Three-Point Boundary Value Problems of Nonlinear Second-Order 𝑞-Difference Equations Involving Different Numbers of 𝑞

Abstract

We study a new class of three-point boundary value problems of nonlinear second-order q-difference equations. Our problems contain different numbers of q in derivatives and integrals. By using a variety of fixed point theorems (such as Banach’s contraction principle, Boyd and Wong fixed point theorem for nonlinear contractions, Krasnoselskii’s fixed point theorem, and Leray-Schauder nonlinear alternative) and Leray-Schauder degree theory, some new existence and uniqueness results are obtained. Illustrative examples are also presented.

1. Introduction

The 𝑞-difference calculus or quantum calculus is an old subject that was initially developed by Jackson [1], Carmichael [2], Mason [3], and Adams [4], in the first quarter of 20th century, has been developed over the years, for instance, see [5–14] and the references therein. In fact, 𝑞-calculus has a rich history, and the details of its basic notions, results, and methods can be found in the text [15]. In recent years, the topic has attracted the attention of several researchers, and a variety of new results can be found in the papers [16–28] and the references cited therein.

In [24], Ahmad et al. studied a boundary value problem of nonlinear 𝑞-difference equations with nonlocal boundary conditions given by 𝐷²_𝑞𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡)),𝑡∈𝐼¹_𝑞,𝛼₁𝑥(0)−𝛽₁𝐷_𝑞𝑥(0)=𝛾₁𝑥𝜂₁,𝛼₂𝑥(1)+𝛽₂𝐷_𝑞𝑥(1)=𝛾₂𝑥𝜂₂, (1) where 𝑓∈𝐶(𝐼¹_𝑞×ℝ,ℝ), 𝐼¹_𝑞={𝑞^𝑛∶𝑛∈ℕ}∪{0,1}, and 𝑞∈(0,1) is a fixed constant. The existence of solutions for problem (1) is shown by means of a variety of fixed point theorems such as Banach's contraction principle, Krasnoselskii’s fixed point theorem, and Leray-Schauder nonlinear alternative.

Yu and Wang [28] considered a boundary value problem with the nonlinear second-order 𝑞-difference equation, 𝐷²_𝑞𝑢(𝑡)+𝑓𝑡,𝑢(𝑡),𝐷_𝑞𝑢(𝑡)=0,𝑡∈𝐼¹_𝑞,𝐷_𝑞𝑢(0)=0,𝐷_𝑞𝑢(1)=𝛼𝑢(1), (2) where 𝑓∈𝐶(𝐼¹_𝑞×ℝ²,ℝ) and 𝛼≠0 is a fixed number. Existence and uniqueness of the solutions are obtained by means of Banach’s contraction principle, Leray-Schauder nonlinear alternative, and Leray-Schauder continuation theorem.

Pongarm et al. [29] considered sequential derivative of nonlinear 𝑞-difference equation with three-point boundary conditions, 𝐷_𝑞𝐷_𝑝+𝜆𝑢(𝑡)=𝑓(𝑡,𝑢(𝑡)),𝑡∈𝐼^𝑇_𝑞=[0,𝑇]∩𝐼¹_𝑞,𝑢(0)=0,𝑢(𝑇)=𝛼^𝜂₀𝑢(𝑠)𝑑_𝑟𝑠, (3) where 0<𝑝,𝑞,𝑟<1, 𝑓∈𝐶(𝐼^𝑇_𝑞×ℝ,ℝ), 0<𝜂<𝑇, and 𝜆, 𝛽 are given constants. Existence results are proved based on Banach’s contraction mapping principle, Krasnoselskii’s fixed point theorem, and Leray-Schauder degree theory.

We note that in the above-mentioned papers [24, 28] the 𝑞-numbers in the equation and the boundary conditions are the same. As far as we know the paper by Pongarm et al. [29] is the first paper which has different values of the 𝑞-numbers in 𝑞-derivative and 𝑞-integral.

In this paper, we discuss the existence of solutions for the following nonlinear 𝑞-difference equation with three-point integral boundary condition 𝐷²_𝑞𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡)),𝑡∈𝐼^𝑇_𝑞,𝛼𝑥𝜂+𝛽𝐷_𝑟𝑥𝜂=0,^𝑇₀𝑥(𝑠)𝑑_𝑝𝑠=0,0<𝜂<𝑇, (4) where 𝑓∈𝐶(𝐼^𝑇_𝑞×ℝ,ℝ), 𝐼^𝑇_𝑞=𝐼¹_𝑞∩[0,𝑇], 𝐼¹_𝑞={𝑞^𝑛∶𝑛∈ℕ}∪{0,1}, 𝑞∈(0,1) is a fixed constant, and 𝜂∈𝐼^𝑇_𝑞⧵{0,𝑇}∶=(0,𝑇)_𝑞. Also, 0<𝑝,𝑞,𝑟<1, and 𝛼, 𝛽 are given constants such that 𝛽≠𝛼((𝑇/(1+𝑝))−𝜂).

It is noteworthy that, in the above problem (4), we have three different values of the 𝑞-numbers, in 𝑞-derivatives and the 𝑞-integral. Moreover, we emphasize the fact that, instead the value 𝑥(0) is usually used in the literature, we use the values of the function and its derivative in an intermediate point 𝜂∈(0,𝑇).

The aim of this paper is to prove some existence and uniqueness results for the boundary value problem (4). Our results are based on Banach’s contraction mapping principle, nonlinear contraction, Krasnoselskii’s fixed point theorem, Leray-Schauder nonlinear alternative, and Leray-Schauder degree theory.

The rest of the paper is organized as follows. In Section 2, we provide some basic definitions, preliminaries facts, and a lemma, which are used later. The main results are given in Section 3. In the end, Section 4, some results illustrating the results established in this paper are also presented.

2. Preliminaries

Let us recall some basic concepts of 𝑞-calculus [15, 18].

Definition 1. For 0<𝑞<1, one defines the 𝑞-derivative of a real valued function 𝑓 as 𝐷_𝑞𝑓(𝑡)=𝑓(𝑡)−𝑓𝑞𝑡1−𝑞𝑡,𝑡∈𝐼¹_𝑞⧵{0},𝐷_𝑞𝑓(0)=lim_𝑡→0𝐷_𝑞𝑓(𝑡). (5)

The higher-order 𝑞-derivatives are given by 𝐷⁰_𝑞𝑓(𝑡)=𝑓(𝑡),𝐷^𝑛_𝑞𝑓(𝑡)=𝐷_𝑞𝐷_𝑞^𝑛−1𝑓(𝑡),𝑛∈ℕ. (6)

For 𝑥≥0 one sets 𝐽_𝑥={𝑥𝑞^𝑛∶𝑛∈ℕ∪{0}}∪{0} and define, the definite 𝑞-integral of a function 𝑓∶𝐽_𝑥→ℝ by 𝐼_𝑞𝑓(𝑥)=^𝑥₀𝑓(𝑠)𝑑_𝑞𝑠=^∞_𝑛=0𝑥1−𝑞𝑞^𝑛𝑓𝑥𝑞^𝑛 (7) provided that the series converges.

