Existence of Positive Solutions for Higher Order (𝑝,𝑞)-Laplacian Two-Point Boundary Value Problems

Abstract

We derive sufficient conditions for the existence of positive solutions to higher order (𝑝,𝑞)-Laplacian two-point boundary value problem, (−1)^{𝑚1+𝑛1−1}[𝜙_𝑝(𝑢^(2𝑚1)(𝑡))]^(𝑛1)=𝑓₁(𝑡,𝑢(𝑡),𝑣(𝑡)), 𝑡∈[0,1], (−1)^{𝑚2+𝑛2−1}[𝜙_𝑞(𝑣^(𝑚2)(𝑡))]^(2𝑛2)=𝑓₂(𝑡,𝑢(𝑡),𝑣(𝑡)), 𝑡∈[0,1], 𝑢^(2𝑖)(0)=0=𝑢^(2𝑖)(1), 𝑖=0,1,2,…,𝑚₁−1, [𝜙_𝑝(𝑢^(2𝑚1)(𝑡))]^(𝑗)_at𝑡=0=0, 𝑗=0,1,…,𝑛₁−2; [𝜙_𝑝(𝑢^(2𝑚1)(1))]=0, [𝜙_𝑞(𝑣^(𝑚2)(𝑡))]^(2𝑖)_at𝑡=0=0=[𝜙_𝑞(𝑣^(𝑚2)(𝑡))]^(2𝑖)_at𝑡=1, 𝑖=0,1,…,𝑛₂−1, 𝑣^(𝑗)(0)=0, 𝑗=0,1,2,…,𝑚₂−2, and 𝑣(1)=0, where 𝑓₁,𝑓₂ are continuous functions from [0,1]×ℝ² to [0,∞), 𝑚₁,𝑛₁,𝑚₂,𝑛₂∈ℕ and 1/𝑝+1/𝑞=1. We establish the existence of at least three positive solutions for the two-point coupled system by utilizing five-functional fixed point theorem. And also, we demonstrate our result with an example.

1. Introduction

The goal of differential equations is to understand the physical phenomena of nature by developing mathematical models. Among all, a class of differential equations governed by nonlinear differential operators, which have wide applications and interest, has been developed to study such type of equations. In this theory, the most investigated operator is the classical 𝑝-Laplacian, given by 𝜙_𝑝(𝑦)∶=𝑦|𝑦|^𝑝−2 with 𝑝>1. These problems have a wide range of applications in physics and related sciences such as biophysics, plasma physics, and chemical reaction design. Due to the importance in both theory and applications, 𝑝-Laplacian boundary value problems have created a great deal of interest in recent years; we mention a few [1–11].

Recently, Prasad and Murali [12] established the existence of positive solutions of 𝑝-Laplacian singular boundary value problem on time scale, 𝜙_𝑝𝑦^Δ(𝑡)^∇+𝑞(𝑡)𝑓𝑡,𝑦(𝑡),𝑦^Δ(𝑡)()=0,𝑡∈0,1_𝕋,𝑦(0)=0=𝑦^Δ(1), (1) by assuming suitable conditions on 𝑓. Till now in the literature of boundary value problems, the theory was not developed to the system of higher order boundary value problems with (𝑝,𝑞)-Laplacian. Mainly, this type of problems arises in radar invention models and microatom invention models.

Due to our interest in the literature, in this paper, we consider two-point higher order (𝑝,𝑞)-Laplacian boundary value problem (BVP) (−1)^{𝑚1+𝑛1−1}𝜙_𝑝𝑢^(2𝑚1)(𝑡)^(𝑛1)=𝑓₁((𝑡)(𝑡,[],()𝑡,𝑢,𝑣))𝑡∈0,1−1^{𝑚2+𝑛2−1}𝜙_𝑞𝑣^(𝑚2)(𝑡)^(2𝑛2)=𝑓₂((𝑡)(𝑡,[],𝑢𝑡,𝑢,𝑣))𝑡∈0,1_(2𝑖)(0)=0=𝑢^(2𝑖)(1),𝑖=0,1,2,…,𝑚₁𝜙−1,_𝑝𝑢^(2𝑚1)(𝑡)^(𝑗)_at𝑡=0=0,𝑗=0,1,2,…,𝑛₁𝜙−2,_𝑝𝑢^(2𝑚1)(𝑡)_at𝑡=1𝜙=0,_𝑞𝑣^(𝑚2)(𝑡)^(2𝑖)_at𝑡=0=0=𝜙_𝑞𝑣^(𝑚2)(𝑡)^(2𝑖)_at𝑡=1,𝑖=0,1,…,𝑛₂𝑣−1,_(𝑗)(0)=0,𝑗=0,1,2,…,𝑚₂−2,𝑣(1)=0, (2) where 𝑓₁,𝑓₂ are continuous functions from [0,1]×ℝ² to [0,∞),  𝑚₁,𝑛₁,𝑚₂,𝑛₂∈ℕ and 1/𝑝+1/𝑞=1.   If we take 𝑝=𝑞 in the above problem then it reduces to 𝑝-Laplacian problem.

