Ravi Agarwal

• Ph.D.
• CEO at Texas A&M University – Kingsville

This article introduces a function space \(\mathcal {S}_{\textbf{A}}(\mathbb {R})\) in the offset linear canonical domain, which is a counterpart of the Schwartz space \(\mathcal {S}(\mathbb {R})\) in the Fourier domain, and studies the properties of offset linear canonical transform (OLCT) on this space. Additionally, the paper introduces a new de...

In this paper, we introduce a novel approach employing two-dimensional uniform and non-uniform Haar wavelet collocation methods to effectively solve the generalized Burgers-Huxley and Burgers-Fisher equations. The demonstrated method exhibits an impressive quartic convergence rate. Several test problems are presented to exemplify the accuracy and e...

In this article, we propose the fourth order non-self-adjoint system of SBVPs to investigate which arise in the theory of epitaxial growth by considering equation $$ \frac{1}{r^{\beta}} \left \{ r^{\beta}\left[\frac{1}{r^{\beta}}(r^{\beta} \Theta')^{'}\right]'\right \}^{'}=\frac{1}{2 r^{\beta}}\left(K_{11}\left( \mu'\Theta'^{2}+2 \mu \Theta^{'} \T...

Taylor series method is a simple analytical method, which is accessible to all non-mathematician, has slow convergence. This paper develops a new Taylor series based numerical method to overcome the shortcoming of the Taylor series while maintaining its simplicity. Some examples are given, showing its reliability and efficiency. The proposed method...

With the increasing importance of the Mittag-Leffler function in physical applications, these days many researchers are studying various generalizations and extensions of the Mittag-Leffler function. In this paper, efforts are made to define the bicomplex extension of the Mittag-Leffler function, and also its analyticity and region of convergence a...

A model of gene regulatory networks with generalized proportional Caputo fractional derivatives is set up, and stability properties are studied. Initially, some properties of absolute value Lyapunov functions and quadratic Lyapunov functions are discussed, and also, their application to fractional order systems and the advantage of quadratic functi...

In this paper, first we prove some new refinements of discrete weighted inequalities with negative powers on finite intervals. Next, by employing these inequalities, we prove that the self-improving property ( backward propagation property ) of the weighted discrete Muckenhoupt classes holds. The main results give exact values of the limit exponent...

In the present paper, we establish an efficient numerical scheme based on weakly L-stable time integration convergent formula and nonstandard finite difference (NSFD) scheme. We solve Burger’s equation with Dirichlet boundary conditions as well as Neumann boundary conditions. We also solve the Fisher equation. We use Hermite approximation polynomia...

In this work, we focus on the non-linear fourth order class of singular boundary value problem which contains the parameter $\lambda$. We convert this non-linear differential equation into third order non-linear differential equation. The third order problem is singular, non self adjoint and nonlinear. Moreover, depending upon $\lambda$, it admits...

In this paper, we prove that the self-improving property of the weighted Gehring class $G_{\lambda }^{p}$ G λ p with a weight λ holds in the non-homogeneous spaces. The results give sharp bounds of exponents and will be used to obtain the self-improving property of the Muckenhoupt class $A^{q}$ A q . By using the rearrangement (nonincreasing rearra...

This article aims to prove the existence of a solution and compute the region of existence for a class of four-point nonlinear boundary value problems (NLBVPs) defined as, -u''(x)=\psi(x,u,u'), \quad x\in (0,1), u'(0)=\lambda_{1}u(\xi), u'(1)=\lambda_{2} u(\eta), where I=[0,1], 0<\xi\leq\eta<1 and \lambda_1,\lambda_2> 0. The non linear source term...

We first prove some new weighted refinements of inequalities of Hardy’s type with negative powers. Next, we prove that any \(A_{\lambda }^{1}\) Muckenhoupt class with a weight \(\lambda \) belongs to some weighted Gehring class \(G_{\lambda }^{p}\) for \(p>1\). We also prove that the self-improving property of the weighted Muckenhoupt class \(A_{\l...