For 𝑎,𝑏∈𝐽_𝑥, one sets ^𝑏_𝑎𝑓(𝑠)𝑑_𝑞𝑠=𝐼_𝑞𝑓(𝑏)−𝐼_𝑞𝑓(𝑎)=1−𝑞^∞_𝑛=0𝑞^𝑛𝑏𝑓𝑏𝑞^𝑛−𝑎𝑓𝑎𝑞^𝑛. (8)

Note that for 𝑎,𝑏∈𝐽_𝑥, one has 𝑎=𝑥𝑞^𝑛1, 𝑏=𝑥𝑞^𝑛2 for some 𝑛₁,𝑛₂∈ℕ; thus, the definite integral ∫^𝑏_𝑎𝑓(𝑠)𝑑_𝑞𝑠 is just a finite sum, so no question about convergence is raised.

One notes that 𝐷_𝑞𝐼_𝑞𝑓(𝑥)=𝑓(𝑥), (9) while if 𝑓 is continuous at 𝑥=0, then 𝐼_𝑞𝐷_𝑞𝑓(𝑥)=𝑓(𝑥)−𝑓(0). (10) In 𝑞-calculus, the product rule and integration by parts formula are 𝐷_𝑞𝑔ℎ(𝑡)=𝐷_𝑞𝑔(𝑡)ℎ(𝑡)+𝑔𝑞𝑡𝐷_𝑞ℎ(𝑡),_𝑥0𝑓(𝑡)𝐷_𝑞𝑔(𝑡)𝑑_𝑞𝑡=𝑓(𝑡)𝑔(𝑡)^𝑥₀−^𝑥₀𝐷_𝑞𝑓(𝑡)𝑔𝑞𝑡𝑑_𝑞𝑡. (11) Further, reversing the order of integration is given by ^𝑡₀^𝑠₀𝑓(𝑟)𝑑_𝑞𝑟𝑑_𝑞𝑠=^𝑡₀^𝑡_𝑞𝑟𝑓(𝑟)𝑑_𝑞𝑠𝑑_𝑞𝑟. (12) In the limit 𝑞→1, the above results correspond to their counterparts in standard calculus.

Lemma 2. Let 0<𝑝,𝑞,𝑟<1 and 𝜂∈(0,𝑇)_𝑞. Then, for any 𝑦∈𝐶(𝐼^𝑇_𝑞,ℝ), the boundary value problem, 𝐷²_𝑞𝑥(𝑡)=𝑦(𝑡),𝑡∈𝐼^𝑇_𝑞,𝛼𝑥𝜂+𝛽𝐷_𝑟𝑥𝜂=0,^𝑇₀𝑥(𝑠)𝑑_𝑝𝑠=0, (13)(14) is equivalent to the integral equation 𝑥(𝑡)=^𝑡₀𝑡−𝑞𝑠𝑦(𝑠)𝑑_𝑞𝑠−1+𝑝𝑡−𝑇Ω×𝛼^𝜂₀𝜂−𝑞𝑠𝑦(𝑠)𝑑_𝑞𝑠+𝛽₀^𝑟𝜂𝑦(𝑠)𝑑_𝑞𝑠+𝛽(1−𝑟)𝜂^𝜂_𝑟𝜂𝜂−𝑞𝑠𝑦(𝑠)𝑑_𝑞𝑠+1+𝑝𝛼𝑇Ω𝑡−𝜂−𝛽^𝑇₀^𝑠₀𝑠−𝑞𝑣𝑦(𝑣)𝑑_𝑞𝑣𝑑_𝑝𝑠, (15) where Ω=𝛼𝜂+𝛽1+𝑝−𝛼𝑇≠0. (16)

Proof. Taking double 𝑞-integral for (13), we have 𝑥(𝑡)=^𝑡₀^𝑠₀𝑦(𝑣)𝑑_𝑞𝑣𝑑_𝑞𝑠+𝑐₁𝑡+𝑐₂. (17) By changing the order of 𝑞-integration, we have 𝑥(𝑡)=^𝑡₀^𝑡_𝑞𝑣𝑦(𝑣)𝑑_𝑞𝑠𝑑_𝑞𝑣+𝑐₁𝑡+𝑐₂=^𝑡₀𝑡−𝑞𝑠𝑦(𝑠)𝑑_𝑞𝑠+𝑐₁𝑡+𝑐₂. (18) In particular, for 𝑡=𝜂, we get 𝑥𝜂=^𝜂₀𝜂−𝑞𝑠𝑦(𝑠)𝑑_𝑞𝑠+𝜂𝑐₁+𝑐₂. (19) Taking 𝑟-derivative for (18), for 𝑡≠0, we obtain 𝐷_𝑟𝑥(𝑡)=𝐷_𝑟^𝑡₀𝑡−𝑞𝑠𝑦(𝑠)𝑑_𝑞𝑠+𝑐₁𝑡+𝑐₂=1(1−𝑟)𝑡^𝑡₀𝑡−𝑞𝑠𝑦(𝑠)𝑑_𝑞𝑠−₀^𝑟𝑡𝑟𝑡−𝑞𝑠𝑦(𝑠)𝑑_𝑞𝑠+𝑐₁=₀^𝑟𝑡𝑦(𝑠)𝑑_𝑞𝑠+^𝑡_𝑟𝑡𝑡−𝑞𝑠()𝑡𝑦(𝑠)𝑑1−𝑟^𝑞𝑠+𝑐₁. (20) For 𝑡=0, we have 𝐷_𝑟𝑥(0)=lim_𝑡→0𝐷_𝑟𝑥(0)=lim_𝑡→0𝑡1−𝑞1−𝑟^∞_𝑛=0𝑞^𝑛1−𝑞^𝑛+1×ℎ𝑡𝑞^𝑛−𝑟²ℎ𝑟𝑡𝑞^𝑛+𝑐₁=𝑐₁. (21) Therefore, 𝐷_𝑟𝑥𝜂=₀^𝑟𝜂𝑦(𝑠)𝑑_𝑞𝑠+^𝜂_𝑟𝜂𝜂−𝑞𝑠()𝜂𝑦(𝑠)𝑑1−𝑟^𝑞𝑠+𝑐₁. (22) Now, using the first condition of (14) with (19), (22), we have 𝛼𝜂+𝛽𝑐₁+𝛼𝑐₂=−𝛼^𝜂₀𝜂−𝑞𝑠𝑦(𝑠)𝑑_𝑞𝑠−𝛽₀^𝑟𝜂𝑦(𝑠)𝑑_𝑞𝑠−𝛽(1−𝑟)𝜂^𝜂_𝑟𝜂𝜂−𝑞𝑠𝑦(𝑠)𝑑_𝑞𝑠. (23) Taking the 𝑝-integral for (18) from 0 to 𝑡, we obtain ^𝑡₀𝑥(𝑠)𝑑_𝑝𝑠=^𝑡₀^𝑠₀𝑠−𝑞𝑣𝑦(𝑣)𝑑_𝑞𝑣𝑑_𝑝𝑡𝑠+²𝑐1+𝑝¹+𝑡𝑐₂. (24) Substituting 𝑡=𝑇 in (24) and using the second condition of (14), we get 𝑇²𝑐1+𝑝¹+𝑇𝑐₂=−^𝑇₀^𝑠₀𝑠−𝑞𝑣𝑦(𝑣)𝑑_𝑞𝑣𝑑_𝑝𝑠. (25) Solving the system of linear equations (23) and (25) for the unknown constants 𝑐₁ and 𝑐₂, we have 𝑐₁=−1+𝑝Ω𝛼^𝜂₀𝜂−𝑞𝑠𝑦(𝑠)𝑑_𝑞𝑠+𝛽₀^𝑟𝜂𝑦(𝑠)𝑑_𝑞𝛽𝑠+()𝜂×1−𝑟_𝜂𝑟𝜂𝜂−𝑞𝑠𝑦(𝑠)𝑑_𝑞𝑠+𝛼1+𝑝𝑇Ω^𝑇₀^𝑠₀𝑠−𝑞𝑣𝑦(𝑣)𝑑_𝑞𝑣𝑑_𝑝𝑐𝑠,₂=−𝛼𝜂+𝛽1+𝑝𝑇Ω^𝑇₀^𝑠₀𝑠−𝑞𝑣𝑦(𝑣)𝑑_𝑞𝑣𝑑_𝑝𝑠+𝑇Ω𝛼^𝜂₀𝜂−𝑞𝑠𝑦(𝑠)𝑑_𝑞𝑠+𝛽₀^𝑟𝜂𝑦(𝑠)𝑑_𝑞𝑠+𝛽(1−𝑟)𝜂^𝜂_𝑟𝜂𝜂−𝑞𝑠𝑦(𝑠)𝑑_𝑞𝑠, (26) where Ω is defined by (16). Substituting the values of 𝑐₁ and 𝑐₂ in (18), we obtain (15). This completes the proof.

