Existence and uniqueness of solutions of Hahn difference equations
© Hamza and Ahmed; licensee Springer. 2013
Received: 25 March 2013
Accepted: 4 September 2013
Published: 8 November 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
Throughout this paper, I is any interval of ℝ containing θ and X is a Banach space.
provided that the series converges at and .
Definition 1.2 
The following results were mentioned in , and we need them in this paper.
for any constant , ,
We notice that (ii) and (iv) are true even if . Also, (i) is true if .
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 2.2 (Gronwall’s inequality)
Theorem 2.3 (Bernoulli’s inequality)
Let . Then, for all .
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 . □
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 .
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.
4 Successive approximations and local results
where a and b are fixed positive numbers.
is continuous at for every .
There is a positive constant A such that the following Lipschitz condition for all is satisfied.
has a unique solution on .
We establish the existence of the solution of (4.1) by the method of successive approximations.
- (i), , . Indeed, we have . Assume that the inequality holds. This implies that
- (ii)Now, we show by induction that(4.3)
We show that converges uniformly to a function on .
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).
The last part of the proof of the existence is to show that is a solution of (4.1).
for all and .
Consequently, we obtain . Clearly, .
Since , then , which completes the proof. □
For , , are continuous at .
- (ii)There is a positive constant A such that, for , , , the following Lipschitz condition is satisfied:
Proof Let and .
First f is continuous at , since each is continuous at .
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). □
This leads us to state the following theorem.
For any values of , f is continuous at .
- (ii)f satisfies a Lipschitz condition
where , , and .
Then the Cauchy problem (4.8) has a unique solution which is valid on .
() 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.
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
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).
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 .
for all . This implies that uniformly on I and uniformly on I.
Consequently, . Clearly, .
Let and be two solutions of (4.1) for all . We show that on I.
Thus, from Gronwall’s inequality, we have . Consequently, for all . □
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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