔 − ( 𝑡 , 𝑟 ) = m a x { 0 , − 𝑔 ( 𝑡 , 𝑟 ) } ; ( 𝐻 5 ) for all 𝑟 2 > 𝑟 1 > 0 , there exists 𝛾 ( ⋅ ) ∈ 𝑀 such that ℎ ( 𝑡 , 𝑟 ) + 𝛾 ( 𝑡 ) 𝑟 is increasing in ( 𝑟 1 , 𝑟 2 ) .

Then there exists 𝜆 ∗ 3 > 0 such that

(i)if 0 < 𝜆 < 𝜆 ∗ 3 , (1.4) has at least one solution 𝑢 ∈ 𝐶 [ 0 , 1 ] ∩ 𝐶 1 ( 0 , 1 ) and 𝑢 > 0 for 𝑡 ∈ ( 0 , 1 ) ;(ii)if 𝜆 > 𝜆 ∗ 3 , (1.4) has no solutions.

Remark 1.5. In [5, 6], the authors consider the boundary value problem (1.4) under the conditions l i m 𝑟 → ∞ ℎ 2 ( 𝑡 , 𝑟 ) 𝑟 = 0 . ( 1 . 1 6 )

In Section 3, we give two examples (see Examples 3.1 and 3.2) which satisfy the conditions in Theorem 1.1 or Theorem 1.2 but they do not satisfy the conditions in [1–5].

2. Proof of Main Results

2.1. Some Lemmas

Lemma 2.1. Consider the following eigenvalue problem − 𝑢 = 𝜏 𝑢 ( 𝑡 ) , 𝑡 ∈ ( 0 , 1 ) , 𝑢 ( 0 ) = 𝑢 ( 1 ) = 0 . ( 2 . 1 ) Then the eigenvalues are 𝜏 𝑚 = ( 𝑚 𝜋 ) 2 f o r 𝑚 = 1 , 2 , … , ( 2 . 2 ) and the corresponding eigenfunctions are 𝜙 𝑚 ( 𝑡 ) = s i n 𝑚 𝜋 𝑡 f o r 𝑚 = 1 , 2 , … . ( 2 . 3 )

Let 𝐺 ( 𝑡 , 𝑠 ) be the Green's function for the BVP:

− 𝑢 = 0 f o r 𝑡 ∈ ( 0 , 1 ) , 𝑢 ( 0 ) = 𝑢 ( 1 ) = 0 . ( 2 . 4 ) Then

𝐺 ( 𝑡 , 𝑠 ) = 𝑠 ( 1 − 𝑡 ) , 0 ≤ 𝑠 < 𝑡 ≤ 1 , 𝑡 ( 1 − 𝑠 ) , 0 ≤ 𝑡 < 𝑠 ≤ 1 . ( 2 . 5 ) Also for all ( 𝑡 , 𝑠 ) ∈ [ 0 , 1 ] × [ 0 , 1 ] , define

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 𝑁 ( 𝑡 , 𝑠 ) = 𝐺 ( 𝑡 , 𝑠 ) 𝜙 1 ( 𝑡 ) i f 𝑡 ≠ 0 , 1 , 1 − 𝑠 𝜋 𝑠 i f 𝑡 = 0 , 𝜋 i f 𝑡 = 1 . ( 2 . 6 ) It follows easily that

0 < 𝐺 ( 𝑡 , 𝑠 ) ≤ 𝑡 ( 1 − 𝑡 ) f o r ( 𝑡 , 𝑠 ) ∈ ( 0 , 1 ) × ( 0 , 1 ) , 𝑠 ( 1 − 𝑠 ) 1 2 𝜋 ≤ 𝑁 ( 𝑡 , 𝑠 ) ≤ 2 f o r ( 𝑡 , 𝑠 ) ∈ ( 0 , 1 ) × ( 0 , 1 ) . ( 2 . 7 )

