A Study on Impulsive Hilfer Fractional Evolution Equations with Nonlocal Conditions

2020 ◽  
Vol 21 (2) ◽  
pp. 205-218
Author(s):  
Haide Gou ◽  
Yongxiang Li
Keyword(s):  
Fractional Derivative ◽  
Measure Of Noncompactness ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Nonlocal Conditions ◽  
Sobolev Type ◽  
Initial Value ◽  
Liouville Fractional Derivative ◽  
Characteristic Solution Operators ◽  
Impulsive Evolution Equations

AbstractIn this paper, we concern with the existence of mild solution to nonlocal initial value problem for nonlinear Sobolev-type impulsive evolution equations with Hilfer fractional derivative which generalized the Riemann–Liouville fractional derivative. At first, we establish an equivalent integral equation for our main problem. Second, by means of the properties of Hilfer fractional calculus, combining measure of noncompactness with the fixed-point methods, we obtain the existence results of mild solutions with two new characteristic solution operators. The results we obtained are new and more general to known results. At last, an example is provided to illustrate the results.

Approximate controllability and existence of mild solutions for Riemann-Liouville fractional stochastic evolution equations with nonlocal conditions of order 1 < α < 2

2019 ◽  
Vol 22 (4) ◽  
pp. 1086-1112 ◽  
Author(s):  
Linxin Shu ◽  
Xiao-Bao Shu ◽  
Jianzhong Mao
Keyword(s):  
Approximate Controllability ◽  
Stochastic Systems ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Nonlocal Conditions ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Mönch Fixed Point Theorem ◽  
Liouville Derivative ◽  
Liouville Fractional Derivative

Abstract In this paper, we consider the existence of mild solutions and approximate controllability for Riemann-Liouville fractional stochastic evolution equations with nonlocal conditions of order 1 < α < 2. As far as we know, there are few articles investigating on this issue. Firstly, the mild solutions to the equations are proved using Laplace transform of the Riemann-Liouville derivative. Moreover, the estimations of resolve operators involving the Riemann-Liouville fractional derivative of order 1 < α < 2 are given. Then, the existence results are obtained via the noncompact measurement strategy and the Mönch fixed point theorem. The approximate controllability of this nonlinear Riemann-Liouville fractional nonlocal stochastic systems of order 1 < α < 2 is concerned under the assumption that the associated linear system is approximately controllable. Finally, the approximate controllability results are obtained by using Lebesgue dominated convergence theorem.

scholarly journals Sobolev Type Fractional Dynamic Equations and Optimal Multi-Integral Controls with Fractional Nonlocal Conditions

2015 ◽  
Vol 18 (1) ◽  
Author(s):  
Amar Debbouche ◽  
Delfim F. M. Torres
Keyword(s):  
Nonlocal Condition ◽  
Mild Solutions ◽  
Semigroup Theory ◽  
Nonlocal Conditions ◽  
Fractional Power ◽  
Integral Solution ◽  
Dynamic Equations ◽  
Sobolev Type ◽  
Gronwall’S Inequality ◽  
Liouville Fractional Derivative

AbstractIn We prove existence and uniqueness of mild solutions to Sobolev type fractional nonlocal dynamic equations in Banach spaces. The Sobolev nonlocal condition is considered in terms of a Riemann-Liouville fractional derivative. A Lagrange optimal control problem is considered, and existence of a multi-integral solution obtained. Main tools include fractional calculus, semigroup theory, fractional power of operators, a singular version of Gronwall's inequality, and Leray-Schauder fixed point theorem. An example illustrating the theory is given.

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.

Topological structure of solution sets for control problems governed by semilinear fractional impulsive evolution equations with nonlocal conditions

2020 ◽  
Vol 37 (4) ◽  
pp. 1089-1113
Author(s):  
Yi-rong Jiang ◽  
Qiong-fen Zhang ◽  
Qi-qing Song
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Control Problem ◽  
Mild Solution ◽  
Point Theorem ◽  
Measure Of Noncompactness ◽  
Evolution Equations ◽  
Nonlocal Conditions ◽  
Solution Set ◽  
Impulsive Evolution Equations

Abstract This article investigates the topological structural of the mild solution set for a control problem monitored by semilinear fractional impulsive evolution equations with nonlocal conditions. The $R_{\delta }$-property of the mild solution set is obtained by applying the measure of noncompactness and a fixed point theorem of condensing maps and a fixed point theorem of nonconvex valued maps. Then this result is applied to prove that the presented control problem has a reachable invariant set under nonlinear perturbations. The obtained results are also applied to characterize the approximate controllability of the presented control problem.