Let 𝒞=𝐶(𝐼^𝑇_𝑞,ℝ) denotes the Banach space of all the continuous functions from 𝐼^𝑇_𝑞 to ℝ endowed with the norm defined by ‖𝑥‖=sup{|𝑥(𝑡)|,𝑡∈𝐼^𝑇_𝑞}. Define an operator 𝐴∶𝒞→𝒞 by (𝐴𝑥)(𝑡)=^𝑡₀𝑡−𝑞𝑠𝑓(𝑠,𝑥(𝑠))𝑑_𝑞𝑠−1+𝑝𝑡−𝑇Ω×𝛼^𝜂₀𝜂−𝑞𝑠𝑓(𝑠,𝑥(𝑠))𝑑_𝑞𝑠+𝛽₀^𝑟𝜂𝑓(𝑠,𝑥(𝑠))𝑑_𝑞𝑠+𝛽(1−𝑟)𝜂^𝜂_𝑟𝜂𝜂−𝑞𝑠𝑓(𝑠,𝑥(𝑠))𝑑_𝑞𝑠+1+𝑝𝛼𝑡−𝜂−𝛽×𝑇Ω_𝑇0𝑠0𝑠−𝑞𝑣𝑓(𝑣,𝑥(𝑣))𝑑_𝑞𝑣𝑑_𝑝𝑠. (27) Observe that the problem (4) has solutions if and only if the operator 𝐴 has fixed points.

For the sake of convenience, we set a constant Λ as 𝑇Λ=²+1+𝑞𝑝𝑇|Ω||𝛼|𝜂²+𝛽+𝛽𝜂()1+𝑞𝑟𝜂1+𝑞2+𝑟+|𝛼|+𝛽𝑇1+𝑞1+𝑝𝑇−𝜂²1+𝑞1+𝑝+𝑝²|Ω|. (28)

3. Main Results

Now, we are in the position to establish the main results. Our first result is based on Banach’s fixed point theorem.

Theorem 3. Assume that 𝑓∶𝐼^𝑇_𝑞×ℝ→ℝ is a continuous function satisfying the conditions(H₁)|𝑓(𝑡,𝑥)−𝑓(𝑡,𝑦)|≤𝐿|𝑥−𝑦|, for all 𝑡∈𝐼^𝑇_𝑞 and 𝑥,𝑦∈ℝ, (H₂)𝐿Λ<1,
where 𝐿 is a Lipschitz constant, and Λ is defined by (28).
Then, the boundary value problem (4) has a unique solution.

