Stability results for a new kind fractional partial differential variational inequalities

Fractional partial differential variational inequality

A class of Erdélyi–Kober fractional differential inequalities with a polynomial nonlinearity and a singular potential function is investigated in this paper. By mean of the test function method, we establish sufficient conditions for the nonexistence of global weak solutions. Some examples are provided to illustrate our obtained results. To the best of our knowledge, the issue of nonexistence of global solutions for fractional differential equations or inequalities, involving Erdélyi–Kober fractional derivatives, was never addressed in the literature.

In this paper, we study a class of partial differential variational inequalities. A general stability result for the partial differential variational inequality is provided in the case the perturbed parameters are involved in both the nonlinear mapping and the set of constraints. The main tools are theory of semigroups, theory of monotone operators, and variational inequality techniques.

This paper is devoted to a generalized evolution system called fractional partial differential variational inequality which consists of a mixed quasi-variational inequality combined with a fractional partial differential equation in a Banach space. Invoking the pseudomonotonicity of multivalued operators and a generalization of the Knaster-Kuratowski-Mazurkiewicz theorem, first, we prove that the solution set of the mixed quasi-variational inequality involved in system is nonempty, closed and convex. Next, the measurability and upper semicontinuity for the mixed quasi-variational inequality with respect to the time variable and state variable are established. Finally, the existence of mild solutions for the system is delivered. The approach is based on the theory of operator semigroups, the Bohnenblust-Karlin fixed point principle for multivalued mappings, and theory of fractional operators.

In this work, we are devoted to discussing a system of fractional stochastic differential variational inequalities with Lévy jumps (SFSDVI with Lévy jumps), that comprises both parts, that is, a system of stochastic variational inequalities (SSVI) and a system of fractional stochastic differential equations(SFSDE) with Lévy jumps. Here it is noteworthy that the SFSDVI with Lévy jumps consists of both sections that possess a mutual symmetry structure. Invoking Picard’s successive iteration process and projection technique, we obtain the existence of only a solution to the SFSDVI with Lévy jumps via some appropriate restrictions. In addition, the major outcomes are invoked to deduce that there is only a solution to the spatial-price equilibria system in stochastic circumstances. The main contributions of the article are listed as follows: (a) putting forward the SFSDVI with Lévy jumps that could be applied for handling different real matters arising from varied domains; (b) deriving the unique existence of solutions to the SFSDVI with Lévy jumps under a few mild assumptions; (c) providing an applicable instance for spatial-price equilibria system in stochastic circumstances affected with Lévy jumps and memory.

The standard solutions of the ℒ 2 -disturbance attenuation and optimal control problems hinge upon the computation of the solution of a Hamilton-Jacobi (HJ), Hamilton-Jacobi-Bellman (HJB) respectively, partial differential equation or inequality, which may be difficult or impossible to obtain in closed-form. Herein we focus on the matched disturbance attenuation and on the optimal control problems for fully actuated mechanical systems. We propose a methodology to avoid the solution of the resulting HJ (HJB, respectively) partial differential inequality by means of a dynamic state feedback. It is shown that for planar mechanical systems the solution of the matched disturbance attenuation and the optimal control problems can be given in closed-form.

Dynamic generalized controllability and observability functions with applications to model reduction and sensor deployment

Dynamic Solution of the Hamilton-Jacobi inequality in the ℒ2-disturbance Attenuation Problem

Using a modification of the nonlinear capacity method, we obtain necessary conditions for the solvability of some nonlinear partial differential equations and inequalities containing the polyharmonic operator and terms that depend on the norm of the gradient of the solution, both in the entire space and in bounded domains; in the latter case the coefficients of the inequality are allowed to have singularities.

In this paper, an interior penalty method is proposed to solve a parabolic complementarity problem involving fractional Black–Scholes operator arising in pricing American options under a geometric Lévy process. The complementarity problem is first reformulated as a fractional partial differential variational inequality problem using the representations of fractional order operators and appropriate mathematical techniques. A penalty equation is then proposed to approximate the variational inequality problem by introducing a novel interior-point based penalty term. The existence and uniqueness of the solution to the penalized problem are proved, and an upper bound on the distance between the solutions to the penalty equation and the variational inequality problem is established. To test our method, we discretize the penalty equation by a finite difference method in space and the Crank–Nicolson method in time. We then present numerical experimental results to demonstrate the usefulness and effectiveness for the interior penalty method.

Partial differential variational inequalities involving nonlocal boundary conditions in Banach spaces

Fractional differential inequalities have emerged as powerful tools for modeling and analyzing dynamic systems with fractional-order derivatives, offering a sophisticated framework to capture the complexities of real-world processes. Among the various analytical techniques, the comparison principle stands out as a fundamental approach in understanding the behavior of solutions to fractional differential inequalities. This study focuses on the development and analysis of comparison principles for some of the fractional differential inequalities involving the variable-order Caputo fractional derivative which is a generalization of the classical Caputo derivative that allows the order of differentiation to vary with respect to time or space. Such flexibility is important for modeling systems whose memory characteristics change over time or space. We formulate both weak and strong versions of the comparison principle with variable order Caputo fractional derivative. Our approach combines analytical techniques from fractional calculus and the theory of differential inequalities to establish some results. To have the applicability and relevance of our theoretical work, we provide an example demonstrating the effectiveness of the proposed comparison theorems. The findings of this paper not only contribute to the theoretical advancement of fractional differential inequalities with variable order but also applicable to systems where dynamic memory effects are prominent.

In this paper, we introduce and study a new class of fractional delay differential mixed variational inequalities (FDDMVI, for short) formulated by a fractional delay evolution inclusion and a mixed variational inequality in infinite Banach spaces. By applying a new measure of noncompactness and fixed point theorem for a condensing set-valued map, we obtain the global solvability of a FDDMVI on the half-line. Furthermore, we apply the obtained results to establish a weakly asymptotic stability result of the zero mild solution to a FDDMVI. Finally, an example is given to demonstrate the main results.

In many practical applications, stability with respect to part of the system's states is often necessary with finite-time convergence to the equilibrium state of interest. Finite-time partial stability involves dynamical systems whose part of the trajectory converges to an equilibrium state in finite time. Since finite-time convergence implies non-uniqueness of system solutions in backward time, such systems possess non- Lipschitzian dynamics. In this paper, we address finite-time partial stability and uniform finite-time partial stability for nonlinear dynamical systems. Specifically, we provide Lyapunov conditions involving a Lyapunov function that is positive definite and decrescent with respect to part of the system state, and satisfies a differential inequality involving fractional powers for guaranteeing finite-time partial stability. In addition, we show that finite-time partial stability leads to uniqueness of solutions in forward time and we establish necessary and sufficient conditions for continuity of the settling-time function of the nonlinear dynamical system.

Existence results of partial differential mixed variational inequalities without Lipschitz continuity