To obtain a solution of the BVP (2), we construct Green's functions for the corresponding homogeneous BVPs. For 𝑛₁≥2, let 𝐺(𝑡,𝑠) be Green's function of the BVP, (−1)^𝑛1−1𝑥^(𝑛1)(𝑡)[],𝑥=0,𝑡∈0,1_(𝑗)(0)=0,𝑗=0,1,2,…,𝑛₁−2,𝑥(1)=0 (3) and it is given by 𝐺()=𝑡𝑡,𝑠^𝑛1−1(1−𝑠)^𝑛1−1𝑛₁!𝑡−1,0≤𝑡≤𝑠<1,_𝑛1−1(1−𝑠)^𝑛1−1𝑛₁!−()−1𝑡−𝑠^𝑛1−1𝑛₁−1!,0<𝑠≤𝑡≤1. (4) Let 𝐻_𝑚1(𝑡,𝑠) be Green's function of the BVP (−1)^𝑚1𝑦^(2𝑚1)(𝑡)[],𝑦=0,𝑡∈0,1_(2𝑖)(0)=0=𝑦^(2𝑖)(1),𝑖=0,1,…,𝑚₁−1, (5) and it can be recursively defined as 𝐻_𝑚1(𝑡,𝑠)=¹₀𝐻_𝑚1−1(𝑡,𝑟)𝐻₁(𝑟,𝑠)𝑑𝑟, (6) where 𝐻₁(𝑡,𝑠) is Green's function of −𝑦^′′(𝑡)=0,𝑦(0)=0=𝑦(1) (7) and is given by 𝐻₁()=𝑡()𝑠()𝑡,𝑠1−𝑠,0≤𝑡≤𝑠<1,1−𝑡,0<𝑠≤𝑡≤1. (8) Then 𝑢(𝑡) can be expressed in the form 𝑢(𝑡)=¹₀𝐻_𝑚1(𝑡,𝑠)𝜙_𝑝⁻¹×¹₀𝐺(𝑠,𝜏)𝑓₁(𝜏,𝑢(𝜏),𝑣(𝜏))𝑑𝜏𝑑𝑠. (9) Since 𝜙_𝑝⁻¹=𝜙_𝑞, we have 𝑢(𝑡)=¹₀𝐻_𝑚1(𝑡,𝑠)𝜙_𝑞¹₀𝐺(𝑠,𝜏)𝑓₁(𝜏,𝑢(𝜏),𝑣(𝜏))𝑑𝜏𝑑𝑠. (10) For 𝑛₁=1, 𝑢(𝑡)=¹₀𝐻_𝑚1(𝑡,𝑠)𝜙_𝑞¹_𝑠𝑓₁(𝜏,𝑢(𝜏),𝑣(𝜏))𝑑𝜏𝑑𝑠. (11) For 𝑚₂≥2, let 𝐻(𝑡,𝑠) be Green’s function of the BVP (−1)^𝑚2−1𝑥^(𝑚2)(𝑡)[],𝑥=0,𝑡∈0,1_(𝑗)(0)=0,𝑗=0,1,2,…,𝑚₂−2,𝑥(1)=0 (12) and it is given by 𝐻()=𝑡𝑡,𝑠^𝑚2−1(1−𝑠)^𝑚2−1𝑚₂!𝑡−1,0≤𝑡≤𝑠<1,_𝑚2−1(1−𝑠)^𝑚2−1𝑚₂!−()−1𝑡−𝑠^𝑚2−1𝑚₂−1!,0<𝑠≤𝑡≤1. (13) Let 𝐺_𝑛2(𝑡,𝑠) be Green's function of the BVP (−1)^𝑛2𝑦^(2𝑛2)(𝑡)[],𝑦=0,𝑡∈0,1_(2𝑖)(0)=0=𝑦^(2𝑖)(1),𝑖=0,1,…,𝑛₂−1, (14) and it can be recursively defined as 𝐺_𝑛2(𝑡,𝑠)=¹₀𝐺_𝑛2−1(𝑡,𝑟)𝐺₁(𝑟,𝑠)𝑑𝑟, (15) where 𝐺₁(𝑡,𝑠) is Green’s function of −𝑦^′′(𝑡)=0,𝑦(0)=0=𝑦(1) (16) and is given by 𝐺₁()=𝑡()𝑠()𝑡,𝑠1−𝑠,0≤𝑡≤𝑠<1,1−𝑡,0<𝑠≤𝑡≤1. (17) Then 𝑣(𝑡) can be expressed in the form 𝑣(𝑡)=¹₀𝐻(𝑡,𝑠)𝜙_𝑞⁻¹×¹₀𝐺_𝑛2(𝑠,𝜏)𝑓₂(𝜏,𝑢(𝜏),𝑣(𝜏))𝑑𝜏𝑑𝑠. (18) Since 𝜙_𝑞⁻¹=𝜙_𝑝, we have 𝑣(𝑡)=¹₀𝐻(𝑡,𝑠)𝜙_𝑝¹₀𝐺_𝑛2(𝑠,𝜏)𝑓₂(𝜏,𝑢(𝜏),𝑣(𝜏))𝑑𝜏𝑑𝑠. (19) For 𝑚₂=1, 𝑣(𝑡)=¹_𝑡𝜙_𝑝¹₀𝐺_𝑛2(𝑠,𝜏)𝑓₂(𝜏,𝑢(𝜏),𝑣(𝜏))𝑑𝜏𝑑𝑠. (20) Further, it is easily seen that 𝐻_𝑚1(𝑡,𝑠), 𝐻(𝑡,𝑠), 𝐺_𝑛2(𝑡,𝑠) and 𝐺(𝑡,𝑠), all are nonnegative on [0,1]×[0,1].

A solution of the BVP (2) is a function (𝑢,𝑣)∈𝐶^2𝑚1[0,1]×𝐶^𝑚2[0,1] such that (𝜙_𝑝∘𝑢^(2𝑚1),𝜙_𝑞∘𝑣^(𝑚2))∈𝐶^𝑛1[0,1]×𝐶^2𝑛2[0,1], and (𝑢,𝑣) satisfies the BVP (2).

A positive solution (𝑢,𝑣) of the BVP (2) is a solution of the BVP (2) such that 𝑢 and 𝑣 are nonnegative on [0,1].

The rest of the paper is organized as follows. In Section 2, we estimate the bounds of Green's functions. In Section 3, we establish the existence of at least three positive solutions for two-point BVP (2) by using Avery's generalization of Leggett-Williams fixed point theorem. And also, we demonstrate our result with an example.

