Upper and Lower Solution Method for Fourth-Order Four-Point Boundary Value Problem on Time Scales

Abstract

We consider a fourth-order four-point boundary value problem for dynamic equations on time scales. By the upper and lower solution method, some results on the existence of solutions of the fourth-order four-point boundary value problem on time scales are obtained. An example is also included to illustrate our results.

1. Introduction

Let 𝕋 be a closed nonempty subset of ℝ, and let 𝕋 have the subspace topology inherited from the Euclidean topology on ℝ. In some of the current literature, 𝕋 is called a time scale (or measure chain). For notation, we shall use the convention that, for each interval 𝐽 of ℝ, 𝐽 will denote the time scales interval, that is, 𝐽∶=𝐽∩𝕋. Some preliminary definitions and theorems on time scales can be found in the books [1, 2], which are excellent references for calculus of time scales.

In this paper, let 𝕋 be a time scale and 𝜎(𝑡) the forward jump function in 𝕋. We are concerned with the following fourth-order four-point boundary value problem on time scales 𝕋: 𝑦^Δ4(𝑡)−𝑞(𝑡)𝑦^Δ2(𝜎(𝑡))=𝑓𝑡,𝑦(𝜎(𝑡)),𝑦^Δ2(𝜎(𝑡,𝑦(𝑎)𝜎))=0,𝑦²(𝑏)+𝜆𝑦^Δ𝜎²(𝑏)=0,𝜁𝑦^Δ2𝜉₁−𝜂𝑦^Δ3𝜉₁=0,𝛾𝑦^Δ2𝜉₂+𝛿𝑦^Δ3𝜉₂=0, (1.1) for 𝑡∈[𝑎,𝑏]⊂𝕋,𝑎≤𝜉₁<𝜉₂≤𝜎(𝑏). We will assume that the following conditions are satisfied.(H1)𝜁,𝛾,𝜂,𝛿≥0,𝜆≥𝜎³(𝑏)−𝜎²(𝑏), (H2)𝑞(𝑡)≥0. If 𝑞(𝑡)≡0, then 𝜁+𝛾>0, (H3)𝑘=𝜁𝛿+𝜂𝛾+𝜁𝛾(𝜉₂−𝜉₁)>0,𝜂−𝜁𝜉₁≥0,𝛿−𝛾(𝜎(𝑏)−𝜉₂)≥0.

The upper and lower solution method has been used to deal with the boundary value problems for dynamic equations in recent years. In most of these studies, two-point boundary value problem for second-order dynamic equations is considered [3–7].

Pang and Bai [8] studied the following fourth-order four-point BVP on time scales: 𝑢^ΔΔΔΔ(𝑡)=𝑓𝑡,𝑢(𝜎(𝑡)),𝑢^ΔΔ(𝑡)[],𝑢(0)𝜎,𝑡∈0,1=𝑢⁴(1)=0,𝛼𝑦^Δ2𝜉₁−𝛽𝑦^Δ3𝜉₁=0,𝛾𝑦^Δ2𝜉₂+𝜂𝑦^Δ3𝜉₂=0 (1.2) for 0≤𝜉₁<𝜉₂≤𝜎(𝑏), and 𝑓∈𝒞([0,1]×ℝ×ℝ,ℝ) and 𝛼,𝛽,𝛾,𝜂 are nonnegative constants satisfying 𝛼𝜂+𝛽𝛾+𝛼𝛾(𝜉₂−𝜉₁)>0. They establish criteria for the existence of a solution by developing the upper and lower solution method and the monotone iterative technique. Our problem is more general than the problems in [8], and our results are even new for the differential equations as well as for dynamic equations on general time scales.

2. Preliminaries

To prove the main results in this paper, we will employ several lemmas. We consider the linear boundary value problem 𝑦^Δ2(𝑡)−𝑞(𝑡)𝑦(𝜎(𝑡))=ℎ(𝑡),𝑡∈𝜉₁,𝜌𝜉₂,𝜁𝑦𝜉₁−𝜂𝑦^Δ𝜉₁=0,𝛾𝑦𝜉₂+𝛿𝑦^Δ𝜉₂=0. (2.1)

Denote by 𝜑 and 𝜓, the solutions of the corresponding homogeneous equation 𝑦^Δ2(𝑡)−𝑞(𝑡)𝑦(𝜎(𝑡))=0,𝑡∈𝜉₁,𝜌𝜉₂ (2.2) under the initial conditions 𝜑𝜉₁=𝜂,𝜑^Δ𝜉₁𝜓𝜉=𝜁,₂=𝛿,𝜓^Δ𝜉₂=−𝛾, (2.3) so that 𝜑 and 𝜓 satisfy the first and second boundary conditions of (2.1), respectively. Let us set 𝐷∶=𝜁𝜓𝜉₁−𝜂𝜓^Δ𝜉₁=𝛿𝜑^Δ𝜉₂+𝛾𝜑𝜉₂. (2.4) Using the initial conditions (2.3), we can deduce from (2.2) for 𝜑 and 𝜓 the following equations: 𝜑(𝑡)=𝜂+𝜁𝑡−𝜉₁+^𝑡_𝜉1^𝜏_𝜉1𝑞(𝑠)𝜑(𝜎(𝑠𝜓(𝑡)𝜉))Δ𝑠Δ𝜏,=𝛿+𝛾₂−𝑡+^𝜉2_𝑡^𝜉2_𝜏𝑞(𝑠)𝜓(𝜎(𝑠))Δ𝑠Δ𝜏. (2.5)(2.6)

Lemma 2.1. Under the conditions (H1) and (H2), the following inequalities 𝜑(𝑡)≥0,𝑡∈𝜉₁,𝜎𝜉₂;𝜓(𝑡)≥0,𝑡∈𝜉₁,𝜉₂;𝜑^Δ(𝑡)≥0,𝑡∈𝜉₁,𝜉₂;𝜓^Δ(𝑡)≤0,𝑡∈𝜉₁,𝜉₂; (2.7) yield.

