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Positive solutions of nonlinear boundary value problems for delayed fractional qdifference systems
Advances in Difference Equations volume 2014, Article number: 51 (2014)
Abstract
In this paper, we investigate the existence and uniqueness of positive solutions to nonlinear boundary value problems for delayed fractional qdifference systems by applying the properties of the Green function and some wellknown fixedpoint theorems. As applications, some examples are presented to illustrate the main results.
MSC:39A13, 34B18, 34A08.
1 Introduction
In the past decades, fractional differential equations have been proved to be valuable tools in the investigation of many phenomena in various fields of science and engineering such as physics, mechanics, chemistry, biology, engineering, etc. Therefore, the subject of fractional differential equations has gained considerable attention by many researchers. Some recent results on fractional boundary value problems can be found in [1–4] and references therein. For example, Ahmad and Nieto [5] dealt with some existence results for a boundary value problem involving a nonlinear fractional order integrodifferential equation with integral boundary conditions based on a contraction mapping principle and Krasnoselskiii’s fixedpoint theorem. Ahmad et al. [6] investigated the existence and uniqueness of solutions for a class of Caputotype fractional boundary value problems involving fourpoint nonlocal RiemannLiouville integral boundary conditions of different order by means of standard tools of fixedpoint theory and LeraySchauder nonlinear alternative. Ouyang et al. [7] considered the following nonlinear system of fractional order differential equations with delays:
where {D}^{{\alpha}_{i}} is the standard RiemannLiouville fractional derivative. By using some fixedpoint theorems and some properties of the Green function, the existence of positive solutions was obtained.
The qdifference calculus or quantum calculus is an old subject that was initially developed by Jackson [8, 9]; basic definitions and properties of qdifference calculus can be found in the book mentioned in [10].
The fractional qdifference calculus had its origin in the works by AlSalam [11] and Agarwal [12]. Recently, maybe due to the explosion in research within the fractional differential calculus setting, new developments in this theory of fractional qdifference calculus were made; for example, qanalogues of the integral and differential fractional operators properties such as the qLaplace transform, qTaylor’s formula, MittageLeffler function [13–16], just to mention some.
More recently, boundary value problems of nonlinear fractional qdifference equations have gained popularity and importance. Many researchers pay attention to the existence and multiplicity of solutions or positive solutions for nonlinear boundary value problems of fractional qdifference equations by means of upper and lower solutions method and some fixedpoint theorems such as the Krasnoselskii fixedpoint theorem, the LeggettWilliams fixedpoint theorem, and the Schauder fixedpoint theorem; for examples, see [17–21] and the references therein. ElShahed and AlAskar [22] studied the existence of multiple positive solutions to the nonlinear qfractional boundary value problems by using GuoKrasnoselskii’s fixedpoint theorem in a cone. Graef and Kong [23] investigated the uniqueness, existence, and nonexistence of positive solutions for the boundary value problem with fractional qderivatives in terms of different ranges of λ. Ma and Yang [24] obtained the existence of solutions for multipoint boundary value problems of nonlinear fractional qdifference equations by means of the Banach contraction principle and Krasnoselskii’s fixedpoint theorem. Zhao et al. [25] showed some existence results of positive solutions to nonlocal qintegral boundary value problems of a nonlinear fractional qderivative equation using the generalized Banach contraction principle, the monotone iterative method, and Krasnoselskii’s fixedpoint theorem. Ferreira [26] and [27] dealt with the existence of positive solutions to nonlinear qdifference boundary value problems,
and
