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In this paper, we will study solution existence and its stability for hybrid fractional DE with fractional integral, fractional differential derivative and Φp-operator in Caputo sense. Our problem deals with two basic types of fractional order derivatives, that is, Riemann-Liouville derivative of order $\delta $ and Caputo fractional derivative of order $\lambda , $ where $n - 1 \lt \lambda ,\sigma \le n, $ and $n \ge 3. $ We will transform the problem into an integral equation by using Green function and examine whether it is increasing or decreasing and positive or negative function. Some fixed point theorems (Krasnoselskii Theorem) are utilized for the existence of a positive solution (EPS). Addition to studying HU-stability technique for our suggested problem. An example is included to apply the results.

Orphaned colonies of Cryptotermes domesticus readily produce replacement reproductives and continue propagation. In this study, we aimed to investigate the production and fecundity of neotenic reproductives in 5-year-old colonies of C. domesticus after orphaning. All 15 experimental colonies were successfully re-established by the neotenic reproductive pair. Three types of neotenic reproductives with various wing-bud lengths were observed: type I with micro wing buds, type II with short wing buds, and type III with long wing buds. Four patterns of pairs made up of these neotenics, namely, type I + type II, type I + type III, type II + type II, and type II + type III, exhibited reproductive capacities similar to those of the primary reproductive pair. We speculated that these neotenic reproductives were derived from various nymphal instars. The 5-year-old colonies had three instars of nymphs, with the majority being in the second instar, followed by the first. Thus, the combination of neotenic reproductives with short wing buds and micro wing buds was the dominant differentiation pathway of the orphaned colonies. After the removal of the original primary reproductive pair, the nymphs matured into neotenic reproductives and took over reproduction in the colony in 107.40 ± 15.18 days. This study highlights the importance of quarantine and routine inspection of wood, as well as the significance of early prevention and control of C. domesticus infestation in wood. Moreover, this study confirms the high differentiation and reproductive capacities of C. domesticus. [ABSTRACT FROM AUTHOR]

In this article, we consider the inverse nodal problem for the Sturm–Liouville operator arisen from acoustic scattering problems. We show that a dense subset of nodal points can uniquely determine the potential on the finite interval [0, 1] and coefficientaof the boundary condition and provide a constructive procedure for the solution of the inverse nodal problem. A corollary of this theorem is to reconstruct the spherically symmetric speed of sound on the interval [0, b] for acoustic scattering problems from nodal data. [ABSTRACT FROM AUTHOR]