Proof. We transform the boundary value problem (4) into a fixed point problem 𝑥=𝐴𝑥, where 𝐴∶𝒞→𝒞 is defined by (27). Assume that sup_{𝑡∈𝐼𝑇𝑞}|𝑓(𝑡,0)|=𝑀, and choose a constant 𝑅 satisfying 𝑅≥𝑀Λ.1−𝐿Λ (29) Now, we will show that 𝐴𝐵_𝑅⊂𝐵_𝑅, where 𝐵_𝑅={𝑥∈𝒞∶‖𝑥‖≤𝑅}. For any 𝑥∈𝐵_𝑅, we have ‖𝐴𝑥‖=sup_{𝑡∈𝐼𝑇𝑞}^𝑡₀𝑡−𝑞𝑠𝑓(𝑠,𝑥(𝑠))𝑑_𝑞𝑠−1+𝑝𝑡−𝑇Ω×𝛼^𝜂₀𝜂−𝑞𝑠𝑓(𝑠,𝑥(𝑠))𝑑_𝑞𝑠+𝛽₀^𝑟𝜂𝑓(𝑠,𝑥(𝑠))𝑑_𝑞𝛽𝑠+()𝜂×1−𝑟_𝜂𝑟𝜂𝜂−𝑞𝑠𝑓(𝑠,𝑥(𝑠))𝑑_𝑞𝑠+1+𝑝𝛼𝑡−𝜂−𝛽×𝑇Ω_𝑇0𝑠0𝑠−𝑞𝑣𝑓(𝑣,𝑥(𝑣))𝑑_𝑞𝑣𝑑_𝑝𝑠≤sup_{𝑡∈𝐼𝑇𝑞}^𝑡₀×𝑓((𝑠(+𝑓(𝑑𝑡−𝑞𝑠𝑠,𝑥))−𝑓𝑠,0)𝑠,0)_𝑞𝑠+1+𝑝𝑡−𝑇|Ω|×|𝛼|^𝜂₀𝑓((𝑠(+𝑓(𝑑𝜂−𝑞𝑠𝑠,𝑥))−𝑓𝑠,0)𝑠,0)_𝑞𝑠+𝛽₀^𝑟𝜂𝑓((𝑠(+𝑓(𝑑𝑠,𝑥))−𝑓𝑠,0)𝑠,0)_𝑞𝑠+𝛽()𝜂×1−𝑟_𝜂𝑟𝜂𝑓((𝑠(+𝑓(𝑑𝜂−𝑞𝑠𝑠,𝑥))−𝑓𝑠,0)𝑠,0)_𝑞𝑠+1+𝑝𝛼𝑡−𝜂−𝛽𝑇|Ω|×^𝑇₀^𝑠₀×𝑓((𝑠(+𝑓(𝑑𝑠−𝑞𝑣𝑣,𝑥))−𝑓𝑣,0)𝑣,0)_𝑞𝑣𝑑_𝑝𝑠≤sup_{𝑡∈𝐼𝑇𝑞}^𝑡₀𝑡−𝑞𝑠(𝐿𝑅+𝑀)𝑑_𝑞𝑠+1+𝑝𝑡−𝑇|Ω|×|𝛼|^𝜂₀𝜂−𝑞𝑠(𝐿𝑅+𝑀)𝑑_𝑞𝛽×𝑠+_0𝑟𝜂(𝐿𝑅+𝑀)𝑑_𝑞𝑠+𝛽(1−𝑟)𝜂^𝜂_𝑟𝜂𝜂−𝑞𝑠(𝐿𝑅+𝑀)𝑑_𝑞𝑠+1+𝑝𝛼𝑡−𝜂−𝛽𝑇|Ω|×^𝑇₀^𝑠₀𝑠−𝑞𝑣(𝐿𝑅+𝑀)𝑑_𝑞𝑣𝑑_𝑝𝑠≤(𝑇𝐿𝑅+𝑀)²+1+𝑞𝑝𝑇|Ω|×|𝛼|𝜂²+𝛽+𝛽𝜂(1+𝑞𝑟𝜂1+𝑞2+𝑟)+|𝛼|+𝛽𝑇1+𝑞1+𝑝𝑇−𝜂²1+𝑞1+𝑝+𝑝²|Ω|=(𝐿𝑅+𝑀)Λ≤𝑅. (30) Therefore, 𝐴𝐵_𝑅⊂𝐵_𝑅.
Next, we will show that 𝐴 is a contraction. For any 𝑥,𝑦∈𝒞 and for each 𝑡∈𝐼^𝑇_𝑞, we have 𝐴𝑥−𝐴𝑦=sup_{𝑡∈𝐼𝑇𝑞}(𝐴𝑥)(𝑡)−𝐴𝑦(𝑡)≤sup_{𝑡∈𝐼𝑇𝑞}^𝑡₀𝑡−𝑞𝑠𝑓(𝑠,𝑥(𝑠))−𝑓𝑠,𝑦(𝑠)𝑑_𝑞𝑠−1+𝑝𝑡−𝑇Ω×𝛼^𝜂₀𝑓((𝑠(𝑠)𝑑𝜂−𝑞𝑠𝑠,𝑥))−𝑓𝑠,𝑦_𝑞𝑠+𝛽₀^𝑟𝜂𝑓(𝑠,𝑥(𝑠))−𝑓𝑠,𝑥₂(𝑠)𝑑_𝑞𝑠+𝛽()𝜂×1−𝑟_𝜂𝑟𝜂𝑓((𝑠)(𝑠)𝑑𝜂−𝑞𝑠𝑠,𝑥−𝑓𝑠,𝑦_𝑞𝑠+1+𝑝𝛼𝑡−𝜂−𝛽×𝑇Ω_𝑇0𝑠0×𝑓((𝑣(𝑣)𝑑𝑠−𝑞𝑣𝑣,𝑥))−𝑓𝑣,𝑦_𝑞𝑣𝑑_𝑝𝑠≤sup_{𝑡∈𝐼𝑇𝑞}𝐿𝑥−𝑦^𝑡₀𝑡−𝑞𝑠𝑑_𝑞×𝑠+𝐿𝑥−𝑦1+𝑝𝑡−𝑇|Ω||𝛼|^𝜂₀𝜂−𝑞𝑠𝑑_𝑞𝑠+𝛽₀^𝑟𝜂𝑑_𝑞𝛽𝑠+()𝜂×1−𝑟_𝜂𝑟𝜂𝜂−𝑞𝑠𝑑_𝑞𝑠+1+𝑝|𝛼|𝑡−𝜂+𝛽𝑇|Ω|𝐿×𝑥−𝑦_𝑇0𝑠0𝑠−𝑞𝑣𝑑_𝑞𝑣𝑑_𝑝𝑠𝑇≤𝐿𝑥−𝑦²+1+𝑞𝑝𝑇|Ω|×|𝛼|𝜂²+𝛽+𝛽𝜂()1+𝑞𝑟𝜂1+𝑞2+𝑟+|𝛼|+𝛽𝑇1+𝑞1+𝑝𝑇−𝜂²1+𝑞1+𝑝+𝑝²|Ω|=𝐿Λ𝑥−𝑦. (31) Since 𝐿Λ<1, 𝐴 is a contraction. Thus, the conclusion of the theorem follows by Banach’s contraction mapping principle. This completes the proof.

Next, we can still deduce the existence and uniqueness of a solution to the boundary value problem (4). We will use nonlinear contraction to accomplish this.

Definition 4. Let 𝐸 be a Banach space and let 𝐹∶𝐸→𝐸 be a mapping. 𝐹 is said to be a nonlinear contraction if there exists a continuous nondecreasing function Ψ∶𝑅⁺→𝑅⁺ such that Ψ(0)=0 and Ψ(𝜌)<𝜌 for all 𝜌>0 with the following property: 𝐹𝑥−𝐹𝑦≤Ψ𝑥−𝑦,∀𝑥,𝑦∈𝐸. (32)

Lemma 5 (Boyd and Wong [30]). Let 𝐸 be a Banach space and let 𝐹∶𝐸→𝐸 be a nonlinear contraction. Then, 𝐹 has a unique fixed point in 𝐸.

Theorem 6. Suppose that (H₃) there exists a continuous function ℎ∶𝐼^𝑇_𝑞→ℝ⁺ such that 𝑓(𝑡,𝑥)−𝑓𝑡,𝑦≤ℎ(𝑡)𝑥−𝑦𝐺+𝑥−𝑦 (33) for all 𝑡∈𝐼^𝑇_𝑞 and 𝑥,𝑦≥0, where 𝐺=^𝑇₀𝑇−𝑞𝑠ℎ(𝑠)𝑑_𝑞𝑠+𝑝𝑇|Ω|×|𝛼|^𝜂₀𝜂−𝑞𝑠ℎ(𝑠)𝑑_𝑞𝑠+𝛽₀^𝑟𝜂ℎ(𝑠)𝑑_𝑞𝑠+𝛽(1−𝑟)𝜂^𝜂_𝑟𝜂𝜂−𝑞𝑠ℎ(𝑠)𝑑_𝑞𝑠+1+𝑝|𝛼|𝑇−𝜂+𝛽𝑇|Ω|×^𝑇₀^𝑠₀𝑠−𝑞𝑣ℎ(𝑣)𝑑_𝑞𝑣𝑑_𝑝𝑠 (34) and Ω is defined in (16).
Then, the boundary value problem (4) has a unique solution.

