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.

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.

This manuscript mainly focused on the nonlocal controllability of Hilfer fractional stochastic differential equations via almost sectorial operators. The key ideas of the study are illustrated by using ideas from fractional calculus, the fixed point technique, and measures of noncompactness. Then, the authors establish new criteria for the mild existence of solutions and derive fundamental characteristics of the nonlocal controllability of a system. In addition, researchers offer theoretical and real-world examples to demonstrate the effectiveness and suitability of our suggested solutions.

In this paper, we focus on the existence of Hilfer fractional stochastic differential systems via almost sectorial operators. The main results are obtained by using the concepts and ideas from fractional calculus, multivalued maps, semigroup theory, sectorial operators, and the fixed-point technique. We start by confirming the existence of the mild solution by using Dhage’s fixed-point theorem. Finally, an example is provided to demonstrate the considered Hilferr fractional stochastic differential systems theory.

The goal of this study is to propose the existence of mild solutions to delay fractional neutral stochastic differential systems with almost sectorial operators involving the Hilfer fractional (HF) derivative in Hilbert space, which generalized the famous Riemann-Liouville fractional derivative. The main techniques rely on the basic principles and concepts from fractional calculus, semigroup theory, almost sectorial operators, stochastic analysis, and the Mönch fixed point theorem via the measure of noncompactness (MNC). Particularly, the existence result of the equation is obtained under some weakly compactness conditions. An example is given at the end of this article to show the applications of the obtained abstract results. Зерттеудiң мақсаты әйгiлi Риман-Лиувилл бөлшек туындысын жалпылайтын Гильберт кеңiстiгiндегi Хильфер бөлшек туындысы қатысатын дерлiк секторлық операторлары бар кешiгетiн бөлшекнейтралды стохастикалық дифференциалдық жүйелер үшiн жұмсақ шешiмдердiң болуын ұсыну. Негiзгi әдiстер бөлшек есептеудiң, жартылай группа теориясының, дерлiк секторлық операторлардың, стохастикалық талдаудың және компактылы емес өлшемi арқылы Мёнхтiң қозғалмайтын нүкте теоремасының негiзгi қағидалары мен тұжырымдамаларына негiзделген. Атап айтқанда, теңдеудiң бар болуының нәтижесi әлсiз компактылықтың белгiлi бiр жағдайында алынды. Мақаланың соңында алынған абстрактылы нәтижелердiң қолдану аясын көрсететiн мысал бар. Цель данного исследования — предложить существование мягких решений для запаздывающих дробнонейтральных стохастических дифференциальных систем с почти секториальными операторами, включающими дробную производную Хильфера в гильбертовом пространстве, которая обобщает знаменитую дробную производную Римана-Лиувилля. Основные методы построены на базовых принципах и концепциях дробного исчисления, теории полугрупп, почти секториальных операторах, стохастическом анализе и теореме Мёнха о неподвижной точке через меру некомпактности. В частности, результат существования уравнения получен при некоторых условиях слабой компактности. В конце статьи приведен пример, демонстрирующий применение полученных абстрактных результатов.