The Experts below are selected from a list of 18021 Experts worldwide ranked by ideXlab platform
Baoqiang Yan - One of the best experts on this subject based on the ideXlab platform.
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The multiplicity solutions for nonlinear Fractional Differential Equations of Riemann-Liouville type
Fractional Calculus and Applied Analysis, 2018Co-Authors: Baoqiang YanAbstract:Abstract Positive and negative definite comparison results for nonlinear q-th Fractional Differential Equations of Riemann-Liouville type are derived without requiring Hölder continuity assumption. Monotone iterative method is then developed to a class of nonlinear boundary value problems for Fractional Differential Equations, using coupled upper and lower solutions. Existence of the multiplicity solutions for the nonlinear Fractional Differential Equations is presented.
A S Vatsala - One of the best experts on this subject based on the ideXlab platform.
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Monotone iterative technique for Fractional Differential Equations with periodic boundary conditions
Opuscula Mathematica, 2009Co-Authors: Josimar Ramirez, A S VatsalaAbstract:In this paper we develop Monotone Method using upper and lower solutions for Fractional Differential Equations with periodic boundary conditions. Initially we develop a comparison result and prove that the solution of the linear Fractional Differential equation with periodic boundary condition exists and is unique. Using this we develop iterates which converge uniformly monotonically to minimal and maximal solutions of the nonlinear Fractional Differential Equations with periodic boundary conditions in the weighted norm.
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basic theory of Fractional Differential Equations
Nonlinear Analysis-theory Methods & Applications, 2008Co-Authors: V Lakshmikantham, A S VatsalaAbstract:Abstract In this paper, the basic theory for the initial value problem of Fractional Differential Equations involving Riemann–Liouville Differential operators is discussed employing the classical approach. The theory of inequalities, local existence, extremal solutions, comparison result and global existence of solutions are considered.
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general uniqueness and monotone iterative technique for Fractional Differential Equations
Applied Mathematics Letters, 2008Co-Authors: V Lakshmikantham, A S VatsalaAbstract:In this paper, the general existence and uniqueness result is proved which exhibits the idea of comparison principle. This result is also valid for Fractional Differential Equations in a Banach space. The well-known monotone iterative technique is then extended for Fractional Differential Equations which provides computable monotone sequences that converge to the extremal solutions in a sector generated by upper and lower solutions.
Zhenlai Han - One of the best experts on this subject based on the ideXlab platform.
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positive solutions to boundary value problems of nonlinear Fractional Differential Equations
Abstract and Applied Analysis, 2011Co-Authors: Yige Zhao, Shurong Sun, Zhenlai HanAbstract:We study the existence of positive solutions for the boundary value problem of nonlinear Fractional Differential Equations
Jinrong Wang - One of the best experts on this subject based on the ideXlab platform.
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Mixed Order Fractional Differential Equations
Mathematics, 2017Co-Authors: Michal Fečkan, Jinrong WangAbstract:This paper studies Fractional Differential Equations (FDEs) with mixed Fractional derivatives. Existence, uniqueness, stability, and asymptotic results are derived.
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Periodic impulsive Fractional Differential Equations
Advances in Nonlinear Analysis, 2017Co-Authors: Michal Fečkan, Jinrong WangAbstract:Abstract This paper deals with the existence of periodic solutions of Fractional Differential Equations with periodic impulses. The first part of the paper is devoted to the uniqueness, existence and asymptotic stability results for periodic solutions of impulsive Fractional Differential Equations with varying lower limits for standard nonlinear cases as well as for cases of weak nonlinearities, equidistant and periodically shifted impulses. We also apply our result to an impulsive Fractional Lorenz system. The second part extends the study to periodic impulsive Fractional Differential Equations with fixed lower limit. We show that in general, there are no solutions with long periodic boundary value conditions for the case of bounded nonlinearities.
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Existence and stability of Fractional Differential Equations with Hadamard derivative
Topological Methods in Nonlinear Analysis, 2013Co-Authors: Jinrong Wang, Yong Zhou, Milan MedveďAbstract:In this paper, we study nonlinear Fractional Differential Equations with Hadamard derivative and Ulam stability in the weighted space of continuous functions. Firstly, some new nonlinear integral inequalities with Hadamard type singular kernel are established, which can be used in the theory of certain classes of Fractional Differential Equations. Secondly, some sufficient conditions for existence of solutions are given by using fixed point theorems via a prior estimation in the weighted space of the continuous functions. Meanwhile, a sufficient condition for nonexistence of blowing-up solutions is derived. Thirdly, four types of Ulam-Hyers stability definitions for Fractional Differential Equations with Hadamard derivative are introduced and Ulam-Hyers stability and generalized Ulam-Hyers-Rassias stability results are presented. Finally, some examples and counterexamples on Ulam-Hyers stability are given.