Define the operator 𝐴 , 𝐵 ∶ 𝑀 → 𝐶 [ 0 , 1 ] by

𝐴 𝑥 ( 𝑡 ) = 1 0 𝐺 ( 𝑡 , 𝑠 ) 𝑥 ( 𝑠 ) 𝑑 𝑠 , 𝐵 𝑥 ( 𝑡 ) = 1 0 𝑁 ( 𝑡 , 𝑠 ) 𝑥 ( 𝑠 ) 𝑑 𝑠 . ( 2 . 8 )

The following four results can be found in [5] (notice l i m 𝑟 → ∞ ( ℎ 2 ( 𝑡 , 𝑟 ) / 𝑟 ) = 0 is not needed in the proofs there).

Lemma 2.2. Suppose ( 𝐺 1 ) and ( 𝐻 1 ) hold. Let 𝑛 0 ∈ 𝑁 . Assume that for every 𝑛 > 𝑛 0 , there exist 𝑎 𝑛 , 𝛿 𝑛 , 𝛿 ∈ 𝑀 such that 0 ≤ 𝑎 𝑛 | | 𝛿 ( 𝑡 ) , 𝑛 | | ( 𝑡 ) ≤ 𝛿 ( 𝑡 ) , l i m 𝑛 → ∞ 𝛿 𝑛 ( 𝑡 ) = 0 , f o r 𝑡 ∈ ( 0 , 1 ) ( 2 . 9 ) and there exist 𝑢 , 𝑢 𝑛 , ̂ 𝑢 𝑛 , ̂ 𝑢 ∈ 𝐶 [ 0 , 1 ] such that 0 < 𝑢 ( 𝑡 ) ≤ 𝑢 𝑛 ( 𝑡 ) ≤ ̂ 𝑢 𝑛 ( 𝑡 ) ≤ ̂ 𝑢 ( 𝑡 ) f o r 𝑡 ∈ ( 0 , 1 ) , ( 2 . 1 0 ) and ̂ 𝑢 ( 0 ) = ̂ 𝑢 ( 1 ) = 0 . If − 𝑢 𝑛 ( 𝑡 ) + 𝑎 𝑛 ( 𝑡 ) 𝑢 𝑛 1 ( 𝑡 ) ≤ 𝑔 𝑡 , 𝑛 + 𝑣 + 𝜆 ℎ ( 𝑡 , 𝑣 ) + 𝛿 𝑛 ( 𝑡 ) + 𝑎 𝑛 ( 𝑡 ) 𝑣 ( 𝑡 ) f o r 𝑡 ∈ ( 0 , 1 ) , − ̂ 𝑢 𝑛 ( 𝑡 ) + 𝑎 𝑛 ( 𝑡 ) ̂ 𝑢 𝑛 1 ( 𝑡 ) ≥ 𝑔 𝑡 , 𝑛 + 𝑣 + 𝜆 ℎ ( 𝑡 , 𝑣 ) + 𝛿 𝑛 ( 𝑡 ) + 𝑎 𝑛 ( 𝑡 ) 𝑣 ( 𝑡 ) f o r 𝑡 ∈ ( 0 , 1 ) , ( 2 . 1 1 ) where 𝜆 ≥ 0 and 𝑣 ∈ 𝐷 ̂ 𝑢 𝑛 𝑢 𝑛 , then (1.4) has a solution 𝑢 ∈ 𝐶 [ 0 , 1 ] ∩ 𝐶 1 ( 0 , 1 ) such that 𝑢 ( 𝑡 ) ≤ 𝑢 ( 𝑡 ) ≤ ̂ 𝑢 ( 𝑡 ) for 𝑡 ∈ [ 0 , 1 ] .

