Point Theorems For Multivalued Mappings Research Articles

Discussion on optimal feedback control for stochastic fractional differential system by hemivariational inequalities

This study focuses on optimal feedback control for stochastic Caputo fractional evolution systems in separable Hilbert spaces. Initially, we deal with the existence of a mild solution, establishing it through the application of the generalised Clarke's subdifferential and a fixed point theorem for multivalued maps. Then, we utilise the Cesari property and the Filippov theorem to demonstrate the existence of feasible pairs. Additionally, we provide pairs of suitable feedback controls under specific conditions. Finally, we present a simple illustration to support our theoretical findings.

Optimal control results for fractional differential hemivariational inequalities of order r (1, 2)

ABSTRACT In this paper, we investigate the optimal control results for fractional differential hemivariational inequalities of order r ∈ ( 1 , 2 ) . Multivalued maps, fractional calculus, and the fixed point theorem are being applied to evaluate the main findings of this study. To begin, a well-known fixed point theorems of multivalued maps and the properties of generalized Clarke subdifferential are used to establish and prove the existence of mild solutions. Furthermore, we obtained the conclusion that the presented control system has optimal control under some mild conditions. Finally, an example is given to illustrate the theory.

Stochastic fractional differential inclusion driven by fractional Brownian motion

Abstract In this paper, we prove the existence result for a mild solution of a fractional stochastic evolution inclusion involving the Caputo derivative in the Hilbert space driven by a fractional Brownian motion with the Hurst parameter H > 1 2 {H>\frac{1}{2}} . The results are obtained by using fractional calculation, operator semigroups and the fixed point theorem for multivalued mappings.

A new conversation on the existence of Hilfer fractional stochastic Volterra–Fredholm integro-differential inclusions via almost sectorial operators

The existence of Hilfer fractional stochastic Volterra–Fredholm integro-differential inclusions via almost sectorial operators is the topic of our paper. The researchers used fractional calculus, stochastic analysis theory, and Bohnenblust–Karlin’s fixed point theorem for multivalued maps to support their findings. To begin with, we must establish the existence of a mild solution. In addition, to show the principle, an application is presented.

Some fixed point theorems for multi-valued mappings via Ishikawa-type iterations in CAT(0) spaces

In this manuscript, we establish strong convergence theorems for the Ishikawa iteration scheme involving quasi-nonexpansive multi-valued maps in the setting of CAT(0) spaces. Our results extend and improve some related results in the literature.

Common fixed point theorems for multi-valued mappings in bicomplex valued metric spaces with application

<abstract><p>The aim of this article is to introduce a generalized Hausdorff distance function in the setting of a bicomplex valued metric space. Using this, we obtain common fixed point theorems for generalized contractions. Our outcomes extend and generalize some conventional fixed point results in the literature. We also furnish a significant example to express the genuineness of the presented results. As an application, we derive some common fixed point results for self mappings, including the leading results of [<italic>Demonstr. Math.</italic>, 54 (2021), 474-487] and [<italic>Int. J. Nonlinear Anal. Appl.</italic>, 12 (2021), 717-727].</p></abstract>

FIXED POINT AND COMMON FIXED POINT THEOREM FOR MULTIVALUED MAPPINGS FOR BANACH SPACE

In this Paper we will investigate some fixed point and common fixed point theorem for multivalued mappings of Banach space. We show that an extended form of many Known results taking multivalued mappings and inequities. The general form of a result proved by Banach Space and Convergence for multivalued mappings in fixed point and common fixed point which contains new generalized form of Theorem.

Optimal Control Problems for Hilfer Fractional Neutral Stochastic Evolution Hemivariational Inequalities

In this paper, we concentrate on a control system with a non-local condition that is governed by a Hilfer fractional neutral stochastic evolution hemivariational inequality (HFNSEHVI). By using concepts of the generalized Clarke sub-differential and a fixed point theorem for multivalued maps, we first demonstrate adequate requirements for the existence of mild solutions to the concerned control system. Then, using limited Lagrange optimal systems, we demonstrate the existence of optimal state-control pairs that are regulated by an HFNSEHVI with a non-local condition. In order to demonstrate the existence of fixed points, the symmetric structure of the spaces and operators that we create is essential. Without considering the uniqueness of the control system’s solutions, the best control results are established. Lastly, an illustration is used to demonstrate the major result.

