Browsing by Author "Agarwal, Ravi P."

Now showing 1 - 20 of 38

Asymptotic Integration of (1 + Α) -Order Fractional Differential Equations
(Pergamon-Elsevier Science Ltd, 2011) Baleanu, Dumitru; Mustafa, Octavian G.; Agarwal, Ravi P.; Bleanu, Dumitru
We establish the long-time asymptotic formula of solutions to the (1+α)-order fractional differential equation 0iOt1+αx+a(t)x=0, t>0, under some simple restrictions on the functional coefficient a(t), where 0iOt1+α is one of the fractional differential operators 0Dtα(x′), (0Dtαx)′= 0Dt1+αx and 0Dtα(tx′-x). Here, 0Dtα designates the Riemann-Liouville derivative of order α∈(0,1). The asymptotic formula reads as [b+O(1)] ·xsmall+c·xlarge as t→+∞ for given b, c∈R, where xsmall and xlarge represent the eventually small and eventually large solutions that generate the solution space of the fractional differential equation 0iOt1+αx=0, t>0
Asymptotic integration of (1+alpha)-order fractional differential equations
(Pergamon-Elsevier Science Ltd, 2011) Baleanu, Dumitru; Mustafa, Octavian G.; Agarwal, Ravi P.
We establish the long-time asymptotic formula of solutions to the (1 + alpha)-order fractional differential equation (i)(0)O(t)(1+alpha)x + a (t)x = 0, t > 0, under some simple restrictions on the functional coefficient a(t), where (i)(0)O(t)(1+alpha)x is one of the fractional differential operators D-0(t)alpha(x'), ((0)D(t)(alpha)x)' = D-0(t)1+alpha x and D-0(t)alpha(tx' - x). Here, D-0(t)alpha designates the Riemann-Liouville derivative of order a E (0, 1). The asymptotic formula reads as [b + O(1)] . x(small) + c . x(large) as t -> +infinity for given b, c E is an element of R, where x(small) and x(large) represent the eventually small and eventually large solutions that generate the solution space of the fractional differential equation (i)(0)O(t)(1+alpha)x = 0, t > 0.
Asymptotic Integration of (1+Alpha)-Order Fractional Differential Equations
(Pergamon-Elsevier Science LTD, 2011) Baleanu, Dumitru; Mustafa, Octavian G.; Agarwal, Ravi P.
We establish the long-time asymptotic formula of solutions to the (1 + alpha)-order fractional differential equation (i)(0)O(t)(1+alpha)x + a (t)x = 0, t > 0, under some simple restrictions on the functional coefficient a(t), where (i)(0)O(t)(1+alpha)x is one of the fractional differential operators D-0(t)alpha(x'), ((0)D(t)(alpha)x)' = D-0(t)1+alpha x and D-0(t)alpha(tx' - x). Here, D-0(t)alpha designates the Riemann-Liouville derivative of order a E (0, 1). The asymptotic formula reads as [b + O(1)] . x(small) + c . x(large) as t -> +infinity for given b, c E is an element of R, where x(small) and x(large) represent the eventually small and eventually large solutions that generate the solution space of the fractional differential equation (i)(0)O(t)(1+alpha)x = 0, t > 0. (C) 2011 Elsevier Ltd. All rights reserved.
Citation - WoS: 28
Citation - Scopus: 30
Asymptotic Integration of (1+α)-Order Fractional Differential Equations
(Pergamon-elsevier Science Ltd, 2011) Mustafa, Octavian G.; Agarwal, Ravi P.; Baleanu, Dumitru
We establish the long-time asymptotic formula of solutions to the (1 + alpha)-order fractional differential equation (i)(0)O(t)(1+alpha)x + a (t)x = 0, t > 0, under some simple restrictions on the functional coefficient a(t), where (i)(0)O(t)(1+alpha)x is one of the fractional differential operators D-0(t)alpha(x'), ((0)D(t)(alpha)x)' = D-0(t)1+alpha x and D-0(t)alpha(tx' - x). Here, D-0(t)alpha designates the Riemann-Liouville derivative of order a E (0, 1). The asymptotic formula reads as [b + O(1)] . x(small) + c . x(large) as t -> +infinity for given b, c E is an element of R, where x(small) and x(large) represent the eventually small and eventually large solutions that generate the solution space of the fractional differential equation (i)(0)O(t)(1+alpha)x = 0, t > 0. (C) 2011 Elsevier Ltd. All rights reserved.
Citation - WoS: 25
Citation - Scopus: 32
Asymptotic Integration of Some Nonlinear Differential Equations With Fractional Time Derivative
(Iop Publishing Ltd, 2011) Agarwal, Ravi P.; Mustafa, Octavian G.; Cosulschi, Mirel; Baleanu, Dumitru
We establish that, under some simple integral conditions regarding the nonlinearity, the (1 + alpha)-order fractional differential equation D-0(t)alpha (x') + f (t, x) = 0, t > 0, has a solution x is an element of C([0, +infinity), R) boolean AND C-1((0, +infinity), R), with lim(t SE arrow 0) [t(1-alpha)x'(t)] is an element of R, which can be expanded asymptotically as a+bt(alpha)+O(t(alpha-1)) when t ->+infinity for given real numbers a, b. Our arguments are based on fixed point theory. Here, D-0(t)alpha designates the Riemann-Liouville derivative of order alpha is an element of (0, 1).
Citation - WoS: 15
Citation - Scopus: 28
Asymptotically Linear Solutions for Some Linear Fractional Differential Equations
(Hindawi Publishing Corporation, 2010) Mustafa, Octavian G.; Agarwal, Ravi P.; Baleanu, Dumitru; Bleanu, Dumitru
We establish here that under some simple restrictions on the functional coefficient a(t) the fractional differential equation 0D(t)(alpha)[tx' - x + x(0)] + a(t)x = 0, t > 0, has a solution expressible as ct + d + o(1) for t -> +infinity, where D-0(t)alpha designates the Riemann-Liouville derivative of order alpha is an element of (0, 1) and c, d is an element of R.
Citation - WoS: 17
Citation - Scopus: 19
Existence and Uniqueness of Positive Solutions To Fractional Boundary Value Problems With Nonlinear Boundary Conditions
(Springer, 2013) Baleanu, Dumitru; Agarwal, Ravi P.; Nyamoradi, Nemat
In this manuscript, we consider two problems of boundary value problems for a fractional differential equation. A fixed point theorem in partially ordered sets and a contraction mapping principle are applied to prove the existence of at least one positive solution for both fractional boundary value problems.
Citation - WoS: 9
Citation - Scopus: 9
The Existence and Uniqueness of Solutions for a Class of Nonlinear Fractional Differential Equations With Infinite Delay
(Hindawi Ltd, 2013) Baleanu, Dumitru; Agarwal, Ravi P.; Babakhani, Azizollah
We prove the existence and uniqueness of solutions for two classes of infinite delay nonlinear fractional order differential equations involving Riemann-Liouville fractional derivatives. The analysis is based on the alternative of the Leray-Schauder fixed-point theorem, the Banach fixed-point theorem, and the Arzela-Ascoli theorem in Omega = {y : (-infinity,b] -> R : y vertical bar(<-infinity, 0]) epsilon B} such that y vertical bar ([0,b]) is continuous and B is a phase space.
Citation - WoS: 38
Citation - Scopus: 42
The Existence of Solutions for Some Fractional Finite Difference Equations Via Sum Boundary Conditions
(Springer, 2014) Baleanu, Dumitru; Rezapour, Shahram; Salehi, Saeid; Agarwal, Ravi P.
In this manuscript we investigate the existence of the fractional finite difference equation (FFDE) Delta(mu)(mu-2)x(t) = g(t + mu - 1, x(t + mu - 1), Delta x(t + mu - 1)) via the boundary condition x(mu - 2) = 0 and the sum boundary condition x(mu + b + 1) = Sigma(alpha)(k=mu-1) x(k) for order 1 < mu <= 2, where g : N-mu-1(mu+b+1) x R x R -> R, alpha is an element of N-mu-1(mu+b), and t is an element of N-0(b+2). Along the same lines, we discuss the existence of the solutions for the following FFDE: Delta(mu)(mu-3)x(t) = g(t + mu - 2, x(t + mu - 2)) via the boundary conditions x(mu - 3) = 0 and x(mu + b + 1) = 0 and the sum boundary condition x(alpha) = Sigma(beta)(k=gamma)x(k) for order 2 < mu <= 3, where g : N-mu-2(mu+b+1) x R -> R, b is an element of N-0, t is an element of N-0(b+3), and alpha, beta,gamma N-mu-2(mu+b) with gamma < beta < alpha.