Lemma 2.3. Let 𝜓 ∶ ( 0 , 1 ) × ( 0 , ∞ ) → ( 0 , ∞ ) be a continuous function with 𝜓 𝜓 ( 𝑡 , ⋅ ) i s s t r i c t l y d e c r e a s i n g , ( ⋅ , 𝑟 ) ∈ 𝑀 ∀ 𝑟 > 0 . ( 2 . 1 2 ) Then the problem − 𝜔 1 ( 𝑡 ) = 𝜓 𝑡 , 𝜔 ( 𝑡 ) + 𝑛 f o r 𝑡 ∈ ( 0 , 1 ) , 𝜔 ( 0 ) = 𝜔 ( 1 ) = 0 ( 2 . 1 3 ) has a solution 𝜔 𝑛 ∈ 𝐶 [ 0 , 1 ] such that 𝜔 𝑛 ( 𝑡 ) ≤ 𝜔 𝑛 + 1 ( 𝑡 ) ≤ 1 + 𝜔 1 ( 𝑡 ) ≤ 1 + 1 0 [ ] 𝜓 ( 𝑠 , 1 ) 𝑑 𝑠 f o r 𝑡 ∈ 0 , 1 , 𝑛 ∈ 𝑁 . ( 2 . 1 4 ) If we let 𝜔 ( 𝑡 ) = l i m 𝑛 → ∞ 𝜔 𝑛 ( 𝑡 ) for 𝑡 ∈ [ 0 , 1 ] , then [ ] 𝜔 ∈ 𝐶 0 , 1 , 𝜔 ( 𝑡 ) > 0 f o r 𝑡 ∈ ( 0 , 1 ) , − 𝜔 ( 𝑡 ) = 𝜓 ( 𝑡 , 𝜔 ( 𝑡 ) ) f o r 𝑡 ∈ ( 0 , 1 ) , 𝜔 ( 0 ) = 𝜔 ( 1 ) = 0 . ( 2 . 1 5 )

Next we consider the boundary value problem

− 𝑢 + 𝑎 ( 𝑡 ) 𝑢 ( 𝑡 ) = 𝑓 ( 𝑡 ) , 𝑡 ∈ ( 0 , 1 ) , 𝑢 ( 0 ) = 0 = 𝑢 ( 1 ) , ( 2 . 1 6 ) where 𝑎 , 𝑓 ∈ 𝑀 , 𝑎 ( 𝑡 ) ≥ 0 for 𝑡 ∈ ( 0 , 1 ) .

Lemma 2.4. The following statements hold: (i)for any 𝑓 ∈ 𝑀 , (2.16) is uniquely solvable and 𝑢 + 𝐴 ( 𝑎 𝑢 ) = 𝐴 ( 𝑓 ) ; ( 2 . 1 7 ) (ii)if 𝑓 ( 𝑡 ) ≥ 0 for 𝑡 ∈ ( 0 , 1 ) , then the solution of (2.16) is nonnegative.

Corollary 2.5. Let Φ ∶ 𝑀 → 𝐶 [ 0 , 1 ] ∩ 𝐶 1 ( 0 , 1 ) be the operator such that Φ ( 𝑓 ) is the solution of (2.16). Then we have (i)if 𝑓 1 ( 𝑡 ) ≤ 𝑓 2 ( 𝑡 ) for 𝑡 ∈ ( 0 , 1 ) , then Φ ( 𝑓 1 ) ( 𝑡 ) ≤ Φ ( 𝑓 2 ) ( 𝑡 ) for 𝑡 ∈ [ 0 , 1 ] ; (ii)let 𝐸 ⊂ 𝑀 and 𝛽 ∈ 𝑀 . If | 𝑓 ( 𝑡 ) | ≤ 𝛽 ( 𝑡 ) , 𝑡 ∈ ( 0 , 1 ) for all 𝑓 ∈ 𝐸 , then Φ ( 𝐸 ) is relatively compact with respect to the topology of 𝐶 [ 0 , 1 ] .