respectively. By applying a fixedpoint theorem in cones, sufficient conditions for the existence of nontrivial solutions were enunciated.
In [28], Liang and Zhang discussed the following nonlinear qfractional threepoint boundary value problem:
By using a fixedpoint theorem in partially ordered sets, the authors obtained sufficient conditions for the existence and uniqueness of positive and nondecreasing solutions to the above boundary value problem.
In [29], Ahmad et al. studied the following nonlocal boundary value problems of nonlinear fractional qdifference equations,
where {}^{c}D_{q}^{\alpha} denotes the Caputo fractional qderivative of order α, and {a}_{i},{b}_{i},{c}_{i},{\eta}_{i}\in \mathbb{R} (i=1,2). The existence of solutions for the problem was shown by applying some wellknown tools of fixedpoint theory, such as Banach contraction principle, Krasnoselskii’s fixedpoint theorem, and LeraySchauder nonlinear alternative.
In [30], Alsaedi et al. were concerned with the following nonlinear fractional qdifference equations with nonlocal integral boundary conditions:
The existence results were obtained by applying some wellknown fixedpoint theorems.
Motivated by the above works, in this paper, we consider the following system of nonlinear fractional qdifference equations with delays:
where {D}_{q}^{{\alpha}_{i}} is the fractional qderivative of the RiemannLiouville type, {\alpha}_{i}\in ({n}_{i}1,{n}_{i}] for some {n}_{i}>2, {\eta}_{i}\ge 0 for i=1,2,\dots ,N, 0\le {\tau}_{ij}(t)\le t for i,j=1,2,\dots ,N, and {f}_{i} is a nonlinear function from [0,1]\times {\mathbb{R}}_{+}^{N} to {\mathbb{R}}_{+}=[0,\mathrm{\infty}). The purpose of this paper is to establish sufficient conditions on the existence of positive solutions for fractional qdifference system (1.1) by using some properties of the Green function and some fixedpoint theorems such as the Banach contraction principle, Krasnoselskii’s fixedpoint theorem, and the LeraySchauder nonlinear alternative. By a positive solution for the fractional qdifference system (1.1) we mean a mapping with positive components on [0,1] such that (1.1) is satisfied. Obviously, (1.1) includes the usual system of fractional qdifference equations when {\tau}_{ij}(t)\equiv t for all i and j. Therefore, the obtained results generalize and include some existing ones.
2 Preliminaries
For convenience of the reader, we present some necessary definitions and lemmas of fractional qcalculus theory to facilitate analysis of problem (1.1). These details can be found in the recent literature; see [10] and references therein.
Let q\in (0,1) and define
The qanalogue of the power {(ab)}^{n} with n\in {\mathbb{N}}_{0} is
More generally, if \alpha \in \mathbb{R}, then
Note that, if b=0, then {a}^{(\alpha )}={a}^{\alpha}. The qgamma function is defined by
and satisfies {\mathrm{\Gamma}}_{q}(x+1)={[x]}_{q}{\mathrm{\Gamma}}_{q}(x).
The qderivative of a function f is here defined by
and qderivatives of higher order by
The qintegral of a function f defined in the interval [0,b] is given by
If a\in [0,b] and f is defined in the interval [0,b], its integral from a to b is defined by
Similarly to what is done for derivatives, an operator {I}_{q}^{n} can be defined, namely,
The fundamental theorem of calculus applies to these operators {I}_{q} and {D}_{q}, i.e.,
and if f is continuous at x=0, then
Basic properties of the two operators can be found in the book [10]. We now point out three formulas that will be used later ({}_{i}D_{q} denotes the derivative with respect to variable i)
We note that if \alpha >0 and a\le b\le t, then {(ta)}^{(\alpha )}\ge {(tb)}^{(\alpha )} [26].
Definition 2.1 ([12])
Let \alpha \ge 0 and f be function defined on [0,1]. The fractional qintegral of the RiemannLiouville type is {I}_{q}^{0}f(x)=f(x) and
Definition 2.2 ([14])
The fractional qderivative of the RiemannLiouville type of order \alpha \ge 0 is defined by {D}_{q}^{0}f(x)=f(x) and
where m is the smallest integer greater than or equal to α.
Definition 2.3 ([14])
The fractional qderivative of the Caputo type of order \alpha \ge 0 is defined by
where m is the smallest integer greater than or equal to α.
Let \alpha ,\beta \ge 0 and f be a function defined on [0,1]. Then the following formulas hold:

