Advances in Difference Equations
VolumeΒ 2009Β (2009), Article IDΒ 623932, 21 pages
doi:10.1155/2009/623932
Research Article

Positive Solution to a Singular 𝑝 -Laplacian BVP with Sign-Changing Nonlinearity Involving Derivative on Time Scales

You-Hui Su,1Β Subei Li,1Β and Can-Yun Huang2

1School of Mathematics and Physics, Xuzhou Institute of Technology, Xuzhou, Jiangsu 221008, China
2Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou 730050, China

Received 3 January 2009; Revised 4 March 2009; Accepted 24 March 2009

Academic Editor: AlbertoΒ Cabada

Copyright Β© 2009 You-Hui Su 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

Let 𝕋 be a time scale such that 0 , 𝑇 ∈ 𝕋 . By using a monotone iterative method, we present some existence criteria for positive solution of a multiple point general Dirichlet-Robin BVP on time scales with the singular sign-changing nonlinearity. These results are even new for the corresponding differential ( 𝕋 = ℝ ) and difference equation ( 𝕋 = β„€ ) as well as in general time scales setting. As an application, an example is given to illustrate the results. The interesting point here is that the sign-changing nonlinear term is involved with the first-order derivative explicitly, and the singularity may occur at 𝑒 = 0 , 𝑑 = 0 , and 𝑑 = 𝑇 .

1. Introduction

Initiated by Hilger in his Ph.D. thesis [1] in 1988, the theory of time scales has been improved greatly ever since. In particular, considerable works have been made in the existence problems of solutions of dynamic systems on time scales, for details, see [2–12] and the references therein. The reason for that lies in two aspects. On one hand, the time scales approach not only unifies differential and difference equations, but also solves other problems that are a mix of stop-start and continuous behavior. On the other hand, the time scales calculus has a tremendous potential for application, for example, Hoffacker et al. have used the theory to model how students suffering from the eating disorder bulimia are influenced by their college friends. With the theory on time scales, they can model how the number of sufferers changes during the continuous college term as well as during long breaks [13]. Moreover, the theory is widely applied to the research of biology, heat transfer, stock market, wound healing and epidemic models [3, 13–16], and so forth.

Here and hereafter, we denote πœ‘ 𝑝 ( 𝑒 ) as 𝑝 -Laplacian operator, that is, πœ‘ 𝑝 ( 𝑒 ) = | 𝑒 | 𝑝 βˆ’ 2 𝑒 for 𝑝 > 1 and ( πœ‘ 𝑝 ) βˆ’ 1 = πœ‘ π‘ž , where 1 / 𝑝 + 1 / π‘ž = 1 . We make the blanket assumption that 0, 𝑇 are points in 𝕋 , by an interval ( 0 , 𝑇 ) 𝕋 we always mean ( 0 , 𝑇 ) ∩ 𝕋 . Other types of interval are defined similarly.

In [17], Su et al. concerned with the π‘š -point singular 𝑝 -Laplacian boundary value problem of the form ξ€· πœ‘ 𝑝 ξ€· 𝑒 Ξ” ( 𝑑 ) ξ€Έ ξ€Έ βˆ‡ + π‘ž ( 𝑑 ) 𝑓 ( 𝑑 , 𝑒 ( 𝑑 ) ) = 0 , 𝑑 ∈ ( 0 , 𝑇 ) 𝕋 , 𝑒 ( 0 ) = 0 , 𝑒 ( 𝑇 ) βˆ’ π‘š βˆ’ 2  𝑖 = 1 πœ“ 𝑖 ξ€· 𝑒 ξ€· πœ‰ 𝑖 ξ€Έ ξ€Έ = 0 , ( 1 . 1 ) and obtained some existence criteria for positive solutions of boundary value problem (1.1). Yet, the singularity of nonlinear term of boundary value problem (1.1) is only occur at 𝑒 = 0 . As a result, they failed to further provide comprehensible results of the singularity that may occur at 𝑒 = 0 , 𝑑 = 0 , or 𝑑 = 𝑇 . Now, it is natural to consider the existence of positive solutions of 𝑝 -Laplacian dynamic equations with the singularity that may occur at 𝑒 = 0 , 𝑑 = 0 , and 𝑑 = 𝑇 in all respects.

For the existence problems of positive solutions of singular 𝑝 -Laplacian boundary value problem with sign changing nonlinearity on time scales, some authors have obtained a few results, for details, see [17–20] and the references therein. It is also noted that the above-mentioned references [17–20] only considered the existence of positive solutions of boundary value problems with nonlinear terms that are not involved with first-order derivative explicitly. Naturally, it is quite necessary to consider that the existence of positive solutions for 𝑝 -Laplacian dynamic equations with the nonlinear term is involved with the first-order derivative explicitly.

Motivated by the above-mentioned ideas, we all-sidedly consider the multiple point singular 𝑝 -Laplacian boundary value problem on time scales of the form ξ€· πœ‘ 𝑝 ξ€· 𝑒 Ξ” ( 𝑑 ) ξ€Έ ξ€Έ βˆ‡ ξ€· + π‘ž ( 𝑑 ) 𝑓 𝑑 , 𝑒 ( 𝑑 ) , 𝑒 Ξ” ξ€Έ ( 𝑑 ) = 0 , 𝑑 ∈ ( 0 , 𝑇 ) 𝕋 , ( 1 . 2 ) 𝑒 ( 0 ) = 0 , π‘š 1  𝑗 = 1 πœ™ 𝑗 ξ€· 𝑒 ξ€· πœ‰ ξ…ž 𝑗 βˆ’ ξ€Έ ξ€Έ π‘š 2  𝑖 = 1 πœ“ 𝑖 ξ€· 𝑒 Ξ” ξ€· πœ‰ 𝑖 ξ€Έ ξ€Έ = 0 , π‘š 1 , π‘š 2 ∈ { 1 , 2 , … } , ( 1 . 3 ) where πœ‘ 𝑝 ( 𝑒 ) = | 𝑒 | 𝑝 βˆ’ 2 𝑒 for 𝑝 > 1 , πœ™ 𝑗 , πœ“ 𝑖 ∢ ℝ β†’ ℝ are continuous, nondecreasing and πœ™ 𝑗 , πœ“ 𝑖 may be nonlinear, 0 < πœ‰ ξ…ž 1 < πœ‰ ξ…ž 2 < β‹― < πœ‰ ξ…ž π‘š 1 βˆ’ 1 < πœ‰ ξ…ž π‘š 1 = 𝑇 , and 0 < πœ‰ 1 < πœ‰ 2 < β‹― < πœ‰ π‘š 2 ≀ 𝑇 . The singularity may occur at 𝑒 = 0 , 𝑑 = 0 ,or 𝑑 = 𝑇 , and the nonlinearity is allowed to change sign and is involved with the first-order derivative explicitly. In particular, the boundary condition (1.3) includes the Dirichlet boundary condition and Robin boundary condition. By applying a monotone iterative method, we obtain some new existence criteria for positive solutions of the boundary value problem (1.2) and (1.3). Our results are even new for the corresponding differential ( 𝕋 = ℝ ) and difference equations ( 𝕋 = β„€ ) as well as in general time scales setting. It has been well known that a second-order dynamic derivative does not approximate a second-order derivative nor a conventional difference; see [21–23]. Thus, it would be interesting that the mathematical results obtained in our article can be conveniently extended for differential or difference equations.

