With the help of the topological degree theory in this manuscript, we develop qualitative theory for a class of multi-terms fractional order differential equations (FODEs) with proportional delay using the Caputo derivative. In the same line, we will also study various forms of Ulam stability results. To clarify our theocratical analysis, we provide three different pertinent examples.
The pantograph equation is a special type of functional differential equations with proportional delay. The present study introduces a compound technique incorporating the perturbation method with an iteration algorithm to solve numerically the delay differential equations of pantograph type. We put forward two types of algorithms, depending upon the order of derivatives in the Taylor series expansion. The crucial convenience of this method when compared with other perturbation methods is that this method does not require a small perturbation parameter. Furthermore, a relatively fast convergence of the iterations to the exact solutions and more accurate results can be achieved. Several illustrative examples are given to demonstrate the efficiency and reliability of the technique, even for nonlinear cases.
Global Exponential Stability Criteria for Proportional Delay High-Order Neural Networks: A Hyper-Exponential Stability Technique