Purpose In this article, the authors consider the following nonlinear singular boundary value problem (SBVP) known as Lane–Emden equations, − u ″( t )-( α / t ) u ′( t ) = g ( t , u ), 0 < t < 1 where α ≥ 1 subject to two-point and three-point boundary conditions. The authors propose to develop a novel method to solve the class of Lane–Emden equati...

In this paper, we will prove some fundamental properties of the discrete power mean operator Mpun=1/n∑k=1n upk1/p,for n∈I⊆ℤ+, of order p, where u is a nonnegative discrete weight defined on I⊆ℤ+ the set of the nonnegative integers. We also establish some lower and upper bounds of the composition of different operators with different powers. Next, w...

In this article, we come up with a novel numerical scheme based on Haar wavelet (HW) along with nonstandard ‎finite difference (NSFD) scheme to solve time-fractional Burgers’ equation with variable diffusion coefficient and ‎time delay. In the solution process, we discretize the fractional time derivative by NSFD ‎ formula and spatial ‎derivative b...

In this article we consider the fourth order non-self-adjoint singular boundary value problem $$ \frac{1}{r}\left[ r \left\lbrace \frac{1}{r} \left(r \phi' \right)^{'} \right\rbrace^{'}\right]^{'}=\frac{\phi' \phi''}{r}+\lambda,$$ with $\lambda$ as a parameter measures the speed at which new particles are deposited. This differential equation is...

In this paper, we prove some basic properties of the discrete Muckenhoupt class Ap\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{A}^{p}$\end{document} and the...

In this paper, we will prove a discrete Rubio De Francia extrapolation theorem in the theory of discrete Ap? Muckenhoupt weights for which the discrete Hardy-Littlewood maximal operator is bounded on ?p w (Z+). The results will be proved by employing the self-improving property of the discrete Ap? Muckenhoupt weights and the Marcinkiewicz Interpola...

Lane-Emden type equations arise various physical phenomena in mathematical and astrophysics like stellar structure, thermionic currents, thermal explosions, radiative cooling, CTC, etc. In this work, we consider a model by considering the equation (x^{\beta}y'(x))'+x^{\beta}f(x,y)=0, 0<x<1, y'(0)=0, b_{1}y(1)+a_{1}y'(1)=c_{1}. For $\beta=1$ and $\b...

In this article, we propose novel coupled nonlinear singular boundary value problems arising in epitaxial growth theory. The coupled equations are nonlinear, non‐self‐adjoint, and singular and have no exact solutions. We derive some qualitative properties of the coupled solutions, which depend on the size of parameters that occur in the coupled sys...

This paper firstly studies an SIR (susceptible-infectious-recovered) epidemic model without demography and with no disease mortality under both total and under partial quarantine of the susceptible subpopulation or of both the susceptible and the infectious ones in order to satisfy the hospital availability requirements on bed disposal and other ne...

In this paper, we propose a $7^{th}$ order weakly $L$-stable time integration scheme. In the process of derivation of the scheme, we use explicit backward Taylor's polynomial approximation of sixth-order and Hermite interpolation polynomial approximation of fifth order. We apply this formula in the vector form to solve Burger's equation which is a...

Several real-life problems are modeled by nonlinear singular differential equations. In this article, we study a class of nonlinear singular differential equations, explore its various aspects, and provide a detailed literature survey. Nonlinear singular differential equations are not easy to solve and their exact solution does not exist in most ca...

Using Hölder's inequality, the chain rule on time scales and the properties of geometrically convex and concave functions we prove some new dynamic inequalities and their converses on time scales. As a special case, we derive the classical Saitoh integral inequality.