Lemma 2.6 (see [2]). Let 𝑓 ∈ 𝑀 , 𝑓 ≥ 0 , 𝑓 ≢ 0 , 𝑢 ∈ 𝐶 [ 0 , 1 ] ∩ 𝐶 1 ( 0 , 1 ) satisfy − 𝑢 = 𝑓 i n ( 0 , 1 ) , 𝑢 ( 0 ) = 𝑢 ( 1 ) = 0 . ( 2 . 1 8 ) Then there exist 𝑚 = 𝑚 ( 𝑓 ) > 0 , 𝑀 = 𝑀 ( 𝑓 ) > 0 such that [ ] . 𝑚 𝑙 ( 𝑡 ) ≤ 𝑢 ( 𝑡 ) ≤ 𝑀 𝑙 ( 𝑡 ) f o r 𝑡 ∈ 0 , 1 ( 2 . 1 9 )

2.2. The Proof of Theorem 1.1

Claim 1 (see [5]). There exists 𝜆 ∗ 1 > 0 , 𝑐 > 0 , independent of 𝜆 , such that for all 𝜆 ≥ 𝜆 ∗ 1 there exist 𝑅 𝜆 > 𝑐 , 𝑢 ∈ 𝐶 ( [ 0 , 1 ] ) , with 𝑐 𝜙 1 ( 𝑡 ) ≤ 𝑢 ( 𝑡 ) ≤ 𝑅 𝜆 𝜙 1 ( 𝑡 ) and − 𝑢 ( 𝑡 ) = − 𝑔 1 𝑡 , 𝑢 ( 𝑡 ) + 𝜆 ℎ 1 𝑡 , 𝑢 ( 𝑡 ) , f o r 𝑡 ∈ ( 0 , 1 ) , 𝑢 ( 0 ) = 𝑢 ( 1 ) = 0 , ( 2 . 2 0 ) with 𝑔 1 ⋅ , 𝑢 ( ⋅ ) , ℎ 1 ⋅ , 𝑢 ( ⋅ ) ∈ 𝑀 . ( 2 . 2 1 )

Let 𝜆 ∗ 1 > 0 ,   𝑐 > 0 and 𝑢 ∈ 𝐶 [ 0 , 1 ] be defined in Claim 1. Define

𝜓 ( 𝑡 , 𝑟 ) = 𝑔 2 ( 𝑡 , 𝑟 ) f o r 𝑡 ∈ ( 0 , 1 ) . ( 2 . 2 2 )

From ( 𝐺 1 ) notice that 𝜓 satisfies the assumptions of Lemma 2.3, so there exist 𝜔 , 𝜔 𝑛 ∈ 𝐶 [ 0 , 1 ] ,   𝜔 𝑛 ( 𝑡 ) > 0 ,   𝜔 ( 𝑡 ) > 0 for 𝑡 ∈ ( 0 , 1 ) such that

− 𝜔 𝑛 ( 𝑡 ) = 𝑔 2 1 𝑡 , 𝑛 + 𝜔 𝑛 𝜔 f o r 𝑡 ∈ ( 0 , 1 ) , 𝑛 ( 0 ) = 𝜔 𝑛 𝜔 ( 1 ) = 0 , 𝑛 ( 𝑡 ) ≤ 𝜔 𝑛 + 1 ( 𝑡 ) ≤ 1 + 𝜔 1 ( 𝑡 ) ≤ 𝑎 1 [ ] f o r 𝑡 ∈ 0 , 1 , 𝑛 ∈ 𝑁 , 𝜔 ( 𝑡 ) = l i m 𝑛 → ∞ 𝜔 𝑛 ( [ ] , 𝑡 ) f o r 𝑡 ∈ 0 , 1 − 𝜔 ( 𝑡 ) = 𝑔 2 ( 𝑡 , 𝜔 ( 𝑡 ) ) f o r 𝑡 ∈ ( 0 , 1 ) , 𝜔 ( 0 ) = 𝜔 ( 1 ) = 0 , ( 2 . 2 3 ) where 𝑎 1 ∫ = 1 + 1 0 𝑔 2 ( 𝑠 , 1 ) 𝑑 𝑠 .