(1)
({I}_{q}^{\beta}{I}_{q}^{\alpha}f)(x)={I}_{q}^{\alpha +\beta}f(x),

(2)
({D}_{q}^{\alpha}{I}_{q}^{\alpha}f)(x)=f(x).
Theorem 2.5 ([26])
Let \alpha >0 and p be a positive integer. Then the following equality holds:
Theorem 2.6 (Banach contraction mapping theorem [31])
Let M be a complete metric space and let T:M\to M be a contraction mapping. Then T has a unique fixed point.
Let C be a closed and convex subset of a Banach space X. Assume that U is a relatively open subset of C with 0\in U and T:\overline{U}\to C is completely continuous. Then at least one of the following two properties holds:

(i)
T has a fixed point in \overline{U},

(ii)
there exist u\in \partial U and \lambda \in (0,1) with u=\lambda Tu.
Theorem 2.8 (Krasnoselskii fixedpoint theorem [31, 34])
Let P be a cone in a Banach space X. Assume that {\mathrm{\Omega}}_{1} and {\mathrm{\Omega}}_{2} are open subsets of X with 0\in {\mathrm{\Omega}}_{1} and {\overline{\mathrm{\Omega}}}_{1}\subseteq {\mathrm{\Omega}}_{2}. Suppose that T:P\cap ({\overline{\mathrm{\Omega}}}_{2}\setminus {\mathrm{\Omega}}_{1}) is a completely continuous operator such that either

(i)
\parallel Tu\parallel \le \parallel u\parallel for u\in P\cap \partial {\mathrm{\Omega}}_{1} and \parallel Tu\parallel \ge \parallel u\parallel for u\in P\cap \partial {\mathrm{\Omega}}_{2}, or

(ii)
\parallel Tu\parallel \ge \parallel u\parallel for u\in P\cap \partial {\mathrm{\Omega}}_{1} and \parallel Tu\parallel \le \parallel u\parallel for u\in P\cap \partial {\mathrm{\Omega}}_{2}.
Then T has a fixed point in {\overline{\mathrm{\Omega}}}_{2}\setminus {\mathrm{\Omega}}_{1}.
3 Existence of positive solutions
Throughout this paper, we let E=C([0,1],{\mathbb{R}}^{N}). Then (E,{\parallel \cdot \parallel}_{E}) is a Banach space, where
In this section, we always assume that f={({f}_{1},\dots ,{f}_{N})}^{T}\in C([0,1]\times {\mathbb{R}}_{+}^{N},{\mathbb{R}}_{+}^{N}).
Lemma 3.1 Fractional qdifference systems (1.1) is equivalent to the following system of qintegral equations:
for i=1,2,\dots ,N, where
Proof It is easy to see that if {({u}_{1},{u}_{2},\dots ,{u}_{N})}^{T} satisfies (3.1), then it also satisfies (1.1). So, assume that {({u}_{1},{u}_{2},\dots ,{u}_{N})}^{T} is a solution to (1.1). In view of Lemma 2.4 and Theorem 2.5, integrating both sides of the first equation of (1.1) of order {\alpha}_{i} with respect to t, we can see that
for 0\le t\le 1, i=1,2,\dots ,N. It follows that
for 0\le t\le 1, i=1,2,\dots ,N. Combining with the boundary conditions in (1.1), this yields
Similarly, one can obtain {c}_{{n}_{i}2,i}={c}_{{n}_{i}3,i}=\cdots ={c}_{2,i}=0, i=1,2,\dots ,N. Also we have
Then it follows from (3.3) and the boundary condition ({D}_{q}^{{n}_{i}1}u)(1)={\eta}_{i} that
Therefore, for i=1,2,\dots ,N,
where {G}_{i} is defined as in (3.2). The proof is completed. □
Some properties of the Green functions {G}_{i}(t,s) needed in the sequel are now stated and proved.
Lemma 3.2 Function {G}_{i}(t,s) defined above satisfies the following conditions:

(a)
{G}_{i}(t,qs)\ge 0 and {G}_{i}(t,qs)\le {G}_{i}(1,qs) for all 0\le t,s\le 1;

