Abstract
This paper addresses the issue of the existence and uniqueness of intuitionistic fuzzy solutions for some classes of partial functional differential equations with state-dependent delay in a new weighted complete metric space. Theorems on the existence and uniqueness of intuitionistic fuzzy solutions for these problems with integral boundary conditions are established under some sufficient assumptions. Some numerical examples of applications of the main result of this work are presented.
1. Introduction
In 1965, Zadeh [1] proposed the concept of fuzzy sets with the purpose to model ambiguity, uncertainty, and vagueness in complicated systems. It can be considered as an extension of the usual (crisp) set theory. It has greater pliability to capture different aspects of incompleteness, imperfection, and uncertainty in data about many situations. The membership of an element of a fuzzy set is a single value between 0 and 1. Therefore, in reality, it may not always be true that the degree of nonmembership of an element in a fuzzy set is equal to 1 minus the membership degree because there may be some uncertainty degree. Thus, since the fuzzy set has no means to characterize the neutral state, neither support nor oppose, Atanassov [2] included the nonmembership function and defined the degree of uncertainty as . He introduced the topic of intuitionistic fuzzy sets (IFSs) as an extension of the standard fuzzy sets [3]. Several applications of IFS theory in diverse fields have been carried out; they are a very necessary and powerful tool in modeling imprecision; IFSs have been flourished in many different fields, an intuitionistic fuzzy approach to artificial intelligence [4, 5], medical diagnosis [6], drug selection [7], along with pattern recognition [8], microelectronic fault analysis [9], and decision-making problems [10, 11].
The concept of intuitionistic fuzzy differential equations has a rich history and is one of the actively developing topics of intuitionistic fuzzy set theory. The first paper on intuitionistic fuzzy differential equations was by Melliani and Chadli [12], and recently, the authors established some intuitionistic fuzzy solutions for such equations. They proved the existence and uniqueness of intuitionistic fuzzy solutions under some assumptions for these intuitionistic fuzzy differential equations using different procedures, see [13, 14]. The topic of intuitionistic fuzzy functional differential equations with delay is very rare: in [15], they first elaborated conditions that can be satisfied by intuitionistic fuzzy functional differential equations with delay and that guarantee the existence and uniqueness of local and global intuitionistic fuzzy solutions for such equations. Numerical algorithms for treatment of IFDEs are still evolving, see [16–20] for applications.
The notion of intuitionistic fuzzy partial differential equations was first studied in 2000 [21], and intuitionistic fuzzy partial differential equations with integral boundary conditions were first introduced in 2019 [22], where they proved existence and uniqueness results and defined a procedure to calculate the intuitionistic fuzzy solutions of such problems. However, according to our knowledge, research on the properties of solutions of intuitionistic fuzzy partial functional differential equations is very limited. The study of intuitionistic fuzzy partial differential equations with state-dependent delay has been initiated recently in 2019 [23], in which they considered the existence and uniqueness of intuitionistic fuzzy solutions for these types of equations with local, nonlocal, and integral boundary conditions, based on some complicated assumptions on the domain. But those hypotheses can be reduced to be milder. In the present paper, our models can be considered as an extension and complementation of the previous results into new weighted metric spaces of intuitionistic fuzzy valued functions. We study state-dependent delay intuitionistic fuzzy partial functional differential equations with integral boundary conditions of the following forms:where is continuous and , , , , and are given functions.
Motivated by this consideration, first, we shall discuss the existence and uniqueness of the intuitionistic fuzzy solutions to those problems by using some appropriate Banach fixed point theorems under suitable conditions. The main contributions of this work are as follows: introducing the integral boundary value problem for intuitionistic fuzzy partial differential equations with the state-dependent delays and defining their solution; further developing theoretical results on the existence and uniqueness of the intuitionistic fuzzy solution; defining the intuitionistic fuzzy solution to these problems with integral boundary conditions through a corresponding parametric problem.
