Sequential Derivatives of Nonlinear 𝑞-Difference Equations with Three-Point 𝑞-Integral Boundary Conditions

Abstract

This paper studies sufficient conditions for the existence of solutions to the problem of sequential derivatives of nonlinear 𝑞-difference equations with three-point 𝑞-integral boundary conditions. Our results are concerned with several quantum numbers of derivatives and integrals. By using Banach's contraction mapping, Krasnoselskii's fixed-point theorem, and Leray-Schauder degree theory, some new existence results are obtained. Two examples illustrate our results.

1. Introduction

The study of 𝑞-calculus or quantum calculus was initiated by the pioneer works of Jackson [1], Carmichael [2], Mason [3], Adams [4], Trjitzinsky [5], and so forth. Since then, in the last few decades, this subject has evolved into a multidisciplinary research area with many applications; for example, see [6–14]. For some recent works, we refer the reader to [15–21] and references therein. However, the theory of boundary value problems for nonlinear 𝑞-difference equations is still in the beginning stages and it needs to be explored further.

In [22], Ahmad investigated the existence of solutions for a nonlinear boundary value problem of third-order 𝑞-difference equation: 𝐷³_𝑞𝑢(𝑡)((𝑡𝑢(0)=𝑓𝑡,𝑢)),0≤𝑡≤1,=0,𝐷_𝑞𝑢(0)=0,𝑢(1)=0. (1) Using Leray-Schauder degree theory and standard fixed-point theorems, some existence results were obtained. Moreover, he showed that if 𝑞→1, then his results corresponded to the classical results. Ahmad et al. [23] studied a boundary value problem of a nonlinear second-order 𝑞-difference equation with nonseparated boundary conditions 𝐷²_𝑞𝑢(𝑡)((𝑡[],𝑢(0)(𝑇)=𝑓𝑡,𝑢)),𝑡∈0,𝑇=𝜂𝑢,𝐷_𝑞𝑢(0)=𝜂𝐷_𝑞𝑢(𝑇). (2) They proved the existence and uniqueness theorems of the problem (2) using the Leray-Schauder nonlinear alternative and some standard fixed-point theorems. For some very recent results on nonlocal boundary value problems of nonlinear 𝑞-difference equations and inclusions, see [24–26].

In this paper, we discuss the existence of solutions for the following nonlinear 𝑞-difference equation with three-point integral boundary condition: 𝐷_𝑞𝐷_𝑝𝑥(𝑡)((𝑡[],𝑥(0)+𝜆=𝑓𝑡,𝑥)),𝑡∈0,𝑇=0,𝛽^𝜂₀𝑥(𝑠)𝑑_𝑟𝑠=𝑥(𝑇), (3) where 0<𝑝, 𝑞, 𝑟<1, 𝑓∈𝐶([0,𝑇]×ℝ, ℝ), 𝛽≠𝑇(1+𝑟)/𝜂², 𝜂∈(0,𝑇) is a fixed point, and 𝜆 is a given constant.

The aim of this paper is to prove some existence and uniqueness results for the boundary value problem (3). Our results are based on Banach’s contraction mapping, Krasnoselskii's fixed-point theorem, and Leray-Schauder degree theory. Since the problem (3) has different values of the quantum numbers of the 𝑞-derivative and the 𝑞-integral, the existence results of such problem are also new.

2. Preliminaries

Let us recall some basic concepts of quantum calculus [15].

For 0<𝑞<1, we define the 𝑞-derivative of a real-valued function 𝑓 as 𝐷_𝑞𝑓(𝑡)=𝑓(𝑡)−𝑓𝑞𝑡1−𝑞𝑡,𝐷_𝑞𝑓(0)=lim_𝑡→0𝐷_𝑞𝑓(𝑡). (4) 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. (5) The 𝑞-integral of a function 𝑓 defined on the interval [0,𝑇] is given by ^𝑡_𝑎𝑓(𝑠)𝑑_𝑞𝑠∶=^∞_𝑛=01−𝑞𝑞^𝑛×𝑡𝑓𝑡𝑞^𝑛−𝑎𝑓𝑞^𝑛𝑎,𝑡∈[0,𝑇], (6) and for 𝑎 = 0, we denote 𝐼_𝑞𝑓(𝑡)=^𝑡₀𝑓(𝑠)𝑑_𝑞𝑠=^∞_𝑛=0𝑡1−𝑞𝑞^𝑛𝑓𝑡𝑞^𝑛, (7) provided the series converges. If 𝑎∈[0,𝑇] and 𝑓 is defined on the interval [0,𝑇], then ^𝑏_𝑎𝑓(𝑠)𝑑_𝑞𝑠=^𝑏₀𝑓(𝑠)𝑑_𝑞𝑠−^𝑎₀𝑓(𝑠)𝑑_𝑞𝑠. (8) Similarly, we have 𝐼⁰_𝑞𝑓(𝑡)=𝑓(𝑡),𝐼^𝑛_𝑞𝑓(𝑡)=𝐼_𝑞𝐼_𝑞^𝑛−1𝑓(𝑡),𝑛∈ℕ. (9) Observe that 𝐷_𝑞𝐼_𝑞𝑓(𝑡)=𝑓(𝑡), (10) and if 𝑓 is continuous at 𝑡=0, then 𝐼_𝑞𝐷_𝑞𝑓(𝑡)=𝑓(𝑡)−𝑓(0). (11) In 𝑞-calculus, the product rule and integration by parts formula are 𝐷_𝑞𝑔ℎ(𝑡)=𝐷_𝑞𝑔(𝑡)ℎ(𝑡)+𝑔𝑞𝑡𝐷_𝑞ℎ(𝑡),_𝑡0𝑓(𝑠)𝐷_𝑞𝑔(𝑠)𝑑_𝑞𝑠=𝑓(𝑠)𝑔(𝑠)^𝑡₀−^𝑡₀𝐷_𝑞𝑓(𝑠)𝑔𝑞𝑠𝑑_𝑞𝑠. (12) In the limit 𝑞→1, the 𝑞-calculus corresponds to the classical calculus. The above results are also true for quantum numbers 𝑝, 𝑟 such that 0<𝑝<1 and 0<𝑟<1.