As an application, an example is given to illustrate these results. In particular, our results improve and generalize some known works of Agarwal et al. [24], O’Regan [25] ( 𝑝 = 2 ) , Lü et al. [26, 27] when 𝕋 = ℝ ; extend and include the results of Lü et al. [28] in the case of 𝕋 = ℝ ; if 𝑓 ( 𝑑 , 𝑒 , 𝑒 Ξ” ) = 𝑓 ( 𝑑 , 𝑒 ) , then the works of [17, 19] are only the special cases of our results.

For the convenience of statements, we present some basic definitions and lemmas concerning the calculus on time scales that one needs to read this paper, which can be found in [2, 3]. One of another excellent sources on dynamical systems on time scales is from the book in [29].

A time scale 𝕋 is a nonempty closed subset of ℝ . It follows that the jump operators 𝜎 , 𝜌 ∢ 𝕋 β†’ 𝕋 defined by 𝜎 ( 𝑑 ) = i n f { 𝜏 ∈ 𝕋 ∢ 𝜏 > 𝑑 } and 𝜌 ( 𝑑 ) = s u p { 𝜏 ∈ 𝕋 ∢ 𝜏 < 𝑑 } (supplemented by i n f βˆ… ∢ = s u p 𝕋 and s u p βˆ… ∢ = i n f 𝕋 ) are well defined. The point 𝑑 ∈ 𝕋 is left dense, left scattered, right dense, right scattered if 𝜌 ( 𝑑 ) = 𝑑 , 𝜌 ( 𝑑 ) < 𝑑 , 𝜎 ( 𝑑 ) = 𝑑 , 𝜎 ( 𝑑 ) > 𝑑 , respectively. If 𝕋 has a right-scattered minimum π‘š , define 𝕋 πœ… = 𝕋 βˆ’ { π‘š } ; otherwise, set 𝕋 πœ… = 𝕋 . If 𝕋 has a left-scattered maximum 𝑀 , define 𝕋 πœ… = 𝕋 βˆ’ { 𝑀 } ; otherwise, set 𝕋 πœ… = 𝕋 . The forward graininess is πœ‡ ( 𝑑 ) ∢ = 𝜎 ( 𝑑 ) βˆ’ 𝑑 . Similarly, the backward graininess is 𝜈 ( 𝑑 ) ∢ = 𝑑 βˆ’ 𝜌 ( 𝑑 ) .

A function 𝑓 ∢ 𝕋 β†’ ℝ is ld-continuous provided that it is continuous at left-dense points in 𝕋 and its right-sided limit exists (finite) at right-dense points in 𝕋 . It is known [3] that if 𝑓 is ld-continuous, then there is a function 𝐹 ( 𝑑 ) such that 𝐹 βˆ‡ ( 𝑑 ) = 𝑓 ( 𝑑 ) . In this case, we define ∫ 𝑏 π‘Ž 𝑓 ( 𝜏 ) βˆ‡ 𝜏 = 𝐹 ( 𝑏 ) βˆ’ 𝐹 ( π‘Ž ) .

Throughout this paper, it is assumed that

(H1) 𝑓 ( 𝑑 , π‘₯ , 𝑦 ) ∢ ( 0 , 𝑇 ) 𝕋 Γ— ( 0 , ∞ ) Γ— ( βˆ’ ∞ , ∞ ) β†’ ℝ is continuous;(H2) π‘ž ( 𝑑 ) ∈ 𝐢 ( ( 0 , 𝑇 ) 𝕋 , ( 0 , ∞ ) ) and ∫ 𝑇 0 π‘ž ( 𝑑 ) βˆ‡ 𝑑 < ∞ ;(H3) πœ™ 𝑗 , πœ“ 𝑖 ∢ ℝ β†’ ℝ are continuous and nondecreasing, where 𝑗 = 1 , 2 , … , π‘š 1 and 𝑖 = 1 , 2 , … , π‘š 2 .

2. Existence Results

Let 𝐸 = 𝐢 ( [ 0 , 𝑇 ] 𝕋 , ℝ ) ∩ 𝐢 Ξ” ( ( 0 , 𝑇 ] 𝕋 πœ… , ℝ ) , and define the norm with β€– ξƒ― 𝑒 β€– = m a x s u p [ ] 𝑑 ∈ 0 , 𝑇 𝕋 | | | | 𝑒 ( 𝑑 ) , s u p [ ] 𝑑 ∈ 0 , 𝑇 𝕋 πœ… | | 𝑒 Ξ” | | ξƒ° , ( 𝑑 ) ( 2 . 1 ) then 𝐸 is a Banach space.

To demonstrate existence of positive solutions to problem (1.2) and (1.3), we firstly approximate the singular problem by means of a sequence of nonsingular problems, and by using the lower and upper solutions for nonsingular problem together with Schauders fixed point theorem, and then we establish the existence of solutions to each approximating problem. We remark here that the singularity of the following results occurs at 𝑒 = 0 , 𝑑 = 0 , or 𝑑 = 𝑇 .

Now we state and prove our main result.

