Positive Solutions for a System of Fourth-Order 𝑝-Laplacian Boundary Value Problems

Abstract

We investigate the existence of positive solutions for the system of fourth-order 𝑝-Laplacian boundary value problems (|𝑢^′′|^𝑝−1𝑢^′′)^′′=𝑓₁(𝑡,𝑢,𝑣),(|𝑣^′′|^𝑞−1𝑣^′′)^′′=𝑓₂(𝑡,𝑢,𝑣),𝑢^(2𝑖)(0)=𝑢^(2𝑖)(1)=0,𝑖=0,1,𝑣^(2𝑖)(0)=𝑣^(2𝑖)(1)=0,𝑖=0,1, where 𝑝,𝑞>0 and 𝑓₁,𝑓₂∈𝐶([0,1]×ℝ²₊,ℝ₊)(ℝ₊∶=[0,∞)). Based on a priori estimates achieved by utilizing Jensen’s integral inequalities and nonnegative matrices, we use fixed point index theory to establish our main results.

1. Introduction

This paper is concerned with the existence of positive solutions for the system of fourth-order 𝑝-Laplacian boundary value problems 𝑢^{′′𝑝−1}𝑢^′′′′=𝑓₁(),𝑣𝑡,𝑢,𝑣_{′′𝑞−1}𝑣^′′′′=𝑓₂(),𝑢𝑡,𝑢,𝑣_(2𝑖)(0)=𝑢^(2𝑖)(1)𝑣=0,𝑖=0,1,_(2𝑖)(0)=𝑣^(2𝑖)(1)=0,𝑖=0,1,(1) where 𝑝,𝑞>0 and 𝑓₁,𝑓₂∈𝐶([0,1]×ℝ²₊,ℝ₊)(ℝ₊∶=[0,∞)).

In recent years, boundary value problems for fourth-order nonlinear ordinary differential equations have been extensively studied, with many papers on this direction published. See [1–4] and the references cited therein. The so-called 𝑝-Laplacian boundary value problems arise in non-Newtonian mechanics, nonlinear elasticity, glaciology, population biology, combustion theory, and nonlinear flow laws; see [5, 6]. That explains why many authors have extensively studied the existence of positive solutions for 𝑝-Laplacian boundary value problems, by using topological degree theory, monotone iterative techniques, coincidence degree theory [7], and the Leggett-Williams fixed point theorem [8] or its variants; see [9–18] and the references therein.

To the best of our knowledge, problem (1) is a new topic in the existing literature. Closely related to our work here is [19] that studies the existence and multiplicity of positive solutions for the system of 𝑝-Laplacian boundary value problems −𝑢^′_𝑖^{𝑝𝑖−1′}=𝑓_𝑖𝑡,𝑢₁,…,𝑢_𝑛,𝑢_𝑖(0)=𝑢^′_𝑖(1)=0,𝑖=1,…,𝑛,(2) where 𝑛⩾2,𝑝_𝑖>1,𝑓_𝑖∈𝐶([0,1]×ℝ^𝑛₊,ℝ₊)(𝑖=1,…,𝑛). Based on a priori estimates achieved by utilizing the Jensen integral inequalities and ℝ^𝑛₊-monotone matrices, the author used fixed point index theory to establish the existence and multiplicity of positive solutions for the previous problem. For more details of the recent progress in the boundary value problems for systems of nonlinear ordinary differential equations, we refer the reader to [19–27] and the references cited therein.

We observe that a close link exists between (1) and the problem below 𝑢⁽⁴⁾=𝑓₁(𝑡,𝑢,𝑣),𝑣⁽⁴⁾=𝑓₂(),𝑢𝑡,𝑢,𝑣_(2𝑖)(0)=𝑢^(2𝑖)(1)𝑣=0,𝑖=0,1,_(2𝑖)(0)=𝑣^(2𝑖)(1)=0,𝑖=0,1,(3) and this link can be established by Jensen's integral inequalities for concave functions and convex functions. In other words, (1) may be regarded as a perturbation of (3). With this perspective, a priori estimates of positive solutions for some problems associated with (1) can be derived by utilizing Jensen's integral inequalities. It is the a priori estimates that permit us to use fixed point index theory to establish our main results.

This paper is organized as follows. In Section 2, we provide some preliminary results. Our main results, namely, Theorems 6 and 7, are stated and proved in Section 3.

2. Preliminaries

We first make the following hypothesis throughout this paper:(H1)𝑓₁,𝑓₂∈𝐶([0,1]×ℝ²₊,ℝ₊).

