On Nonlinear Boundary Value Problems for Functional Difference Equations with 𝑝-Laplacian

Abstract

Sufficient conditions for the existence of solutions of nonlinear boundary value problems for higher-order functional difference equations with 𝑝-Laplacian are established by making of continuation theorems. We allow 𝑓 to be at most linear, superlinear, or sublinear in obtained results.

1. Introduction

The existence of solutions of boundary value problems for finite difference equations were studied by many authors, one may see the text books [1, 2], the papers [3–5] and the references therein. We present some representative ones, which are the motivations of this paper.

In papers [3, 4], using Krasnoselskii fixed point theorem and Leggett-Williams fixed point theorem, respectively, Karakostas studied the existence of three positive solutions of the problems consisting of the functional differential equation

[Φ(𝑥^′)]^′+𝑐(𝑡)𝑓(𝑡,𝑥(𝑔₁(𝑡)),𝑥(𝑔₂(𝑡)),…,𝑥(𝑔_𝑛(𝑡)))=0,a.a.𝑡∈[0,1] (1.1) and one of the following three pairs of conditions:

𝑥(0)−𝐵₀(𝑥^′(0))=0,𝑥(1)+𝐵₁(𝑥^′(1))=0,𝑥(0)−𝐵₀(𝑥^′(0))=0,𝑥^′𝑥(1)=0,_′(0)=0,𝑥(1)+𝐵₁(𝑥^′(1))=0. (1.2) Here Φ is a Sup-Multiplicative-Like function, see [3, 4]. In [4] to get the main existence theorems the author assumes the validity of

(H1)for each 𝑖=0,1 the function 𝐵_𝑖 is continuous nondecreasing and such that 𝛼𝐵_𝑖(𝛼)≥0 and at least one of the following:(H2)limsup_𝛼→0+(𝐵₀(𝛼)/𝛼)<+∞,(H3)limsup_𝛼→0+(−𝐵₁(−𝛼)/𝛼)<+∞,(H4)sup_𝛼>0(𝐵₀(𝛼)/𝛼)<+∞,(H5)sup_𝛼>0(−𝐵₁(−𝛼)/𝛼)<+∞.

The discrete simulation of BVPs studied in [3, 4] is as follows:

Δ[Φ(Δ𝑥(𝑘))]+𝑐(𝑘)𝑓(𝑘,𝑥(𝑔₁(𝑘)),…,𝑥(𝑔_𝑛(𝑘)))=0,𝑘∈[0,𝑇] (1.3) subject to one of the following boundary conditions:

𝑥(0)−𝐵₀(Δ𝑥(0))=0,𝑥(𝑇+2)+𝐵₁(Δ𝑥(𝑇+1))=0,𝑥(0)−𝐵₀(Δ𝑥(0))=0,Δ𝑥(𝑇+1)=0,Δ𝑥(0)=0,𝑥(𝑇+2)+𝐵₁(Δ𝑥(𝑇+1))=0. (1.4) The question follows: under what conditions above BVP (1.3) has solutions if (H1)–(H5) are not satisfied?

Particular significance lies in the fact that when a BVP is discretized, strange and interesting changes can occur in the solutions. For example, properties such as existence, uniqueness and multiplicity of solutions may not be shared between the continuous differential equation and its related discrete difference equation. Moreover, when investigating difference equations, as opposed to differential equations, basic ideas from calculus are not necessarily available to use, such as the intermediate value theorem, the mean value theorem and Rolle′s theorem. Thus, new challenges are faced and innovation is required [5].

In recent paper [5], Liu studied the solvability of the following problem consisting of the higher-order functional difference equation and boundary conditions

Δ²𝑥(𝑛)=𝑓(𝑛,𝑥(𝑛+1),𝑥(𝑛−𝜏₁(𝑛)),…,𝑥(𝑛−𝜏_𝑚(𝑛))),𝑛∈[0,𝑇−1],𝑎𝑥(0)−𝑏Δ𝑥(0)=0,𝑐𝑥(𝑇+1)+𝑑Δ𝑥(𝑇)=0,𝑥(𝑖)=𝜙(𝑖),𝑖∈[−𝜏,−1],𝑥(𝑖)=𝜓(𝑖),𝑖∈[𝑇+2,𝑇+𝛿], (1.5) where 𝑇≥1, 𝑎,𝑏,𝑐,𝑑∈𝑅 with 𝑎²+𝑏²≠0 and 𝑐²+𝑑²≠0, 𝜏_𝑖(𝑛), 𝑖=1,…,𝑚, are sequences,

