Hindawi Publishing Corporation Advances in Difference Equations Volume 2009, Article ID 123565, 20 pages doi:10.1155/2009/123565 Research Article Existence of Solutions for Nonlinear Four-Point p-Laplacian Boundary Value Problems on Time Scales S Gulsan Topal, O Batit Ozen, and Erbil Cetin Department of Mathematics, Ege University, Bornova, 35100 Izmir, Turkey Correspondence should be addressed to S Gulsan Topal, f.serap.topal@ege.edu.tr Received 16 March 2009; Accepted 20 July 2009 Recommended by Alberto Cabada We are concerned with proving the existence of positive solutions of a nonlinear second-order fourpoint boundary value problem with a p-Laplacian operator on time scales The proofs are based on the fixed point theorems concerning cones in a Banach space Existence result for p-Laplacian boundary value problem is also given by the monotone method Copyright q 2009 S Gulsan Topal 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 Introduction Let T be any time scale such that 0, be subset of T The concept of dynamic equations on time scales can build bridges between differential and difference equations This concept not only gives us unified approach to study the boundary value problems on discrete intervals with uniform step size and real intervals but also gives an extended approach to study on discrete case with non uniform step size or combination of real and discrete intervals Some basic definitions and theorems on time scales can be found in 1, In this paper, we study the existence of positive solutions for the following nonlinear four-point boundary value problem with a p-Laplacian operator: φp xΔ αφp x ρ ∇ t − Ψ φp xΔ ξ h t f t, x t 0, 0, t ∈ 0, , γφp x σ |s|p−2 s for p > 1, φp where φp s is an operator, that is, φp s 1/q 1, α, γ > 0, δ ≥ 0, ξ, η ∈ ρ , σ with ξ < η: 1.1 δφp xΔ η −1 s 0, 1.2 φq s , where 1/p Advances in Difference Equations H1 the function f ∈ C 0, × 0, ∞ , 0, ∞ , H2 the function h ∈ Cld T, 0, ∞ and does not vanish identically on any closed σ subinterval of ρ , σ and < ρ h t ∇t < ∞, H3 Ψ : R → R is continuous and satisfies that there exist B2 ≥ B1 > such that B1 s ≤ Ψ s ≤ B2 s, for s ∈ 0, ∞ In recent years, the existence of positive solutions for nonlinear boundary value problems with p-Laplacians has received wide attention, since it has led to several important s is linear, mathematical and physical applications 3, In particular, for p or φp s the existence of positive solutions for nonlinear singular boundary value problems has been obtained 5, p-Laplacian problems with two-, three-, and m-point boundary conditions for ordinary differential equations and difference equations have been studied in 7–9 and the references therein Recently, there is much attention paid to question of positive solutions of boundary value problems for