scholarly journals Existence of Mild Solutions for a Class of Fractional Evolution Equations with Compact Analytic Semigroup

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
He Yang
Keyword(s):  
Analytic Semigroup ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Fractional Power ◽  
Growth Conditions ◽  
Local Growth ◽  
Fractional Evolution Equations ◽  
Nonlinear Part ◽  
Compact Analytic Semigroup

This paper deals with the existence of mild solutions for a class of fractional evolution equations with compact analytic semigroup. We prove the existence of mild solutions, assuming that the nonlinear part satisfies some local growth conditions in fractional power spaces. An example is also given to illustrate the applicability of abstract results.

scholarly journals Positive Solutions for the Initial Value Problem of Fractional Evolution Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
He Yang ◽  
Yue Liang
Keyword(s):  
Differential Equation ◽  
Fixed Point Theorems ◽  
Analytic Semigroup ◽  
Evolution Equations ◽  
Existence Theorems ◽  
Mild Solutions ◽  
Parabolic Type ◽  
Initial Value ◽  
Fractional Evolution Equations ◽  
The Cauchy Problem

By using the fixed point theorems and the theory of analytic semigroup, we investigate the existence of positive mild solutions to the Cauchy problem of Caputo fractional evolution equations in Banach spaces. Some existence theorems are obtained under the case that the analytic semigroup is compact and noncompact, respectively. As an example, we study the partial differential equation of the parabolic type of fractional order.

scholarly journals Existence of positive S-asymptotically periodic solutions of the fractional evolution equations in ordered Banach spaces

2021 ◽  
Vol 26 (5) ◽  
pp. 928-946
Author(s):  
Qiang Li ◽  
Lishan Liu ◽  
Mei Wei
Keyword(s):  
Evolution Equations ◽  
Mild Solutions ◽  
Upper And Lower Solutions ◽  
Periodic Problem ◽  
Growth Conditions ◽  
Monotone Iterative Method ◽  
Monotone Iterative ◽  
Asymptotically Periodic ◽  
Asymptotically Periodic Solutions ◽  
Ordered Banach Spaces

In this paper, we discuss the asymptotically periodic problem for the abstract fractional evolution equation under order conditions and growth conditions. Without assuming the existence of upper and lower solutions, some new results on the existence of the positive S-asymptotically ω-periodic mild solutions are obtained by using monotone iterative method and fixed point theorem. It is worth noting that Lipschitz condition is no longer needed, which makes our results more widely applicable.

scholarly journals Periodic mild solutions of impulsive fractional evolution equations

2020 ◽  
Vol 5 (1) ◽  
pp. 497-506 ◽  
Author(s):  
Lulu Ren ◽  
◽  
JinRong Wang ◽  
Michal Fečkan ◽  
◽  
...  
Keyword(s):  
Evolution Equations ◽  
Mild Solutions ◽  
Fractional Evolution Equations

scholarly journals Upper and lower solution method for Hilfer fractional evolution equations with nonlocal conditions

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Haide Gou ◽  
Yongxiang Li
Keyword(s):  
Banach Space ◽  
Solution Method ◽  
Measure Of Noncompactness ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Nonlocal Conditions ◽  
Lower Solution ◽  
Ordered Banach Space ◽  
Fractional Evolution Equations ◽  
Lower Solution Method

AbstractThis paper is concerned with the existence of extremal mild solutions for Hilfer fractional evolution equations with nonlocal conditions in an ordered Banach space E. By employing the method of lower and upper solutions, the measure of noncompactness, and Sadovskii’s fixed point theorem, we obtain the existence of extremal mild solutions for Hilfer fractional evolution equations with noncompact semigroups. Finally, an example is provided to illustrate the feasibility of our main results.

Nonlocal problem for fractional stochastic evolution equations with solution operators

2016 ◽  
Vol 19 (6) ◽  
Author(s):  
Pengyu Chen ◽  
Xuping Zhang ◽  
Yongxiang Li
Keyword(s):  
Hilbert Spaces ◽  
Evolution Equations ◽  
Initial Conditions ◽  
Mild Solutions ◽  
Nonlocal Problem ◽  
Growth Conditions ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Nonlocal Initial Conditions ◽  
Analysis Theory

AbstractIn this article, we are concerned with a class of fractional stochastic evolution equations with nonlocal initial conditions in Hilbert spaces. The existence of mild solutions is obtained under the situation that the nonlinear term satisfies some appropriate growth conditions by using fractional calculations, Schauder fixed point theorem, stochastic analysis theory,

On the existence of mild solutions for a class of fractional differential equations with nonlocal conditions in the α-norm

2014 ◽  
Vol 51 (2) ◽  
pp. 141-154
Author(s):  
Mohamed Abbas
Keyword(s):  
Cauchy Problem ◽  
Mild Solutions ◽  
Linear Part ◽  
Nonlocal Conditions ◽  
Fractional Power ◽  
Existence Results ◽  
Nonlinear Part ◽  
Fractional Differential ◽  
Fractional Cauchy Problem ◽  
Lipschitz Conditions

This paper concerns the existence of mild solutions for some fractional Cauchy problem with nonlocal conditions in the α-norm. The linear part of the equations is assumed to generate an analytic compact bounded semigroup, and the nonlinear part satisfies some Lipschitz conditions with respect to the fractional power norm of the linear part. By using a fixed point theorem of Sadovskii, we establish some existence results which generalize ones in the case of fractional order derivative.

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