𝜏=max{0,max_{𝑛∈[0,𝑇−1]}{𝜏_𝑖(𝑛)}∶𝑖=1,…,𝑚},𝛿=−min{0,min_{𝑛∈[0,𝑇−1]}{𝜏_𝑖(𝑛)}∶𝑖=1,…,𝑚}, (1.6)𝑓(𝑛,𝑢) is continuous in 𝑢=(𝑥₀,…,𝑥_𝑚,𝑥_𝑚+1) for each 𝑛. Two cases, that is, 𝑏𝑐+𝑎𝑑+𝑎𝑐(𝑇+1)=0 or 𝑏𝑐+𝑎𝑑+𝑎𝑐(𝑇+1)≠0 are considered in [6].

Motivated by [3–5], we study the nonlinear boundary value problems for higher-order functional difference equation with 𝑝-Laplacian, that is, the equation

Δ[𝜙(Δ𝑥(𝑛))]=𝑓(𝑛,𝑥(𝑛+1),𝑥(𝑛−𝜏₁(𝑛)),…,𝑥(𝑛−𝜏_𝑚(𝑛))),𝑛∈[0,𝑇−1], (1.7) subject to the following boundary conditions

𝑥(0)−𝐵₀(Δ𝑥(0))=0,𝐵₁(𝑥(𝑇+1))+Δ𝑥(𝑇)=0,𝑥(𝑖)=𝛾(𝑖),𝑖∈[−𝜏,−1],𝑥(𝑖)=𝜓(𝑖),𝑖∈[𝑇+2,𝑇+𝛿], (1.8) where [0,𝑇−1]={0,1,…,𝑇−1}, 𝜙(𝑥)=|𝑥|^𝑝−2𝑥 for 𝑥≠0 and 𝜙(0)=0 with 𝑝>1, the inverse function is denoted by 𝜙⁻¹, 𝑇≥1 an integer, 𝐵₀,𝐵₁ are continuous and satisfy 𝑥𝐵_𝑖(𝑥)≥0 for all 𝑥∈𝑅, 𝑖=0,1, {𝜏_𝑖(𝑛)}, 𝑖=1,…,𝑚, are integer vectors,

𝜏=max{0,max_{𝑛∈[0,𝑇−1]}{𝜏_𝑖(𝑛)}∶𝑖=1,…,𝑚},𝛿=−min{0,min_{𝑛∈[0,𝑇−1]}{𝜏_𝑖(𝑛)}∶𝑖=1,…,𝑚}, (1.9)𝑓(𝑛,𝑢) is continuous about 𝑢=(𝑥₀,…,𝑥_𝑚) for each 𝑛. Boundary condition (1.8) is called nonlinear Sturm-Liouville type conditions.

The purposes of this paper are to establish sufficient conditions for the existence of at least one solutions of BVP (1.7)-(1.8). It is interesting that we allow that 𝑓 to be sublinear, at most linear or superlinear. We do not need the assumptions (H2)–(H5) imposed on 𝐵₀,𝐵₁.

This paper is organized as follows. In Section 2, we give the main results of this paper, and in Section 3, examples to illustrate the main results will be presented.

2. Main Results

To get existence results for solutions of BVP (1.7)-(1.8), we need the following fixed point theorem, which was used to solve multi-point boundary value problems for differential equations in many papers but not used to solve boundary value problems for difference equations.

Let 𝑋 and 𝑌 be real Banach spaces, let 𝐿∶Dom𝐿⊂𝑋→𝑌 be a Fredholm operator of index zero. If Ω is an open bounded subset of 𝑋, Dom𝐿∩Ω≠∅, the map 𝑁∶𝑋→𝑌 will be called 𝐿-compact on Ω if 𝑁(Ω) is bounded and compact.