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nonlinear impulsive problems for Fractional Differential Equations and ulam stability
Computers & Mathematics With Applications, 2012Co-Authors: Jinrong Wang, Yong Zhou, Michal FečkanAbstract:In this paper, the first purpose is treating Cauchy problems and boundary value problems for nonlinear impulsive Differential Equations with Caputo Fractional derivative. We introduce the concept of piecewise continuous solutions for impulsive Cauchy problems and impulsive boundary value problems respectively. By using a new fixed point theorem, we obtain many new existence, uniqueness and data dependence results of solutions via some generalized singular Gronwall inequalities. The second purpose is discussing Ulam stability for impulsive Fractional Differential Equations. Some new concepts in stability of impulsive Fractional Differential Equations are offered from different perspectives. Some applications of our results are also provided.
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On some impulsive Fractional Differential Equations in Banach spaces
Opuscula Mathematica, 2010Co-Authors: Jinrong Wang, W. Wei, Yanlong YangAbstract:This paper deals with some impulsive Fractional Differential Equations in Banach spaces. Utilizing the Leray-Schauder fixed point theorem and the impulsive nonlinear singular version of the Gronwall inequality, the existence of \(PC\)-mild solutions for some Fractional Differential Equations with impulses are obtained under some easily checked conditions. At last, an example is given for demonstration.
Ravi P. Agarwal - One of the best experts on this subject based on the ideXlab platform.
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Practical stability with respect to initial time difference for Caputo Fractional Differential Equations
Communications in Nonlinear Science and Numerical Simulation, 2017Co-Authors: Ravi P. Agarwal, Snezhana Hristova, Donal O'regan, M. CicekAbstract:Abstract 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 practical stability with initial time difference is generalized to Caputo Fractional Differential Equations. A definition of the derivative of Lyapunov like function along the given nonlinear Caputo Fractional Differential equation is given. Comparison results using this definition and scalar Fractional Differential Equations are proved. Sufficient conditions for several types of practical stability with initial time difference for nonlinear Caputo Fractional Differential Equations are obtained via Lyapunov functions. Some examples are given to illustrate the results.
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A survey of Lyapunov functions, stability and impulsive Caputo Fractional Differential Equations
Fractional Calculus and Applied Analysis, 2016Co-Authors: Ravi P. Agarwal, Snezhana Hristova, Donal O'reganAbstract:AbstractWe 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 Fractional Differential Equations is an appropriate definition of its derivative among the Differential equation of Fractional order. A brief overview of those used in the literature is given, and we discuss their advantages and disadvantages. One type of derivative, the so called Caputo Fractional Dini derivative, is generalized to impulsive Fractional Differential Equations. We apply it to study stability and uniform stability. Some examples are given to illustrate the results.
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Lyapunov functions and strict stability of Caputo Fractional Differential Equations
Advances in Difference Equations, 2015Co-Authors: Ravi P. Agarwal, Snezhana Hristova, Donal O'reganAbstract:One of the main properties studied in the qualitative theory of Differential Equations is the stability of solutions. The stability of Fractional order systems is quite recent. There are several approaches in the literature to study stability, one of which is the Lyapunov approach. However, the Lyapunov approach to Fractional Differential Equations causes many difficulties. In this paper a new definition (based on the Caputo Fractional Dini derivative) for the derivative of Lyapunov functions to study a nonlinear Caputo Fractional Differential equation is introduced. Comparison results using this definition and scalar Fractional Differential Equations are presented, and sufficient conditions for strict stability and uniform strict stability are given. Examples are presented to illustrate the theory.
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On the oscillation of Fractional Differential Equations
Fractional Calculus and Applied Analysis, 2012Co-Authors: Said R. Grace, Ravi P. Agarwal, Patricia J. Y. Wong, Agacik ZaferAbstract:In this paper we initiate the oscillation theory for Fractional Differential Equations. Oscillation criteria are obtained for a class of nonlinear Fractional Differential Equations of the form $$D_a^q x + f_1 (t,x) = v(t) + f_2 (t,x),\mathop {\lim }\limits_{t \to a} J_a^{1 - q} x(t) = b_1 $$ , where D a q denotes the Riemann-Liouville Differential operator of order q, 0 < q ≤ 1. The results are also stated when the Riemann-Liouville Differential operator is replaced by Caputo’s Differential operator.