(b)
{G}_{i}(t,qs)\ge {g}_{i}(t){G}_{i}(1,qs) for all 0\le t,s\le 1 with g(t)={t}^{{\alpha}_{i}1}.
Proof We start by defining the two functions
and
It is clear that {\psi}_{i}(t,qs)\ge 0 and {\psi}_{i}(0,qs)=0. On the other hand, for t\ne 0
Therefore, {G}_{i}(t,qs)\ge 0. Moreover, for fixed s\in [0,1],
i.e., {\phi}_{i}(t,qs) is an increasing function of t. Obviously, {\psi}_{i}(t,qs) is increasing in t, therefore {G}_{i}(t,qs) is an increasing function of t for fixed s\in [0,1]. This concludes the proof of (a).
Suppose now that t\ge qs. Then we have
On the other hand, if t\le qs, then we have
and this finishes the proof of (b). □
Now, we are ready to present the main results.
Theorem 3.3 Suppose that there exist functions {\lambda}_{ij}(t)\in C([0,1],{\mathbb{R}}_{+}), i,j=1,2,\dots ,N, such that
for t\in [0,1], {({u}_{1},{u}_{2},\dots ,{u}_{N})}^{T},{({v}_{1},{v}_{2},\dots ,{v}_{N})}^{T}\in {\mathbb{R}}_{+}^{N}. If
then (1.1) has a unique positive solution.
Proof Let
It is easy to see that Ω is a complete metric space. Define an operator T on Ω by
where G(t,s)=diag({G}_{1}(t,s),{G}_{2}(t,s),\dots ,{G}_{N}(t,s)) and
Because of the continuity of G and f, it follows easily from Lemma 3.2 that T maps Ω into itself. To finish the proof, we only need to show that T is a contraction. Indeed, for u,v\in \mathrm{\Omega}, by (3.4), we have
This, combined with Theorem 2.6 and (3.5), immediately implies that T:\mathrm{\Omega}\to \mathrm{\Omega} is a contraction. Therefore, the proof is complete with the help of Lemma 3.1 and Theorem 2.6. □
The following result can be proved in the same spirit as that for Theorem 3.3.
Theorem 3.4 Suppose that there exist functions {\lambda}_{i}(t)\in C([0,1],{\mathbb{R}}_{+}), i,j=1,2,\dots ,N, and nonnegative constants {p}_{i1},{p}_{i2},\dots ,{p}_{iN} such that {\sum}_{j=1}^{N}{p}_{ij}=1 and
for t\in [0,1], {({u}_{1},{u}_{2},\dots ,{u}_{N})}^{T},{({v}_{1},{v}_{2},\dots ,{v}_{N})}^{T}\in {\mathbb{R}}_{+}^{N}. If
then (1.1) has a unique positive solution.
Theorem 3.5 Suppose that there exist nonnegative realvalued functions {m}_{i},{n}_{i1},\dots ,{n}_{iN}\in L[0,1], i,j=1,2,\dots ,N, such that
for almost every t\in [0,1] and all {({u}_{1},{u}_{2},\dots ,{u}_{N})}^{T}\in {\mathbb{R}}_{+}^{N}. If
then (1.1) has at least one positive solution.
Proof Let Ω and T:\mathrm{\Omega}\to \mathrm{\Omega} be defined by (3.6) and (3.7), respectively. We first show that T is completely continuous through the following three steps.
Step 1. Show that T:\mathrm{\Omega}\to \mathrm{\Omega} is continuous. Let \{{u}^{k}(t)\} be a sequence in Ω such that {u}^{k}(t)\to u(t)\in \mathrm{\Omega}. Then {\mathrm{\Omega}}_{0}=[0,1]\times \{u(t){u}^{k}(t)\in \mathrm{\Omega},t\in [0,1],k\ge 1\} is bounded in [0,1]\times {\mathbb{R}}_{+}^{N}. Since f is continuous, it is uniformly continuous on any compact set. In particular, for any \epsilon >0, there exists a positive integer {K}_{0} such that
for t\in [0,1] and k\ge {K}_{0}, i=1,2,\dots ,N. Then, for t\in [0,1] and k\ge {K}_{0}, i=1,2,\dots ,N, we have
Therefore, \parallel T{u}^{k}(t)Tu(t)\parallel \le \epsilon for k\ge {K}_{0}, which implies that T is continuous.
Step 2. Show that T maps bounded sets of Ω into bounded sets. Let A be a bounded subset of Ω. Then [0,1]\times \{u(t)t\in [0,1],u\in A\}\subseteq [0,1]\times {\mathbb{R}}_{+}^{N} is bounded. Since f is continuous, there exists an M>0 such that
It follows that, for u\in A, t\in [0,1] and 1\le i\le N,
Immediately, we can easily see that TA is a bounded subset of Ω.
Step 3. Show that T maps bounded sets of Ω into equicontinuous sets. Let B be a bounded subset of Ω. Similarly as in Step 2, there exists L>0 such that
Then, for any u\in B and {t}_{1},{t}_{2}\in [0,1] and 1\le i\le N,
Now the equicontinuity of T on B follows easily from the fact that {G}_{i} is continuous and hence uniformly continuous on [0,1]\times [0,1].
Now we have shown that T is completely continuous. To apply Theorem 2.7, let
Fix r>\mu and define U=\{u\in \mathrm{\Omega}:\parallel u\parallel <r\}. We claim that there is no u\in U such that u=\lambda Tu for some \lambda \in (0,1). Otherwise, assume that there exist \lambda \in (0,1) and u\in \partial U such that u=\lambda Tu. Then
Therefore, \parallel u\parallel <r, a contradiction to u\in \partial U. This proves the claim. Applying Theorem 2.7, we know that T has a fixed point in \overline{U}, which is a positive solution to (1.1) by Lemma 3.1. Therefore, the proof is complete. □
Corollary 3.6 If all {f}_{i}, i,j=1,2,\dots ,N, are bounded, then (1.1) has at least one positive solution.
To state the last result of this section, we introduce
Theorem 3.7 Suppose that there exist {M}_{2}\in (0,{M}_{1}) and positive constants 0<{r}_{1}<{r}_{2} with {r}_{2}\ge {max}_{1\le i\le N}\{{\eta}_{i}/{[{\alpha}_{i}1]}_{q}\cdots {[{\alpha}_{i}{n}_{i}+1]}_{q}\}/(1{M}_{2}/{M}_{1}) such that

