Abstract
In this paper, we study a nonlocal boundary value problem for a second-order Hahn difference equation. Our problem contains two Hahn difference operators with different numbers of q and ω. An existence and uniqueness result is proved by using the Banach fixed point theorem, and the existence of a positive solution is established by using the Krasnoselskii fixed point theorem.
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1 Introduction
The quantum calculus, also known as the calculus without considering limits, deals with sets of nondifferentiable functions. There are many different types of quantum difference operators, for example, the Jackson q-difference operator, the forward (delta) difference operator, the backward (nabla) difference operator, and so on. These operators are found in many applications of mathematical areas such as orthogonal polynomials, basic hypergeometric functions, combinatorics, the calculus of variations, the theory of relativity, hypergeometric series, complex analysis, particle physics, and quantum mechanics. For some recent results and applications of the quantum calculus, see [1–9] and the references therein.
In 1949, Hahn [10] introduced the Hahn difference operator \(D_{q,\omega}\),
The Hahn difference operator is generalized to two well-known difference operators, the forward difference operator and the Jackson q-difference operator. Notice that, under appropriate conditions,
The Hahn difference operator has been employed in many pieces of literature to construct families of orthogonal polynomials and to investigate some approximation problems; see [11–13] and the references therein.
Unfortunately, in the past, no one was interested in finding the right inverse of the Hahn difference operator. Until in 2009, Aldwoah [14, 15] (Ph.D. thesis supervised by M.H. Annaby and A.E. Hamza) defined the right inverse of \(D_{q,\omega}\) in terms of both the Jackson q-integral containing the right inverse of \(D_{q}\) [16] and Nörlund sum involving the right inverse of \(\Delta_{\omega}\) [16].
In 2010, Malinowska and Torres [17, 18] introduced the Hahn quantum variational calculus. In 2013, Malinowska and Martins [19] studied the generalized transversality conditions for the Hahn quantum variational calculus. In the same year, Hamza et al. [20, 21] studied the theory of linear Hahn difference equations and investigated the existence and uniqueness results for the initial value problems for Hahn difference equations by using the method of successive approximations; moreover, they proved Gronwall’s and Bernoulli’s inequalities with respect to the Hahn difference operator and also established mean value theorems for this calculus.
In particular, the boundary value problem for Hahn difference equations has not been studied. The results mentioned are the motivation for this research. In this paper, we consider a nonlinear Hahn difference equation with nonlocal boundary value conditions of the form
where \(0< q<1\), \(0<\omega<T\), \(\omega_{0}:=\frac{\omega}{1-q}\), \(1\leq \lambda<\frac{T-\omega_{0}}{\eta-\omega_{0}}\), \(p=q^{m}\), \(m\in {\mathbb{N}}\), \(\theta=\omega(\frac{1-p}{1-q} )\), \(f: [\omega_{0},T]_{q,\omega}\times{\mathbb{R}}\times{\mathbb {R}}\rightarrow{\mathbb{R}}\) is a given function, and \(\varphi: C ([\omega_{0},T]_{q,\omega},{\mathbb{R}} )\rightarrow {\mathbb{R}}\) is a given functional.
In the next section, we briefly recall some definitions and lemmas used in this research. In Section 3, we prove the existence and uniqueness of a solution to problem (1.1) by the Banach fixed point theorem. In Sections 4-5, we establish some properties of the Green function and the existence of a positive solution to problem (1.1) by using the Krasnoselskii fixed point theorem. Finally, we provide an example to illustrate our results in the last section.
The following theorem is Krasnoselskii’s fixed point theorem in a cone.
Theorem 1.1
([22])
Let E be a Banach space, and let \(K\subset E\) be a cone. Let \(\Omega_{1}\), \(\Omega_{2}\) be open subsets of E with \(0\in \Omega_{1}\), \(\overline{\Omega}_{1}\subset\Omega_{2}\), and let
be a completely continuous operator such that
-
(i)
\(\Vert Au\Vert \leqslant \Vert u\Vert \), \(u\in K\cap \partial\Omega_{1}\), and \(\Vert Au\Vert \geqslant \Vert u\Vert \), \(u\in K\cap\partial \Omega_{2}\); or
-
(ii)
\(\Vert Au\Vert \geqslant \Vert u\Vert \), \(u\in K\cap \partial\Omega_{1}\), and \(\Vert Au\Vert \leqslant \Vert u\Vert \), \(u\in K\cap\partial \Omega_{2}\).
Then A has a fixed point in \(K\cap(\overline{\Omega}_{2}\setminus\Omega_{1})\).
2 Preliminaries
We now give the notation, definitions, and lemmas used in the main results.
