In this paper, we study the local and global existence, and uniqueness of mild solution to initial value problems for fractional semilinear evolution equations with compact and noncompact semigroup in Banach spaces. In particular, we derive the form of fundamental solution in terms of semigroup induced by resolvent and ψ-function from Caputo fractional derivatives. These results generalize previous work where the classical Caputo fractional derivative is considered. Moreover, we prove the Mittag-Leffler–Ulam–Hyers stability result. Finally, we give examples of time-fractional heat equation to illustrate the result.

Gifted students need a form of special education through extracurricular and learning experiences because they have extraordinary potential in terms of intelligence, creativity, social and mentality, compared to other students. This study aims to investigate various teaching and learning approaches designed for gifted students, and to constitute appropriate techniques that enhance their creative thinking in mathematics. The participants of this study were first-year gifted students enrolled in fundamental mathematics courses towards their undergraduate engineering programs at a science and technology-based university in Thailand. This study discovered that an adequate use of challenge-based learning, problem solving process, project-based learning, well-designed questions and in-depth learning style in the classroom effectively fostered their insightful and creative thought. Moreover, an online tool such as Facebook could be used as a learning platform outside the classroom to engage them in online discussion and collaboration and to challenge their thinking. Ultimately, based on students’ feedbacks and evaluation, this study found that a combination of these techniques and methods built up a more comfortable and motivating atmosphere that helped them for creative thinking and corresponded to their needs and satisfaction.

Biological pest control has a strong advantage in its non-chemical effects on the environment. In this study, a fixed-time synergetic control scheme for the biological pest control problems represented by the n-dimensional Lotka-Volterra model was proposed. The proof of stability shows that the proposed controller can regulate the biological pest control systems with the characteristic of fixed-time convergence. The performance of the proposed scheme was demonstrated through simulation studies. The simulation results show that the pre-specified bound of the settling time can be satisfied regardless of the initial conditions, confirming a desired fixed-time convergence characteristic. Moreover, unlike the existing control policy based on the sliding mode control, the control inputs of the proposed policy are free from chattering.