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The solution of pseudo initial value differential equations, either ordinary or partial (including those of fractional nature), requires the development of adequate analytical methods, complementing those well established in the ordinary…
In this paper, we introduce the concepts of S-asymptotically $\omega$-periodic solutions in distribution for a class of stochastic fractional functional differential equations. The existence and uniqueness results for the S-asymptotically…
We shall study backward stochastic differential equations and we will present a new approach for the existence of the solution. This type of equation appears very often in the valuation of financial derivatives in complete markets.…
This paper studies a nonlinear fractional implicit differential equation (FIDE) with boundary conditions involving a HilferHadamard type fractional derivative. We establish the equivalence between the Cauchy-type problem (FIDE) and its…
The goal of this work is to discuss how should we impose initial values in fractional problems to ensure that they have exactly one smooth unique solution, where smooth simply means that the solution lies in a certain suitable space of…
Given the abstract evolution equation \[ y'(t)=Ay(t),\ t\ge 0, \] with scalar type spectral operator $A$ in a complex Banach space, found are conditions necessary and sufficient for all weak solutions of the equation, which a priori need…
We study a stochastic linear evolution equation $dX+A(t)Xdt=F(t)dt+ G(t)dw_t$ in a Banach space of M-type 2. We construct unique strict solutions to the equation on the basis of the theory of deterministic linear evolution equations. The…
In this paper, we present a study on the Ulam-Hyers and Ulam-Hyers-Rassias stabilities of the solution of the fractional functional differential equation using the Banach fixed point theorem.
In this paper we analyze a nonlinear abstract evolution equation with an infinite number of time-dependent time delays and a Lipschitz continuous nonlinear term. By using a fixed point argument we prove the existence of a mild solution.…
The aim of the present paper is twofold:(1) We carry on with developing an abstract method for deriving decay estimates on the semigroup associated to non-symmetric operators in Banach spaces as introduced in [10]. We extend the method so…
We investigate a class of fractional neutral evolution equations on Banach spaces involving Caputo derivatives. Main results establish conditions for the controllability of the fractional-order system and conditions for existence of a…
In this paper, we extend the concept of split variational inequality problems from Hilbert spaces to Banach spaces. Then we apply the Fan-KKM theorem to prove the existence of solutions to some split variational inequality problems and some…
We consider evolution differential equations in Fr\'echet spaces that possess unconditional Schauder basis and construct a version of the majorant functions method to obtain existence theorems for Cauchy problems. Applications to PDE and…
We prove results of existence of a solution (resp. existence and uniqness of a solution) for nonlinear differential equations of type $x'(t) +G(x,t) x(t) = F(x,t),$ in an abstract Banach subspace $X$ of the space of bounded real-valued…
This paper introduces a novel steepest descent flow in Banach spaces. This extends previous works on generalized gradient descent, notably the work of Charpiat et al., to the setting of Finsler metrics. Such a generalized gradient allows…
We give a unified proof of the Yamada-Watanabe-Engelbert theorem for various notions of solutions for SPDEs in Banach spaces with cylindrical Wiener noise. We use Kurtz' generalization of the theorems of Yamada, Watanabe and Engelbert. In…
We consider a class of nonlinear fractional Volterra integrodifferential equation with fractional integrable impulses and investigate the existence and uniqueness results in the Bielecki's normed Banach spaces. Further, Bielecki--Ulam type…
The $m$-point nonlocal problem for the first order differential equation with an operator coefficient in a Banach space $X$ is considered. An exponentially convergent algorithm is proposed and justified provided that the operator…
This work introduces and analyzes a finite element scheme for evolution problems involving fractional-in-time and in-space differentiation operators up to order two. The left-sided fractional-order derivative in time we consider is employed…
In this paper, we prove several fixed point theorems on both of normal partially ordered Banach spaces and regular partially ordered Banach spaces by using the normality, regularity, full regularity, and chain -complete property. Then, by…