Proof. Let the operator 𝐴∶𝒞→𝒞 be defined as (27). We define a continuous nondecreasing function Ψ∶ℝ⁺→ℝ⁺ by Ψ𝜌=𝐺𝜌𝐺+𝜌,∀𝜌≥0, (35) such that Ψ(0)=0 and Ψ(𝜌)<𝜌, for all 𝜌>0.
Let 𝑥,𝑦∈𝒞. Then, we get 𝑓((𝑠(𝑠)≤ℎ(𝑠)𝑠,𝑥))−𝑓𝑠,𝑦𝐺Ψ𝑥−𝑦. (36) Thus, (𝑡)(𝑡≤𝐴𝑥−𝐴𝑦)_𝑡0ℎ(𝑠)𝑥(𝑠)(𝑠𝑡−𝑞𝑠−𝑦)𝑥(𝑠)(𝑠𝑑𝐺+−𝑦)^𝑞𝑠+1+𝑝𝑡−𝑇|Ω||𝛼|^𝜂₀(𝑠)𝑥(𝑠)(𝑠𝜂−𝑞𝑠×ℎ−𝑦)𝑥(𝑠)(𝑠𝑑𝐺+−𝑦)^𝑞𝑠+𝛽₀^𝑟𝜂ℎ(𝑠)𝑥(𝑠)−𝑦(𝑠)𝑥(𝑠)(𝑠𝑑𝐺+−𝑦)^𝑞𝑠+𝛽(1−𝑟)𝜂^𝜂_𝑟𝜂(𝑠)𝑥(𝑠)(𝑠𝜂−𝑞𝑠×ℎ−𝑦)𝑥(𝑠)(𝑠𝑑𝐺+−𝑦)^𝑞𝑠+1+𝑝𝛼𝑡−𝜂−𝛽𝑇|Ω|×^𝑇₀^𝑠₀(𝑣)𝑥(𝑣)(𝑣𝑠−𝑞𝑣×ℎ−𝑦)𝐺^∗+𝑥(𝑣)(𝑣𝑑−𝑦)^𝑞𝑣𝑑_𝑝𝑠≤^𝑇₀𝑇−𝑞𝑠ℎ(𝑠)𝑑_𝑞𝑠+𝑝𝑇|Ω|×|𝛼|^𝜂₀𝜂−𝑞𝑠ℎ(𝑠)𝑑_𝑞𝑠+𝛽₀^𝑟𝜂ℎ(𝑠)𝑑_𝑞𝑠+𝛽(1−𝑟)𝜂^𝜂_𝑟𝜂𝜂−𝑞𝑠ℎ(𝑠)𝑑_𝑞𝑠+1+𝑝|𝛼|𝑇−𝜂+𝛽𝑇|Ω|×^𝑇₀^𝑠₀𝑠−𝑞𝑣ℎ(𝑣)𝑑_𝑞𝑣𝑑_𝑝𝑠×𝑥−𝑦=𝐺𝐺+𝑥−𝑦𝑥−𝑦𝐺+𝑥−𝑦,∀𝑡∈𝐼^𝑇_𝑞. (37) This implies that ‖𝐴𝑥−𝐴𝑦‖≤Ψ(‖𝑥−𝑦‖). Hence, 𝐴 is a nonlinear contraction. Therefore, by Lemma 5, the operator 𝐴 has a unique fixed point in 𝒞, which is a unique solution of problem (4).

The third result is based on the following Krasnoselskii fixed point theorem [31].

Theorem 7. Let 𝐾 be a bounded closed convex and nonempty subset of a Banach space 𝑋. Let 𝐴, 𝐵 be operators such that: (i)𝐴𝑥+𝐵𝑦∈𝐾 whenever 𝑥,𝑦∈𝐾, (ii)𝐴 is compact and continuous, (iii)𝐵 is a contraction mapping.
Then, there exists 𝑧∈𝐾 such that 𝑧=𝐴𝑧+𝐵𝑧.

Theorem 8. Assume that (𝐻₁) and (𝐻₂) hold. In addition one supposes that: (H₄)|𝑓(𝑡,𝑥)|≤𝜇(𝑡), for all (𝑡,𝑥)∈𝐼^𝑇_𝑞×ℝ, with 𝜇∈𝐿¹(𝐼^𝑇_𝑞,ℝ⁺).
If Λ<1, (38) where Λ is given by (28), then the boundary value problem (4) has at least one solution on 𝐼^𝑇_𝑞.

Proof. Setting max_𝑡∈𝐼^𝑇_𝑞|𝜇(𝑡)|=‖𝜇‖ and choosing a constant 𝑅≥𝜇Λ, (39) we consider 𝐵_𝑅={𝑥∈𝒞∶‖𝑥‖≤𝑅}.
In view of Lemma 2, we define the operators ℱ₁ and ℱ₂ on the ball 𝐵_𝑅 as ℱ₁𝑥(𝑡)=^𝑡₀𝑡−𝑞𝑠𝑓(𝑠,𝑥(𝑠))𝑑_𝑞ℱ𝑠,₂𝑥(𝑡)=−1+𝑝𝑡−𝑇Ω𝛼^𝜂₀𝜂−𝑞𝑠𝑓(𝑠,𝑥(𝑠))𝑑_𝑞𝑠+𝛽₀^𝑟𝜂𝑓(𝑠,𝑥(𝑠))𝑑_𝑞𝑠+𝛽()𝜂×1−𝑟_𝜂𝑟𝜂𝜂−𝑞𝑠𝑓(𝑠,𝑥(𝑠))𝑑_𝑞𝑠+1+𝑝𝛼𝑡−𝜂−𝛽×𝑇Ω_𝑇0𝑠0𝑠−𝑞𝑣𝑓(𝑣,𝑥(𝑣))𝑑_𝑞𝑣𝑑_𝑝𝑠. (40) For 𝑥,𝑦∈𝐵_𝑅, by computing directly, we have ℱ₁𝑥+ℱ₂𝑦≤𝜇^𝑡₀𝑡−𝑞𝑠𝑑_𝑞𝑠+𝜇1+𝑝𝑡−𝑇|Ω|×|𝛼|^𝜂₀𝜂−𝑞𝑠𝑑_𝑞𝑠+𝛽₀^𝑟𝜂𝑑_𝑞𝑠+𝛽(1−𝑟)𝜂^𝜂_𝑟𝜂𝜂−𝑞𝑠𝑑_𝑞𝑠+𝜇1+𝑝𝛼𝑡−𝜂−𝛽𝑇|Ω|×^𝑇₀^𝑠₀𝑠−𝑞𝑣𝑑_𝑞𝑣𝑑_𝑝𝑠≤𝜇Λ≤𝑅. (41) Therefore, ℱ₁𝑥+ℱ₂𝑦∈𝐵_𝑅. Condition (38) implies that ℱ₂ is a contraction mapping. Next, we will show that ℱ₁ is compact and continuous. Continuity of 𝑓 coupled with the assumption (𝐻₃) implies that the operator ℱ₁ is continuous and uniformly bounded on 𝐵_𝑅. We define sup_{(𝑡,𝑥)∈𝐼𝑇𝑞×𝐵𝑅}|𝑓(𝑡,𝑥)|=𝑓_max<∞. For 𝑡₁,𝑡₂∈𝐼^𝑇_𝑞 with 𝑡₁≤𝑡₂ and 𝑥∈𝐵_𝑅, we have ℱ₁𝑥𝑡₂−ℱ₁𝑥𝑡₁=^𝑡2₀𝑡₂−𝑞𝑠𝑓(𝑠,𝑥(𝑠))𝑑_𝑞𝑠−^𝑡1₀𝑡₁−𝑞𝑠𝑓(𝑠,𝑥(𝑠))𝑑_𝑞𝑠=^𝑡1₀𝑡₂−𝑡₁𝑓(𝑠,𝑥(𝑠))𝑑_𝑞𝑠+^𝑡2_𝑡1𝑡₂−𝑞𝑠𝑓(𝑠,𝑥(𝑠))𝑑_𝑞𝑠≤𝑡²₂−𝑡²₁1+2𝑞𝑓1+𝑞^max. (42) Actually, as 𝑡₂−𝑡₁→0, the right-hand side of the above inequality tends to be zero. So, ℱ₁ is relatively compact on 𝐵_𝑅. Hence, by the Arzelá-Ascoli Theorem, ℱ₁ is compact on 𝐵_𝑅. Therefore, all the assumptions of Theorem 7 are satisfied, and the conclusion of Theorem 7 implies that the boundary value problem (4) has at least one solution on 𝐼^𝑇_𝑞. This completes the proof.