Lemma 1. Let 𝑇(1+𝑟)≠𝛽𝜂², 0<𝑝, 𝑞, 𝑟<1, and let 𝜆 be a constant. Then for any ℎ∈𝐶[0,𝑇], the boundary value problem 𝐷_𝑞𝐷_𝑝+𝜆𝑥(𝑡)=ℎ(𝑡),𝑡∈[0,𝑇],𝑥(0)=0,𝛽^𝜂₀𝑥(𝑠)𝑑_𝑟𝑠=𝑥(𝑇),0<𝜂<𝑇, (13)(14) is equivalent to the integral equation 𝑥(𝑡)=^𝑡₀^𝑠₀ℎ(𝑢)𝑑_𝑞𝑢𝑑_𝑝𝑠−𝜆^𝑡₀𝑥(𝑠)𝑑_𝑝𝑠+𝛽(1+𝑟)𝑡𝑇(1+𝑟)−𝛽𝜂²×^𝜂₀^𝑣₀^𝑠₀ℎ(𝑢)𝑑_𝑞𝑢−𝜆𝑥(𝑠)𝑑_𝑝𝑠𝑑_𝑟𝑣+𝜆(1+𝑟)𝑡𝑇(1+𝑟)−𝛽𝜂^2𝑇₀𝑥(𝑠)𝑑_𝑝𝑠−(1+𝑟)𝑡𝑇(1+𝑟)−𝛽𝜂^2𝑇₀^𝑠₀ℎ(𝑢)𝑑_𝑞𝑢𝑑_𝑝𝑠. (15)

Proof. For 𝑡∈[0,𝑇], 𝑞-integrating (13) from 0 to 𝑡, we obtain 𝐷_𝑝+𝜆𝑥(𝑡)=^t₀ℎ(𝑠)𝑑_𝑞𝑠+𝑐₁. (16) Equation (16) can be written as 𝐷_𝑝𝑥(𝑡)=^𝑡₀ℎ(𝑠)𝑑_𝑞𝑠−𝜆𝑥(𝑡)+𝑐₁. (17) For 𝑡∈[0,𝑇], 𝑝-integrating (17) from 0 to 𝑡, we have 𝑥(𝑡)=^𝑡₀^𝑠₀ℎ(𝑢)𝑑_𝑞𝑢𝑑_𝑝𝑠−𝜆^𝑡₀𝑥(𝑠)𝑑_𝑝𝑠+𝑐₁𝑡+𝑐₂. (18) From the first condition of (14), it follows that 𝑐₂=0. For 𝑡∈[0,𝑇], 𝑟-integrating equation (18) from 0 to 𝑡, we get ^𝑡₀𝑥(𝑣)𝑑_𝑟𝑣=^𝑡₀^𝑣₀^𝑠₀ℎ(𝑢)𝑑_𝑞𝑢𝑑_𝑝𝑠𝑑_𝑟𝑣−𝜆^𝑡₀^𝑣₀𝑥(𝑠)𝑑_𝑝𝑠𝑑_𝑟𝑣+𝑐₁𝑡².1+𝑟 (19) The second boundary condition (14) implies that 𝛽^𝜂₀𝑥(𝑣)𝑑_𝑟𝑣=𝛽^𝜂₀^𝑣₀^𝑠₀ℎ(𝑢)𝑑_𝑞𝑢𝑑_𝑝𝑠𝑑_𝑟𝑣−𝛽𝜆^𝜂₀^𝑣₀𝑥(𝑠)𝑑_𝑝𝑠𝑑_𝑟𝑣+𝑐₁𝛽𝜂²1+𝑟=𝛽^𝜂₀^𝑣₀^𝑠₀ℎ(𝑢)𝑑_𝑞𝑢−𝜆𝑥(𝑠)𝑑_𝑝𝑠𝑑_𝑟𝑣+𝑐₁𝛽𝜂²=1+𝑟_𝑇0𝑠0ℎ(𝑢)𝑑_𝑞𝑢𝑑_𝑝𝑠−𝜆^𝑇₀𝑥(𝑠)𝑑_𝑝𝑠+𝑐₁𝑇. (20) Therefore, 𝑐₁=𝛽(1+𝑟)𝑇(1+𝑟)−𝛽𝜂^2𝜂₀^𝑣₀^𝑠₀ℎ(𝑢)𝑑_𝑞𝑢−𝜆𝑥(𝑠)𝑑_𝑝𝑠𝑑_𝑟𝑣+𝜆(1+𝑟)𝑇(1+𝑟)−𝛽𝜂^2𝑇₀𝑥(𝑠)𝑑_𝑝𝑠−1+𝑟𝑇(1+𝑟)−𝛽𝜂^2𝑇₀^𝑠₀ℎ(𝑢)𝑑_𝑞𝑢𝑑_𝑝𝑠. (21) Substituting the values of 𝑐₁ and 𝑐₂ in (18), we obtain (15). This completes the proof.

For the forthcoming analysis, let 𝒞=𝐶([0,𝑇],ℝ) denote the Banach space of all continuous functions from [0,𝑇] to ℝ endowed with the norm defined by ‖𝑥‖=sup{|𝑥(𝑡)|,𝑡∈[0,𝑇]}.

In the following, for the sake of convenience, we set 1Ω=×𝑇1+𝑝₂+𝛽𝑇𝜂³(1+𝑟)𝑇(1+𝑟)−𝛽𝜂²1+𝑟+𝑟²+𝑇³(1+𝑟)𝑇(1+𝑟)−𝛽𝜂²,|𝜆|𝛽𝜆|Φ=𝑇+|𝑇𝜂²𝑇(1+𝑟)−𝛽𝜂²+|𝜆|𝑇²(1+𝑟)𝑇(1+𝑟)−𝛽𝜂². (22)(23)

3. Main Results

Now, we are in the position to establish the main results. We transform the boundary value problem (3) into a fixed-point problem. In view of Lemma 1, for 𝑡∈[0,𝑇], 𝑥∈𝒞, we define the operator 𝐴∶𝒞→𝒞 as (𝑡)=𝐴𝑥)(_𝑡0𝑠0𝑓(𝑢,𝑥(𝑢))𝑑_𝑞𝑢𝑑_𝑝𝑠−𝜆^𝑡₀𝑥(𝑠)𝑑_𝑝𝑠+𝛽(1+𝑟)𝑡𝑇(1+𝑟)−𝛽𝜂²×^𝜂₀^𝑣₀^𝑠₀𝑓(𝑢,𝑥(𝑢))𝑑_𝑞𝑢−𝜆𝑥(𝑠)𝑑_𝑝𝑠𝑑_𝑟𝑣+𝜆(1+𝑟)𝑡𝑇(1+𝑟)−𝛽𝜂^2𝑇₀𝑥(𝑠)𝑑_𝑝𝑠−(1+𝑟)𝑡𝑇(1+𝑟)−𝛽𝜂^2𝑇₀^𝑠₀𝑓(𝑢,𝑥(𝑢))𝑑_𝑞𝑢𝑑_𝑝𝑠. (24) Note that the problem (3) has solutions if and only if the operator equation 𝐴𝑥=𝑥 has fixed points.

Our first result is based on Banach’s fixed-point theorem.

Theorem 2. Assume that 𝑓∶[0,𝑇]×ℝ→ℝ is a jointly continuous function satisfying the conditions (𝐻₁)|𝑓(𝑡,𝑥)−𝑓(𝑡,𝑦)|≤𝐿|𝑥−𝑦|, for all 𝑡∈[0,𝑇], 𝑥,𝑦∈ℝ; (𝐻₂)Λ∶=(Φ+𝐿Ω)<1,where 𝐿 is a Lipschitz constant, Ω and Φ are defined by (22) and (23), respectively.
Then, the boundary value problem (3) has a unique solution.