The regions of existence are established for a class of two point nonlinear diffusion type boundary value problems (NDBVP) \begin{eqnarray*} &&\label{abst-intr-1} -s''(x)-ns'(x)-\frac{m}{x}s'(x)=f(x,s), \qquad m>0,~n\in \mathbb{R},\qquad x\in(0,1),\\ &&\label{abst-intr-2} s'(0)=0, \qquad a_{1}s(1)+a_{2}s'(1)=C, \end{eqnarray*} where $a_{1}>0,$ $a_{...

Motivated by the work of Eltayeb and Killicman in this paper we generalize complex double Laplace transform to bicomplex double Laplace transform. Also, we derive some of its basic properties and inversion theorem in bicomplex space. Applications of bicomplex Double Laplace transform have been discussed in finding the solution of two-dimensional ti...

In this paper, we consider a non-self-adjoint, singular, nonlinear fourth order boundary value problem which arises in the theory of epitaxial growth. It is possible to reduce the fourth order equation to a singular boundary value problem of second order given by w''-(1/r)w'=(w^2/2r^2)+(1/2)\lambda r^2 The problem depends on the parameter λ and a...

In this article, we propose a novel modification to Quasi-Newton method, which is now a days popularly known as variation iteration method (VIM) and use it to solve the following class of nonlinear singular differential equations which arises in chemistry \[-y''(x)-\frac{\alpha}{x}y'(x)=f(x,y),~x\in(0,1),\] where $\alpha\geq1$, subject to certain t...

In this paper, we prove some new dynamic inequalities of Opial type on time scales. By employing these new inequalities we establish some new Lyapunov type inequalities for a second order dynamic equation with a damping term. These new Lyapunov inequalities give lower bounds on the distance between zeros of a solution and/or its derivative.

This paper presents a multicontroller structure for an SEIADR (susceptible-exposed-symptomatic infectious-asymptomatic infectious-dead lying bodies recovered by immunity, or immune, subpopulations) epidemic model, which has three different controls involving feedback, namely, the vaccination on the susceptible subpopulation, the antiviral treatment...

The existence of three nontrivial solutions is established for a class of fourth-order elliptic equations. Our technical approach is based on Linking Theorem and (∇)-Theorem. © 2018 Juliusz Shauder Centre for Nonlinear Studies Niolaus Copernius University.

One of the main properties of solutions of nonlinear Caputo fractional neural networks is stability and often the direct Lyapunov method is used to study stability properties (usually these Lyapunov functions do not depend on the time variable). In connection with the Lyapunov fractional method we present a brief overview of the most popular fracti...

In this paper, a new giving up smoking model is proposed by incorporating the continuous age-structure in the chain smokers class, which is known as a class of age-structured giving up smoking model. Smoking generation number is defined and proved to be a classic threshold parameter. Two steady states of the model are found. The corresponding chara...

This paper presents new existence results for singular discrete initial value problems. In particular our nonlinearity may be singular in its dependent variable and is allowed to change sign.

The case of differential equations with instantaneous impulses is studied in the literature; so we begin with a brief overview of its statements and later we will compare it with the case of non-instantaneous impulses.

Fractional calculus is the theory of integrals and derivatives of arbitrary non-integer order, which unifies and generalizes the concepts of ordinary differentiation and integration. For more details on geometric and physical interpretations of fractional derivatives and for a general historical perspective we refer the reader to the monographs [42...

In some real world phenomena a process may change instantaneously at uncertain moments and act non instantaneously on finite intervals. In modeling such processes it is necessarily to combine deterministic differential equations with random variables at the moments of impulses. The presence of randomness in the jump condition changes the solutions...

This monograph is the first published book devoted to the theory of differential equations with non-instantaneous impulses. It aims to equip the reader with mathematical models and theory behind real life processes in physics, biology, population dynamics, ecology and pharmacokinetics. The authors examine a wide scope of differential equations with...

