ANALYSIS SEQUENTIAL FRACTIONAL DIFFERENCES AND RELATED MONOTONICITY RESULTS

This paper employs the monotonicity analysis for non-negativity to derive a class of sequential fractional backward differences of Riemann-Liouville type a+1RL∇ν aRL∇αu (t) based on a certain subspace in the parameter space (0, 1) × (0, 1). Auxiliary and restriction conditions are included in the monotonicity results obtained in this paper and they confirm the monotonicity of the function on {a + 2,a + 3,...}. A non-monotonicity result is also established based on the main conditions together with further dual conditions, and this confirms that the main theorem is almost sharp. Furthermore, we recast the dual conditions in a sing condition, and then we represent the sharpness result in a new corollary. Finally, numerical results via MATLAB software are used to illustrate the main mathematical results for some special cases.