Existence of Mild Solutions for Sobolev-Type Hilfer Fractional Nonautonomous Evolution Equations with Delay

2018 ◽  
Vol 19 (5) ◽  
pp. 481-492
Author(s):  
Haide Gou ◽  
Baolin Li
Keyword(s):  
Existence Result ◽  
Measure Of Noncompactness ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Sobolev Type ◽  
Operator Family ◽  
Hilfer Fractional Derivative ◽  
Definition Of ◽  
Sadovskii Fixed Point Theorem ◽  
Equations With Delay

AbstractThis paper treats the existence of mild solutions for Sobolev-type Hilfer fractional nonautonomous evolution equations with delay in Banach spaces. We first characterize the definition of mild solutions for the studied problem which was given based on an operator family generated by the operator pair (A,B) and probability density function. And then via Hilfer fractional derivative and combining the techniques of fractional calculus, measure of noncompactness and Sadovskii fixed-point theorem, we obtain new existence result of mild solutions for Sobolev-type Hilfer fractional nonautonomous evolution equations. Particularly, the existence or compactness of an operator $B^{-1} $ is not necessarily needed in our results. Furthermore, our results obtained improve and extend some related conclusions on this topic. At last, an example is given to illustrate our main results.

scholarly journals Existence and Attractivity for Fractional Evolution Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Yong Zhou ◽  
Jia Wei He ◽  
Bashir Ahmad ◽  
Ahmed Alsaedi
Keyword(s):  
Fractional Derivative ◽  
Evolution Equations ◽  
Global Attractivity ◽  
Mild Solutions ◽  
Sufficient Conditions ◽  
Cauchy Problems ◽  
Fractional Evolution Equations ◽  
Liouville Fractional Derivative

We study the existence and attractivity of solutions for fractional evolution equations with Riemann-Liouville fractional derivative. We establish sufficient conditions for the global attractivity of mild solutions for the Cauchy problems in the case that semigroup is compact.

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 Nonlocal Problem for Fractional Evolution Equations of Mixed Type with the Measure of Noncompactness

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Pengyu Chen ◽  
Yongxiang Li
Keyword(s):  
Mixed Type ◽  
Measure Of Noncompactness ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Nonlocal Problem ◽  
Nonlocal Conditions ◽  
Fractional Evolution Equations ◽  
Infinite Dimensional ◽  
Equations Of Mixed Type ◽  
Condensing Operator

A general class of semilinear fractional evolution equations of mixed type with nonlocal conditions on infinite dimensional Banach spaces is concerned. Under more general conditions, the existence of mild solutions and positive mild solutions is obtained by utilizing a new estimation technique of the measure of noncompactness and a new fixed point theorem with respect to convex-power condensing operator.

scholarly journals Existence of solutions for delay evolution equations with nonlocal conditions

2017 ◽  
Vol 15 (1) ◽  
pp. 616-627 ◽  
Author(s):  
Xuping Zhang ◽  
Yongxiang Li
Keyword(s):  
Fixed Point ◽  
Measure Of Noncompactness ◽  
Evolution Equations ◽  
Existence Of Solutions ◽  
Fixed Point Theory ◽  
Mild Solutions ◽  
Nonlocal Conditions ◽  
Point Theory ◽  
Existence Results ◽  
Kuratowski Measure Of Noncompactness

Abstract In this paper, we are devoted to study the existence of mild solutions for delay evolution equations with nonlocal conditions. By using tools involving the Kuratowski measure of noncompactness and fixed point theory, we establish some existence results of mild solutions without the assumption of compactness on the associated semigroup. Our results improve and generalize some related conclusions on this issue. Moreover, we present an example to illustrate the application of the main results.

scholarly journals Existence of Solutions for Nonlinear Mixed Type Integrodifferential Functional Evolution Equations with Nonlocal Conditions

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Shengli Xie
Keyword(s):  
Mixed Type ◽  
Measure Of Noncompactness ◽  
Evolution Equations ◽  
Existence Of Solutions ◽  
Mild Solutions ◽  
A Priori ◽  
Nonlocal Conditions ◽  
Functional Evolution ◽  
Mönch Fixed Point Theorem ◽  
A Priori Estimation

Using Mönch fixed point theorem, this paper proves the existence and controllability of mild solutions for nonlinear mixed type integrodifferential functional evolution equations with nonlocal conditions in Banach spaces, some restricted conditions on a priori estimation and measure of noncompactness estimation have been deleted, our results extend and improve many known results. As an application, we have given a controllability result of the system.

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