Fractional Inequalities Research Articles

Foundations of generalized Prabhakar-Hilfer fractional calculus with applications

Here we introduce the generalized Prabhakar fractional calculus and we also combine it with the generalized Hilfer calculus. We prove that the generalized left and right side Prabhakar fractional integrals preserve continuity and we find tight upper bounds for them. We present several left and right side generalized Prabhakar fractional inequalities of Hardy, Opial and Hilbert-Pachpatte types. We apply these in the setting of generalized Hilfer calculus.

On Multivalent Analytic Functions Considered by a Multi-Arbitrary Differential Operator in a Complex Domain

(1) Background: There is an increasing amount of information in complex domains, which necessitates the development of various kinds of operators, such as differential, integral, and linear convolution operators. Few investigations of the fractional differential and integral operators of a complex variable have been undertaken. (2) Methods: In this effort, we aim to present a generalization of a class of analytic functions based on a complex fractional differential operator. This class is defined by utilizing the subordination and superordination theory. (3) Results: We illustrate different fractional inequalities of starlike and convex formulas. Moreover, we discuss the main conditions to obtain sandwich inequalities involving the fractional operator. (4) Conclusion: We indicate that the suggested class is a generalization of recent works and can be applied to discuss the upper and lower bounds of a special case of fractional differential equations.

Discrete Fractional Boundary Value Problems and Inequalities

Abstract In this paper, a general nonlinear discrete fractional boundary value problem is considered, of order between one and two. The main result is an existence theorem, proving the existence of at least one solution to the boundary value problem, subject to validity of a certain key inequality that allows unrestricted growth in the problem. The proof of this existence theorem is accomplished by using Brouwer's fixed point theorem as well as two other main results of this paper, namely, first, a result showing that the solutions of the boundary value problem are exactly the solutions to a certain equivalent integral representation, and, second, the establishment of solutions satisfying certain a priori bounds provided the key inequality holds. In order to establish the latter result, several novel discrete fractional inequalities are developed, each of them interesting in itself and of potential future use in different contexts. We illustrate the usefulness of our existence results by presenting two examples.

New Fractional Dynamic Inequalities via Conformable Delta Derivative on Arbitrary Time Scales

Building on the work of Josip Pečarić in 2013 and 1982 and on the work of Srivastava in 2017. We prove some new α-conformable dynamic inequalities of Steffensen-type on time scales. In the case when α=1, we obtain some well-known time scale inequalities due to Steffensen inequalities. For some specific time scales, we further show some relevant inequalities as special cases: α-conformable integral inequalities and α-conformable discrete inequalities. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities.

New Fractional Hermite–Hadamard–Mercer Inequalities for Harmonically Convex Function

In 2003, Mercer presented an interesting variation of Jensen’s inequality called Jensen–Mercer inequality for convex function. In the present paper, by employing harmonically convex function, we introduce analogous versions of Hermite–Hadamard inequalities of the Jensen–Mercer type via fractional integrals. As a result, we introduce several related fractional inequalities connected with the right and left differences of obtained new inequalities for differentiable harmonically convex mappings. As an application viewpoint, new estimates regarding hypergeometric functions and special means of real numbers are exemplified to determine the pertinence and validity of the suggested scheme. Our results presented here provide extensions of others given in the literature. The results proved in this paper may stimulate further research in this fascinating area.

Generalizations of the Nonlinear Henry Inequality and the Leray–Schauder Type Fixed Point Theorem with Applications to Fractional Differential Inclusions

The authors give some singular versions of the Gronwall–Bihari–Henry inequalities. They also establish a multivalued version of the Leray–Schauder alternative in strictly star-shaped sets. Based on these new fractional inequalities and fixed point theorem, they study an initial value problem for fractional differential inclusions with delay.

Adaptive Synchronization of Fractional-Order Complex-Valued Chaotic Neural Networks with Time-Delay and Unknown Parameters

The purpose of this paper is to study and analyze the concept of fractional-order complex-valued chaotic networks with external bounded disturbances and uncertainties. The synchronization problem and parameter identification of fractional-order complex-valued chaotic neural networks (FOCVCNNs) with time-delay and unknown parameters are investigated. Synchronization between a driving FOCVCNN and a response FOCVCNN, as well as the identification of unknown parameters are implemented. Based on fractional complex-valued inequalities and stability theory of fractional-order chaotic complex-valued systems, the paper designs suitable adaptive controllers and complex update laws. Moreover, it scientifically estimates the uncertainties and external disturbances to establish the stability of controlled systems. The computer simulation results verify the correctness of the proposed method. Not only a new method for analyzing FOCVCNNs with time-delay and unknown complex parameters is provided, but also a sensitive decrease of the computational and analytical complexity.