Theorem 2.1. Let 𝑛 0 ∈ { 1 , 2 , … } be fixed. Assume that (H1)–(H3) hold and the following conditions are satisfied: (A1)for each 𝑛 ∈ { 𝑛 0 , 𝑛 0 + 1 , … } ≑ β„• 1 , there exists a constant sequence 𝜌 𝑛 such that { 𝜌 𝑛 } is a strictly monotone decreasing sequence with l i m 𝑛 β†’ ∞ 𝜌 𝑛 = 0 , and π‘ž ( 𝑑 ) 𝑓 ( 𝑑 , 𝜌 𝑛 , 0 ) β‰₯ 0 for 𝑑 ∈ [ 1 / 2 𝑛 + 1 , 𝑇 βˆ’ ( 1 / 2 𝑛 + 1 ) ] 𝕋 ;(A2)there exists a function 𝛼 ( 𝑑 ) ∈ 𝐢 [ 0 , 𝑇 ] 𝕋 ∩ 𝐢 Ξ” ( 0 , 𝑇 ] 𝕋 , πœ‘ 𝑝 ( 𝛼 Ξ” ( 𝑑 ) ) ∈ 𝐢 βˆ‡ ( 0 , 𝑇 ) 𝕋 such that 𝛼 ( 0 ) = 0 , 𝛼 ( 𝑑 ) > 0 on ( 0 , 𝑇 ] 𝕋 , βˆ‘ π‘š 1 𝑗 = 1 πœ™ 𝑗 ( 𝛼 ( πœ‰ ξ…ž 𝑗 βˆ‘ ) ) βˆ’ π‘š 2 𝑖 = 1 πœ“ 𝑖 ( 𝛼 Ξ” ( πœ‰ 𝑖 ) ) ≀ 0 together with βˆ’ ( πœ‘ 𝑝 ( 𝛼 Ξ” ( 𝑑 ) ) ) βˆ‡ ≀ π‘ž ( 𝑑 ) 𝑓 ( 𝑑 , 𝛼 ( 𝑑 ) , 𝛼 Ξ” ( 𝑑 ) ) for 𝑑 ∈ ( 0 , 𝑇 ) 𝕋 ;(A3)there exists a function 𝛽 ( 𝑑 ) ∈ 𝐢 [ 0 , 𝑇 ] 𝕋 ∩ 𝐢 Ξ” ( 0 , 𝑇 ] 𝕋 πœ… , πœ‘ 𝑝 ( 𝛽 Ξ” ( 𝑑 ) ) ∈ 𝐢 βˆ‡ ( 0 , 𝑇 ) 𝕋 such that 𝛽 ( 𝑑 ) β‰₯ 𝛼 ( 𝑑 ) and 𝛽 ( 𝑑 ) β‰₯ 𝜌 𝑛 0 for 𝑑 ∈ [ 0 , 𝑇 ] 𝕋 , βˆ‘ π‘š 1 𝑗 = 1 πœ™ 𝑗 ( 𝛽 ( πœ‰ ξ…ž 𝑗 βˆ‘ ) ) βˆ’ π‘š 2 𝑖 = 1 πœ“ 𝑖 ( 𝛽 Ξ” ( πœ‰ 𝑖 ) ) > 0 and βˆ’ ( πœ‘ 𝑝 ( 𝛽 Ξ” ( 𝑑 ) ) ) βˆ‡ β‰₯ π‘ž ( 𝑑 ) 𝑓 ( 𝑑 , 𝛽 ( 𝑑 ) , 𝛽 Ξ” ( 𝑑 ) ) for 𝑑 ∈ [ 1 / 2 𝑛 0 + 1 , 𝑇 βˆ’ ( 1 / 2 𝑛 0 + 1 ) ] 𝕋 , with βˆ’ ξ€· πœ‘ 𝑝 ξ€· 𝛽 Ξ” ( 𝑑 ) ξ€Έ ξ€Έ βˆ‡ ξ‚΅ 1 β‰₯ π‘ž ( 𝑑 ) 𝑓 2 𝑛 0 + 1 , 𝛽 ( 𝑑 ) , 𝛽 Ξ” ξ‚Ά ( 𝑑 ) f o r ξ‚΅ 1 𝑑 ∈ 0 , 2 𝑛 0 + 1 ξ‚Ά 𝕋 , βˆ’ ξ€· πœ‘ 𝑝 ξ€· 𝛽 Ξ” ( 𝑑 ) ξ€Έ ξ€Έ βˆ‡ ξ‚΅ 1 β‰₯ π‘ž ( 𝑑 ) 𝑓 𝑇 βˆ’ 2 𝑛 0 + 1 , 𝛽 ( 𝑑 ) , 𝛽 Ξ” ξ‚Ά ( 𝑑 ) f o r ξ‚΅ 1 𝑑 ∈ 𝑇 βˆ’ 2 𝑛 0 + 1 ξ‚Ά , 𝑇 𝕋 . ( 2 . 2 ) Then the boundary value problem (1.2) and (1.3) has a positive solution 𝑒 ( 𝑑 ) ∈ 𝐢 [ 0 , 𝑇 ] 𝕋 ∩ 𝐢 Ξ” ( 0 , 𝑇 ] 𝕋 πœ… , πœ‘ 𝑝 ( 𝑒 Ξ” ( 𝑑 ) ) ∈ 𝐢 βˆ‡ ( 0 , 𝑇 ) 𝕋 , with 𝑒 ( 𝑑 ) β‰₯ 𝛼 ( 𝑑 ) for 𝑑 ∈ [ 0 , 𝑇 ] 𝕋 .

