Superpositional measurability of a multivalued function under generalized Caratheodory conditions

2021 ◽  
pp. 305-314
Author(s):  
Irina D. Serova
Keyword(s):  
Multivalued Mapping ◽  
Multivalued Function ◽  
Multivalued Mappings ◽  
Upper Semicontinuous ◽  
Carathéodory Conditions ◽  
Measurable Section

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]).

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

2020 ◽  
Vol 30 (3) ◽  
pp. 429-443
Author(s):  
V.I. Ukhobotov ◽  
V.N. Ushakov
Keyword(s):  
Control Problem ◽  
Multivalued Mapping ◽  
Control Process ◽  
Multivalued Function ◽  
Convex Compact ◽  
Compact Sets ◽  
Stable Bridge ◽  
Multivalued Functions ◽  
The Given ◽  
Convex Compact 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.

scholarly journals Common fixed point of g-approximative multivalued mapping in partially ordered metric space

Filomat ◽  
2013 ◽  
Vol 27 (7) ◽  
pp. 1173-1182 ◽  
Author(s):  
Mujahid Abbas ◽  
Ali Erduran
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Metric Space ◽  
Multivalued Mapping ◽  
Metric Spaces ◽  
Multivalued Mappings ◽  
Ordered Metric Space ◽  
Partially Ordered Metric Space ◽  
Ordered Metric Spaces ◽  
Partially Ordered

In this paper, we introduce g-approximative multivalued mappings. Based on this definition, we gave some new definitions. Further, common fixed point results for g-approximative multivalued mappings satisfying generalized contractive conditions are obtained in the setup of ordered metric spaces. Our results generalize Theorems 2.6-2.9 given in ([1]).

scholarly journals Fixed point index approach for solutions of variational inequalities

2005 ◽  
Vol 2005 (12) ◽  
pp. 1879-1887 ◽  
Author(s):  
Yisheng Lai ◽  
Yuanguo Zhu ◽  
Yinbing Deng
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Variational Inequalities ◽  
Fixed Point Index ◽  
Reflexive Banach Space ◽  
The Other ◽  
Multivalued Mappings ◽  
Point Index ◽  
Upper Semicontinuous ◽  
Index Approach

By using fixed point index approach for multivalued mappings, the existence of nonzero solutions for a class of generalized variational inequalities is studied in reflexive Banach space. One of the mappings concerned here is coercive or monotone and the other is set-contractive or upper semicontinuous.

The set of regularity of a multivalued mapping in a space with a vector-valued metric

2019 ◽  
pp. 39-46
Author(s):  
Tatiana Zhukovskaia ◽  
Elena Pluzhnikova
Keyword(s):  
Normed Space ◽  
Multivalued Mapping ◽  
Linear Normed Space ◽  
Multivalued Mappings ◽  
Vector Valued

We consider multivalued mappings acting in spaces with a vector-valued metric. A vector-valued metric is understood as a mapping satisfying the axioms “of an ordinary metric” with values in the cone of a linear normed space. The concept of the regularity set of a multivalued mapping is defined. A set of regularity is used in the study of inclusions in spaces with a vector-valued metric.

A General Framework for Upper and Lower Possibilities and Necessities

1998 ◽  
Vol 06 (01) ◽  
pp. 1-33 ◽  
Author(s):  
Elena Tsiporkova ◽  
Bernard de Baets
Keyword(s):  
Multivalued Mapping ◽  
General Framework ◽  
Multivalued Mappings ◽  
Possibility Measure ◽  
Confidence Measure ◽  
Triangular Norms ◽  
Necessity Measure ◽  
The Given ◽  
Equivalent Expressions ◽  
Fuzzy Subsets

We show that a (fuzzy) multivalued mapping carries a possibility and necessity measure defined over (fuzzy) subsets of a first universe into a system of upper and lower possibilities and necessities defined over (fuzzy) subsets of a second universe. Upper possibilities (resp. lower necessities) form again a possibility measure (resp. necessity measure), while lower possibilities and upper necessities both form a confidence measure. The approach presented is based on possibilistic conditioning, in particular on the definitions of conditional possibilities and necessities in a general framework based on triangular norms and conorms. In case of a multivalued mapping, upper and lower possibilities and necessities can be expressed equivalently in terms of conditional possibilities and necessities of lower and upper inverse images under the given multivalued mapping, or in terms of a basic possibility assignment and a basic necessity assignment. In case of a fuzzy multivalued mapping, such equivalent expressions cannot be established in general. Upper and lower possibilities and necessities can then be introduced in two alternative ways. For normalized fuzzy multivalued mappings or crisp multivalued mappings, interesting relationships are obtained.

scholarly journals Fixed points of asymptotically regular multivalued mappings

1992 ◽  
Vol 53 (3) ◽  
pp. 313-326 ◽  
Author(s):  
Ismat Beg ◽  
Akbar Azam
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Multivalued Mapping ◽  
Metric Spaces ◽  
Common Fixed Points ◽  
Multivalued Mappings ◽  
Coincidence Points ◽  
Asymptotically Regular

AbstractSome results on fixed point of asymptotically regular multivalued mapping are obtained in metric spaces. The structure of common fixed points and coincidence points of a pair of compatible multivalued mappings is also discussed. Our work generalizes known results of Aubin and Siegel, Dube, Dube and Singh, Hardy and Rogers, Hu, Iseki, Jungck, Kaneko, Nadler, Ray and Shiau, Tan and Wong.

scholarly journals On systems of parabolic variational inequalities with multivalued terms

2020 ◽  
Author(s):  
Siegfried Carl ◽  
Vy. K. Le
Keyword(s):  
Variational Inequalities ◽  
Analytical Framework ◽  
Multivalued Function ◽  
Time Dependent ◽  
Existence Theory ◽  
Hemivariational Inequalities ◽  
Cylindrical Domain ◽  
Upper Semicontinuous ◽  
Quasilinear Elliptic ◽  
Lattice Condition

AbstractIn 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.

scholarly journals About One of a Fixed Point Theorem

2019 ◽  
pp. 539-543
Author(s):  
Husham Rahman Mohammed ◽  
Hala Abbas Mehdi
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fixed Points ◽  
Point Theorem ◽  
Multivalued Mapping ◽  
Multivalued Mappings

In this paper, we give an overview of the main directions in the theory of fixed points of multivalued mappings. We prove a fixed point theorem of multivalued mapping and the following lemma has important role in the proof of main theorem.

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