second-order dynamic equations on time scales, see 10–13 In particular, we would like to mention some results of Agarwal and O’Regan 14 , Chyan and Henderson , Song and Weng 15 , Sun and Li 16 , and Liu 17 , which motivate us to consider the p-Laplacian boundary value problem on time scales The aim of this paper is to establish some simple criterions for the existence of positive solutions of the p-Laplacian BVP 1.1 - 1.2 This paper is organized as follows In Section we first present the solution and some properties of the solution of the linear p-Laplacian BVP corresponding to 1.1 - 1.2 Consequently we define the Banach space, cone and the integral operator to prove the existence of the solution of 1.1 - 1.2 In Section 3, we state the fixed point theorems in order to prove the main results and we get the existence of at least one and two positive solutions for nonlinear p-Laplacian BVP 1.1 - 1.2 Finally, using the monotone method, we prove the existence of solutions for p-Laplacian BVP in Section Preliminaries and Lemmas In this section, we will give several fixed point theorems to prove existence of positive solutions of nonlinear p-Laplacian BVP 1.1 - 1.2 Also, to state the main results in this paper, we employ the following lemmas These lemmas are based on the linear dynamic equation: φp xΔ ∇ t y t 2.1 Lemma 2.1 Suppose condition (H2) holds, then there exists a constant θ ∈ ρ , σ − ρ /2 that satisfies σ −θ 0< h t ∇t < ∞ 2.2 θ Furthermore, the function t A t t φq θ s h u ∇u Δs σ −θ s φq t t h u ∇u Δs, t ∈ θ, σ − θ 2.3 Advances in Difference Equations is a positive continuous function, therefore, A t has a minimum on θ, σ − θ , hence one supposes that there exists L > such that A t ≥ L for t ∈ θ, σ − θ Proof It is easily seen that A t is continuous on θ, σ − θ Let t t φq A1 t θ σ −θ h u ∇u Δs, A2 t s φq s t h u ∇u Δs 2.4 t Then, from condition H2 , we have that the function A1 t is strictly monoton nondecreasing 0, the function A2 t is strictly monoton nonincreasing on on θ, σ − θ and A1 θ 0, which implies L mint∈ θ,σ −θ A t > θ, σ − θ and A2 σ − θ C 0, , then E is a Banach space with the norm x Throughout this paper, let E supt∈ 0,1 |x t | Let K {x ∈ E : x t ≥ 0, x t concave function on 0, } 2.5 Lemma 2.2 Let x t ∈ K and θ be as in Lemma 2.1, then x t ≥ Proof Suppose τ cases θ x , σ −ρ inf{ς ∈ ρ , σ ∀t ∈ θ, σ − θ : supt∈ ρ ,σ x t 2.6 x ς } We have three different i τ ∈ ρ , θ It follows from the concavity of x t that each point on the chard between τ, x τ and σ , x σ is below the graph of x t , thus x t ≥x τ x σ −x τ σ −τ t−τ , t ∈ θ, σ − θ , 2.7 then x t ≥ t∈ θ,σ −θ x τ x σ −x τ σ −τ x τ x σ −x τ σ −τ σ −θ−τ x σ σ −τ ≥ t−τ σ −θ−τ θ x τ σ −τ θ x τ , σ −ρ this means x t ≥ θ/ σ − ρ x for t ∈ θ, σ − θ 2.8 Advances in Difference Equations ii τ ∈ θ, σ − θ If t ∈ θ, τ , similarly, we have x t ≥x τ x τ −x ρ τ −ρ t−τ ≥x τ x τ −x ρ τ −ρ θ−τ θ−ρ x τ τ −ρ ≥ 2.9 τ −θ x ρ τ −ρ θ−ρ θ x τ ≥ x τ σ −ρ σ −ρ If t ∈ τ, σ − θ , similarly, we have x t ≥x τ ≥ x σ −x τ σ −τ t∈ θ,σ −θ x σ −x τ σ −τ x τ t−τ σ −τ −θ x σ σ −τ θ x τ σ −τ ≥ t−τ 2.10 θ x τ , σ −ρ this means x t ≥ θ/ σ − ρ x for t ∈ θ, σ − θ iii τ ∈ σ − θ, σ Similarly we have x t ≥x τ x τ −x ρ τ −ρ t−τ , t ∈ θ, σ − θ , 2.11 then x t ≥ t∈ θ,σ −θ θ−ρ x τ τ −ρ ≥ x τ −x ρ τ −ρ x τ τ −θ x ρ τ −ρ t−τ 2.12 θ x τ , σ −ρ this means x t ≥ θ/ σ − ρ x for t ∈ θ, σ − θ From the above, we Advances in Difference Equations know x t ≥ θ x , σ −ρ t ∈ θ, σ − θ 2.13 Lemma 2.3 Suppose that condition (H3) holds Let y ∈ C ρ , σ p-Laplacian BVP 2.1 - 1.2 has a solution ⎧ ⎪ ⎪φ ⎪ ⎪ q ⎪ ⎨ x t τ Ψ α ⎪ ⎪ ⎪ ⎪ ⎪φq δ ⎩ γ η t y r ∇r τ φq ξ ρ σ y r ∇r s φq τ t y r ∇r Δs, and y t ≥ Then ρ ≤ t ≤ τ; s 2.14 y r ∇r Δs, τ ≤t≤σ , τ where τ is a solution of the following equation V1 t V2 t , t ∈ ρ ,σ , 2.15 where V1 t φq t Ψ α t y r ∇r t φq ξ ρ y r ∇r Δs, s 2.16 V2 t φq δ γ η y r ∇r t σ s φq t y r ∇r Δs t Proof Obviously V1 ρ < and V1 σ > 0, beside these V2 ρ > and V2 σ < So, there must be an intersection point between ρ and σ for V1 t and V2 t , which is a 0, since V1 t and V2 t are continuous It is easy to verify that x t is solution V1 t − V2 t −y t ≤ a solution of 2.1 - 1.2 If 2.1 has a solution, denoted by x, then φ xΔ ∇ t If it does not hold, without loss There exists a constant τ ∈ ρ , σ such that xΔ τ of generality, one supposes that xΔ t > for ρ , σ From the boundary conditions, we have φp x ρ > 0, φp x σ which is a contradiction Ψ φp xΔ ξ α δ − φp xΔ η γ < 0, 2.17 Advances in Difference Equations Integrating 2.1 on τ, t , we get φp xΔ t − t y s ∇s 2.18 τ Then, we have xΔ t t φq − y s ∇s t −φq τ y s ∇s , τ 2.19 x τ − x t t s φq τ y r ∇r Δs τ Using the second boundary condition and the formula 2.18 for t x σ φq δ γ η η, we have y s ∇s 2.20 τ Also, using the formula 2.18 , we have x t φq φq δ γ η σ τ τ τ δ γ η σ s y s ∇s s φq y s ∇s φq τ t y r ∇r Δs − t s φq τ y r ∇r Δs τ 2.21 y r ∇r Δs τ Similarly, integrating 2.1 on t, τ , we get x t φq Ψ α τ y s ∇s ξ t τ φq ρ y r ∇r Δs 2.22 s Throughout this paper, we assume that τ ∈ ρ , σ ∩ T Lemma 2.4 Suppose that the conditions in Lemma 2.3 hold Then there exists a constant A such that the solution x t of p-Laplacian BVP 2.1 - 1.2 satisfies max t∈ ρ ,σ |x t | ≤ A max t∈ ρ ,σ xΔ t 2.23 Advances in Difference Equations Proof It is clear that x t satisfies t x t xΔ s Δs x ρ ρ t Ψ φp xΔ ξ α φq xΔ s Δs ρ ≤ φq B2 φp max xΔ t α t∈ ρ ,σ ≤ φq B2 α φq max t∈ ρ ,σ B2 α max t∈ ρ ,σ xΔ t xΔ t max t∈ ρ ,σ σ −ρ max t∈ ρ ,σ xΔ t t−ρ 2.24 σ −ρ xΔ t Similarly, x t − x σ σ xΔ s Δs t δ φq − φp xΔ η γ ≤ φq δ γ t∈ ρ ,σ ≤ φq δ γ t∈ ρ ,σ φq If we define A max max δ γ min{φq B2 /α − σ xΔ s Δs t xΔ t max t∈ ρ ,σ xΔ t σ −ρ max t∈ ρ ,σ max t∈ ρ ,σ σ − ρ , φq δ/γ max t∈ ρ ,σ |x t | ≤ A max t∈ ρ ,σ xΔ t σ −t xΔ t σ −ρ 2.25 xΔ t σ − ρ }, we get xΔ t 2.26 Advances in Difference Equations Now, we define a mapping T : K → E given by ⎧ ⎪ ⎪ ⎪φq ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ T x t τ Ψ α h r f r, x r ∇r ξ t τ φq ρ ⎪ ⎪ ⎪ ⎪φ δ ⎪ q ⎪ ⎪ γ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ η h r f r, x r ∇r Δs, ρ ≤ t ≤ τ; s 2.27 h r f r, x r ∇r τ σ s φq t h r f r, x r ∇r Δs, τ p for all u ∈ ∂P μ, p , then A has at least two fixed points u1 and u2 such that p < μ u1 with θ u1 < q, q < θ u2 with φ u2 < r 2.34 Main Results In this section, we will prove the existence of at least one and two positive solution of pLaplacian BVP 1.1 - 1.2 In the following theorems we will make use of Krasnoselskii, Schauder, and Avery-Henderson fixed point theorems, respectively 10 Advances in Difference Equations Theorem 3.1 Assume that (H1)–(H3) are satisfied In addition, suppose that f satisfies A1 f t, x ≥ φp mk1 for θk1 / σ − ρ ≤ x ≤ k1 , A2 f t, x ≤ φp Mk2 for ≤ x ≤ k2 , where m ∈ R1 , ∞ and M ∈ 0, R2 Then the p-Laplacian BVP 1.1 - 1.2 has a positive solution x t such that k1 ≤ x ≤ k2 Proof Without loss of generality, we suppose k1 < k2 For any x ∈ K, by Lemma 2.2, we have x t ≥ Ω2 θ x , σ −ρ ∀t ∈ θ, σ − θ We define two open subsets Ω1 and Ω2 of E such that Ω1 {x ∈ K : x < k2 } For x ∈ ∂Ω1 , by 3.1 , we have x ≥x t ≥ k1 θ θ x ≥ k1 , σ −ρ σ −ρ 3.1 {x ∈ K : x < k1 } and t ∈ θ, σ − θ 3.2 For t ∈ θ, σ − θ , if A1 holds, we will discuss it from three perspectives i If τ ∈ θ, σ − θ , thus for x ∈ ∂Ω1 , by A1 and Lemma 2.1, we have Tx Tx τ ≥ τ τ φq ρ ≥ mk1 σ h r f r, x r ∇r Δs s τ τ τ φq θ s φq σ −θ h r ∇r Δs mk1 s s φq τ ≥ mk1 A τ ≥ mk1 L ≥ R1 k1 L 2k1 h r f r, x r ∇r Δs τ 3.3 h r ∇r Δs τ x ii If τ ∈ σ − θ, σ , thus for x ∈ ∂Ω1 , by A1 and Lemma 2.1, we have Tx Tx τ ≥ τ τ φq ρ ≥ mk1 h r f r, x r ∇r Δs s σ −θ σ −θ φq θ 3.4 h r ∇r Δs s ≥ mk1 A σ − θ ≥ mk1 L ≥ 2k1 > k1 x Advances in Difference Equations 11 iii If τ ∈ ρ , θ , thus for x ∈ ∂Ω1 , by A1 and Lemma 2.1, we have Tx Tx τ ≥ σ s φq h r f r, x r ∇r Δs τ τ ≥ mk1 σ −θ s φq θ 3.5 h r ∇r Δs θ ≥ mk1 A θ ≥ mk1 L ≥ 2k1 > k1 Therefore, we have T x ≥ x , ∀x ∈ ∂Ω1 On the other hand, as x ∈ ∂Ω2 , we have x t ≤ x Tx x k2 , by A2 , we know Tx τ φq ≤ φq B2 α ≤ Mk2 φq Mk2 φq Mk2 φq Mk2 τ Ψ α τ h r f r, x r ∇r ξ σ ρ σ φq ρ ρ B2 φq α σ B2 φq α σ h r ∇r ρ h r f r, x r ∇r Δs ρ σ σ φq ρ h r ∇r h r f r, x r ∇r Δs s σ h r f r, x r ∇r B2 α τ φq σ − ρ φq ρ σ 3.6 h r ∇r ρ σ −ρ 1 < Mk2 R2 M h r ∇r Δs ρ σ φq h r ∇r ρ k2 x Then, T has a fixed point x ∈ Ω2 \ Ω1 Obviously, x is a positive solution of the p-Laplacian BVP 1.1 - 1.2 and k1 ≤ x ≤ k2 Existence of at least one positive solution is also proved using Schauder fixed point theorem Theorem 2.7 Then we have the