Citation - WoS: 103
Citation - Scopus: 105
An Existence Result for a Superlinear Fractional Differential Equation
(Pergamon-elsevier Science Ltd, 2010) Mustafa, Octavian G.; Agarwal, Ravi P.; Baleanu, Dumitru
We establish the existence and uniqueness of solution for the boundary value problem D-0(t)alpha(x') + a(t)x(lambda) = 0, t > 0, x' (0) = 0, lim(t ->+infinity) x(t) = 1, where D-0(t)alpha designates the Riemann-Liouville derivative of order alpha epsilon (0, 1) and lambda > 1. Our result might be useful for establishing a non-integer variant of the Atkinson classical theorem on the oscillation of Emden-Fowler equations. (C) 2010 Elsevier Ltd. All rights reserved.
Citation - WoS: 19
Citation - Scopus: 25
Extended Suprametric Spaces and Stone-Type Theorem
(Amer inst Mathematical Sciences-aims, 2023) Agarwal, Ravi P.; Karapinar, Erdal; Panda, Sumati Kumari
Extended suprametric spaces are defined, and the contraction principle is established using elementary properties of the greatest lower bound instead of the usual iteration procedure. Thereafter, some topological results and the Stone-type theorem are derived in terms of suprametric spaces. Also, we have shown that every suprametric space is metrizable. Further, we prove the existence of a solution of Ito-Doob type stochastic integral equations using our main fixed point theorem in extended suprametric spaces.
Citation - WoS: 52
Citation - Scopus: 62
F-Contraction Mappings on Metric-Like Spaces in Connection With Integral Equations on Time Scales
(Springer-verlag Italia Srl, 2020) Aksoy, Umit; Karapinar, Erdal; Erhan, Inci M.; Agarwal, Ravi P.
In this paper we investigate the existence and uniqueness of fixed points of certain (phi,F)-type contractions in the frame of metric-like spaces. As an application of the theorem we consider the existence and uniqueness of solutions of nonlinear Fredholm integral equations of the second kind on time scales. We also present a particular example which demonstrates our theoretical results.
Fixed Point Theory in Generalized Metric Spaces
(2022) Karapınar, Erdal; Agarwal, Ravi P.
This book presents fixed point theory, one of the crucial tools in applied mathematics, functional analysis, and topology, which has been used to solve distinct real-world problems in computer science, engineering, and physics. The authors begin with an overview of the extension of metric spaces. Readers are introduced to general fixed-point theorems while comparing and contrasting important and insignificant metric spaces. The book is intended to be self-contained and serves as a unique resource for researchers in various disciplines.
Citation - WoS: 10
Citation - Scopus: 10
Fixed-Point Results for Meir-Keeler Type Contractions in Partial Metric Spaces: a Survey
(Mdpi, 2022) Agarwal, Ravi P.; Yesilkaya, Seher Sultan; Wang, Chao; Karapinar, Erdal
In this paper, we aim to review Meir-Keeler contraction mappings results on various abstract spaces, in particular, on partial metric spaces, dislocated (metric-like) spaces, and M-metric spaces. We collect all significant results in this direction by involving interesting examples. One of the main reasons for this work is to help young researchers by giving a framework for Meir Keeler's contraction.
Citation - WoS: 114
Citation - Scopus: 104
Fractional Calculus in the Sky
(Springer, 2021) Agarwal, Ravi P.; Baleanu, Dumitru
Fractional calculus was born in 1695 on September 30 due to a very deep question raised in a letter of L'Hospital to Leibniz. The prophetical answer of Leibniz to that deep question encapsulated a huge inspiration for all generations of scientists and is continuing to stimulate the minds of contemporary researchers. During 325 years of existence, fractional calculus has kept the attention of top level mathematicians, and during the last period of time it has become a very useful tool for tackling the dynamics of complex systems from various branches of science and engineering. In this short manuscript, we briefly review the tremendous effect that the main ideas of fractional calculus had in science and engineering and briefly present just a point of view for some of the crucial problems of this interdisciplinary field.