Proof. We apply the induction principle for time scales to the statement 𝐴(𝑡)∶𝜑(𝑡)≥0,𝜑^Δ(𝑡)≥0, (2.8) where 𝑡∈[𝜉₁,𝜉₂].
(I) The statement 𝐴(𝜉₁) is true, since 𝜑(𝜉₁)=𝜂 and 𝜑^Δ(𝜉₁)=𝜁.
(II) Let 𝑡 be right-scattered and let𝐴(𝑡) be true, that is, 𝜑(𝑡)≥0 and 𝜑^Δ(𝑡)≥0. We need to show that 𝜑(𝜎(𝑡))≥0 and 𝜑^Δ(𝜎(𝑡))≥0. By the definition of Δ-derivative, we have 𝜑(𝜎(𝑡))=𝜑(𝑡)+[𝜎(𝑡)−𝑡]𝜑^Δ(𝑡). (2.9) Further, by the definition of Δ-derivative and (2.2) for 𝜑(𝑡), we have 𝜑^Δ(𝜎(𝑡))=𝜑^Δ(𝑡)+[𝜎(𝑡)−𝑡]𝜑^Δ2(𝑡)=𝜑^Δ(𝑡)+[𝜎(𝑡)−𝑡]𝑞(𝑡)𝜑(𝜎(𝑡)). (2.10) From (2.9), we get 𝜑(𝜎(𝑡))≥0, and then from (2.10), we get 𝜑^Δ(𝜎(𝑡))≥0.
(III) Let 𝑡₀ be right-dense, 𝐴(𝑡₀) be true and 𝑡₁∈[𝜉₁,𝜉₂] such that 𝑡₁>𝑡₀ and is sufficiently close to 𝑡₀. We need to prove that 𝐴(𝑡) is true for 𝑡∈[𝑡₀,𝑡₁].
From (2.2) with 𝑦(𝑡)=𝜑(𝑡), the equations 𝜑^Δ(𝑡)=𝜑^Δ𝑡₀+^𝑡_𝑡0𝑞(𝑠)𝜑(𝜎(𝑠𝜑(𝑡)𝑡))Δ𝑠,=𝜑₀+𝜑^Δ𝑡₀𝑡−𝑡₀+^𝑡_𝑡0^𝜏_𝑡0𝑞(𝑠)𝜑(𝜎(𝑠))Δ𝑠Δ𝜏 (2.11)(2.12) follow. To investigate the function 𝜑(𝑡) appearing in (2.12), we consider the equation 𝑦(𝑡)=𝜑𝑡₀+𝜑^Δ𝑡₀𝑡−𝑡₀+^𝑡_𝑡0^𝜏_𝑡0𝑞(𝑠)𝑦(𝜎(𝑠))Δ𝑠Δ𝜏, (2.13) where 𝑦(𝑡) is the desired solution. Our aim is to show that with𝑡₁ sufficiently close to 𝑡₀, (2.13) has a unique continuous solution 𝑦(𝑡) satisfying inequality 𝑦(𝑡)≥𝜑𝑡₀+𝜑^Δ𝑡₀𝑡−𝑡₀,𝑡∈𝑡₀,𝑡₁. (2.14) We solve (2.13) by the method of successive approximations, setting 𝑦₀(𝑡)=𝜑𝑡₀+𝜑^Δ𝑡₀𝑡−𝑡₀,𝑦_𝑗(𝑡)=^𝑡_𝑡0^𝜏_𝑡0𝑞(𝑠)𝑦_𝑗−1(𝜎(𝑠))Δ𝑠Δ𝜏,𝑗=1,2,3,…. (2.15) If the series ∑^∞_𝑗=0𝑦_𝑗(𝑡) converges uniformly with respect to 𝑡∈[𝑡₀,𝑡₁], then its sum will be, obviously, a continuous solution of (2.13). To prove the uniform convergence of this series, we let 𝑀₀=𝜑𝑡₀+𝜑^Δ𝑡₀𝑡₁−𝑡₀,𝑀₁=^𝑡1_𝑡0^𝜏_𝑡0𝑞(𝑠)Δ𝑠Δ𝜏. (2.16) Then the estimate 0≤𝑦_𝑗(𝑡)≤𝑀₀𝑀^𝑗₁,𝑡∈𝑡₀,𝑡₁,𝑗=0,1,2,… (2.17) can easily be obtained. Indeed, (2.17) evidently holds for 𝑗=0. Let it also hold for 𝑗=𝑛. Then from (2.15), we get for 𝑡∈[𝑡₀,𝑡₁], 0≤𝑦_𝑛+1(𝑡)≤^𝑡1_𝑡0^𝜏_𝑡0𝑞(𝑠)𝑦_𝑛(𝜎(𝑠≤))Δ𝑠Δ𝜏_𝑡1𝑡0max_{𝑡0(𝜏)≤𝑠≤𝜌}𝑦_𝑛(𝜎(𝑠))^𝜏_𝑡0𝑞(𝑠)Δ𝑠Δ𝜏≤max_{𝑡0≤𝑠≤𝜌2𝑡1}𝑦_𝑛(𝜎(𝑠))^𝑡1_𝑡0^𝜏_𝑡0𝑞(𝑠)Δ𝑠Δ𝜏≤𝑀₀𝑀^𝑛₁𝑀₁=𝑀₀𝑀₁^𝑛+1. (2.18) Therefore, by the usual mathematical induction principle, (2.17) holds for all 𝑗=0,1,2,….