2. Bounds of Green's Functions

In this section, we state some lemmas to estimate bounds on Green’s functions which are needed in later discussions. The following lemma is included in the paper to prove the remaining lemmas.

Lemma 1. Let 𝑤∈(0,(𝑏−𝑎)/2] and 𝐼=[𝑎+𝑤,𝑏−𝑤]. Then, for any 𝑗∈ℕ𝐺_𝑗()≥𝑤𝑡,𝑠𝑏−𝑎^𝑗𝜙_𝑤^𝑗−1𝐺₁(𝑠,𝑠),∀(𝑡,𝑠)∈𝐼×[𝑎,𝑏], (21) where 𝜙_𝑤=∫^𝑏−𝑤_𝑎+𝑤𝐺₁(𝑟,𝑟)∇𝑟>0.

Proof. The proof is by induction. First, for 𝑗=1 inequality (21) is obvious. Next, for fixed 𝑗, assume that (21) is true; from recursive formula we have, for each (𝑡,𝑠)∈𝐼×[𝑎,𝑏], 𝐺_𝑗+1()=𝑡,𝑠_𝑏𝑎𝐺_𝑗(𝑡,𝑟)𝐺₁()≥𝑟,𝑠∇𝑟_𝑏𝑎𝑤𝑏−𝑎^𝑗𝜙_𝑤^𝑗−1𝐺₁(𝑟,𝑟)⋅𝐺₁()≥𝑤𝑟,𝑠∇𝑟𝑏−𝑎^𝑗𝜙_𝑤^{𝑗−1𝑏−𝑤}_𝑎+𝑤𝐺₁(𝑟,𝑟)⋅𝐺₁()≥𝑤𝑟,𝑠∇𝑟𝑏−𝑎^𝑗𝜙_𝑤^{𝑗−1𝑏−𝑤}_𝑎+𝑤𝐺₁()⋅𝑤𝑟,𝑟𝐺𝑏−𝑎¹()=𝑤𝑠,𝑠∇𝑟𝑏−𝑎^𝑗+1𝜙_𝑤^𝑗𝐺₁(𝑠,𝑠). (22) Hence, by induction the proof is complete.

In the previous lemma by choosing 𝑎=0, 𝑏=1, and 𝑤=1/4, 𝑤=1/3, and 𝑤=1/8, we get 𝐼=[1/4,3/4], 𝐼₁=[1/3,2/3], and 𝐼₂=[1/8,7/8]. We know that 𝐺₁(𝑠,𝑠)=𝑠(1−𝑠).

Lemma 2. For (𝑡,𝑠)∈𝐼×[0,1], one has 𝐺_𝑗(𝑡,𝑠)≥116^𝑗−114^3𝑗−2(1−𝑠)𝑠. (23) For (𝑡,𝑠)∈𝐼₁×[0,1], one has 𝐺_𝑗()≥1𝑡,𝑠313486^𝑗−1(1−𝑠)𝑠. (24) For (𝑡,𝑠)∈𝐼₂×[0,1], one has 𝐺_𝑗()≥1𝑡,𝑠8392048^𝑗−1(1−𝑠)𝑠, (25) where 𝑗=1,2,…,𝑛₂.

Lemma 3. For 𝑡,𝑠∈[0,1], one has 𝐺_𝑗()≤1𝑡,𝑠6^𝑗−1(1−𝑠)𝑠, (26) where 𝑗=1,2,…,𝑛₂.

For details refer to [9].

In Lemma 1, replacing 𝐺_𝑗(⋅,⋅) by 𝐻_𝑗(⋅,⋅), we have the following.

Lemma 4. For (𝑡,𝑠)∈𝐼×[0,1], one has 𝐻_𝑗(𝑡,𝑠)≥116^𝑗−114^3𝑗−2(1−𝑠)𝑠. (27) For (𝑡,𝑠)∈𝐼₁×[0,1], one has 𝐻_𝑗()≥1𝑡,𝑠313486^𝑗−1(1−𝑠)𝑠. (28) For (𝑡,𝑠)∈𝐼₂×[0,1], one has 𝐻_𝑗()≥1𝑡,𝑠8392048^𝑗−1(1−𝑠)𝑠, (29) where 𝑗=1,2,…,𝑚₁.

Lemma 5. For 𝑡,𝑠∈[0,1], one has 𝐻_𝑗()≤1𝑡,𝑠6^𝑗−1(1−𝑠)𝑠, (30) where 𝑗=1,2,…,𝑚₁.

For details refer to [9].

Lemma 6. For (𝑡,𝑠)∈𝐼×[0,1], one has 𝐻()≥1𝑡,𝑠4^𝑚−1𝐻(𝑠,𝑠). (31) For (𝑡,𝑠)∈𝐼₁×[0,1], one has 𝐻()≥1𝑡,𝑠3^𝑚−1𝐻(𝑠,𝑠). (32) For (𝑡,𝑠)∈𝐼₂×[0,1], one has 𝐻()≥1𝑡,𝑠8^𝑚−1𝐻(𝑠,𝑠). (33)

For details refer to [11].

Lemma 7. For 𝑡,𝑠∈[0,1], one has 𝐻(𝑡,𝑠)≤𝐻(𝑠,𝑠). (34)

For details refer to [11].

Lemma 8. For (𝑡,𝑠)∈𝐼×[0,1], one has 𝐺()≥1𝑡,𝑠4^𝑛−1𝐺(𝑠,𝑠). (35) For (𝑡,𝑠)∈𝐼₁×[0,1], one has 𝐺()≥1𝑡,𝑠3^𝑛−1𝐺(𝑠,𝑠). (36) For (𝑡,𝑠)∈𝐼₂×[0,1], one has 𝐺()≥1𝑡,𝑠8^𝑛−1𝐺(𝑠,𝑠). (37)

For details refer to [11].

