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 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 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.

scholarly journals Existence Results for Stochastic Semilinear Differential Inclusions with Nonlocal Conditions

2011 ◽  
Vol 2011 ◽  
pp. 1-17
Author(s):  
A. Vinodkumar ◽  
A. Boucherif
Keyword(s):  
Differential Inclusions ◽  
Fixed Point Theorems ◽  
Mild Solutions ◽  
A Priori ◽  
Nonlocal Conditions ◽  
Sufficient Conditions ◽  
Existence Results ◽  
A Priori Bounds ◽  
Stochastic Differential Inclusions ◽  
Multivalued Operators

We discuss existence results of mild solutions for stochastic differential inclusions subject to nonlocal conditions. We provide sufficient conditions in order to obtain a priori bounds on possible solutions of a one-parameter family of problems related to the original one. We, then, rely on fixed point theorems for multivalued operators to prove our main results.

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 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 Existence Theorems for Fractional Semilinear Integrodifferential Equations with Noninstantaneous Impulses and Delay

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Bo Zhu ◽  
Minhui Zhu
Keyword(s):  
Fixed Point ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Fixed Point Theorems ◽  
Existence Theorems ◽  
Mild Solutions ◽  
Semigroup Theory ◽  
Integrodifferential Equations ◽  
Partial Differential ◽  
Noninstantaneous Impulses

In this paper, we consider a class of fractional semilinear integrodifferential equations with noninstantaneous impulses and delay. By the semigroup theory and fixed point theorems, we establish various theorems for the existence of mild solutions for the problem. An example involving partial differential equations with noninstantaneous impulses is given to show the application of our main results.

Nonlocal fractional semilinear differential inclusions with noninstantaneous impulses and of order α ∈ (1, 2)

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
JinRong Wang ◽  
Ahmed G. Ibrahim ◽  
Donal O’Regan ◽  
Adel A. Elmandouh
Keyword(s):  
Differential Inclusions ◽  
Mild Solutions ◽  
Multivalued Function ◽  
Linear Operators ◽  
Solution Set ◽  
Cosine Family ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Noninstantaneous Impulses

AbstractIn this paper, we establish the existence of mild solutions for nonlocal fractional semilinear differential inclusions with noninstantaneous impulses of order α ∈ (1,2) and generated by a cosine family of bounded linear operators. Moreover, we show the compactness of the solution set. We consider both the case when the values of the multivalued function are convex and nonconvex. Examples are given to illustrate the theory.

scholarly journals Solvability of control problem for fractional nonlinear differential inclusions with nonlocal conditions

2019 ◽  
Vol 24 (4) ◽  
Author(s):  
Alka Chadha ◽  
Rathinasamy Sakthivel ◽  
Swaroop Nandan Bora
Keyword(s):  
Differential Inclusions ◽  
Approximate Controllability ◽  
Measure Of Noncompactness ◽  
Nonlocal Conditions ◽  
Sufficient Conditions ◽  
Fractional Differential Inclusions ◽  
Multivalued Maps ◽  
Nonlinear Differential Inclusions ◽  
Approximately Controllable ◽  
Fractional Differential

In this paper, we study the approximate controllability of nonlocal fractional differential inclusions involving the Caputo fractional derivative of order q ∈ (1,2) in a Hilbert space. Utilizing measure of noncompactness and multivalued fixed point strategy, a new set of sufficient conditions is obtained to ensure the approximate controllability of nonlocal fractional differential inclusions when the multivalued maps are convex. Precisely, the results are developed under the assumption that the corresponding linear system is approximately controllable.  

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