In this paper, a notion of partially Hausdorff measure of noncompactness in partially ordered Banach spaces is introduced, and some Krasnoselskii-type fixed point theorems under certain mixed conditions are proved. Some applications of the obtained fixed point theorems are given to a class of fractional hybrid evolution equations for proving the ex...

Recent modeling of real world phenomena give rise to Caputo type fractional order differential equations with non-instantaneous impulses. The main goal of the survey is to highlight some basic points in introducing non-instantaneous impulses in Caputo fractional differential equations. In the literature there are two approaches in interpretation of...

Fractional differential equations with random impulses arise in modeling real world phenomena where the state changes instantaneously at uncertain moments. Using queuing theory and the usual distribution for waiting time, we study the case of exponentially distributed random variables between two consecutive moments of impulses. The p-moment expone...

Practical stability of a nonlinear Caputo fractional differential equation with noninstantaneous impulses is studied using Lyapunov like functions. We present a new definition of the derivative of a Lyapunov like function along the given fractional differential equation with noninstantaneous impulses. Sufficient conditions for practical stability,...

The aim of this paper is to investigate the solution of a generalized space-time fractional reaction-diffusion equation associated with the Hilfer-Prabhakar time fractional derivative and the space fractional Laplacian operator. The solution of the equation in terms of the three parameter Mittag-Leffler function, is obtained by applying the Laplace...

Recently, various forms and improvements of Opial dynamic inequalities have been given in the literature. In this paper, we give refinements of Opial inequalities on time scales that reduce in the continuous case to classical inequalities named after Beesack and Shum. These refinements are new in the important discrete case.

Some new iterative algorithms with errors for approximating common zero point of an infinite family of m-accretive mappings in a real Banach space are presented. A path convergence theorem and some new weak and strong convergence theorems are proved by means of some new techniques, which extend the corresponding works by some authors. As applicatio...

A new three-step iterative algorithm for approximating the zero point of the sum of an infinite family of m-accretive mappings and an infinite family of μ i -inversely strongly accretive mappings in a real q-uniformly smooth and uniformly convex Banach space is presented. The computational error in each step is being considered. A strong convergenc...

In this paper, we present the abstract results for the existence and uniqueness of the solution of nonlinear elliptic systems, parabolic systems and integro-differential systems involving the generalized (p,q)-Laplacian operator. Our method makes use of the characteristics of the ranges of linear and nonlinear maximal monotone operators and the sub...

In this short note we point out that the recently announced notion, the C ∗ -valued metric, does not bring about a real extension in metric fixed point theory. Besides, fixed point results in the C ∗ -valued metric can be derived from the desired Banach mapping principle and its famous consecutive theorems.

In this short note, we announce that all the presented fixed point results in the setting of multiplicative metric spaces can be derived from the corresponding existing results in the context of standard metric spaces in the literature. MSC: 47H10, 26A33, 45G10.

In this paper, a new impulsive Lasota-Wazewska model with patch structure and forced perturbed terms is proposed and analyzed on almost periodic time scales. For this, we introduce the concept of matrix measure on time scales and obtain some of its properties. Then, sufficient conditions are established which ensure the existence and exponential st...

In this paper, we present an asymptotic formula for Bateman’s G -function ${G(x)}$ and deduce the double inequality $\frac{1}{2x^{2}+3/2}<G(x)-\frac{1}{x}<\frac{1}{2x^{2}},\quad x>0.$ We apply this result to find estimates for the error term of the alternating series $\sum_{k=1}^{\infty}\kern-2.0pt\frac{(-1)^{k-1}}{k+h}$ , $h\kern-1.0pt\neq\kern-1....

One parabolic p-Laplacian-like differential equation with mixed boundaries is studied in this paper, where the item in the corresponding studies is replaced by , which makes it more general. The sufficient condition of the existence and uniqueness of non-trivial solution in L² (0, T; L² (Ω)) is presented by employing the techniques of splitting the...