following result Theorem 3.2 Assume that (H1)–(H3) are satisfied If R satisfies Q ≤ R, R2 3.7 12 Advances in Difference Equations where Q satisfies for t ∈ 0, , φp Q ≥ max f t, x t x ≤R 3.8 then the p-Laplacian BVP 1.1 - 1.2 has at least one positive solution Proof Let KR : {x ∈ K : x ≤ R} Note that KR is closed, bounded, and convex subset of E to which the Schauder fixed point theorem is applicable Define T : KR → E as in 2.27 for t ∈ ρ , σ It can be shown that T : KR → E is continuous Claim that T : KR → KR Let x ∈ KR By using the similar methods used in the proof of Theorem 3.1, we have Tx Tx τ φq ≤ φq B2 α ≤ Q φq Q τ Ψ α τ h r f r, x r ∇r ξ σ τ φq ρ σ h r f r, x r ∇r σ φq ρ ρ B2 α σ σ −ρ φq h r f r, x r ∇r Δs s h r f r, x r ∇r Δs ρ 3.9 h r ∇r ρ ≤ R, R2 which implies T x ∈ KR The compactness of the operator T : KR → KR follows from the Arzela-Ascoli theorem Hence T has a fixed point in KR Corollary 3.3 If f is continuous and bounded on 0, × R , then the p-Laplacian BVP 1.1 - 1.2 has a positive solution Now we will give the sufficient conditions to have at least two positive solutions for p-Laplacian BVP 1.1 - 1.2 Set t P t : φq h r ∇r σ −θ φq θ h r ∇r , t ∈ θ, σ − θ 3.10 t The function P t is positive and continuous on θ, σ − θ Therefore, P t has a minimum on θ, σ − θ Hence we suppose there exists N > such that P t ≥ N Also, we define the nonnegative, increasing continuous functions Υ, Φ, and Γ by Υ x Φx x θ x σ −θ , max t∈ ρ ,θ ∪ σ −θ,σ Γx max t∈ ρ ,σ x t , x t 3.11 Advances in Difference Equations 13 We observe here that, for every x ∈ K, Υ x ≤ Φ x ≤ Γ x and from Lemma 2.2, x ≤ σ − ρ /θ Υ x Also, for ≤ λ ≤ 1, Φ λx λΦ x Theorem 3.4 Assume that (H1)–(H3) are satisfied Suppose that there exist positive numbers a < b < c such that the function f satisfies the following conditions: i f t, x ≥ φp ma for x ∈ 0, a , ii f t, x ≤ φp Mb for x ∈ 0, σ − ρ /θ b , iii f t, x ≥ φp 2/θn c for x ∈ θ/ σ − ρ c, σ − ρ /θ c , for positive constants m ∈ R1 , ∞ , M ∈ 0, R2 , and n ∈ 0, N Then the p-Laplacian BVP 1.1 1.2 has at least two positive solutions x1 , x2 such that a< b< max x1 t t∈ ρ ,σ max t∈ ρ ,θ ∪ σ −θ,σ with x2 t max t∈ ρ ,θ ∪ σ −θ,σ with x1 t < b, 3.12 x2 θ x2 σ − θ < c Proof Define the cone as in 2.5 From Lemmas 2.2 and 2.3 and the conditions H1 and H2 , we can obtain T K ⊂ K Also from Lemma 2.5, we see that T : K → K is completely continuous We now show that the conditions of Theorem 2.8 are satisfied To fulfill property i of Theorem 2.8, we choose x ∈ ∂P Υ, c , thus Υ x 1/2 x θ σ − ρ /θ c, we have x σ −θ c Recalling that x ≤ σ − ρ /θ Υ x σ −ρ θ x ≤x t ≤ c σ −ρ θ 3.13 Then assumption iii implies f t, x > φp 2/θn c for t ∈ θ, σ − θ We have three different cases a If τ ∈ σ − θ, σ , we have Tx θ Υ Tx ≥ Tx θ ≥ θ τ τ ρ h r φp θ φq c ∇r Δs θn 2 cP θ θ ≥ cP θ ≥ 2c θn N σ −θ θ ρ σ −θ τ φq ξ h r f r, x r ∇r Δs ≥ φq θ h r f r, x r ∇r s θ ρ ≥ Ψ α φq φq ρ ≥ Tx σ − θ h r f r, x r ∇r Δs s h r f r, x r ∇r Δs θ cφq θn σ −θ h r ∇r θ−ρ θ 3.14 14 Advances in Difference Equations Thus we have Υ T x ≥ c b If τ ∈ ρ , θ , we have Tx θ Υ Tx η δ γ φq Tx σ − θ s φq h r f r, x r ∇r Δs τ σ −θ h r f r, x r ∇r Δs ≥ φq φq τ h r f r, x r ∇r σ −σ θ θ σ −θ h r φp θ ≥ s σ −θ τ σ −θ ≥ φq σ h r f r, x r ∇r σ ≥ ≥ Tx σ − θ c ∇r θn θ ≥ σ −θ cφq θn h r ∇r θ θ 2 cP θ ≥ cP θ ≥ 2c n N 3.15 Thus we have Υ T x ≥ c c If τ ∈ θ, σ − θ , we have 2Υ T x Tx σ − θ Tx θ ≥ θ τ φq ρ ≥ ≥ ≥ ≥ c θn σ h r f r, x r ∇r Δs σ −θ s θ τ φq ρ σ −θ τ c φq θn τ h r ∇r φq φq θ h r ∇r Δs τ σ −θ θ−ρ h r f r, x r ∇r Δs τ σ −θ σ h r ∇r Δs θ c φq θn s φq h r ∇r θ τ h r ∇r θ 2 cP τ ≥ cN n N σ −θ φq h r ∇r θ τ 2c 3.16 Thus we have Υ T x ≥ c and condition i of Theorem 2.8 holds Next we will show condition b ii of Theorem 2.8 is satisfied If x ∈ ∂P Φ, b , then maxt∈ ρ ,θ ∪ σ −θ,σ x t Noting that x ≤ we have ≤ x t ≤ σ −ρ σ −ρ Υ x ≤ Φx θ θ σ − ρ /θ b, for t ∈ ρ , σ σ −ρ b, θ 3.17 Advances in Difference Equations 15 Then ii yields f t, x ≤ φp Mb for t ∈ ρ , σ As T x ∈ K, so Φ Tx max t∈ ρ ,θ ∪ σ −θ,σ φq ≤ φq ≤ φq τ Ψ α B2 α τ h r f r, x r ∇r σ ρ h r f r, x r ∇r Δs ρ σ h r φp bM ∇r σ φq ρ B2 φq α σ σ φq ρ σ h r f r, x r ∇r Δs s σ h r f r, x r ∇r ρ bMφq τ φq ξ B2 φq α bMφq Tx t ≤ Tx τ ρ σ bMφq ρ h r ∇r h r φp bM ∇r Δs ρ σ h r ∇r 3.18 h r ∇r σ −ρ ρ B2 α φq ρ σ −ρ ≤ bR2 R2 b So condition ii of Theorem 2.8 holds a/2, t ∈ ρ , σ is To fulfill property iii of Theorem 2.8, we note x∗ t a/2, so P Γ, a / ∅ Now choose x ∈ ∂P Γ, a , then a member of P Γ, a and Γ x∗ a and this implies that ≤ x t ≤ a for t ∈ ρ , σ It follows Γx maxt∈ ρ ,σ x t from the assumption i , we have f t, x ≥ φp ma for t ∈ ρ , σ As before we obtain the following cases a If τ < θ, we have Γ Tx max t∈ ρ ,σ ≥ Tx t σ s φq τ ≥ h r f r, x r ∇r Δs τ σ −θ s φq θ ≥ h r f r, x r ∇r Δs τ σ −θ s φq θ ≥ Tx τ h r f r, x r ∇r Δs θ σ −θ s φq θ h r φp ma ∇r Δs θ σ −θ s φq ma θ maA θ ≥ R1 aL Thus we have Γ T x ≥ a h r ∇r Δs θ 2a ≥ a 3.19 16 Advances in Difference Equations b If τ ∈ θ, σ − θ , we have 2Γ T x 2T x τ ≥ τ τ φq ρ ≥ τ φq θ τ s φq τ τ τ φq s φq s h r φp ma ∇r Δs 3.20 τ σ −θ h r ∇r Δs h r f r, x r ∇r Δs τ σ −θ h r φp ma ∇r Δs s θ s φq s τ ≥ ma σ h r f r, x r ∇r Δs τ h r ∇r Δs τ maA τ ≥ R1 aL ≥ a Thus we have Γ T x ≥ a c If τ > σ − θ, we have Γ Tx Tx τ ≥ τ τ φq ρ ≥ h r f r, x r ∇r Δs s τ τ φq ρ h r φp ma ∇r Δs 3.21 s σ −θ ≥ ma σ −θ φq θ h r ∇r Δs s maA σ − θ ≥ R1 aL ≥ a Thus we have Γ T x ≥ a Therefore, condition iii of Theorem 2.8 holds Since all conditions of Theorem 2.8 are satisfied, the p-Laplacian BVP 1.1 - 1.2 has at least two positive solutions x1 , x2 such that a< b< max t∈ ρ ,σ max x1 t t∈ ρ ,θ ∪ σ −θ,σ with x2 t max t∈ ρ ,θ ∪ σ −θ,σ 1 with x2 θ x1 t < b, 