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In this study, we investigate the effect of Clarke's subdifferential type on the optimal control results for fractional differential systems of order 1

We present straightforward conditions which ensure that a strongly elliptic linear operator L generates an analytic semigroup on Hölder spaces on an arbitrary complete manifold of bounded geometry. This is done by establishing the equivalent property that L is ‘sectorial,’ a condition that specifies the decay of the resolvent (λ I - L) - 1 as λ diverges from the Hölder spectrum of L. A key step is that we prove existence of this resolvent if λ is sufficiently large using a geometric microlocal version of the semiclassical pseudodifferential calculus. The properties of L and e - t L we obtain can then be used to prove well-posedness of a wide class of nonlinear flows. We illustrate this by proving well-posedness on Hölder spaces of the flow associated with the ambient obstruction tensor on complete manifolds of bounded geometry. This new result for a higher-order flow on a noncompact manifold exhibits the broader applicability of our technique. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the approximate controllability of fractional stochastic Sobolev-type hemivariational differential systems of order r ∈ (1, 2) 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. [ABSTRACT FROM AUTHOR]

The impulsive effect on the fractional stochastic integrodifferential system of order ϱ∈(1,2)$$ \varrho \in \left(1,2\right) $$ with infinite delay via sectorial operator is investigated in this work. The primary findings of this study are tested by referring to multivalued functions, sectorial operators, fractional derivatives, and the fixed point approach. Initially, the presence of a mild solution for the systems class was introduced. The existence outcomes of semilinear control problems were then established. Eventually, an application is given for deriving the theory of the key findings. [ABSTRACT FROM AUTHOR]

The purpose of this work is to develop a new model of fractional operators called Hilfer-fractional random nonlinear integro-differential equations. In this paradigm, a further discussion is encouraged under almost sectorial operators. The results are supported by fractional calculus, stochastic analysis theory, and Bohnenblust–Karlin’s fixed point theorem for multi-valued mappings. In addition, a mild solution to the model under consideration is presented. Ultimately, an example is provided to support our results. [ABSTRACT FROM AUTHOR]

In this work, we discuss the existence and approximate controllability results for impulsive fractional differential systems of order 1

We obtain necessary and sufficient conditions for the strongly L p well-posedness of three abstract evolution equations, arising from fractional Moore-Gibson-Thompson type equations which have recently appeared in the literature. We use Fourier multiplier techniques to derive new characterizations in terms of the R -boundedness of the operator-valued symbol associated to each abstract model, when endowed with the time-fractional Liouville-Grünwald derivative. As a consequence of our characterization, we give new insights into the differences between the models based on the structure of the respective operator-valued symbols and show novel applications by including several classes of operators other than the Laplacian. [ABSTRACT FROM AUTHOR]

In this paper, we first define the phases of a sectorial operator based on the numerical range. We are interested in the estimation of the spectrum phases of AB based on the phases of two sectorial operators A and B. Motivated by the classical small gain theorem, we formulate an operator small phase theorem with necessity for the invertibility of I + A B , which plays a crucial role in feedback stability analysis. Afterwards, we consider the special class of sectorial operators of the form P + K , where P is strictly positive and K is compact. More properties of the phases for those operators are studied, including those of compressions, Schur complements, operator means and products. Finally, for the special class of sectorial operators, we further establish a majorization relation between the phases of the spectrum of AB and the phases of two operators A and B. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the effect of hemivariational inequalities on the approximate controllability of Caputo fractional differential systems. The main results of this study are tested by using multivalued maps, sectorial operators of type (P; η r; β), fractional calculus, and the fixed point theorem. Initially, we introduce the idea of mild solution for fractional hemivariational inequalities. Next, the approximate controllability results of semilinear control problems were then established. Moreover, we will move on to the system involving nonlocal conditions. Finally, an example is provided in support of the main results we acquired. [ABSTRACT FROM AUTHOR]

Originating with the famous monograph by Dan Henry, the semigroup approach to evolution problems having a positive sectorial operator in the main part allows us to settle them at various levels of the fractional power scale associated with the main linear operator. This translates into different regularity properties of local solutions to such equations. Specific applications of the abstract results to the 2D surface quasi-geostrophic equation or the fractional chemotaxis system are presented. [ABSTRACT FROM AUTHOR]

