scholarly journals Analysis and Optimal Control of φ-Hilfer Fractional Semilinear Equations Involving Nonlocal Impulsive Conditions

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2084
Author(s):  
Sarra Guechi ◽  
Rajesh Dhayal ◽  
Amar Debbouche ◽  
Muslim Malik
Keyword(s):  
Mild Solutions ◽  
Semigroup Theory ◽  
Nonlocal Conditions ◽  
Optimal Controls ◽  
Impulsive Conditions ◽  
Semilinear Equations ◽  
New Class ◽  
Fractional Differential ◽  
The Fixed Point Theorem ◽  
Fractional Control

The goal of this paper is to consider a new class of φ-Hilfer fractional differential equations with impulses and nonlocal conditions. By using fractional calculus, semigroup theory, and with the help of the fixed point theorem, the existence and uniqueness of mild solutions are obtained for the proposed fractional system. Symmetrically, we discuss the existence of optimal controls for the φ-Hilfer fractional control system. Our main results are well supported by an illustrative example.

scholarly journals Neutral delay Hilfer fractional integrodifferential equations with fractional brownian motion

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yousef Alnafisah ◽  
Hamdy M. Ahmed
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Existence And Uniqueness ◽  
Mild Solutions ◽  
Semigroup Theory ◽  
Sufficient Conditions ◽  
Integrodifferential Equations ◽  
Neutral Delay ◽  
Fractional Differential ◽  
The Fixed Point Theorem

<p style='text-indent:20px;'>In this paper, we study the existence and uniqueness of mild solutions for neutral delay Hilfer fractional integrodifferential equations with fractional Brownian motion. Sufficient conditions for controllability of neutral delay Hilfer fractional differential equations with fractional Brownian motion are established. The required results are obtained based on the fixed point theorem combined with the semigroup theory, fractional calculus and stochastic analysis. Finally, an example is given to illustrate the obtained results.</p>

scholarly journals Global attracting solutions to Hilfer fractional differential inclusions of Sobolev type with noninstantaneous impulses and nonlocal conditions

2019 ◽  
Vol 24 (5) ◽  
Author(s):  
JinRong Wang ◽  
Ahmed Gamal Ibrahim ◽  
Donal O’Regan
Keyword(s):  
Differential Inclusions ◽  
Fixed Point Theorems ◽  
Mild Solutions ◽  
Semigroup Theory ◽  
Nonlocal Conditions ◽  
Continuous Functions ◽  
Sobolev Type ◽  
Unbounded Interval ◽  
Fractional Differential ◽  
Noninstantaneous Impulses

In this paper, we establish the existence of decay mild solutions on an unbounded interval of nonlocal fractional semilinear differential inclusions with noninstantaneous impulses and involving the Hilfer derivative. Our argument uses fixed point theorems, semigroup theory, multi-functions and a measure of noncompactness on the space of piecewise weighted continuous functions defined on an unbounded interval. An example is provided to illustrate our results.

scholarly journals Hilfer-type fractional differential switched inclusions with noninstantaneous impulsive and nonlocal conditions

2018 ◽  
Vol 23 (6) ◽  
pp. 921-941 ◽  
Author(s):  
JinRong Wang ◽  
Ahmed Gamal Ibrahim ◽  
Donal O’Regan
Keyword(s):  
Hausdorff Measure ◽  
Measure Of Noncompactness ◽  
Mild Solutions ◽  
Nonlocal Conditions ◽  
Continuous Functions ◽  
Inclusion Problem ◽  
Nonlocal Problems ◽  
Hausdorff Measure Of Noncompactness ◽  
New Class ◽  
Fractional Differential

In this paper, we study a new class of nonlocal problems for noninstantaneous impulsive Hilfer-type fractional differential switched inclusions in Banach spaces. First, we introduce a mild solution formula for this noninstantaneous impulsive inclusion problem. Second, we show the existence of mild solutions using the Hausdorff measure of noncompactness on the space of piecewise weighted continuous functions. Finally, an example is provided to illustrate the theory.