Lemma 9. For 𝑡,𝑠∈[0,1], one has 𝐺(𝑡,𝑠)≤𝐺(𝑠,𝑠). (38)

For details refer to [11].

Denote ℳ=min116^𝑗−114^3𝑗−2⋅𝜙_𝑝⁻¹14^𝑛1−1,𝑗=1,…,𝑚₁1−1,4^𝑚2−1⋅𝜙_𝑞⁻¹116^𝑗−114^3𝑗−2,𝑗=1,…,𝑛₂−1. (39)

3. Existence of Multiple Positive Solutions

In this section, we establish the existence of at least three positive solutions for the system of BVP (2), by using Avery's generalization of the Leggett-Williams fixed point theorem.

Let 𝐵 be a real Banach space with cone 𝑃. We define the nonnegative continuous convex functionals 𝛾, 𝛽, and 𝜃 and nonnegative continuous concave functionals 𝛼, 𝜓 on 𝑃, for nonnegative numbers 𝑎^′, 𝑏^′, 𝑐^′, 𝑑^′, and ℎ^′; we define the following sets: 𝑃𝛾,𝑐^′=𝑦∈𝑃∣𝛾𝑦<𝑐^′,𝑃𝛾,𝛼,𝑎^′,𝑐^′=𝑦∈𝑃∣𝑎^′≤𝛼𝑦,𝛾𝑦≤𝑐^′,𝑄𝛾,𝛽,𝑑^′,𝑐^′=𝑦∈𝑃∣𝛽𝑦≤𝑑^′,𝛾𝑦≤𝑐^′,𝑃𝛾,𝜃,𝛼,𝑎^′,𝑏^′,𝑐^′=𝑦∈𝑃∣𝑎^′≤𝛼𝑦,𝜃𝑦≤𝑏^′,𝛾𝑦≤𝑐^′,𝑄𝛾,𝛽,𝜓,ℎ^′,𝑑^′,𝑐^′=𝑦∈𝑃∣ℎ^′≤𝜓𝑦,𝛽𝑦≤𝑑^′,𝛾𝑦≤𝑐^′. (40)

In obtaining multiple positive solutions of the BVP (2), the following so-called five-functionals fixed point theorem will be fundamental.

Theorem 10 (see [4]). Let 𝑃 be a cone in a real Banach space 𝐵. Suppose 𝛼 and 𝜓 are nonnegative continuous concave functionals on 𝑃 and 𝛾; 𝛽 and 𝜃 are nonnegative continuous convex functionals on 𝑃 such that, for some positive numbers 𝑐^′ and 𝑘, 𝛼𝑦≤𝛽𝑦,𝑦≤𝑘𝛾𝑦∀𝑦∈𝑃(𝛾,𝑐^′). (41) Suppose further that 𝑇∶𝑃(𝛾,𝑐^′)→𝑃(𝛾,𝑐^′) is completely continuous, and there exist constants ℎ^′,𝑑^′,𝑎^′,𝑏^′≥0 with 0<𝑑^′<𝑎^′ such that each of the following is satisfied (B1)  {𝑦∈𝑃(𝛾,𝜃,𝛼,𝑎^′,𝑏^′,𝑐^′)∣𝛼(𝑦)>𝑎^′}≠∅ and  𝛼(𝑇𝑦)>𝑎^′ for 𝑦∈𝑃(𝛾,𝜃,𝛼,𝑎^′,𝑏^′,𝑐^′),(B2){𝑦∈𝑄(𝛾,𝛽,𝜓,ℎ^′,𝑑^′,𝑐^′)∣𝛽(𝑦)<𝑑^′}≠∅ and  𝛽(𝑇𝑦)<𝑑^′ for 𝑦∈𝑄(𝛾,𝛽,𝜓,ℎ^′,𝑑^′,𝑐^′),(B3)𝛼(𝑇𝑦)>𝑎^′ provided 𝑦∈𝑃(𝛾,𝛼,𝑎^′,𝑐^′) with 𝜃(𝑇𝑦)>𝑏^′, (B4)𝛽(𝑇𝑦)<𝑑^′ provided 𝑦∈𝑄(𝛾,𝛽,𝑑^′,𝑐^′) with 𝜓(𝑇𝑦)<ℎ^′.
Then 𝑇 has at least three fixed points 𝑦₁,𝑦₂,𝑦₃∈𝑃(𝛾,𝑐^′) such that 𝛽𝑦₁<𝑑^′,𝑎<𝛼𝑦₂,𝑑^′<𝛽𝑦₃with𝛼𝑦₃<𝑎^′. (42)

Let 𝐸={𝑦∣𝑦∈𝐶[0,1]}, and denote 𝐵=𝐸×𝐸 by the norm defined as ‖(𝑢,𝑣)‖=‖𝑢‖₀+‖𝑣‖₀, where 𝑦₀=max_𝑡∈[0,1]𝑦(𝑡). (43) Then the set 𝐵 with the norm ‖⋅‖ is a complete normed linear space.