Proof. Let πœ‰ = m i n { πœ‰ 1 , πœ‰ ξ…ž 1 } . Without loss of generality, fix 𝑛 ∈ β„• 1 , we suppose that m i n 𝑑 ∈ [ πœ‰ , 𝑇 ] 𝕋 𝛼 ( 𝑑 ) β‰₯ 𝜌 𝑛 , let 𝑑 𝑛 ∈ ( 0 , πœ‰ ) 𝕋 be such that 𝛼 ξ€· 𝑑 𝑛 ξ€Έ = 𝜌 𝑛 , 𝛼 ( 𝑑 ) ≀ 𝜌 𝑛 f o r ξ€Ί 𝑑 ∈ 0 , 𝑑 𝑛 ξ€» 𝕋 . ( 2 . 3 ) Let 𝛼 𝑛 ξƒ― 𝜌 ( 𝑑 ) = 𝑛 , i f ξ€Ί 𝑑 ∈ 0 , 𝑑 𝑛 ξ€» 𝕋 , 𝛼 ( 𝑑 ) , i f ξ€Ί 𝑑 𝑑 ∈ 𝑛 ξ€» , 𝑇 𝕋 , h e r e 𝛼 ξ€· 𝑑 𝑛 ξ€Έ = 𝜌 𝑛 . ( 2 . 4 ) Assume that 𝑒 𝑛 = [ 1 / 2 𝑛 + 1 , 𝑇 βˆ’ ( 1 / 2 𝑛 + 1 ) ] 𝕋 , πœ” 𝑛 ( ⎧ βŽͺ ⎨ βŽͺ ⎩ ξ‚» 1 𝑑 ) = m a x 2 𝑛 + 1 ξ‚Ό , , 𝑑 f o r ξ‚Έ 1 𝑑 ∈ 0 , 𝑇 βˆ’ 2 𝑛 + 1 ξ‚Ή 𝕋 , ξ‚» 1 m i n 𝑇 βˆ’ 2 𝑛 + 1 ξ‚Ό , , 𝑑 f o r ξ‚Έ 1 𝑑 ∈ 𝑇 βˆ’ 2 𝑛 + 1 ξ‚Ή , 𝑇 𝕋 , 𝑓 𝑛 ξ€½ ξ€· πœ” ( 𝑑 , π‘₯ , 𝑦 ) = m a x 𝑓 ( 𝑑 , π‘₯ , 𝑦 ) , 𝑓 𝑛 . ( 𝑑 ) , π‘₯ , 𝑦 ξ€Έ ξ€Ύ ( 2 . 5 ) We define a sequence β„Ž 𝑛 0 ( 𝑑 , π‘₯ , 𝑦 ) = 𝑓 𝑛 0 ( 𝑑 , π‘₯ , 𝑦 ) and β„Ž 𝑛 ξ€½ 𝑓 ( 𝑑 , π‘₯ , 𝑦 ) = m i n 𝑛 0 ( 𝑑 , π‘₯ , 𝑦 ) , … , 𝑓 𝑛 ξ€Ύ ( 𝑑 , π‘₯ , 𝑦 ) , 𝑛 = 𝑛 0 + 1 , 𝑛 0 + 2 , … . ( 2 . 6 ) Then 𝑓 ( 𝑑 , π‘₯ , 𝑦 ) ≀ β‹― ≀ β„Ž 𝑛 + 1 ( 𝑑 , π‘₯ , 𝑦 ) ≀ β„Ž 𝑛 ( 𝑑 , π‘₯ , 𝑦 ) ≀ β‹― ≀ β„Ž 𝑛 0 ( 𝑑 , π‘₯ , 𝑦 ) , f o r ( 𝑑 , π‘₯ , 𝑦 ) ∈ ( 0 , 𝑇 ) 𝕋 β„Ž Γ— ( 0 , ∞ ) Γ— ( βˆ’ ∞ , + ∞ ) , 𝑛 ( 𝑑 , π‘₯ , 𝑦 ) = 𝑓 ( 𝑑 , π‘₯ , 𝑦 ) , f o r ( 𝑑 , π‘₯ , 𝑦 ) ∈ 𝑒 𝑛 Γ— ( 0 , ∞ ) Γ— ( βˆ’ ∞ , ∞ ) . ( 2 . 7 ) Consider the boundary value problem ξ€· πœ‘ 𝑝 ξ€· 𝑒 Ξ” ( 𝑑 ) ξ€Έ ξ€Έ βˆ‡ + π‘ž ( 𝑑 ) β„Ž βˆ— 𝑛 0 ξ€· 𝑑 , 𝑒 ( 𝑑 ) , 𝑒 Ξ” ξ€Έ ( 𝑑 ) = 0 , 𝑑 ∈ ( 0 , 𝑇 ) 𝕋 , ( 2 . 8 ) 𝑒 ( 0 ) = 𝜌 𝑛 0 , π‘š 1  𝑗 = 1 πœ™ βˆ— 𝑗 ξ€· 𝑒 ξ€· πœ‰ ξ…ž 𝑗 βˆ’ ξ€Έ ξ€Έ π‘š 2  𝑖 = 1 πœ“ βˆ— 𝑖 ξ€· 𝑒 Ξ” ξ€· πœ‰ 𝑖 ξ€Έ ξ€Έ = 𝜌 𝑛 0 , ( 2 . 9 ) where β„Ž βˆ— 𝑛 0 ξ€· 𝑑 , 𝑒 ( 𝑑 ) , 𝑒 Ξ” ξ€Έ = ⎧ βŽͺ ⎨ βŽͺ ⎩ β„Ž ( 𝑑 ) 𝑛 0 ξ€· 𝑑 , 𝛼 𝑛 0 ( 𝑑 ) , 𝛼 Ξ” 𝑛 0 ( ξ€Έ ξ€· 𝛼 𝑑 ) + π‘Ÿ 𝑛 0 ( ξ€Έ 𝑑 ) βˆ’ 𝑒 ( 𝑑 ) , 𝑒 ( 𝑑 ) ≀ 𝛼 𝑛 0 ( β„Ž 𝑑 ) , 𝑛 0 ξ€· 𝑑 , 𝑒 ( 𝑑 ) , 𝑒 Ξ” ξ€Έ ( 𝑑 ) , 𝛼 𝑛 0 β„Ž ( 𝑑 ) ≀ 𝑒 ( 𝑑 ) ≀ 𝛽 ( 𝑑 ) , 𝑛 0 ξ€· 𝑑 , 𝛽 ( 𝑑 ) , 𝛽 Ξ” ξ€Έ ( 𝑑 ) + π‘Ÿ ( 𝛽 ( 𝑑 ) βˆ’ 𝑒 ( 𝑒 ) ) , 𝑒 ( 𝑑 ) β‰₯ 𝛽 ( 𝑑 ) , ( 2 . 