Proof. Assume that sup_{𝑡∈[0,𝑇]}|𝑓(𝑡,0)|=𝑀₀; we choose a constant 𝑀𝑅≥⁰Ω.1−Λ (25) 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∈0,𝑇}^𝑡₀^𝑠₀𝑓(𝑢,𝑥(𝑢))𝑑_𝑞𝑢𝑑_𝑝𝑠−𝜆^𝑡₀𝑥(𝑠)𝑑_𝑝𝑠+𝛽(1+𝑟)𝑡𝑇(1+𝑟)−𝛽𝜂²×^𝜂₀^𝑣₀^𝑠₀𝑓(𝑢,𝑥(𝑢))𝑑_𝑞𝑢−𝜆𝑥(𝑠)𝑑_𝑝𝑠𝑑_𝑟𝑣+𝜆(1+𝑟)𝑡𝑇(1+𝑟)−𝛽𝜂^2𝑇₀𝑥(𝑠)𝑑_𝑝𝑠−(1+𝑟)𝑡𝑇(1+𝑟)−𝛽𝜂^2𝑇₀^𝑠₀𝑓(𝑢,𝑥(𝑢))𝑑_𝑞𝑢𝑑_𝑝𝑠≤sup_{[]𝑡∈0,𝑇}^𝑡₀^𝑠₀𝑓(𝑢,𝑥(𝑢))−𝑓(𝑢,0)+𝑓(𝑢,0)𝑑_𝑞𝑢𝑑_𝑝𝑠+|𝜆|^𝑡₀|𝑥(𝑠)|𝑑_𝑝𝑠+𝛽(1+𝑟)𝑡𝑇(1+𝑟)−𝛽𝜂²×^𝜂₀^𝑣₀^𝑠₀𝑓(𝑢,𝑥(𝑢))−𝑓(𝑢,0)+𝑓(𝑢,0)×𝑑_𝑞𝑢𝑑_𝑝𝑠𝑑_𝑟𝑣+|𝜆|^𝜂₀^𝑣₀|𝑥(𝑠)|𝑑_𝑝𝑠𝑑_𝑟𝑣+|𝜆|(1+𝑟)𝑡𝑇(1+𝑟)−𝛽𝜂^2𝑇₀|𝑥(𝑠)|𝑑_𝑝𝑠+(1+𝑟)𝑡𝑇(1+𝑟)−𝛽𝜂²×^𝑇₀^𝑠₀𝑓((𝑢(+𝑓(𝑑𝑢,𝑥))−𝑓𝑢,0)𝑢,0)_𝑞𝑢𝑑_𝑝𝑠≤sup_{[]𝑡∈0,𝑇}^𝑡₀^𝑠₀𝐿|𝑥(𝑢)|+𝑓(𝑢,0)𝑑_𝑞𝑢𝑑_𝑝𝑠+|𝜆|^𝑡₀|𝑥(𝑠)|𝑑_𝑝𝛽)𝑡𝑠+(1+𝑟𝑇(1+𝑟)−𝛽𝜂²×^𝜂₀^𝑣₀^𝑠₀𝐿|𝑥(𝑢)|+𝑓(𝑢,0)𝑑_𝑞𝑢𝑑_𝑝𝑠𝑑_𝑟𝑣+|𝜆|^𝜂₀^𝑣₀|𝑥(𝑠)|𝑑_𝑝𝑠𝑑_𝑟𝑣+|𝜆|(1+𝑟)𝑡𝑇(1+𝑟)−𝛽𝜂^2𝑇₀|𝑥(𝑠)|𝑑_𝑝𝑠+(1+𝑟)𝑡𝑇(1+𝑟)−𝛽𝜂²×^𝑇₀^𝑠₀𝐿|𝑥(𝑢)|+𝑓(𝑢,0)𝑑_𝑞𝑢𝑑_𝑝𝑠≤sup_{[]𝑡∈0,𝑇}𝐿‖𝑥‖+𝑀₀^𝑡₀^𝑠₀𝑑_𝑞𝑢𝑑_𝑝𝑠+|𝜆|‖𝑥‖^𝑡₀𝑑_𝑝𝛽)𝑡𝑠+(1+𝑟𝑇(1+𝑟)−𝛽𝜂²×𝐿‖𝑥‖+𝑀₀^𝜂₀^𝑣₀^𝑠₀𝑑_𝑞𝑢𝑑_𝑝𝑠𝑑_𝑟𝑣+|𝜆|‖𝑥‖^𝜂₀^𝑣₀𝑑_𝑝𝑠𝑑_𝑟𝑣+|𝜆|‖𝑥‖(1+𝑟)𝑡𝑇(1+𝑟)−𝛽𝜂^2𝑇₀𝑑_𝑝𝑠+(1+𝑟)𝑡𝑇(1+𝑟)−𝛽𝜂²𝐿‖𝑥‖+𝑀₀^𝑇₀^𝑠₀𝑑_𝑞𝑢𝑑_𝑝𝑠=sup_{[]𝑡∈0,𝑇}𝐿‖𝑥‖+𝑀₀𝑡²+|𝜆𝑥‖𝑡+𝛽)𝑡1+𝑝|‖(1+𝑟𝑇(1+𝑟)−𝛽𝜂²×𝐿‖𝑥‖+𝑀₀(1−𝑟)𝜂³1+𝑝1−𝑟³+|𝜆𝑥‖𝜂|‖²+|𝜆𝑥)𝑡1+𝑟|‖‖(1+𝑟𝑇(1+𝑟)−𝛽𝜂²𝑇+(1+𝑟)𝑡𝑇(1+𝑟)−𝛽𝜂²𝐿‖𝑥‖+𝑀₀𝑇²1+𝑝=sup_{𝑡∈[0,𝑇]}𝑥‖|𝜆|𝛽𝜆|𝜂‖𝑡+|²𝑡𝑇(1+𝑟)−𝛽𝜂²+|𝜆|𝑇(1+𝑟)𝑡𝑇(1+𝑟)−𝛽𝜂²+𝐿‖𝑥‖+𝑀₀×𝑡1+𝑝₂+𝛽(1+𝑟)𝜂³𝑡𝑇(1+𝑟)−𝛽𝜂²1+𝑟+𝑟²+𝑇²(1+𝑟)𝑡𝑇(1+𝑟)−𝛽𝜂²𝜆|𝛽𝜆|≤𝑅|𝑇+|𝑇𝜂²𝑇(1+𝑟)−𝛽𝜂²+|𝜆|𝑇²(1+𝑟)𝑇(1+𝑟)−𝛽𝜂²+𝐿𝑅+𝑀₀×𝑇1+𝑝₂+𝛽𝑇𝜂³(1+𝑟)𝑇(1+𝑟)−𝛽𝜂²1+𝑟+𝑟²+𝑇³(1+𝑟)𝑇(1+𝑟)−𝛽𝜂²=𝑅Φ+(𝐿𝑅+𝑀)Ω≤𝑅. (26)
Next, we will show that 𝐴 is a contraction. For any 𝑥,𝑦∈𝒞 and for each 𝑡∈[0,𝑇], we have (𝐴𝑥)−𝐴𝑦=sup_{[]𝑡∈0,𝑇}(𝐴𝑥)(𝑡)−𝐴𝑦(𝑡)=sup_{𝑡∈[0,𝑇]}^𝑡₀^𝑠₀𝑓(𝑢,𝑥(𝑢))−𝑓𝑢,𝑦(𝑢)𝑑_𝑞𝑢𝑑_𝑝𝑠−𝜆^𝑡₀𝑥(𝑠)−𝑦(𝑠)𝑑_𝑝𝑠+𝛽(1+𝑟)𝑡𝑇(1+𝑟)−𝛽𝜂²×^𝜂₀^𝑣₀^𝑠₀𝑓(𝑢,𝑥(𝑢))−𝑓𝑢,𝑦(𝑢)𝑑_𝑞𝑢−𝜆𝑥(𝑠)−𝑦(𝑠)𝑑_𝑝𝑠𝑑_𝑟𝑣+𝜆(1+𝑟)𝑡𝑇(1+𝑟)−𝛽𝜂^2𝑇₀𝑥(𝑠)−𝑦(𝑠)𝑑_𝑞𝑠−(1+𝑟)𝑡𝑇(1+𝑟)−𝛽𝜂²×^𝑇₀^𝑠₀𝑓(𝑢,𝑥(𝑢))−𝑓𝑢,𝑦(𝑢)𝑑_𝑞𝑢𝑑_𝑝𝑠≤sup_{𝑡∈[0,𝑇]}𝐿𝑥−𝑦^𝑡₀^𝑠₀𝑑_𝑞𝑢𝑑_𝑝𝑠+|𝜆|𝑥−𝑦^𝑡₀𝑑_𝑝𝑠+𝛽(1+𝑟)𝑡𝑇(1+𝑟)−𝛽𝜂²×𝐿𝑥−𝑦^𝜂₀^𝑣₀^𝑠₀𝑑_𝑞𝑢𝑑_𝑝𝑠𝑑_𝑟𝑣+|𝜆|𝑥−𝑦^𝜂₀^𝑣₀𝑑_𝑝𝑠𝑑_𝑟𝑣+|𝜆|(1+𝑟)𝑡𝑇(1+𝑟)−𝛽𝜂²×𝑥−𝑦_𝑇0𝑑_𝑞()𝑡𝑠+1+𝑟𝑇(1+𝑟)−𝛽𝜂²×𝐿𝑥−𝑦^𝑇₀^𝑠₀𝑑_𝑞𝑢𝑑_𝑝𝑠=sup_{𝑡∈[0,𝑇]}𝐿𝑡𝑥−𝑦²+|𝜆𝑡+𝛽)𝑡1+𝑝|𝑥−𝑦(1+𝑟𝑇(1+𝑟)−𝛽𝜂²×𝐿()𝜂𝑥−𝑦1−𝑟³1+𝑝1−𝑟³+|𝜆𝜂|𝑥−𝑦²+|𝜆()𝑡1+𝑟|𝑥−𝑦1+𝑟𝑇(1+𝑟)−𝛽𝜂²𝑇+(1+𝑟)𝑡𝑇(1+𝑟)−𝛽𝜂²𝐿𝑇𝑥−𝑦²≤|𝜆|𝛽𝜆|1+𝑝𝑥−𝑦𝑇+|𝑇𝜂²𝑇(1+𝑟)−𝛽𝜂²+|𝜆|𝑇²(1+𝑟)𝑇(1+𝑟)−𝛽𝜂²+𝐿𝑥−𝑦𝑇1+𝑝²+𝛽𝜂³𝑇(1+𝑟)𝑇(1+𝑟)−𝛽𝜂²1+𝑟+𝑟²+𝑇³(1+𝑟)𝑇(1+𝑟)−𝛽𝜂²=(.Φ+𝐿Ω)𝑥−𝑦≤Λ𝑥−𝑦 (27) Since Λ<1, 𝐴 is a contraction. Thus, the conclusion of the theorem follows by Banach's contraction mapping principle. This completes the proof.