Some fixed points of multivalued maps in multiplicative metric spaces

In this manuscript, new fixed point theorems for multivalued maps, particularly α−ϕ contractions, in complete multiplicative metric spaces are proved. We further prove a fixed point theorem for multivalued mappings on a graph-equipped multiplicative metric space. Our results extend the result of Dosenovic et al. [4], and generalize the result of Ali et al. [2]. We included an example to validate our result.

A generalized Suzuki–Berinde contraction that characterizes Banach spaces

Abstract We introduce a large class of contractive mappings, called Suzuki–Berinde type contraction. We show that any Suzuki–Berinde type contraction has a fixed point and characterizes the completeness of the underlying normed space. A fixed point theorem for multivalued mappings is also obtained. These results unify, generalize and complement various known comparable results in the literature.

New discussion regarding approximate controllability for Sobolev-type fractional stochastic hemivariational inequalities of order [formula omitted

In this paper, we deal with the approximate controllability of Sobolev-type fractional stochastic evolution hemivariational inequalities of order r∈(1,2). Firstly, by using stochastic analysis, fractional calculus, sine and cosine family operators, and fixed point theorems of multivalued maps, we show the existence of mild solutions for the fractional stochastic evolution systems. Then we provide a sufficient condition to guarantee the approximate controllability of the stochastic evolution systems. Next, our results cover problems involving nonlocal conditions. Finally, we present theoretical and practical applications to support the validity of the study.

Approximate controllability for a new class of stochastic functional differential inclusions with infinite delay

Abstract This manuscript investigates the approximate controllability for a wide range of infinite-delayed semilinear stochastic differential inclusions. First, we construct the expression for a mild solution in terms of the fundamental solution. Then, employing the fixed point theorem for multivalued maps, we formulate a set of sufficient conditions to assure the existence of a solution for the aforementioned system. Further, the approximate controllability for the semilinear stochastic differential inclusion is investigated under the condition that the associated linear deterministic control system is approximately controllable. The discussed results are more general and a continuation of the ongoing research on this issue. Finally, an example is included to highlight the applicability of the considered results.

Analysis of Fractional Differential Inclusion Models for COVID-19 via Fixed Point Results in Metric Space

We examine in this paper some new problems on coincidence point and fixed point theorems for multivalued mappings in metric space. By applying the characterizations of a modified M T ~ -function, under the name D -function, a few novel fixed point results different from the existing fixed point theorems are launched. It is well-known that differential equation of either integer or fractional order is not sufficient to capture ambiguity, since the derivative of a solution to any differential equation inherits all the regularity properties of the mapping involved and of the solution itself. This does not hold in the case of differential inclusions. In particular, fractional-order differential inclusion models are more suitable for describing epidemics. Thus, as a generalization of a newly launched existence result for fractional-order model for COVID-19, using Banach and Shauder fixed point theorems, we investigate solvability criteria of a novel Caputo-type fractional-order differential inclusion model for COVID-19 by applying a standard fixed point theorem of multivalued contraction. Stability analysis of the proposed model in the framework of Ulam-Hyers is also discussed. Nontrivial comparative illustrations are constructed to show that our ideas herein complement, unify and, extend a significant number of existing results in the corresponding literature.

Hilfer Fractional Neutral Stochastic Volterra Integro-Differential Inclusions via Almost Sectorial Operators

In our paper, we mainly concentrate on the existence of Hilfer fractional neutral stochastic Volterra integro-differential inclusions with almost sectorial operators. The facts related to fractional calculus, stochastic analysis theory, and the fixed point theorem for multivalued maps are used to prove the result. In addition, an illustration of the principle is provided.

Common fixed point theorems for multivalued mappings via cone-valued measure of noncompactness

Common fixed point theorems for multivalued mappings via cone-valued measure of noncompactness

Approximate controllability for Hilfer fractional stochastic evolution inclusion with nonlocal conditions

In this paper, the approximate controllability of Hilfer fractional stochastic evolution inclusion with nonlocal conditions is investigated. By using fractional calculus, semigroups theory, stochastic analysis and the fixed point theorem for multi-valued maps, a new sufficient condition for the existence of mild solution of the Hilfer fractional stochastic system in the space of weighted continuous functions is derived. Furthermore, a novel sufficient condition for the approximate controllability of the Hilfer fractional stochastic evolution inclusion with nonlocal conditions is established. Finally, an example is presented to illustrate the main results.