1 0 ) πœ™ βˆ— 𝑗 ξ€· 𝑧 ξ…ž 𝑗 ξ€Έ = ⎧ βŽͺ ⎨ βŽͺ ⎩ πœ™ 𝑗 ξ€· 𝛼 ξ€· πœ‰ ξ…ž 𝑗 ξ€Έ ξ€Έ , 𝑧 ξ…ž 𝑗 ≀ 𝛼 𝑛 0 ξ€· πœ‰ ξ…ž 𝑗 ξ€Έ ξ€· πœ‰ = 𝛼 ξ…ž 𝑗 ξ€Έ , πœ™ 𝑗 ξ€· 𝑧 ξ…ž 𝑗 ξ€Έ , 𝛼 𝑛 0 ξ€· πœ‰ ξ…ž 𝑗 ξ€Έ ≀ 𝑧 ξ…ž 𝑗 ξ€· πœ‰ ≀ 𝛽 ξ…ž 𝑗 ξ€Έ , πœ™ 𝑗 ξ€· 𝛽 ξ€· πœ‰ ξ…ž 𝑗 ξ€Έ ξ€Έ , 𝑧 ξ…ž 𝑗 ξ€· πœ‰ β‰₯ 𝛽 ξ…ž 𝑗 ξ€Έ , 𝑗 = 1 , … , π‘š 1 βˆ’ 1 , ( 2 . 1 1 ) πœ“ βˆ— 𝑖 ξ€· 𝑧 𝑖 ξ€Έ = ⎧ βŽͺ ⎨ βŽͺ ⎩ πœ“ 𝑖 ξ€· 𝛼 Ξ” ξ€· πœ‰ 𝑖 ξ€Έ ξ€Έ , 𝑧 𝑖 ≀ 𝛼 Ξ” 𝑛 0 ξ€· πœ‰ 𝑖 ξ€Έ = 𝛼 Ξ” ξ€· πœ‰ 𝑖 ξ€Έ , πœ“ 𝑖 ξ€· 𝑧 𝑖 ξ€Έ , 𝛼 Ξ” 𝑛 0 ξ€· πœ‰ 𝑖 ξ€Έ ≀ 𝑧 𝑖 ≀ 𝛽 Ξ” ξ€· πœ‰ 𝑖 ξ€Έ , πœ“ 𝑖 ξ€· 𝛽 Ξ” ξ€· πœ‰ 𝑖 ξ€Έ ξ€Έ , 𝑧 𝑖 β‰₯ 𝛽 Ξ” ξ€· πœ‰ 𝑖 ξ€Έ , 𝑖 = 1 , … , π‘š 2 , ( 2 . 1 2 ) and π‘Ÿ ( 𝑒 ) ∢ ℝ β†’ [ βˆ’ 1 , 1 ] is the radial retraction function defined by ⎧ βŽͺ ⎨ βŽͺ ⎩ 𝑒 π‘Ÿ ( 𝑒 ) = 𝑒 , | 𝑒 | ≀ 1 , | 𝑒 | , | 𝑒 | > 1 . ( 2 . 1 3 )
Assume that 𝐢 0 [ ] 0 , 𝑇 𝕋 = ξ€½ [ ] 𝑒 ∈ 𝐢 0 , 𝑇 𝕋 ξ€Ύ ∢ 𝑒 ( 0 ) = 0 , 𝐢 Ξ” 𝜌 𝑛 0 [ ] 0 , 𝑇 𝕋 πœ… = ξ€½ 𝑒 ∈ 𝐢 Ξ” [ ] 0 , 𝑇 𝕋 πœ… ∢ 𝑒 ( 0 ) = 𝜌 𝑛 0 ξ€Ύ . ( 2 . 1 4 ) We define the mappings 𝐿 𝑝 , 𝐹 ∢ 𝐢 Ξ” 𝜌 𝑛 0 [ 0 , 𝑇 ] 𝕋 πœ… β†’ 𝐢 0 [ 0 , 𝑇 ] 𝕋 Γ— ℝ to be such that 𝐿 𝑝  πœ‘ 𝑒 ( 𝑑 ) = 𝑝 ξ€· 𝑒 Ξ” ξ€Έ ( 𝑑 ) βˆ’ πœ‘ 𝑝 ξ€· 𝑒 Ξ” ξ€Έ , ( 0 ) π‘š 1  𝑗 = 1 πœ™ βˆ— 𝑗 ξ€· 𝑒 ξ€· πœ‰ ξ…ž 𝑗 ξ€Έ ξƒͺ ,  βˆ’ ξ€œ 𝐹 𝑒 ( 𝑑 ) = 𝑑 0 π‘ž ( π‘₯ ) β„Ž βˆ— 𝑛 0 ξ€· π‘₯ , 𝑒 ( π‘₯ ) , 𝑒 Ξ” ξ€Έ ( π‘₯ ) βˆ‡ π‘₯ , π‘š 2  𝑖 = 1 πœ“ βˆ— 𝑖 ξ€· 𝑒 Ξ” ξ€· πœ‰ 𝑖 ξ€Έ ξ€Έ + 𝜌 𝑛 0 ξƒͺ . ( 2 . 1 5 ) It follows from the Arzela-Ascoli theorem on time scales [30] that 𝐹 is continuous and compact. Also if 𝐿 𝑝 𝑣 = ( 𝑒 , 𝛾 ) , f o r 𝑒 ∈ 𝐢 0 [ ] 0 , 𝑇 𝕋 , 𝛾 = π‘š 1  𝑖 = 1 πœ™ βˆ— 𝑗  𝜌 𝑛 0 + ξ€œ πœ‰ β€² 𝑗 0 πœ‘ π‘ž  𝑒 ( π‘₯ ) βˆ’ π‘š 1  𝑗 = 1 πœ™ βˆ— 𝑗 ξ€· 𝑒 ξ€· πœ‰ ξ…ž 𝑗 ξ€Έ ξƒͺ ξƒͺ , Ξ” π‘₯ ( 2 . 1 6 ) then we have 𝑣 ( 𝑑 ) = 𝜌 𝑛 0 + ∫ 𝑑 0 πœ‘ π‘ž βˆ‘ ( 𝑒 ( π‘₯ ) βˆ’ π‘š 1 𝑗 = 1 πœ™ βˆ— 𝑗 ( 𝑒 ( πœ‰ ξ…ž 𝑗 ) ) ) Ξ” π‘₯ . Hence 𝐿 𝑝 βˆ’ 1 exists and is continuous. It is clear that solving the boundary value problem (2.8) and (2.9) is equivalent to finding a fixed point of 𝑒 = 𝐿 𝑝 βˆ’ 1 𝐹 𝑒 ≑ 𝑁 𝑒 , where 𝑁 = 𝐿 𝑝 βˆ’ 1 𝐹 ∢ 𝐢 Ξ” 𝜌 𝑛 0 [ 0 , 𝑇 ] 𝕋 πœ… β†’ 𝐢 Ξ” 𝜌 𝑛 0 [ 0 , 𝑇 ] 𝕋 πœ… is compact. It follows from Schauder’s fixed point theorem that the boundary value problem (2.8) and (2.9) has a solution 𝑒 𝑛 0 ( 𝑑 ) ∈ 𝐢 Ξ” [ 0 , 𝑇 ] 𝕋 πœ… with πœ‘ 𝑝 ( 𝑒 Ξ” 𝑛 0 ( 𝑑 ) ) ∈ 𝐢 βˆ‡ ( 0 , 𝑇 ) 𝕋 .
