Existence of a class of fractional difference equations with two point boundary value problem
© Ibrahim and Jalab 2015
Received: 27 June 2015
Accepted: 15 August 2015
Published: 2 September 2015
This paper studies a fractional difference equation of two point boundary value problem (BVP) type, which is recognized as the ‘discrete’ BVP. Certain cases are expressed under which the discrete boundary value problems (DBVP) will have a single solution. The novelty hither comprises a method selection of metric and employment of Hölder’s inequality. This attitude allows the related functions to be contractive, which were earlier non-contractive in classical regularities. This consequently qualifies an enhanced application of Banach’s fixed point theorem for classifying a more extensive framework of issues than those which appeared in the current designs.
In , Diaz and Osler concluded that the fractional difference by choice is a normal method of letting the index of differences, in the criterion appearance for the nth difference, to be any number (real or complex). Farther along, Hirota  employed the fractional difference operator (FDO) for any real number utilizing a Taylor’s series. In , Nagai assumed different description of FDO by adapting Hirota’s  concept. Newly, in , Deekshitulu and Mohan improved Nagai’s definition . In , Jumarie suggested another formula of fractional difference operator, of which the leading features are a new fractional Taylor series and its companion Rolle’s formula which are employed to non-differentiable functions. In [6, 7], the authors generalized Gâteaux derivative by employing a fractional discrete operator for a Jumarie fractional operator. The method of finding solutions was based on critical point theorems of finite dimensional Banach spaces. Other techniques can be found in  and .
Investigators in the areas of control systems, neural networks, computer science, food processing and economics rely on mathematical modeling because it surely affects nonlinear difference equations. Consequently, many novelists have widely established various procedures and patterns, such as fixed point theorems, upper and lower solutions and Brouwer degree, to consider discrete problems [10, 11]. The investigation of fractional differential equations (FDE) was started to establish the existence and uniqueness of findings for various classes of FDE. Moreover, the theory of integro-differential equations has been almost settled parallel to the theory of FDE. However, the theory of fractional difference equations led to a very minor development of itself.
This paper aims to study a fractional difference equation of two point BVP type, which was realized as the discrete BVP. Certain classes are formulated in which the discrete boundary value problems will have a single solution. The novelty hither comprises a method selection of metric and employment of Hölder’s inequality. This investigation allows the related functions to be contractive, which were earlier non-contractive in classical regularities. This work, unlike those which appeared in the current designs, consequently grants the enhanced applications of Banach’s fixed point theorem for classifying an extensive framework of issues.
2 Main methods
This section concerns some concepts as well as preliminaries.
The above formula is hard to study the properties of findings (solutions). Therefore, Mohan and Deekshitulu  introduced the following formula for \(\kappa=k=1\).
3 Main findings
The main result is as follows.
As a special case, for \(p=q=2\), we have the following result.
The authors would like to thank the referees for giving very useful suggestions for improving the work. This research is supported by Project No. RG312-14AFR from the University of Malaya.
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