Our second result is based on the following Krasnoselskii’s fixed-point theorem [27].

Theorem 3. 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 4. Assume that (𝐻₁) and (𝐻₂) hold with
(𝐻₃)|𝑓(𝑡,𝑥)|≤𝜇(𝑡), for all (𝑡,𝑥)∈[0,𝑇]×ℝ, with 𝜇∈𝐿¹([0,𝑇],ℝ⁺).
If 𝛽|𝜆|𝑇𝜂²𝑇(1+𝑟)−𝛽𝜂²+|𝜆|𝑇²(1+𝑟)𝑇(1+𝑟)−𝛽𝜂²+1𝛽𝜂1+𝑝³𝑇(1+𝑟)𝑇(1+𝑟)−𝛽𝜂²1+𝑟+𝑟²+𝑇³(1+𝑟)𝑇(1+𝑟)−𝛽𝜂²<1, (28) then the boundary value problem (3) has at least one solution on [0,𝑇].

Proof. Setting max_{𝑡∈[0,𝑇]}|𝜇(𝑡)|=||𝜇|| and choosing a constant 𝜇Ω𝑅≥,1−Φ (29) where Ω and Φ are given by (22) and (23), respectively, we consider that 𝐵_𝑅={𝑥∈𝒞∶||𝑥||≤𝑅}.
In view of Lemma 1, we define the operators 𝐹₁ and 𝐹₂ on the set 𝐵_𝑅 as 𝐹₁𝑥(𝑡)=^𝑡₀^𝑠₀𝑓(𝑢,𝑥(𝑢))𝑑_𝑞𝑢𝑑_𝑝𝑠−𝜆^𝑡₀𝑥(𝑠)𝑑_𝑝𝐹𝑠,₂𝑥(𝑡)=𝛽(1+𝑟)𝑡𝑇(1+𝑟)−𝛽𝜂²×^𝜂₀^𝑣₀^𝑠₀𝑓(𝑢,𝑥(𝑢))𝑑_𝑞𝑢−𝜆𝑥(𝑠)𝑑_𝑝𝑠𝑑_𝑟𝑣+𝜆(1+𝑟)𝑡𝑇(1+𝑟)−𝛽𝜂^2𝑇₀𝑥(𝑠)𝑑_𝑝𝑠−(1+𝑟)𝑡𝑇(1+𝑟)−𝛽𝜂^2𝑇₀^𝑠₀𝑓(𝑢,𝑥(𝑢))𝑑_𝑞𝑢𝑑_𝑝𝑠, (30) for 𝑥,𝑦∈𝐵_𝑅. By computing directly, we have 𝐹₁𝑥+𝐹₂𝑦≤𝜇^𝑡₀^𝑠₀𝑑_𝑞𝑢𝑑_𝑝𝑠+|𝜆|‖𝑥‖^𝑡₀𝑑_𝑝𝑠+𝛽(1+𝑟)𝑡𝑇(1+𝑟)−𝛽𝜂²×𝜇^𝜂₀^𝑣₀^𝑠₀𝑑_𝑞𝑢𝑑_𝑝𝑠𝑑_𝑟𝑣+|𝜆|𝑦^𝜂₀^𝑣₀𝑑_𝑝𝑠𝑑_𝑟𝑣+|𝜆|(1+𝑟)𝑡𝑇(1+𝑟)−𝛽𝜂²𝑦^𝑇₀𝑑_𝑝𝑠+(1+𝑟)𝑡𝑇(1+𝑟)−𝛽𝜂²𝜇^𝑇₀^𝑠₀𝑑_𝑞𝑢𝑑_𝑝𝜇𝑠,≤𝑅Φ+Ω≤𝑅. (31) Therefore, (𝐹₁𝑥)+(𝐹₂y)∈𝐵_𝑅. The condition (28) 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)∈[0,𝑇]×𝐵𝑅}|𝑓(𝑡,𝑥)|=𝑓_max<∞. For 𝑡₁,𝑡₂∈[0,𝑇] with 𝑡₂<𝑡₁ and 𝑥∈𝐵_𝑅, we have 𝐹₁𝑥𝑡₁−𝐹₂𝑥𝑡₂≤sup_{()∈[]𝑡,𝑥0,𝑇×𝐵𝑅}^𝑡1₀^𝑠₀𝑓(𝑢,𝑥(𝑢))𝑑_𝑞𝑢𝑑_𝑝𝑠−𝜆^𝑡1₀𝑥(𝑠)𝑑_𝑝𝑠−^𝑡2₀^𝑠₀𝑓(𝑢,𝑥(𝑢))𝑑_𝑞𝑢𝑑_𝑝𝑠+𝜆^𝑡2₀𝑥(𝑠)𝑑_𝑝𝑠=sup_{()∈[]𝑡,𝑥0,𝑇×𝐵𝑅}^𝑡1_𝑡2^𝑠₀𝑓(𝑢,𝑥(𝑢))𝑑_𝑞𝑢𝑑_𝑝𝑠−𝜆^𝑡1_𝑡2𝑥(𝑠)𝑑_𝑝𝑠≤𝑓_max𝑡²₁−𝑡²₂+|𝜆|𝑡1+𝑝¹−𝑡₂𝑅. (32) Actually, as 𝑡₁−𝑡₂→0, the right-hand side of the above inequality tends to 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 5 are satisfied and the conclusion of Theorem 5 implies that the three-point integral boundary value problem (3) has at least one solution on [0,𝑇]. This completes the proof.