Now choosing 𝑡₁ appropriately, we obtain 𝑀₁<1. Then (2.13) will have a continuous solution 𝑦(𝑡)=^∞_𝑗=0𝑦_𝑗(𝑡)for𝑡∈𝑡₀,𝑡₁. (2.19) Since 𝑦_𝑗(𝑡)≥0, it follows that 𝑦(𝑡)≥𝑦₀(𝑡) thereby proving the validity of inequality (2.14). To prove uniqueness of solution of (2.13) for 𝑡∈[𝑡₀,𝑡₁], suppose it has two solutions 𝑦₁ and 𝑦₂, and passing on to the modulus, we get 𝑦₁(𝑡)−𝑦₂(𝑡)≤^𝑡_𝑡0^𝜏_𝑡0𝑞(𝑠)𝑦₁(𝜎(𝑠))−𝑦₂(𝜎(𝑠))Δ𝑠Δ𝜏≤max_{𝑡0≤𝑠≤𝜌2𝑡1}𝑦₁(𝜎(𝑠))−𝑦₂(𝜎(𝑠))^𝑡1_𝑡0^𝜏_𝑡0𝑞(𝑠)Δ𝑠Δ𝜏. (2.20) Thus, 𝑦₁(𝑡)−𝑦₂(𝑡)≤𝑀₁max_{𝑡0≤𝑠≤𝜌2𝑡1}𝑦₁(𝜎(𝑠))−𝑦₂(𝜎(𝑠)),𝑡∈𝑡₀,𝑡₁. (2.21) Since 𝑀₁<1, hence it follows that 𝑦₁(𝑡)=𝑦₂(𝑡) for 𝑡∈[𝑡₀,𝑡₁].
From (2.12) and (2.13) in view of the uniqueness of solution, we get that 𝜑(𝑡)=𝑦(𝑡),𝑡∈[𝑡₀,𝑡₁]. Therefore,𝜑(𝑡)≥𝜑𝑡₀+𝜑^Δ𝑡₀𝑡−𝑡₀,𝑡∈𝑡₀,𝑡₁. (2.22) Hence by making use of the induction hypothesis 𝐴(𝑡₀) being true, we obtain 𝜑(𝑡)≥0 for 𝑡∈[𝑡₀,𝑡₁]. Taking this into account, from (2.11), we also get 𝜑^Δ(𝑡)≥0 for 𝑡∈[𝑡₀,𝑡₁]. Thus, 𝐴(𝑡) is true for all 𝑡∈[𝑡₀,𝑡₁].
(IV) Let 𝑡∈(𝜉₁,𝜉₂] and assume 𝑡 is left-dense and such that 𝐴(𝑠) is true for all 𝑠<𝑡, that is, 𝜑(𝑠)≥0,𝜑^Δ(𝑠)≥0,∀𝑠∈𝜉₁,𝑡. (2.23) Passing on here to the limit as 𝑠→𝑡, we get by the continuity of 𝜑(𝑠) and 𝜑^Δ(𝑠) that 𝜑(𝑡)≥0 and 𝜑^Δ(𝑡)≥0, thereby verifying the validity of 𝐴(𝑡).
Consequently, by the induction principle on time scales, (2.8) holds for all 𝑡∈[𝜉₁,𝜉₂].
From (2.9) and (2.8) for 𝑡=𝜉₂, we also get 𝜑(𝜎(𝜉₂))≥0. So the statements (2.7) for 𝜑 are proved.
We can prove the statements of the lemma for 𝜓 similarly applying the backward induction principle on time scales. The lemma is proved.

