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Existence and uniqueness of solutions of Hahn difference equations
Advances in Difference Equations volume 2013, Article number: 316 (2013)
Hahn introduced the difference operator in 1949, where and are fixed real numbers. This operator extends the classical difference operator as the Jackson q-difference operator .
In this paper, we present new results of the calculus based on the Hahn difference operator. Also, we establish an existence and uniqueness result of solutions of Hahn difference equations by using the method of successive approximations.
1 Introduction and preliminaries
Hahn introduced his difference operator which is defined by
at and the usual derivative at , where and are fixed real numbers [1, 2]. This operator unifies and generalizes two well-known difference operators. The first is Jackson q-difference operator defined by
where q is fixed. Here f is supposed to be defined on a q-geometric set , for which whenever , see [3–10]. The second operator is the forward difference operator
In , Annaby et al. gave a rigorous analysis of the calculus associated with . They stated and proved some basic properties of such a calculus. For instance, they defined a right inverse of in terms of both the Jackson q-integral; see , which contains the right inverse of and Nörlund sum; cf. , which involves the right inverse of . Then they proved a fundamental theorem of Hahn’s calculus. An essential function which plays an important role in this calculus is . This function is normally taken to be defined on an interval I, which contains the number . One can see that the k th order iteration of is given by
The sequence is uniformly convergent to θ on I. Here is defined by
Throughout this paper, I is any interval of ℝ containing θ and X is a Banach space.
Definition 1.1 Assume that is a function, and let . The -integral of f from a to b is defined by
provided that the series converges at and .
Definition 1.2 
For certain , the -exponential functions and are defined by
To guarantee the convergence of the infinite product in (1.1) with , we assume additionally that
see [16, 17]. For a fixed , (1.2) converges for all , defining an entire function of order zero. For the proofs of the equalities in (1.1) and (1.2), see [, Section 1.3] and . Here the q-shifted factorial for a complex number b and is defined to be
The following results were mentioned in , and we need them in this paper.
Lemma 1.3 Let be q, ω differentiable at . Then the following statements are true:
for any constant , ,
We notice that (ii) and (iv) are true even if . Also, (i) is true if .
Lemma 1.4 For a fixed , the parametric -exponential functions and are the unique solutions of the first order initial value problems
Lemma 1.5 Let , , and be -integrable on I. If for all , then for , ,
Theorem 1.6 Let be continuous at θ. Define
Then F is continuous at θ. Furthermore, exists for every and
Lemma 1.5 and Theorem 1.6 are also true if f is a function with values in X with replacing the norm instead of the modulus .
The aim of this paper is to establish an existence and uniqueness result of solutions of the first order abstract Hahn difference equations by using the method of successive approximations. This method is a very powerful tool that dates back to the works of Liouville  and Picard . It is based on defining a sequence of functions and showing that will successively approximate the solution ϕ in the sense that the ‘error’ between the two monotonically decreases as k increases. Also, it differs from fixed point methods and topological ideas that were used by some researchers to develop the existence and uniqueness of solutions.
We organize this paper as follows.
In Section 2, we prove Gronwall’s and Bernoulli’s inequalities with respect to the Hahn difference operator. In Section 3, we establish mean value theorems in the calculus based on this operator. In Sections 4 and 5, we apply the method of successive approximations to obtain the local and global existence and uniqueness theorem of first order Hahn difference equations in Banach spaces. Hence, we deduce this theorem for the n th order Hahn difference equations.
2 Gronwall’s and Bernoulli’s inequalities
In this section, I is a subinterval of , where , and f is a real valued function defined on I.
Theorem 2.1 Assume that , and y, f are continuous at θ. Then
for all implies that
Proof We have
Integrating both sides from θ to t in the inequality above, we obtain
This implies that
Theorem 2.2 (Gronwall’s inequality)
Let , and y, f be continuous at θ. Then
for all implies that
Proof Set . Then , and . Consequently,
Theorem 2.3 (Bernoulli’s inequality)
Let . Then, for all .
Proof Let . Then,
Since , we have by Theorem 2.1
Therefore, . □
3 Mean value theorems
Theorem 3.1 Let , be q, ω differentiable on I and .
Assume that for all . Then for all , .
Proof The inequality implies that for all . □
Corollary 3.2 Suppose that is q, ω differentiable. The following statements are true:
For every , the inequality holds for all , .
If for all , then f is a constant function.
If for all . Then for all , where c is a constant in X.
Proof (i) Let . Then for all .
By Theorem 3.1,
Statement (i) implies that for every and . Letting , we obtain for all .
Can be deduced immediately from (ii). □
As a direct consequence of Theorem 1.6, we get the following result.
Theorem 3.3 Let be continuous at θ. Then
4 Successive approximations and local results
Throughout the remainder of the paper,
and R is the rectangle
where a and b are fixed positive numbers.
Theorem 4.1 Assume that satisfies the following conditions:
is continuous at for every .
There is a positive constant A such that the following Lipschitz condition for all is satisfied.
Then there is such that the following Cauchy problem
has a unique solution on .
Proof Since f is continuous at , there exists such that for all t with . Hence,
for all , such that and . Define the following constants K, h to be
We establish the existence of the solution of (4.1) by the method of successive approximations.
We consider the sequence defined as follows:
The proof of the existence consists of four steps.
, , . Indeed, we have . Assume that the inequality holds. This implies that
i.e., , . Also, each is continuous at .
