Abstract
In this paper, we discuss the existence and uniqueness of solutions for Langevinimpulsive q-difference equations with boundary conditions. Our studyrelies on Banach’s and Schaefer’s fixed point theorems. Illustrativeexamples are also presented.
MSC: 26A33, 39A13, 34A37.
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1 Introduction and preliminaries
In recent years, the boundary value problems of fractional order differentialequations have emerged as an important area of research, since these problems haveapplications in various disciplines of science and engineering such as mechanics,electricity, chemistry, biology, economics, control theory, signal and imageprocessing, polymer rheology, regular variation in thermodynamics, biophysics,aerodynamics, viscoelasticity and damping, electro-dynamics of complex medium, wavepropagation, blood flow phenomena, etc.[1–5]. Many researchers have studied the existence theory for nonlinearfractional differential equations with a variety of boundary conditions, forinstance, see the papers [6–18], and the references therein.
The Langevin equation (first formulated by Langevin in 1908) is found to be aneffective tool to describe the evolution of physical phenomena in fluctuatingenvironments [19]. For some new developments on the fractional Langevin equation, see, forexample, [20–27].
Nowadays there is a significant increase of activities in the area ofq-calculus due to its applications in various fields such as mathematics,mechanics, and physics. The book by Kac and Cheung [28] covers many of the fundamental aspects of the quantum calculus.A variety of new results can be found in the papers [29–41] and the references cited therein.
Impulsive differential equations serve as basic models to study the dynamics ofprocesses that are subject to sudden changes in their states. Recent development inthis field has been motivated by many applied problems, such as control theory,population dynamics and medicine. For some recent works on the theory of impulsivedifferential equations, we refer the interested reader to the monographs [42–44].
Recently in [45] the notions of -derivative and -integral on finite intervals were introduced. Let usrecall here these notions. For a fixed let be an interval and be a constant. We define -derivative of a function at a point as follows.
Definition 1.1 Assume is a continuous function and let. Then the expression
is called the -derivative of function f at t.
We say that f is -differentiable on provided exists for all . Note that if and in (1.1), then , where is the well-known q-derivative of thefunction defined by
In addition, we should define the higher -derivative of functions.
Definition 1.2 Let be a continuous function, we call the second-order-derivative provided is -differentiable on with . Similarly, we define higher order-derivative .
The -integral is defined as follows.
Definition 1.3 Assume is a continuous function. Then the-integral is defined by
for . Moreover, if then the definite -integral is defined by
Note that if and , then (1.3) reduces to q-integral of afunction , defined by for .
For the basic properties of the -derivative and -integral we refer to [45].
In this paper we combine all the above subjects and investigate the nonlinearsecond-order impulsive -difference Langevin equation with boundary conditionsof the form
where , is a continuous function, λ is a givenconstant, , for , , for , and α, β,γ, η are given constants.
The rest of this paper is organized as follows. In Section 2, we present apreliminary result which will be used in this paper. In Section 3, we willconsider the existence results for problem (1.4) while in Section 4, we willgive examples to illustrate our main results.
2 An auxiliary lemma
In this section, we present an auxiliary lemma which will be used throughout thispaper. Let , , for .
Lemma 2.1 Let. The unique solution of problem (1.4) isgiven by
with, where
Proof For using -integral for the first equation of (1.4), we get
Setting and , we have
which leads to
For we obtain by -integrating (2.2),
In particular, for
For , -integrating (1.4), we have
From the second impulsive equations of (1.4), we have
Applying -integral to (2.5) for , we obtain
Using the second impulsive equation of (1.4) with (2.4) and (2.6), one has
Repeating the above process, for , we get
For , we get
It is easy to see that
For and using , we have
Applying the boundary conditions of (1.4) with (2.8) and (2.9), it follows that
and
Substituting the values of A and B into (2.7), we get (2.1) asrequired. The proof is completed. □
3 Main results
Let = {: is continuous everywhere except for some at which and exist and , }. is a Banach space with the norm.