As the third result, we prove the existence of solutions of (3) by using Leray-Schauder degree theory.

Theorem 5. Let 𝑓∶[0,𝑇]×ℝ→ℝ. Assume that there exist constants 0≤𝜅<(1−Φ)Ω⁻¹, where Ω and Φ are given by (22) and (23), respectively, and 𝑀>0 such that |𝑓(𝑡,𝑥)|≤𝜅|𝑥|+𝑀 for all 𝑡∈[0,𝑇], 𝑥∈𝒞. Then, the boundary value problem (3) has at least one solution.

Proof. Let us define an operator 𝐴∶𝒞→𝒞 as in (24). We will prove that there exists at least one solution 𝑥∈𝒞 of the fixed-point equation 𝑥=𝐴𝑥. (33) We define a ball 𝐵_𝑅⊂𝒞, with a constant radius 𝑅>0, given by 𝐵_𝑅=𝑥∈𝒞∶max_{[]𝑡∈0,𝑇}|𝑥(𝑡)|<𝑅. (34) Then, it is sufficient to show that 𝐴∶𝐵_𝑅→𝒞 satisfies 𝑥≠𝜃𝐴𝑥,∀𝑥∈𝜕𝐵_𝑅,∀𝜃∈[0,1]. (35) Now, we set 𝐻(𝜃,𝑥)=𝜃𝐴𝑥,𝑥∈𝒞,𝜃∈[0,1]. (36) Then, by the Arzelá-Ascoli theorem, we get that ℎ_𝜃(𝑥)=𝑥−𝐻(𝜃,𝑥)=𝑥−𝜃𝐴𝑥 is completely continuous. If (35) holds, then the following Leray-Schauder degrees are well defined. From the homotopy invariance of topological degree, it follows that degℎ_𝜃,𝐵_𝑅,0=deg𝐼−𝜃𝐴,𝐵_𝑅,0=degℎ,𝐵_𝑅ℎ,0=deg₀,𝐵_𝑅,0=deg𝐼,𝐵_𝑅,0=1≠0,0∈𝐵_𝑅, (37) 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=𝜃𝐴𝑥 for some 𝜃∈[0,1]. Then, for all 𝑡∈[0,𝑇], we obtain |𝑥(𝑡)|=|𝜃(𝐴𝑥)(𝑡)|≤^𝑡₀^𝑠₀𝑓(𝑢,𝑥(𝑢))𝑑_𝑞𝑢𝑑_𝑝𝑠+|𝜆|^𝑡₀|𝑥(𝑠)|𝑑_𝑝𝑠+𝛽(1+𝑟)𝑡𝑇(1+𝑟)−𝛽𝜂²×^𝜂₀^𝑣₀^𝑠₀𝑓(𝑢,𝑥(𝑢))𝑑_𝑞𝑢+|𝜆𝑥(𝑠)|𝑑_𝑝𝑠𝑑_𝑟𝑣+|𝜆|(1+𝑟)𝑡𝑇(1+𝑟)−𝛽𝜂²×^𝑇₀|𝑥(𝑠)|𝑑_𝑝()𝑡𝑠+1+𝑟𝑇(1+𝑟)−𝛽𝜂²×^𝑇₀^𝑠₀𝑓(𝑢,𝑥(𝑢))𝑑_𝑞𝑢𝑑_𝑝𝑠≤(𝜅|𝑥|+𝑀)^𝑡₀^𝑠₀𝑑_𝑞𝑢𝑑_𝑝𝑠+|𝜆||𝑥|^𝑡₀𝑑_𝑝𝑠+𝛽(1+𝑟)𝑡𝑇(1+𝑟)−𝛽𝜂²×(𝜅|𝑥|+𝑀)^𝜂₀^𝑣₀^𝑠₀𝑑_𝑞𝑢𝑑_𝑝𝑠𝑑_𝑟𝑣+|𝜆||𝑥|^𝜂₀^𝑣₀𝑑_𝑝𝑠𝑑_𝑟𝑣+|𝜆||𝑥|(1+𝑟)𝑡𝑇(1+𝑟)−𝛽𝜂²×^𝑇₀𝑑_𝑝()𝑡𝑠+1+𝑟𝑇(1+𝑟)−𝛽𝜂²(𝜅|𝑥|)×+𝑀_𝑇0𝑠0𝑑_𝑞𝑢𝑑_𝑝𝑠≤(𝜅|𝑥|)𝑇+𝑀²+|𝜆𝑥|𝑇+𝛽)𝑇1+𝑝||(1+𝑟𝑇(1+𝑟)−𝛽𝜂²×(𝜅|𝑥|)()𝜂+𝑀1−𝑟³1+𝑝1−𝑟³+|𝜆𝑥|𝜂||²+|𝜆𝑥)𝑇1+𝑟|||(1+𝑟²𝑇(1+𝑟)−𝛽𝜂²+(𝜅|𝑥|)()𝑇+𝑀1+𝑟³𝑇(1+𝑟)−𝛽𝜂²=|𝑥||𝜆|𝛽𝜆|1+𝑝𝑇+|𝑇𝜂²𝑇(1+𝑟)−𝛽𝜂²+|𝜆|𝑇²(1+𝑟)𝑇(1+𝑟)−𝛽𝜂²+(𝜅|𝑥|+𝑀)𝑇1+𝑝²+𝛽𝑇𝜂³(1+𝑟)𝑇(1+𝑟)−𝛽𝜂²1+𝑟+𝑟²+𝑇³(1+𝑟)𝑇(1+𝑟)−𝛽𝜂²=|𝑥|Φ+(𝜅|𝑥|+𝑀)Ω. (38) Taking norm sup_{𝑡∈[0,𝑇]}|𝑥(𝑡)|=‖𝑥‖ and solving for ‖𝑥‖, this yields ‖𝑥‖≤𝑀Ω().1−Φ+𝜅Ω (39) Let 𝑅=𝑀Ω/(1−(Φ+𝜅Ω))+1, then (35) holds. This completes the proof.