Let 𝐼=[1/4,3/4], 𝐼₁=[1/3,2/3], and 𝐼₂=[1/8,7/8]. Define the cone 𝑃⊂𝐵 by 𝑃=(𝑢,𝑣)∈𝐵∣𝑢(𝑡)≥0,𝑣(𝑡)≥0,min_𝑡∈𝐼(𝑢(𝑡)+𝑣(𝑡))≥ℳ‖(𝑢,𝑣)‖. (44) Now, define the nonnegative continuous concave functionals 𝛼,𝜓 and the nonnegative continuous convex functionals 𝛽,𝜃, and 𝛾 on 𝑃 by 𝛾(𝑢,𝑣)=max_𝑡∈𝐼2(𝑢(𝑡)(𝑡,𝜓()+𝑣))𝑢,𝑣=min_𝑡∈𝐼1(𝑢(𝑡)(𝑡,𝛽()+𝑣))𝑢,𝑣=max_𝑡∈𝐼1(𝑢(𝑡)(𝑡,𝛼()+𝑣))𝑢,𝑣=min_𝑡∈𝐼(𝑢(𝑡)(𝑡,𝜃()+𝑣))𝑢,𝑣=max_𝑡∈𝐼(𝑢(𝑡)+𝑣(𝑡)). (45) We observe that, for any (𝑢,𝑣)∈𝑃, 𝛼(𝑢,𝑣)=min_𝑡∈𝐼(𝑢(𝑡)+𝑣(𝑡))≤max_𝑡∈𝐼1(𝑢(𝑡)+𝑣(𝑡))=𝛽(𝑢,𝑣),=1‖(𝑢,𝑣)‖ℳmin_𝑡∈𝐼(𝑢(𝑡)(𝑡≤1+𝑣))ℳmax_𝑡∈𝐼2(𝑢(𝑡)(𝑡=1+𝑣))ℳ𝛾(𝑢,𝑣). (46)(47)

We derive growth conditions on 𝑓₁,𝑓₂ so that the BVP (2) has at least three positive solutions. We are now ready to present the main result of this section.

We denote 𝒮=¹₀𝐺(𝜏,𝜏)𝑑𝜏;ℛ=¹₀𝐻(𝑠,𝑠)𝑑𝑠. (48)

Theorem 11. Suppose there exists 0<𝑎^′<𝑏^′<𝑏^′/ℳ≤𝑐^′ such that 𝑓₁ and 𝑓₂ satisfy the following conditions: (A1)|𝑓₁(𝑡,𝑢,𝑣)|<𝜙_𝑝(6^𝑚1𝑎^′/2𝒮) and |𝑓₂(𝑡,𝑢,𝑣)|<6^𝑛2𝜙_𝑞(𝑎^′/2ℛ), for all 𝑡∈[0,1] and 𝑢,𝑣∈[ℳ𝑎^′,𝑎^′],(A2)|𝑓₁(𝑡,𝑢,𝑣)|>4^𝑛1−1𝜙_𝑝(𝑏^′6^𝑚14^3𝑚1−2/𝒮11^𝑚1−1) for all 𝑡∈[1/4,3/4] and 𝑢,𝑣∈[𝑏^′,𝑏^′/ℳ] or |𝑓₂(𝑡,𝑢,𝑣)|>(6^𝑛24^3𝑛2−2/11^𝑛2−1)𝜙_𝑞(𝑏^′4^𝑚2/ℛ) for all 𝑡∈[1/4,3/4] and 𝑢,𝑣∈[𝑏^′,𝑏^′/ℳ], (A3)|𝑓₁(𝑡,𝑢,𝑣)|<𝜙_𝑝(6^𝑚1𝑐^′/2𝒮) and |𝑓₂(𝑡,𝑢,𝑣)|<6^𝑛2𝜙_𝑞(𝑐^′/2ℛ), for all 𝑡∈[0,1] and 𝑢,𝑣∈[0,𝑐^′].
Then the BVP (2) has at least three positive solutions.