Lemma 2.1 (see [6]). Let 𝑋 and 𝑌 be Banach spaces. Suppose 𝐿∶Dom𝐿⊂𝑋→𝑌 is a Fredholm operator of index zero with Ker𝐿={0}, 𝑁∶𝑋→𝑌 is 𝐿-compact on any open bounded subset of 𝑋. If Ω⊂𝑋 is an open bounded set with 0∈Ω and 𝐿𝑥≠𝜆𝑁𝑥,∀𝑥∈Dom𝐿∩𝜕Ω,𝜆∈[0,1], (2.1) then there is at least one 𝑥∈Ω so that 𝐿𝑥=𝑁𝑥.
Let 𝑋=𝑅^{𝑇+𝜏+𝛿+1}×𝑅^𝑇+1 be endowed with the norm ‖(𝑥,𝑦)‖=max{max_{𝑛∈[1,𝑇+𝜏+𝛿+1]}|𝑥(𝑛)|,max_{𝑛∈[0,𝑇]}|𝑦(𝑛)|}for(𝑥,𝑦)∈𝑋. (2.2)
Let 𝑌=𝑅^𝑇+1×𝑅^𝑇×𝑅²×𝑅^𝜏×𝑅^𝛿−1 be endowed with the norm ‖(𝑢,𝑣,𝑎,𝑏,𝑠,𝑡)‖=max{max_{𝑛∈[0,𝑇]}|𝑢(𝑛)|,max_{𝑛∈[0,𝑇−1]}|𝑣(𝑛)|,|𝑎|,|𝑏|,max_{𝑛∈[−𝜏,−1]}|𝑠(𝑛)|,max_{𝑛∈[𝑇+2,𝑇+𝛿]}|𝑡(𝑛)|} (2.3) for (𝑢,𝑣,𝑎,𝑏,𝑠,𝑡)∈𝑌.
It is easy to see that 𝑋 and 𝑌 are real Banach spaces.
Let 𝐿∶𝑋→𝑌, and 𝐿=𝑥(𝑛)𝑦(𝑛)Δ𝑥(𝑛),𝑛∈[0,𝑇]Δ𝑦(𝑛),𝑛∈[0,𝑇−1]𝑥(1)𝑦(𝑇)𝑥(𝑖),𝑖∈[−𝜏,−1]𝑥(𝑗),𝑗∈[𝑇+2,𝑇+𝛿], (2.4) and 𝑁∶𝑋→𝑌 by 𝑁=𝜙𝑥(𝑛)𝑦(𝑛)⁻¹(𝑦(𝑛)),𝑛∈[0,𝑇]𝑓(𝑛,𝑥(𝑛+1),𝑥(𝑛−𝜏₁(𝑛)),…,𝑥(𝑛−𝜏_𝑚𝐵(𝑛))),𝑛∈[0,𝑇−1]₀(𝜙⁻¹(𝑦(0)))−𝜙(𝐵₁(𝑥(𝑇+1)))𝛾(𝑖),𝑖∈[−𝜏,−1]𝜓(𝑗),𝑗∈[𝑇+2,𝑇+𝛿] (2.5) for all (𝑥,𝑦)∈𝑋.
Since 𝑓, 𝐵₀,𝐵₁ are continuous, it is easy to show that(i)(𝑥,𝑦)∈Dom𝐿 is a solution of 𝐿(𝑥,𝑦)=𝑁(𝑥,𝑦) implies that 𝑥 is a solution of BVP (1.7)-(1.8),(ii)Ker𝐿={0},(iii)𝐿 is a Fredholm operator of index zero and 𝑁 is 𝐿-compact on each Ω with Ω being an open bounded subset of 𝑋.

Lemma 2.2. (∑^𝑚_𝑖=1𝑎_𝑖)^𝜎≤𝑙_𝜎^𝑚−1(∑^𝑚_𝑖=1𝑎^𝜎_𝑖) for all 𝑎_𝑖≥0 and 𝜎>0, where 𝑙_𝜎 is defined by 𝑙_𝜎=2^𝜎−1 for 𝜎≥1 and 𝑙_𝜎=1 for 𝜎∈(0,1).

Proof. We have the following cases.

Case 1 (𝑚=2). Without loss of generality, suppose 𝑎₁≥𝑎₂. Let 𝑔(𝑥)=(1+𝑥)^𝜎−𝑙_𝜎(1+𝑥^𝜎),𝑥∈[1,+∞), (2.6) then 𝑔(1)=2^𝜎−2𝑙_𝜎=2^𝜎−2^𝜎2=0,𝜎≥1,_𝜎+1−2≤0,𝜎∈(0,1) (2.7) and for 𝑥∈[1,∞), we get 𝑔^′(𝑥)=𝜎𝑥^𝜎−11[(1+𝑥)^𝜎−1−𝑙_𝜎]=𝜎𝑥^𝜎−11[(1+𝑥)^𝜎−1−2^𝜎−1]≤0,𝜎≥1,𝜎𝑥^𝜎−11[(1+𝑥)^𝜎−1−1]≤0,𝜎∈(0,1). (2.8) We get that 𝑔(𝑥)≤𝑔(1) for all 𝑥≥1 and so (1+𝑥)^𝜎≤𝑙_𝜎(1+𝑥^𝜎) for all 𝑥∈[1,+∞). Hence (𝑎₁+𝑎₂)^𝜎=𝑎^𝜎₂(1+(𝑎₁/𝑎₂))^𝜎≤𝑙_𝜎𝑎^𝜎₂[1+(𝑎₁/𝑎₂)^𝜎)=𝑙_𝜎(𝑎^𝜎₁+𝑎^𝜎₂).

