A new class of fuzzy fractional differential inclusions driven by variational inequalities

Abstract

The goal of this paper is to consider a new fuzzy differential system consisting of a fuzzy fractional differential inclusion combined with a variational inequality, which is called fuzzy fractional differential variational inequality (FFDVI). Such a model captures the desired features of both fuzzy fractional differential inclusions and fractional differential variational inequalities within the same framework. By employing the set-valued version of the Krasnoselskii fixed point theorem, an existence of solutions is obtained for the FFDVI under some mild conditions. Moreover, an approximating algorithm is provided to find a solution of the FFDVI. Finally, two numerical examples are given to illustrate our main results.

Introduction

It is well known that fuzzy differential equations (inclusions) have many important applications for modeling uncertain phenomena in various fields of science, such as artificial intelligence, population dynamics, petroleum engineering, mechanics, medicine, etc. We would like to mention that Hüllermeier [29] was motivated by application in knowledge-based systems, then he introduced and studied the following a class of fuzzy differential inclusions

$$\{\begin{array}{c}{x}^{\prime }(t)\in {[F_{(t,x(t))}]}_{\alpha },\alpha \in [0,1],\hfill \\ x(0)\in {[x_0]}_{\alpha }.\hfill \end{array}$$

Guo et al. [20] established some existence results for the fuzzy impulsive functional differential inclusions and provided an application to the fuzzy population models. Min et al. [52] investigated a class of implicit fuzzy differential inclusions and also presented an application in drilling petroleum engineering dynamics. Majumdar et al. [50] discussed the application of fuzzy differential inclusions in atmospheric and medical cybernetics. Liu et al. [47] further discussed the fuzzy delay differential inclusions. Xiao et al. [74] established some existence results for the semilinear fuzzy differential inclusion. Dai et al. [18] investigated a class of universal oscillator fuzzy differential equations. Liu et al. [45] provided some basic concepts of fuzzy process, hybrid process and uncertain process, they also developed a fuzzy calculus and proposed a new class of fuzzy differential equations. Recently, Hung et al. [31] studied a class of fuzzy differential inclusions with resolvent operators in Banach spaces.

Moreover, fractional differential equations are found to be more adequate than integer-order as fractional-order models provide an excellent tool for the description of hereditary and memory properties of various processes and materials. In recent years, much effort has been made in this interesting field. For instance, Wu et al. [72] established some existence and uniqueness results for a class of global fractional-order projective dynamical system with delay and perturbation and provided an application to the idealized traveler information systems for day-to-day adjustments processes. Ahmad et al. [4] studied a new class of boundary value problems of nonlinear fractional differential inclusions of order $q\in (1,2]$ with fractional separated boundary conditions. Bai et al. [6] established the existence and multiplicity of positive solutions for boundary value problem of nonlinear fractional differential equation of order $1<\alpha \le 2$. Muslim et al. [54] discussed the exact controllability of a control system governed by a fractional differential equation of order $\alpha \in (1,2]$ with non-instantaneous impulses. In 2010, Agarwal et al. [3] firstly introduced fractional calculus into the fuzzy differential equations, they proposed the concept of fuzzy fractional differential equation. Allahviranloo et al. [1] examined the existence and uniqueness of the solution for a class of fuzzy Caputo fractional differential equation. Ahmadian et al. [2] investigated a fuzzy fractional kinetic model to calculate the concentration and yield of xylose from the Oil Palm Frond. Malinowski [51] studied a class of random fuzzy fractional integral and differential equations. Ngo [58] investigated the existence and uniqueness for fuzzy fractional functional integral equations. Long et al. [46] considered a class of fuzzy fractional partial differential equations. Salahshour et al. [64] considered the properties of Caputo-Fabrizio derivative for interval-valued functions under uncertainty and studied a class of fractional differential equations under this notion. Recently, Wu et al. [73] investigated a class of fuzzy fractional differential inclusions with projection operators in finite dimensional Euclidean spaces as follows

