Mild Solutions Research Articles

Heteroatom Substitution Strategy Modulates Thermodynamics Towards Chemically Recyclable Polyesters and Monomeric Unit Sequence by Temperature Switching

AbstractIn this paper, we proposed a heteroatom substitution strategy (HSS) in the δ‐valerolactone (VL) system to modulate thermodynamics toward chemically recyclable polyesters. Three VL‐based monomers containing different heteroatoms (M1 (N), M2 (S), and M3 (O)), instead of C‐5 carbon, were designed and synthesized to verify our proposed HSS. All three monomers undergo organocatalytic living/controlled ROP and controllable depolymerization. Impressively, the resulting P(M1) achieved over 99 % monomer recovery under both mild solution depolymerization and high vacuum pyrolysis conditions without any side reactions, and the recycled monomers can be polymerized again forming new polymers. The systematic study of the relationship between heteroatom substitution and recyclability shows that introducing heteroatoms does change the thermodynamics of the monomers (ΔHpo, ΔSpo and Tc values), thereby adjusting the polymerizability and depolymerizability. DFT calculations found that the introduction of heteroatoms adjusts the ring strain by changing the angular strain of the monomers, and the order of their angular strain (M2>M1>M3) is consistent with the order of the experimentally obtained enthalpy change. Notably, the one‐pot/one‐step copolymerization of two of each of the three monomers enables the synthesis of sequence‐controlled copolymers from gradient to random to block structures, by simply switching the copolymerization temperature.

Abstract random differential equations with state-dependent delay using measures of noncompactness

This paper is devoted to the existence of random mild solutions for a general class of second-order abstract random differential equations with state-dependent delay. The technique used is a generalization of the classical Darbo fixed point theorem for Fréchet spaces associated with the concept of measures of noncompactness. An application related to partial random differential equations with state-dependent delay is presented.

Existence of a Class of Doubly Perturbed Stochastic Functional Differential Equations with Poisson Jumps

In this paper, we use successive approximations and Picard iterative method to establish the existence and uniqueness of mild solution for a class of doubly perturbed impulsive neutral stochastic functional differential equations with Poisson jumps in Hilbert spaces. An example of a doubly perturbed stochastic differential equation with delays is given to illustrate our main results.

New Study on the Controllability of Non-Instantaneous Impulsive Hilfer Fractional Neutral Stochastic Evolution Equations with Non-Dense Domain

The purpose of this work is to investigate the controllability of non-instantaneous impulsive (NII) Hilfer fractional (HF) neutral stochastic evolution equations with a non-dense domain. We construct a new set of adequate assumptions for the existence of mild solutions using fractional calculus, semigroup theory, stochastic analysis, and the fixed point theorem. Then, the discussion is driven by some suitable assumptions, including the Hille–Yosida condition without the compactness of the semigroup of the linear part. Finally, we provide examples to illustrate our main result.

Measure pseudoasymptotically Bloch periodic functions in the sense of Stepanov and applications

UDC 517.9 We focus on the measures of Stepanov-like pseudoasymptotically Bloch τ -periodicity and its applications. First, we define a new notion of measures of Stepanov-like pseudoasymptotically Bloch periodic functions and diccuss some of its fundamental properties. Then the obtained results are applied to investigate the existence and uniqueness of the measure Stepanov-like pseudoasymptotically Bloch periodic mild solutions to semilinear delay differential equation in Banach spaces. Finally, an application is presented to illustrate the efficiency of the results.

Existence and Ulam stability of mild solutions for nonlinear fractional integro-differential equations in a Banach space

Existence and Ulam stability of mild solutions for nonlinear fractional integro-differential equations in a Banach space

Periodic solutions of the parabolic–elliptic Keller–Segel system on whole spaces

AbstractIn this paper, we investigate to the existence and uniqueness of periodic solutions for the parabolic–elliptic Keller–Segel system on whole spaces detailized by Euclidean space and real hyperbolic space . We work in framework of critical spaces such as on weak‐Lorentz space to obtain the results for the Keller–Segel system on and on for to obtain those on . Our method is based on the dispersive and smoothing estimates of the heat semigroup and fixed point arguments. This work provides also a fully comparison between the asymptotic behaviors of periodic mild solutions of the Keller–Segel system obtained in and the one in .

On solutions of a semilinear measure‐driven evolution equation with nonlocal conditions on infinite interval

AbstractThis paper studies the existence and asymptotic properties of solutions for a semilinear measure‐driven evolution equation with nonlocal conditions on an infinite interval. The existence result of the solutions for the considered equation is established by Schauder's fixed point theorem. Then, the asymptotic stability of solutions is further proved to show that all the solutions may converge to the unique solution of the corresponding Cauchy problem. In addition, under some conditions the existence of global attracting sets and quasi‐invariant sets of mild solutions is investigated as well. Finally, an example is provided to illustrate the applications of the obtained results.