In the following, we will show that 𝛼 𝑛 0 ( 𝑑 ) ≀ 𝑒 𝑛 0 ( 𝑑 ) , f o r [ ] 𝑑 ∈ 0 , 𝑇 𝕋 . ( 2 . 1 7 ) Assume that (2.17) is not true, then the function 𝑒 𝑛 0 ( 𝑑 ) βˆ’ 𝛼 𝑛 0 ( 𝑑 ) has a negative minimum for some 𝜏 ∈ ( 0 , 𝑇 ] 𝕋 . We consider two cases, that is, 𝜏 ∈ ( 0 , 𝑇 ) 𝕋 and 𝜏 = 𝑇 .
Case 1. Assume that 𝜏 ∈ ( 0 , 𝑇 ) 𝕋 , then we claim ξ€· πœ‘ 𝑝 ξ€· 𝑒 Ξ” 𝑛 0 ξ€Έ ξ€Έ βˆ‡ ξ€· πœ‘ ( 𝜏 ) β‰₯ 𝑝 ξ€· 𝛼 Ξ” 𝑛 0 ξ€Έ ξ€Έ βˆ‡ ( 𝜏 ) . ( 2 . 1 8 ) Since 𝑒 𝑛 0 ( 𝑑 ) βˆ’ 𝛼 𝑛 0 ( 𝑑 ) has a negative minimum for some 𝜏 ∈ ( 0 , 𝑇 ) 𝕋 , we have 𝑒 Ξ” 𝑛 0 ( 𝜏 ) βˆ’ 𝛼 Ξ” 𝑛 0 ( 𝜏 ) β‰₯ 0 , and there exists a 𝛿 with 𝜏 βˆ’ 𝛿 ∈ [ 0 , 𝜏 ) 𝕋 such that 𝑒 Ξ” 𝑛 0 ( 𝑑 ) βˆ’ 𝛼 Ξ” 𝑛 0 ( 𝑑 ) ≀ 0 for 𝑑 ∈ [ 𝜏 βˆ’ 𝛿 , 𝜏 ) 𝕋 . Thus πœ‘ 𝑝 ξ€· 𝑒 Ξ” 𝑛 0 ( ξ€Έ 𝑑 ) βˆ’ πœ‘ 𝑝 ξ€· 𝛼 Ξ” 𝑛 0 ( ξ€Έ 𝑑 ) ≀ πœ‘ 𝑝 ξ€· 𝑒 Ξ” 𝑛 0 ( ξ€Έ 𝜏 ) βˆ’ πœ‘ 𝑝 ξ€· 𝛼 Ξ” 𝑛 0 ( ξ€Έ , 𝜏 ) f o r 𝑑 ∈ [ 𝜏 βˆ’ 𝛿 , 𝜏 ) 𝕋 , ( 2 . 1 9 ) which leads to πœ‘ 𝑝 ξ€· 𝑒 Ξ” 𝑛 0 ( ξ€Έ 𝑑 ) βˆ’ πœ‘ 𝑝 ξ€· 𝑒 Ξ” 𝑛 0 ( ξ€Έ 𝜏 ) β‰₯ πœ‘ 𝑑 βˆ’ 𝜏 𝑝 ξ€· 𝛼 Ξ” 𝑛 0 ( ξ€Έ 𝑑 ) βˆ’ πœ‘ 𝑝 ξ€· 𝛼 Ξ” 𝑛 0 ( ξ€Έ 𝜏 ) , 𝑑 βˆ’ 𝜏 f o r 𝑑 ∈ [ 𝜏 βˆ’ 𝛿 , 𝜏 ) 𝕋 . ( 2 . 2 0 )
If 𝜏 is left dense, then ξ€· πœ‘ 𝑝 ξ€· 𝑒 Ξ” 𝑛 0 ξ€Έ ξ€Έ βˆ‡ ( 𝜏 ) = l i m 𝑑 ∈ [ 𝜏 βˆ’ 𝛿 , 𝜏 ) β†’ 𝜏 πœ‘ 𝑝 ξ€· 𝑒 Ξ” 𝑛 0 ( ξ€Έ 𝑑 ) βˆ’ πœ‘ 𝑝 ξ€· 𝑒 Ξ” 𝑛 0 ( ξ€Έ 𝜏 ) 𝑑 βˆ’ 𝜏 β‰₯ l i m 𝑑 ∈ [ 𝜏 βˆ’ 𝛿 , 𝜏 ) β†’ 𝜏 πœ‘ 𝑝 ξ€· 𝛼 Ξ” 𝑛 0 ξ€Έ ( 𝑑 ) βˆ’ πœ‘ 𝑝 ξ€· 𝛼 Ξ” 𝑛 0 ξ€Έ ( 𝜏 ) = ξ€· πœ‘ 𝑑 βˆ’ 𝜏 𝑝 ξ€· 𝛼 Ξ” 𝑛 0 ξ€Έ ξ€Έ βˆ‡ ( 𝜏 ) . ( 2 . 2 1 )
If 𝜏 is left scattered, then, by means of (2.20) we obtain ξ€· πœ‘ 𝑝 ξ€· 𝑒 Ξ” 𝑛 0 ξ€Έ ξ€Έ βˆ‡ πœ‘ ( 𝜏 ) = 𝑝 ξ€· 𝑒 Ξ” 𝑛 0 ( ξ€Έ 𝜏 ) βˆ’ πœ‘ 𝑝 ξ€· 𝑒 Ξ” 𝑛 0 ( ξ€Έ 𝜌 ( 𝜏 ) ) β‰₯ πœ‘ 𝜏 βˆ’ 𝜌 ( 𝜏 ) 𝑝 ξ€· 𝛼 Ξ” 𝑛 0 ξ€Έ ( 𝜏 ) βˆ’ πœ‘ 𝑝 ξ€· 𝛼 Ξ” 𝑛 0 ξ€Έ ( 𝜌 ( 𝜏 ) ) = ξ€· πœ‘ 𝜏 βˆ’ 𝜌 ( 𝜏 ) 𝑝 ξ€· 𝛼 Ξ” 𝑛 0 ξ€Έ ξ€Έ βˆ‡ ( 𝜏 ) . ( 2 . 2 2 ) Hence, (2.18) is true.