Let 𝐸∶=𝐶[0,1] and {(𝑡)[,‖𝑢‖𝑃∶=𝑢∈𝐸∶𝑢⩾0,∀𝑡∈0,1]}∶=max_𝑡∈[0,1]|𝑢(𝑡)|;(4) then (𝐸,‖⋅‖) is a real Banach space and 𝑃 is a cone in 𝐸. It is easy to see that (1) is equivalent to the system of nonlinear integral equations 𝑢(𝑡)=¹₀𝑘(𝑡,𝑠)¹₀𝑘(𝑠,𝜏)𝑓₁(𝜏,𝑢(𝜏),𝑣(𝜏))𝑑𝜏^1/𝑝𝑣(𝑡)=𝑑𝑠,₁₀𝑘(𝑡,𝑠)¹₀𝑘(𝑠,𝜏)𝑓₂(𝜏,𝑢(𝜏),𝑣(𝜏))𝑑𝜏^1/𝑞𝑑𝑠,(5) where 𝑘()𝑡()𝑠()𝑡,𝑠∶=1−𝑠,0⩽𝑡⩽𝑠⩽1,1−𝑡,0⩽𝑠⩽𝑡⩽1.(6) Define the operators 𝐴_𝑖(𝑖=1,2)∶𝑃→𝑃 and 𝐴∶𝑃×𝑃→𝑃×𝑃 by 𝐴₁(𝑢,𝑣)(𝑡)∶=¹₀𝑘(𝑡,𝑠)¹₀𝑘(𝑠,𝜏)𝑓₁(𝜏,𝑢(𝜏),𝑣(𝜏))𝑑𝜏^1/𝑝𝐴𝑑𝑠,₂(𝑢,𝑣)(𝑡)∶=¹₀𝑘(𝑡,𝑠)¹₀𝑘(𝑠,𝜏)𝑓₂(𝜏,𝑢(𝜏),𝑣(𝜏))𝑑𝜏^1/𝑞𝐴(𝑡)𝐴𝑑𝑠,𝑢,𝑣)(∶=₁,𝐴₂(𝑢,𝑣)(𝑡).(7) Obviously, under condition (H1), the operators 𝐴_𝑖(𝑖=1,2)∶𝑃→𝑃 and 𝐴∶𝑃×𝑃→𝑃×𝑃 are completely continuous operators. In our setting, the existence of positive solutions for (1) is equivalent to that of positive fixed points of 𝐴.

Below are some elementary inequalities.

Lemma 1. Let 𝜑(𝑡)∶=sin𝜋𝑡, then 𝜑∈𝑃⧵{0} and ¹₀𝑘()𝜑(𝑡)1𝑡,𝑠𝑑𝑡=𝜋²𝜑(𝑠),₁₀𝑘()𝜑(𝑠)1𝑡,𝑠𝑑𝑠=𝜋²𝜑(𝑡).(8)

Lemma 2. Suppose that 𝜓∈𝑃 is concave on [0,1]. Then ¹₀𝜓(𝑡)2sin𝜋𝑡𝑑𝑡⩾𝜋²𝜓.(9)

Lemma 3. Let 𝑎∈ℝ₊,𝑏∈ℝ₊. If 𝜎∈(0,1], then (𝑎+𝑏)^𝜎⩾2^𝜎−1𝑎^𝜎+𝑏^𝜎.(10) If 𝜎∈[1,+∞), then (𝑎+𝑏)^𝜎⩽2^𝜎−1𝑎^𝜎+𝑏^𝜎.(11)

Lemma 4 (see [28]). Let 𝐸 be a real Banach space, and 𝐾 is a cone in 𝐸. Suppose that Ω⊂𝐸 is a bounded open set and 𝑇∶Ω∩𝐾→𝐾 is a completely continuous operator. If there exists 𝑢₀∈𝐾⧵{0} such that 𝑢−𝑇𝑢≠𝜆𝑢₀,∀𝜆>0,𝑢∈𝜕Ω∩𝐾,(12) then 𝑖(𝑇,Ω∩𝐾,𝐾)=0, where 𝑖 indicates the fixed point index on 𝐾.

Lemma 5 (see [28]). Let 𝐸 be a real Banach space, and 𝐾 is a cone in 𝐸. Suppose that Ω⊂𝐸 is a bounded open set, 0∈Ω, and 𝑇∶Ω∩𝐾→𝐾 is a completely continuous operator. If 𝑢−𝜆𝑇𝑢≠0,∀𝜆∈[0,1],𝑢∈𝜕Ω∩𝐾,(13) then 𝑖(𝑇,Ω∩𝐾,𝐾)=1.

3. Main Results

Let 𝑝^∗∶=max{1,𝑝}, 𝑝_∗∶=min{1,𝑝}, 𝑞^∗∶=max{1,𝑞}, 𝑞_∗∶=min{1,𝑞}. Then the previous constants satisfy the simple relations 0<𝑝_∗⩽1⩽𝑝^∗,0<𝑞_∗⩽1⩽𝑞^∗,𝑝_∗𝑝^∗=𝑝,𝑞_∗𝑞^∗=𝑞.(14) We now list our hypotheses on 𝑓₁ and 𝑓₂.