Some new uniqueness and Ulam–Hyers type stability results for nonlinear fractional neutral hybrid differential equations with time-varying lags

Abstract This paper deals with some qualitative properties of solutions to nonlinear neutral hybrid differential equations connected to ψ-Caputo fractional derivative with time-varying lags. First, we demonstrate the problem possesses a mild solution uniquely where the source function may have temporal singularities. Second, in some cases, we indicate that the problem possesses a unique mild solution under some weaker conditions than the previous one. Third, we also obtain a result on a global mild solution for the problem. Finally, the results are further enriched by studying a new type of Ulam–Hyers stability for the main equation. The main results are obtained by applying the nice inequality, first proposed and proven in this paper. Some befit examples are given to justify the applicability of the main results.

Fractional nonlinear heat equations and characterizations of some function spaces in terms of fractional Gauss–Weierstrass semi–groups

AbstractWe present a new proof of the caloric smoothing related to the fractional Gauss–Weierstrass semi–group in Triebel-Lizorkin spaces. This property will be used to prove existence and uniqueness of mild and strong solutions of the Cauchy problem for a fractional nonlinear heat equation.

Global attracting set of stochastic differential equations with unbounded delay driven by fractional Ornstein–Uhlenbeck process

Abstract In this paper, we have studied stochastic differential equations with unbounded delay in fractional power spaces perturbed by fractional Ornstein–Uhlenbeck process Y H , ξ ⁢ ( t ) {{Y^{H,\xi}}(t)} with H ∈ ( 1 2 , 1 ) {H\in(\frac{1}{2},1)} . Subsequently, the existence and uniqueness of mild solution of the considered equation have been proved with fixed-point theorem. Finally, we obtain the global attracting set of the considered equations by some stochastic analysis and inequality technique.

On bilinear estimates and critical uniqueness classes for Navier-Stokes equations

We are concerned with bilinear estimates and uniqueness of mild solutions for the Navier-Stokes equations in critical spaces. For that, we construct general settings in which estimates for the bilinear term of the mild formulation hold true without using auxiliary norms such as Kato time-weighted ones. We first obtain necessary conditions in abstract critical spaces and then consider further structures to obtain the estimates in general classes of Besov, Morrey and Besov-Morrey spaces based on Banach spaces. Examples of applications are provided in different spaces as well as for other PDEs. In particular, as far as we know, the bilinear estimate and the uniqueness property obtained in the framework of Besov-weak-Herz spaces are not available in the existing literature. The proofs are mainly based on characterizations and estimates on the corresponding predual spaces.

An analysis on time optimal feedback control for Caputo fractional neutral systems with delay

ABSTRACT This paper aims to provide a systematic approach to investigate the optimal feedback control systems governed by the fractional neutral evolution system with fixed delay in the sense of Caputo fractional derivatives in separable reflexive Banach spaces. Initially, we study the existence result of mild solutions for the system by applying fixed point theorem for a condensing map due to Martelli. The existence of feasible pairs is obtained by using the Filippov theorem and the Cesari property. Furthermore, the existence of optimal feedback control and time optimal control are studied for the corresponding control problem. Finally, an example is given to validate our findings.

The 3-D time-dependent Navier–Stokes equations on multi-connected domains with inhomogeneous boundary conditions and spectral hyperviscosity

We consider the time-dependent 3-D Navier–Stokes equations (NSE) on a multi-connected bounded domain Ω⊂R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega \\subset \\mathbb {R^{\ extrm{3}}}$$\\end{document} with inhomogeneous boundary data β∈H1/2(Γ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta \\in H^{1/2}(\\Gamma )$$\\end{document} on ∂Ω=Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial \\Omega =\\Gamma $$\\end{document}, where Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document} is a union of Lipschitz continuous surfaces Γ0,Γ1,⋯,Γl\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma _{0},\\Gamma _{1},\\dots ,\\Gamma _{l}$$\\end{document}. This assumption includes the particular case when the Γi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma _{i}$$\\end{document} are disjoint, the stationary version of which is classically known as Leray’s problem. Existence results for Leray’s problem have either assumed flux conditions beyond the general flux condition necessitated by compatibility constraints, or required size restrictions on the data. Here we incorporate a spectral hyperviscosity term in the time-dependent case and obtain existence and foundational results, assuming only the general flux condition and without imposing size restrictions on the boundary data β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document}. For any such β∈H1/2(Γ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta \\in H^{1/2}(\\Gamma )$$\\end{document} we establish global existence and uniqueness of mild solutions. Then on any interval [0, T] on which these solutions and the corresponding NSE solution share a common H1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^{1}$$\\end{document}-bound (as is present on local intervals of existence of strong solutions, in certain special cases, or as is commonly assumed in achieving strong convergence results in numerical studies) we show for slightly more regular β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document} that the spectrally-hyperviscous solutions converge strongly and uniformly in H1(Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^{1}(\\Omega )$$\\end{document} to the NSE solution as the spectral hyperviscosity term vanishes in the limit of key parameters. To achieve this robust sense of approximation of the NSE system, an involved setup and specially-adapted semigroup techniques assume essential roles, and the exposition is new for the case β=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta =0$$\\end{document} as well. Our final results adapt the NSE reformulation in Liu et al. (J Comput Phys 229(9):3428–3453, 2010) to recast our approximating system in a form potentially more adaptable to computation.

