Multivalued Mappings Research Articles

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.

An analysis on the approximate controllability outcomes for fractional stochastic Sobolev‐type hemivariational inequalities of order 1<r<2$$ 1<r<2 $$ using sectorial operators

In this paper, we deal with the approximate controllability of fractional stochastic Sobolev‐type hemivariational differential systems of order with sectorial operators. Firstly, by using stochastic analysis, fractional calculus, sine and cosine family operators, sectorial operators, and the 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.

A study of nonlocal fractional neutral stochastic integrodifferential inclusions of order 1

This paper considers a class of nonlocal fractional neutral stochastic integrodifferential inclusions of order 1 < α < 2 with impulses in a Hilbert space. We study the existence of the mild solution for the cases when the multi-valued map has convex and non-convex values. The results are obtained by combining fixed-point theorems with the fractional order cosine family, semigroup theory, and stochastic techniques. A new set of sufficient conditions is developed to demonstrate the approximate controllability of the system. Finally, an example is given to illustrate the obtained results.

New exploration on approximate controllability of fractional neutral‐type delay stochastic differential inclusions with non‐instantaneous impulse

This paper aims to derive a new set of sufficient conditions for the existence and approximate controllability of neutral‐type fractional stochastic integrodifferential inclusions with infinite delay and non‐instantaneous impulse in a separable Hilbert space using the Atangana–Baleanu Caputo fractional derivative. We investigate the existence of a mild solution for the Atangana–Baleanu Caputo fractional neutral‐type delay integrodifferential stochastic system while taking into account the non‐instantaneous impulses. For this purpose, the Atangana–Baleanu Caputo fractional neutral‐type impulsive delay stochastic system is transferred into an equivalent fixed point problem via an integral operator, and then, the Bohnenblust–Karlin fixed point approach is applied. Further, the approximate controllability results of the proposed nonlinear stochastic impulsive control system are established under the consideration that the corresponding linear system is approximately controllable. The set of sufficient conditions is established by using the concepts of stochastic analysis, fractional calculus, fixed point technique, semigroup theory of bounded linear operators, and the theory of multivalued maps. To illustrate the abstract results, we provide an example at the end of the paper.

A class of time-optimal feedback control for fractional neutral evolution hemivariational inequalities with fixed delay

ABSTRACT In this paper, we study a class of optimal feedback control for a neutral fractional system with hemivariational inequalities. Initially, we derive some existence results of mild solutions for the system by applying the Marteli's fixed point theorem of multivalued maps, properties of generalized Clarke's subdifferential and semigroup operators. Then, by using the Filippov theorem and the Cesari property, a new set of sufficient conditions are formulated to ensure the existence results of a feasible pair for the feedback control systems. Also, we apply our main results to obtain the optimal feedback control pair. Further, we investigate the time-optimal feedback control for our control system. In the end, an example is given to illustrate our theory.

On second order differential inclusion driven by quasi-variational–hemivariational inequalities

In this paper we investigate an evolution problem (SDQHVI) which constitutes of the second order differential inclusion driven by a quasi-variational–hemivariational inequality (QHVI) with perturbation operator in Banach spaces. Unlike the existing literature on differential inclusions, on the one hand, the second order differential operator is not assumed to be compact; on the other hand, the solvability of the inequality is proved under non-coercive condition. More precisely, first, it is shown that the solution set of the QHVI with perturbation operator is nonempty, bounded, closed and convex under the KKM theorem and the Minty formulation. Then, an associated multivalued map with the solution set of the QHVI with perturbation operator is introduced, and we prove that it is upper semicontinuous and measurable. Finally, by utilizing Kakutani–Fan–Glicksberg fixed point theorem, a weak topology technique and properties of strong continuous cosine operators, we prove the existence of mild solutions for SDQHVI.

Fixed point and endpoint theorems of multivalued mappings in convex $ b $-metric spaces with an application

&lt;abstract&gt;&lt;p&gt;In this paper, we investigated several new fixed points theorems for multivalued mappings in the framework $ b $-metric spaces. We first generalized $ S $-iterative schemes for multivalued mappings to above spaces by means of a convex structure and then we developed the existence of fixed points and approximate endpoints of the multivalued contraction mappings using iteration techniques. Furthermore, we introduced the modified $ S $-iteration process for approximating a common endpoint of a multivalued $ \alpha_{s} $-nonexpansive mapping and a multivalued mapping satisfying conditon $ (E^{'}) $. We also showed that this new iteration process converges faster than the $ S $-iteration process in the sense of Berinde. Some convergence results for this iterative procedure to a common endpoint under some certain additional hypotheses were proved. As an application, we applied the $ S $-iteration process in finding the solution to a class of nonlinear quadratic integral equations.&lt;/p&gt;&lt;/abstract&gt;

Common Fixed Points of Generalized α-Nonexpansive Multivalued Mappings via Modified S-Type Iteration

In this paper, we introduce a class of generalized α-nonexpansive multivalued mapping and study some of its important properties. In particular, this class is a multivalued version of the single-valued nonexpansive mapping, called generalized α-nonexpansive mapping proposed by Pant and Shukla [11]. A modified S-type iteration scheme is proposed to approximate the common fixed point of two multivalued mappings. Our algorithm provides a multivalued extension of the method given by Khan et al. [6]. Strong and weak convergence of the iterative process are also proved under suitable assumptions.