(H2) There exist five constants 𝑎_𝑖1, 𝑏_𝑖1⩾0, 𝑐>0 such that 𝑓₁𝑡,𝑥,𝑦⩾𝑎₁₁𝑥^{𝑝∗𝑞∗}+𝑏₁₁𝑦^{𝑝∗𝑞∗}𝑓−𝑐,₂𝑡,𝑥,𝑦⩾𝑎₂₁𝑥^{𝑞∗𝑝∗}+𝑏₂₁𝑦^{𝑞∗𝑞∗}−𝑐(15) for all 𝑥,𝑦∈ℝ⁺, 𝑡∈[0,1], and the matrix 𝐴₁∶=^{𝑎1−1𝑏1}_{𝑎2𝑏2−1} is invertible with 𝐴₁⁻¹ nonnegative, where 𝑎₁∶=2^{𝑝∗/𝑝−1}𝑎^𝑝∗/𝑝₁₁𝜋⁻⁴,𝑏₁∶=2^{𝑝∗/𝑝−1}𝑏^𝑝∗/𝑝₁₁𝜋⁻⁴,𝑎₂∶=2^{𝑞∗/𝑞−1}𝑎^𝑞∗/𝑞₂₁𝜋⁻⁴,𝑏₂∶=2^{𝑞∗/𝑞−1}𝑏^𝑞∗/𝑞₂₁𝜋⁻⁴.(16)

(H3) There exist five constants 𝑐_𝑖1,𝑑_𝑖1⩾0, 𝑟>0 such that 𝑓₁𝑡,𝑥,𝑦⩽𝑐₁₁𝑥^{𝑝∗𝑝∗}+𝑑₁₁𝑦^{𝑝∗𝑞∗},𝑓₂𝑡,𝑥,𝑦⩽𝑐₂₁𝑥^{𝑞∗𝑝∗}+𝑐₂₁𝑦^{𝑞∗𝑞∗}(17) for all 𝑥,𝑦∈[0,𝑟], 𝑡∈[0,1], and the matrix 𝐴₂∶=^{1−𝑐1−𝑑1}_{−𝑐21−𝑑2} is invertible with 𝐴₂⁻¹ nonnegative, where 𝑐₁∶=2^{𝑝∗/𝑝−1}𝑐^𝑝∗/𝑝₁₁𝜋⁻⁴,𝑑₁∶=2^{𝑝∗/𝑝−1}𝑑^𝑝∗/𝑝₁₁𝜋⁻⁴,𝑐₂∶=2^{𝑞∗/𝑞−1}𝑐^𝑞∗/𝑞₂₁𝜋⁻⁴,𝑑₂∶=2^{𝑞∗/𝑞−1}𝑑^𝑞∗/𝑞₂₁𝜋⁻⁴.(18)

(H4) There exist five constants 𝑙_𝑖1,𝑚_𝑖1⩾0 and, a real number 𝑟>0 such that 𝑓₁𝑡,𝑥,𝑦⩾𝑙₁₁𝑥^{𝑝∗𝑝∗}+𝑚₁₁𝑦^{𝑝∗𝑞∗},𝑓₂𝑡,𝑥,𝑦⩾𝑙₂₁𝑥^{𝑞∗𝑝∗}+𝑚₂₁𝑦^{𝑞∗𝑞∗}(19) for all 𝑥,𝑦∈[0,𝑟], 𝑡∈[0,1], and the matrix 𝐴₃∶=^{1−𝑙1−𝑚1}_{−𝑙21−𝑚2} is invertible with 𝐴₃⁻¹ nonnegative, where 𝑙₁∶=2^{𝑝∗/𝑝−1}𝑙^𝑝∗/𝑝₁₁𝜋⁻⁴,𝑚₁∶=2^{𝑝∗/𝑝−1}𝑚^𝑝∗/𝑝₁₁𝜋⁻⁴,𝑙₂∶=2^{𝑞∗/𝑞−1}𝑙^𝑞∗/𝑞₂₁𝜋⁻⁴,𝑚₂∶=2^{𝑞∗/𝑞−1}𝑚^𝑞∗/𝑞₂₁𝜋⁻⁴.(20)

(H5) There exist five constants 𝑝_𝑖1,𝑞_𝑖1⩾0, 𝑐>0 such that 𝑓₁𝑡,𝑥,𝑦⩽𝑝₁₁𝑥^{𝑝∗𝑝∗}+𝑞₁₁𝑦^{𝑝∗𝑞∗}𝑓+𝑐,₂𝑡,𝑥,𝑦⩽𝑝₂₁𝑥^{𝑞∗𝑝∗}+𝑞₂₁𝑦^{𝑞∗𝑞∗}+𝑐(21) for all 𝑥,𝑦∈ℝ⁺, 𝑡∈[0,1], and the matrix 𝐴₄∶=^{𝑝1−1𝑞1}_{𝑝2𝑞2−1} is invertible with 𝐴₄⁻¹ nonnegative, where 𝑝₁∶=4^{𝑝∗/𝑝−1}𝑝^𝑝∗/𝑝₁₁𝜋⁻⁴,𝑞₁∶=4^{𝑝∗/𝑝−1}𝑞^𝑝∗/𝑝₁₁𝜋⁻⁴,𝑝₂∶=4^{𝑞∗/𝑞−1}𝑝^𝑞∗/𝑞₂₁𝜋⁻⁴,𝑞₂∶=4^{𝑞∗/𝑞−1}𝑞^𝑞∗/𝑞₂₁𝜋⁻⁴.(22) In the sequel, we set Ω_𝜌∶={(𝑢,𝑣)∈𝐸²∶‖(𝑢,𝑣)‖<𝜌} for all 𝜌>0.

Theorem 6. If (H1)–(H3) hold, then (1) has at least one positive solution.