scholarly journals Solvability and Optimal Controls of Semilinear Riemann-Liouville Fractional Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Xue Pan ◽  
Xiuwen Li ◽  
Jing Zhao
Keyword(s):  
Differential Equations ◽  
Control Systems ◽  
Fractional Derivatives ◽  
Broad Class ◽  
Mild Solutions ◽  
Optimal Controls ◽  
Infinite Dimensional ◽  
Mild Conditions ◽  
Fractional Differential ◽  
Fractional Control

We consider the control systems governed by semilinear differential equations with Riemann-Liouville fractional derivatives in Banach spaces. Firstly, by applying fixed point strategy, some suitable conditions are established to guarantee the existence and uniqueness of mild solutions for a broad class of fractional infinite dimensional control systems. Then, by using generally mild conditions of cost functional, we extend the existence result of optimal controls to the Riemann-Liouville fractional control systems. Finally, a concrete application is given to illustrate the effectiveness of our main results.

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.

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.

Existence of mild solutions for semilinear differential equations with nonlocal and impulsive conditions

2014 ◽  
Vol 12 (4) ◽  
Author(s):  
Leszek Olszowy
Keyword(s):  
Differential Equations ◽  
Mild Solutions ◽  
Linear Part ◽  
Nonlocal Conditions ◽  
Continuous Functions ◽  
Evolution System ◽  
Impulsive Conditions ◽  
Measures Of Noncompactness ◽  
Piecewise Continuous ◽  
Unbounded Intervals

AbstractThis paper is concerned with the existence of mild solutions for impulsive semilinear differential equations with nonlocal conditions. Using the technique of measures of noncompactness in Banach and Fréchet spaces of piecewise continuous functions, existence results are obtained both on bounded and unbounded intervals, when the impulsive functions and the nonlocal item are not compact in the space of piecewise continuous functions but they are continuous and Lipschitzian with respect to some measure of noncompactness, and the linear part generates only a strongly continuous evolution system.

scholarly journals Approximate Controllability of Fractional Integrodifferential Evolution Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
R. Ganesh ◽  
R. Sakthivel ◽  
N. I. Mahmudov ◽  
S. M. Anthoni
Keyword(s):  
Control System ◽  
Control Systems ◽  
Approximate Controllability ◽  
Evolution Equations ◽  
Semigroup Theory ◽  
Nonlocal Conditions ◽  
Sufficient Conditions ◽  
Approximately Controllable ◽  
Fractional Control ◽  
Lipschitz Conditions

This paper addresses the issue of approximate controllability for a class of control system which is represented by nonlinear fractional integrodifferential equations with nonlocal conditions. By using semigroup theory,p-mean continuity and fractional calculations, a set of sufficient conditions, are formulated and proved for the nonlinear fractional control systems. More precisely, the results are established under the assumption that the corresponding linear system is approximately controllable and functions satisfy non-Lipschitz conditions. The results generalize and improve some known results.

scholarly journals Impulsive Fractional Semilinear Integrodifferential Equations with Nonlocal Conditions

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Xue Wang ◽  
Bo Zhu
Keyword(s):  
Fixed Point ◽  
Resolvent Operator ◽  
Fixed Point Theorems ◽  
Initial Conditions ◽  
Mild Solutions ◽  
Semigroup Theory ◽  
Nonlocal Conditions ◽  
Integrodifferential Equations ◽  
Existence Theory ◽  
Nonlocal Initial Conditions

This paper is devoted to a class of impulsive fractional semilinear integrodifferential equations with nonlocal initial conditions. Based on the semigroup theory and some fixed point theorems, the existence theory of PC-mild solutions is established under the condition of compact resolvent operator. Furthermore, the uniqueness of PC-mild solutions is proved in the case of the noncompact resolvent operator.

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