scholarly journals Relaxation of nonlocal m-dissipative differential inclusions

2019 ◽  
Vol 27 (3) ◽  
pp. 45-63
Author(s):  
S. Bilal ◽  
O. Cârjă ◽  
T. Donchev ◽  
N. Javaid ◽  
A. I. Lazu
Keyword(s):  
Differential Inclusion ◽  
Differential Inclusions ◽  
Solution Set ◽  
Limit Solution ◽  
Limit Solutions ◽  
Dissipative Differential Inclusions ◽  
Integral Solutions

AbstractWe show here that the set of the integral solutions of a nonlocal differential inclusion is dense in the set of the solution set of the corresponding relaxed differential inclusion. We further define a notion of limit solution and show that the set of limit solutions is closed and is the closure of the set of integral solutions. An illustrative example is provided.

scholarly journals Evolution Inclusions in Banach Spaces under Dissipative Conditions

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 750
Author(s):  
Tzanko Donchev ◽  
Shamas Bilal ◽  
Ovidiu Cârjă ◽  
Nasir Javaid ◽  
Alina I. Lazu
Keyword(s):  
Banach Spaces ◽  
Differential Inclusions ◽  
Qualitative Properties ◽  
Evolution Inclusions ◽  
Fully Nonlinear ◽  
Limit Solution ◽  
Nonlinear Differential Inclusions ◽  
Limit Solutions ◽  
Lipschitz Conditions ◽  
Integral Solutions

We develop a new concept of a solution, called the limit solution, to fully nonlinear differential inclusions in Banach spaces. That enables us to study such kind of inclusions under relatively weak conditions. Namely we prove the existence of this type of solutions and some qualitative properties, replacing the commonly used compact or Lipschitz conditions by a dissipative one, i.e., one-sided Perron condition. Under some natural assumptions we prove that the set of limit solutions is the closure of the set of integral solutions.

scholarly journals One Sided Lipschitz Evolution Inclusions in Banach Spaces

Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3265
Author(s):  
Ali N. A. Koam ◽  
Tzanko Donchev ◽  
Alina I. Lazu ◽  
Muhammad Rafaqat ◽  
Ali Ahmad
Keyword(s):  
Banach Spaces ◽  
Lipschitz Condition ◽  
Differential Inclusions ◽  
Initial Conditions ◽  
Lipschitz Constant ◽  
Evolution Inclusions ◽  
Nonlocal Initial Conditions ◽  
Limit Solution ◽  
The Right ◽  
Dissipative Differential Inclusions

Using the notion of limit solution, we study multivalued perturbations of m-dissipative differential inclusions with nonlocal initial conditions. These solutions enable us to work in general Banach spaces, in particular L1. The commonly used Lipschitz condition on the right-hand side is weakened to a one-sided Lipschitz one. No compactness assumptions are required. We consider the cases of an arbitrary one-sided Lipschitz condition and the case of a negative one-sided Lipschitz constant. Illustrative examples, which can be modifications of real models, are provided.

scholarly journals Regular and singular perturbations of upper semicontinuous differential inclusion

1997 ◽  
Vol 20 (4) ◽  
pp. 699-706 ◽  
Author(s):  
Tzanko Donchev ◽  
Vasil Angelov
Keyword(s):  
Differential Inclusion ◽  
Singular Perturbations ◽  
Differential Inclusions ◽  
Existence Result ◽  
Lower Semicontinuous ◽  
Solution Set ◽  
Singularly Perturbed ◽  
Solution Sets ◽  
Upper Semicontinuous ◽  
Continuity Properties

In the paper we study the continuity properties of the solution set of upper semicontinuous differential inclusions in both regularly and singularly perturbed case. Using a kind of dissipative type of conditions introduced in [1] we obtain lower semicontinuous dependence of the solution sets. Moreover new existence result for lower semicontinuous differential inclusions is proved.

Optimal control of higher order differential inclusions with functional constraints

2020 ◽  
Vol 26 ◽  
pp. 37 ◽  
Author(s):  
Elimhan N. Mahmudov
Keyword(s):  
Optimal Control ◽  
Optimality Conditions ◽  
Differential Inclusion ◽  
Differential Inclusions ◽  
Higher Order ◽  
Second Order ◽  
Sufficient Optimality Conditions ◽  
Functional Constraints ◽  
Complementary Slackness ◽  
Complementary Slackness Conditions

The present paper studies the Mayer problem with higher order evolution differential inclusions and functional constraints of optimal control theory (PFC); to this end first we use an interesting auxiliary problem with second order discrete-time and discrete approximate inclusions (PFD). Are proved necessary and sufficient conditions incorporating the Euler–Lagrange inclusion, the Hamiltonian inclusion, the transversality and complementary slackness conditions. The basic concept of obtaining optimal conditions is locally adjoint mappings and equivalence results. Then combining these results and passing to the limit in the discrete approximations we establish new sufficient optimality conditions for second order continuous-time evolution inclusions. This approach and results make a bridge between optimal control problem with higher order differential inclusion (PFC) and constrained mathematical programming problems in finite-dimensional spaces. Formulation of the transversality and complementary slackness conditions for second order differential inclusions play a substantial role in the next investigations without which it is hardly ever possible to get any optimality conditions; consequently, these results are generalized to the problem with an arbitrary higher order differential inclusion. Furthermore, application of these results is demonstrated by solving some semilinear problem with second and third order differential inclusions.

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.

Uncertainty in public health models treated by differential inclusions

2020 ◽  
Vol 11 (04) ◽  
pp. 2050040
Author(s):  
Stanislaw Raczynski
Keyword(s):  
Public Health ◽  
Differential Inclusion ◽  
Differential Inclusions ◽  
Epidemic Models ◽  
Reachable Sets ◽  
Model Construction ◽  
Model Parameters ◽  
Uncertain Parameters ◽  
Epidemic Spread ◽  
Epidemic Modeling

An application of differential inclusions in the epidemic spread models is presented. Some mostly used epidemic models are discussed here, and a brief survey of epidemic modeling is given. Most of the models are some modifications of the Susceptible–Infected–Recovered model. Simple simulations are carried out. Then, we consider the influence of some uncertain parameters. It is pointed out that the presence of some fluctuating model parameters can be treated by differential inclusions. The solution to such differential inclusion is given in the form of reachable sets for model variables. Here, we focus on the differential inclusion application rather than the model construction.

Sign in / Sign up

Export Citation Format

Share Document