Proof. Let ℳ₁∶=(𝑢,𝑣)∈𝑃×𝑃∶(𝑢,𝑣)=𝐴(𝑢,𝑣)+𝜆𝜑,𝜑,𝜆⩾0,(23) where 𝜑∈𝑃⧵{0} is defined by 𝜑(𝑡)∶=sin𝜋𝑡. We want to show that ℳ₁ is bounded. Indeed, if (𝑢,𝑣)∈ℳ₁, then there exists 𝜆⩾0 such that (𝑢,𝑣)=𝐴(𝑢,𝑣)+𝜆𝜑,𝜑,(24) and thus (𝑢,𝑣)⩾𝐴(𝑢,𝑣). Note that 𝑢,𝑣∈ℳ₁ implies that 𝑢,𝑣 are concave. So are 𝑢^𝑝∗,𝑣^𝑞∗∈𝑃 with 𝑝_∗,𝑞_∗∈(0,1]. Now Lemma 2 implies that 𝑢^𝑝∗⩽𝜋²2¹₀𝑢^𝑝∗(𝑡)𝑣sin𝜋𝑡𝑑𝑡,_𝑞∗⩽𝜋²2¹₀𝑣^𝑞∗(𝑡)sin𝜋𝑡𝑑𝑡(25) hold for every (𝑢,𝑣)∈ℳ₁. Note that 𝑘(𝑡,𝑠)∈[0,1] for all 𝑡,𝑠∈[0,1] and 𝑝_∗,𝑝_∗/𝑝∈(0,1]. By (H2), Jensen's integral inequalities, and Lemma 3, we obtain 𝑢^𝑝∗(𝑡)⩾¹₀𝑘(𝑡,𝑠)¹₀𝑘(𝑠,𝜏)𝑓₁(𝜏,𝑢(𝜏),𝑣(𝜏))𝑑𝜏^1/𝑝𝑑𝑠^𝑝∗⩾¹₀𝑘(𝑡,𝑠)𝑘(𝑠,𝜏)𝑓^𝑝∗₁^/𝑝((𝜏)(𝜏⩾𝜏,𝑢,𝑣))𝑑𝜏𝑑𝑠₁₀𝑘()𝑘()×𝑎𝑡,𝑠𝑠,𝜏₁₁𝑢^{𝑝∗𝑝∗}(𝜏)+𝑏₁₁𝑣^{𝑝∗𝑞∗}(𝜏)^𝑝∗/𝑝−𝑐^𝑝∗/𝑝𝑑𝜏𝑑𝑠⩾2^{𝑝∗/𝑝−11}₀𝑘()𝑘()×𝑎𝑡,𝑠𝑠,𝜏_{𝑝∗/𝑝11}𝑢^𝑝∗(𝜏)+𝑏^𝑝∗/𝑝₁₁𝑣^𝑞∗(𝜏)−𝑐^𝑝∗/𝑝𝑑𝜏𝑑𝑠(26) for all 𝑢,𝑣∈ℳ₁,𝑡∈[0,1]. Note 𝑞_∗,𝑞_∗/𝑞∈(0,1]. The same argument for the previous inequality can be used to obtain 𝑣^𝑞∗(𝑡)⩾2^{𝑞∗/𝑞−11}₀𝑘()𝑘()×𝑎𝑡,𝑠𝑠,𝜏_{𝑞∗/𝑞21}𝑢^𝑝∗(𝜏)+𝑏^𝑞∗/𝑞₂₁𝑣^𝑞∗(𝜏)−𝑐^𝑞∗/𝑞𝑑𝜏𝑑𝑠(27) for all 𝑢,𝑣∈ℳ₁,𝑡∈[0,1]. Multiply the last two inequalities by 𝜑(𝑡), integrate over [0,1], and use Lemma 1 and (H2) to obtain ¹₀𝑢^𝑝∗(𝑡)sin𝜋𝑡𝑑𝑡⩾𝑎₁¹₀𝑢^𝑝∗(𝑡)sin𝜋𝑡𝑑𝑡+𝑏₁¹₀𝑣^𝑞∗(𝑡)sin𝜋𝑡𝑑𝑡−ℎ₁,₁₀𝑣^𝑞∗(𝑡)sin𝜋𝑡𝑑𝑡⩾𝑎₂¹₀𝑢^𝑝∗(𝑡)sin𝜋𝑡𝑑𝑡+𝑏₂¹₀𝑣^𝑞∗(𝑡)sin𝜋𝑡𝑑𝑡−ℎ₂,(28) for every (𝑢,𝑣)∈ℳ₁, where ℎ₁∶=2^𝑝∗/𝑝𝑐^𝑝∗/𝑝𝜋⁻⁵,ℎ₂∶=2^𝑞∗/𝑞𝑐^𝑞∗/𝑞𝜋⁻⁵. The last two inequalities can be written as 𝑎₁−1𝑏₁𝑎₂𝑏₂−1¹⁰𝑢^𝑝∗(𝑡)sin𝜋𝑡𝑑𝑡₁₀𝑣^𝑞∗(𝑡)sin𝜋𝑡𝑑𝑡=𝐴₁¹⁰𝑢^𝑝∗(𝑡)sin𝜋𝑡𝑑𝑡₁₀𝑣^𝑞∗(𝑡)⩽ℎsin𝜋𝑡𝑑𝑡ℎ¹₂.(29) Now (H2) implies ¹₀𝑢^𝑝∗(𝑡)sin𝜋𝑡𝑑𝑡₁₀𝑣^𝑞∗(𝑡)sin𝜋𝑡𝑑𝑡⩽𝐴₁⁻¹ℎ₁ℎ₂.(30) Consequently, there exists a constant 𝑀>0 such that ¹₀𝑢^𝑝∗(𝑡)sin𝜋𝑡𝑑𝑡⩽𝑀,¹₀𝑣^𝑞∗(𝑡)sin𝜋𝑡𝑑𝑡⩽𝑀.(31) Recalling (25), we obtain 𝑢^𝑝∗⩽𝜋²𝑀2,𝑣^𝑞∗⩽𝜋²𝑀2,∀(𝑢,𝑣)∈ℳ₁.