In this article, we look into the important requirements for exact controllability of fractional impulsive differential systems of order 1

In this study, we investigate nonlocal problems for a class of the Sobolev type fractional delay impulsive integrodifferential inclusions of order r ∈ (1 , 2) via sectorial operator in Hilbert space. The results are obtained by engaging of mixed Volterra-Fredholm integrodifferential systems, sectorial operator of type (P , κ , r , γ) and Martelli's fixed point theorem under the assumption that the corresponding linear system is approximately controllable. Finally, an example is given to illustrate the effectiveness of the results obtained. [ABSTRACT FROM AUTHOR]

The mainspring of the study is to investigate the out‐turn of stochastic Volterra–Fredholm integro‐differential inclusion of order μ∈(1,2)$$ \mu \in \left(1,2\right) $$ with sectorial operator of the type (P,η,ϱ,γ)$$ \left(P,\eta, \varrho, \gamma \right) $$. The existence results of our proposed problem is derived by employing Martelli's fixed point approach. We do not limit the theoretical results of fractional stochastic equation to local condition but extend to nonlocal condition, and physical interpretation of our obtained results is proved with an appropriate illustration. [ABSTRACT FROM AUTHOR]

The study deals with the findings of the outcome of the approximate controllability results of inclusion type fractional stochastic system in Banach space with the order of the fractional system varrho in (1,2). At first, we implement Bohnenblust-Karlin's fixed point technique to deduce the required conditions on which the fractional system with inital conditions is approximately controllable, and there by, we postulate the sufficient conditions for extending the obtained results to the system with nonlocal conditions.

In this article, we look into the important requirements for exact controllability of fractional impulsive differential systems of order 1 < r < 2 . Definitions of mild solutions are given for fractional integrodifferential equations with impulses. In addition, applying fixed point methods, fractional derivatives, essential conditions, mixed Volterra-Fredholm integrodifferential type, for exact controllability of the solutions are produced. Lastly, a case study is supplied to show the illustration of the primary theorems. In this article, we look into the important requirements for exact controllability of fractional impulsive differential systems of order . Definitions of mild solutions are given for fractional integrodifferential equations with impulses. In addition, applying fixed point methods, fractional derivatives, essential conditions, mixed Volterra-Fredholm integrodifferential type, for exact controllability of the solutions are produced. Lastly, a case study is supplied to show the illustration of the primary theorems. [ABSTRACT FROM AUTHOR]

In this article, we look into the important requirements for exact controllability of fractional impulsive differential systems of order 1 < r < 2. Definitions of mild solutions are given for fractional integrodifferential equations with impulses. In addition, applying fixed point methods, fractional derivatives, essential conditions, mixed Volterra-Fredholm integrodifferential type, for exact controllability of the solutions are produced. Lastly, a case study is supplied to show the illustration of the primary theorems. [ABSTRACT FROM AUTHOR]

This research investigates the existence of nonlocal impulsive fractional integrodifferential equations of order 1 < r < 2 with infinite delay. To begin with, we discuss the existence of a mild solution for the fractional derivatives by using the sectorial operators, the nonlinear alternative of the Leray-Schauder fixed point theorem, mixed Volterra-Fredholm integrodifferential types, and impulsive systems. Furthermore, we develop the optimal control results for the given system. The application of our findings is demonstrated with the help of an example. [ABSTRACT FROM AUTHOR]

We introduce some families of generalized Black–Scholes equations which involve the Riemann–Liouville and Weyl space-fractional derivatives. We prove that these generalized Black–Scholes equations are well-posed in (L 1 - L ∞) -interpolation spaces. More precisely, we show that the elliptic-type operators involved in these equations generate holomorphic semigroups. Then, we give explicit integral expressions for the associated solutions. In the way to obtain well-posedness, we prove a new connection between bisectorial-like operators and sectorial operators in an abstract setting. Such a connection extends the scaling property of sectorial operators to a wider family of both operators and the functions involved. [ABSTRACT FROM AUTHOR]

This research investigates the existence of nonlocal impulsive fractional integrodifferential equations of order 1 < r < 2 with infinite delay. To begin with, we discuss the existence of a mild solution for the fractional derivatives by using the sectorial operators, the nonlinear alternative of the Leray–Schauder fixed point theorem, mixed Volterra–Fredholm integrodifferential types, and impulsive systems. Furthermore, we develop the optimal control results for the given system. The application of our findings is demonstrated with the help of an example.

