scholarly journals Exact Null Controllability for Fractional Nonlocal Integrodifferential Equations via Implicit Evolution System

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Amar Debbouche ◽  
Dumitru Baleanu
Keyword(s):  
Banach Space ◽  
Strong Solutions ◽  
Existence Results ◽  
Null Controllability ◽  
Evolution System ◽  
Integrodifferential System ◽  
Exact Null Controllability ◽  
Nonlinear Integrodifferential System ◽  
Nonlocal Nonlinear ◽  
Controllability Result

We introduce a new concept called implicit evolution system to establish the existence results of mild and strong solutions of a class of fractional nonlocal nonlinear integrodifferential system, then we prove the exact null controllability result of a class of fractional evolution nonlocal integrodifferential control system in Banach space. As an application that illustrates the abstract results, two examples are provided.

scholarly journals Null controllability of a nonlinear heat equation

2002 ◽  
Vol 7 (7) ◽  
pp. 375-383 ◽  
Author(s):  
G. Aniculăesei ◽  
S. Aniţa
Keyword(s):  
Heat Equation ◽  
Dirichlet Boundary ◽  
Null Controllability ◽  
Carleman Estimates ◽  
Nonlinear Heat Equation ◽  
Homogeneous Dirichlet Boundary Condition ◽  
Linearized System ◽  
Kakutani Fixed Point ◽  
Exact Null Controllability ◽  
Kakutani Fixed Point Theorem

We study the internal exact null controllability of a nonlinear heat equation with homogeneous Dirichlet boundary condition. The method used combines the Kakutani fixed-point theorem and the Carleman estimates for the backward adjoint linearized system. The result extends to the case of boundary control.

scholarly journals Variational and Topological Methods for a Class of Nonlinear Equations which Involves a Duality Mapping

2020 ◽  
pp. 56-71
Author(s):  
Jenica Cringanu
Keyword(s):  
Banach Space ◽  
Variational Method ◽  
Critical Points ◽  
Mountain Pass Theorem ◽  
Growth Conditions ◽  
Existence Results ◽  
Duality Mapping ◽  
Topological Methods ◽  
Carathéodory Function ◽  
Direct Variational Method

The purpose of this paper is to show the existence results for the following abstract equation Jpu = Nfu,where Jp is the duality application on a real reflexive and smooth X Banach space, that corresponds to the gauge function φ(t) = tp-1, 1 < p < ∞. We assume that X is compactly imbedded in Lq(Ω), where Ω is a bounded domain in RN, N ≥ 2, 1 < q < p∗, p∗ is the Sobolev conjugate exponent.Nf : Lq(Ω) → Lq′(Ω), 1/q + 1/q′ = 1, is the Nemytskii operator that Caratheodory function generated by a f : Ω × R → R which satisfies some growth conditions. We use topological methods (via Leray-Schauder degree), critical points methods (the Mountain Pass theorem) and a direct variational method to prove the existence of the solutions for the equation Jpu = Nfu.

Lack of null controllability of viscoelastic flows

2019 ◽  
Vol 25 ◽  
pp. 60
Author(s):  
Debayan Maity ◽  
Debanjana Mitra ◽  
Michael Renardy
Keyword(s):  
Approximate Controllability ◽  
Momentum Equation ◽  
Null Controllability ◽  
Viscoelastic Flow ◽  
Viscoelastic Flows ◽  
Relaxation Mode ◽  
Maxwell Fluids ◽  
Linear Viscoelastic ◽  
Exact Null Controllability

We consider controllability of linear viscoelastic flow with a localized control in the momentum equation. We show that, for Jeffreys fluids or for Maxwell fluids with more than one relaxation mode, exact null controllability does not hold. This contrasts with known results on approximate controllability.