(32) Therefore, ℳ₁ is bounded, as required. Taking 𝑅>sup{‖(𝑢,𝑣)‖∶(𝑢,𝑣)∈ℳ₁}, we have (𝑢,𝑣)≠𝐴(𝑢,𝑣)+𝜆𝜑,𝜑,∀(𝑢,𝑣)∈𝜕𝑄_𝑅∩(𝑃×𝑃),𝜆⩾0.(33) Now Lemma 4 yields 𝑖𝐴,𝑄_𝑅∩(𝑃×𝑃),𝑃×𝑃=0.(34) Let ℳ₂∶=(𝑢,𝑣)∈𝑄_𝑟∩(𝑃×𝑃)∶(𝑢,𝑣)=𝜆𝐴(𝑢,𝑣),𝜆∈[0,1].(35) Now we want to show that ℳ₂={0}. Indeed, if (𝑢,𝑣)∈ℳ₂, then there exists 𝜆∈[0,1] such that (𝑢,𝑣)=𝜆𝐴(𝑢,𝑣), and thus (𝑢,𝑣)⩽𝐴(𝑢,𝑣) for all (𝑢,𝑣)∈ℳ₂. Note that 𝑘(𝑡,𝑠)∈[0,1] for all 𝑡,𝑠∈[0,1] and 𝑝^∗,𝑝^∗/𝑝⩾1. By (H3), Jensen’s integral inequalities, and Lemma 3, we obtain 𝑢^𝑝∗(𝑡)⩽¹₀𝑘(𝑡,𝑠)¹₀𝑘(𝑠,𝜏)×𝑓₁(𝜏,𝑢(𝜏),𝑣(𝜏))𝑑𝜏^1/𝑝𝑑𝑠^𝑝∗⩽¹₀𝑘(𝑡,𝑠)𝑘(𝑠,𝜏)𝑓^𝑝∗₁^/𝑝((𝜏)(𝜏⩽𝜏,𝑢,𝑣))𝑑𝜏𝑑𝑠₁₀𝑘()𝑘()×𝑐𝑡,𝑠𝑠,𝜏₁₁𝑢^{𝑝∗𝑝∗}(𝜏)+𝑑₁₁𝑣^{𝑝∗𝑞∗}(𝜏)^𝑝∗/𝑝𝑑𝜏𝑑𝑠⩽2^{𝑝∗/𝑝−11}₀𝑘()𝑘()×𝑐𝑡,𝑠𝑠,𝜏_{𝑝∗/𝑝11}𝑢^𝑝∗(𝜏)+𝑑^𝑝∗/𝑝₁₁𝑣^𝑞∗(𝜏)𝑑𝜏𝑑𝑠(36) for all (𝑢,𝑣)∈ℳ₂,𝑡∈[0,1]. Note that 𝑞^∗,𝑞^∗/𝑞⩾1. The same argument for the previous inequality can be used to obtain 𝑣^𝑞∗(𝑡)⩽2^{𝑞∗/𝑞−11}₀𝑘()𝑘()×𝑐𝑡,𝑠𝑠,𝜏_{𝑞∗/𝑞21}𝑢^𝑝∗(𝜏)+𝑑^𝑞∗/𝑞₂₁𝑣^𝑞∗(𝜏)𝑑𝜏𝑑𝑠(37) for all (𝑢,𝑣)∈ℳ₂,𝑡∈[0,1]. Multiply the last two inequalities by 𝜑(𝑡), integrate over [0,1], and use Lemma 1 and (H3) to obtain ¹₀𝑢^𝑝∗(𝑡)sin𝜋𝑡𝑑𝑡⩽𝑐₁¹₀𝑢^𝑝∗(𝑡)sin𝜋𝑡𝑑𝑡+𝑑₁¹₀𝑣^𝑞∗(𝑡)sin𝜋𝑡𝑑𝑡,₁₀𝑣^𝑞∗(𝑡)sin𝜋𝑡𝑑𝑡⩽𝑐₂¹₀𝑢^𝑝∗(𝑡)sin𝜋𝑡𝑑𝑡+𝑑₂¹₀𝑣^𝑞∗(𝑡)sin𝜋𝑡𝑑𝑡(38) for all (𝑢,𝑣)∈ℳ₂. The last two inequalities can be written as 1−𝑐₁−𝑑₁−𝑐₂1−𝑑₂¹⁰𝑢^𝑝∗(𝑡)sin𝜋𝑡𝑑𝑡₁₀𝑣^𝑞∗(𝑡)sin𝜋𝑡𝑑𝑡=𝐴₂¹⁰𝑢^𝑝∗(𝑡)sin𝜋𝑡𝑑𝑡₁₀𝑣^𝑞∗(𝑡)⩽00.sin𝜋𝑡𝑑𝑡(39) Now (H3) implies ¹₀𝑢^𝑝∗(𝑡)sin𝜋𝑡𝑑𝑡₁₀𝑣^𝑞∗(𝑡)sin𝜋𝑡𝑑𝑡⩽𝐴₂⁻¹00=00,(40) so that ¹₀𝑢^𝑝∗(𝑡)sin𝜋𝑡𝑑𝑡=0,¹₀𝑣^𝑞∗(𝑡)sin𝜋𝑡𝑑𝑡=0,(41) whence 𝑢≡0,𝑣≡0, and ℳ₂={0}, as required. Thus we have (𝑢,𝑣)≠𝜆𝐴(𝑢,𝑣),∀(𝑢,𝑣)∈𝜕𝑄_𝑟∩(𝑃×𝑃),𝜆∈[0,1].(42) Now Lemma 5 yields 𝑖𝐴,𝑄_𝑟∩(𝑃×𝑃),𝑃×𝑃=1.(43) Combining (34) and (43) gives 𝑖𝐴,𝑄_𝑅⧵𝑄_𝑟∩(𝑃×𝑃),𝑃×𝑃=0−1=−1.(44) Consequently, 𝐴 has at least one fixed point on (𝑄_𝑅⧵𝑄_𝑟)∩(𝑃×𝑃). Thus (1) has at least one positive solution. This completes the proof.