Case 2 (𝑚>2). It is easy to see that (^𝑚_𝑖=1𝑎_𝑖)^𝜎=(𝑎₁+^𝑚_𝑖=2𝑎_𝑖)^𝜎≤𝑙_𝜎(𝑎^𝜎₁+(^𝑚_𝑖=2𝑎_𝑖)^𝜎)≤𝑙_𝜎[𝑎^𝜎₁+𝑙_𝜎(𝑎^𝜎₂+(^𝑚_𝑖=3𝑎_𝑖)^𝜎)]≤𝑙²_𝜎(𝑎^𝜎₁+𝑎^𝜎₂+(^𝑚_𝑖=3𝑎_𝑖)^𝜎)≤⋯≤𝑙_𝜎^𝑚−1(^𝑚_𝑖=1𝑎^𝜎_𝑖). (2.9) The proof is complete.

Theorem 2.3. Suppose that(A)𝑓, 𝐵₀,𝐵₁ are continuous, and 𝑥𝐵_𝑖(𝑥)≥0 for all 𝑥∈𝑅 and 𝑖=0,1;(B)there exist numbers 𝛽>0, 𝜃>1, nonnegative sequences 𝑝_𝑖(𝑛),𝑟(𝑛)(𝑖=0,…,𝑚), functions 𝑔(𝑛,𝑥₀,…,𝑥_𝑚), ℎ(𝑛,𝑥₀,…,𝑥_𝑚) such that𝑓(𝑛,𝑥₀,…,𝑥_𝑚)=𝑔(𝑛,𝑥₀,…,𝑥_𝑚)+ℎ(𝑛,𝑥₀,…,𝑥_𝑚),𝑔(𝑛,𝑥₀,𝑥₁,…,𝑥_𝑚)𝑥₀≥𝛽|𝑥₀|^𝜃+1,|ℎ(𝑛,𝑥₀,…,𝑥_𝑚)|≤^𝑚_𝑖=0𝑝_𝑖(𝑛)|𝑥_𝑖|^𝜃+𝑟(𝑛), (2.10) for all 𝑛∈{1,…,𝑇}, (𝑥₀,𝑥₁,…,𝑥_𝑚)∈𝑅^𝑚+1.
Then BVP (1.7)-(1.8) has at least one solution if ‖𝑝₀‖+𝑇^{𝑚𝜃/(𝜃+1)}_𝑖=1‖𝑝_𝑖‖<𝛽. (2.11)