However, by (2.4), (2.10), and 𝑒 𝑛 0 ( 𝜏 ) < 𝛼 𝑛 0 ( 𝜏 ) , we obtain ξ€· πœ‘ 𝑝 ξ€· 𝑒 Ξ” 𝑛 0 ( 𝜏 ) ξ€Έ ξ€Έ βˆ‡ βˆ’ ξ€· πœ‘ 𝑝 ξ€· 𝛼 Ξ” 𝑛 0 ( 𝜏 ) ξ€Έ ξ€Έ βˆ‡  = βˆ’ π‘ž ( 𝜏 ) β„Ž 𝑛 0 ξ€· 𝜏 , 𝛼 𝑛 0 ( 𝜏 ) , 𝛼 Ξ” 𝑛 0 ξ€Έ ξ€· 𝛼 ( 𝜏 ) + π‘ž ( 𝜏 ) π‘Ÿ 𝑛 0 ( 𝜏 ) βˆ’ 𝑒 𝑛 0 ξ€Έ + ξ€· πœ‘ ( 𝜏 ) 𝑝 ξ€· 𝛼 Ξ” 𝑛 0 ( 𝜏 ) ξ€Έ ξ€Έ βˆ‡ ξ‚„ = ξƒ― βˆ’  π‘ž ( 𝜏 ) β„Ž 𝑛 0 ξ€· 𝜏 , 𝛼 ( 𝜏 ) , 𝛼 Ξ” ξ€Έ ξ€· ( 𝜏 ) + π‘ž ( 𝜏 ) π‘Ÿ 𝛼 ( 𝜏 ) βˆ’ 𝑒 𝑛 0 ξ€Έ + ξ€· πœ‘ ( 𝜏 ) 𝑝 ξ€· 𝛼 Ξ” ( 𝜏 ) ξ€Έ ξ€Έ βˆ‡ ξ‚„ ξ€Ί 𝑑 , 𝜏 ∈ 𝑛 0 ξ€Έ , 𝑇 𝕋 , βˆ’ ξ€Ί π‘ž ( 𝜏 ) β„Ž 𝑛 0 ξ€· 𝜏 , 𝜌 𝑛 0 ξ€Έ ξ€· 𝜌 , 0 + π‘ž ( 𝜏 ) π‘Ÿ 𝑛 0 βˆ’ 𝑒 𝑛 0 ξ€· ( 𝜏 ) ξ€Έ ξ€» , 𝜏 ∈ 0 , 𝑑 𝑛 0 ξ€Έ 𝕋 . ( 2 . 2 3 )
Assume that 𝜏 ∈ [ 1 / 2 𝑛 0 + 1 , 𝑇 βˆ’ ( 1 / 2 𝑛 0 + 1 ) ] 𝕋 , then β„Ž 𝑛 0 ( 𝜏 , π‘₯ , 𝑦 ) = 𝑓 ( 𝜏 , π‘₯ , 𝑦 ) for ( π‘₯ , 𝑦 ) ∈ ( 0 , ∞ ) Γ— ( βˆ’ ∞ , + ∞ ) . It follows from (A1) and (A2) that ξ€· πœ‘ 𝑝 ξ€· 𝑒 Ξ” 𝑛 0 ( 𝜏 ) ξ€Έ ξ€Έ βˆ‡ βˆ’ ξ€· πœ‘ 𝑝 ξ€· 𝛼 Ξ” 𝑛 0 ( 𝜏 ) ξ€Έ ξ€Έ βˆ‡ = ξƒ― βˆ’  ξ€· π‘ž ( 𝜏 ) 𝑓 𝜏 , 𝛼 ( 𝜏 ) , 𝛼 Ξ” ξ€Έ ξ€· ( 𝜏 ) + π‘ž ( 𝜏 ) π‘Ÿ 𝛼 ( 𝜏 ) βˆ’ 𝑒 𝑛 0 ξ€Έ + ξ€· πœ‘ ( 𝜏 ) 𝑝 ξ€· 𝛼 Ξ” ( 𝜏 ) ξ€Έ ξ€Έ βˆ‡ ξ‚„ ξ€Ί 𝑑 , 𝜏 ∈ 𝑛 0 ξ€Έ , 𝑇 𝕋 , βˆ’ ξ€Ί ξ€· π‘ž ( 𝜏 ) 𝑓 𝜏 , 𝜌 𝑛 0 ξ€Έ ξ€· 𝜌 , 0 + π‘ž ( 𝜏 ) π‘Ÿ 𝑛 0 βˆ’ 𝑒 𝑛 0 ξ€· ( 𝜏 ) ξ€Έ ξ€» , 𝜏 ∈ 0 , 𝑑 𝑛 0 ξ€Έ 𝕋 , < 0 . ( 2 . 2 4 ) This is a contraction.
Assume that 𝜏 ∈ ( 0 , ( 1 / 2 𝑛 0 + 1 ) ) 𝕋 βˆͺ ( 𝑇 βˆ’ ( 1 / 2 𝑛 0 + 1 ) , 𝑇 ) 𝕋 , then β„Ž 𝑛 0 ( 𝜏 , π‘₯ , 𝑦 ) = 𝑓 𝑛 0 ξ€½ 𝑓 ξ€· πœ” ( 𝜏 , π‘₯ , 𝑦 ) = m a x ( 𝜏 , π‘₯ , 𝑦 ) , 𝑓 𝑛 0 ξ‚» 𝑓 ξ‚΅ 1 ( 𝜏 ) , π‘₯ , 𝑦 ξ€Έ ξ€Ύ = m a x 2 𝑛 0 + 1 ξ‚Ά ξ‚΅ 1 , π‘₯ , 𝑦 , 𝑓 𝑇 βˆ’ 2 𝑛 0 + 1 ξ‚Ά ξ‚Ό , , π‘₯ , 𝑦 , 𝑓 ( 𝜏 , π‘₯ , 𝑦 ) ( 2 . 