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.

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.

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 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.

scholarly journals Existence of Solutions for Fractional Impulsive Integrodifferential Equations in Banach Spaces

2016 ◽  
Vol 2016 ◽  
pp. 1-11
Author(s):  
Haide Gou ◽  
Baolin Li
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Banach Spaces ◽  
Fixed Point Theorems ◽  
Measure Of Noncompactness ◽  
Evolution Equations ◽  
Existence Of Solutions ◽  
Nonlocal Conditions ◽  
Integrodifferential Equations ◽  
Fractional Evolution Equations

We investigate the existence of solutions for a class of impulsive fractional evolution equations with nonlocal conditions in Banach space by using some fixed point theorems combined with the technique of measure of noncompactness. Our results improve and generalize some known results corresponding to those obtained by others. Finally, two applications are given to illustrate that our results are valuable.

Fractional evolution equations with nonlocal conditions in partially ordered Banach space

2018 ◽  
Vol 34 (3) ◽  
pp. 379-390
Author(s):  
HEMANT KUMAR NASHINE ◽  
◽  
HE YANG ◽  
RAVI P. AGARWAL ◽  
◽  
...  
Keyword(s):  
Fixed Point ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Nonlocal Conditions ◽  
Coupled Fixed Point ◽  
Positive Cone ◽  
Bounded Linear ◽  
Partially Ordered ◽  
C0 Semigroup ◽  
Uniformly Bounded

In the present work, we discuss the existence of mild solutions for the initial value problem of fractional evolution equation of the form where C Dσ t denotes the Caputo fractional derivative of order σ ∈ (0, 1), −A : D(A) ⊂ X → X generates a positive C0-semigroup T(t)(t ≥ 0) of uniformly bounded linear operator in X, b > 0 is a constant, f is a given functions. For this, we use the concept of measure of noncompactness in partially ordered Banach spaces whose positive cone K is normal, and establish some basic fixed point results under the said concepts. In addition, we relaxed the conditions of boundedness, closedness and convexity of the set at the expense that the operator is monotone and bounded. We also supply some new coupled fixed point results via MNC. To justify the result, we prove an illustrative example that rational of the abstract results for fractional parabolic equations.

scholarly journals A priori estimation of memory effects in reduced-order models of nonlinear systems using the Mori–Zwanzig formalism

2017 ◽  
Vol 473 (2205) ◽  
pp. 20170385 ◽  
Author(s):  
Ayoub Gouasmi ◽  
Eric J. Parish ◽  
Karthik Duraisamy
Keyword(s):  
Evolution Equations ◽  
Nonlinear Dynamical Systems ◽  
A Priori ◽  
Time History ◽  
Original System ◽  
Memory Kernel ◽  
Nonlinear Dynamical ◽  
History Of ◽  
Consistent Basis ◽  
A Priori Estimation

Reduced models of nonlinear dynamical systems require closure, or the modelling of the unresolved modes. The Mori–Zwanzig procedure can be used to derive formally closed evolution equations for the resolved physics. In these equations, the unclosed terms are recast as a memory integral involving the time history of the resolved variables. While this procedure does not reduce the complexity of the original system, these equations can serve as a mathematically consistent basis to develop closures based on memory approximations. In this scenario, knowledge of the memory kernel is paramount in assessing the validity of a memory approximation. Unravelling the memory kernel requires solving the orthogonal dynamics, which is a high-dimensional partial differential equation that is intractable, in general. A method to estimate the memory kernel a priori , using full-order solution snapshots, is proposed. The key idea is to solve a pseudo orthogonal dynamics equation, which has a convenient Liouville form, instead. This ersatz arises from the assumption that the semi-group of the orthogonal dynamics is a composition operator for one observable. The method is exact for linear systems. Numerical results on the Burgers and Kuramoto–Sivashinsky equations demonstrate that the proposed technique can provide valuable information about the memory kernel.

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