Proof. To apply Lemma 2.1, we divide the proof into two steps.
Let Ω₁={(𝑥,𝑦)∶𝐿(𝑥,𝑦)=𝜆𝑁(𝑥,𝑦),((𝑥,𝑦),𝜆)∈[Dom𝐿]×[0,1]}. For (𝑥,𝑦)∈Ω₁, we have 𝐿(𝑥,𝑦)=𝜆𝑁(𝑥,𝑦), 𝜆∈[0,1], then Δ𝑥(𝑛)=𝜆𝜙⁻¹(𝑦(𝑛)),𝑛∈[0,𝑇],Δ𝑦(𝑛)=𝜆𝑓(𝑛,𝑥(𝑛+1),𝑥(𝑛−𝜏₁(𝑛)),…,𝑥(𝑛−𝜏_𝑚(𝑛))),𝑛∈[0,𝑇−1],𝑥(0)=𝜆𝐵₀(𝜙⁻¹(𝑦(0))),𝑦(𝑇)=−𝜆𝜙(𝐵₁(𝑥(𝑇+1))),𝑥(𝑖)=𝜆𝛾(𝑖),𝑖∈[−𝜏,−1],𝑥(𝑗)=𝜆𝜓(𝑗),𝑗∈[𝑇+2,𝑇+𝛿]. (2.12)
Step 1. We will show that if 𝐿(𝑥,𝑦)=𝜆𝑁(𝑥,𝑦), for some 𝜆∈[0,1], then 𝑥 is bounded. Indeed, we see that Δ[𝜙(Δ𝑥(𝑛))]𝑥(𝑛+1)=𝜆𝜙(𝜆)𝑓(𝑛,𝑥(𝑛+1),𝑥(𝑛−𝜏₁(𝑛)),…,𝑥(𝑛−𝜏_𝑚(𝑛)))𝑥(𝑛+1). (2.13) Since 𝑥(0)=𝜆𝐵₀(𝜙⁻¹(𝑦(0))),𝑦(𝑇)=−𝜆𝜙(𝐵₁(𝑥(𝑇+1))), (2.14) we get 𝑥(0)=𝜆𝐵₀(Δ𝑥(0)𝜆),Δ𝑥(𝑇)𝜆=−𝜙⁻¹(𝜆)𝐵₁(𝑥(𝑇+1)). (2.15)
It is easy to see from (2.12) and the definition of 𝐵₀,𝐵₁ and 𝜙 that
^{𝑇−1𝑛=0}=Δ[𝜙(Δ𝑥(𝑛))]𝑥(𝑛+1)_{𝑇−1𝑛=0}=[𝜙(Δ𝑥(𝑛+1))−𝜙(Δ𝑥(𝑛))][𝑥(𝑛+2)−Δ𝑥(𝑛+1)]_{𝑇−1𝑛=0}[𝜙(Δ𝑥(𝑛+1))𝑥(𝑛+2)−𝜙(Δ𝑥(𝑛))𝑥(𝑛+1)]−^𝑇−1_𝑛=0𝜙(Δ𝑥(𝑛+1))Δ𝑥(𝑛+1)=𝜙(Δ𝑥(𝑇))𝑥(𝑇+1)−𝜙(Δ𝑥(0))𝑥(1)−^𝑇−1_𝑛=0𝜙(Δ𝑥(𝑛+1))Δ𝑥(𝑛+1)=−𝜙(𝜆𝜙⁻¹(𝜆)𝐵₁−(𝑥(𝑇+1)))𝑥(𝑇+1)−𝜙(Δ𝑥(0))Δ𝑥(0)−𝜙(Δ𝑥(0))𝑥(0)_{𝑇−1𝑛=0}𝜙(Δ𝑥(𝑛+1))Δ𝑥(𝑛+1)=−𝜙(𝜆𝜙⁻¹(𝜆)𝐵₁(𝑥(𝑇+1)))𝑥(𝑇+1)−𝜙(Δ𝑥(0))𝜆𝐵₀(Δ𝑥(0)𝜆)−^𝑇−1_𝑛=0𝜙(Δ𝑥(𝑛+1))Δ𝑥(𝑛+1)−𝜙(Δ𝑥(0))Δ𝑥(0)≤0. (2.16) So we get ^𝑇−1_𝑛=0𝑓(𝑛,𝑥(𝑛+1),𝑥(𝑛−𝜏₁(𝑛)),…,𝑥(𝑛−𝜏_𝑚(𝑛)))𝑥(𝑛+1)≤0. (2.17) It follows from the assumptions that 𝛽^𝑇−1_𝑛=0|𝑥(𝑛+1)|^𝜃+1≤^𝑇−1_𝑛=0𝑔(𝑛,𝑥(𝑛+1),𝑥(𝑛−𝜏₁(𝑛)),…,𝑥(𝑛−𝜏_𝑚(𝑛))𝑥(𝑛+1)≤−^𝑇−1_𝑛=0ℎ(𝑛,𝑥(𝑛+1),𝑥(𝑛−𝜏₁(𝑛)),…,𝑥(𝑛−𝜏_𝑚≤(𝑛))𝑥(𝑛+1)_{𝑇−1𝑛=0}|ℎ(𝑛,𝑥(𝑛+1),𝑥(𝑛−𝜏₁(𝑛)),…,𝑥(𝑛−𝜏_𝑚≤(𝑛))||𝑥(𝑛+1)|_{𝑇−1𝑛=0}𝑝₀(𝑛)|𝑥(𝑛+1)|^𝜃+1+^𝑚_𝑖=1^𝑇−1_𝑛=0𝑝_𝑖(𝑛)|𝑥(𝑛−𝜏_𝑖(𝑛))|^𝜃|𝑥(𝑛+1)|+^𝑇−1_𝑛=0𝑟(𝑛)|𝑥(𝑛+1)|≤‖𝑝₀‖^𝑇−1_𝑛=0|𝑥(𝑛+1)|^𝜃+1+^𝑚_𝑖=1‖𝑝_𝑖‖^𝑇−1_𝑛=0|𝑥(𝑛−𝜏_𝑖(𝑛))|^𝜃|𝑥(𝑛+1)|+‖𝑟‖^𝑇−1_𝑛=0|𝑥(𝑛+1)|. (2.18) For 