VIABILITY FOR DELAY FRACTIONAL DIFFERENTIAL INCLUSIONS

Tóm tắt

Viability theory designs and develops mathematical and algorithmic methods that can be found in many domains such as living beings, biological evolution, economics, environmental sciences, financial markets or control theory, and robotics. In the paper, we provide sufficient conditions assuring the existence of viable solutions of differential inclusions with fractional derivatives with delay: ????????????????????????????????(????????)∈????????(????????,????????????????),0<????????<1,????????∈????????≔[0,????????].
We were inheriting the ideas of Carja, Donchev, Rafaqat, and Ahmed (2014) and Girejko (2018), we give the condition of tangency and the concept of approximate solution, these are compatible with our problem. Thanks to Brezis-Browder Theorem, we prove the existence of an approximate solution to the interval [0,T]. Then, passing to the limit, the sequence of approximate solutions convergent to the viable solution. These results generalize the corresponding results by Carja et al. (2014), Girejko, Mozyrska, and Wyrwas (2011), Vasundgaradevi and Lakshmikantham (2009).