Lemma 2.2. Under the conditions (H1) and (H2), the inequality 𝐷>0 holds.

Proof. By (2.4) and (2.5), we have 𝐷=𝜁𝛿+𝜂𝛾+𝜁𝛾𝜉₂−𝜉₁+𝛿^𝜉2_𝜉1𝑞(𝑠)𝜑(𝜎(𝑠))Δ𝑠+𝛾^𝜉2_𝜉1^𝜏_𝜉1𝑞(𝑠)𝜑(𝜎(𝑠))Δ𝑠Δ𝜏. (2.24) Since 𝜑(𝑡)≥0for 𝑡∈[𝜉₁,𝜎(𝜉₂)], from (2.8), we have 𝐷≥𝜁𝛿+𝜂𝛾+𝜁𝛾𝜉₂−𝜉₁. (2.25) If 𝑞(𝑡)≡0, then in (2.25) the equality holds. From the condition (H2), we get 𝐷>0. This proof is completed.

Lemma 2.3. Assume that the conditions (H1) and (H2) are satisfied. If ℎ∈𝒞[𝜉₁,𝜌(𝜉₂)], then the boundary value problem 𝑦^Δ2(𝑡)−𝑞(𝑡)𝑦(𝜎(𝑡))=ℎ(𝑡),𝑡∈𝜉₁,𝜌𝜉₂,𝜁𝑦𝜉₁−𝜂𝑦^Δ𝜉₁𝜉=0,𝛾𝑦₂+𝛿𝑦^Δ𝜉₂=0 (2.26) has a unique solution 𝑦(𝑡)=−^𝜉2_𝜉1𝐺(𝑡,𝑠)ℎ(𝑠)Δ𝑠, (2.27) where 𝐺()=1𝑡,𝑠𝐷𝜓(𝜎(𝑠𝜑(𝑡)𝜓(𝑡)𝜑(𝜎(𝑠(𝑠).)),𝑡≤𝑠,)),𝑡≥𝜎 (2.28) Here 𝐷,𝜑,𝜓 are as in (2.4), (2.5), and (2.6), respectively.