Definition 2.1
([10])
For \(0< q<1\), \(\omega>0\), and f defined on an interval \(I\subseteq{\mathbb{R}}\) containing \(\omega_{0}:=\frac{\omega}{1-q}\), the Hahn difference of f is defined by
and \(D_{q,\omega}f(\omega_{0})=f'(\omega_{0})\), provided that f is differentiable at \(\omega_{0}\). We call \(D_{q,\omega}f\) the \(q,\omega \)-derivative of f and say that f is \(q,\omega\)-differentiable on I.
Let \(a,b\in I \subseteq\mathbb{R} \) with \(a<\omega_{0}<b\), and \([k]_{q}=\frac{1-q^{k}}{1-q}\), \(k\in{\mathbb{N}}_{0}:={\mathbb{N}}\cup\{0\} \). We define the \(q,\omega\)-interval by
Observe that, for each \(s\in[a,b]_{q,\omega}\), the sequence \(\{ q^{k} s+\omega[k]_{q} \}_{k=0}^{\infty}\) is uniformly convergent to \(\omega_{0}\).
If f is \(q,\omega\)-differentiable n times on a \(q,\omega\)-interval \(I_{q,\omega}\), then we define the higher-order derivatives by
where \(D^{0}_{q,\omega}f(s):=f(s)\), \(s\in I_{q,\omega}\subset{\mathbb{R}}\).
Next, we introduce the right inverse of the operator \(D_{q,\omega}\), which call the \(q,\omega\)-integral operator.
Definition 2.2
([14])
Let I be any closed interval of \(\mathbb{R}\) containing a, b, and \(\omega_{0}\). For a function \(f:I\rightarrow\mathbb{R}\), we define the \(q,\omega \)-integral of f from a to b by
where
provided that the series converges at \(x=a\) and \(x=b\); we say that f is \(q,\omega\)-integrable on \([a,b]\), and the sum on the right-hand side of the above equation is called the Jackson-Norlund sum.
Note that the actual domain of definition of f is \([a,b]_{q,\omega }\subset I\).
The following lemma is the fundamental theorem of Hahn calculus.
Lemma 2.1
([14])
Let \(f:I\rightarrow\mathbb {R}\) be continuous at \(\omega_{0}\). Define
Then F is continuous at \(\omega_{0}\). Furthermore, \(D_{q,\omega _{0}}F(x)\) exists for every \(x\in I\), and
Conversely,
Next, we give some auxiliary lemmas for simplifying calculations.
Lemma 2.2
Let \(0< q<1\), \(\omega>0\), and \(f:I\rightarrow\mathbb{R}\) be continuous at \(\omega_{0}\). Then
Proof
Using the definition of the \(q,\omega\)-integral, we have
□
Lemma 2.3
Let \(0< q<1\) and \(\omega>0\). Then
Proof
Using the definition of the \(q,\omega\)-integral, we have
and
The proof is complete. □
The following lemma deals with the linear version of problem (1.1) and gives a representation of the solution.
Lemma 2.4
Let \(1\leq\lambda<\frac{T-\omega_{0}}{\eta-\omega_{0}}\), \(h\in C ([\omega_{0},T]_{q,\omega}, {\mathbb{R}} )\) be a given function, and \(\varphi: C ([\omega_{0},T]_{q,\omega }, {\mathbb{R}} )\rightarrow{\mathbb{R}}\) be a given functional. Then the problem
has the unique solution
where
Proof
By Lemmas 2.1 and 2.2 a general solution for (2.1) can be written as
for \(t\in[\omega_{0},T]_{q,\omega}\).
From the conditions (2.1) we obtain
where Λ is defined by (2.3).
Substituting the constants \(C_{1}\), \(C_{2}\) into (2.4), we obtain (2.2). □
Lemma 2.5
Problem (2.1) has the unique solution of the from
where
with \(g_{i}(t,s)\), \(1\leq i \leq4\), defined as
Proof
Suppose \(t>\eta\). The unique solution of problem (2.1) can be written as
and similarly for \(t<\eta\). The unique solution of problem (2.1) can be written as
This completes the proof. □
3 Existence and uniqueness of a solution for problem (1.1)
In this section, we present an existence and uniqueness result for problem (1.1). Let \({\mathcal{C}}=C ([\omega_{0},T]_{q,\omega}, {\mathbb{R}} )\) be the Banach space of all continuous functions x with the norm
where \(\Vert x\Vert = \max_{t\in[\omega_{0},T]_{q,\omega}} \vert x(t)\vert \) and \(\Vert D_{p,\theta}x \Vert = \max_{t\in[\omega_{0},T]_{q,\omega}} \vert D_{p,\theta}x(pt+\theta)\vert \). Also, define the operator \({\mathcal{F}}:{\mathcal{C}}\rightarrow{\mathcal{C}}\) by
where \(\Lambda\neq0\) is defined by (2.3), \(p=q^{m}\), \(m\in {\mathbb{N}}\), and \(\theta=\omega(\frac{1-p}{1-q} )\).