Existence and approximate controllability of non-autonomous functional impulsive evolution inclusions in Banach spaces

In this paper, we are concerned with the approximate controllability results for a class of impulsive functional differential control systems involving time dependent operators in Banach spaces. First, we show the existence of a mild solution for non-autonomous functional impulsive evolution inclusions in separable reflexive Banach spaces with the help of the evolution family and a generalization of the Leray-Schauder fixed point theorem for multi-valued maps. In order to establish sufficient conditions for the approximate controllability of our problem, we first consider a linear-quadratic regulator problem and obtain the optimal control in the feedback form, which contains the resolvent operator consisting of duality mapping. With the help of this optimal control, we prove the approximate controllability of the linear system and hence derive sufficient conditions for the approximate controllability of our problem. Moreover, in this paper, we rectify several shortcomings of the related works available in the literature, namely, proper identification of resolvent operator in Banach spaces, characterization of phase space in the presence of impulsive effects, and lack of compactness of the operator h(⋅)↦∫0⋅U(⋅,s)h(s)ds:L1([0,T];Y)→C([0,T];Y), where Y is a Banach space and U(⋅,⋅) is the evolution family, etc. Finally, we provide a concrete example to illustrate the efficiency of our results.

Multidimensional common fixed point theorems for multivalued mappings in dislocated metric spaces

Multidimensional common fixed point theorems for multivalued mappings in dislocated metric spaces

Results on approximate controllability of Sobolev-type fractional neutral differential inclusions of Clarke subdifferential type

The basic motivation of this research paper is the approximate controllability of Sobolev-type fractional neutral differential inclusions of Clarke subdifferential type which generalized the Riemann-Liouville fractional derivative. Combining with the techniques of fractional calculus, fixed point theorem for multi-value dmaps and Sobolev-type and Clarke subdifferential to the approximate controllability system we obtain new existence result of mild solutions. Next, we demonstrate linear and semilinear systems for approximate controllability. Finally, to verify our theoretical results, an illustration is also included.

Recent advances in the Lefschetz fixed point theory for multivalued mappings

In 1923 S. Lefschetz proved the famous fixed point theorem which is now known as the Lefschetz fixed point theorem (comp. [5], [9], [20], [21]. The multivalued case was considered for the first time in 1946 by S. Eilenberg and D. Montgomery ([10]). They proved the Lefschetz fixed point theorem for acyclic mappings of compact ANR-spaces (absolute neighborhood retracts (see [4] or [13]) using Vietoris mapping theorem (see [4], [13], [16]) as the main tool. In 1970 Eilenberg, Montgomery's result was generalized for acyclic mappings of complete ANR-s (see [17]). Next, a class of admissible multivalued mappings was introduced ([13] or [16]).
 Note that the class of admissible mappings is quite large and contains as a special case not only acyclic mappings but also infinite compositions of acyclic mappings. For this class of multivalued mappings several versions of the Lefschetz fixed point theorem were proved (comp. [11], [13]-15], [18], [19], [27]). In 1982 G. Skordev and W. Siegberg ([26]) introduced the class of multivalued mappings so-called now (1 − n)-acyclic mappings. Note that the class (1−n)-acyclic mappings contain as a special case n-valued mappings considered in [6], [12], [28]. We recommend [8] for the most important results connected with (1 − n)-acyclic mappings. Finally, the Lefschetz fixed point theorem was considered for spheric mappings (comp. [3], [2], [7], [23]) and for random multivalued mappings (comp. [1], [2], [13]). Let us remark that the main classes of spaces for which the Lefschetz fixed point theorem was formulated are the class of ANR-spaces ([4]) and MANR-spaces (multi absolute neighborhood retracts (see [27]). The aim of this paper is to recall the most important results concerning the Lefschetz fixed point theorem for multivalued mappings and to prove new versions of this theorem mainly for AANR-spaces (approximative absolute neighborhood retracts (see [4] or [13]) and for MANR-s. We believe that this article will be useful for analysts applying topological fixed point theory for multivalued mappings in nonlinear analysis, especially in differential inclusions.