scholarly journals Existence Results for an Impulsive Neutral Fractional Integrodifferential Equation with Infinite Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Alka Chadha ◽  
Dwijendra N. Pandey
Keyword(s):  
Banach Space ◽  
Mild Solution ◽  
Hausdorff Measure ◽  
Integrodifferential Equation ◽  
Measure Of Noncompactness ◽  
Infinite Delay ◽  
Existence Results ◽  
Hausdorff Measure Of Noncompactness ◽  
Arbitrary Banach Space ◽  
Fractional Integrodifferential Equation

We consider an impulsive neutral fractional integrodifferential equation with infinite delay in an arbitrary Banach spaceX. The existence of mild solution is established by using solution operator and Hausdorff measure of noncompactness.

scholarly journals Existence Results for Differential Inclusions with Nonlinear Growth Conditions in Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Messaoud Bounkhel
Keyword(s):  
Banach Space ◽  
Hilbert Spaces ◽  
Differential Inclusions ◽  
Existence Of Solutions ◽  
Growth Conditions ◽  
Existence Results ◽  
Nonlinear Growth ◽  
Upper Semicontinuous ◽  
Set Valued Mapping ◽  
Sweeping Processes

In the Banach space setting, the existence of viable solutions for differential inclusions with nonlinear growth; that is,ẋ(t)∈F(t,x(t))a.e. onI,x(t)∈S,∀t∈I,x(0)=x0∈S, (*), whereSis a closed subset in a Banach space𝕏,I=[0,T],(T>0),F:I×S→𝕏, is an upper semicontinuous set-valued mapping with convex values satisfyingF(t,x)⊂c(t)x+xp𝒦,∀(t,x)∈I×S, wherep∈ℝ, withp≠1, andc∈C([0,T],ℝ+). The existence of solutions for nonconvex sweeping processes with perturbations with nonlinear growth is also proved in separable Hilbert spaces.

scholarly journals Method of semidiscretization in time for quasilinearintegrodifferential equations

2004 ◽  
Vol 2004 (9) ◽  
pp. 469-478 ◽  
Author(s):  
D. Bahuguna ◽  
Reeta Shukla
Keyword(s):  
Banach Space ◽  
Initial Data ◽  
Continuous Dependence ◽  
Reflexive Banach Space ◽  
Strong Solutions ◽  
Integrodifferential Equations

We consider a class of quasilinear integrodifferential equations in a reflexive Banach space. We apply the method of semidiscretization in time to establish the existence, uniqueness, and continuous dependence on the initial data of strong solutions.

scholarly journals A Class of Fractionalp-Laplacian Integrodifferential Equations in Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Yiliang Liu ◽  
Liang Lu
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Boundary Value Problems ◽  
Banach Spaces ◽  
Fixed Point Theorems ◽  
Nonlocal Boundary ◽  
Existence Results ◽  
Integrodifferential Equations ◽  
Laplacian Operator ◽  
Laplacian Equations

We study a class of nonlinear fractional integrodifferential equations withp-Laplacian operator in Banach space. Some new existence results are obtained via fixed point theorems for nonlocal boundary value problems of fractionalp-Laplacian equations. An illustrative example is also discussed.

scholarly journals Brezis pseudomonotone bifunctions and quasi equilibrium problems via penalization

2021 ◽  
Author(s):  
M. Bianchi ◽  
G. Kassay ◽  
R. Pini
Keyword(s):  
Banach Space ◽  
Variational Inequalities ◽  
Equilibrium Problem ◽  
Penalty Method ◽  
Real Banach Space ◽  
Equilibrium Problems ◽  
Existence Results ◽  
Quasi Equilibrium ◽  
Coercivity Conditions ◽  
Usual Equilibrium

AbstractIn this paper we investigate quasi equilibrium problems in a real Banach space under the assumption of Brezis pseudomonotonicity of the function involved. To establish existence results under weak coercivity conditions we replace the quasi equilibrium problem with a sequence of penalized usual equilibrium problems. To deal with the non compact framework, we apply a regularized version of the penalty method. The particular case of set-valued quasi variational inequalities is also considered.

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