Obviously, problem (1.1) has solutions if and only if the operator \(\mathcal{F}\) has fixed points.
Theorem 3.1
Assume that the following conditions hold:
- \((H_{1})\) :
-
There exist constants \(\gamma_{1},\gamma_{2}>0\) such that
$$\begin{aligned} & \bigl\vert f \bigl(t,x(t),D_{p,\theta}x(pt+\theta) \bigr)-f \bigl(t,y(t),D_{p,\theta }y(pt+\theta) \bigr) \bigr\vert \\ &\quad \leq \gamma_{1} \bigl\vert x(t)-y(t) \bigr\vert + \gamma_{2} \bigl\vert D_{p,\theta }x(pt+\theta)-D_{p,\theta}y(pt+ \theta) \bigr\vert \end{aligned}$$for all \(t\in[\omega_{0},T]_{q,\omega}\) and \(x, y\in\mathcal{C}\).
- \((H_{2})\) :
-
There exists a constant \(\ell>0\) such that
$$\bigl\vert \varphi(x)-\varphi(y) \bigr\vert \leq\ell \bigl\Vert x(t)-y(t) \bigr\Vert _{\mathcal{C}} $$for each \(x, y\in\mathcal{C}\).
- \((H_{3})\) :
-
\(\mathfrak{S}:=\gamma\Omega+\ell\Phi< 1\), where
$$ \begin{aligned} &\gamma = \max\{\gamma_{1}, \gamma_{2}\}, \\ &\Omega = \frac{(T-\omega_{0})}{\vert \Lambda \vert } \biggl[\frac {(T-\omega _{0})^{2}-\lambda(\eta-\omega_{0})^{2}}{(1+q)} \biggr] + \frac{(T-\omega_{0})^{2}}{1+q}, \\ &\Phi = 1+\frac{(\lambda-1)(T-\omega_{0})}{\vert \Lambda \vert }.\end{aligned} $$(3.2)
Then problem (1.1) has a unique solution in \([\omega _{0},T]_{q,\omega}\).
Proof
Denote \({\mathcal{H}}\vert x-y\vert (t):=\vert f(t,x(t),D_{p,\theta }x(pt+\theta))-f(t,y(t),D_{p,\theta}y(pt+\theta))\vert \). Using Lemma 2.3, for all \(t\in[\omega_{0},T]_{q,\omega }\) and \(x,y\in{\mathcal{C}}\), we have
Taking the \(p,\theta\)-derivative for (3.1) where \(p=q^{m}\), \(m\in{\mathbb{N}}\), and \(\theta=\omega(\frac{1-p}{1-q} )\), we obtain
This implies that \({\mathcal{F}}\) is a contraction. Therefore, by the Banach fixed point theorem, \({\mathcal{F}}\) has a fixed point, which is a unique solution of problem (1.1) on \(t\in[\omega_{0},T]_{q,\omega}\). □
4 Properties of Green’s function for problem (1.1)
We next prove that Green’s function \(G(t,s)\) in (2.8) satisfies a variety of properties that are necessary for considering the existence of a positive solution to problem (1.1). Firstly, we prove some necessary preliminary lemmas.
Theorem 4.1
([21], Mean Value Theorem)
Let \(f:I\rightarrow X\) be \(q,\omega\)-differentiable on I. For every \(s \in I\),
for all \(a,b \in \{sq^{k}+\omega[k]_{q} \}_{k=0}^{\infty}\) and \(a< b\).
Theorem 4.2
Let f be \(q,\omega\)-differentiable on \((a,b)_{q,\omega}\) and continuous on \([a,b]_{q,\omega}\). The following statements are true:
-
(i)
If \(D_{q,\omega}f(t)>0\) for all \(t\in(a,b)_{q,\omega}\), then f is an increasing function on \([a,b]_{q,\omega}\).
-
(ii)
If \(D_{q,\omega}f(t)<0\) for all \(t\in(a,b)_{q,\omega}\), then f is a decreasing function on \([a,b]_{q,\omega}\).
-
(iii)
If \(D_{q,\omega}f(t)=0\) for all \(t\in(a,b)_{q,\omega}\), then f is a constant function on \([a,b]_{q,\omega}\).
Proof
Let \(t_{1},t_{2}\in[a,b]_{q,\omega}\), \(t_{1}< t_{2}\). Since f is \(q,\omega\)-differentiable on \((a,b)_{q,\omega}\) and continuous on \([a,b]_{q,\omega}\), we have that f is a continuous function on \((a,b)_{q,\omega}\).
By Theorem 4.1 there exists \(t^{*}\in(a,b)_{q,\omega}\) such that \(D_{q,\omega}f(t^{*})=\frac{\Vert f(b)-f(a)\Vert }{b-a}\).