This paper focuses on the approximate controllability of Hilfer fractional neutral Volterra integro-differential inclusions via almost sectorial operators. Almost sectorial operators, fractional differential, Leray-Schauder fixed point theorem and multivalued maps are used to prove the result. We start by emphasizing the existence of a mild solution and demonstrate the approximate controllability of the fractional system. In addition, an example is presented to demonstrate the principle. [ABSTRACT FROM AUTHOR]

This manuscript focuses on the existence of a mild solution Hilfer fractional neutral integro-differential inclusion with almost sectorial operator. By applying the facts related to fractional calculus, semigroup, and Martelli's fixed point theorem, we prove the primary results. In addition, the application is provided to demonstrate how the major results might be applied. [ABSTRACT FROM AUTHOR]

The goal of this research is to investigate the issue of existence and optimal control results for fractional mixed Volterra–Fredholm type integrodifferential systems of order 1

Abstract In this paper, we prove two results concerning the existence of S-asymptotically ω-periodic solutions for non-instantaneous impulsive semilinear differential inclusions of order 1 < α < 2 $1

In this work we consider the abstract Cauchy problem with Caputo fractional time derivative of order α ∈ (0 , 1 ] , and discuss the continuity of the respective solutions regarding the parameter α . We also present a study about the continuity of the Mittag-Leffler families of operators (for α ∈ (0 , 1 ] ), when they are induced by sectorial operators. [ABSTRACT FROM AUTHOR]

Given a complex, elliptic coefficient function we investigate for which values of p the corresponding second-order divergence form operator, complemented with Dirichlet, Neumann or mixed boundary conditions, generates a strongly continuous semigroup on L p (Ω) . Additional properties like analyticity of the semigroup, H ∞ -calculus and maximal regularity are also discussed. Finally, we prove a perturbation result for real coefficients that gives the whole range of p's for small imaginary parts of the coefficients. Our results are based on the recent notion of p-ellipticity, reverse Hölder inequalities and Gaussian estimates for the real coefficients. [ABSTRACT FROM AUTHOR]

Starting from a bi-continuous semigroup in a Banach space X (which might actually be strongly continuous), we investigate continuity properties of the semigroup that is induced in real interpolation spaces between X and the domain D(A) of the generator. Of particular interest is the case (X , D (A)) θ , ∞ . We obtain topologies with respect to which the induced semigroup is bi-continuous, among them topologies induced by a variety of norms. We illustrate our results with applications to a nonlinear Schrödinger equation and to the Navier–Stokes equations on R d . [ABSTRACT FROM AUTHOR]

This paper focuses on the existence results for nonlocal fractional integrodifferential equations with interval impulses and measure of noncompactness using Mönch fixed point theorem and sectorial operator. At the end of this paper, an example is given to illustrate the applications of the abstract results.

In this paper, we prove two results concerning the existence of S-asymptotically ω-periodic solutions for non-instantaneous impulsive semilinear differential inclusions of order 1 < α < 2 and generated by sectorial operators. In the first result, we apply a fixed point theorem for contraction multivalued functions. In the second result, we use a compactness criterion in the space of bounded piecewise continuous functions defined on the unbounded interval J = [ 0 , ∞) . We adopt the fractional derivative in the sense of the Caputo derivative. We provide three examples illustrating how the results can be applied. [ABSTRACT FROM AUTHOR]

In the present work the optimal regularity, in the sense of Hölder continuity, of linear and semi-linear abstract fractional differential equations is investigated in the framework of complex Banach spaces. This framework has been considered by the authors as the most convenient to provide a posteriori error estimates for the time discretizations of such a kind of abstract differential equations. In the spirit of the classical a posteriori error estimates, under certain assumptions, the error is bounded in terms of computable quantities, in our case measured in the norm of Hölder continuous and weighted Hölder continuous functions. [ABSTRACT FROM AUTHOR]

Abstract This paper deals with the existence and uniqueness of PC-mild solutions for fractional impulsive evolution equation involving nonlocal conditions and sectorial operators. We also study the nonlocal controllability of the control system governed by fractional impulsive evolution equation.