$$\{\begin{array}{c}{F}_{(t,x(t))}({}_0{}^{C}D_t^{q}x(t)+g(x(t))-P_K[g(x(t))-\rho M(x(t))])\ge \alpha (x(t)),\hfill \\ \text{i.e.},{}_0{}^{C}D_t^{q}x(t)+g(x(t))-P_K[g(x(t))-\rho M(x(t))]\in {\left[F_{(t,x(t))}\right]}_{\alpha (x(t))},\text{for}\mathit{\text{a.e.}}t\in [0,T],\hfill \\ x\left(0\right)=x_0,\hfill \end{array}$$

where ${}_0{}^{C}D_t^q$ is the Caputo fractional derivative of order $q\in (0,1)$, $F:[0,T]\times R^n\to E^n$ is a fuzzy mapping, $g,M:R^n\to R^n$ are two mappings, $\alpha :R^n\to [0,1]$ is a function, $\rho >0$, $K\subset R^n$, $P_K$ is a projection operator. We would like to mention that, if $K=R^n$, then (1.1) reduces to the following fuzzy fractional differential inclusions

$$\{\begin{array}{c}{F}_{(t,x(t))}({}_0{}^{C}D_t^{q}x(t)+\rho M(x(t)))\ge \alpha (x(t)),\hfill \\ \text{i.e.},{}_0{}^{C}D_t^{q}x(t)\in {\left[F_{(t,x(t))}\right]}_{\alpha (x(t))}-\rho M(x(t)),\text{for}\mathit{\text{a.e.}}t\in [0,T],\hfill \\ x\left(0\right)=x_0.\hfill \end{array}$$

Variational inequality theory is an important part of optimization theory and nonlinear analysis, which has been widely applied into linear and nonlinear programming, economics, mechanics, transportation and so on (see, e.g. the monograph [8], [27], [28], [57]). Differential variational inequalities (DVIs) are a class of dynamic systems consisting of variational inequalities and ordinary differential equations, which extend the notion of projected dynamical systems [17], evolution variational inequalities [21], differential algebraic equations [33], differential complementarity problems [65] and so on. In 2008, Pang and Stewart [61] firstly introduced and systematically studied a class of differential variational inequalities in finite dimensional Euclidean spaces. Due to the importance of DVIs in the fields of dynamic traffic networks [63], dynamic oligopolistic network competition [26], dynamic Nash equilibrium problem [11], microbial fermentation processes [62], spatial price equilibrium control problem [41], frictional contact problems [48], [49], compliant visco-plastic particle contact model [67], smoothed particle hydrodynamics [12] and so on, many efforts have been made in this interesting field in recent decades. For example, Gwinner [24] introduced a new class of DVIs and exhibited the relationship of projected dynamical systems to DVIs. Chen et al. [10] provided convergence analysis of regularized time-stepping methods for DVIs. Li et al. [39], [40] extended the DVIs to differential mixed variational inequalities and impulsive differential variational inequalities, they obtained some new existence results and numerical method by using some results on differential inclusions and discrete Euler time-dependent procedure. Anh et al. [5] studied asymptotic behavior of solutions to a class of DVIs with finite delays by employing measure of noncompactness and fixed point theorem. Li et al. [42] investigated the existence and stability for a generalized differential mixed quasi-variational inequality. Wang et al. [70] studied the existence result for DVIs with relaxing the convexity condition. Wang et al. also introduced and studied a class of differential fuzzy variational inequalities in [69], which consist of a differential equation and a fuzzy variational inequality. Nguyen et al. [56] discussed the solvability and existence of a global attractor for a class of DVIs of parabolic-elliptic type by using measure of noncompactness and fixed point theorem. Liu et al. [43], [44] examined several existence results for partial differential variational inequalities and differential mixed variational inequalities in Banach spaces. Guo et al. [22] established stability result for the partial differential variational inequality. In 2015, Ke et al. [34] firstly introduced fractional calculus into DVIs. They considered a class of fractional delay differential variational inequalities, which is composed of a fractional delay differential equation and a variational inequality. Loi et al. [35] studied some new classes of (fractional) differential variational inequalities by using topological methods. Zeng et al. [75] investigated a class of fractional differential hemivariational inequalities in Banach spaces. Recently, Migórski et al. [53] established an existence result of solutions for fractional differential mixed variational inequalities via approximation approach in Banach spaces.