Proof. Taking 𝑧(𝑡)1=−𝐷^𝑡_𝜉1𝜑(𝜎(𝑠))𝜓(𝑡)−𝜑(𝑡)𝜓(𝜎(𝑠))ℎ(𝑠)Δ𝑠, (2.29) we have 𝑧^Δ2(𝑡)1=−𝐷𝜑(𝜎(𝑡))𝜓^Δ(𝜎(𝑡))−𝜑^Δ(𝜎(𝑡𝜓(𝜎(𝑡ℎ(𝑡)−1))))𝐷^𝑡_𝜉1𝜑(𝜎(𝑠))𝜓^Δ2(𝑡)−𝜑^Δ2(𝑡)𝜓(𝜎(𝑠))ℎ(𝑠)Δ𝑠. (2.30) By Corollary  3.14 in [1], since the Wronskian of any two solutions of (2.2) is independent of 𝑡, we get 𝐷=𝑊𝜓,𝜑=𝜓(𝜎(𝑡))𝜑^Δ(𝜎(𝑡))−𝜑(𝜎(𝑡))𝜓^Δ(𝜎(𝑡)). (2.31) Hence we get 𝑧^Δ2(𝑡)1=−𝐷−𝐷ℎ(𝑡)+𝑞(𝑡)^𝑡_𝜉1𝜑(𝜎(𝑠𝜓(𝜎(𝑡(𝜎(𝑡𝜓(𝜎(𝑠ℎ(𝑠)(𝑡)(𝑡−1))))−𝜑))))Δ𝑠=ℎ+𝑞)𝐷^𝑡_𝜉1𝜑(𝜎(𝑠𝜓(𝜎(𝑡(𝜎(𝑡𝜓(𝜎(𝑠ℎ(𝑠)(𝑡)(𝑡−1))))−𝜑))))Δ𝑠=ℎ+𝑞)𝐷_𝜉^𝜎(𝑡)₁𝜑(𝜎(𝑠𝜓(𝜎(𝑡(𝜎(𝑡𝜓(𝜎(𝑠ℎ(𝑠)(𝑡)(𝑡)𝑧(𝜎(𝑡.))))−𝜑))))Δ𝑠=ℎ+𝑞)) (2.32) So the general solution of equation 𝑦^Δ2(𝑡)−𝑞(𝑡)𝑦(𝜎(𝑡))=ℎ(𝑡),𝑡∈𝜉₁,𝜌𝜉₂, (2.33) has the form 𝑦(𝑡)=𝑐₁𝜑(𝑡)+𝑐₂𝜓(𝑡)−1𝐷^𝑡_𝜉1𝜑(𝜎(𝑠))𝜓(𝑡)−𝜑(𝑡)𝜓(𝜎(𝑠))ℎ(𝑠)Δ𝑠, (2.34) where 𝑐₁ and 𝑐₂ are arbitrary constants. Substituting this expression for 𝑦(𝑡) in the boundary conditions of BVP (2.26), we can evaluate 𝑐₁ and 𝑐₂. After some easy calculations, we can get (2.27) and (2.28).

Lemma 2.4. Under the conditions (H1) and (H2), the Green’s function of BVP (2.26) possesses the following property: 𝐺(𝑡,𝑠)≥0,(𝑡,𝑠)∈𝜉₁,𝜉₂×𝜉₁,𝜌𝜉₂. (2.35)

Proof. The lemma follows from (2.28), Lemmas 2.1 and 2.2 immediately.

Lemma 2.5. Assume that the conditions (H1) and (H2) are satisfied. If ℎ∈𝒞[𝑎,𝑏], then the boundary value problem 𝑦^Δ4(𝑡)−𝑞(𝑡)𝑦^Δ2(𝜎(𝑡(𝑡)[],𝑦(𝑎)𝜎))=ℎ,𝑡∈𝑎,𝑏=0,𝑦²(𝑏)+𝜆𝑦^Δ𝜎²(𝑏)=0,𝜁𝑦^Δ2𝜉₁−𝜂𝑦^Δ3𝜉₁=0,𝛾𝑦^Δ2𝜉₂+𝛿𝑦^Δ3𝜉₂=0 (2.36) has a unique solution 𝑦(𝑡)=^𝜎2_𝑎^(𝑏)𝐺₁(𝑡,𝜉)^𝜉2_𝜉1𝐺₂(𝜉,𝑠)ℎ(𝑠)Δ𝑠Δ𝜉, (2.37) where 𝐺₁()=1𝑡,𝑠𝜎²(𝑏)𝜎−𝑎+𝜆²(𝑏)(𝑠)()𝜎−𝜎+𝜆𝑡−𝑎,𝑡≤𝑠,₂(𝑏)−𝑡+𝜆(𝜎(𝑠)−𝑎),𝑡≥𝜎(𝑠), (2.38)𝐺₂()=1𝑡,𝑠𝐷𝜓(𝜎(𝑠𝜑(𝑡)𝜓(𝑡)𝜑(𝜎(𝑠(𝑠).)),𝑡≤𝑠,)),𝑡≥𝜎 (2.39) Here 𝐷,𝜑,𝜓 are as in (2.4), (2.5), and (2.6), respectively.