As the fourth result, we prove the existence of solutions of (4) by using Leray-Schauder nonlinear alternative.

Theorem 9 (Nonlinear Alternative for Single Valued Maps [32]). Let 𝐸 be a Banach space, 𝐶 a closed convex subset of 𝐸, 𝑈 an open subset of 𝐶, and 0∈𝑈. Suppose that 𝐹∶𝑈→𝐶 is a continuous, compact (that is, 𝐹(𝑈) is a relatively compact subset of 𝐶) map. Then, either (i)𝐹 has a fixed point in 𝑈 or (ii)there is a 𝑢∈𝜕𝑈 (the boundary of 𝑈 in 𝐶) and 𝜆∈(0,1) with 𝑢=𝜆𝐹(𝑢).

Theorem 10. Assume that: (H₅) there exists a continuous nondecreasing function 𝜓∶[0,∞)→(0,∞) and a function 𝑧∈𝐿¹(𝐼^𝑇_𝑞,ℝ⁺) such that 𝑓(𝑡,𝑢)≤𝑧(𝑡)𝜓(‖𝑢‖),𝑓𝑜𝑟𝑒𝑎𝑐ℎ(𝑡,𝑢)∈𝐼^𝑇_𝑞×ℝ; (43)(H₆) there exists a constant 𝑀>0 such that 𝑀𝜓(𝑀)‖𝑧‖_𝐿¹Λ>1. (44)Then, the boundary value problem (4) has at least one solution on 𝐼^𝑇_𝑞.