Delay differential equations have been used in many fields for a long time. Researchers have considered state-dependent delay differential equations (SDDEs) for the last 50 years. However, more precisely integral initial boundary value problems with state-dependent delays constitute a very interesting and important class of applied problems; they have been used in modeling scientific phenomena in various applications in different fields such as biology, neural networks, control theory, physics, and medicine, see [24–33], for details. Linking the two aspects introduced, intuitionistic fuzzy mathematics and partial functional differential equations with state-dependent delays, we obtain intuitionistic fuzzy partial functional differential equations (IFPFDEs) with state-dependent delays, which will attract great interest both in mathematics and in applications. No result has been reported on the existing models of state-dependent delay IFPFDEs with integral boundary initial conditions, and among the purposes of this work is to fill this gap. This paper initiates the study of such equations: solving state-dependent delay IFPFDEs with integral boundary initial conditions, providing some definitions for intuitionistic fuzzy solutions of such problems, progressing the main results on the existence and uniqueness of intuitionistic fuzzy solutions under suitable hypothesizes in new weighted metric spaces, without any restraints in the data, displaying rigorous well-posedness and regularity analysis of the state-dependent delay intuitionistic fuzzy PFDEs, and establishing numerical examples that can be considered as a contribution to the subject of our present work and will be introduced elsewhere.
The rest of the paper is organised as follows: in Section 2, we will recall some basic definitions and preliminary facts which will be used throughout the following sections. In Section 3, we provide the existence and uniqueness of intuitionistic fuzzy solutions for partial functional differential equations with integral boundary conditions under suitable conditions in a new weighted metric space. In Section 4, we illustrate our abstract results by numerical applications. Finally, Section 5 gives the main conclusions and possible future works in this area.
2. Preliminaries
We define as follows:
We called as an intuitionistic fuzzy number if it verifies the following assumptions:(a) is normal, i.e., there exists and such that and (b) is fuzzy convex, and is fuzzy concave(c) is upper semicontinuous, and is lower semicontinuous(d) is bounded
We define the upper and lower -cuts of with by the following forms:
We define as
Let and and ; then,
For and and , therefore
Definition 1. Let and ; the following sets are defined by
Remark 1.
Proposition 1 (see [34]). For all and and ,(a)(b) and are nonempty compact convex sets in (c)if , therefore and (d)If , therefore and
Let ; we denote (any set) as follows:
Lemma 1 (see [34]). Let and be two families of subsets of ; verify the assumptions of Proposition 1 if and are defined as follows:
Then, .
We define the following metric bywhere is the usual Euclidean norm in .
Theorem 1 (see [34]). is a metric on .
Theorem 2 (see [34]). is a complete metric space.
We denote(i), , , , and (ii): the space of all continuous mappings defined over into (i = c, d, 0, and r)
For arbitrary, it can be shown that is a complete metric space, where the supremum weighted metric on is defined by
Definition 2. A mapping is continuous at point provided for any arbitrary ; there exists an such thatwhenever .
Definition 3. A mapping is called continuous at point provided for any arbitrary ; there exists an such thatwhenever and , .
Definition 4. A mapping is called strongly measurable if for , the set-valued mappings defined by and defined by are (Lebesgue) measurable.
Definition 5. is integrably bounded if there exists an integrable function such that holds for any
Definition 6. Suppose is integrably bounded and strongly measurable for each ; then,
Let be integrable and . The integral has the elementary properties as follows:(i)(ii)(iii)
Definition 7. Let and ; we call the Hukuhara difference as an intuitionistic fuzzy number ; if it exists, such that
Definition 8. Let be differentiable, and suppose that the derivative is integrable over ; if the Hukuhara difference exists in , then we have .
Definition 9. Let . The intuitionistic fuzzy partial derivative of with respect to at the point is the intuitionistic fuzzy quantity such that sufficiently small, the H-differences and exist in and the limitsThe intuitionistic fuzzy partial derivative of with respect to at the point and higher order of intuitionistic fuzzy partial derivative of are defined similarly.
3. The Main Results
We prove the existence-uniqueness of the intuitionistic fuzzy solution for the following intuitionistic fuzzy functional partial differential equation with integral boundary conditions:
Assume that and is continuous; , , , and are given functions.
For each , we define the state-dependent delays by
In the 2nd part of this section, we will prove the existence-uniqueness of intuitionistic fuzzy solution for the following state-dependent delay intuitionistic fuzzy functional partial differential equation of the general model:where , , , , and are as in problem (19), and .