3.22 x2 σ − θ < c Advances in Difference Equations 17 Monotone Method In this section, we will prove the existence of solution of p-Laplacian BVP 1.1 - 1.2 by using upper and lower solution method We define the set D: x : φp xΔ ∇ is continuous on 0, Definition 4.1 A real-valued function u t ∈ D on ρ , σ 1.2 if φp uΔ ∇ t h t f t, u t > 0, 4.1 is a lower solution for 1.1 - t ∈ 0, , 4.2 αφp u ρ Δ − Ψ φp u ξ ≤ 0, γφp u σ Similarly, a real-valued function v t ∈ D on ρ , σ δφp u Δ η ≤ is an upper solution for 1.1 - 1.2 if φp vΔ ∇ t h t f t, v t < 0, t ∈ 0, , 4.3 αφp v ρ − Ψ φp v Δ ξ ≥ 0, γφp v σ δφp v Δ η ≥ We will prove when the lower and the upper solutions are given in the well order, that is, u ≤ v, the p-Laplacian BVP 1.1 - 1.2 admits a solution lying between both functions Theorem 4.2 Assume that (H1)–(H3) are satisfied and u and v are, respectively, lower and upper solutions for the p-Laplacian BVP 1.1 - 1.2 such that u ≤ v on ρ , σ Then the p-Laplacian BVP 1.1 - 1.2 has a solution x t ∈ u t , v t on ρ , σ Proof Consider the p-Laplacian BVP: φp xΔ ∇ t h t F t, x t 0, t ∈ 0, , 4.4 αφp x ρ − Ψ φp x Δ ξ 0, γφp x σ δφp x Δ η 0, where F t, x t for t ∈ 0, ⎧ ⎪f t, v t , ⎪ ⎪ ⎨ f t, x t , ⎪ ⎪ ⎪ ⎩ f t, u t , x t >v t , u t ≤x t ≤v t , x t has a positive there exists a t0 ∈ ρ , σ such that z t0 maximum Consequently, we know that zΔ t0 ≤ and there exists t1 ∈ ρ , t0 such that zΔ t ≥ on t1 , t0 On the other hand by the continuity of z t at t0 , we know there exists t2 ∈ ρ , t0 such that z t > on t2 , t0 Let t max{t1 , t2 }, then we have zΔ t ≥ on t, t0 Thus we get zΔ t ≥ ⇒ φp xΔ t ≥ φp vΔ t , zΔ t0 ≤ ⇒ φp xΔ t0 ≤ φp vΔ t0 4.6 Therefore, ≥ φp xΔ t0 t0 − φp vΔ t0 − φp xΔ t ∇ φp xΔ − φp vΔ vΔ − φp t t ∇t t t0 φp x ∇ Δ t − φp v Δ ∇ 4.7 t ∇t t t0 > −h t f t, v t h t f t, v t ∇t 0, t which is a contradiction and thus t0 cannot be an element of ρ , σ If t0 ρ , from the boundary conditions, we have 1−q αφp x ρ ≤ B2 φp xΔ ξ ⇒ φp α1−q x ρ ≤ φp B2 xΔ ξ αφp v ρ ≥ B1 φp vΔ ξ ⇒ φp α1−q v ρ ≥ φp B1 vΔ ξ 1−q , 4.8 Thus we get α1−q x ρ 1−q ≤ B2 x Δ ξ , α1−q v ρ 1−q ≥ B1 v Δ ξ 4.9 From this inequalities, we have α1−q x − v ρ 1−q α1−q z ρ which is a contradiction 1−q 1−q ≤ B2 x Δ ξ − B v Δ ξ ≤ B 1−q ≤ B1 zΔ ξ ≤ 0, x−v Δ ξ , 4.10 Advances in Difference Equations If t0 19 σ , from the boundary conditions, we have γφp x σ −δφp xΔ η ⇒ φp γ 1−q x σ −φp δ1−q xΔ η γφp v σ φp −δ1−q xΔ η , 4.11 ≥ −δφp vΔ η ⇒ φp γ 1−q v σ ≥ −φp δ1−q vΔ η φp −δ1−q vΔ η Thus we get γ 1−q x σ δ1−q xΔ η , γ 1−q v σ ≥ −δ1−q vΔ η 4.12 From this inequalities, we have γ 1−q x − v σ γ 1−q z σ ≤ −δ1−q x − v Δ ≤ −δ1−q zΔ η ≤ 0, η , 4.13 which is a contradiction Thus we have x t ≤ v t on ρ , σ Similarly, we can get u t ≤ x t on ρ , σ Thus x t is a solution of p-Laplacian BVP 1.1 - 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