In this article, we utilize recent generalized fractional operators to establish some fractional inequalities in Hermite–Hadamard and Minkowski settings. It is obvious that many previously published inequalities can be derived as particular cases from our outcomes. Moreover, we articulate some flaws in the proofs of recently affiliated formulas by revealing the weak points and introducing more rigorous proofs amending and expanding the results.
A new model for the growth of a single-species with a two-stage structure is developed employing a state-dependent age of maturity. The paper discusses some remarks in a previous model where the time to maturity is state dependent. Then, we present a new model which is mainly based on making the maturation period of juveniles depend on the total population size not at the present time t but at an earlier time stage, namely, the time when they were born. The paper considers both biological and mathematical aspects. Emphasis is given to the basic theory of the solutions, such as local and global properties of existence, uniqueness, positivity and boundedness.
This paper is devoted to proving some new fractional inequalities via recent generalized fractional operators. These inequalities are in the Hermite–Hadamard and Minkowski settings. Many previously documented inequalities may clearly be deduced as specific examples from our findings. Moreover, we give some comparative remarks to show the advantage and novelty of the obtained results.
We explore a general type of stable Bessel beams in graded index media. The proposed axially symmetric medium is characterized by an "α" index profile. Explicit solutions for the radial envelope of the field E(r) are derived in terms of generalized Bessel functions. Emphasis is given on illustrating how far the conditions of the proposed modified structure permit only a Bessel function of the first kind to be uniquely retained in the solution. This paper considers both the optical and mathematical aspects. Some numerical examples corroborating our theoretical results are included, showing the stability, propagation, and diffraction of such Bessel beams.
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