Asymptotic behavior of solutions of nonlinear fractional differential equations with Caputo-Type Hadamard derivatives

2017 ◽  
Vol 20 (1) ◽  
Author(s):  
John R. Graef ◽  
Said R. Grace ◽  
Ercan Tunç
Keyword(s):  
Asymptotic Behavior ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Higher Order ◽  
Asymptotic Behavior Of Solutions ◽  
Nonlinear Fractional Differential Equations ◽  
Fractional Differential ◽  
Derivatives Of

AbstractIn this paper, the authors study the asymptotic behavior of solutions of higher order fractional differential equations with Caputo-type Hadamard derivatives of the formwhere

Asymptotic behavior of solutions of fractional differential equations with Hadamard fractional derivatives

2021 ◽  
Vol 24 (2) ◽  
pp. 483-508
Author(s):  
Mohammed D. Kassim ◽  
Nasser-eddine Tatar
Keyword(s):  
Asymptotic Behavior ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Sufficient Conditions ◽  
Asymptotic Behavior Of Solutions ◽  
Logarithmic Function ◽  
Fractional Differential ◽  
Derivatives Of ◽  
L’Hôpital’S Rule

Abstract The asymptotic behaviour of solutions in an appropriate space is discussed for a fractional problem involving Hadamard left-sided fractional derivatives of different orders. Reasonable sufficient conditions are determined ensuring that solutions of fractional differential equations with nonlinear right hand sides approach a logarithmic function as time goes to infinity. This generalizes and extends earlier results on integer order differential equations to the fractional case. Our approach is based on appropriate desingularization techniques and generalized versions of Gronwall-Bellman inequality. It relies also on a kind of Hadamard fractional version of l'Hopital’s rule which we prove here.

scholarly journals ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS

2016 ◽  
Vol 21 (5) ◽  
pp. 610-629 ◽  
Author(s):  
Mohammed D. Kassim ◽  
Khaled M. Furati ◽  
Nasser-Eddine Tatar
Keyword(s):  
Asymptotic Behavior ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fractional Differential Equations ◽  
Higher Order ◽  
Asymptotic Behavior Of Solutions ◽  
Power Type ◽  
Fractional Damping ◽  
Integer Order ◽  
Fractional Differential

It is known that, under certain conditions, solutions of some ordinary differential equations of first, second or even higher order are asymptotic to polynomials as time goes to infinity. We generalize and extend some of the existing results to differential equations of non-integer order. Reasonable conditions and appropriate underlying spaces are determined ensuring that solutions of fractional differential equations with nonlinear right hand sides approach power type functions as time goes to infinity. The case of fractional differential problems with fractional damping is also considered. Our results are obtained by using generalized versions of GronwallBellman inequality and appropriate desingularization techniques.

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