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.

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 Mild Solutions for Impulsive Integro-Differential Equations Involving Hilfer Fractional Derivative with almost Sectorial Operators

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 313
Author(s):  
Kulandhaivel Karthikeyan ◽  
Panjaiyan Karthikeyan ◽  
Nichaphat Patanarapeelert ◽  
Thanin Sitthiwirattham
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Jump Conditions ◽  
Hilfer Fractional Derivative ◽  
Sectorial Operators ◽  
Almost Sectorial Operators ◽  
Suitable Definition ◽  
Definition Of

In this manuscript, we establish the mild solutions for Hilfer fractional derivative integro-differential equations involving jump conditions and almost sectorial operator. For this purpose, we identify the suitable definition of a mild solution for this evolution equations and obtain the existence results. In addition, an application is also considered.

scholarly journals Existence and optimal controls for nonlocal fractional evolution equations of order (1,2) in Banach spaces

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Denghao Pang ◽  
Wei Jiang ◽  
Azmat Ullah Khan Niazi ◽  
Jiale Sheng
Keyword(s):  
Banach Spaces ◽  
Mild Solution ◽  
Measure Of Noncompactness ◽  
Evolution Equations ◽  
Well Posedness ◽  
Optimal Controls ◽  
Lagrange Problem ◽  
Fractional Differential ◽  
Fractional Evolution Systems ◽  
Definition Of

AbstractIn this paper, we mainly investigate the existence, continuous dependence, and the optimal control for nonlocal fractional differential evolution equations of order (1,2) in Banach spaces. We define a competent definition of a mild solution. On this basis, we verify the well-posedness of the mild solution. Meanwhile, with a construction of Lagrange problem, we elaborate the existence of optimal pairs of the fractional evolution systems. The main tools are the fractional calculus, cosine family, multivalued analysis, measure of noncompactness method, and fixed point theorem. Finally, an example is propounded to illustrate the validity of our main results.

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 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.

scholarly journals Existence of Mild Solutions for a Class of Impulsive Hilfer Fractional Coupled Systems

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Karim Guida ◽  
Khalid Hilal ◽  
Lahcen Ibnelazyz ◽  
Ming Mei
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fractional Derivative ◽  
Banach Spaces ◽  
Point Theorem ◽  
Measure Of Noncompactness ◽  
Mild Solutions ◽  
Existence Results ◽  
Coupled Systems ◽  
Hilfer Fractional Derivative

The aim of this paper is to give existence results for a class of coupled systems of fractional integrodifferential equations with Hilfer fractional derivative in Banach spaces. We first give some definitions, namely the Hilfer fractional derivative and the Hausdorff’s measure of noncompactness and the Sadovskii’s fixed point theorem.

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.

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