Proof. We will show that 𝐴 maps bounded sets (balls) into bounded sets in 𝒞. For a positive number 𝜌, let 𝐵_𝜌={𝑥∈𝐶(𝐼^𝑇_𝑞,ℝ)∶‖𝑥‖≤𝜌} be a bounded ball in 𝐶(𝐼^𝑇_𝑞,ℝ). Then, for 𝑡∈𝐼^𝑇_𝑞, we have 𝑡≤|(𝐴𝑥)()|_𝑡0𝑡−𝑞𝑠𝑓(𝑠,𝑥(𝑠))𝑑_𝑞𝑠+1+𝑝𝑡−𝑇|Ω|×|𝛼|^𝜂₀𝜂−𝑞𝑠𝑓(𝑠,𝑥(𝑠))𝑑_𝑞𝑠+𝛽₀^𝑟𝜂𝑓(𝑠,𝑥(𝑠))𝑑_𝑞𝛽𝑠+(1−𝑟)𝜂×^𝜂_𝑟𝜂𝜂−𝑞𝑠𝑓(𝑠,𝑥(𝑠))𝑑_𝑞𝑠+1+𝑝𝛼𝑡−𝜂−𝛽𝑇|Ω|×^𝑇₀^𝑠₀𝑠−𝑞𝑣𝑓(𝑣,𝑥(𝑣))𝑑_𝑞𝑣𝑑_𝑝𝑠≤𝜓(‖𝑥‖)^𝑡₀𝑡−𝑞𝑠𝑧(𝑠)𝑑_𝑞𝜓𝑥𝑠+(‖‖)1+𝑝𝑡−𝑇|Ω|×|𝛼|^𝜂₀𝜂−𝑞𝑠𝑧(𝑠)𝑑_𝑞𝑠+𝛽₀^𝑟𝜂𝑧(𝑠)𝑑_𝑞𝑠+𝛽(1−𝑟)𝜂^𝜂_𝑟𝜂𝜂−𝑞𝑠𝑧(𝑠)𝑑_𝑞𝑠+𝜓(‖𝑥‖)1+𝑝𝛼𝑡−𝜂−𝛽𝑇|Ω|×^𝑇₀^𝑠₀𝑠−𝑞𝑣𝑧(𝑠)𝑑_𝑞𝑣𝑑_𝑝𝑠≤𝜓(‖𝑥‖)‖𝑧‖_𝐿^1𝑡₀𝑡−𝑞𝑠𝑑_𝑞𝑠+𝜓(‖𝑥‖)‖𝑧‖_𝐿¹1+𝑝𝑡−𝑇|Ω|×|𝛼|^𝜂₀𝜂−𝑞𝑠𝑑_𝑞𝑠+𝛽₀^𝑟𝜂𝑑_𝑞𝑠+𝛽(1−𝑟)𝜂^𝜂_𝑟𝜂𝜂−𝑞𝑠𝑑_𝑞𝑠+𝜓(‖𝑥‖)‖𝑧‖_𝐿¹1+𝑝𝛼𝑡−𝜂−𝛽𝑇|Ω|×^𝑇₀^𝑠₀𝑠−𝑞𝑣𝑑_𝑞𝑣𝑑_𝑝𝑠≤𝜓(‖𝑥‖)‖𝑧‖_𝐿¹𝑇²+𝜓𝑥𝑧‖1+𝑞(‖‖)‖_𝐿¹𝑝𝑇|Ω||𝛼|𝜂²+𝛽+𝛽𝜂()1+𝑞𝑟𝜂1+𝑞2+𝑟+𝜓𝑥𝑧‖1+𝑞(‖‖)‖_𝐿¹1+𝑝|𝛼|𝑇−𝜂+𝛽𝑇²1+𝑞1+𝑝+𝑝²|Ω|=𝜓(‖𝑥‖)‖𝑧‖_𝐿¹Λ. (45)
Consequently, ‖𝐴𝑥‖≤𝜓(‖𝑥‖)‖𝑧‖_𝐿¹Λ. (46)
Next, we will show that 𝐴 maps bounded sets into equicontinuous sets of 𝐶(𝐼^𝑇_𝑞,ℝ). Let 𝑡₁,𝑡₂∈𝐼^𝑇_𝑞 with 𝑡₁≤𝑡₂ and 𝑥∈𝐵_𝜌. Then, we have (𝐴𝑥)𝑡₂−(𝐴𝑥)𝑡₁≤^𝑡2₀𝑡₂−𝑞𝑠𝑓(𝑠,𝑥(𝑠))𝑑_𝑞𝑠−^𝑡1₀𝑡₁−𝑞𝑠𝑓(𝑠,𝑥(𝑠))𝑑_𝑞𝑠+1+𝑝𝑡₂−𝑡₁|Ω||𝛼|^𝜂₀𝜂−𝑞𝑠𝑓(𝑠,𝑥(𝑠))𝑑_𝑞𝑠+𝛽₀^𝑟𝜂𝑓(𝑠,𝑥(𝑠))𝑑_𝑞𝑠+𝛽(1−𝑟)𝜂^𝜂_𝑟𝜂×𝑓((𝑠𝑑𝜂−𝑞𝑠𝑠,𝑥))_𝑞𝑠+1+𝑝|𝛼|𝑡₂−𝑡₁𝑇|Ω|×^𝑇₀^𝑠₀𝑠−𝑞𝑣𝑓(𝑣,𝑥(𝑣))𝑑_𝑞𝑣𝑑_𝑝𝑠≤^𝑡1₀𝑡₂−𝑡₁𝑧(𝑠)𝜓𝜌𝑑_𝑞𝑠+^𝑡2_𝑡1𝑡₂−𝑞𝑠𝑧(𝑠)𝜓𝜌𝑑_𝑞𝑠+1+𝑝𝑡₂−𝑡₁|Ω||𝛼|^𝜂₀𝜂−𝑞𝑠𝑧(𝑠)𝜓𝜌𝑑_𝑞𝑠+𝛽₀^𝑟𝜂𝑧(𝑠)𝜓𝜌𝑑_𝑞𝛽𝑠+(1−𝑟)𝜂×^𝜂_𝑟𝜂𝜂−𝑞𝑠𝑧(𝑠)𝜓𝜌𝑑_𝑞𝑠+1+𝑝|𝛼|𝑡₂−𝑡₁𝑇|Ω|×^𝑇₀^𝑠₀𝑠−𝑞𝑣𝑧(𝑠)𝜓𝜌𝑑_𝑞𝑣𝑑_𝑝𝑠. (47) As 𝑡₂−𝑡₁→0, the right-hand side of the above inequality tends to zero independently of 𝑥∈𝐵_𝜌. As 𝐴 satisfies the above assumptions; therefore, it follows by the Arzelá-Ascoli theorem that 𝐴∶𝐶(𝐼^𝑇_𝑞,ℝ)→𝐶(𝐼^𝑇_𝑞,ℝ) is completely continuous.
Let 𝑥 be a solution. Then, for 𝑡∈𝐼^𝑇_𝑞 and following the similar computations as in the first step, we have |𝑥(𝑡)|≤𝜓(‖𝑥‖)‖𝑧‖_𝐿¹Λ. (48)
Consequently, we have ‖𝑥‖𝜓(‖𝑥‖)‖𝑧‖_𝐿¹Λ≤1. (49) In view of (𝐻₅), there exists 𝑀 such that ‖𝑥‖≠𝑀. Let us set 𝑈=𝑥∈𝐶𝐼^𝑇_𝑞,ℝ∶‖𝑥‖<𝑀. (50) Note that the operator 𝐴∶𝑈→𝐶(𝐼^𝑇_𝑞,ℝ) is continuous and completely continuous. From the choice of 𝑈, there is no 𝑥∈𝜕𝑈 such that 𝑥=𝜆𝐴𝑥 for some 𝜆∈(0,1). Consequently, by the nonlinear alternative of Leray-Schauder type (Theorem 9), we deduce that 𝐴 has a fixed point 𝑥∈𝑈 which is a solution of the problem (4). This completes the proof.

Finally, we prove that problem (4) has at least one solution on 𝐼^𝑇_𝑞 by using Leray-Schauder degree theory.

Theorem 11. Let 𝑓∶𝐼^𝑇_𝑞×ℝ→ℝ be a continuous function. Assume that: (H₇) there exist constants 0≤𝜅<Λ⁻¹, where Λ is given by (28) and 𝑁>0 such that |𝑓(𝑡,𝑥)|≤𝜅|𝑥|+𝑁 for all 𝑡∈𝐼^𝑇_𝑞, 𝑥∈𝒞. Then, the boundary value problem (4) has at least one solution.