Nonexistence Results for Higher Order Fractional Differential Inequalities with Nonlinearities Involving Caputo Fractional Derivative

Higher order fractional differential equations are important tools to deal with precise models of materials with hereditary and memory effects. Moreover, fractional differential inequalities are useful to establish the properties of solutions of different problems in biomathematics and flow phenomena. In the present work, we are concerned with the nonexistence of global solutions to a higher order fractional differential inequality with a nonlinearity involving Caputo fractional derivative. Namely, using nonlinear capacity estimates, we obtain sufficient conditions for which we have no global solutions. The a priori estimates of the structure of solutions are obtained by a precise analysis of the integral form of the inequality with appropriate choice of test function.

Nonlocal Lazer\u2013McKenna-type problem perturbed by the Hardy\u2019s potential and its parabolic equivalence

In this paper, we study the effect of Hardy potential on the existence or nonexistence of solutions to the following fractional problem involving a singular nonlinearity: \t\t\t{(−Δ)su=λu|x|2s+μuγ+fin Ω,u>0in Ω,u=0in (RN∖Ω).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} \\textstyle\\begin{cases} (-\\Delta )^{s} u = \\lambda \\frac{u}{ \\vert x \\vert ^{2s}} + \\frac{\\mu }{u^{\\gamma }}+f & \\text{in } \\Omega, \\\\ u>0 & \\text{in } \\Omega, \\\\ u=0 & \\text{in } (\\mathbb{R}^{N} \\setminus \\Omega ). \\end{cases}\\displaystyle \\end{aligned}$$ \\end{document} Here 0 < s<1, lambda >0, gamma >0, and Omega subset mathbb{R}^{N} (N > 2s) is a bounded smooth domain such that 0 in Omega . Moreover, 0 leq mu,f in L^{1}(Omega ). For 0< lambda leq Lambda _{N,s}, Lambda _{N,s} being the best constant in the fractional Hardy inequality, we find a necessary and sufficient condition for the existence of a positive weak solution to the problem with respect to the data μ and f. Also, for a regular datum of f, under suitable assumptions, we obtain some existence and uniqueness results and calculate the rate of growth of solutions. Moreover, we mention a nonexistence and a complete blowup result for the case lambda > Lambda _{N,s}. Besides, we consider the parabolic equivalence of the above problem in the case mu equiv 1 and some suitable f(x,t), that is, \t\t\t{ut+(−Δ)su=λu|x|2s+1uγ+f(x,t)in Ω×(0,T),u>0in Ω×(0,T),u=0in (RN∖Ω)×(0,T),u(x,0)=u0in RN,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} \\textstyle\\begin{cases} u_{t}+(-\\Delta )^{s} u = \\lambda \\frac{u}{ \\vert x \\vert ^{2s}} + \\frac{1}{u^{\\gamma }}+f(x,t) & \\text{in } \\Omega \\times (0,T), \\\\ u>0 & \\text{in } \\Omega \\times (0,T), \\\\ u =0 & \\text{in } (\\mathbb{R}^{N} \\setminus \\Omega ) \\times (0,T), \\\\ u(x,0)=u_{0} & \\text{in } \\mathbb{R}^{N}, \\end{cases}\\displaystyle \\end{aligned}$$ \\end{document} where u_{0} in X_{0}^{s}(Omega ) satisfies an appropriate cone condition. In the case 0<gamma leq 1 or gamma >1 with 2s(gamma -1)<(gamma +1), we show the existence of a unique solution for any 0< lambda < Lambda _{N,s} and prove a stabilization result for certain range of λ.

Impulsive Fractional Cohen-Grossberg Neural Networks: Almost Periodicity Analysis

In this paper, a fractional-order Cohen–Grossberg-type neural network with Caputo fractional derivatives is investigated. The notion of almost periodicity is adapted to the impulsive generalization of the model. General types of impulsive perturbations not necessarily at fixed moments are considered. Criteria for the existence and uniqueness of almost periodic waves are proposed. Furthermore, the global perfect Mittag–Leffler stability notion for the almost periodic solution is defined and studied. In addition, a robust global perfect Mittag–Leffler stability analysis is proposed. Lyapunov-type functions and fractional inequalities are applied in the proof. Since the type of Cohen–Grossberg neural networks generalizes several basic neural network models, this research contributes to the development of the investigations on numerous fractional neural network models.