Let 𝜆 ≥ 𝜆 ∗ 1 , 𝑛 ∈ 𝑁 be fixed. We consider the following boundary value problem:

− 𝑣 ( 𝑡 ) = 𝜆 ℎ 2 𝑡 , 𝑣 + 𝜔 𝑛 + 𝜆 ℎ 1 𝑡 , 𝑢 f o r 𝑡 ∈ ( 0 , 1 ) , 𝑣 ( 0 ) = 𝑣 ( 1 ) = 0 . ( 2 . 2 4 )

By ( 𝐻 4 ) , there exist { 𝑅 𝑗 } ∞ 𝑗 = 1 such that l i m 𝑗 → ∞ 𝑅 𝑗 = ∞ and

l i m 𝑗 → ∞ ℎ 2 𝑡 , 𝑅 𝑗 + 𝑎 1 𝑅 𝑗 = 0 f o r 𝑡 ∈ ( 0 , 1 ) , ( 2 . 2 5 ) so

l i m 𝑗 → ∞ 𝜆 ℎ 2 𝑡 , 𝑅 𝑗 + 𝑎 1 + 𝜆 ℎ 1 𝑡 , 𝑢 ( 𝑡 ) 𝑅 𝑗 = 0 f o r 𝑡 ∈ ( 0 , 1 ) . ( 2 . 2 6 ) There exists 𝑗 0 ∈ 𝑁 such that

𝜆 ℎ 2 𝑡 , 𝑅 𝑗 0 + 𝑎 1 + 𝜆 ℎ 1 𝑡 , 𝑢 ( 𝑡 ) ≤ 𝑅 𝑗 0 . ( 2 . 2 7 ) If 𝑣 ∈ 𝐶 [ 0 , 1 ] and 0 ≤ 𝑣 ( 𝑡 ) ≤ 𝑅 𝑗 0 𝜙 1 ( 𝑡 ) for 𝑡 ∈ [ 0 , 1 ] , then

1 0 𝑁 ( 𝑡 , 𝑠 ) 𝜆 ℎ 2 𝑠 , 𝑣 ( 𝑠 ) + 𝜔 𝑛 ( 𝑠 ) + 𝜆 ℎ 1 𝑠 , 𝑢 ≤ 𝑑 𝑠 1 0 𝑁 ( 𝑡 , 𝑠 ) 𝜆 ℎ 2 𝑠 , 𝑣 ( 𝑠 ) + 𝑎 1 + 𝜆 ℎ 1 𝑠 , 𝑢 ≤ 𝑑 𝑠 1 0 𝑁 ( 𝑡 , 𝑠 ) 𝜆 ℎ 2 𝑠 , 𝑅 𝑗 0 𝜙 1 ( 𝑠 ) + 𝑎 1 + 𝜆 ℎ 1 𝑠 , 𝑢 ≤ 𝑑 𝑠 1 0 𝑁 ( 𝑡 , 𝑠 ) 𝜆 ℎ 2 𝑠 , 𝑅 𝑗 0 + 𝑎 1 + 𝜆 ℎ 1 𝑠 , 𝑢 ≤ 𝑅 𝑑 𝑠 𝑗 0 2 , f o r 𝑡 ∈ ( 0 , 1 ) , ( 2 . 2 8 ) and so

0 ≤ 1 0 𝐺 ( 𝑡 , 𝑠 ) 𝜆 ℎ 2 𝑠 , 𝑣 ( 𝑠 ) + 𝜔 𝑛 ( 𝑠 ) + 𝜆 ℎ 1 𝑠 , 𝑢 ( 𝑠 ) 𝑑 𝑠 ≤ 𝑅 𝑗 0 𝜙 1 [ ] . ( 𝑡 ) f o r 𝑡 ∈ 0 , 1 ( 2 . 2 9 )