(a)
{f}_{i}(t,{u}_{1},\dots ,{u}_{N})\le {M}_{2}{r}_{2} for (t,{u}_{1},\dots ,{u}_{N})\in [0,1]\times {B}_{{r}_{2}}, i=1,2,\dots ,N, and

(b)
{f}_{i}(t,{u}_{1},\dots ,{u}_{N})\ge {M}_{1}{r}_{1} for (t,{u}_{1},\dots ,{u}_{N})\in [0,1]\times {B}_{{r}_{1}}, i=1,2,\dots ,N,
where {B}_{{r}_{i}}=\{u={({u}_{1},\dots ,{u}_{N})}^{T}\in {\mathbb{R}}_{+}^{N}{max}_{1\le i\le N}{u}_{i}\le {r}_{i}\}, i=1,2. Then (1.1) has at least a positive solution.
Proof Let Ω be defined by (3.5) and {\mathrm{\Omega}}_{i}=\{u\in E\parallel u\parallel <{r}_{i}\}, i=1,2. Obviously, Ω is a cone in E. From the proof of Theorem 3.5, we know that the operator T defined by (3.6) is completely continuous on Ω. For any u\in \mathrm{\Omega}\cap \partial {\mathrm{\Omega}}_{1}, it follows from Theorem 2.8 and condition (b) that
that is, {\parallel Tu\parallel}_{E}\ge {\parallel u\parallel}_{E} for u\in \mathrm{\Omega}\cap \partial {\mathrm{\Omega}}_{1}.
On the other hand, for any u\in \mathrm{\Omega}\cap \partial {\mathrm{\Omega}}_{2}, it follows from Lemma 3.2 and condition (a) that, for t\in [0,1],
that is, {\parallel Tu\parallel}_{E}\le {\parallel u\parallel}_{E} for u\in \mathrm{\Omega}\cap \partial {\mathrm{\Omega}}_{2}. Therefore, we have verified condition (b) of Theorem 2.8. It follows that T has a fixed point in \mathrm{\Omega}\cap ({\overline{\mathrm{\Omega}}}_{2}\setminus {\mathrm{\Omega}}_{1}), which is a positive solution to (1.1). This completes the proof. □
4 Some examples
In this section, we demonstrate the feasibility of some of the results obtained in Section 3.
Example 4.1 Consider the following fractional qdifference system:
Here {n}_{1}={n}_{2}=3, {\alpha}_{1}={\alpha}_{2}=2.5, q={\eta}_{1}={\eta}_{2}=0.5, {\tau}_{11}(t)=t/2, {\tau}_{12}(t)={\tau}_{22}(t)=sint, {\tau}_{21}(t)={t}^{2},
One can easily see that (3.4) is satisfied with
Moreover,
and hence [27]
It follows from Theorem 3.3 that (4.1) has a unique positive solution on [0,1].
Example 4.2 Consider the following fractional qdifference system:
Here {n}_{1}={n}_{2}=3, {\alpha}_{1}={\alpha}_{2}=2.5, q={\eta}_{1}={\eta}_{2}=0.5, {\eta}_{1}={\eta}_{2}=1/2,
where
One can easily see that (3.8) is satisfied. Moreover,
and hence [27]
It follows from Theorem 3.5 that (4.2) has at least one positive solution on [0,1].
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Yuan, Q., Yang, W. Positive solutions of nonlinear boundary value problems for delayed fractional qdifference systems. Adv Differ Equ 2014, 51 (2014). https://doi.org/10.1186/16871847201451
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DOI: https://doi.org/10.1186/16871847201451