scholarly journals Hilfer fractional evolution hemivariational inequalities with nonlocal initial conditions and optimal controls

2019 ◽  
Vol 24 (2) ◽  
pp. 189-209
Author(s):  
Yatian Pei ◽  
Yong-Kui Chang
Keyword(s):  
Control System ◽  
Initial Conditions ◽  
Mild Solutions ◽  
Sufficient Conditions ◽  
Hemivariational Inequality ◽  
Hemivariational Inequalities ◽  
Optimal Controls ◽  
Initial Condition ◽  
Uniqueness Of Solutions ◽  
Nonlocal Initial Condition

In this paper, we mainly consider a control system governed by a Hilfer fractional evolution hemivariational inequality with a nonlocal initial condition. We first establish sufficient conditions for the existence of mild solutions to the addressed control system via properties of generalized Clarke subdifferential and a fixed point theorem for condensing multivalued maps. Then we present the existence of optimal state-control pairs of the limited Lagrange optimal systems governed by a Hilfer fractional evolution hemivariational inequality with a nonlocal initial condition. The optimal control results are derived without uniqueness of solutions for the control system.

scholarly journals On fractional Langevin equation involving two fractional orders in different intervals

2019 ◽  
Vol 24 (6) ◽  
Author(s):  
Hamid Baghani ◽  
J. Nieto
Keyword(s):  
Langevin Equation ◽  
Existence And Uniqueness ◽  
Initial Conditions ◽  
Sufficient Conditions ◽  
Uniqueness Of Solutions ◽  
Numerical Examples ◽  
Fractional Equations ◽  
Fractional Langevin Equation ◽  
Nonlinear Langevin Equation ◽  
Fractional Orders

In this paper, we study a nonlinear Langevin equation involving two fractional orders  α ∈ (0; 1] and β ∈ (1; 2] with initial conditions. By means of an interesting fixed point theorem, we establish sufficient conditions for the existence and uniqueness of solutions for the fractional equations. Some illustrative numerical examples are also discussed. 

scholarly journals Boundary Value Problems of Fractional Order Differential Equation with Integral Boundary Conditions and Not Instantaneous Impulses

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Peiluan Li ◽  
Changjin Xu
Keyword(s):  
Boundary Conditions ◽  
Fractional Order ◽  
Fixed Point Theorems ◽  
Order Differential Equation ◽  
Mild Solutions ◽  
Sufficient Conditions ◽  
Integral Boundary Conditions ◽  
Fractional Order Differential Equation ◽  
Uniqueness Of Solutions ◽  
Integral Boundary

We investigate the existence of mild solutions for fractional order differential equations with integral boundary conditions and not instantaneous impulses. By some fixed-point theorems, we establish sufficient conditions for the existence and uniqueness of solutions. Finally, two interesting examples are given to illustrate our theory results.

scholarly journals Partial differential hemivariational inequalities

2018 ◽  
Vol 7 (4) ◽  
pp. 571-586 ◽  
Author(s):  
Zhenhai Liu ◽  
Shengda Zeng ◽  
Dumitru Motreanu
Keyword(s):  
Evolution Equations ◽  
Existence Of Solutions ◽  
Mild Solutions ◽  
Upper Semicontinuity ◽  
Solution Set ◽  
Hemivariational Inequality ◽  
Hemivariational Inequalities ◽  
Well Posedness ◽  
Partial Differential ◽  
New Class

AbstractThe aim of this paper is to introduce and study a new class of problems called partial differential hemivariational inequalities that combines evolution equations and hemivariational inequalities. First, we introduce the concept of strong well-posedness for mixed variational quasi hemivariational inequalities and establish metric characterizations for it. Then we show the existence of solutions and meaningful properties such as measurability and upper semicontinuity for the solution set of the mixed variational quasi hemivariational inequality associated to the partial differential hemivariational inequality. Relying, on these properties we are able to prove the existence of mild solutions for partial differential hemivariational inequalities. Furthermore, we show the compactness of the set of the corresponding mild trajectories.

scholarly journals Uniqueness and non-uniqueness of solutions of Vlasov-McKean equations

1987 ◽  
Vol 43 (2) ◽  
pp. 246-256 ◽  
Author(s):  
Michael Scheutzow
Keyword(s):  
Lipschitz Continuity ◽  
Initial Conditions ◽  
Sufficient Conditions ◽  
Expected Value ◽  
Deterministic Case ◽  
Uniqueness Of Solutions ◽  
Random Initial Conditions

AbstractWe study the equation dY(t)/dt = f(Y(t), Eh(Y(t))) for random initial conditions, where E denotes the expected value. It turns out that in contrast to the deterministic case local Lipschitz continuity of f and h are not sufficient to ensure uniqueness of the solutions. Finally we also state some sufficient conditions for uniqueness.

scholarly journals Subdifferential inclusions and quasi-static hemivariational inequalities for frictional viscoelastic contact problems

2012 ◽  
Vol 10 (6) ◽  
Author(s):  
Stanisław Migórski
Keyword(s):  
Mathematical Modeling ◽  
Boundary Conditions ◽  
Existence And Uniqueness ◽  
Contact Problems ◽  
Hemivariational Inequality ◽  
Hemivariational Inequalities ◽  
Uniqueness Of Solutions ◽  
Viscoelastic Contact

AbstractWe survey recent results on the mathematical modeling of nonconvex and nonsmooth contact problems arising in mechanics and engineering. The approach to such problems is based on the notions of an operator subdifferential inclusion and a hemivariational inequality, and focuses on three aspects. First we report on results on the existence and uniqueness of solutions to subdifferential inclusions. Then we discuss two classes of quasi-static hemivariational ineqaulities, and finally, we present ideas leading to inequality problems with multivalued and nonmonotone boundary conditions encountered in mechanics.