Proof. Define the completely continuous operator 𝑇∶𝑃→𝐵 by 𝑇(𝑢,𝑣)∶=¹₀𝐻_𝑚1(𝑡,𝑠)𝜙_𝑞¹₀𝐺(𝑠,𝜏)𝑓₁(𝜏,𝑢(𝜏),𝑣(𝜏))𝑑𝜏𝑑𝑠,₁₀𝐻(𝑡,𝑠)𝜙_𝑝¹₀𝐺_𝑛2(𝑠,𝜏)𝑓₂(𝜏,𝑢(𝜏),𝑣(𝜏))𝑑𝜏𝑑𝑠. (49) And also we denote 𝑇₁(𝑢,𝑣)∶=¹₀𝐻_𝑚1(𝑡,𝑠)𝜙_𝑞×¹₀𝐺(𝑠,𝜏)𝑓₁((𝜏)(𝜏𝑇𝜏,𝑢,𝑣))𝑑𝜏𝑑𝑠,₂(𝑢,𝑣)∶=¹₀𝐻(𝑡,𝑠)𝜙_𝑝×¹₀𝐺_𝑛2(𝑠,𝜏)𝑓₂(𝜏,𝑢(𝜏),𝑣(𝜏))𝑑𝜏𝑑𝑠. (50)
It is obvious that a fixed point of 𝑇 is a solution of the BVP (2). We seek three fixed points (𝑥₁,𝑥₂), (𝑦₁,𝑦₂), and (𝑧₁,𝑧₂)∈𝑃 of 𝑇. First, we show that 𝑇∶𝑃→𝑃. Let (𝑢,𝑣)∈𝑃. Clearly, 𝑇₁(𝑢,𝑣)(𝑡)≥0,  𝑇₂(𝑢,𝑣)(𝑡)≥0 for 𝑡∈[0,1]. Consider min_𝑡∈𝐼𝑇(𝑢,𝑣)=min_𝑡∈𝐼𝑇₁(𝑢,𝑣)(𝑡)+𝑇₂(𝑢,𝑣)(𝑡)=min_𝑡∈𝐼𝑇₁(𝑢,𝑣)(𝑡)+min_𝑡∈𝐼𝑇₂(𝑢,𝑣)(𝑡)≥min_𝑡∈𝐼ℳ‖𝑇(𝑢,𝑣)‖₀+min_𝑡∈𝐼ℳ‖𝑇(𝑢,𝑣)‖₀=ℳ‖𝑇(𝑢,𝑣)‖. (51) Thus, 𝑇∶𝑃→𝑃. Next, for all (𝑢,𝑣)∈𝑃, by (46), (47), respectively, we have 𝛼(𝑢,𝑣)≤𝛽(𝑢,𝑣) and ‖(𝑢,𝑣)‖≤(1/ℳ)𝛾(𝑢,𝑣). To show that 𝑇∶𝑃(𝛾,𝑐^′)→𝑃(𝛾,𝑐^′), let (𝑢,𝑣)∈𝑃(𝛾,𝑐^′). This implies ‖(𝑢,𝑣)‖≤(1/ℳ)𝑐^′. We may now use condition (A3) to obtain 𝛾𝑇₁(𝑢,𝑣),𝑇₂(𝑢,𝑣)=max_𝑡∈𝐼2𝑇₁(𝑢,𝑣)(𝑡)+𝑇₂(𝑢,𝑣)(𝑡)=max_𝑡∈𝐼2¹₀𝐻_𝑚1(𝑡,𝑠)𝜙_𝑝⁻¹×¹₀𝐺(𝑠,𝜏)𝑓₁(𝜏,𝑢(𝜏),𝑣(𝜏))𝑑𝜏𝑑𝑠+max_𝑡∈𝐼2¹₀𝐻(𝑡,𝑠)𝜙_𝑞⁻¹×¹₀𝐺_𝑛2(𝑠,𝜏)𝑓₂(𝜏,𝑢(𝜏),𝑣(𝜏))𝑑𝜏𝑑𝑠<max_𝑡∈𝐼216^𝑚1−11₀𝑠(1−𝑠)𝜙_𝑝⁻¹×¹₀𝐺(𝜏,𝜏)𝑓₁(𝜏,𝑢(𝜏),𝑣(𝜏))𝑑𝜏𝑑𝑠+max_𝑡∈𝐼2¹₀𝐻(𝑠,𝑠)𝜙_𝑞⁻¹×¹₀16^𝑛2−1𝜏(1−𝜏)𝑓₂((𝜏)(𝜏=1𝜏,𝑢,𝑣))𝑑𝜏𝑑𝑠6^𝑚1−1𝜙_𝑝⁻¹¹₀𝐺(𝜏,𝜏)𝑓₁(𝜏,𝑢(𝜏),𝑣(𝜏))𝑑𝜏×¹₀𝑠()+1−𝑠𝑑𝑠₁₀𝐻(𝑠,𝑠)𝜙_𝑞⁻¹×16^𝑛2−11₀𝜏(1−𝜏)𝑓₂((𝜏)(𝜏=1𝜏,𝑢,𝑣))𝑑𝜏𝑑𝑠6^𝑚1𝜙_𝑝⁻¹¹₀𝐺(𝜏,𝜏)𝑓₁((𝜏)(𝜏+𝜏,𝑢,𝑣))𝑑𝜏₁₀𝐻(𝑠,𝑠)𝜙_𝑞⁻¹×16^𝑛2−11₀𝜏(1−𝜏)𝑓₂(𝜏,𝑢(𝜏),𝑣(𝜏))𝑑𝜏𝑑𝑠≤𝑐^′. (52) Therefore, 𝑇∶𝑃(𝛾,𝑐^′)→𝑃(𝛾,𝑐^′).
We first verify that condition (B1) of Theorem 10 is satisfied. The constant function is 𝑏^′+𝑏^′/ℳ2∈(𝑢,𝑣)∈𝑃𝛾,𝜃,𝛼,𝑏^′,𝑏^′ℳ,𝑐^′∣𝛼(𝑢,𝑣)>𝑏^′≠∅. (53) Next, let (𝑢,𝑣)∈𝑃(𝛾,𝜃,𝛼,𝑏^′,𝑏^′/ℳ,𝑐^′). It follows that from (A2) 𝛼𝑇₁(𝑢,𝑣),𝑇₂(𝑢,𝑣)=min_𝑡∈𝐼𝑇₁(𝑢,𝑣)(𝑡)+𝑇₂(𝑢,𝑣)(𝑡)≥min_𝑡∈𝐼𝑇₁(𝑢,𝑣)(𝑡)=min_𝑡∈𝐼¹₀𝐻_𝑚1(𝑡,𝑠)𝜙_𝑝⁻¹×¹₀𝐺(𝑠,𝜏)𝑓₁(𝜏,𝑢(𝜏),𝑣(𝜏))𝑑𝜏𝑑𝑠≥¹₀11^𝑚1−1𝑠(1−𝑠)6^𝑚1−14^3𝑚1−2𝜙_𝑝⁻¹×¹₀14^𝑛1−1𝐺(𝜏,𝜏)𝑓₁((𝜏)(𝜏=𝜏,𝑢,𝑣))𝑑𝜏𝑑𝑠11^𝑚1−16^𝑚1−14^3𝑚1−2𝜙_𝑝⁻¹×¹₀14^𝑛1−1𝐺(𝜏,𝜏)𝑓₁((𝜏)(𝜏×𝜏,𝑢,𝑣))𝑑𝜏₁₀𝑠()≥1−𝑠𝑑𝑠11^𝑚1−16^𝑚14^3𝑚1−2𝜙_𝑝⁻¹14^𝑛1−1𝜙_𝑝⁻¹×¹₀𝐺(𝜏,𝜏)𝑓₁(𝜏,𝑢(𝜏),𝑣(𝜏))𝑑𝜏=𝑏^′. (54) Similarly, min_𝑡∈𝐼𝑇₂(𝑢,𝑣)(𝑡)=min_𝑡∈𝐼¹₀𝐻(𝑡,𝑠)𝜙_𝑞⁻¹×¹₀𝐺_𝑛2(𝑠,𝜏)𝑓₂((𝜏)(𝜏≥𝜏,𝑢,𝑣))𝑑𝜏𝑑𝑠₁₀14^𝑚2−1𝐻(𝑠,𝑠)𝜙_𝑞⁻¹×¹₀11^𝑛2−1𝜏(1−𝜏)6^𝑛2−14^3𝑛2−2𝑓₂((𝜏)(𝜏≥1𝜏,𝑢,𝑣))𝑑𝜏𝑑𝑠4^𝑚2−1𝜙_𝑞⁻¹×11^𝑛2−16^𝑛2−14^3𝑛2−21₀𝜏(1−𝜏)𝑓₂((𝜏)(𝜏×𝜏,𝑢,𝑣))𝑑𝜏₁₀𝐻(𝑠,𝑠)𝑑𝑠=𝑏^′, (55) and, hence, we have 𝛼(𝑇₁(𝑢,𝑣),𝑇₂(𝑢,𝑣))≥𝑏^′.
