New Study of Existence and Dimension of the Set of Solutions for Nonlocal Impulsive Differential Inclusions With Sectorial Operator

In this article, we are interested in a new generic class of nonlocal fractional impulsive differential inclusions with linear sectorial operator and Lipschitz multivalued function in the setting of finite dimensional Banach spaces. By modifying the definition of PC-mild solutions initiated by Shu, we succeeded to determine new conditions that sufficiently guarantee the existence of the solutions. The results are obtained by combining techniques of fractional calculus and fixed point theorem for contraction maps. We also characterize the topological structure of the set of solutions. Finally, we provide a demonstration to address the applicability of the theoretical results.

Tykhonov triples, well-posedness and convergence results

"In this paper we present a unified theory of convergence results in the study of abstract problems. To this end we introduce a new mathematical object, the so-called Tykhonov triple $\cT=(I,\Omega,\cC)$, constructed by using a set of parameters $I$, a multivalued function $\Omega$ and a set of sequences $\cC$. Given a problem $\cP$ and a Tykhonov triple $\cT$, we introduce the notion of well-posedness of problem $\cP$ with respect to $\cT$ and provide several preliminary results, in the framework of metric spaces. Then we show how these abstract results can be used to obtain various convergences in the study of a nonlinear equation in reflexive Banach spaces. "

Superpositional measurability of a multivalued function under generalized Caratheodory conditions

For a multivalued mapping F:[a; b] × R^m → comp(R^m), the problem of superpositional measurability and superpositional selectivity is considered. As it is known, for superpositional measurability it is sufficient that the mapping F satisfies the Caratheodory conditions, for superpositional selectivity it is sufficient that F(•,x) has a measurable section and F(t; •) is upper semicontinuous. In this paper, we propose generalizations of these conditions based on the replacement, in the definitions of continuity and semicontinuity, of the limit of the sequence of coordinates of points in the images of multivalued mappings to a one-sided limit. It is shown that under such weakened conditions the multivalued mapping F possesses the required properties of superpositional measurability / superpositional selectivity. Illustrative examples are given, as well as examples of the significance of the proposed conditions. For single-valued mappings, the proposed conditions coincide with the generalized Caratheodory conditions proposed by I.V. Shragin (see [Bulletin of the Tambov University. Series: natural and technical sciences, 2014, 19:2, 476–478]).

On systems of parabolic variational inequalities with multivalued terms

In this paper we present an analytical framework for the following system of multivalued parabolic variational inequalities in a cylindrical domain $$Q=\varOmega \times (0,\tau )$$ Q = Ω × ( 0 , τ ) : For $$k=1,\dots , m$$ k = 1 , ⋯ , m , find $$u_k\in K_k$$ u k ∈ K k and $$\eta _k\in L^{p'_k}(Q)$$ η k ∈ L p k ′ ( Q ) such that $$\begin{aligned}&u_k(\cdot ,0)=0\ \text{ in } \varOmega ,\ \ \eta _k(x,t)\in f_k(x,t,u_1(x,t), \dots , u_m(x,t)), \\&\langle u_{kt}+A_k u_k, v_k-u_k\rangle +\int _Q \eta _k\, (v_k-u_k)\,dxdt\ge 0,\ \ \forall \ v_k\in K_k, \end{aligned}$$ u k ( · , 0 ) = 0 in Ω , η k ( x , t ) ∈ f k ( x , t , u 1 ( x , t ) , ⋯ , u m ( x , t ) ) , ⟨ u kt + A k u k , v k - u k ⟩ + ∫ Q η k ( v k - u k ) d x d t ≥ 0 , ∀ v k ∈ K k , where $$K_k $$ K k is a closed and convex subset of $$L^{p_k}(0,\tau ;W_0^{1,p_k}(\varOmega ))$$ L p k ( 0 , τ ; W 0 1 , p k ( Ω ) ) , $$A_k$$ A k is a time-dependent quasilinear elliptic operator, and $$f_k:Q\times \mathbb {R}^m\rightarrow 2^{\mathbb {R}}$$ f k : Q × R m → 2 R is an upper semicontinuous multivalued function with respect to $$s\in {\mathbb R}^m$$ s ∈ R m . We provide an existence theory for the above system under certain coercivity assumptions. In the noncoercive case, we establish an appropriate sub-supersolution method that allows us to get existence and enclosure results. As an application, a multivalued parabolic obstacle system is treated. Moreover, under a lattice condition on the constraints $$K_k$$ K k , systems of evolutionary variational-hemivariational inequalities are shown to be a subclass of the above system of multivalued parabolic variational inequalities.

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

In 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.

On one control problem with disturbance and vectograms depending linearly on given sets

A control problem with a given end time is considered, in which the control vectograms and disturbance depend linearly on the given convex compact sets. A multivalued mapping of the phase space of the control problem to the linear normed space E is given. The goal of constructing a control is that at the end of the control process the fixed vector of the space E belongs to the image of the multivalued mapping for any admissible realization of the disturbance. A stable bridge is defined in terms of multivalued functions. The presented procedure constructs, according to a given multivalued function which is a stable bridge, a control that solves the problem. Explicit formulas are obtained that determine a stable bridge in the considered control problem. Conditions are found under which the constructed stable bridge is maximal. Some problems of group pursuit can be reduced to the considered control problem with disturbance. The article provides such an example.

Controllability of Fractional Evolution Inclusions with Noninstantaneous Impulses

This paper is concerned with the controllability issue of fractional semilinear evolution inclusions with noninstantaneous impulses. Using weak sequentially closed graph operators, we establish sufficient conditions to guarantee controllability results. We do not assume that the semigroup is compact or we do not assume a compactness-type condition on the multivalued function. Finally, two examples are given to illustrate our theory.

Remarks on Numerical Experiments of the Allen-Cahn Equations with Constraint via Yosida Approximation

We consider a one-dimensional Allen-Cahn equation with constraint from the viewpoint of numerical analysis. The constraint is provided by the subdifferential of the indicator function on the closed interval, which is the multivalued function. Therefore, it is very difficult to perform a numerical experiment of our equation. In this paper we approximate the constraint by the Yosida approximation. Then, we study the approximating system of the original model numerically. In particular, we give the criteria for the standard forward Euler method to give the stable numerical experiments of the approximating equation. Moreover, we provide the numerical experiments of the approximating equation.

Convex Sweeping Processes with Noncompact Perturbations and with Delay in Banach Spaces

We prove two results concerning the existence of solutions for functional differential inclusions that are governed by sweeping processes, with noncompact valued perturbations in Banach spaces. Indeed, we have two goals. The first is to give a technique that allows considering sweeping processes with noncompact valued perturbations and associated with a multivalued function depending on time. The second is to give a technique to overcome the arising problem from the nonlinearity of the normalized mappings, when we deal with sweeping processes with noncompact valued perturbations and associated with a multivalued function depending on time and state.