Applying fixed point techniques to solve fractional differential inclusions under new boundary conditions

&lt;abstract&gt;&lt;p&gt;Many scholars have lately explored fractional-order boundary value issues with a variety of conditions, including classical, nonlocal, multipoint, periodic/anti-periodic, fractional-order, and integral boundary conditions. In this manuscript, the existence and uniqueness of solutions to sequential fractional differential inclusions via a novel set of nonlocal boundary conditions were investigated. The existence results were presented under a new class of nonlocal boundary conditions, Carathéodory functions, and Lipschitz mappings. Further, fixed-point techniques have been applied to study the existence of results under convex and non-convex multi-valued mappings. Ultimately, to support our findings, we analyzed an illustrative example.&lt;/p&gt;&lt;/abstract&gt;

Application of fixed point theory to synaptic delay differential equations in neural networks

&lt;p&gt;The objective of this research is to propose a new concept known as rational ($ \alpha \eta $-$ \psi) $-contractions in the framework of $ \mathcal{F} $-metric spaces and to establish several fixed point theorems. These theorems help to generalize and unify various established fixed point results from the existing literature. To demonstrate the practical effectiveness of our approach, we provide a significant example that confirms our findings. In addition, we introduce a generalized multivalued ($ \alpha $-$ \psi) $-contraction concept in $ \mathcal{F} $-metric spaces and use it to prove fixed point theorems specifically designed for multivalued mappings. To demonstrate the practical utility of our findings, we apply our main results to the solution of synaptic delay differential equations in neural networks.&lt;/p&gt;

Approximate controllability for some retarded integrodifferential inclusions using a version of the Leray-Schauder fixed point theorem for the multivalued maps

Approximate controllability for some retarded integrodifferential inclusions using a version of the Leray-Schauder fixed point theorem for the multivalued maps

On a Class of Condensing Multivalued Maps

On a Class of Condensing Multivalued Maps

Strict fixed point and Ulam-Hyers stability of multivalued asymptotically regular mappings

In this paper, we establish the existence and uniqueness of strict fixed point for an asymptotically regular multivalued mapping in a metric space. We also study the Ulam-Hyers stability, well-posedness and data dependence of the associated strict fixed point problem. We give an example to illustrate our results. Our work extends and complements important results existing in the literature.

Ishikawa iteration convergence to fixed points of a multi-valued mapping in modular function spaces

Ishikawa iteration convergence to fixed points of a multi-valued mapping in modular function spaces

СТРАТЕГІЇ КЕРУВАННЯ У ПРОБЛЕМІ ЗБЛИЖЕННЯ КОНФЛІКТНО-КЕРОВАНИХ ОБ’ЄКТІВ

The authors propose a new approach to forming control strategies in the problem of approach of conflict-controlled objects. Modifications of the first direct method is developed for the stroboscopic strategy in the class of counter-controls where the classical Pontryagin’s condition is not satisfied. The lower resolving function is considered, which plays a key role in the formulation of the results and in the general case can be determined using the Minkowski functional of a multivalued mapping. An upper resolving function is introduced and a modified scheme of the method of resolving functions is proposed, which guarantees the termination of the conflict-controlled process in the class of quasi-strategies and counter-controls where the classical Pontryagin’s condition is not satisfied. The guaranteed times for different schemes of the considered methods are compared. The theoretical results are illustrated on a second-order model example with a special non-convex control region of the pursuer. Keywords: control strategy, resolving function, dynamic game problem, problem of approach of controlled objects.

СТРАТЕГІЇ КЕРУВАННЯ У ПРОБЛЕМІ ЗБЛИЖЕННЯ КОНФЛІКТНО-КЕРОВАНИХ ОБ’ЄКТІВ

The authors propose a new approach to forming control strategies in the problem of approach of conflict-controlled objects. Modifications of the first direct method is developed for the stroboscopic strategy in the class of counter-controls where the classical Pontryagin’s condition is not satisfied. The lower resolving function is considered, which plays a key role in the formulation of the results and in the general case can be determined using the Minkowski functional of a multivalued mapping. An upper resolving function is introduced and a modified scheme of the method of resolving functions is proposed, which guarantees the termination of the conflict-controlled process in the class of quasi-strategies and counter-controls where the classical Pontryagin’s condition is not satisfied. The guaranteed times for different schemes of the considered methods are compared. The theoretical results are illustrated on a second-order model example with a special non-convex control region of the pursuer. Keywords: control strategy, resolving function, dynamic game problem, problem of approach of controlled objects.

APPROXIMATION METHOD FOR COMMON FIXED POINT OF A COUNTABLE FAMILY OF MULTI-VALUED QUASI-ϕ-NONEXPANSIVE MAPPINGS IN BANACH SPACES

In this paper, we study a new iterative algorithm to approximate a common fixed point of acountable family of multi-valued quasi-φ nonexpansive mappings in a uniformly smooth and uniformlyconvex real Banach space. We prove strong convergence theorem which improve and generalize recentlyannounced results in the literature. We also give some example to illustrate our theorem and we alsogive application to zero point problem of maximal monotone mappings

Multiple solutions of nonlinear Neumann inclusions

We prove new results on the existence of multiple solutions for elliptic inclusions with nonlinear boundary conditions of Neumann type. Our approach is topological and relies on the fixed point index for multivalued map.

Fixed Points Theorems of Multi-Valued Mappings in F-Metric spaces

Fixed Points Theorems of Multi-Valued Mappings in F-Metric spaces

Coincidence point theorems in some generalized metric spaces

"Let (X, d) be a complete dislocated metric space, (Y, ρ) be a semimetric space and f, g : X → Y be two mappings. Several coincidence point results are obtained for singlevalued and multivalued mappings. Keywords: Dislocated metric space, semimetric space, singlevalued and multivalued mapping, comparison function, comparison pair, lower semi-continuity, coincidence point displacement functional, iterative approximation of coincidence point, weakly Picard mapping, pre-weakly Picard mapping."