In this paper we investigate a new kind of nonlocal multi-point boundary value problem of Caputo type sequential fractional integro-differential equations involving Riemann-Liouville integral boundary conditions. Several existence and uniqueness results are obtained via suitable fixed point theorems. Some illustrative examples are also presented. T...

This paper deals with the exact controllability for a class of first-order integro-differential evolution equations involving nonlocal initial conditions. By using Sadovskii fixed point theorem, exact controllability results are obtained without assuming the compactness and Lipschitz conditions for nonlocal functions. An example is given to illustr...

This chapter considers time scale versions of Copson type inequalities and their converses. We prove extensions of Copson type inequalities proved by Walsh on discrete time scales and we also consider converses of these inequalities. This chapter is divided into four sections and is organized as follows. In Sect. 2.1, we prove a time scale version...

In this chapter, we prove some dynamic Hardy-type inequalities on time scales with two different weight functions. This chapter is divided into two sections. In Sect. 5.1, we prove some weight inequalities which as special cases contain the results due to Copson, Bliss, Flett and Bennett by a suitable choice of weight functions. In Sect. 5.2, we pr...

This chapter considers Hardy-Knopp type inequalities on an arbitrary time scale \( \mathbb{T} \). One-dimensional, two-dimensional and multidimensional versions Hardy-Knopp type inequalities are considered. Moreover, Hardy-Knopp type inequalities for several functions and refinement inequalities of Hardy-Knopp type with general kernels and Hardy-Kn...

This chapter considers time scale versions of classical Hardy-type inequalities and time scale versions of Hardy and Littlewood type inequalities. We present extensions of Hardy-type inequalities on time scales. These dynamic inequalities not only contain the integral and discrete inequalities but can be extended to different types of time scales....

This chapter (with four sections) considers time scale versions of Leindler type inequalities.

In this chapter (which consists of four sections) we consider inequalities of Littlewood type. The aim is to extend these inequalities to time scales. In Sect. 4.1 we give a generalization and a time scale version of his inequality (which he proved in 1967). In Sects. 4.2 and 4.3 we present time scale versions of extensions of Littlewood’s results...

This chapter considers time scale versions of Levinson, Chang and Pachpatte type inequalities. The chapter is divided into six sections and is organized as follows. In Sects. 6.1 and 6.2 we present a variety of dynamic inequalities of Levinson type on time scales. In Sect. 6.3 we consider dynamic inequalities of Pachpatte type via convexity. Sectio...

The book is devoted to dynamic inequalities of Hardy type and extensions and generalizations via convexity on a time scale T. In particular, the book contains the time scale versions of classical Hardy type inequalities, Hardy and Littlewood type inequalities, Hardy-Knopp type inequalities via convexity, Copson type inequalities, Copson-Beesack typ...

Calvert and Gupta’s results concerning the perturbations on the ranges of m-accretive mappings have been employed widely in the discussion of the existence of solutions of nonlinear elliptic differential equation with Neumann boundary. In this paper, we shall focus our attention on certain hyperbolic differential equation with mixed boundaries. By...

In some real world phenomena a process may change instantaneously at uncertain moments and act non instantaneously on finite intervals. In modeling such processes it is necessarily to combine deterministic differential equations with random variables at the moments of impulses. The presence of randomness in the jump condition changes the solutions...

In this paper, we present some new Lyapunov and Hartman type inequalities for Riemann-Liouville fractional differential equations of the form ((a)D(alpha)x)(t) + p(t) vertical bar x(t) vertical bar(mu-1) x(t) + q(t) vertical bar x(t) vertical bar(gamma-1) x(t) = f(t), where p, q, f are real-valued functions and 0 < gamma < 1 < mu < 2. No sign restr...

In this paper, we establish the existence of infinitely many periodic solutions for a class of new superquadratic second-order Hamiltonian systems. Our technique is based on the Fountain Theorem due to Bartsch.