4. Examples

In this section, we give two examples to illustrate our results.

Example 6. Consider the following nonlinear 𝑞-difference equation with boundary value problem: 𝐷_1/2𝐷_1/3−27𝑥(𝑡)=1(𝑡+2)²⋅|𝑥||𝑥|[],𝑥(0)(1)+2+1,𝑡∈0,1=0,𝑥3₀^3/4𝑥(𝑠)𝑑_1/4𝑠=0. (40)
Set 𝑞=1/2, 𝑝=1/3, 𝑟=1/4, 𝑇=1, 𝜆=−2/7, 𝜂=3/4, 𝛽=−2/3, and 𝑓(𝑡,𝑥)=(1/(𝑡+2)²)(‖𝑥‖/(1+‖𝑥‖)). Since |𝑓(𝑡,𝑥)−𝑓(𝑡,𝑦)|≤(1/4)‖𝑥−𝑦‖, then, (𝐻₁) and (𝐻₂) are satisfied with 𝑇(1+𝑟)−𝛽𝜂²=13/8≠0, 1Ω=𝑇1+𝑝²+𝛽𝑇𝜂³(1+𝑟)𝑇(1+𝑟)−𝛽𝜂²1+𝑟+𝑟²+𝑇³(1+𝑟)𝑇(1+𝑟)−𝛽𝜂²=132,|𝜆|𝛽𝜆|91Φ=𝑇+|𝑇𝜂²𝑇(1+𝑟)−𝛽𝜂²+|𝜆|𝑇²(1+𝑟)𝑇(1+𝑟)−𝛽𝜂²=47, (41)𝐿=1/4. Hence, Λ=∶Φ+𝐿Ω=85/91<1. Therefore, by Theorem 2, the boundary value problem (40) has a unique solution on [0,1].

Example 7. Consider the following nonlinear 𝑞-difference equation with boundary value problem: 𝐷_2/3𝐷_4/5+19𝑥(𝑡)=sin(5𝜋𝑥)+|𝑥|25𝜋|𝑥|3+1,𝑡∈0,2,𝑥(0)3=0,𝑥2=14¹₀𝑥(𝑠)𝑑_1/2𝑠. (42)
Set 𝑞=2/3, 𝑝=4/5, 𝑟=1/2, 𝑇=3/2, 𝜆=1/9, 𝜂=1, and 𝛽=1/4. Here, |𝑓(𝑡,𝑥)|=|sin(5𝜋𝑥)/25𝜋+|𝑥|/(1+|𝑥|)|≤(|𝑥|/5)+1. So, 𝑀=1, 𝑇(1+𝑟)−𝛽𝜂²=2≠0, and 1Ω=×𝑇1+𝑝₂+𝛽𝑇𝜂³(1+𝑟)𝑇(1+𝑟)−𝛽𝜂²1+𝑟+𝑟²+𝑇³(1+𝑟)𝑇(1+𝑟)−𝛽𝜂²=615,|𝜆|𝛽𝜆|224Φ=𝑇+|𝑇𝜂²𝑇(1+𝑟)−𝛽𝜂²+|𝜆|𝑇²(1+𝑟)𝑇(1+𝑟)−𝛽𝜂²=38,1𝜅=5<(1−Φ)Ω⁻¹=28.123 (43) Hence, by Theorem 5, the boundary value problem (42) has at least one solution on [0,3/2].

Acknowledgments

The authors would like to thank the reviewers for their valuable comments and suggestions on the paper. This research is supported by King Mongkut’s University of Technology North Bangkok, Thailand.