Mild Solutions Research Articles

Synthesis, Corrosion Inhibition Efficiency in Acidic Media, and Quantum Chemical Studies of Some Hydrazine Derivatives

In this work, four hydrazone Schiff base derivatives N-(2,4-Dinitro-phenyl)-N’-(1H-pyrrol-2-ylmethylene)-hydrazine (1a), N-Benzo [1,3] dioxol-5-ylmethylene-N’-(2,4-dinitro-phenyl)-hydrazine (1b), (E)-5-((2-(2,4-dinitrophenyl)hydrazono)methyl)-2-hydroxybenzoic acid (1c) and (E)-1-(2,4-dinitrophenyl)-2-(2-methoxybenzylidene)hydrazine (1d) were synthesized by reaction of four aldehydes namely pyrrole-2-carboxaldehyde, piperonal, 5-formylsalicylic acid, and o-vanillin with 2,4-dinitrophenyl hydrazine to produce the final compounds 1a, 1b, 1c, and 1d, respectively. These four compounds were investigated as corrosion inhibitors in aqueous mild acidic static solution. FTIR, HNMR, and elemental analysis were used to elucidate the chemical structure of the synthesized inhibitors. Using potential dynamic polarization measurements, these inhibitors’ efficiency in preventing C-steel corrosion in 1.00 M HCl was studied. The results of the experiments revealed that 1×10−3 M is the ideal concentration for 1a, 1b, 1c, and 1d, and that the corresponding inhibition efficiencies for these subunits were 80.70%, 91.30%, 91.34, and 88.80%, respectively. The best corrosion inhibitors were compounds 1b and 1c. Furthermore, studies suggested that these substances are mixed-type inhibitors and that the efficiency of the inhibition is strongly correlated with their quantity. Quantum paraments included Dipole moment, energy band gap (ΔE), value of energy of lowermost unoccupied molecular orbital (ELUMO), and energy of high most occupied molecular orbital (EHOMO) using Molecular Operating Environment MOE, Gaussian, and HyperChem software packages were determined which demonstrated strong agreement between algorithmic and practical findings.

Existence and optimal controls of non-autonomous for impulsive evolution equation without Lipschitz assumption

In this paper, we investigate the existence of mild solutions as well as optimal controls for non-autonomous impulsive evolution equations with nonlocal conditions. Using the Schauder’s fixed-point theorem as well as the theory of evolution family, we prove the existence of mild solutions for the concerned problem. Furthermore, without the Lipschitz continuity of the nonlinear term, the optimal control result is derived by setting up minimizing sequences twice. An example is given of the application of the results.

A Lifshitz-Slyozov type model for adipocyte size dynamics: limit from Becker-Döring system and numerical simulation.

Biological data show that the size distribution of adipose cells follows a bimodal distribution. In this work, we introduce a Lifshitz-Slyozov type model, based on a transport partial differential equation, for the dynamics of the size distribution of adipose cells. We prove a new convergence result from the related Becker-Döring model, a system composed of several ordinary differential equations, toward mild solutions of the Lifshitz-Slyozov model using distribution tail techniques. Then, this result allows us to propose a new advective-diffusive model, the second-order diffusive Lifshitz-Slyozov model, which is expected to better fit the experimental data. Numerical simulations of the solutions to the diffusive Lifshitz-Slyozov model are performed using a well-balanced scheme and compared to solutions to the transport model. Those simulations show that both bimodal and unimodal profiles can be reached asymptotically depending on several parameters. We put in evidence that the asymptotic profile for the second-order system does not depend on initial conditions, unlike for the transport Lifshitz-Slyozov model.

An analysis on the approximate controllability outcomes for fractional stochastic Sobolev‐type hemivariational inequalities of order 1<r<2$$ 1<r<2 $$ using sectorial operators

In this paper, we deal with the approximate controllability of fractional stochastic Sobolev‐type hemivariational differential systems of order 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.

A study of nonlocal fractional neutral stochastic integrodifferential inclusions of order 1

This paper considers a class of nonlocal fractional neutral stochastic integrodifferential inclusions of order 1 < α < 2 with impulses in a Hilbert space. We study the existence of the mild solution for the cases when the multi-valued map has convex and non-convex values. The results are obtained by combining fixed-point theorems with the fractional order cosine family, semigroup theory, and stochastic techniques. A new set of sufficient conditions is developed to demonstrate the approximate controllability of the system. Finally, an example is given to illustrate the obtained results.