Now, we show by induction that(4.3)
Assume that (4.3) is true. Hence,
Thus, inequality (4.3) is true for every .
We show that converges uniformly to a function on .
We write as the following sum
provided that (4.4) converges.
Thus, the convergence of the sequence of functions is equivalent to the convergence of the right-hand side of (4.4).
To prove the latter, we use estimate (4.3), and then apply the Weierstrass M-test. The convergence of the series implies the uniform convergence of the series on . This implies that the right-hand side of (4.4) is uniform convergent to some function . That is, there exists a function such that . Obviously, since .
The last part of the proof of the existence is to show that is a solution of (4.1).
From the Lipschitz condition,
for all and .
Thus, for every , such that
This implies that
uniformly on . Taking in (4.2), it follows that
Consequently, we obtain . Clearly, .
To prove the uniqueness, let us assume that is another solution which is valid in . We have
From which we deduce
Taking , we get
By taking the limit as , we get
Since , then , which completes the proof. □
Corollary 4.2 Let I be an interval containing such that the following conditions are satisfied:
For , , are continuous at .
There is a positive constant A such that, for , , , the following Lipschitz condition is satisfied:
Then there exists a unique solution of the initial value problem
Proof Let and .
System (4.6) yields
First f is continuous at , since each is continuous at .
Second f satisfies a Lipschitz condition because for
Applying Theorem 4.1, then there exist such that the initial value problem (4.7) has a unique solution on . It is easy to show that (4.7) is equivalent to the initial value problem (4.6). □
The following corollary shows that Corollary 4.2 can be applied to indicate the required conditions for the existence and uniqueness of solutions of the Cauchy problem
The proof of this corollary depends on converting the n th order Hahn difference equation (4.8) to a first order system. Clearly, the Cauchy problem (4.8) is equivalent to the first order system
in the sense that is a solution of (4.9) if and only if is a solution of (4.8). Here,
This leads us to state the following theorem.
Theorem 4.3 Let be a function defined on such that the following conditions are satisfied:
For any values of , f is continuous at .
f satisfies a Lipschitz condition
where , , and .
Then the Cauchy problem (4.8) has a unique solution which is valid on .
The following corollary gives us the required conditions for the existence of solutions of the Cauchy problem
Corollary 4.4 Assume that the functions () and satisfy the following conditions:
() and are continuous at θ with .
is bounded on I, .
Then, for any elements , equation (4.10) has a unique solution on a subinterval containing θ.
Proof Dividing by , we get the equation
where , and . Since and are continuous at , then the function defined by
is continuous at . Furthermore, is bounded on I. Consequently, there is such that , . We can easily see that f satisfies a Lipschitz condition with Lipschitz constant A. Thus, the function satisfies the conditions of Theorem 4.3. Hence, there exists a unique solution of (4.11) on a subinterval J of I containing θ. □
5 Nonlocal results
Theorem 4.1 is called a local existence theorem, because it guarantees the existence of a solution defined for , which is close to the initial point θ. However, in many situations, a solution will actually exist on the entire interval . We now show that if f satisfies a Lipschitz condition on a strip
rather than on the rectangle R, which is given in Section 4, then solutions will exist on the entire interval .
Theorem 5.1 Let f be continuous on the strip S, and suppose that there exists a constant such that for all , where . Then the successive approximations that are given in (4.2) exist on the entire interval and converge there uniformly to the unique solution of (4.1).
Proof There is a constant M such that for all . We will show that the inequality
holds for every . Inequality (5.1) is true at . Indeed, we have
Assume that inequality (5.1) is true. Now, we have
We see that
which converges to , . This convergence enables us to apply the Weierstrass M-test to conclude that a series
converges uniformly on I. Also, we can write as the following sum
Consequently, the absolute uniform convergence of the series in (5.3) on I implies that converges uniformly to some function on I.
Our objective now is to show that is a solution of (4.1) for all .
From the Lipschitz condition
for all . This implies that uniformly on I and uniformly on I.
Taking in (4.2), it follows that
Consequently, . Clearly, .
Let and be two solutions of (4.1) for all . We show that on I.
For , consider
Thus, from Gronwall’s inequality, we have . Consequently, for all . □
Corollary 5.2 Let f be continuous on the half-plane
Assume that f satisfies a Lipschitz condition
on each strip
where is a constant that may depend on θ and a. Then, the initial value problem (4.1) has a unique solution that exists on the whole half-line .
Proof The proof involves showing that the conditions of Theorem 5.1 hold on every strip of and is omitted for brevity. □
6 Conclusion and future directions
This article was devoted to establish the method of successive approximations in proving the existence and uniqueness of solutions of the initial value problems associated with Hahn difference operators. Also, some new results of the calculus based on this operator like a mean value theorem were obtained. In one direction, one should ask about the q-ω Taylor’s theorem. In this respect, we point out that q-Taylor’s theorem has been established in . Another direction, is to study in more details the theory of Hahn difference equations, based on Hahn difference operator, and the stability of its solutions.
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The authors declare that they have no competing interests.
All authors contributed equally in writing this paper. All authors read and approved the final manuscript.
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Hamza, A.E., Ahmed, S.M. Existence and uniqueness of solutions of Hahn difference equations. Adv Differ Equ 2013, 316 (2013). https://doi.org/10.1186/1687-1847-2013-316
- Hahn difference operator
- Jackson q-difference operator