In this paper, we consider variational relation problems involving a binary relation. The framework presented is more general than that in [J. Optim. Theory Appl. 138 (2008) , 65–76] and in other recent papers which deal with this subject.

We study a boundary value problem of sequential fractional differential equations equipped with nonlocal integral boundary conditions (strip conditions of finite arbitrary size) involving the first-order derivative of the unknown function. As a variant problem, we discuss a case when nonlocal integral boundary conditions are governed by the unknown...

In the case of oscillatory potentials, we present Lyapunov type inequalities for nth order forced differential equations of the form x((n))(t) + Sigma(m)(j=1) qj (t)vertical bar x(t)vertical bar(alpha j-1)x(t)= f(t) satisfying the boundary conditions x(a(i)) = x(1)(a(i)) = x(11)(ai) = center dot center dot center dot = x((ki))(ai) = 0; i = 1, 2,......

In this manuscript, we consider a control system represented by a second order nonlinear impulsive differential systems with deviated argument in a Banach space X. We used the strongly continuous cosine family of linear operators and fixed point method to study the exact controllability. Also, we study the trajectory controllability of the control...

In this paper, we introduce the concept of Δ-sub-derivative on time scales to define ε-equivalent impulsive functional dynamic equations on almost periodic time scales. To obtain the existence of solutions for this type of dynamic equation, we establish some new theorems to characterize the compact sets in regulated function space on noncompact int...

Paper 22: Ravi Agarwal, Hans Agarwal and Syamal K. Sen, “Birth, growth and computation of pi to ten trillion digits,” Advances in Di erence Equations, 2013:100, p. 1–59. Synopsis: This paper presents one of the most complete and up-to-date chronologies of the analysis and computation of π through the ages, from approximations used by Indian and Bab...

Stability of Caputo fractional differential equations with impulses occurring at random moments and with non-instantaneous time of their action is studied. Using queuing theory and the usual distribution for waiting time, we study the case of exponentially distributed random variables between two consecutive moments of impulses. The p-moment expone...

In this paper, we presented the Raabe's integral and Hermite's formula for q-gamma function γq(x) 0 < q < 1. We deduced new proofs of the formulas γ q(x)/ γq(x) and q-Gauss's multiplication using the Hermite's formula of γq(x) and H. Jack's technique [11]. Also, we deduced new double inequality of γq(x).

Using a measure of non-compactness argument, we study in this paper the existence of solutions for a class of functional equations involving a fractional integral with respect to another function. Some examples are presented to illustrate the obtained results.

In this paper, we establish the existence and multiplicity of solutions for a class of superlinear elliptic systems without Ambrosetti and Rabinowitz growth condition. Our results are based on minimax methods in critical point theory.

We consider the Sturm-Liouville boundary value problem {y(m)(t)+F(t,y(t),y′(t),…,y(q)(t))=0,t∈[0,1],y(k)(0)=0,0≤k≤m−3,ζy(m−2)(0)−θy(m−1)(0)=0,ρy(m−2)(1)+δy(m−1)(1)=0,$\left \{ \textstyle\begin{array}{@{}l} y^{(m)}(t)+ F (t,y(t),y'(t),\ldots,y^{(q)}(t))=0, \quad t\in[0,1],\\ y^{(k)}(0)=0,\quad 0\leq k\leq m-3, \\ \zeta y^{(m-2)}(0)-\theta y^{(m-1)}(...

In this paper, we investigate the existence and uniqueness of (

The aim of this paper is to investigate the solutions of Time-space fractional advection-dispersion equation with Hilfer composite fractional derivative and the space fractional Laplacian operator. The solution of the equation is obtained by applying the Laplace and Fourier transforms, in terms of Mittag-le�er function.

In this paper, new forms of Ostrowski type inequalities are established for a general class of fractional integral operators. The main results are used to derive Ostrowski type inequalities involving the familiar Riemann-Liouville fractional integral operators and other important integral operators. We further obtain similar types of inequalities f...