Proof. Let us define an operator 𝐴∶𝒞→𝒞 as (27). We wish to prove that there exists at least one solution 𝑥∈𝒞 of the fixed point equation 𝑥=𝐴𝑥. (51) We define a ball 𝐵_𝑅⊂𝒞, with a constant radius 𝑅>0 given by 𝐵_𝑅=𝑥∈𝒞∶max_{𝑡∈𝐼𝑇𝑞}|𝑥(𝑡)|<𝑅. (52) Then, it is sufficient to show that 𝐴∶𝐵_𝑅→𝒞(𝐼^𝑇_𝑞) satisfies 𝑥≠𝜆𝐴𝑥,∀𝑥∈𝜕𝐵_𝑅,∀𝜆∈[0,1]. (53) Now, we set 𝐻(𝜆,𝑥)=𝜆𝐴𝑥,𝑥∈𝒞,𝜆∈[0,1]. (54) Then, by the Arzelá-Ascoli theorem, we conclude that a continuous map ℎ_𝜆 defined by ℎ_𝜆(𝑥)=𝑥−𝐻(𝜆,𝑥)=𝑥−𝜆𝐴𝑥 is completely continuous. If (53) holds, then the following Leray-Schauder degrees are well defined. From the homotopy invariance of topological degree, it follows that degℎ_𝜆,𝐵_𝑅,0=deg𝐼−𝜆𝐴,𝐵_𝑅,0=degℎ₁,𝐵_𝑅ℎ,0=deg₀,𝐵_𝑅,0=deg𝐼,𝐵_𝑅,0=1≠0,0∈𝐵_𝑅, (55) where 𝐼 denotes the unit operator. By the nonzero property of Leray-Schauder degree, ℎ₁(𝑥)=𝑥−𝐴𝑥=0 for at least one 𝑥∈𝐵_𝑅. Let us assume that 𝑥=𝜆𝐴𝑥 for some 𝜆∈[0,1]. Then, for all 𝑡∈𝐼^𝑇_𝑞, we obtain |𝑥(𝑡=|𝜆(𝑡≤)|𝐴𝑥)()|_𝑡0𝑡−𝑞𝑠𝑓(𝑠,𝑥(𝑠))𝑑_𝑞𝑠+1+𝑝𝑡−𝑇|Ω|×|𝛼|^𝜂₀𝜂−𝑞𝑠𝑓(𝑠,𝑥(𝑠))𝑑_𝑞𝑠+𝛽₀^𝑟𝜂𝑓(𝑠,𝑥(𝑠))𝑑_𝑞𝛽𝑠+()𝜂×1−𝑟_𝜂𝑟𝜂𝜂−𝑞𝑠𝑓(𝑠,𝑥(𝑠))𝑑_𝑞𝑠+1+𝑝𝛼𝑡−𝜂−𝛽𝑇|Ω|^𝑇₀^𝑠₀×𝑦((𝑣𝑑𝑠−𝑞𝑣𝑣,𝑥))_𝑞𝑣𝑑_𝑝𝑠≤(𝜅|𝑥|+𝑁)^𝑡₀𝑡−𝑞𝑠𝑑_𝑞(𝜅|𝑥|)×𝑠++𝑁1+𝑝𝑡−𝑇|Ω|×|𝛼|^𝜂₀𝜂−𝑞𝑠𝑑_𝑞𝑠+𝛽₀^𝑟𝜂𝑑_𝑞𝑠+𝛽(1−𝑟)𝜂^𝜂_𝑟𝜂𝜂−𝑞𝑠𝑑_𝑞𝑠+(𝜅|𝑥|)𝛼+𝑁1+𝑝𝑡−𝜂−𝛽𝑇|Ω|×^𝑇₀^𝑠₀𝑠−𝑞𝑣𝑑_𝑞𝑣𝑑_𝑝𝑠≤(𝜅|𝑥|𝑇+𝑁)²+1+𝑞𝑝𝑇|Ω||𝛼|𝜂²+𝛽+𝛽𝜂()1+𝑞𝑟𝜂1+𝑞2+𝑟+|𝛼+𝛽𝑇1+𝑞1+𝑝|𝑇−𝜂²1+𝑞1+𝑝+𝑝²Ω|=(𝜅|𝑥|)|+𝑁Λ. (56) Taking norm sup_{𝑡∈𝐼𝑇𝑞}|𝑥(𝑡)|=‖𝑥‖ and solving it for ‖𝑥‖, this yields ‖𝑥‖≤𝑁Λ.1−𝜅Λ (57) Let 𝑅=(𝑁Λ/(1−𝜅Λ))+1, then (53) holds. This completes the proof

4. Examples

In this section, we illustrate our main results with some examples. Let us consider the following boundary value problem of nonlinear second-order 𝑞-difference equations with three-point boundary conditions 𝐷²_1/2𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡)),𝑡∈𝐼^1/2_1/2=𝐼¹_1/2∩10,2,23𝑥18−13𝐷_3/4𝑥18=0,₀^1/2𝑥(𝑠)𝑑_1/4𝑠=0. (58) Here, we have 𝑞=1/2, 𝑝=1/4, 𝑟=3/4, 𝑇=1/2, 𝛼=2/3, 𝛽=−1/3, and 𝜂=1/8. We find that 𝑇Λ=²+1+𝑞𝑝𝑇|Ω||𝛼|𝜂²+𝛽𝛽𝜂()1+𝑞𝑟𝜂+1+𝑞2+𝑟+|𝛼|+𝛽𝑇1+𝑞1+𝑝𝑇−𝜂²1+𝑞1+𝑝+𝑝²|Ω|=16+6131+1144+3219+2882801953≈0.33019713. (59)(a) Let 𝑓∶𝐼^1/2_1/2×ℝ→ℝ be a continuous function given by 𝑓()=𝑒𝑡,𝑥^−sin2𝑡1+𝑒^cos2𝑡⋅|𝑥(𝑡)||𝑥(𝑡.1+)| (60) Since, |𝑓(𝑡,𝑥)−𝑓(𝑡,𝑦)|≤(1/2)|𝑥−𝑦|, then (𝐻₁) is satisfied with 𝐿=1/2. We can find that 𝐿Λ≈0.16509857<1. (61) Hence, by Theorem 3, problem (58) with 𝑓(𝑡,𝑥) given by (60) has a unique solution on 𝐼^1/2_1/2.(b) If 𝑓∶𝐼^1/2_1/2×ℝ→ℝ is a continuous function given by 𝑓()=(𝑥|𝑡,𝑥𝑡+1)||𝑥|.1+ (62) Choosing ℎ(𝑡)=𝑡+1, we find that 𝐺=^𝑇₀𝑇−𝑞𝑠ℎ(𝑠)𝑑_𝑞𝑠+𝑝𝑇|Ω|×|𝛼|^𝜂₀𝜂−𝑞𝑠ℎ(𝑠)𝑑_𝑞𝑠+𝛽₀^𝑟𝜂(𝑠+1)𝑑_𝑞𝛽𝑠+()𝜂×1−𝑟_𝜂𝑟𝜂𝜂−𝑞𝑠(𝑠+1)𝑑_𝑞𝑠+1+𝑝|𝛼|𝑇−𝜂+𝛽𝑇|Ω|×^𝑇₀^𝑠₀𝑠−𝑞𝑣(𝑣+1)𝑑_𝑞𝑣𝑑_𝑝𝑠≈0.40987235. (63) Here, 𝑓()≤(𝑡,𝑥−𝑓𝑡,𝑦1+𝑡)𝑥−𝑦.0.40987235+𝑥−𝑦 (64) Therefore, by Theorem 6, the problem (58) with 𝑓(𝑡,𝑥) given by (62) has a unique solution on 𝐼^1/2_1/2.(c) Consider a continuous function 𝑓∶𝐼^1/2_1/2×ℝ→ℝ given by 𝑓()3𝑡,𝑥=sin2𝑥+𝑒^−𝑥2.+𝑡+2 (65) We can show that 𝑓(=3𝑡,𝑥)sin2𝑥+𝑒^−𝑥2‖𝑥‖+3+𝑡+2≤22, (66) with 1𝜅=2<Λ≈3.02849389, (67) and 𝑁=3/2. By Theorem 11, the problem (58) with the 𝑓(𝑡,𝑥) given by (65) has at least one solution on 𝐼^1/2_1/2.

Acknowledgment

This research of T. Sitthiwirattham and J. Tariboon is supported by King Mongkut’s University of Technology North Bangkok, Thailand. Sotiris K. Ntouyas is a Member of Nonlinear Analysis and Applied Mathematics (NAAM), Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.