In this paper, regularity properties, Strichartz type estimates for solution of integral problem for linear and nonlinear abstract Schrödinger equations in vector-valued function spaces are obtained. The equation includes a linear operator A defined in a Banach space E , in which by choosing E and A we can obtain numerous classis of initial value problems for Schrödinger equations which occur in a wide variety of physical systems. [ABSTRACT FROM AUTHOR]

In this paper we study the abstract semilinear parabolic problem of the form du/dt + Au = ƒ(u), as the limit of the corresponding fractional approximations du/dt + Aαu = ƒ(u), in a Banach space X, where the operator A : D(A) ⊂ X → X is a sectorial operator in the sense of Henry [22]. Under suitable assumptions on nonlinearities ƒ : Xα → X (Xα : = D(Aα)), we prove the continuity with rate (with respect to the parameter α) for the global attractors (as seen in Babin and Vishik [4] Chapter 8, Theorem 2.1). As an application of our analysis we consider a fractional approximation of the strongly damped wave equations and we study the convergence with rate of solutions of such approximations. [ABSTRACT FROM AUTHOR]

Let ${\mathcal{D}}$ be a Schauder decomposition on some Banach space $X$. We prove that if ${\mathcal{D}}$ is not $R$ -Schauder, then there exists a Ritt operator $T\in B(X)$ which is a multiplier with respect to ${\mathcal{D}}$ such that the set $\{T^{n}:n\geq 0\}$ is not $R$ -bounded. Likewise, we prove that there exists a bounded sectorial operator $A$ of type $0$ on $X$ which is a multiplier with respect to ${\mathcal{D}}$ such that the set $\{e^{-tA}:t\geq 0\}$ is not $R$ -bounded. [ABSTRACT FROM AUTHOR]

We consider densely defined sectorial operators A ± that can be written in the form A ± = ± i S + V with D (A ±) = D (S) = D (V) , where both S and V ≥ ε > 0 are assumed to be symmetric. We develop an analog to the Birmin–Kreĭn–Vishik–Grubb (BKVG) theory of selfadjoint extensions of a given strictly positive symmetric operator, where we will construct all maximally accretive extensions A D of A + with the property that A + ¯ ⊂ A D ⊂ A - ∗ . Here, D is an auxiliary operator from ker (A - ∗) to ker (A + ∗) that parametrizes the different extensions A D . After this, we will give a criterion for when the quadratic form ψ ↦ Re ⟨ ψ , A D ψ ⟩ is closable and show that the selfadjoint operator V ^ that corresponds to the closure is an extension of V. We will show how V ^ depends on D, which—using the classical BKVG-theory of selfadjoint extensions—will allow us to define a partial order on the real parts of A D depending on D. All of our results will be presented in a way that emphasizes their connection to the classical BKVG-theory and that shows how the BKVG formulas generalize when considering sectorial operators A ± = ± i S + V instead of just a strictly positive symmetric operator V. Applications to second order ordinary differential operators are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence of mild solutions and approximate controllability for Riemann-Liouville fractional stochastic evolution equations with nonlocal conditions of order 1 < α < 2. As far as we know, there are few articles investigating on this issue. Firstly, the mild solutions to the equations are proved using Laplace transform of the Riemann-Liouville derivative. Moreover, the estimations of resolve operators involving the Riemann-Liouville fractional derivative of order 1 < α < 2 are given. Then, the existence results are obtained via the noncompact measurement strategy and the Mönch fixed point theorem. The approximate controllability of this nonlinear Riemann-Liouville fractional nonlocal stochastic systems of order 1 < α < 2 is concerned under the assumption that the associated linear system is approximately controllable. Finally, the approximate controllability results are obtained by using Lebesgue dominated convergence theorem. [ABSTRACT FROM AUTHOR]

In this paper we investigate the asymptotically periodic behavior of solutions of fractional evolution equations of order 1 < α < 2 and in particular existence and uniqueness results are established. Two examples are given to illustrate our results. [ABSTRACT FROM AUTHOR]

In this note we comment on an error present in the recent and extensive literature on exact controllability of abstract control differential problems. Our observations are valid for first order, second order, integro-differential and fractional differential control problems.

