Derivatives In Banach Spaces Research Articles

On the controllability results of semilinear delayed evolution systems involving fractional derivatives in Banach spaces

<abstract><p>The paper investigated the exact controllability of delayed fractional evolution systems of order $ \alpha\in (1, 2) $ in abstract spaces. At first, the exact controllability result is obtained when the nonlinear term $ f $ is locally Lipschitz continuous. Then, the certain compactness conditions and the measure of noncompactness conditions were applied to demonstrate the exact controllability of the concerned problem. The discussion was based on the fixed point theorems and the cosine family theory.</p></abstract>

Existence of mild solutions for perturbed fractional neutral differential equations through deformable derivatives in Banach spaces

The deformable derivative [[Formula: see text]] is used in this work to give the necessary restrictions for the existence of mild solutions for perturbed fractional neutral differential equations [PFNDE] in Banach spaces. Several novel existence results are made using fixed point and semigroup techniques. In the end, two numerical examples are given to illustrate the application of the theoretical concepts discussed.

Approximate controllability of fractional order non-instantaneous impulsive functional evolution equations with state-dependent delay in Banach spaces

Abstract This paper deals with the control problems governed by fractional impulsive functional evolution equations with state-dependent delay involving Caputo fractional derivatives in Banach spaces. The main objective of this work is to formulate sufficient conditions for the approximate controllability of the considered system in separable reflexive Banach spaces. We have exploited the resolvent operator technique and Schauder’s fixed point theorem in the proofs to achieve this goal. The approximate controllability of linear system is discussed in detail, which lacks in the existing literature. Moreover, we point out some shortcomings of the existing works in the context of characterization of mild solution, phase space, and approximate controllability of fractional order impulsive systems in Banach spaces. Finally, we investigate the approximate controllability of the fractional order heat equation with non-instantaneous impulses and delay by using the developed results.

Existence and controllability of higher‐order nonlinear fractional integrodifferential systems via fractional resolvent

This work analyzes the existence of solution and approximate controllability for higher‐order nonlinear fractional integrodifferential systems with Riemann–Liouville derivatives in Banach spaces. First, the definition of mild solution for the system is derived. Then a set of sufficient conditions for the existence of mild solution and approximate controllability of the system is obtained. The discussions are based on fixed point approach, and the theory of convolution and fractional resolvent. To illustrate the feasibility of developed theory, an example is given.

Higher-order efficiency conditions for constrained vector equilibrium problems

ABSTRACT In this paper, we investigate some optimality conditions of higher-order for nonsmooth nonconvex vector equilibrium problems with constraints (or without constraints) in terms of the higher-order upper and lower Studniarski derivatives in Banach spaces. The calculus rule of all the data involved in the problem is taken into account. Using the notion of higher-order upper Studniarski derivative with the class of m-stable/m-steady functions, we first provide the higher-order necessary optimality conditions for the local weak efficient solution of vector equilibrium problem without constraints and then we present the higher-order necessary and sufficient optimality conditions for the local strict minimum of order m to such problem. We second obtain the necessary and sufficient optimality conditions for those efficient solutions of vector equilibrium problem with cone and equality constraints through Lagrange multiplier rules in finite-dimensional spaces. Final, the higher-order necessary and sufficient optimality conditions in terms of the higher-order upper and lower Studniarski derivatives for the local weak efficient solution of vector equilibrium problem with set, cone and equality constraints in Banach spaces are also established. Some examples are proposed to demonstrate our findings.

Existence of Weak Solutions for Caputo’s Fractional Derivatives in Banach Spaces

The objective of this project is to represent the existence of solutions for Caputo’s fractional derivatives in Banach spaces. The result is based on some well-known fixed point theorems. To show the efficiency of the stated result some examples will be demonstrated
 GANIT J. Bangladesh Math. Soc.Vol. 39 (2019) 111-118

Optimality Conditions for the Efficient Solutions of Vector Equilibrium Problems with Constraints in Terms of Directional Derivatives and Applications

The aim of this paper is to establish the Kuhn–Tucker optimality conditions for the efficient solution types of constrained vector equilibrium problems in terms of directional derivatives in Banach spaces. Under the suitable assumptions on generalized convexity of objective and constraint functions, the Kuhn–Tucker necessary and sufficient optimality conditions for efficient solution, weakly efficient solution, Henig efficient solution, globally efficient solution and superefficient solution of vector equilibrium problem with set and cone constraints are established. Some applications to the constrained vector variational inequality problem and the constrained vector optimization problem are also given. Besides, the Karush–Kuhn–Tucker necessary and sufficient optimality conditions for weakly efficient solutions to the model of transportation–production and Nash–Cournot equilibria problems are obtained. We also provide several examples to illustrate our results.

Monotone iterative technique for periodic problem involving Riemann\u2013Liouville fractional derivatives in Banach spaces

In this paper, we use a monotone iterative technique in the presence of lower and upper solutions to discuss the existence and uniqueness of periodic solutions for a class of fractional differential equations in an ordered Banach space E. Under some monotonicity conditions and noncompactness measure conditions of nonlinearity, we obtain the existence of extremal solutions and a unique solution between lower and upper solutions.