Theorem 7. If (H1), (H4), and (H5) hold, then (1) has at least one positive solution.

Proof. Let ℳ₃∶=(𝑢,𝑣)∈𝑄_𝑟∩()∶()().𝑃×𝑃𝑢,𝑣=𝐴𝑢,𝑣+𝜆𝜑,𝜑,𝜆⩾0(45) We now assert that ℳ₃⊂{0}. Indeed, if (𝑢,𝑣)∈ℳ₃, then there exists 𝜆∈[0,1] such that (𝑢,𝑣)=𝐴(𝑢,𝑣)+𝜆(𝜑,𝜑). Thus (𝑢,𝑣)⩾𝐴(𝑢,𝑣). Note that 𝑘(𝑡,𝑠)∈[0,1] for all 𝑡,𝑠∈[0,1] and 𝑝_∗,𝑝_∗/𝑝∈(0,1]. By (H4), Jensen's integral inequalities, and Lemma 3, we obtain 𝑢^𝑝∗(𝑡)⩾¹₀𝑘(𝑡,𝑠)¹₀𝑘(𝑠,𝜏)×𝑓₁(𝜏,𝑢(𝜏),𝑣(𝜏))𝑑𝜏^1/𝑝𝑑𝑠^𝑝∗⩾¹₀𝑘(𝑡,𝑠)𝑘(𝑠,𝜏)𝑓^𝑝∗₁^/𝑝((𝜏)(𝜏⩾𝜏,𝑢,𝑣))𝑑𝜏𝑑𝑠₁₀𝑘()𝑘()×𝑙𝑡,𝑠𝑠,𝜏₁₁𝑢^{𝑝∗𝑝∗}(𝜏)+𝑚₁₁𝑣^{𝑝∗𝑞∗}(𝜏)^𝑝∗/𝑝𝑑𝜏𝑑𝑠⩾2^{𝑝∗/𝑝−11}₀𝑘()𝑘()×𝑙𝑡,𝑠𝑠,𝜏_{𝑝∗/𝑝11}𝑢^𝑝∗(𝜏)+𝑚^𝑝∗/𝑝₁₁𝑣^𝑞∗(𝜏)𝑑𝜏𝑑𝑠(46) for all (𝑢,𝑣)∈ℳ₃,𝑡∈[0,1]. Note that 𝑞_∗,𝑞_∗/𝑞∈(0,1]. The same argument for the previous inequality can be used to obtain 𝑣^𝑞∗(𝑡)⩾2^{𝑞∗/𝑞−11}₀𝑘()𝑘()×𝑙𝑡,𝑠𝑠,𝜏_{𝑞∗/𝑞21}𝑢^𝑝∗(𝜏)+𝑚^𝑞∗/𝑞₂₁𝑣^𝑞∗(𝜏)𝑑𝜏𝑑𝑠(47) for all (𝑢,𝑣)∈ℳ₃,𝑡∈[0,1]. Multiply the last two inequalities by 𝜑(𝑡), integrate over [0,1], and use Lemma 1 and (H4) to obtain ¹₀𝑢^𝑝∗(𝑡)sin𝜋𝑡𝑑𝑡⩾𝑙₁¹₀𝑢^𝑝∗(𝑡)sin𝜋𝑡𝑑𝑡+𝑚₁¹₀𝑣^𝑞∗(𝑡)sin𝜋𝑡𝑑𝑡,₁₀𝑣^𝑞∗(𝑡)sin𝜋𝑡𝑑𝑡⩾𝑙₂¹₀𝑢^𝑝∗(𝑡)sin𝜋𝑡𝑑𝑡+𝑚₂¹₀𝑣^𝑞∗(𝑡)sin𝜋𝑡𝑑𝑡(48) for all (𝑢,𝑣)∈ℳ₃. The last two inequalities can be written as 𝑙₁−1𝑚₁𝑙₂𝑚₂−1¹⁰𝑢^𝑝∗(𝑡)sin𝜋𝑡𝑑𝑡₁₀𝑣^𝑞∗(𝑡)sin𝜋𝑡𝑑𝑡=𝐴₃¹⁰𝑢^𝑝∗(𝑡)sin𝜋𝑡𝑑𝑡₁₀𝑣^𝑞∗(𝑡)⩽00.sin𝜋𝑡𝑑𝑡(49) Now (H4) implies ¹₀𝑢^𝑝∗(𝑡)sin𝜋𝑡𝑑𝑡₁₀𝑣^𝑞∗(𝑡)sin𝜋𝑡𝑑𝑡⩽𝐴₃⁻¹00=00,(50) so that ¹₀𝑢^𝑝∗(𝑡)sin𝜋𝑡𝑑𝑡=0,¹₀𝑣^𝑞∗(𝑡)sin𝜋𝑡𝑑𝑡=0,(51) whence 𝑢≡0,𝑣≡0, and ℳ₃⊂{0}, as asserted. As a result of this, we obtain (𝑢,𝑣)≠𝐴(𝑢,𝑣)+𝜆𝜑,𝜑,∀(𝑢,𝑣)∈𝜕𝑄_𝑟∩(𝑃×𝑃),𝜆⩾0.