Proof. Let us consider the following BVP: 𝑦^Δ2(𝑡)=−^𝜉2_𝜉1𝐺₂(𝑡,𝑠)ℎ(𝑠)Δ𝑠,𝑡∈𝑎,𝜎²(𝑏,𝑦(𝑎)𝜎)=0,𝑦²(𝑏)+𝜆𝑦^Δ𝜎²(𝑏)=0. (2.40) The Green’s function associated with the BVP (2.40) is 𝐺₁(𝑡,𝑠). This completes the proof.

Lemma 2.6. Assume that the conditions (H1)–(H3) are satisfied. If 𝑦 satisfies 𝑦^Δ4−𝑞(𝑡)𝑦^Δ2(𝜎(𝑡[],𝑦(𝑎)𝜎))≥0,𝑡∈𝑎,𝑏≥0,𝑦²(𝑏)+𝜆𝑦^Δ𝜎²(𝑏)≥0,𝜁𝑦^Δ2𝜉₁−𝜂𝑦^Δ3𝜉₁≤0,𝛾𝑦^Δ2𝜉₂+𝛿𝑦^Δ3𝜉₂≤0, (2.41) then 𝑦(𝑡)≥0,𝑡∈[𝑎,𝜎²(𝑏)] and 𝑦^Δ2(𝑡)≤0,𝑡∈[𝑎,𝜎(𝑏)].

Proof. Let 𝑦^Δ4(𝑡)−𝑞(𝑡)𝑦^Δ2(𝜎(𝑡(𝑡)[],𝑦(𝑎)))=ℎ,𝑡∈𝑎,𝑏=𝑡₀,𝑦𝜎²(𝑏)+𝜆𝑦^Δ𝜎²(𝑏)=𝑡₁,𝜁𝑦^Δ2𝜉₁−𝜂𝑦^Δ3𝜉₁=𝑡₂,𝛾𝑦^Δ2𝜉₂+𝛿𝑦^Δ3𝜉₂=𝑡₃, (2.42) where 𝑡₀≥0,𝑡₁≥0,𝑡₂≤0,𝑡₃≤0,ℎ≥0.
It is easy to check that 𝑦 and 𝑦^Δ2 can be given by the expression𝑦(𝑡)=𝑆(𝑡)−^𝜎2_𝑎^(𝑏)𝐺₁(𝑡,𝜉)𝑅(𝜉)Δ𝜉+^𝜎2_𝑎^(𝑏)𝐺₁(𝑡,𝜉)^𝜉2_𝜉1𝐺₂()ℎ(𝑠)𝑦𝜉,𝑠Δ𝑠Δ𝜉,_Δ2(𝑡)=𝑅(𝑡)−^𝜉2_𝜉1𝐺₂(𝑡,𝑠)ℎ(𝑠)Δ𝑠, (2.43) where 𝑆(𝑡)=1𝜎²(𝑏)𝜎−𝑎+𝜆²(𝑏)+𝜆−𝑡𝑡₀+(𝑡−𝑎)𝑡₁,𝑅(𝑡)=1𝑘𝜁𝑡−𝜉₁+𝜂𝑡₃+𝛾𝜉₂−𝑡+𝛿𝑡₂, (2.44) and 𝐺₁(𝑡,𝑠),𝐺₂(𝑡,𝑠)are as in (2.38) and (2.39), respectively. The hypothesis of the lemma implies that 𝑆(𝑡)≥0 for 𝑡∈[𝑎,𝜎²(𝑏)], 𝑅(𝑡)≤0 for 𝑡∈[𝑎,𝜎(𝑏)], 𝐺₁(𝑡,𝑠)≥0 for (𝑡,𝑠)∈[𝑎,𝜎²(𝑏)]×[𝑎,𝜎(𝑏)], and 𝐺₂(𝑡,𝑠)>0 for (𝑡,𝑠)∈[𝑎,𝜎(𝑏)]×[𝑎,𝑏]. Therefore, we get 𝑦(𝑡)≥0 for 𝑡∈[𝑎,𝜎²(𝑏)] and 𝑦^Δ2(𝑡)≤0 for 𝑡∈[𝑎,𝜎(𝑏)]. The proof is completed.

3. Upper and Lower Solution Method

In this section, we present existence results for the BVP (1.1) by using the method of upper and lower solutions. We define the set 𝐷∶=𝑦∶𝑦^Δ𝑛∈𝒞𝑎,𝜎⁴(𝑏)^𝑘𝑛,𝑘=0,1,2,3,4. (3.1)Definition 3.1. Letting 𝛼(𝑡)∈𝐷 on [𝑎,𝜎⁴(𝑏)], we say 𝛼 is a lower solution for the problem (1.1) if 𝛼 satisfies 𝛼^Δ4(𝑡)−𝑞(𝑡)𝛼^Δ2(𝜎(𝑡))≤𝑓𝑡,𝛼(𝜎(𝑡)),𝛼^Δ2(𝜎(𝑡,𝛼(𝑎)𝜎))≤0,𝛼²(𝑏)+𝜆𝛼^Δ𝜎²(𝑏)≤0,𝜁𝛼^Δ2𝜉₁−𝜂𝛼^Δ3𝜉₁≥0,𝛾𝛼^Δ2𝜉₂+𝛿𝛼^Δ3𝜉₂≥0. (3.2)