Let Φ ∶ 𝐶 [ 0 , 1 ] → 𝐶 [ 0 , 1 ] be the operator defined by

( Φ 𝑣 ) ( 𝑡 ) ∶ = 1 0 𝐺 ( 𝑡 , 𝑠 ) 𝜆 ℎ 2 𝑠 , 𝑣 ( 𝑠 ) + 𝜔 𝑛 ( 𝑠 ) + 𝜆 ℎ 1 𝑠 , [ ] [ ] . 𝑢 ( 𝑠 ) 𝑑 𝑠 f o r 𝑣 ∈ 𝐶 0 , 1 , 𝑡 ∈ 0 , 1 ( 2 . 3 0 ) It is easy to see that Φ is a continuous and completely continuous operator. Also if 0 ≤ 𝑣 ( 𝑡 ) ≤ 𝑅 𝑗 0 𝜙 1 ( 𝑡 ) for 𝑡 ∈ [ 0 , 1 ] , then 0 ≤ Φ ( 𝑣 ) ( 𝑡 ) ≤ 𝑅 𝑗 0 𝜙 1 ( 𝑡 ) for 𝑡 ∈ [ 0 , 1 ] , so Schauder's fixed point theorem guarantees that there exists ̃ 𝑣 ∈ [ 0 , 𝑅 𝑗 0 𝜙 1 ] such that ̃ ̃ Φ ( 𝑣 ) = 𝑣 , that is,

− ̃ 𝑣 ( 𝑡 ) = 𝜆 ℎ 2 ̃ 𝑣 𝑡 , ( 𝑡 ) + 𝜔 𝑛 ( 𝑠 ) + 𝜆 ℎ 1 𝑡 , 𝑢 , ̃ ̃ ( 𝑡 ) 𝑣 ( 1 ) = 𝑣 ( 1 ) = 0 . ( 2 . 3 1 )

Let

̂ 𝑢 𝑛 ( 𝑡 ) = 𝜔 𝑛 ̃ 𝑣 ( 𝑡 ) + 𝑛 [ ] . ( 𝑡 ) f o r 𝑡 ∈ 0 , 1 ( 2 . 3 2 ) Then ̂ 𝑢 𝑛 ∈ 𝐶 [ 0 , 1 ] , ̂ 𝑢 𝑛 ( 1 ) = ̂ 𝑢 𝑛 ( 1 ) = 0 , and

− ̂ 𝑢 𝑛 ( 𝑡 ) = − 𝜔 𝑛 ̃ 𝑣 ( 𝑡 ) − 𝑛 ( 𝑡 ) = 𝑔 2 1 𝑡 , 𝑛 + 𝜔 𝑛 + 𝜆 ℎ 2 𝑡 , 𝜔 𝑛 + ̃ 𝑣 𝑛 + 𝜆 ℎ 1 𝑡 , 𝑢 ≥ 𝑔 2 1 𝑡 , 𝑛 + ̂ 𝑢 𝑛 + 𝜆 ℎ 1 𝑡 , 𝑢 + 𝜆 ℎ 2 𝑡 , ̂ 𝑢 𝑛 f o r 𝑡 ∈ ( 0 , 1 ) . ( 2 . 3 3 ) Let

̂ 𝑢 ( 𝑡 ) = 𝜔 ( 𝑡 ) + 𝑅 𝑗 0 𝜙 1 [ ] , ( 𝑡 ) f o r 𝑡 ∈ 0 , 1 ( 2 . 3 4 ) so

0 ≤ ̂ 𝑢 𝑛 [ ] . ( 𝑡 ) ≤ ̂ 𝑢 ( 𝑡 ) f o r 𝑡 ∈ 0 , 1 ( 2 . 3 5 )