(52) By Lemma 4, we have 𝑖𝐴,𝑄_𝑟∩(𝑃×𝑃),𝑃×𝑃=0.(53) Let ℳ₄∶={(𝑢,𝑣)∈𝑃×𝑃∶(𝑢,𝑣)=𝜆𝐴(𝑢,𝑣),𝜆∈[0,1]}.(54) Now we want to show that ℳ₄ is bounded. Indeed, if (𝑢,𝑣)∈ℳ₄, then there exists 𝜆∈[0,1] such that (𝑢,𝑣)=𝜆𝐴(𝑢,𝑣). This implies (𝑢,𝑣)⩽𝐴(𝑢,𝑣). Note that 𝑘(𝑡,𝑠)∈[0,1] for all 𝑡,𝑠∈[0,1] and 𝑝^∗,𝑝^∗/𝑝⩾1. By (H5), Jensen's integral inequalities, and Lemma 3, we obtain 𝑢^𝑝∗(𝑡)⩽¹₀𝑘(𝑡,𝑠)𝑘(𝑠,𝜏)𝑓^𝑝∗₁^/𝑝×((𝜏)(𝜏⩽𝜏,𝑢,𝑣))𝑑𝜏𝑑𝑠₁₀𝑘(𝑡,𝑠)𝑘(𝑠,𝜏)𝑝₁₁𝑢^{𝑝∗𝑝∗}(𝜏)+𝑞₁₁𝑣^{𝑝∗𝑞∗}(𝜏)+𝑐^𝑝∗/𝑝𝑑𝜏𝑑𝑠⩽2^{𝑝∗/𝑝−11}₀𝑘()𝑘()×𝑝𝑡,𝑠𝑠,𝜏₁₁𝑢^{𝑝∗𝑝∗}(𝜏)+𝑞₁₁𝑣^{𝑝∗𝑞∗}(𝜏)^𝑝∗/𝑝+𝑐^𝑝∗/𝑝𝑑𝜏𝑑𝑠⩽2^{𝑝∗/𝑝−11}₀𝑘()𝑘()×2𝑡,𝑠𝑠,𝜏_{𝑝∗/𝑝−1}𝑝^𝑝∗/𝑝₁₁𝑢^𝑝∗(𝜏)+𝑞^𝑝∗/𝑝₁₁𝑣^𝑞∗(𝜏)+𝑐^𝑝∗/𝑝𝑑𝜏𝑑𝑠=4^{𝑝∗/𝑝−11}₀𝑘()𝑘()×𝑝𝑡,𝑠𝑠,𝜏_{𝑝∗/𝑝11}𝑢^𝑝∗+𝑞^𝑝∗/𝑝₁₁𝑣^𝑞∗𝑑𝜏𝑑𝑠+2^{𝑝∗/𝑝−11}₀𝑘(𝑡,𝑠)𝑘(𝑠,𝜏)𝑐^𝑝∗/𝑝𝑑𝜏𝑑𝑠(55) for all (𝑢,𝑣)∈ℳ₄,𝑡∈[0,1]. Note that 𝑞^∗,𝑞^∗/𝑞⩾1. The same argument for the previous inequality can be used to obtain 𝑣^𝑞∗(𝑡)⩽4^{𝑞∗/𝑞−11}₀𝑘()𝑘()×𝑝𝑡,𝑠𝑠,𝜏_{𝑞∗/𝑞21}𝑢^𝑝∗+𝑞^𝑞∗/𝑞₂₁𝑣^𝑞∗𝑑𝜏𝑑𝑠+2^{𝑞∗/𝑞−11}₀𝑘(𝑡,𝑠)𝑘(𝑠,𝜏)𝑐^𝑞∗/𝑞𝑑𝜏𝑑𝑠(56) for all (𝑢,𝑣)∈ℳ₄,𝑡∈[0,1]. Multiply the last two inequalities by 𝜑(𝑡), integrate over [0,1], and use Lemma 1 and (H5) to obtain ¹₀𝑢^𝑝∗(𝑡)sin𝜋𝑡𝑑𝑡⩽𝑝₁¹₀𝑢^𝑝∗(𝑡)sin𝜋𝑡𝑑𝑡+𝑞₁¹₀𝑣^𝑞∗(𝑡)sin𝜋𝑡𝑑𝑡+ℎ₃,₁₀𝑣^𝑞∗(𝑡)sin𝜋𝑡𝑑𝑡⩽𝑝₂¹₀𝑢^𝑝∗(𝑡)sin𝜋𝑡𝑑𝑡+𝑞₂¹₀𝑣^𝑞∗(𝑡)sin𝜋𝑡𝑑𝑡+ℎ₄(57) for all (𝑢,𝑣)∈ℳ₄, where ℎ₃∶=2^𝑝∗/𝑝𝑐^𝑝∗/𝑝𝜋⁻⁵,ℎ₄∶=2^𝑞∗/𝑞𝑐^𝑞∗/𝑞𝜋⁻⁵. The last two inequalities can be written as 1−𝑝₁−𝑞₁−𝑝₂1−𝑞₂¹⁰𝑢^𝑝∗(𝑡)sin𝜋𝑡𝑑𝑡₁₀𝑣^𝑞∗(𝑡)sin𝜋𝑡𝑑𝑡=𝐴₄¹⁰𝑢^𝑝∗(𝑡)sin𝜋𝑡𝑑𝑡₁₀𝑣^𝑞∗(𝑡)⩽ℎsin𝜋𝑡𝑑𝑡ℎ³₄.(58) Now (H3) implies ¹₀𝑢^𝑝∗(𝑡)sin𝜋𝑡𝑑𝑡₁₀𝑣^𝑞∗(𝑡)sin𝜋𝑡𝑑𝑡⩽𝐴₄⁻¹ℎ₃ℎ₄,(59) and thus there exists a constant 𝑀>0 such that ¹₀𝑢^𝑝∗(𝑡)sin𝜋𝑡𝑑𝑡⩽𝑀,¹₀𝑣^𝑞∗(𝑡)sin𝜋𝑡𝑑𝑡⩽𝑀(60) for every (𝑢,𝑣)∈ℳ₄. The concavity of 𝑢,𝑣 and Lemma 2 imply ‖𝑢‖⩽𝜋²2¹₀𝑢(𝑡)‖𝑣‖⩽𝜋sin𝜋𝑡𝑑𝑡,²2¹₀𝑣(𝑡)sin𝜋𝑡𝑑𝑡.(61) The fact that 𝑝^∗,𝑞^∗⩾1, together with Jensen's inequalities, implies ¹₀𝑢(𝑡)sin𝜋𝑡𝑑𝑡^𝑝∗⩽¹₀𝑢^𝑝∗(𝑡)(sin𝜋𝑡)^𝑝∗⩽𝑑𝑡₁₀𝑢^𝑝∗(𝑡)sin𝜋𝑡𝑑𝑡⩽𝑀,₁₀𝑣(𝑡)sin𝜋𝑡𝑑𝑡^𝑞∗⩽¹₀𝑣^𝑞∗(𝑡)(sin𝜋𝑡)^𝑞∗⩽𝑑𝑡₁₀𝑣^𝑞∗(𝑡)sin𝜋𝑡𝑑𝑡⩽𝑀,(62) and, in turn, ¹₀𝑢(𝑡)sin𝜋𝑡𝑑𝑡⩽𝑀^1/𝑝∗,¹₀𝑣(𝑡)sin𝜋𝑡𝑑𝑡⩽𝑀^1/𝑞∗.(63) Recalling (61), we obtain ‖𝑢‖⩽𝜋²𝑀^1/𝑝∗2,‖𝑣‖⩽𝜋²𝑀^1/𝑞∗2(64) for all (𝑢,𝑣)∈ℳ₄. This proves the boundedness of ℳ₄. Taking 𝑅>sup{‖(𝑢,𝑣)‖∶(𝑢,𝑣)∈ℳ₄}, we have (𝑢,𝑣)≠𝜆𝐴(𝑢,𝑣),∀(𝑢,𝑣)∈𝜕𝑄_𝑅∩(𝑃×𝑃),𝜆∈[0,1].(65) Now Lemma 3 yields 𝑖𝐴,𝑄_𝑅∩(𝑃×𝑃),𝑃×𝑃=1.(66) Combining (53) with (66) gives 𝑖𝐴,𝑄_𝑅⧵𝑄_𝑟∩(𝑃×𝑃),𝑃×𝑃=1−0=1.(67) Hence, 𝐴 has at least one fixed point on (𝑄_𝑅⧵𝑄_𝑟)∩(𝑃×𝑃). Thus (1) has at least one positive solution. This completes the proof.