New exploration on approximate controllability of fractional neutral‐type delay stochastic differential inclusions with non‐instantaneous impulse

This paper aims to derive a new set of sufficient conditions for the existence and approximate controllability of neutral‐type fractional stochastic integrodifferential inclusions with infinite delay and non‐instantaneous impulse in a separable Hilbert space using the Atangana–Baleanu Caputo fractional derivative. We investigate the existence of a mild solution for the Atangana–Baleanu Caputo fractional neutral‐type delay integrodifferential stochastic system while taking into account the non‐instantaneous impulses. For this purpose, the Atangana–Baleanu Caputo fractional neutral‐type impulsive delay stochastic system is transferred into an equivalent fixed point problem via an integral operator, and then, the Bohnenblust–Karlin fixed point approach is applied. Further, the approximate controllability results of the proposed nonlinear stochastic impulsive control system are established under the consideration that the corresponding linear system is approximately controllable. The set of sufficient conditions is established by using the concepts of stochastic analysis, fractional calculus, fixed point technique, semigroup theory of bounded linear operators, and the theory of multivalued maps. To illustrate the abstract results, we provide an example at the end of the paper.

Approximate controllability of third order dispersion systems

The current paper focuses on the approximate controllability of Volterra-Fredholm type integrodifferential third order dispersion system. The Banach fixed point theorem and semigroup operator are used to obtain the existence and uniqueness of mild solutions to the considered system. In order to examine the approximate controllability results, a different approach is used, namely, the sequential method.

The solution of stochastic evolution equation with the fractional derivative

In this paper, we consider the stochastic evolution equation driven by the Gaussian noise with white time and colored space, where the noise coefficient is the Marchaud fractional derivative. The key idea is that we transform our model into a stochastic space-fractional equation by taking the Marchaud fractional derivative, and then use Chaos expansion to prove the mild solution. There are three main results in this paper. First, we apply Chaos expansion to obtain the existence, uniqueness and Lyapunove exponent of the solution of the transformed equation. Second, we prove that there exists an unique mild solution of the original equation, the approach is taking the fractional integral operator into the transformed equation. Finally, we explore Hölder continuity of the mild solution.

On the S-asymptotically ω-periodic mild solutions for multi-term time fractional measure integro-differential equations

This paper deals with a class of nonlocal problem for multi-term time-fractional measure integro-differential evolution equations of mixed type in Hilbert spaces. Firstly, we introduce the concept of S-asymptotically ω-periodic mild solution, on the premise of by utilizing -resolvent family and measure functional (Henstock–Lebesgue–Stieltjes integral). Secondly, combining regulated functions and fixed point theorem, the existence S-asymptotically ω-periodic mild solution for a classes of measure-driven differential equation is discussed. Finally, as samples of applications, we consider the nonlocal problem to a class of time-fractional measure partial differential equation with nonlocal condition at the end of this paper.

A Study on Approximate Controllability of Ψ-Caputo Fractional Differential Equations with Impulsive Effects

In this article, we studied the approximate controllability of Ψ-Caputo fractional differential systems. We prove the sufficient conditions for an abstract Cauchy problem invloving infinite delay, impulsive and nonlocal conditions. The result is shown by means of the infinitesimal operator, semigroup theory, fractional calculus, and Schauder’s fixed point theorem. First, we prove the existence of the mild solution and demonstrate that the Ψ-Caputo fractional system is approximately controllable. Finally, an example is given to analyse the obtained results.

Existence Results of Random Impulsive Integrodifferential Inclusions with Time-Varying Delays

This study examines the existence of mild solutions for nonlinear random impulsive integrodifferential inclusions with time-varying delays under sufficient conditions. Our study is based on the Martelli fixed point theorem, Pachpatte’s inequality, and the fixed point theorem due to Covitz and Nadler. Besides, we generalize, extend, and develop some well-known results in the existing literature.

Controllability of Hilfer Fractional Semilinear Integro-Differential Equation of Order α ∊ (0, 1), β ∊ [0, 1

In this paper, Hilfer fractional differential equation is studied. Firstly, we used Laplace transform and semigroup theory to find the mild solution of the system. Then exact controllability of proposed system is established using the Arzela-Ascoli theorem and Schauder fixed point theorem. To illustrate the developed theory we provide an example at the end.

