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.

In this paper, we study a modified extragradient method for computing a common solution to the split equilibrium problem and fixed point problem of a nonexpansive semigroup in real Hilbert spaces. The weak and strong convergence characteristics of the proposed algorithm are investigated by employing suitable control conditions in such a setting of spaces. As a consequence, we provide a simplified analysis of various existing results concerning the extragradient method in the current literature. We also provide a numerical example to strengthen the theoretical results and the applicability of the proposed algorithm.

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.

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