-
(i)
If \(D_{q,\omega}f(t)>0\) for all \(t\in(a,b)_{q,\omega}\), then \(D_{q,\omega}f(t^{*})>0\), which implies that
$$f(t_{2})-f(t_{1})=(t_{2}-t_{1})D_{q,\omega}f \bigl(t^{*}\bigr)>0. $$So \(f(t_{2})>f(t_{1})\) for all \(t_{1}\), \(t_{2}\), and hence f is increasing on \([a,b]_{q,\omega}\).
-
(ii)
If \(D_{q,\omega}f(t)<0\) for all \(t\in(a,b)_{q,\omega}\), then \(D_{q,\omega}f(t^{*})<0\), which implies that
$$f(t_{2})-f(t_{1})=(t_{2}-t_{1})D_{q,\omega}f \bigl(t^{*}\bigr)< 0. $$So \(f(t_{2})< f(t_{1})\) for all \(t_{1}\), \(t_{2}\), and hence f is decreasing on \([a,b]_{q,\omega}\).
-
(iii)
If \(D_{q,\omega}f(t)=0\) for all \(t\in(a,b)_{q,\omega}\), then \(D_{q,\omega}f(t^{*})=0\), which implies that
$$f(t_{2})-f(t_{1})=(t_{2}-t_{1})D_{q,\omega}f \bigl(t^{*}\bigr)=0. $$So \(f(t_{2})=f(t_{1})\) for all \(t_{1}\), \(t_{2}\), and hence f is constant on \([a,b]_{q,\omega}\). □
Lemma 4.1
We have that \(\Lambda>0 \) and \(1+\frac{(\lambda-1)(t-\omega _{0})}{\Lambda}\) is positive and strictly decreasing in t for \(t\in [\omega_{0},T]_{q,\omega}\). In addition,
Proof
Considering Λ in (2.3) and \(1\leq \lambda< \frac{T-\omega_{0}}{\eta-\omega_{0}}\), we obtain
For the proof that \(1+\frac{(\lambda-1)(t-\omega_{0})}{\Lambda}>0\), \(t\in[\omega_{0},T]_{q,\omega}\), it is sufficient to show that
Next, we prove that \(1+\frac{(\lambda-1)(t-\omega_{0})}{\Lambda}\) is strictly decreasing in \(t\in[\omega_{0},T]_{q,\omega}\). Note that the \({q,\omega}\)-derivative with respect to t for \(1+\frac{(\lambda -1)(t-\omega_{0})}{\Lambda}\) is
By Theorem 4.2 we have that \(1+\frac{(\lambda-1)(t-\omega _{0})}{\Lambda}\) is strictly decreasing in \(t\in[\omega_{0},T]_{q,\omega}\).
Finally, observe that
The proof is complete. □
Next, we show that Green’s function given in (2.8) is positive.
Lemma 4.2
Let \(G(t,s)\) be Green’s function given in (2.8). Then \(G(t,s)\geq0 \) for each \((t,s)\in[\omega_{0},T]_{q,\omega} \times[\omega _{0},T]_{q,\omega}\).
Proof
We aim to show that \(g_{i}(t,qs+\omega) >0\) for all i, \(1\leq i \leq 4\), and for each admissible pair \((t,s)\).
Firstly, we consider the function \(g_{4}(t,qs+\omega)=\frac{1}{\Lambda }(t-\omega_{0}) [T-(qs+\omega) ]\), \(s\in[t,T]_{q,\omega}\cap [\eta,T]_{q,\omega}\). To guarantee that \(g_{4}(t,qs+\omega)>0\), it suffices to show that
Thus, we conclude that \(g_{4}(t,qs+\omega)>0\) on their respective domains.
Next, we consider the function \(g_{2}(t,qs+\omega) \) for \(s\in[\eta ,T]_{q,\omega}\) and \(t\in[\eta,T]_{q,\omega}\):
To guarantee that \(g_{2}(t,qs+\omega)>0\), it suffices to show that
So, we conclude that \(g_{2}(t,qs+\omega)>0\) on their respective domains.
We next consider the function \(g_{3}(t,qs+\omega) \) for \(s\in[t,\eta ]_{q,\omega}\) and \(t\in[\omega_{0},\eta]_{q,\omega}\):
To guarantee that \(g_{3}(t,qs+\omega)>0\), it suffices to show that
Hence, \(g_{3}(t,qs+\omega)>0\), as claimed.