Locally risk minimizing pricing of Asian option in a semi-Markov modulated market

We consider a regime-switching model where the stock volatility dynamics is a semi-Markov process. Under this model assumption, we find the locally risk-minimizing price of some Asian options with European-style exercise. The price function is shown to satisfy a non-local degenerate system of parabolic PDEs in dimension two with a terminal condition. We show this by deriving the F-S decomposition of the discounted contingent claim. The related Cauchy problem involving the PDE is shown to be equivalent to an integral equation (IE). The existence and uniqueness of the classical solution to the PDE are determined by studying the IE and using semigroup theory. To be more precise, we first obtain the mild solution of the PDE and then we show that the mild solution has sufficient regularity. The locally risk-minimizing hedging for the option has also been identified in this work. Finally, the computational aspects of Asian option prices have been discussed by solving the equation numerically.

On the Cauchy problem for the generalized double dispersion equation with logarithmic nonlinearity

Abstract In this paper, we investigate the global existence and finite time blow-up of solution for the Cauchy problem of one-dimensional fifth-order Boussinesq equation with logarithmic nonlinearity. Fist we prove the existence and uniqueness of local mild solutions in the energy space by means of the contraction mapping principle. Further under some restriction on the initial data, we establish the results on existence and uniqueness of global solutions and finite time blow-up of solutions by using the potential well method. Moreover, the sufficient and necessary conditions of global existence and finite time blow-up of solutions are given.

A class of time-optimal feedback control for fractional neutral evolution hemivariational inequalities with fixed delay

ABSTRACT In this paper, we study a class of optimal feedback control for a neutral fractional system with hemivariational inequalities. Initially, we derive some existence results of mild solutions for the system by applying the Marteli's fixed point theorem of multivalued maps, properties of generalized Clarke's subdifferential and semigroup operators. Then, by using the Filippov theorem and the Cesari property, a new set of sufficient conditions are formulated to ensure the existence results of a feasible pair for the feedback control systems. Also, we apply our main results to obtain the optimal feedback control pair. Further, we investigate the time-optimal feedback control for our control system. In the end, an example is given to illustrate our theory.

On second order differential inclusion driven by quasi-variational–hemivariational inequalities

In this paper we investigate an evolution problem (SDQHVI) which constitutes of the second order differential inclusion driven by a quasi-variational–hemivariational inequality (QHVI) with perturbation operator in Banach spaces. Unlike the existing literature on differential inclusions, on the one hand, the second order differential operator is not assumed to be compact; on the other hand, the solvability of the inequality is proved under non-coercive condition. More precisely, first, it is shown that the solution set of the QHVI with perturbation operator is nonempty, bounded, closed and convex under the KKM theorem and the Minty formulation. Then, an associated multivalued map with the solution set of the QHVI with perturbation operator is introduced, and we prove that it is upper semicontinuous and measurable. Finally, by utilizing Kakutani–Fan–Glicksberg fixed point theorem, a weak topology technique and properties of strong continuous cosine operators, we prove the existence of mild solutions for SDQHVI.

Mild solution and finite-approximate controllability of higher-order fractional integrodifferential equations with nonlocal conditions

Mild solution and finite-approximate controllability of higher-order fractional integrodifferential equations with nonlocal conditions

On the Existence and Uniqueness of Fuzzy Mild Solution of Fractional Evolution Equations

In this paper, the nonlocal Cauchy problem is discussed for the fuzzy fractional evolution equations in an arbitrary Banach space for order q ∈ (1, 2) and the criteria on the existence and uniqueness of mild fuzzy solutions are obtained by using Schauder’s fixed point theorem. An example to illustrate the applications of main results is also given.

Analyzing the continuity of the mild solution in finite element analysis of semilinear stochastic subdiffusion problems

&lt;abstract&gt;&lt;p&gt;This paper aimed to further introduce the finite element analysis of non-smooth data for semilinear stochastic subdiffusion problems driven by fractionally integrated additive noise. The mild solution of this stochastic model consisted of three different Mittag-Leffler functions. We analyzed the smoothness of the solution and utilized complex integration to approximate the error of the solution operator under non-smooth data. Consequently, optimal convergence estimates were obtained, and we also obtained the continuity conditions of the mild solution. Finally, the influence of the fractional parameters $ \alpha $ and $ \gamma $ on the convergence rates were accurately demonstrated through numerical examples.&lt;/p&gt;&lt;/abstract&gt;

Solving delay integro-differential inclusions with applications

&lt;abstract&gt;&lt;p&gt;This work primarily delves into three key areas: the presence of mild solutions, exploration of the topological and geometrical makeup of solution sets, and the continuous dependency of solutions on a second-order semilinear integro-differential inclusion. The Bohnenblust-Karlin fixed-point method has been integrated with Grimmer's theory of resolvent operators. Ultimately, the study delves into a mild solution for a partial integro-differential inclusion to showcase the achieved outcomes.&lt;/p&gt;&lt;/abstract&gt;