From Claim 1, we obtain

− 𝑢 ( 𝑡 ) = − 𝑔 1 𝑡 , 𝑢 + 𝜆 ℎ 1 𝑡 , 𝑢 ≤ 𝜆 ℎ 1 𝑡 , 𝑢 ≤ 𝜆 ℎ 1 𝑡 , 𝑢 + 𝑔 2 1 𝑡 , 𝑛 + ̂ 𝑢 𝑛 + 𝜆 ℎ 2 𝑡 , ̂ 𝑢 𝑛 ≤ − ̂ 𝑢 𝑛 ( 𝑡 ) f o r 𝑡 ∈ ( 0 , 1 ) , ( 2 . 3 6 ) that is,

− 𝑢 − ̂ 𝑢 𝑛 ( 𝑡 ) ≤ 0 f o r 𝑡 ∈ ( 0 , 1 ) . ( 2 . 3 7 ) A standard argument yields

𝑢 ( 𝑡 ) ≤ ̂ 𝑢 𝑛 [ ] . ( 𝑡 ) f o r 𝑡 ∈ 0 , 1 ( 2 . 3 8 )

From ( 𝐺 2 ) , there exists 𝛾 ∈ 𝑀 such that 𝑟 → 𝑔 2 ( 𝑡 , 1 / 𝑛 + 𝑟 ) + 𝛾 ( 𝑡 ) 𝑟 is increasing on ( 0 , | ̂ 𝑢 | ∞ ) . Let 𝑢 𝑛 = 𝑢 . From (2.35) and (2.38) , we have

0 < 𝑢 ( 𝑡 ) ≤ 𝑢 𝑛 ( 𝑡 ) ≤ ̂ 𝑢 𝑛 ( 𝑡 ) ≤ ̂ 𝑢 ( 𝑡 ) f o r 𝑡 ∈ ( 0 , 1 ) . ( 2 . 3 9 ) Also for 𝑣 ∈ 𝐷 ̂ 𝑢 𝑛 𝑢 𝑛 we have

− 𝑢 𝑛 ( 𝑡 ) + 𝛾 ( 𝑡 ) 𝑢 𝑛 ( 𝑡 ) = − 𝑔 1 𝑡 , 𝑢 𝑛 + 𝜆 ℎ 1 𝑡 , 𝑢 𝑛 + 𝛾 ( 𝑡 ) 𝑢 𝑛 ( 𝑡 ) ≤ − 𝑔 1 ( 𝑡 , 𝑣 ) + 𝜆 ℎ 1 ( 𝑡 , 𝑣 ) + 𝛾 ( 𝑡 ) 𝑣 ( 𝑡 ) ≤ − 𝑔 1 1 𝑡 , 𝑛 + 𝑣 + 𝜆 ℎ 1 1 ( 𝑡 , 𝑣 ) + 𝛾 ( 𝑡 ) 𝑣 ( 𝑡 ) ≤ 𝑔 𝑡 , 𝑛 + 𝑣 + 𝜆 ℎ ( 𝑡 , 𝑣 ) + 𝛾 ( 𝑡 ) 𝑣 ( 𝑡 ) f o r 𝑡 ∈ ( 0 , 1 ) , − ̂ 𝑢 𝑛 ( 𝑡 ) + 𝛾 ( 𝑡 ) ̂ 𝑢 𝑛 ( 𝑡 ) ≥ 𝑔 2 1 𝑡 , 𝑛 + ̂ 𝑢 𝑛 + 𝜆 ℎ 1 𝑡 , 𝑢 + 𝜆 ℎ 2 𝑡 , ̂ 𝑢 𝑛 + 𝛾 ( 𝑡 ) ̂ 𝑢 𝑛 ( 𝑡 ) ≥ 𝑔 2 1 𝑡 , 𝑛 + ̂ 𝑢 𝑛 + 𝛾 ( 𝑡 ) ̂ 𝑢 𝑛 ( 𝑡 ) + 𝜆 ℎ 2 𝑡 , ̂ 𝑢 𝑛 ≥ 𝑔 2 1 𝑡 , 𝑛 + 𝑣 + 𝛾 ( 𝑡 ) 𝑣 ( 𝑡 ) + 𝜆 ℎ