In this paper, we study almost periodic and changing-periodic time scales considered byWang and Agarwal in 2015. Some improvements of almost periodic time scales are made. Furthermore, we introduce a new concept of periodic time scales in which the invariance for a time scale is dependent on an translation direction. Also some new results on period...

In this paper, we present some new Lyapunov and Hartman type inequalities for second order forced impulsive differential equations with mixed nonlinearities: x″(t)+p(t)|x(t)|β-1x(t)+q(t)|x(t)|γ-1x(t)=f(t),t≠θi;Δx'(t)+pi|x(t)|β-1x(t)+qi|x(t)|γ-1x(t)=fi,t=θi, where p, q, f are real-valued functions, {pi}, {qi}, {fi} are real sequences and 0 < γ < 1 <...

Practical stability with initial data difference for nonlinear Caputo fractional differential equations is studied. This type of stability generalizes known concepts of stability in the literature. It enables us to compare the behavior of two solutions when both initial values and initial intervals are different. In this paper the concept of practi...

In this short note, we present a new general dynamic inequality of Opial type. This inequality is new even in both the continuous and discrete cases. The inequality is proved by making use of a recently introduced new technique for Opial dynamic inequalities, the time scales integration by parts formula, the time scales chain rule, and classical as...

In this paper, we establish sufficient conditions for the existence and uniqueness of solutions for boundary value problems of Hadamard-type fractional functional differential equations and inclusions involving both retarded and advanced arguments. We make use of the standard tools of fixed point theory to obtain the main results.

In this paper, we define bicomplex Fourier-Stieltjes transform which is more generalized form of bicomplex Fourier transform. Also, we derive some of its basic properties and generalize the classical Bochner theorem in the framework of bicomplex analysis. Applications of bicomplex Fourier transform in finding the solution of initial value heat equa...

This paper is concerned with the existence of solutions for boundary value problems of fractional differential equations and inclusions supplemented with nonlocal and average-valued (integral) boundary conditions. The existence results for the single-valued case (equations) are obtained by means of fixed point theorems due to O’Regan and Sadovski,...

In this paper we present an existence result for causal functional evolution equations. The result is obtained using the Schauder fixed point theorem. An application to partial differential equations is given to illustrate our main result.

We present an overview of the literature on solutions to impulsive Caputo fractional differential equations. Lyapunov direct method is used to obtain sufficient conditions for stability properties of the zero solution of nonlinear impulsive fractional differential equations. One of the main problems in the application of Lyapunov functions to fract...

Stability of the solutions to a nonlinear impulsive Caputo fractional differential equation is studied using Lyapunov like functions. The derivative of piecewise continuous Lyapunov functions among the nonlinear impulsive Caputo differential equation of fractional order is defined. This definition is a natural generalization of the Caputo fractiona...

In this paper, we study Sturm-Liouville boundary value problems for second order difference equations on a half line. By using the discrete upper and lower solutions, the Schäuder fixed point theorem, and the degree theory, the existence of one and three solutions are investigated. An interesting feature of our existence theory is that the solution...

In this article, we investigate the existence of solutions for boundary value problems of fractional differential equations and inclusions with semiperiodic and three-point boundary conditions. The existence results for equations are obtained by applying Banach’s contraction mapping principle, Schaefer-type fixed point theorem, Leray-Schauder degre...

We study oscillatory behavior of a class of nonlinear second-order neutral differential equations. A new criterion is established that amends related results reported in the literature.

Two new ergodic convergence theorems for approximating the common element of the set of zero points of an m-accretive mapping and the set of fixed points of an infinite family of non-expansive mappings in a real smooth and uniformly convex Banach space are obtained, which improves some of the previous work. The computational experiments to demonstr...

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 di¤erent weight functions. The proofs use Hölder's inequality, the reverse Hölder's inequality, the chain rule, and a power rule of integration on time scal...