On fractional heat equations with non-local initial conditions

2015 ◽  
Vol 59 (1) ◽  
pp. 65-76 ◽  
Author(s):  
Bruno de Andrade ◽  
Claudio Cuevas ◽  
Herme Soto
Keyword(s):  
Initial Conditions ◽  
Mild Solutions ◽  
Sufficient Conditions ◽  
Heat Equations ◽  
Non Local

AbstractIn this paper we consider the problem of existence of mild solutions to semilinear fractional heat equations with non-local initial conditions. We provide sufficient conditions for existence and regularity of such solutions.

scholarly journals The uniqueness of bounded or measurable solutions of some functional equations

1970 ◽  
Vol 11 (2) ◽  
pp. 186-190
Author(s):  
T. D. Howroyd
Keyword(s):  
Banach Algebra ◽  
Measurable Function ◽  
Functional Equations ◽  
Positive Measure ◽  
Continuous Solution ◽  
Initial Conditions ◽  
Initial Condition ◽  
Real Solution ◽  
Uniqueness Of Solutions ◽  
Image Position

In this paper we are concerned with the uniqueness of solutions of functional equations of the form Some conditions for (1) or (2) to have at most one real continuous solution f which satisfies two given initial conditions are contained in [2], [3], [4] and [7]. Conditions sufficient for the equation to determine at most one continuous solution f with values in a Banach algebra are contained in [5]. It is well known (see [1] ch. 2) that one initial condition suffices for Cauchy's equation or two for Jensen's equation to uniquely determine a real solution f which is bounded on an interval or majorized on a set of positive measure by a measurable function. We place conditions on F and H so that similar statements can be made about solutions of (1) or (2). The corresponding results for solutions which are functions of many real variables follow as for Cauchy's and Jensen's equations (see [1] ch. 5).

THE MINIMIZING TOTAL VARIATION FLOW WITH MEASURE INITIAL CONDITIONS

2004 ◽  
Vol 06 (03) ◽  
pp. 431-494 ◽  
Author(s):  
F. ANDREU ◽  
J. M. MAZÓN ◽  
J. S. MOLL ◽  
V. CASELLES
Keyword(s):  
Total Variation ◽  
Initial Conditions ◽  
Singular Part ◽  
Dimensional Manifold ◽  
Initial Condition ◽  
Uniqueness Of Solutions ◽  
Limit Solution ◽  
Total Variation Flow ◽  
The Cauchy Problem ◽  
Limit Solutions

In this paper we obtain existence and uniqueness of solutions for the Cauchy problem for the minimizing total variation flow when the initial condition is a Radon measure in ℝN. We study limit solutions obtained by weakly approximating the initial measure μ by functions in L1(ℝN). We are able to characterize limit solutions when the initial condition μ=h+μs, where h∈L1(ℝN)∩L∞(ℝN), and μs=αℋk⌊ S,α≥0,k is an integer and S is a k-dimensional manifold with bounded curvatures. In case k<N-1 we prove that the singular part of the solution does not move, it remains equal to μs for all t≥0. In particular, u(t)=δ0 when u(0)=δ0. In case k=N-1 we prove that the singular part of the limit solution is [Formula: see text] and we also characterize its absolutely continuous part. This explicit behaviour permits to characterize limit solutions. We also give an entropy condition characterization of the solution which is more satisfactory when k<N-1. Finally, we describe some distributional solutions which do not have the behaviour characteristic of limit solutions.

scholarly journals Optimal control of feedback control systems governed by systems of evolution hemivariational inequalities

Filomat ◽  
2018 ◽  
Vol 32 (15) ◽  
pp. 5205-5220 ◽  
Author(s):  
Lu-Chuan Ceng ◽  
Zhenhai Liu ◽  
Jen-Chih Yao ◽  
Yonghong Yao
Keyword(s):  
Optimal Control ◽  
Control System ◽  
Feedback Control ◽  
Control Systems ◽  
Existence Result ◽  
Sufficient Conditions ◽  
Hemivariational Inequalities ◽  
Feedback Control System ◽  
Feedback Control Systems ◽  
Evolution Hemivariational Inequalities

In this paper, we introduce and consider a feedback control system governed by the system of evolution hemivariational inequalities. Several sufficient conditions are formulated by virtue of the properties of multimaps and partial Clarke?s subdifferentials such that the existence result of feasible pairs of the feedback control systems is guaranteed. Moreover, an existence result of optimal control pairs for an optimal control system is also established.

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