2 5 ) in view of (A1), (A2) and π‘ž ( 𝜏 ) > 0 , we have ξ€· πœ‘ 𝑝 ξ€· 𝑒 Ξ” 𝑛 0 ( 𝜏 ) ξ€Έ ξ€Έ βˆ‡ βˆ’ ξ€· πœ‘ 𝑝 ξ€· 𝛼 Ξ” 𝑛 0 ( 𝜏 ) ξ€Έ ξ€Έ βˆ‡ ≀ ⎧ βŽͺ βŽͺ βŽͺ βŽͺ ⎨ βŽͺ βŽͺ βŽͺ βŽͺ ⎩ βˆ’  ξ€· π‘ž ( 𝜏 ) 𝑓 𝜏 , 𝛼 ( 𝜏 ) , 𝛼 Ξ” ξ€Έ + ξ€· πœ‘ ( 𝜏 ) 𝑝 ξ€· 𝛼 Ξ” ( 𝜏 ) ξ€Έ ξ€Έ βˆ‡ ξ‚„ ξ€· βˆ’ π‘ž ( 𝜏 ) π‘Ÿ 𝛼 ( 𝜏 ) βˆ’ 𝑒 𝑛 0 ξ€Έ ξ‚Έ 𝑑 ( 𝜏 ) , 𝜏 ∈ 𝑛 0 1 , 𝑇 βˆ’ 2 𝑛 0 + 1 ξ‚Ά 𝕋 , βˆ’ ξ‚Έ ξ‚΅ 1 π‘ž ( 𝜏 ) 𝑓 2 𝑛 0 + 1 , 𝜌 𝑛 0 ξ‚Ά ξ€· 𝜌 , 0 + π‘ž ( 𝜏 ) π‘Ÿ 𝑛 0 βˆ’ 𝑒 𝑛 0 ξ€Έ ξ‚Ή ξ€· ( 𝜏 ) , 𝜏 ∈ 0 , 𝑑 𝑛 0 ξ€Έ 𝕋 , βˆ’ ξ‚Έ ξ‚΅ 1 π‘ž ( 𝜏 ) 𝑓 𝑇 βˆ’ 2 𝑛 0 + 1 , 𝜌 𝑛 0 ξ‚Ά ξ€· 𝜌 , 0 + π‘ž ( 𝜏 ) π‘Ÿ 𝑛 0 βˆ’ 𝑒 𝑛 0 ξ€Έ ξ‚Ή ξ‚Έ 1 ( 𝜏 ) , 𝜏 ∈ 𝑇 βˆ’ 2 𝑛 0 + 1 ξ‚Ά , 𝑇 𝕋 , < 0 . ( 2 . 2 6 ) This is a contraction.Case 2. Assume that 𝜏 = 𝑇 . That is, 𝛼 𝑛 0 ( 𝑇 ) βˆ’ 𝑒 𝑛 0 ( 𝑇 ) > 0 , this implies πœ™ π‘š 1 ( 𝛼 𝑛 0 ( 𝑇 ) ) βˆ’ πœ™ π‘š 1 ( 𝑒 𝑛 0 ( 𝑇 ) ) > 0 .
From (2.4), (2.9), (2.11), (2.12), and βˆ‘ π‘š 1 𝑗 = 1 πœ™ 𝑗 ( 𝛼 ( πœ‰ ξ…ž 𝑗 βˆ‘ ) ) ≀ π‘š 2 𝑖 = 1 πœ“ 𝑗 ( 𝛼 Ξ” ( πœ‰ 𝑖 ) ) , we have the following three subcases.
(a) If 𝑒 𝑛 0 ( πœ‰ ξ…ž 𝑗 ) ≀ 𝛼 ( πœ‰ ξ…ž 𝑗 ) for 𝑗 = 1 , 2 , … , π‘š 1 βˆ’ 1 and 𝑒 Ξ” 𝑛 0 ( πœ‰ 𝑖 ) ≀ 𝛼 Ξ” ( πœ‰ 𝑖 ) for 𝑖 = 1 , 2 , … , π‘š 2 , then 0 < πœ™ π‘š 1 ξ€· 𝛼 𝑛 0 ξ€Έ ( 𝑇 ) βˆ’ πœ™ π‘š 1 ξ€· 𝑒 𝑛 0 ξ€Έ ( 𝑇 ) = πœ™ π‘š 1 ( 𝛼 ( 𝑇 ) ) βˆ’ πœ™ π‘š 1 ξ€· 𝑒 𝑛 0 ξ€Έ ( 𝑇 ) ≀ βˆ’ π‘š 1 βˆ’ 1  𝑗 = 1 πœ™ 𝑗 ξ€· 𝛼 ξ€· πœ‰ ξ…ž 𝑗 + ξ€Έ ξ€Έ π‘š 2  𝑗 = 1 πœ“ 𝑖 ξ€· 𝛼 Ξ” ξ€· πœ‰ 𝑖 + ξ€Έ ξ€Έ π‘š 1 βˆ’ 1  𝑗 = 1 πœ™ βˆ— 𝑗 ξ€· 𝑒 𝑛 0 ξ€· πœ‰ ξ…ž 𝑗 βˆ’ ξ€Έ ξ€Έ π‘š 2  𝑖 = 1 πœ“ βˆ— 𝑖 ξ€· 𝑒 Ξ” 𝑛 0 ξ€· πœ‰ 𝑖 ξ€Έ ξ€Έ βˆ’ 𝜌 𝑛 0 < βˆ’ π‘š 1 βˆ’ 1  𝑗 = 1 πœ™ 𝑗 ξ€· 𝛼 ξ€· πœ‰ ξ…ž 𝑗 + ξ€Έ ξ€Έ π‘š 2  𝑖 = 1 πœ“ 𝑖 ξ€· 𝛼 Ξ” ξ€· πœ‰ 𝑖 + ξ€Έ ξ€Έ π‘š 1 βˆ’ 1  𝑗 = 1 πœ™ 𝑗 ξ€· 𝛼 ξ€· πœ‰ ξ…ž 𝑗 βˆ’ ξ€Έ ξ€Έ π‘š 2  𝑖 = 1 πœ“ 𝑖 ξ€· 𝛼 Ξ” ξ€· πœ‰ 𝑖 ξ€Έ ξ€Έ = 0 , ( 2 . 2 7 ) this is a contradiction.
(b) If 𝛼 ( πœ‰ ξ…ž 𝑗 ) ≀ 𝑒 𝑛 0 ( πœ‰ ξ…ž 𝑗 ) for 𝑗 = 1 , 2 , … , π‘š 1 βˆ’ 1 and 𝛼 Ξ” ( πœ‰ 𝑖 ) ≀ 𝑒 Ξ” 𝑛 0 ( πœ‰ 𝑖 ) for 𝑖 = 1 , 2 , … , π‘š 2 , then we discuss the four subcases.
Assume that 𝑒 𝑛 0 ( πœ‰ ξ…ž 𝑗 ) ≀ 𝛽 ( πœ‰ ξ…ž 𝑗 ) f o r 𝑗 = 1 , 2 , … , π‘š 1 βˆ’ 1 , then π‘š 1 βˆ’ 1  𝑗 = 1 πœ™ βˆ— 𝑗 ξ€· 𝑒 𝑛 0 ξ€· πœ‰ ξ…ž 𝑗 = ξ€Έ ξ€Έ π‘š 1 βˆ’ 1  𝑗 = 1 πœ™ 𝑗 ξ€· 𝑒 𝑛 0 ξ€· πœ‰ ξ…ž 𝑗 . ξ€Έ ξ€Έ ( 2 . 2 8 )
Assume that