Next, we show that (B2) is fulfilled. The constant function is 𝑎^′ℳ+𝑎^′2∈(𝑢,𝑣)∈𝑄𝛾,𝛽,𝜓,𝑎^′ℳ,𝑎^′,𝑐^′∣𝛽𝑣₁,𝑣₂<𝑎^′≠∅. (56) Let (𝑢,𝑣)∈𝑄(𝛾,𝛽,𝜓,𝑎^′ℳ,𝑎^′,𝑐^′). And from (A1), 𝛽𝑇₁(𝑢,𝑣),𝑇₂(𝑢,𝑣)=max_𝑡∈𝐼1𝑇₁(𝑢,𝑣)(𝑡)+𝑇₂(𝑢,𝑣)(𝑡)=max_𝑡∈𝐼1¹₀𝐻_𝑚(𝑡,𝑠)𝜙_𝑝⁻¹×¹₀𝐺(𝑠,𝜏)𝑓₁(𝜏,𝑢(𝜏),𝑣(𝜏))𝑑𝜏𝑑𝑠+max_𝑡∈𝐼1¹₀𝐻(𝑡,𝑠)𝜙_𝑞⁻¹×¹₀𝐺_𝑛(𝑠,𝜏)𝑓₂(𝜏,𝑢(𝜏),𝑣(𝜏))𝑑𝜏𝑑𝑠≤max_𝑡∈𝐼116^𝑚1−11₀𝑠(1−𝑠)𝜙_𝑝⁻¹×¹₀𝐺(𝜏,𝜏)𝑓₁(𝜏,𝑢(𝜏),𝑣(𝜏))𝑑𝜏𝑑𝑠+max_𝑡∈𝐼1¹₀𝐻(𝑠,𝑠)𝜙_𝑞⁻¹×¹₀16^𝑛2−1𝜏(1−𝜏)𝑓₂((𝜏)(𝜏=1𝜏,𝑢,𝑣))𝑑𝜏𝑑𝑠6^𝑚1−1𝜙_𝑝⁻¹¹₀𝐺(𝜏,𝜏)𝑓₁((𝜏)(𝜏×𝜏,𝑢,𝑣))𝑑𝜏₁₀𝑠()+1−𝑠𝑑𝑠₁₀𝐻(𝑠,𝑠)𝜙_𝑞⁻¹×16^𝑛2−11₀𝜏(1−𝜏)𝑓₂((𝜏)(𝜏=1𝜏,𝑢,𝑣))𝑑𝜏𝑑𝑠6^𝑚1𝜙_𝑝⁻¹¹₀𝐺(𝜏,𝜏)𝑓₁(𝜏,𝑢(𝜏),𝑣(𝜏))𝑑𝜏+¹₀𝐻(𝑠,𝑠)𝜙_𝑞⁻¹×16^𝑛2−11₀𝜏(1−𝜏)𝑓₂(𝜏,𝑢(𝜏),𝑣(𝜏))𝑑𝜏𝑑𝑠=𝑎^′. (57) To see that (B3) is satisfied, let (𝑣₁,𝑣₂)∈𝑃(𝛾,𝛼,𝑏^′,𝑐^′) with 𝜃𝑇₁𝑣₁,𝑣₂,𝑇₂𝑣₁,𝑣₂>𝑏^′ℳ. (58) Using (23), (27), (31), and (35), we get 𝛼𝑇₁(𝑢,𝑣),𝑇₂(𝑢,𝑣)=min_𝑡∈𝐼𝑇₁(𝑢,𝑣)(𝑡)+𝑇₂(𝑢,𝑣)(𝑡)=min_𝑡∈𝐼¹₀𝐻_𝑚1(𝑡,𝑠)𝜙_𝑝⁻¹×¹₀𝐺(𝑠,𝜏)𝑓₁(𝜏,𝑢(𝜏),𝑣(𝜏))𝑑𝜏𝑑𝑠+min_𝑡∈𝐼¹₀𝐻(𝑡,𝑠)𝜙_𝑞⁻¹×¹₀𝐺_𝑛2(𝑠,𝜏)𝑓₂((𝜏)(𝜏≥𝜏,𝑢,𝑣))𝑑𝜏𝑑𝑠₁₀11^𝑚1−1𝑠(1−𝑠)6^𝑚1−14^3𝑚1−2𝜙_𝑝⁻¹×¹₀14^𝑛1−1𝐺(𝜏,𝜏)𝑓₁((𝜏)(𝜏+𝜏,𝑢,𝑣))𝑑𝜏𝑑𝑠₁₀14^𝑚2−1𝐻(𝑠,𝑠)𝜙_𝑞⁻¹×¹₀11^𝑛2−1𝜏(1−𝜏)6^𝑛2−14^3𝑛2−2𝑓₂((𝜏)(𝜏=𝜏,𝑢,𝑣))𝑑𝜏𝑑𝑠11^𝑚1−16^𝑚1−14^3𝑚1−2𝜙_𝑝⁻¹×¹₀14^𝑛1−1𝐺(𝜏,𝜏)𝑓₁((𝜏)(𝜏×𝜏,𝑢,𝑣))𝑑𝜏₁₀𝑠()+11−𝑠𝑑𝑠4^𝑚2−1𝜙_𝑞⁻¹×11^𝑛2−16^𝑛2−14^3𝑛2−21₀𝜏(1−𝜏)𝑓₂((𝜏)(𝜏×𝜏,𝑢,𝑣))𝑑𝜏₁₀𝐻()𝑇𝑠,𝑠𝑑𝑠=ℳ𝜃₁(𝑢,𝑣),𝑇₂(𝑢,𝑣)>𝑏^′. (59)
Finally, we show that (B4) holds. Let (𝑢,𝑣)∈𝑄(𝛾,𝛽,𝑎^′,𝑐^′) with 𝜓(𝑇₁(𝑢,𝑣),𝑇₂(𝑢,𝑣))<𝑎^′/ℳ. Using (23), (27), (31), and (35), we have 𝛽𝑇₁(𝑢,𝑣),𝑇₂(𝑢,𝑣)=max_𝑡∈𝐼1𝑇₁(𝑢,𝑣)(𝑡)+𝑇₂(𝑢,𝑣)(𝑡)=max_𝑡∈𝐼1¹₀𝐻_𝑚1(𝑡,𝑠)𝜙_𝑝⁻¹×¹₀𝐺(𝑠,𝜏)𝑓₁(𝜏,𝑢(𝜏),𝑣(𝜏))𝑑𝜏𝑑𝑠+max_𝑡∈𝐼1¹₀𝐻(𝑡,𝑠)𝜙_𝑞⁻¹×¹₀𝐺_𝑛2(𝑠,𝜏)𝑓₂((𝜏)(𝜏≥𝜏,𝑢,𝑣))𝑑𝜏𝑑𝑠₁₀11^𝑚1−1𝑠(1−𝑠)6^𝑚1−14^3𝑚1−2𝜙_𝑝⁻¹×¹₀14^𝑛1−1𝐺(𝜏,𝜏)𝑓₁(𝜏,𝑢(𝜏),𝑣(𝜏))𝑑𝜏𝑑𝑠+¹₀14^𝑚2−1𝐻(𝑠,𝑠)𝜙_𝑞⁻¹×¹₀11^𝑛2−1𝜏(1−𝜏)6^𝑛2−14^3𝑛2−2𝑓₂((𝜏)(𝜏=𝜏,𝑢,𝑣))𝑑𝜏𝑑𝑠11^𝑚1−16^𝑚1−14^3𝑚1−2𝜙_𝑝⁻¹×¹₀14^𝑛1−1𝐺(𝜏,𝜏)𝑓₁(𝜏,𝑢(𝜏),𝑣(𝜏))𝑑𝜏×¹₀𝑠()+11−𝑠𝑑𝑠4^𝑚2−1𝜙_𝑞⁻¹×11^𝑛2−16^𝑛2−14^3𝑛2−21₀𝜏(1−𝜏)𝑓₂((𝜏)(𝜏×𝜏,𝑢,𝑣))𝑑𝜏₁₀𝐻()𝑇𝑠,𝑠𝑑𝑠=ℳ𝜓₁(𝑢,𝑣),𝑇₂(𝑢,𝑣)>𝑎^′. (60)
We have proved that all the conditions of Theorem 10 are satisfied, and so there exist at least three positive solutions. Therefore the BVP (2) has at least three positive solutions. This completed the proof of the theorem.

