Browsing by Author "Agarwal, Ravi P."

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Citation - WoS: 7
Citation - Scopus: 13
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.
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.
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: 16
Citation - Scopus: 18
On a Multipoint Boundary Value Problem for a Fractional Order Differential Inclusion on an Infinite Interval
(Hindawi Ltd, 2013) Baleanu, Dumitru; Agarwal, Ravi P.; Nyamoradi, Nemat
We investigate the existence of solutions for the following multipoint boundary value problem of a fractional order differential inclusion D(0+)(alpha)u(t) + F(t,u(t),u'(t)) (sic) 0, 0 < t < +infinity,u(0) = u'(0) = 0, D(alpha-1)u(+infinity) - Sigma(m-2)(i-1) beta(i)u(xi(i)) = 0, where D-0+(alpha) is the standard Riemann-Liouville fractional derivative, 2 < alpha < 3, 0 < xi(1) < xi(2) < center dot center dot center dot < xi(m-2) < +infinity, satisfies 0 < Sigma(m-2)(i=1) beta(i)xi(alpha-1)(i) < Gamma(alpha), and F : [0, +infinity) x R x R (sic) P(R) is a set-valued map. Several results are obtained by using suitable fixed point theorems when the right hand side has convex or nonconvex values.
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.
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: 20
Citation - Scopus: 28
On the Existence of Solution for Fractional Differential Equations of Order 3 < Δ1 ≤ 4
(Springer, 2015) Agarwal, Ravi P.; Khan, Rahmat Ali; Jafari, Hossein; Baleanu, Dumitru; Khan, Hasib
In this paper, we deal with a fractional differential equation of order delta(1) is an element of (3,4] with initial and boundary conditions, D-delta 1 psi(x) = -H(x,psi(x)), D-alpha 1 psi(1) = 0 = I3-delta 1 psi(0) = I4-delta 1 psi(0), psi(1) = Gamma(delta(1)-alpha(1))/Gamma(nu(1)) I delta 1-alpha 1 H(x,psi(x))(1), where x is an element of [0, 1], alpha(1) is an element of (1, 2], addressing the existence of a positive solution (EPS), where the fractional derivatives D-delta 1, D-alpha 1 are in the Riemann-Liouville sense of the order delta(1), alpha(1), respectively. The function H is an element of C([0, 1] x R, R) and I delta 1-alpha 1 H(x, psi(x))(1) = 1/Gamma(delta(1)-alpha(1)) integral(1)(0) (1 -z)(delta 1-alpha 1-1) H(z,psi(z)) dz. To this aim, we establish an equivalent integral form of the problem with the help of a Green's function. We also investigate the properties of the Green's function in the paper which we utilize in our main result for the EPS of the problem. Results for the existence of solutions are obtained with the help of some classical results.
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
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: 164
Citation - Scopus: 177
A Survey on Fuzzy Fractional Differential and Optimal Control Nonlocal Evolution Equations
(Elsevier, 2018) Baleanu, Dumitru; Nieto, Juan J.; Torres, Delfim F. M.; Zhou, Yong; Agarwal, Ravi P.
We survey some representative results on fuzzy fractional differential equations, controllability, approximate controllability, optimal control, and optimal feedback control for several different kinds of fractional evolution equations. Optimality and relaxation of multiple control problems, described by nonlinear fractional differential equations with nonlocal control conditions in Banach spaces, are considered. (C) 2017 Elsevier B.V. All rights reserved.
Citation - WoS: 78
Citation - Scopus: 97
Some Existence Results for a Nonlinear Fractional Differential Equation on Partially Ordered Banach Spaces
(Springeropen, 2013) Agarwal, Ravi P.; Mohammadi, Hakimeh; Rezapour, Shahram; Baleanu, Dumitru
By using fixed point results on cones, we study the existence and uniqueness of positive solutions for some nonlinear fractional differential equations via given boundary value problems. Examples are presented in order to illustrate the obtained results.
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.
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: 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: 8
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: 113
Citation - Scopus: 103
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: 13
Citation - Scopus: 15
A Remark on the Fractional Integral Operators and the Image Formulas of Generalized Lommel-Wright Function
(Frontiers Media Sa, 2018) Jain, Sonal; Agarwal, Ravi P.; Baleanu, Dumitru; Agarwal, Ritu
In this paper, the operators of fractional integration introduced by Marichev-Saigo-Maeda involving Appell's function F-3(center dot) are applied, and several new image formulas of generalized Lommel-Wright function are established. Also, by implementing some integral transforms on the resulting formulas, few more image formulas have been presented. We can conclude that all derived results in our work generalize numerous well-known results and are capable of yielding a number of applications in the theory of special functions. Primary: 44A20 Transforms of special functions; 65R10 For numerical methods; 26A33 Fractional derivatives and integrals; Secondary: 33C20 Generalized hypergeometric series, pFq; 33E50 Special functions in characteristic p (gamma functions, etc.); 2010 AMS classification by MathSciNet
Citation - WoS: 11
Citation - Scopus: 10
Solution of Maxwell's Wave Equations in Bicomplex Space
(Editura Acad Romane, 2017) Agarwal, Ritu; Baleanu, Dumitru; Goswami, Mahesh Puri; Agarwal, Ravi P.; Venkataratnam, Kamma Kesa; Baleanu, Dumitru; Matematik
The concept of bicomplex numbers is introduced in Electromagnetics, with direct applications to the solution of Maxwell's equations. In this paper, we discuss the technique to find the analytic solution of the electromagnetic wave equation in vacuum with the help of bicomplex analysis as tool. Also, we find the solution of Gaussian pulse wave using bicomplex vector field.
Citation - WoS: 24
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).
Oscillation criteria for even order dynamic equations on time-scales
(2011) Grace, Said R.; Agarwal, Ravi P.; Kaymakçalan, Billur
Some new criteria for the oscillation of even order linear dynamic equations on time-scales of the form xΔn(t) + q(t)x(t) = 0 are established.
Citation - WoS: 66
Citation - Scopus: 72
On L^p-solutions for a Class of Sequential Fractional Differential Equations
(Elsevier Science inc, 2011) Mustafa, Octavian G.; Agarwal, Ravi P.; Baleanu, Dumitru
Under some simple conditions on the coefficient a( t), we establish that the initial value problem ((0)D(t)(alpha)x)' + a(t)x = 0; t > 0; lim(t SE arrow 0)[t(1-alpha)x(t)] = 0 has no solution in L-p((1, +infinity), R), where p-1/p > alpha > 1/p and D-0(t)alpha designates the Riemann-Liouville derivative of order alpha Our result might be useful for developing a non-integer variant of H. Weyl's limit-circle/limit-point classification of differential equations. (C) 2011 Elsevier Inc. All rights reserved.
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
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.