Inspired and motivated by fuzzy fractional differential inclusions and fractional differential variational inequalities, in this paper, we will investigate a new class of fuzzy fractional differential inclusions driven by variational inequalities, which is called fuzzy fractional differential variational inequalities (FFDVIs). It is worth mentioning that the FFDVIs are needed in practice situation. For example, Raghunathan et al. [62] pointed out that the fermentation dynamics can be described by employing a DVI. They showed that differential equations described the evolution of extracellular metabolites, variational inequalities represented the links between cell metabolism and its environment. But in practical situation, microbial metabolism process is easily affected by the uncertainty environmental factors such as the subtle change in temperature and the unpredictable nutrient availability in culture medium. Moreover, as pointed out by Toledo-Hernandez [68], fermentations will result in a dynamic behavior with memory. Therefore, it makes FFDVIs more appropriate than DVIs to describe fermentation dynamics. However, to our best knowledge, there is no paper to study FFDVIs. Consequently, it would be important and interesting to consider FFDVIs in finite/infinite-dimensional spaces. We note that the fixed point method is a useful tool to study the existence of solutions for (fractional) differential variational inequalities (see, e.g. [5], [34], [35], [43], [56]). Up to our knowledge, no attempt has been made to study the (fractional) differential variational inequalities by using Krasnoselskii fixed point theorem. In addition, due to applications of DVIs, some excellent work relating numerical method of DVIs have already presented (see, e.g. [10], [39], [40], [61], [69], [70]). Because of the non-locality of fractional derivative, the numerical computation of fractional differential equations is more complicated than integer-order. There are very few numerical results for fractional differential variational inequalities. The motivation of the present work is to make an attempt in these directions.

The main purpose of this paper is to make an attempt to consider the following FFDVI in finite dimensional Euclidean spaces:

$$\{\begin{array}{c}{}_0{}^{C}D_t^{q}x(t)\in {\left[F_{(t,x(t))}\right]}_{\alpha }+B(t,x(t))u(t),\mathit{\text{a.e.}}t\in [0,T],\hfill \\ 〈G(t,x(t))+Q(u(t)),v-u(t)〉\ge 0,\forall v\in K,\mathit{\text{a.e.}}t\in [0,T],\hfill \\ x\left(0\right)=x_0,\hfill \end{array}$$

where $0<q<1$, $0\le \alpha \le 1$, $x(t)\in R^n$, $u(t)\in K$, $K\subset R^m$ is a nonempty closed convex set, $F:[0,T]\times R^n\to E^n$ is a fuzzy mapping, $(B,G):[0,T]\times R^n\to R^{n\times m}\times R^m$, and $Q:R^m\to R^m$ are given mappings, the notation of $〈\cdot ,\cdot 〉$ denotes the standard inner product of vectors in $R^m$.

Some special cases of (1.3) are as follows:

• If $F_{(t,x(t))}={\chi }_{W(t,x(t))}$, where $W:[0,T]\times R^n\to 2^{R^n}$ is a set-valued mapping with nonempty convex compact values, ${\chi }_A$ is the characteristic function of a set A, then (1.3) reduces to a class of fractional set-valued differential variational inequalities investigated by Loi et al. [35].

• If $q=1$, $F_{(t,x(t))}={\chi }_{W(t,x(t))}$, where $W:[0,T]\times R^n\to R^n$ is a single-valued mapping, ${\chi }_A$ is the characteristic function of a set A, then (1.3) reduces to a class of DVIs introduced by Pang and Stewart [61].

• If $q=1$, $F_{(t,x(t))}={\chi }_{W(t,x(t))}=Hx(t)$, $B(t,x(t))=B$, where H and B are $n\times n$ matrix and $n\times m$ matrix, then (1.3) reduces to a class of DVIs considered by Wang et al. [70].

The rest of this paper is organized as follows. Section 2 presents some definitions, notations and useful lemmas. In Section 3, we show the existence of solution for the system (1.3) by using the set-valued version of the Krasnoselskii fixed point theorem due to Dhage [14]. Section 4 introduces a new numerical algorithm for approximating the solution of (1.3). Before summarizing this paper in Section 6, we give two interesting numerical examples in Section 5 to illustrate the validity of our results.