Citation - WoS: 2
Citation - Scopus: 2
Fractional Evolution Equation With Cauchy Data in L^p Spaces
(Springer, 2022) Baleanu, Dumitru; Agarwal, Ravi P.; Le Dinh Long; Nguyen Duc Phuong; Long, Le Dinh; Phuong, Nguyen Duc
In this paper, we consider the Cauchy problem for fractional evolution equations with the Caputo derivative. This problem is not well posed in the sense of Hadamard. There have been many results on this problem when data is noisy in L-2 and H-s,H- However, there have not been any papers dealing with this problem with observed data in L-p with p not equal 2. We study three cases of source functions: homogeneous case, inhomogeneous case, and nonlinear case. For all of them, we use a truncation method to give an approximate solution to the problem. Under different assumptions on the smoothness of the exact solution, we get error estimates between the regularized solution and the exact solution in L-p. To our knowledge, L-p evaluations for the inverse problem are very limited. This work generalizes some recent results on this problem.
Citation - WoS: 8
Citation - Scopus: 14
Fractional Hybrid Differential Equations and Coupled Fixed-Point Results for Α-Admissible F(Ψ1, Ψ2)-Contractions in M-Metric Spaces
(Hindawi Ltd, 2020) Karapinar, Erdal; Moustafa, Shimaa, I; Shehata, Ayman; Agarwal, Ravi P.; Al-Mdallal, Qasem M.
In this paper, we investigate the existence of a unique coupled fixed point for alpha-admissible mapping which is of F(psi(1),psi(2))-contraction in the context ofM-metric space. We have also shown that the results presented in this paper would extend many recent results appearing in the literature. Furthermore, we apply our results to develop sufficient conditions for the existence and uniqueness of a solution for a coupled system of fractional hybrid differential equations with linear perturbations of second type and with three-point boundary conditions.
Citation - WoS: 19
Citation - Scopus: 69
Fractional Sums and Differences With Binomial Coefficients
(Hindawi Ltd, 2013) Agarwal, Ravi P.; Abdeljawad, Thabet; Baleanu, Dumitru; Jarad, Fahd
In fractional calculus, there are two approaches to obtain fractional derivatives. The first approach is by iterating the integral and then defining a fractional order by using Cauchy formula to obtain Riemann fractional integrals and derivatives. The second approach is by iterating the derivative and then defining a fractional order by making use of the binomial theorem to obtain Grunwald-Letnikov fractional derivatives. In this paper we formulate the delta and nabla discrete versions for left and right fractional integrals and derivatives representing the second approach. Then, we use the discrete version of the Q-operator and some discrete fractional dual identities to prove that the presented fractional differences and sums coincide with the discrete Riemann ones describing the first approach.
FractionalHybridDifferential EquationsandCoupled Fixed-Point Results for α-Admissible F(ψ1, ψ2) − Contractions in M − Metric Spaces
(2020) Karapınar, Erdal; Shimaa I., Moustafa; Shehata, Ayman; Agarwal, Ravi P.
In this paper, we investigate the existence of a unique coupled fixed point for admissible mapping which is of contraction in the context of metric space. We have also shown that the results presented in this paper would extend many recent results appearing in the literature. Furthermore, we apply our results to develop sufficient conditions for the existence and uniqueness of a solution for a coupled system of fractional hybrid differential equations with linear perturbations of second type and with three-point boundary conditions.
Citation - WoS: 9
Citation - Scopus: 10
Identifying the Source Function for Time Fractional Diffusion With Non-Local in Time Conditions
(Springer Heidelberg, 2021) Baleanu, Dumitru; Agarwal, Ravi P.; Long, Le Dinh; Luc, Nguyen Hoang
The diffusion equation has many applications in fields such as physics, environment, and fluid mechanics. In this paper, we consider the problem of identifying an unknown source for a time-fractional diffusion equation in a general bounded domain from the nonlocal integral condition. The problem is non-well-posed in the sense of Hadamard, i.e, if the problem has only one solution, the solution will not depend continuously on the input data. To get a stable solution and approximation, we need to offer the regularization methods. The first contribution to the paper is to provide a regularized solution using the modified fractional Landweber method. Two choices are proposed including a priori and a posteriori parameter choice rules, to estimate the convergence rate of the regularized methods. The second new contribution is to use truncation to give an estimate of L-p for the convergence rate.