scholarly journals Existence of mild solutions for fractional nonautonomous evolution equations of Sobolev type with delay

2017 ◽  
Vol 2017 (1) ◽  
Author(s):  
Haide Gou ◽  
Baolin Li
Keyword(s):  
Evolution Equations ◽  
Mild Solutions ◽  
Sobolev Type

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.

scholarly journals Existence Results of Mild Solutions for the Fractional Stochastic Evolution Equations of Sobolev Type

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 1031
Author(s):  
He Yang
Keyword(s):  
Operator Theory ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Existence Results ◽  
Sobolev Type ◽  
Analysis Method ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Abstract Spaces ◽  
The Resolvent Operator

In this paper, by utilizing the resolvent operator theory, the stochastic analysis method and Picard type iterative technique, we first investigate the existence as well as the uniqueness of mild solutions for a class of α ∈ ( 1 , 2 ) -order Riemann–Liouville fractional stochastic evolution equations of Sobolev type in abstract spaces. Then the symmetrical technique is used to deal with the α ∈ ( 1 , 2 ) -order Caputo fractional stochastic evolution equations of Sobolev type in abstract spaces. Two examples are given as applications to the obtained results.

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 Approximate controllability of nonlocal non-autonomous Sobolev type evolution equations

2019 ◽  
Vol 9 (3) ◽  
pp. 86-94
Author(s):  
Arshi Meraj ◽  
Dwijendra Narain Pandey
Keyword(s):  
Fixed Point ◽  
Linear System ◽  
Approximate Controllability ◽  
Evolution Equations ◽  
Nonlocal Condition ◽  
Mild Solutions ◽  
Sobolev Type ◽  
Evolution System ◽  
Fixed Point Technique ◽  
Sobolev Type Differential Equations

The aim of this article is to investigate the existence of mild solutions as well as approximate controllability of non-autonomous Sobolev type differential equations with the nonlocal condition. To prove our results, we will take the help of Krasnoselskii fixed point technique, evolution system and controllability of the corresponding linear system.

scholarly journals Existence and approximate controllability of Sobolev type fractional stochastic evolution equations

2014 ◽  
Vol 62 (2) ◽  
pp. 205-215 ◽  
Author(s):  
N.I. Mahmudov
Keyword(s):  
Linear System ◽  
Hilbert Spaces ◽  
Approximate Controllability ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Sufficient Conditions ◽  
Sobolev Type ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Approximately Controllable

Abstract We study the existence of mild solutions and the approximate controllability concept for Sobolev type fractional semilinear stochastic evolution equations in Hilbert spaces. We prove existence of a mild solution and give sufficient conditions for the approximate controllability. In particular, we prove that the fractional linear stochastic system is approximately controllable in [0, b] if and only if the corresponding deterministic fractional linear system is approximately controllable in every [s, b], 0 ≤ s < b. An example is provided to illustrate the application of the obtained results.

Existence results of mild solutions for impulsive fractional integro-differential evolution equations with infinite delay

2014 ◽  
Vol 17 (4) ◽  
Author(s):  
Shengli Xie
Keyword(s):  
Differential Equations ◽  
Differential Evolution ◽  
Existence Theorem ◽  
Banach Spaces ◽  
Existence And Uniqueness ◽  
Evolution Equations ◽  
Infinite Delay ◽  
Mild Solutions ◽  
Existence Results ◽  
Order Case

AbstractIn this paper we prove the existence and uniqueness of mild solutions for impulsive fractional integro-differential evolution equations with infinite delay in Banach spaces. We generalize the existence theorem for integer order differential equations to the fractional order case. The results obtained here improve and generalize many known results.

scholarly journals On Almost Automorphic Mild Solutions for Nonautonomous Stochastic Evolution Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-25 ◽  
Author(s):  
Jing Cui ◽  
Litan Yan
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Banach Fixed Point Theorem ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Almost Automorphic ◽  
Composition Theorem

We consider a class of nonautonomous stochastic evolution equations in real separable Hilbert spaces. We establish a new composition theorem for square-mean almost automorphic functions under non-Lipschitz conditions. We apply this new composition theorem as well as intermediate space techniques, Krasnoselskii fixed point theorem, and Banach fixed point theorem to investigate the existence of square-mean almost automorphic mild solutions. Some known results are generalized and improved.

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.

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