New dynamic Hilbert-type inequalities in two independent variables involving Fenchel\u2013Legendre transform

By using the theory of calculus on time scales and some mathematical methods, several dynamic inequalities on time scales are established. Based on these results, we derive some sufficient conditions for permanence of predator‐prey system incorporating a prey refuge on time scales. Finally, examples and numerical simulations are presented to illustrate the feasibility and effectiveness of the results.

The purpose of this paper is to investigate some new dynamic inequalities on time scales. We establish some new dynamic inequalities; the results unify and extend some continuous inequalities and their corresponding discrete analogues. The inequalities given here can be used as tools in the qualitative theory of certain dynamic equations. Some examples are given in the end of this paper.

<abstract><p>In this work, we prove several new $ (\gamma, a) $-nabla Bennett and Leindler dynamic inequalities on time scales. The results proved here generalize some known dynamic inequalities on time scales, unify and extend some continuous inequalities and their corresponding discrete analogues. Our results will be proved by using integration by parts, chain rule and Hölder inequality for the $ (\gamma, a) $-nabla-fractional derivative on time scales.</p></abstract>

In this article, we will prove some new diamond alpha Hilbert-type dynamic inequalities on time scales which are defined as a linear combination of the nabla and delta integrals. These inequalities extend some known dynamic inequalities on time scales, and unify and extend some continuous inequalities and their corresponding discrete analogues. Our results will be proven by using some algebraic inequalities, diamond alpha Hölder inequality, and diamond alpha Jensen’s inequality on time scales.

In this article, we obtain some new dynamic inequalities of Hardy type on time scales. The main results are derived using Fubini’s theorem and the chain rule on time scales. We apply the main results to the continuous calculus, discrete calculus, and q-calculus as special cases.

In the present article, we investigate some new inequalities of Steffensen type on an arbitrary time scale using the diamond‐α dynamic integrals, which are defined as a linear combination of the delta and nabla integrals. The obtained inequalities extend some known dynamic inequalities on time scales and unify and extend some continuous inequalities and their discrete analogues.

We prove some new inequalities of Mitrinović–Pečarić inequalities for convex functions on an arbitrary time scale using delta integrals. These inequalities extend and improve some known dynamic inequalities in the literature. The main results will be proved by using Hölder and Jensen inequalities and a simple consequence of Keller’s and Poetzsche’s chain rules on time scales.

In this paper, we present extensions of dynamic Lyapunov's inequalities and their reverse versions on time scales by using Specht's and Kantorovich's ratios. Our approach unifies and extends some continuous inequalities and their corresponding discrete and quantum analogues.

We propose a hybrid Leslie–Gower predator-prey system with feedback control and hunting cooperation among predators on nonuniform time domain. Our proposed model commutes between a continuous-time dynamical system and discrete-time dynamical system. Using comparison theorems and some dynamic inequality on the time scale, we study the permanence results for the proposed system. Also, with the help of a suitable Lyapunov function, we establish some sufficient conditions that will ensure the global attractivity of the proposed model. Furthermore, simulation studies are provided to verify the theoretical results, as well as to further investigate the effects of feedback control strategy and hunting cooperation on the predator-prey system. These findings suggest that hunting cooperation and feedback control strategy have a substantial influence on controlling the population of a predator-prey dynamical system.

In this paper, we prove some new Opial-type inequalities for two unknowns and general kernels on time scales. Also, we prove some new dynamic Opial-type inequalities of higher-order involving two different weight functions. The proofs use Holder’s inequality, the reverse Holder’s inequality, the chain rule, and a power rule of integration on time scales.

<abstract><p>In the current manuscript, new alpha delta dynamic Hardy-Hilbert inequalities on time scales are discussed. These inequalities combine and expand a number of continuous inequalities and their corresponding discrete analogues in the literature. We shall illustrate our results using Hölder's inequality on time scales and a few algebraic inequalities.</p></abstract>

In this paper, we consider generalized conformable integrals to establish new Opial-type inequalities. The obtained results mainly depend on Hölder’s inequality, some algebraic inequalities, and a simple consequence of Keller’s chain rule on time scales. Our obtained results unify and extend some continuous and discrete inequalities. In the special case α=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\alpha = 1$\\end{document}, our results cover some well-known inequalities of Opial-type on time scales.