Finally, we consider the function \(g_{1}(t,qs+\omega)\) for \(s\in[\omega _{0},t]_{q,\omega} \cap[\omega_{0},\eta]_{q,\omega}\):
To guarantee that \(g_{1}(t,qs+\omega)>0\), it suffices to show that
We observe that
is increasing in λ for \(1<\lambda< \frac{T-\omega_{0}}{\eta -\omega_{0}}\). Note that \(\mathcal{I}(\lambda)\) is increasing for λ if and only if
Clearly, (4.6) implies that (4.4) also holds, and hence \(g_{1}(t,qs+\omega)>0\).
Consequently, from this it follows that \(g_{i}(t,qs+\omega)>0\) for each i, \(1\leq i\leq4\). Therefore, \(G(t,qs+\omega)>0\). □
Lemma 4.3
Let \(G(t,s)\) be Green’s function given in (2.8). Then for given \(\eta\in(\omega_{0},T)_{q,\omega} \) and \(1\leq\lambda< \frac {T-\omega_{0}}{\eta-\omega_{0}}\), it follows that
Proof
Our strategy is to consider the following two cases.
Case 1: \(t<\eta\). We aim to show that \({}_{t}D_{q,\omega} g_{1}(t,s), {}_{t}D_{q,\omega} g_{2}(t,s)<0\) and \({}_{t}D_{q,\omega} g_{4}(t,s)>0\). Theorem 4.2 implies that \(g_{1}\), \(g_{2}\) are decreasing and \(g_{4}\) is increasing in t, so \(G(t,qs+\omega)\leq G(qs+\omega ,qs+\omega)\) for all \((t,s)\in[\omega_{0},T]_{q,\omega} \times [\omega_{0},T]_{q,\omega}\).
Case 2: \(t>\eta\). We aim to show that \({}_{t}D_{q,\omega} g_{1}(t,s)<0\) and \({}_{t}D_{q,\omega} g_{3}(t,s) , {}_{t}D_{q,\omega} g_{4}(t,s)>0\). Theorem 4.2 implies that \(g_{1}\) is decreasing and \(g_{3}\), \(g_{4}\) are increasing in t, so \(G(t,qs+\omega)\leq G(qs+\omega ,qs+\omega)\) for all \((t,s)\in[\omega_{0},T]_{q,\omega} \times [\omega_{0},T]_{q,\omega}\).
Firstly, for \(g_{4}(t,qs+\omega)\), we have that
for all \(s\in[t,T]_{q,\omega}\cap[\eta,T]_{q,\omega}\) and \(t\in (\omega_{0},T]_{q,\omega}\).
Later, for \(g_{3}(t,qs+\omega)\), we have that
From (4.3) we obtain that \({}_{t}D_{q,\omega}g_{3}(t,qs+\omega )>0\) for all \(s\in[t,\eta]_{q,\omega}\) and \(t\in(\omega_{0},\eta ]_{q,\omega}\).
Next, we consider \(g_{2}(t,s)\) and claim that \({}_{t}D_{q,\omega} g_{2}(t,qs+\omega)<0\) for each admissible pair \((t,s)\). To this end, noting that
we obtain
So, \({}_{t}D_{q,\omega} g_{2}(t,qs+\omega)\) is nonpositive when
In addition, (4.12) is true if and only if
Clearly, (4.13) implies that (4.12) also holds. Hence, \({}_{t}D_{q,\omega} g_{2}(t,qs+\omega)<0\) for all \(s\in[\eta ,t]_{q,\omega}\) and \(t\in(\omega_{0},T]_{q,\omega}\), as desired.
Finally, to claim that \({}_{t}D_{q,\omega} g_{1}(t,qs+\omega)<0\) on its domain, we have that
for all \(s\in[\omega_{0},t]_{q,\omega} \cap[\omega_{0},\eta ]_{q,\omega}\) and \(t\in(\omega_{0},T]_{q,\omega}\).
Now, note that
Consequently, this implies that
Observe that \(G(qs+\omega,qs+\omega)=g_{4}(qs+\omega,qs+\omega )=q(s+\omega) [T-(qs+\omega) ]\).
Thus, by the discussion in the first paragraph of this proof we deduce that (4.7) holds. The proof is complete. □
Lemma 4.4
Let \(G(t,s)\) be Green’s function given in (2.8). Then it follows that
where σ satisfies the inequality \(0<\sigma<1\), and
Proof
We define
where \(k=3\) if \(i=1,3\) and \(k=4\) if \(i=2,4\).