𝑥_𝑖≥0,𝑦_𝑖≥0, we have Holder’s inequality ^𝑠_𝑖=1𝑥_𝑖𝑦_𝑖≤(^𝑠_𝑖=1𝑥^𝑝_𝑖)^1/𝑝(^𝑠_𝑖=1𝑦^𝑞_𝑖)^1/𝑞,1𝑝+1𝑞=1,𝑞>0,𝑝>0. (2.19) It follows from Lemma 2.2 that 𝛽^𝑇−1_𝑛=0|𝑥(𝑛+1)|^𝜃+1≤‖𝑝₀‖^𝑇−1_𝑛=0|𝑥(𝑛+1)|^𝜃+1+‖𝑟‖𝑇^{𝜃/(𝜃+1)}(^𝑇−1_𝑛=0|𝑥(𝑛+1)|^𝜃+1)^1/(𝜃+1)+^𝑚_𝑖=1‖𝑝_𝑖‖(^𝑇−1_𝑛=0|𝑥(𝑛−𝜏_𝑖(𝑛))|^𝜃+1)^{𝜃/(𝜃+1)}(^𝑇−1_𝑛=0|𝑥(𝑛+1)|^𝜃+1)^1/(𝜃+1)=‖𝑝₀‖^𝑇−1_𝑛=0|𝑥(𝑛+1)|^𝜃+1+‖𝑟‖𝑇^{𝜃/(𝜃+1)}(^𝑇−1_𝑛=0|𝑥(𝑛+1)|^𝜃+1)^1/(𝜃+1)+^𝑚_𝑖=1‖𝑝_𝑖‖(_{𝑢∈{𝑛−𝜏𝑖(𝑛)−1∶𝑛=0,…,𝑇−1}}|𝑥(𝑢+1)|^𝜃+1)^{𝜃/(𝜃+1)}(^𝑇−1_𝑛=0|𝑥(𝑛+1)|^𝜃+1)^1/(𝜃+1)≤‖𝑝₀‖^𝑇−1_𝑛=0|𝑥(𝑛+1)|^𝜃+1+‖𝑟‖𝑇^{𝜃/(𝜃+1)}(^𝑇−1_𝑛=0|𝑥(𝑛+1)|^𝜃+1)^1/(𝜃+1)+[𝑇^{𝑚𝜃/(𝜃+1)}_𝑖=1‖𝑝_𝑖‖(^𝑇−1_𝑛=0|𝑥(𝑛+1)|^𝜃+1)^{𝜃/(𝜃+1)}+𝑇^{𝑚𝜃/(𝜃+1)}_𝑖=1‖𝑝_𝑖‖(^𝑇+𝛿_𝑛=𝑇|𝜆𝜓(𝑛+1)|^𝜃+1)^{𝜃/(𝜃+1)}+𝑇^{𝑚𝜃/(𝜃+1)}_𝑖=1‖𝑝_𝑖‖(⁻¹_𝑛=−𝜏|𝜆𝛾(𝑛+1)|^𝜃+1)^{𝜃/(𝜃+1)}](^𝑇−1_𝑛=0|𝑥(𝑛+1)|^𝜃+1)^1/(𝜃+1)≤‖𝑝₀‖^𝑇−1_𝑛=0|𝑥(𝑛+1)|^𝜃+1+‖𝑟‖𝑇^{𝜃/(𝜃+1)}(^𝑇−1_𝑛=0|𝑥(𝑛+1)|^𝜃+1)^1/(𝜃+1)+𝑇^{𝑚𝜃/(𝜃+1)}_𝑖=1‖𝑝_𝑖‖^𝑇−1_𝑛=0|𝑥(𝑛+1)|^𝜃+1+𝑇^{𝑚𝜃/(𝜃+1)}_𝑖=1‖𝑝_𝑖‖(^𝑇+𝛿_𝑛=𝑇|𝜓(𝑛+1)|^𝜃+1)^{𝜃/(𝜃+1)}(^𝑇−1_𝑛=0|𝑥(𝑛+1)|^𝜃+1)^1/(𝜃+1)+𝑇^{𝑚𝜃/(𝜃+1)}_𝑖=1‖𝑝_𝑖‖(⁻¹_𝑛=−𝜏|𝛾(𝑛+1)|^𝜃+1)^{𝜃/(𝜃+1)}(^𝑇−1_𝑛=0|𝑥(𝑛+1)|^𝜃+1)^1/(𝜃+1). (2.20)(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13) It follows from (2.11) that there is 𝑀₁>0 such that ∑^𝑇−1_𝑢=0|𝑥(𝑢+1)|^𝜃+1≤𝑀₁.
Hence |𝑥(𝑛+1)|≤𝑀₁^1/(𝜃+1) for all 𝑛∈{0,…,𝑇−1}, which proves Step 1.Step 2. We will show that the set Ω₁ is bounded. We first prove that there exists 𝑘∈[0,𝑇−1] such that 𝑦(𝑘)𝑦(𝑘+1)≤0. In fact, if 𝑦(𝑖)>0 for all 𝑖∈[0,𝑇], then Δ𝑥(𝑛)=𝜆𝜙⁻¹(𝑦(𝑛)),𝑛∈[0,𝑇] (2.21) implies that 𝑥(𝑖) is increasing on [0,𝑇+1], so 𝑥(0)<𝑥(𝑇+1). Then assumption (A) and 𝑥(0)=𝜆𝐵₀(𝜙⁻¹(𝑦(0))),𝑦(𝑇)=−𝜆𝜙(𝐵₁(𝑥(𝑇+1))) (2.22) imply that 𝑥(0)>0 and 𝑥(𝑇+1)<0. This contradicts 𝑥(0)<𝑥(𝑇+1). If 𝑦(𝑖)<0 for all 𝑖∈[0,𝑇], the similar contradiction can be deduced. Hence there exists 𝑘∈[0,𝑇−1] such that 𝑦(𝑘)𝑦(𝑘+1)≤0.