New Results on the Solvability of Abstract Sequential Caputo Fractional Differential Equations with a Resolvent-Operator Approach and Applications

This paper aims to establish the existence and uniqueness of mild solutions to abstract sequential fractional differential equations. The approach employed involves the utilization of resolvent operators and the fixed-point theorem. Additionally, we investigate a specific example concerning a partial differential equation incorporating the Caputo fractional derivative.

Controllability of Mild Solution to Hilfer Fuzzy Fractional Differential Inclusion with Infinite Continuous Delay

This work investigates the solvability of the generalized Hilfer fractional inclusion associated with the solution set of a controlled system of minty type–fuzzy mixed quasi-hemivariational inequality (FMQHI). We explore the assumed inclusion via the infinite delay and the semi-group arguments in the area of solid continuity that sculpts the compactness area. The conformable Hilfer fractional time derivative, the theory of fuzzy sets, and the infinite delay arguments support the solution set’s controllability. We explain the existence due to the convergence properties of Mittage–Leffler functions (Eα,β), that is, hatching the existing arguments according to FMQHI and the continuity of infinite delay, which has not been presented before. To prove the main results, we apply the Leray–Schauder nonlinear alternative thereom in the interpolation of Banach spaces. This problem seems to draw new extents on the controllability field of stochastic dynamic models.

New existence and stability results of mild solutions for fuzzy fractional differential variational inequalities

The aim of this work is to study the stability for a class of fuzzy fractional differential variational inequalities modeled by variational inequalities coupled with fuzzy fractional differential inclusions. By introducing the parametric problem with the help of two parameters in the involved mapping and constraint set, we first establish an existence result for the parametric fuzzy fractional differential variational inequality (PFFDVI), and we also establish the compactness and continuous dependence on the initial value for the PFFDVI. Furthermore, when the parameters are perturbed, we obtain the closedness of solution mapping for the PFFDVI. Then we further investigate the upper semicontinuity of solution mapping for the PFFDVI under some mild conditions. Finally, in order to illustrate the obtained results, we give a numerical example at the end of this paper.

Controllability of Mild Solutions for Second-Order Neutral Evolution Equations with State-Dependent Delay

Controllability of Mild Solutions for Second-Order Neutral Evolution Equations with State-Dependent Delay

Fuzzy fractional delay differential inclusions driven by hemivariational inequalities in Banach spaces

This paper investigates a novel class of nonlinear dynamical fuzzy systems referred to as fuzzy fractional delay differential hemivariational inequalities (FFDDHVIs) in Banach spaces. These systems integrate fractional fuzzy differential inclusions with delay and hemivariational inequalities. The existence theorem for the HVIs is established based on the KKM theorem. Moreover, specific propositions are proved, covering the superpositional measurability and the upper semicontinuity for the HVIs. Next, by using the fixed points theorem, we establish the existence and compactness of mild solution sets for the FFDDHVIs under certain mild conditions. Finally, As an illustrative application, we investigate a frictional quasistatic contact problem for viscoelastic materials, in which the friction and contact conditions are described by the Clarke generalized gradient of nonconvex and nonsmooth functionals.

Analysis of coupled system of q$$ q $$‐fractional Langevin differential equations with q$$ q $$‐fractional integral conditions

In this dissertation, we study the coupled system of ‐fractional Langevin differential equations involving ‐Caputo derivative having ‐fractional integral conditions. With the help of some adequate conditions, we investigate the uniqueness and existence of mild solution of the aforementioned system. We also analyze various kinds of Ulam's stability. Banach fixed point theorem and Leray–Schauder of cone type are used to illustrate the existence and uniqueness results. We also used non‐linear functional analysis methods to explore variety of stability types. An example is provided to clearly demonstrate our theoretical outcomes.