Abstract
Our manuscript is devoted to investigating a class of impulsive boundary value problems under the concept of the Riemann-Liouville fractional order derivative. The subject problem is of implicit type. We develop some adequate conditions for the existence and puniness of a solution to the proposed problem. For our required results, we utilize the classical fixed point theorems from Banach and Scheafer. It is to be noted that the impulsive boundary value problem under the fractional order derivative of the Riemann-Liouville type has been very rarely considered in literature. Finally, to demonstrate the obtained results, we provide some pertinent examples.
Highlights
The fractional order differential equations are the generalization of the ordinary differential equations of the integer order
In 1823, another mathematician by the name of Lacroix, introduced the fractional derivative [1] of simple power function. This area has been studied by many researchers because it has significant applications in various fields of science and technology in mathematical modeling of different fields of Science and Technology
Some phenomena including the diffusion process [2], some chemical processes of electrochemistry [3], infectious disease in biology [4], signal and image processing [5], dynamic processes [6], and systems control theory [7] can be excellently described by using fractional order differential equations (FODEs) instead of the ordinary derivative
Summary
The fractional order differential equations (abbreviated as FODEs) are the generalization of the ordinary differential equations of the integer order. In 1823, another mathematician by the name of Lacroix, introduced the fractional derivative [1] of simple power function This area has been studied by many researchers because it has significant applications in various fields of science and technology in mathematical modeling of different fields of Science and Technology. We remark that a very basic and important qualitative problem in the investigation of IDEs with a fractional order concerns the existence theory of solutions For these purposes, researchers have used the classical fixed point theory and some tools of nonlinear analysis. In [42], the authors have applied fixed point results to develop the corresponding existence theory of solutions by using the Caputo derivative of the fractional order. For the demonstration of our results, we provide some concrete examples
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