For \(t<\eta\), we find that
If \(t>\eta\), then we consider two cases of \(\tilde{g}_{1}(t,qs+\omega )\) and \(\tilde{g}_{2}(t,qs+\omega)\):
and
Observe that \(\mathbf{\mathcal{J}}(\lambda)=\mathbf{\mathcal {I}}^{-1}(\lambda)\), which implies that \(\mathbf{\mathcal {J}}(\lambda)\) is decreasing in λ, and we have
Finally, note that since \(\sigma_{1}>1\), \(0<\sigma_{2}<\sigma_{1}\), and \(0<\sigma_{3}<1\), it follows that
We can conclude that \(\min_{t\in[\eta,T]_{q,\omega}} G(t,qs+\omega)\geq\sigma\max_{t\in [\omega_{0},T]_{q,\omega}}G(t,qs+\omega)\). □
Lemma 4.5
Let φ be a nonnegative function. Then there exists \(\sigma ^{*}\in(0,1)\) such that
Proof
Observe that by Lemma 4.4 there exists a constant \(\sigma\in(0,1)\) such that
Next, by Lemma 4.1 there exists a constant \(\mathcal{S}>0\) such that
In particular, putting (4.25) and (4.26) together implies that by taking
it follows that
Finally, defining
we obtain (4.23). This completes the proof. □
Lemma 4.6
Let G be Green’s function given in (2.8). Then
Proof
Using the definition of the \(q,\omega\)-integral, for \(s\in [\omega_{0},T]_{q,\omega}\), we obtain
This completes the proof. □
5 Existence of a positive solution for problem (1.1)
In this section, we consider the existence of at least one positive solution for problem (1.1) by appealing to the Krasnoselskii fixed point theorem in a cone.
Define the cone \({\mathcal{P}}\subseteq{\mathcal{C}}\) by
Consider nonlinear equation (1.1); then x solves (1.1) if and only if x is a fixed point of the operator \({\mathcal{A}}:{\mathcal {P}}\rightarrow{\mathcal{P}}\) defined by
where G is Green’s function for problem (1.1), and \({\mathcal{C}}\) is the Banach space defined in Section 3.
Lemma 5.1
Suppose that \(f:[\omega_{0},T]_{q,\omega}\times[0,\infty)\times [0,\infty)\rightarrow[0,\infty)\) and \(\varphi:C ([\omega _{0},T]_{q,\omega}\), \([0,\infty) )\rightarrow[0,\infty)\) are continuous. Then the operator \(\mathcal{A}:{\mathcal{P}}\rightarrow{\mathcal{P}}\) is completely continuous.
Proof
Since \(G(t,qs+\omega)\geq0\) for all \((t,s)\in[\omega _{0},T]_{q,\omega}\times[\omega_{0},T]_{q,\omega}\), we have \({\mathcal {A}}\geq0\) for all \(x\in{\mathcal{P}}\). For a constant \(L>0\), we define
and let \(M= \max_{(t,x)\in{[\omega_{0},T]_{q,\omega}}\times B_{L}}\vert f (t,x(t),D_{p,\theta}x(pt+\theta) )\vert \), \(N= \sup_{x\in B_{L}} \vert \varphi(x)\vert \). Then, for \(x\in B_{L}\), we obtain
Similarly to the proof above and Theorem 3.1, we obtain
Therefore, \(\Vert ({\mathcal{A}}x )(t)\Vert _{\mathcal {C}}= \mathcal{K}\), and hence \({\mathcal{A}}(B_{L})\) is uniformly bounded.
Next, we shall show that \({\mathcal{A}}(B_{L})\) is equicontinuous. For \(x\in B_{L}\) and \(t_{1},t_{2}\in[\omega_{0},T]_{q,\omega}\) with \(t_{1}< t_{2}\), there are three cases to consider.
Case 1: If \(\eta\leq t_{1}< t_{2}\), then by (2.7), letting \(g_{i}(t,qs+\omega)=\frac{(t-\omega_{0})}{\Lambda} {\mathfrak {g}}_{i}(t,qs+\omega)\), we obtain
Therefore, there exists a constant \(\delta_{1}>0\) such that
Case 2: If \(t_{1}< t_{2}\leq\eta\), then by (2.7) we obtain
Therefore, there exists a constant \(\delta_{2}>0\) such that
Case 3: If \(t_{1}<\eta<t_{2}\) with \(\vert t_{2}-t_{1}\vert <\delta=\min\{ \delta_{1},\delta_{2}\}\), then from (5.4)-(5.5) it follows that
Similarly to the proof above, by (5.3) we obtain
Hence, we conclude that \(\vert ({\mathcal{A}}x )(t_{2})- ({\mathcal{A}}x )(t_{1})\vert <\epsilon\) if \(\vert t_{2}-t_{1}\vert <\delta=\min\{\delta_{1},\delta_{2}\}\) for \(t_{2},t_{1}\in [\omega_{0},T]_{q,\omega}\), that is, \({\mathcal{A}}(B_{L})\) is equicontinuous. By the Arzelà-Ascoli theorem, \(\mathcal{A}:\mathcal {C}\rightarrow\mathcal{C}\) is a completely continuous operator.