It follows that there is a constant 𝜉∈[𝑘,𝑘+1] such that
𝑦(𝑘−1)−𝑦(𝑘)=𝑘+1−𝑘0−𝑦(𝑘).𝜉−𝑘 (2.23) Then |𝑦(𝑘)|≤|Δ𝑦(𝑘)|. Hence |𝑦(𝑘)|≤|Δ𝑦(𝑘)|=𝜆|𝑓(𝑘,𝑥(𝑘+1),𝑥(𝑘−𝜏₁(𝑘)),…,𝑥(𝑘−𝜏_𝑚(𝑘)))|≤max_{𝑛∈[0,𝑇−1],|𝑥𝑖|≤𝑀11/(𝜃+1),𝑖=0,…,𝑚}|𝑓(𝑛,𝑥₀,…,𝑥_𝑚)|=∶𝑀₂. (2.24) Then for 𝑛∈[𝑘+1,𝑇] one sees that |𝑦(𝑛)|=|𝑦(𝑘)+^𝑛−1_𝑠=𝑘Δ𝑦(𝑠)|≤𝑀₂+^𝑛−1_𝑠=𝑘|Δ𝑦(𝑠)|≤𝑀₂+𝑇max_{𝑛∈[0,𝑇−1],|𝑥𝑖|≤𝑀11/(𝜃+1),𝑖=0,…,𝑚}|𝑓(𝑛,𝑥₀,…,𝑥_𝑚)|=(𝑇+1)𝑀₂. (2.25) For 𝑛∈[0,𝑘−1], we get |𝑦(𝑛)|=|𝑦(𝑘)−^𝑘−1_𝑠=𝑛Δ𝑦(𝑠)|≤𝑀₂+^𝑘−1_𝑠=𝑛|Δ𝑦(𝑠)|≤𝑀₂+𝑇max_{𝑛∈[0,𝑇−1],|𝑥𝑖|≤𝑀11/(𝜃+1),𝑖=0,…,𝑚}|𝑓(𝑛,𝑥₀,…,𝑥_𝑚)|=(𝑇+1)𝑀₂. (2.26) It follows that ‖𝑦‖≤(𝑇+1)𝑀₂. (2.27) Then |𝑥(0)|=|𝜆𝐵₀(𝜙⁻¹(𝑦(0)))|≤|𝐵₀(𝜙⁻¹(𝑦(0)))|≤max_{|𝑥|≤(𝑇+1)𝑀2}|𝐵₀(𝜙⁻¹(𝑥))|. (2.28) On the other hand, since Δ𝑥(𝑇)=𝜆𝜙⁻¹(𝑦(𝑇)) implies that |Δ𝑥(𝑇)|≤𝜙⁻¹(|𝑦(𝑇)|)≤𝜙⁻¹((𝑇+1)𝑀₂), (2.29) then |𝑥(𝑇+1)|=|Δ𝑥(𝑇)+𝑥(𝑇)|≤|Δ𝑥(𝑇)|+|𝑥(𝑇)|≤𝜙⁻¹((𝑇+1)𝑀₂)+𝑀₁^1/(𝜃+1). (2.30) It follows that ‖𝑥‖≤max{max_{𝑖∈[𝑇+2,𝑇+𝛿]}|𝜓(𝑖)|,max_{𝑖∈[−𝜏,−1]}|𝛾(𝑖)|,𝜙⁻¹((𝑇+1)𝑀₂)+𝑀₁^1/(𝜃+1)}. (2.31) Hence ‖(𝑥,𝑦)‖≤max{(𝑇+1)𝑀₂,max_{𝑖∈[𝑇+2,𝑇+𝛿]}|𝜓(𝑖)|,max_{𝑖∈[−𝜏,−1]}|𝛾(𝑖)|,𝜙⁻¹((𝑇+1)𝑀₂)+𝑀₁^1/(𝜃+1)}. (2.32)Step 2 is proved, namely Ω₁ is bounded. Let Ω⊃Ω₁ be an open bounded subset of 𝑋 centered at zero, it is easy to see that 𝐿(𝑥,𝑦)≠𝜆𝑁(𝑥,𝑦) for all (𝑥,𝑦)∈𝜕Ω and 𝜆∈(0,1). It follows from Lemma 2.1 that equation 𝐿(𝑥,𝑦)=𝑁(𝑥,𝑦) has at least one solution (𝑥,𝑦), then 𝑥 is a solution of BVP (1.7)-(1.8). The proof is complete.