Normalized fractional inequalities for continuous random variables

We introduce new normalized fractional integral concepts on random variables. We also introduce the fractional coupled random variables. The Jensen integral inequality is generalized. Also, some fractional bounds estimating the expectation and variance are delivered. At the end, some other results related to the coupled random variables are proved. For this work, some results of the papers [On some statistical and probability inequalities. Journal of Inequalities and Special Functions, 2016] and [Some inequalities for the expectation and variance of a random variable whose PDFs are absolutely continuous using a pre-Chebychev inequality. Tamkang Journal of Mathematics, 2001] are deduced as some special cases.

On Opial-Type Inequalities for Superquadratic Functions and Applications in Fractional Calculus

We have studied the Opial-type inequalities for superquadratic functions proved for arbitrary kernels. These are estimated by applying mean value theorems. Furthermore, by analyzing specific functions, the fractional integral and fractional derivative inequalities are obtained.

A non-local coupling model involving three fractional Laplacians

In this paper, we study a non-local diffusion problem that involves three different fractional Laplacian operators acting on two domains. Each domain has an associated operator that governs the diffusion on it, and the third operator serves as a coupling mechanism between the two of them. The model proposed is the gradient flow of a non-local energy functional. In the first part of the paper, we provide results about existence of solutions and the conservation of mass. The second part encompasses results about the [Formula: see text] decay of the solutions. The third part is devoted to study, the asymptotic behavior of the solutions of the problem when the two domains are a ball and its complementary. Exterior fractional Sobolev and Nash inequalities of independent interest are also provided in Appendix A.

Discrete Fractional Inequalities Pertaining a Fractional Sum Operator with Some Applications on Time Scales

This content replicates some discrete nonlinear fractional inequalities by virtue of the fractional sum operator Ψ ¯ on time scales. Through the recognition of the principle of discrete fractional calculus, we are able to recover the precise estimates for unknown functions of inequalities of the Gronwall type. The resultant inequalities are of unique structure comparative with the latest reviewing disclosures and can be described as a complementary tool for numerically testing the solutions of discrete partial differential equations. The foremost consequences are probably confirmed via handling of assessment procedure and technique of mean value speculation. We display few examples of the proposed inequalities to represent the incentives of our effort.

Complex left Caputo fractional inequalities

"Here we present several complex left Caputo type fractional inequalities of well known kinds, such as of Ostrowski, Poincare, Sobolev, Opial and Hilbert- Pachpatte."

Some New Harmonically Convex Function Type Generalized Fractional Integral Inequalities

In this article, we established a new version of generalized fractional Hadamard and Fejér–Hadamard type integral inequalities. A fractional integral operator (FIO) with a non-singular function (multi-index Bessel function) as its kernel and monotone increasing functions is utilized to obtain the new version of such fractional inequalities. Our derived results are a generalized form of several proven inequalities already existing in the literature. The proven inequalities are useful for studying the stability and control of corresponding fractional dynamic equations.

Fractional Trapezium-Like Inequalities Involving Generalized Relative Semi-(m, h1, h2)-Preinvex Mappings on an m-Invex Set

Fractional Trapezium-Like Inequalities Involving Generalized Relative Semi-(m, h1, h2)-Preinvex Mappings on an m-Invex Set

New generalization of reverse Minkowski's inequality for fractional integral

In this research, we introduce some new fractional integral inequalities of Minkowski’s type by using Riemann-Liouville fractional integral operator. We replace the constants that appear on Minkowski’s inequality by two positive functions. Further, we establish some new fractional inequalities related to the reverse Minkowski type inequalities via Riemann-Liouville fractional integral. Using this fractional integral operator, some special cases of reverse Minkowski type are also discussed.

Some generalized fractional integrals inequalities for a class of functions

In this work,some fractional integrals inequalities are obtained by using a more general fractional integral operator acting on a convex function.

Inequalities for Riemann–Liouville Fractional Integrals of Strongly s , m -Convex Functions

The results of this paper provide two Hadamard-type inequalities for strongly s , m -convex functions via Riemann–Liouville fractional integrals and error estimations of well-known fractional Hadamard inequalities. Their special cases are given and connected with the results of some published papers.