Finally, we apply Lemmas 4.2-4.4 to obtain
and, for \(f\in\mathcal{P}\),
Hence,
Consequence, it follows that \({\mathcal{A}}: {\mathcal{P}}\rightarrow {\mathcal{P}}\) is a completely continuous operator. □
The following notation is used in the sequel:
Next, we introduce some assumptions that will be helpful in the sequel.
- \((A_{1})\) :
-
There exists a constant \(r_{1}>0\) such that
$$\begin{aligned} f \bigl(t,x(t),D_{p,\theta}x(pt+\theta) \bigr)\leq\frac {1}{2} \diamondsuit r_{1} \end{aligned}$$(5.10)for all \(t\in[\omega_{0},T]_{q,\omega}\) and \(0\leq x \leq r_{1}\).
- \((A_{2})\) :
-
There exists a constant \(r_{2}>0\) with \(r_{2}< r_{1}\) such that
$$\begin{aligned} f \bigl(t,x(t),D_{p,\theta}x(pt+\theta) \bigr)\geq\frac {1}{2}\Theta r_{2} \end{aligned}$$(5.11)for all \(t\in[\omega_{0},T]_{q,\omega}\) and \(\sigma^{*}r_{2} \leq x \leq r_{2}\), where \(\sigma^{*} \) is defined in (4.28).
- \((A_{3})\) :
-
There exists a constant \(r_{1}>0\) such that
$$\begin{aligned} \varphi(x)\leq\frac{1}{2}\Psi_{1} r_{1} \end{aligned}$$(5.12)for all \(x\in\mathcal{P}\) and \(0\leq \Vert x\Vert _{\mathcal{C}}\leq r_{1}\).
- \((A_{4})\) :
-
There exists a constant \(r_{2}>0\) such that
$$\begin{aligned} \varphi(x)\geq\frac{1}{2}\Psi_{2} r_{2} \end{aligned}$$(5.13)for all \(x\in\mathcal{P}\) and \(\sigma^{*}r_{2}\leq \Vert x\Vert _{\mathcal {C}}\leq r_{2}\).
Now, we can prove the existence of at least one positive solution.
Theorem 5.1
Suppose that conditions \((A_{1})\)-\((A_{4})\) hold. Let \(f(t,x)\in C ([\eta,T]_{q,\omega}\times[0,\infty )\times[0,\infty), [0,\infty) )\) and \(\varphi(x):C ([\eta ,T]_{q,\omega},[0,\infty) )\rightarrow[0,\infty)\). Then problem (1.1) has at least one positive solution, say \(x^{*}\), where \(r_{2}\leq \Vert x^{*}\Vert _{\mathcal{C}} \leq r_{1}\).
Proof
Set \(\Psi_{1}=\{x\in C ([\omega_{0},T]_{q,\omega} ):\Vert x\Vert _{\mathcal{C}}< r_{1} \}\). Then, for \(x\in{\mathcal {P}}\cap \partial\Psi_{1}\), we have
Since \(\vert (D_{p,\theta}{\mathcal{A}}x)(t)\vert <\vert ({\mathcal{A}}x)(t)\vert \leq r_{1}\), we have
Further, let \(\Psi_{2}=\{x\in C ([\omega_{0},T]_{q,\omega} ):\Vert x\Vert _{\mathcal{C}}< r_{2} \}\). Then, for \(x\in{\mathcal {P}}\cap \partial\Psi_{2}\), using Lemma 4.4, we find that
Since \(\vert (D_{p,\theta}{\mathcal{A}}x)(t)\vert <\vert ({\mathcal{A}}x)(t)\vert \leq r_{2}\), we have
We conclude by Theorem 1.1 that the operator \(\mathcal{A}\) has a fixed point. This implies that problem (1.1) has a positive solution, say \(x^{*}\), where \(r_{2}\leq \Vert x^{*}\Vert _{\mathcal{C}} \leq r_{1}\). □
6 Example
In this section, to illustrate our results, we consider an example.
Example
Consider the following boundary value problem for the second-order Hahn difference equation
where \(t\in[\frac{9}{4},10 ]_{\frac{1}{3},\frac {3}{2}}\), and \(C_{i}\) are given positive constants with \(\frac {1}{e^{2}}\leq\sum_{i=0}^{\infty}C_{i}\leq\frac{2}{e^{2}}\).