3. An Example

In this section, we present an example to illustrate the main results in Section 2.

Example 3.1. Consider the following BVP: Δ(𝜙(Δ𝑥(𝑛)))=𝛽[𝑥(𝑛+1)]^2𝑘+1+^𝑚_𝑖=1𝑝_𝑖(𝑛)[𝑥(𝑛−𝑖)]^2𝑘+1+𝑟(𝑛),𝑛∈[0,𝑇−1],𝑥(0)−5[Δ𝑥(0)]⁵=0,6[𝑥(𝑇+1)]³+Δ𝑥(𝑇)=0,𝑥(𝑛)=𝛾(𝑛),𝑛∈[−𝑚,−1], (3.1) where 𝜙(𝑥)=|𝑥|^𝑝−2𝑥 for 𝑥≠0 and 𝜙(0)=0 with 𝑝>1 a constant, 𝑇≥1 is an integer, 𝛽>0, 𝑝_𝑖(𝑛),𝑟(𝑛) are sequences. Corresponding to the assumptions of Theorem 2.3, we set 𝑔(𝑛,𝑥₀,…,𝑥_𝑚)=𝛽[𝑥₀]^2𝑘+1,ℎ(𝑛,𝑥₀,…,𝑥_𝑚)=^𝑚_𝑖=1𝑝_𝑖(𝑛)𝑥_𝑖^2𝑘+1𝐵+𝑟(𝑛),𝜃=2𝑘+1,₀(𝑥)=5𝑥⁵,𝐵₁(𝑥)=6𝑥³. (3.2) It is easy to see that assumptions (A) and (B) in Theorem 2.3 hold. It follows from Theorem 2.3 that (3.1) has at least one solution if ‖𝑝₀‖+𝑇^{𝜃/(𝜃+1)}∑^𝑚_𝑖=1‖𝑝_𝑖‖<𝛽.

Remark 3.2. The BVP in Example 3.1 cannot be solved by the theorems in [3–5] since the difference equation in (3.1) is a 𝑝-Laplacian equation and the boundary conditions superlinear.

Acknowledgments

The second author was partially supported by Natural Science Foundation of Hunan province, China (no. 06JJ5008) and Natural Science Foundation of Guangdong province (no. 7004569).