Definition 3.2. Letting 𝛽(𝑡)∈𝐷, on [𝑎,𝜎⁴(𝑏)], we say 𝛽 is an upper solution for the problem (1.1) if 𝛽 satisfies 𝛽^Δ4(𝑡)−𝑞(𝑡)𝛽^Δ2(𝜎(𝑡))≥𝑓𝑡,𝛽(𝜎(𝑡)),𝛽^Δ2(𝜎(𝑡,𝛽(𝑎)𝜎))≥0,𝛽²(𝑏)+𝜆𝛽^Δ𝜎²(𝑏)≥0,𝜁𝛽^Δ2𝜉₁−𝜂𝛽^Δ3𝜉₁≤0,𝛾𝛽^Δ2𝜉₂+𝛿𝛽^Δ3𝜉₂≤0. (3.3)
We assume that the function 𝑓(𝑡,𝑦,𝑦^Δ2) satisfies the following condition.
(H4)𝑓∶[𝑎,𝑏]×ℝ×ℝ→ℝ is continuous and satisfies 𝑓𝑡,𝑦₂,𝑧−𝑓𝑡,𝑦₁,𝑧≥0,for𝛼(𝜎(𝑡))≤𝑦₁≤𝑦₂(𝜎(𝑡[],𝑓≤𝛽)),𝑧∈ℝ,𝑡∈𝑎,𝑏𝑡,𝑦,𝑧₂−𝑓𝑡,𝑦,𝑧₁≤0,for𝛽^Δ2(𝜎(𝑡))≤𝑧₁≤𝑧₂≤𝛼^Δ2(𝜎(𝑡)),𝑦∈ℝ,𝑡∈[𝑎,𝑏], (3.4)where 𝛼, 𝛽 are lower and upper solutions, respectively, for the BVP (1.1), and satisfy 𝛼≤𝛽,𝛼^Δ2≥𝛽^Δ2.

Theorem 3.3. Assume that the conditions (H1)–(H4) are satisfied. Then the problem (1.1) has a solution 𝑦(𝑡) with 𝛼(𝑡)≤𝑦(𝑡)≤𝛽(𝑡),𝛼^Δ2(𝑡)≥𝑦^Δ2(𝑡)≥𝛽^Δ2(𝑡) (3.5) for 𝑡∈[𝑎,𝜎³(𝑏)] and 𝑡∈[𝑎,𝜎(𝑏)], respectively.

Proof. Consider the BVP, 𝑦^Δ4(𝑡)−𝑞(𝑡)𝑦^Δ2(𝜎(𝑡))=𝐹𝑡,𝑦(𝜎(𝑡)),𝑦^Δ2(𝜎(𝑡[],𝑦(𝑎)𝜎)),𝑡∈𝑎,𝑏=0,𝑦²(𝑏)+𝜆𝑦^Δ𝜎²(𝑏)=0,𝜁𝑦^Δ2𝜉₁−𝜂𝑦^Δ3𝜉₁=0,𝛾𝑦^Δ2𝜉₂+𝛿𝑦^Δ3𝜉₂=0, (3.6) where 𝐹=𝑓𝑡,𝑥,𝑦^∗(𝜎(𝑡(𝜎(𝑡,𝑓𝑡,𝛼)),𝑦,𝑥<𝛼))_∗(𝜎(𝑡(𝜎(𝑡,𝑓𝑡,𝑥,𝑦,𝛼))≤𝑥≤𝛽))_∗(𝜎(𝑡(𝜎(𝑡,𝑓𝑡,𝛽)),𝑦,𝑥>𝛽))_∗=𝑓𝑡,𝑥,𝑦𝑡,𝑥,𝛽^Δ2(𝜎(𝑡)),𝑦<𝛽^Δ2(𝜎(𝑡,𝑓))𝑡,𝑥,𝑦,𝛽^Δ2(𝜎(𝑡))≤𝑦≤𝛼^Δ2(𝜎(𝑡,𝑓))𝑡,𝑥,𝛼^Δ2(𝜎(𝑡)),𝑦>𝛼^Δ2(𝜎(𝑡)). (3.7) By Lemma 2.5, it is clear that the solutions of the BVP (3.6) are the fixed points of the operator 𝐴𝑦(𝑡)=^𝜎2_𝑎^(𝑏)𝐺₁(𝑡,𝜉)^𝜉2_𝜉1𝐺₂(𝜉,𝑠)𝐹𝑠,𝑦(𝜎(𝑠)),𝑦^Δ2(𝜎(𝑠))Δ𝑠Δ𝜉,𝑡∈𝑎,𝜎⁴(𝑏), (3.8) where 𝐺₁(𝑡,𝑠) and 𝐺₂(𝑡,𝑠) are as in (2.38) and (2.39), respectively. It is clear that 𝐴 is continuous. Since the function 𝑓(𝑡,𝑦,𝑦^Δ2) satisfies the conditions (3.4), 𝑓𝑡,𝛼(𝜎(𝑡)),𝛼^Δ2(𝜎(𝑡))≤𝐹𝑡,𝑦,𝑦^Δ2≤𝑓𝑡,𝛽(𝜎(𝑡)),𝛽^Δ2(𝜎(𝑡))for𝑡∈[𝑎,𝑏]. (3.9) Thus, there exists a positive constant 𝑀 such that |𝐹(𝑡,𝑦,𝑦^Δ2)|≤𝑀, which implies that the operator 𝐴 is uniformly bounded. Moreover, the operator 𝐴 is equicontinuous. Therefore, from the Arzela-Ascoli theorem, the operator 𝐴 is a compact operator. Thus, by Schauder’s fixed point theorem, there exists a solution 𝑦 of the BVP (3.6).
Suppose 𝑦^∗ is a solution of the BVP (3.6). Since 𝑓(𝑡,𝑦,𝑦^Δ2) satisfies the conditions (3.4), we know that 𝑓𝑡,𝛼(𝜎(𝑡)),𝛼^Δ2(𝜎(𝑡))≤𝐹𝑡,𝑦^∗,𝑦^∗Δ2≤𝑓𝑡,𝛽(𝜎(𝑡)),𝛽^Δ2(𝜎(𝑡))for𝑡∈[𝑎,𝑏]. (3.10) Thus, 𝛽−𝑦^∗Δ4(𝑡)−𝑞(𝑡)𝛽−𝑦^∗Δ2(𝜎(𝑡(𝜎(𝑡))≥𝑓𝑡,𝛽)),𝛽²(𝜎(𝑡))−𝐹𝑡,𝑦^∗(𝜎(𝑡)),𝑦^Δ2(𝜎(𝑡))≥0,𝑡∈[𝑎,𝑏],𝛽−𝑦^∗(𝑎)≥0,𝛽−𝑦^∗𝜎²(𝑏)+𝜆𝛽−𝑦^∗Δ𝜎²(𝑏)𝜁≥0,𝛽−𝑦^∗Δ2𝜉₁−𝜂𝛽−𝑦^∗Δ3𝜉₁≤0,𝛾𝛽−𝑦^∗Δ2𝜉₂+𝛿𝛽−𝑦^∗Δ3𝜉₂≤0. (3.11) By virtue of Lemma 2.6, 𝑦^∗(𝑡)≤𝛽(𝑡) for 𝑡∈[𝑎,𝜎²(𝑏)] and 𝑦^∗Δ2(𝑡)≥𝛽^Δ2(𝑡) for 𝑡∈[𝑎,𝜎(𝑏)]. If 𝜎²(𝑏) is right-scattered, by using the inequality (𝛽−𝑦^∗)(𝜎²(𝑏))+𝜆(𝛽−𝑦^∗)^Δ(𝜎²(𝑏))≥0, we get the inequality (𝜆/(𝜎³(𝑏)−𝜎²(𝑏)))(𝛽−𝑦^∗)(𝜎³(𝑏))≥((𝜆/(𝜎³(𝑏)−𝜎(𝑏)))−1)(𝛽−𝑦^∗)(𝜎²(𝑏)). So we get 𝑦^∗(𝑡)≤𝛽(𝑡) on [𝑎,𝜎³(𝑏)]. If 𝜎²(𝑏) is right-dense, it is trivial that the inequality 𝑦^∗(𝑡)≤𝛽(𝑡) holds on [𝑎,𝜎³(𝑏)]. Similarly, one can show that 𝛼(𝑡)≤𝑦^∗(𝑡) for 𝑡∈[𝑎,𝜎³(𝑏)] and 𝛼^Δ2(𝑡)≥𝑦^∗Δ2(𝑡) for 𝑡∈[𝑎,𝜎(𝑏)]. This completes the proof.