Motivated by a two‐specie competition model with cross‐diffusion in population ecology with time‐dependent environmental capacity, we study the semilinear parabolic evolution equation of the form x˙+Ax=f(t,x) where the linear operator −A generates an analytic semigroup. Our main task is to prove the existence of inertial manifolds for mild solutions to such an evolution equation under the conditions that the linear partial differential operator −A has spectral gap being sufficiently large, and the nonlinear term f satisfies ϕ‐Lipschitz condition, ie, ‖f(t,x) − f(t,y)‖ ≤ ϕ(t)‖Aθ(x − y)‖ for ϕ belonging to some admissible space. We then apply the obtained result to study asymptotic behavior of the above‐mentioned competition model. [ABSTRACT FROM AUTHOR]

This work is concerned with a mathematical study of ill-posed backward evolution equations associated with densely defined linear differential operators in Banach spaces. A general approach is presented to investigate the convergence and stability of a class of regularized solutions for ill-posed backward evolution equations associated with sectorial or half-strip operators. Generalized concepts of qualification pairs and index functions are introduced to characterize the explicit convergence rates of the concerned regularized solutions. Applications of our results to general backward evolution equations are also investigated. [ABSTRACT FROM AUTHOR]

We show that the Stokes operator A on the Helmholtz space $${L^p_\sigma(\Omega)}$$ for a bounded Lipschitz domain $${\Omega\subset\mathbb{R}^d}$$ , $${d \ge 3}$$ , has a bounded $${H^\infty}$$ -calculus if $${\left|\frac{1}{p}-\frac{1}{2} \right|\le\frac{1}{2d}}$$ . Our proof uses a new comparison theorem for A and the Dirichlet Laplace $${-\Delta}$$ on $${L^p(\Omega)^d}$$ , which is based on 'off-diagonal' estimates of the Littlewood-Paley decompositions of A and $${-\Delta}$$ . This comparison theorem can be formulated for rather general sectorial operators and is well suited to extrapolate the $${H^\infty}$$ -calculus from L ( U) to the L ( U)-scale or part of it. It also gives some information on coincidence of domains of fractional powers. [ABSTRACT FROM AUTHOR]

We study the existence of mild solutions for the two-term fractional order abstract differential equation $${D}^{\alpha+1}_t u(t) + \mu {D}_t^{\beta} u(t) - Au(t) = D^{\alpha-1}_t f(t,u(t)), \quad t\in [0,1], \quad 0 < \alpha \leq \beta \leq 1, \mu \geq 0,$$ with nonlocal initial conditions and where $A$ is a linear operator of sectorial type. To achieve our goal, we use a new mixed method, which combines a generalization of the theory of $C_0$-semigroups, Hausdorff measure of noncompactness and a fixed point argument.

In this paper, we consider a class of fractional integro-differential inclusions in Hilbert spaces. This paper deals with the approximate controllability for a class of fractional integro-differential control systems. First, we establishes a set of sufficient conditions for the approximate controllability for a class of fractional semilinear integro-differential inclusions in Hilbert spaces. We use Bohnenblust-Karlin's fixed point theorem to prove our main result. Further, we extend the result to study the approximate controllability concept with nonlocal conditions. An example is also given to illustrate our main result. [ABSTRACT FROM AUTHOR]

In this note we comment on an error present in the recent and extensive literature on exact controllability of abstract control difierential problems. Our observations are valid for first order, second order, integro-difierential and fractional difierential control problems. [ABSTRACT FROM AUTHOR]

The abstract results and applications presented in 'Controllability of fractional neutral stochastic functional differential systems, Z. Angew. Math. Phys. 65 (2014), no. 5, 941-959, are not correct. Moreover, the class of differential control problems studied in [1] is not H-controllable. [ABSTRACT FROM AUTHOR]