From Lemma 2.1, we define an operator by
where constants , , , , and Ω are defined as in Lemma 2.1. Itshould be noticed that problem (1.4) has solutions if and only if the operator has fixed points.
Our first result is an existence and uniqueness result for the impulsive boundaryvalue problem (1.4) by using the Banach contraction mapping principle.
For convenience, we set
and
Theorem 3.1 Assume that the following conditions hold:
(H1) is a continuous function and there exists a constantsuch that
for eachand.
(H2) The functionsare continuous and there exist constantssuch that
for each, .
If
whereis defined by (3.2), then the impulsive-difference Langevin boundary value problem(1.4) has a unique solution on J.
Proof Firstly, we transform the impulsive -difference Langevin boundary value problem (1.4) intoa fixed point problem, , where the operator is defined by (3.1).Applying the Banach contraction mapping principle, we shall show that has a fixed point which is the unique solution of theboundary value problem (1.4).
Let , , and be nonnegative constants such that, , and . We choose a suitable constant ρ by
where and defined by (3.3). Now, we will show that, where a set is defined as . For , we have
which implies that .
For any and for each , we have
which implies that . As , is a contraction.Therefore, by the Banach contraction mapping principle, we find that has a fixed point which is the unique solution of problem(1.4). □
The second existence result is based on Schaefer’s fixed point theorem.
Theorem 3.2 Assume that the following conditions hold:
(H3) is a continuous function and there exists a constantsuch that
for eachand all.
(H4) The functionsare continuous and there exist constantssuch that
for all, .
If
then the impulsive-difference Langevin boundary value problem(1.4) has at least one solution on J.
Proof We shall use Schaefer’s fixed point theorem to prove that theoperator defined by (3.1) has a fixed point. We divide the proof intofour steps.
Step 1: Continuity of.
Let be a sequence such that in . Since f is a continuous function on and , are continuous functions on ℝ for, we have
for , as .
Then, for each , we get
which gives as . This means that is continuous.
Step 2: maps bounded sets into bounded sets in.
Let us prove that for any , there exists a positive constant σsuch that for each , we have . For any , we have
Hence, we deduce that .
Step 3: maps bounded sets into equicontinuous sets of.
Let for some , , be a bounded set of as in Step 2, and let . Then we have
The right-hand side of the above inequality is independent of x and tendsto zero as . As a consequence of Steps 1 to 3, together with theArzelá-Ascoli theorem, we deduce that is completely continuous.
Step 4: We show that the set
is bounded.
Let . Then for some . Thus, for each , we have
This implies by (H3) and (H4) that for each, we have
Setting
we have
which yields
This shows that the set E is bounded. As a consequence of Schaefer’sfixed point theorem, we conclude that has a fixed point whichis a solution of the impulsive -difference Langevin boundary value problem(1.4). □
4 Examples
Example 4.1 Consider the following boundary value problem for the second-orderimpulsive -difference Langevin equation:
Here , , , , , , , , , , , and . Since
then (H1) and (H2) are satisfied with , , . We can find that , , , , and thus
Hence, by Theorem 3.1, the boundary value problem (4.1) has a unique solutionon .
Example 4.2 Consider the following boundary value problem for the second-orderimpulsive -difference Langevin equation:
Here , , , , , , , , , , , and . Clearly,
and
We can find that
where , and .
Hence, by Theorem 3.2, the boundary value problem (4.2) has at least onesolution on .
Authors’ information
Sotiris K Ntouyas is a member of Nonlinear Analysis and Applied Mathematics (NAAM) -Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.
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Acknowledgements
We would like to thank the reviewers for their valuable comments and suggestionson the manuscript. The research of J Tariboon is supported by KingMongkut’s University of Technology North Bangkok, Thailand.
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Tariboon, J., Ntouyas, S.K. Nonlinear second-order impulsive q-difference Langevin equation with boundary conditions. Bound Value Probl 2014, 85 (2014). https://doi.org/10.1186/1687-2770-2014-85
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DOI: https://doi.org/10.1186/1687-2770-2014-85