4. Example

Considering the higher order (𝑝,𝑞)-Laplacian two-point boundary value problem, (−1)⁴𝜙_𝑝𝑢^′′(𝑡)^′′′=𝑓₁((𝑡)(𝑡,[],()𝑡,𝑢,𝑣))𝑡∈0,1−1²𝜙_𝑞𝑣^′(𝑡)^′′=𝑓₂((𝑡)(𝑡,1𝑡,𝑢,𝑣))𝑝+1𝑞𝑢(0)(1),𝜙=1,=0=𝑢_𝑝𝑢^′′(𝑡)^(𝑗)_at𝑡=0𝜙=0,𝑗=0,1,_𝑝𝑢^′′(𝑡)_at𝑡=1𝜙=0,_𝑞𝑣^′(𝑡)_at𝑡=0=0=𝜙_𝑞𝑣^′(𝑡)_at𝑡=1,𝑣(1)=0, (61) where 𝑓₁((𝑡)(𝑡=𝑡,𝑢,𝑣))12,098.134𝑢𝑣²[],[],[],𝑓−577.134,𝑢,𝑣∈0.015625,10.267𝑢+11,520.73,𝑢,𝑣∈1,16283.56𝑢+16,061.89,𝑢,𝑣∈16,43₂((𝑡)(𝑡=𝑡,𝑢,𝑣))21.34𝑢²[],[],[],𝑣+3.67,𝑢,𝑣∈0.015625,10.34𝑣+24.67,𝑢,𝑣∈1,16170.85𝑣−2,703.63,𝑢,𝑣∈16,43 (62) are continuous functions from [0,1]×ℝ² to ℝ. A simple calculation shows that ℳ=0.0625, (𝑝=2), 𝒮=1/30, and ℛ=1. If we choose 𝑎^′=1/4, 𝑏^′=1, and 𝑐^′=43, then conditions (A1)–(A3) are satisfied. Therefore, it follows from Theorem 11 that the BVP (61) has at least three positive solutions.

Acknowledgment

One of the authors (Dr. Penugurthi Murali) is thankful to CSIR, India, for awarding him a Research Associate.