Set \(q=\frac{1}{3}\), \(\omega=\frac{3}{2}\), \(\omega_{0}=\frac{9}{4}\), \(p=\frac{1}{3^{5}}=\frac{1}{243}\), \(\theta= \frac{3}{2} [\frac{1-(\frac{1}{3})^{5}}{1-\frac{1}{3}} ]=\frac{121}{54}\), \(T=10\), \(\eta=10 (\frac{1}{3} )^{5}+\frac{3}{2}[5]_{\frac {1}{3}}=\frac{1109}{486}\), \(\lambda=\frac{4}{3}\), \(\varphi(x)=\sum_{i=0}^{\infty}\frac{C_{i}\vert x(t_{i})\vert }{1+\vert x(t_{i})\vert }\), and
I. The existence and uniqueness of solution to problem ( 6.1 ). We can show that
Clearly,
so that \((H_{1})\) holds with \(\gamma_{1}=0.002\), \(\gamma_{2}=0.0016\), and \(\gamma=\max\{\gamma_{1},\gamma_{2}\}=0.002\), and
so that \((H_{2})\) holds with \(\ell=\frac{2}{e^{2}}\).
Also, we can show that
Hence, by Theorem 3.1 problem (6.1) has a unique solution.
II. The existence of at least one positive solution to problem ( 6.1 ). We can show that
Clearly,
Therefore, conditions \((A_{1})\)-\((A_{3})\) are satisfied. Consequently, by Theorem 5.1 problem (6.1) has at least one positive solution \(x^{*}\) such that \(r_{2}=0.000097 \leq \Vert x^{*}\Vert _{\mathcal{C}} \leq0.0036=r_{1}\).
References
Annaby, MH, Mansour, ZS: q-Fractional Calculus and Equations. Springer, Berlin (2012)
Kac, V, Cheung, P: Quantum Calculus. Springer, New York (2002)
Jagerman, DL: Difference Equations with Applications to Queues. Dekker, New York (2000)
Aldowah, KA, Malinowska, AB, Torres, DFM: The power quantum calculus and variational problems. Dyn. Contin. Discrete Impuls. Syst., Ser. B, Appl. Algorithms 19, 93-116 (2012)
Birto da Cruz, AMC, Martins, N, Torres, DFM: Symmetric differentiation on time scales. Appl. Math. Lett. 26(2), 264-269 (2013)
Cruz, B, Artur, MC: Symmetric quantum calculus. Ph.D. thesis, Aveiro University (2012)
Wu, GC, Baleanu, D: New applications of the variational iteration method from differential equations to q-fractional difference equations. Adv. Differ. Equ. 2013, 21 (2013)
Tariboon, J, Ntouyas, SK: Quantum calculus on finite intervals and applications to impulsive difference equations. Adv. Differ. Equ. 2013, 282 (2013)
Álvarez-Nodarse, R: On characterization of classical polynomials. J. Comput. Appl. Math. 196, 320-337 (2006)
Hahn, W: Über Orthogonalpolynome, die q-Differenzenlgleichungen genügen. Math. Nachr. 2, 4-34 (1949)
Costas-Santos, RS, Marcellán, F: Second structure relation for q-semiclassical polynomials of the Hahn Tableau. J. Math. Anal. Appl. 329, 206-228 (2007)
Foupouagnigni, M: Laguerre-Hahn orthogonal polynomials with respect to the Hahn operator: fourth-order difference equation for the rth associated and the Laguerre-Freud equations recurrence coefficients. Ph.D. thesis, Université Nationale du Bénin, Bénin (1998)
Kwon, KH, Lee, DW, Park, SB, Yoo, BH: Hahn class orthogonal polynomials. Kyungpook Math. J. 38, 259-281 (1998)
Aldwoah, KA: Generalized time scales and associated difference equations. Ph.D. thesis, Cairo University (2009)
Annaby, MH, Hamza, AE, Aldwoah, KA: Hahn difference operator and associated Jackson-Nörlund integrals. J. Optim. Theory Appl. 154, 133-153 (2012)
Jackson, FH: Basic integration. Q. J. Math. 2, 1-16 (1951)
Malinowska, AB, Torres, DFM: The Hahn quantum variational calculus. J. Optim. Theory Appl. 147, 419-442 (2010)
Malinowska, AB, Torres, DFM: Quantum Variational Calculus. Spinger Briefs in Electrical and Computer Engineering-Control, Automation and Robotics. Springer, Berlin (2014)
Malinowska, AB, Martins, N: Generalized transversality conditions for the Hahn quantum variational calculus. Optimization 62(3), 323-344 (2013)
Hamza, AE, Ahmed, SM: Theory of linear Hahn difference equations. J. Adv. Math. 4(2), 441-461 (2013)
Hamza, AE, Ahmed, SM: Existence and uniqueness of solutions of Hahn difference equations. Adv. Differ. Equ. 2013, 316 (2013)
Krasnoselskii, MA: Positive Solutions of Operator Equations. Noordhoof, Gronignen (1964)
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Sitthiwirattham, T. On a nonlocal boundary value problem for nonlinear second-order Hahn difference equation with two different \(q,\omega\)-derivatives. Adv Differ Equ 2016, 116 (2016). https://doi.org/10.1186/s13662-016-0842-2
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DOI: https://doi.org/10.1186/s13662-016-0842-2