scholarly journals Perturbation Results and Monotone Iterative Technique for Fractional Evolution Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-18
Author(s):  
Jia Mu
Keyword(s):  
Existence And Uniqueness ◽  
Evolution Equations ◽  
Monotone Iterative Technique ◽  
Iterative Technique ◽  
Ordered Banach Space ◽  
Fractional Evolution Equations ◽  
Existence And Uniqueness Results ◽  
Monotone Iterative ◽  
Fractional Evolution Equation ◽  
Uniqueness Results

We mainly study the fractional evolution equation in an ordered Banach space , , , . Using the monotone iterative technique based on lower and upper solutions, the existence and uniqueness results are obtained. The necessary perturbation results for accomplishing this approach are also developed.

scholarly journals Monotone Iterative Technique for the Initial Value Problems of Impulsive Evolution Equations in Ordered Banach Spaces

2010 ◽  
Vol 2010 ◽  
pp. 1-11 ◽  
Author(s):  
He Yang
Keyword(s):  
Existence And Uniqueness ◽  
Initial Value Problems ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Monotone Iterative Technique ◽  
Iterative Technique ◽  
Initial Value ◽  
Monotone Iterative ◽  
Uniqueness Results ◽  
Impulsive Evolution Equations

This paper deals with the existence and uniqueness of mild solutions for the initial value problems of abstract impulsive evolution equations in an ordered Banach spaceE:u′(t)+Au(t)=f(t,u(t),Gu(t)),t∈[0,a],t≠tk,Δu|t=tk=Ik(u(tk)),0<t1<t2<⋯<tm<a,u(0)=u0, whereA:D(A)⊂E→Eis a closed linear operator, andf:[0,a]×E×E→Eis a nonlinear mapping. Under wide monotone conditions and measure of noncompactness conditions of nonlinearityf, some existence and uniqueness results are obtained by using a monotone iterative technique in the presence of lower and upper solutions.

scholarly journals Positive Mild Solutions of Periodic Boundary Value Problems for Fractional Evolution Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Jia Mu ◽  
Hongxia Fan
Keyword(s):  
Existence And Uniqueness ◽  
Periodic Boundary ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Boundary Value ◽  
Fractional Evolution Equations ◽  
Existence And Uniqueness Results ◽  
Monotone Iterative ◽  
Uniqueness Results ◽  
Periodic Boundary Value Problems

The periodic boundary value problem is discussed for a class of fractional evolution equations. The existence and uniqueness results of mild solutions for the associated linear fractional evolution equations are established, and the spectral radius of resolvent operator is accurately estimated. With the aid of the estimation, the existence and uniqueness results of positive mild solutions are obtained by using the monotone iterative technique. As an application that illustrates the abstract results, an example is given.

Asymptotically periodic behavior of solutions of fractional evolution equations of order 1 < α < 2

2019 ◽  
Vol 69 (3) ◽  
pp. 599-610 ◽  
Author(s):  
Lulu Ren ◽  
Jinrong Wang ◽  
Donal O’Regan
Keyword(s):  
Existence And Uniqueness ◽  
Evolution Equations ◽  
Fractional Evolution Equations ◽  
Periodic Behavior ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Asymptotically Periodic

Abstract In this paper we investigate the asymptotically periodic behavior of solutions of fractional evolution equations of order 1 < α < 2 and in particular existence and uniqueness results are established. Two examples are given to illustrate our results.

scholarly journals On the Existence and Uniqueness Results for Fuzzy Linear and Semilinear Fractional Evolution Equations Involving Caputo Fractional Derivative

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Ali El Mfadel ◽  
Said Melliani ◽  
M’hamed Elomari
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Existence And Uniqueness ◽  
Evolution Equations ◽  
Existence Theorems ◽  
Fractional Evolution Equations ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Banach Contraction

In this manuscript, we establish new existence and uniqueness results for fuzzy linear and semilinear fractional evolution equations involving Caputo fractional derivative. The existence theorems are proved by using fuzzy fractional calculus, Picard’s iteration method, and Banach contraction principle. As application, we conclude this paper by giving an illustrative example to demonstrate the applicability of the obtained results.

scholarly journals Monotone Iterative Technique for Fractional Evolution Equations in Banach Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
Jia Mu
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Evolution Equations ◽  
Upper And Lower Solutions ◽  
Monotone Iterative Technique ◽  
Iterative Technique ◽  
Initial Value ◽  
Fractional Evolution Equations ◽  
Monotone Iterative ◽  
Noncompactness Measure

We investigate the initial value problem for a class of fractional evolution equations in a Banach space. Under some monotone conditions and noncompactness measure conditions of the nonlinearity, the well-known monotone iterative technique is then extended for fractional evolution equations which provides computable monotone sequences that converge to the extremal solutions in a sector generated by upper and lower solutions. An example to illustrate the applications of the main results is given.

scholarly journals Periodic Solutions andS-Asymptotically Periodic Solutions to Fractional Evolution Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-12 ◽  
Author(s):  
Jia Mu ◽  
Yong Zhou ◽  
Li Peng
Keyword(s):  
Periodic Solutions ◽  
Existence And Uniqueness ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Fractional Evolution Equations ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Asymptotically Periodic ◽  
Liouville Fractional Derivative ◽  
Asymptotically Periodic Solutions

This paper deals with the existence and uniqueness of periodic solutions,S-asymptotically periodic solutions, and other types of bounded solutions for some fractional evolution equations with the Weyl-Liouville fractional derivative defined for periodic functions. Applying Fourier transform we give reasonable definitions of mild solutions. Then we accurately estimate the spectral radius of resolvent operator and obtain some existence and uniqueness results.

scholarly journals Monotone Iterative Technique for a Class of Slanted Cantilever Beam Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Mei Wei ◽  
Qiang Li
Keyword(s):  
Cantilever Beam ◽  
Existence And Uniqueness ◽  
Elastic Beam ◽  
Monotone Iterative Technique ◽  
Iterative Sequence ◽  
Order Ordinary Differential Equation ◽  
Existence And Uniqueness Results ◽  
Monotone Iterative ◽  
Uniqueness Results ◽  
Beam Equations

In this paper, we deal with the existence and uniqueness of the solutions of two-point boundary value problem of fourth-order ordinary differential equation: u4(t)=f(t,u(t),u′(t)),  t∈[0,1],  u(0)=u′(0)=u′′(1)=u′′′(1)=0, where f:[0,1]×R2→R is a continuous function. The problem describes the static deformation of an elastic beam whose left end-point is fixed and right is freed, which is called slanted cantilever beam. Under some weaker assumptions, we establish a new maximum principle by the perturbation of positive operator and construct the monotone iterative sequence of the lower and upper solutions, and, based on this, we obtain the existence and uniqueness results for the slanted cantilever beam.

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