In this research, we aim to explore generalizations of Hardy-type inequalities using nabla Hölder’s inequality, nabla Jensen’s inequality, chain rule on nabla calculus and leveraging the properties of convex and submultiplicative functions. Nabla calculus on time scales provides a unified framework that unifies continuous and discrete calculus, making it a powerful tool for studying various mathematical problems on time scales. By utilizing this approach, we seek to extend Hardy-type inequalities beyond their classical continuous or discrete settings to a more general time scale domain. As specific instances of our discoveries, we have the integral inequalities previously established in the existing literature.

This work presents new results concerning weighted Hardy-type inequalities within the framework of delta conformable fractional integrals on arbitrary time scales. The proposed approach unifies the treatment of inequalities across continuous and discrete domains, enabling the derivation of original forms in both settings. The obtained results exhibit symmetry with classical inequalities, and several integral and discrete inequalities arise as special cases. These findings extend and generalize known results and enrich the theory of integral inequalities in fractional and dynamic calculus, providing a versatile platform for further developments in symmetric and weighted inequality analysis.

Three coastal cliff erosion systems on the east coast of England are used to illustrate the importance of spatial and temporal scales. These systems are described in terms of location, dominant processes and erosion rates. The time scales suggested by Schumm and Lichty (1965) are applied to these systems, and the existence of dynamic equilibrium conditions is emphasized. The relative importance of the catastrophic event (storm surge) and the moderate storm is discussed, and is shown to be a function of the time scale under consideration. SCHUMM and Lichty (1965) have examined the concept of scale in geomorphology, and have shown that the distinction between cause and effect in the development of landforms is a function of time and space. They applied these concepts to the study of river systems, and showed that the factors that determine the character of landforms can be either dependent or independent variables as the limits of time and space change. It is the purpose of this paper to apply the time scales and related concepts proposed by Schumm and Lichty to coastal erosion systems. The soft rock cliffs on the east coast of England will be used as examples; these cliffs provide a good starting point, for their short-term changes are appreciable. Current emphasis in geomorphology has indicated that systems analysis can provide a useful framework in which to set our abstractions of the real world (Harvey, 1969; Chorley and Kennedy, 1971). One of the many advantages is that a particular system may be divided into sub-systems which may be studied separately. This paper concentrates on a particular type of coastal system, the coastal cliff erosion system. This particular system may be divided into three sub-systems: the cliff, the beach and the offshore sub-system; all these will be discussed but particular emphasis will be placed on the cliff sub-system. The cliff sub-system may be defined as the zone extending io m landward of the cliff-top edge to the cliff toe, and the beach subsystem extends from the cliff toe to the mean low-water mark of spring tides. The offshore sub-system has been taken as extending from mean low-water mark of spring tides to a depth of 20 m, a distance of approximately 7 km along most of this coast. The main variable in the present study is the retreat of the cliff-top edge over a period of years. Inevitably the actual method used to measure the retreat rate is dependent upon the time scale; for instance in the short term (i to 2 years) cliff retreat may be measured in the field, while over the longer term (ioo years) cliff retreat rates may be obtained by comparing different map editions. Since sea level rose to near the present position (approximately 5000 years B.P., Jelgersma, 1971) no long-term value for cliff retreat exists. A study of the retreat rates for any length of cliff will show considerable variation in both spatial and temporal dimensions, and it is this variation that has yet to be explained in geomorphology. Location of the coastal systems The areas studied are all lengths of unconsolidated Quaternary cliffs (the term consolidation here is not used in the strict soil mechanics sense where it refers to a specific overburden weight), and are all on the east coast of England, bordering the North Sea (see Fig. i). The Holderness