Boundary value problems for fractional differential inclusions with Hadamard type derivatives in Banach spaces

The authors establish sufficient conditions for the existence of solutions to boundary value problems for fractional differential inclusions involving the Hadamard type fractional derivative of order $\alpha \in (1,2]$ in Banach spaces. Their approach uses M\onch's fixed point theorem and the Kuratowski measure of noncompacteness.

New optimality conditions for unconstrained vector equilibrium problem in terms of contingent derivatives in Banach spaces

This article presents necessary and sufficient optimality conditions for weakly efficient solution, Henig efficient solution, globally efficient solution and superefficient solution of vector equilibrium problem without constraints in terms of contingent derivatives in Banach spaces with stable functions. Using the steadiness and stability on a neighborhood of optimal point, necessary optimality conditions for efficient solutions are derived. Under suitable assumptions on generalized convexity, sufficient optimality conditions are established. Without assumptions on generalized convexity, a necessary and sufficient optimality condition for efficient solutions of unconstrained vector equilibrium problem is also given. Many examples to illustrate for the obtained results in the paper are derived as well.

Existence of Mild Solutions to Nonlocal Fractional Cauchy Problems via Compactness

We obtain characterizations of compactness for resolvent families of operators and as applications we study the existence of mild solutions to nonlocal Cauchy problems for fractional derivatives in Banach spaces. We discuss here simultaneously the Caputo and Riemann-Liouville fractional derivatives in the cases0<α<1and1<α<2.

Approximate Controllability of Fractional Evolution Systems with Riemann--Liouville Fractional Derivatives

In this paper, we deal with the control systems governed by fractional evolution differential equations involving Riemann--Liouville fractional derivatives in Banach spaces. Our main purpose in this article is to establish suitable assumptions to guarantee the existence and uniqueness results of mild solutions. Under these conditions, the approximate controllability of the associated fractional evolution systems involving Riemann--Liouville fractional derivatives is formulated and proved.

Fractional Differential Equations with Initial Conditions at Inner Points in Banach Spaces

This paper is concerned with nonlinear fractional differential equations with the Caputo fractional derivatives in Banach spaces. Local existence results are obtained for initial value problems with initial conditions at inner points for the cases that the nonlinear parts are Lipschitz and non-Lipschitz, respectively. Hausdorff measure of non-compactness and Darbo-Sadovskii fixed point theorem are employed to deal with the non-Lipschitz case. The results obtained in this paper extend the classical Peano’s existence theorem for first order differential equations partly to fractional cases.

Solvability of a Third-Order Singular Generalized Left Focal Problem in Banach Spaces

We consider the existence of positive solution for a third-order singular generalized left focal boundary value problem with full derivatives in Banach spaces. Green’s function and its properties, explicit a priori, estimates will be presented. By means of the theories of the fixed point in cones, we establish some new and general results on the existence of single and multiple positive solutions to the third-order singular generalized left focal boundary value problem. Our results are generalizations and extensions of the results of the focal boundary value problem. An example is included to illustrate the results obtained.

Solvability and Optimal Controls of Semilinear Riemann-Liouville Fractional Differential Equations

We consider the control systems governed by semilinear differential equations with Riemann-Liouville fractional derivatives in Banach spaces. Firstly, by applying fixed point strategy, some suitable conditions are established to guarantee the existence and uniqueness of mild solutions for a broad class of fractional infinite dimensional control systems. Then, by using generally mild conditions of cost functional, we extend the existence result of optimal controls to the Riemann-Liouville fractional control systems. Finally, a concrete application is given to illustrate the effectiveness of our main results.

Positive solutions to the singular p-Laplacian BVPs with sign-changing nonlinearities and higher-order derivatives in Banach spaces on time scales

Abstract. In this paper, in order to establish the existence criteria for positive solutions of a multiple-point Dirichlet-Robin BVPs in Banach spaces on time scales, first we prove a fixed point theorem on monotone operators and the Ascoli-Arzela’s theorem on time scales. Then using the monotone operator method, we explore the existence criteria of positive solutions for a general multiple-point Dirichlet-Robin BVPs in Banach spaces on time scales with the singular sign-changing nonlinearities and higher-order derivatives. Finally an example is illustrated to indicate the application of our main results, which generalize some well-known results in the literature.

Direct Computation of the Simultaneous Stone-Weierstrass Approximation of a Function and Its Partial Derivatives in Banach Spaces, and Combination with Hermite Interpolation

We present a new variant of the simultaneous Stone-Weierstrass approximation of a function and its partial derivatives, when the function takes its values in a Banach space, and provide an explicit and direct computation of this approximation. In the particular case of approximation by means of polynomials, we show that the simultaneous approximation can be required to be exact at a finite number of prescribed points.