Theorem 3.4. Assume that the conditions (H1)–(H4) are satisfied. Then there exist two monotone sequences {𝛼_𝑛} and {𝛽_𝑛}, nonincreasing and nondecreasing, respectively, with 𝛼₀=𝛼 and 𝛽₀=𝛽, which converge to the extremal solutions in [𝛽,𝛼] of the problem (1.1).

Proof. For any function 𝜗 which satisfies 𝛼(𝑡)≤𝜗(𝑡)≤𝛽(𝑡) for 𝑡∈[𝑎,𝜎²(𝑏)], consider the following problem: 𝑦^Δ4(𝑡)−𝑞(𝑡)𝑦^Δ2(𝜎(𝑡))=𝑓𝑡,𝜗(𝜎(𝑡)),𝜗^Δ2(𝜎(𝑡[],𝑦(𝑎)𝜎)),𝑡∈𝑎,𝑏=0,𝑦²(𝑏)+𝜆𝑦^Δ𝜎²(𝑏)=0,𝜁𝑦^Δ2𝜉₁−𝜂𝑦^Δ3𝜉₁=0,𝛾𝑦^Δ2𝜉₂+𝛿𝑦^Δ3𝜉₂=0. (3.12) Clearly, this problem is type (2.12). Obviously, it has a unique solution given by the expression𝑦(𝑡)=^𝜎2_𝑎^(𝑏)𝐺₂(𝑡,𝜉)^𝜉2_𝜉1𝐺₁(𝜉,𝑠)𝑓𝑠,𝜗(𝜎(𝑠)),𝜗^Δ2(𝜎(𝑠))Δ𝑠Δ𝜉≡𝐴𝜗(𝑡),𝑡∈𝑎,𝜎⁴(𝑏). (3.13)
Step 1. 𝛼≤𝐴𝛼,𝐴𝛽≤𝛽 on [𝑎,𝜎³(𝑏)] and 𝛼^Δ2≥(𝐴𝛼)^Δ2,(𝐴𝛽)^Δ2≥𝛽^Δ2 on [𝑎,𝜎(𝑏)].
Let 𝐴𝛼(𝑡)=𝑤(𝑡). Thus(𝑤−𝛼)^Δ4(𝑡)−𝑞(𝑡)(𝑤−𝛼)^Δ2(𝜎(𝑡(𝜎(𝑡))≥𝑓𝑡,𝛼)),𝛼^Δ2(𝜎(𝑡(𝜎(𝑡))−𝑓𝑡,𝛼)),𝛼^Δ2(𝜎(𝑡[],(𝑎)()𝜎))=0,𝑡∈𝑎,𝑏𝑤−𝛼)(≥0,𝑤−𝛼²(𝑏)+𝜆(𝑤−𝛼)^Δ𝜎²(𝑏)𝜁()≥0,𝑤−𝛼^Δ2𝜉₁−𝜂(𝑤−𝛼)^Δ3𝜉₁≤0,𝛾(𝑤−𝛼)^Δ2𝜉₂+𝛿(𝑤−𝛼)^Δ3𝜉₂≤0. (3.14) Using Lemma 2.6, we obtain that 𝛼≤𝐴𝛼,𝛼^Δ2≥(𝐴𝛼)^Δ2 on [𝑎,𝜎²(𝑏)] and [𝑎,𝜎(𝑏)], respectively. In case that 𝜎²(𝑏) is right scattered, the assumptions 𝜆≥𝜎³(𝑏)−𝜎²(𝑏) and (𝑤−𝛼)(𝜎²(𝑏))+(𝑤−𝛼)^Δ(𝜎²(𝑏))≥0 imply that 𝛼(𝜎³(𝑏))≤(𝐴𝛼)(𝜎³(𝑏)). Similarly, we show that (𝐴𝛽)(𝑡)≤𝛽(𝑡),(𝐴𝛽)^Δ2(𝑡)≥𝛽^Δ2(𝑡) on [𝑎,𝜎³(𝑏)] and [𝑎,𝜎(𝑏)], respectively.Step 2. If 𝑦₁,𝑦₂∈[𝛼,𝛽]; 𝑦^Δ2₁,𝑦^Δ2₂∈[𝛽^Δ2,𝛼^Δ2], 𝑦₁≤𝑦₂ and 𝑦^Δ2₂≤𝑦^Δ2₁, then we have 𝐴𝑦₁≤𝐴𝑦₂ and (𝐴𝑦₁)^Δ2≥(𝐴𝑦₂)^Δ2.
Let 𝐴𝑦₁=𝑤₁ and 𝐴𝑦₂=𝑤₂. Thus𝑤₂−𝑤₁^Δ4(𝑡)−𝑞(𝑡)𝑤₂−𝑤₁^Δ2(𝜎(𝑡))=𝑓𝑡,𝑦₂(𝜎(𝑡)),𝑦^Δ2₂(𝜎(𝑡))−𝑓𝑡,𝑦₁(𝜎(𝑡)),𝑦^Δ2₁(𝜎(𝑡[],𝑤))≥0,𝑡∈𝑎,𝑏₂−𝑤₁(𝑎)=0,𝑤₂−𝑤₁𝜎³(𝑏)+𝜆𝑤₂−𝑤₁^Δ𝜎³(𝑏)𝜁𝑤=0,₂−𝑤₁^Δ2𝜉₁−𝜂𝑤₂−𝑤₁^Δ3𝜉₁=0,𝛾𝑤₂−𝑤₁^Δ2𝜉₂+𝛿𝑤₂−𝑤₁^Δ3𝜉₂=0. (3.15) Hence we get that 𝐴𝑦₁(𝑡)≤𝐴𝑦₂(𝑡) and (𝐴𝑦₁)^Δ2(𝑡)≥(𝐴𝑦₂)^Δ2(𝑡) on [𝑎,𝜎³(𝑏)] and [𝑎,𝜎(𝑏)], respectively.
Now, we define the sequences {𝛼_𝑛(𝑡)} and {𝛽_𝑛(𝑡)} by 𝛼₀(𝑡)=𝛼(𝑡),𝛼_𝑛+1(𝑡)=𝐴𝛼_𝑛(𝑡),𝛽₀(𝑡)=𝛽(𝑡),𝛽_𝑛+1(𝑡)=𝐴𝛽_𝑛(𝑡),for𝑛≥0. (3.16) From the properties of 𝐴, we have 𝛼=𝛼₀≤𝛼₁≤⋯≤𝛽_𝑛≤⋯≤𝛽₁≤𝛽₀𝛼=𝛽,_Δ2=𝛼^Δ2₀≤𝛼^Δ2₁≤⋯≤𝛽^Δ2_𝑛≤⋯≤𝛽^Δ2₁≤𝛽^Δ2₀=𝛽^Δ2. (3.17) But then 𝛼^∗ and 𝛽^∗ defined by 𝛼^∗=lim_𝑛→∞𝛼_𝑛,𝛽^∗=lim_𝑛→∞𝛽_𝑛,𝛼^∗Δ2=lim_𝑛→∞𝛼^Δ2_𝑛,𝛽^∗Δ2=lim_𝑛→∞𝛽_𝑛 (3.18) are extremal solutions of (1.1).

Example 3.5. Consider the BVP 𝑦^Δ4(𝑡)−𝑡²𝑦^Δ2(𝜎(𝑡))=−𝑦^Δ2(𝜎(𝑡3)),𝑡∈2,𝑦3,42=0,𝑦𝜎²(4)+𝑦^Δ𝜎²(4𝑦)=0,_Δ2(2)−3𝑦^Δ3(2)=0,7𝑦^Δ2(3)+8𝑦^Δ3(3)=0, (3.19) where 𝜎(4)≤29/7 and 𝜎³(4)−𝜎²(4)≤1. It is easy to check that 𝛼(𝑡)=0, 𝛽(𝑡)=𝑡 are lower and upper solutions of the BVP (3.19), respectively, and that all assumptions of Theorem 3.3 are fulfilled. So the BVP (3.19) has a solution 𝑦(𝑡) satisfying 0≤𝑦(𝑡)≤𝑡 for 𝑡∈[3/2,𝜎³(4)], 𝑦^Δ2(𝑡)=0 for 𝑡∈[3/2,𝜎(4)].