Definition 10. A function can be a solution of model (19) if verifies the following equation:if where and , if .
Let and . By applying Banach fixed point theorem, we will prove the following results.
Theorem 3. Suppose that(1) is continuous(2)For , , and , we havewhere is a given constant. Moreover, for all satisfying
Therefore, problem (19) has a unique intuitionistic fuzzy solution in with the metric .
Proof. Consider the operator defined as follows:where .
We will prove that is a contraction operator.
Therefore, consider and , and for , we haveWe have the following inequality:Multiplying (27) by , then we getThe same, we haveSimilarly, we getFrom (28)–(31), we obtainIf ,Hence, for each ,So, for all satisfying (31), we have . Then, is a contraction operator.
Therefore, has a unique fixed point, which is the solution of integral boundary problem (21).
Definition 11. A function is called an intuitionistic fuzzy solution of problem (21) if verifies the following integral equation:if where and if .
Let , , and . By applying the Banach fixed point theorem, we will prove the following result.
Theorem 4. Suppose that(1)A map is continuous(2)For , , and , we havewhere is a given constant. Then, for all satisfying
Therefore, problem (21) has a unique intuitionistic fuzzy solution in with the metric .
Proof. To prove the existence-uniqueness result, we will show that has a unique fixed point.
Define operator bywhereWe prove that is a contraction operator. Then, consider and .
We can see that if . On the contrary, on , we haveFrom the properties of supremum metric, we haveSet , , and . We get the following assessment:Multiplying (42) by , we getSimilarly, we obtainFrom (30) and (31), we implyand from (28), we also getFrom (43)–(48), we haveIf ,Hence, for each ,Then, for all verifying (36), it implies ; therefore, is a contraction operator.
Thus, has a unique fixed point, which is the solution of (21).
4. Examples
We propose the following two examples: to illustrate the usefulness of our main results and to find the intuitionistic fuzzy solution, we will use the method of steps proposed in [22, 23].
Example 1. Now, we present a first example to illustrate Theorem 3; consider the following state-dependent delay intuitionistic fuzzy partial functional differential equations for some in the following form:And, the integral boundary initial conditions arewhere is a triangular intuitionistic fuzzy number, and the initial condition isfor .
Sinceit satisfies hypothesises (1) and (2) of Theorem 3.
Indeed, it is easy to see that is continuous and
Therefore, satisfies 2 with an positive number , , and .
Then, from Theorem 3, we see that if we choose positive weighted number satisfying , Hence, all the assumptions of Theorem 3 are satisfied; therefore, we have a unique intuitionistic fuzzy solution of problems (45)–(47).
We try to calculate the intuitionistic fuzzy solution of this problem by using the method of steps proposed in [22, 23].
We write the deterministic solution of the crisp equation as follows: .
Assume that the parametric form of corresponding intuitionistic fuzzy number iswhere it verifies the conditions of Lemma 1.
The function is defined as
The -cuts of are
If we denote
Then,
Therefore, we solve the following state-dependent delay partial functional differential equations for :with initial conditions for :and for , we have
We get
Then,
Now we denote the families and as follows:and the families and by
It is easy to see that and satisfy the conditions of Proposition 1, and by applying Lemma 1, we can build the intuitionistic fuzzy solution for equations (52)–(54) for every as follows:
Then, is an intuitionistic fuzzy solution which satisfies the initial conditions (53)–(54) and is written as follows:
Figures 1 and 2 show the membership and nonmembership functions of triangular intuitionistic fuzzy number and the simulation of -cuts of the intuitionistic fuzzy solution at some values of with .
In Figure 3 we present the surface of the intuitionistic fuzzy solution with triangular intuitionistic fuzzy numbers and .
Example 2. Now, we present a second example to illustrate Theorem 4. Consider the following state-dependent delay intuitionistic fuzzy partial functional differential equations in the general form:And, the integral boundary initial conditions arewhere are triangular intuitionistic fuzzy numbers and , and for , the initial condition is
Sinceit verifies conditions 1 and 2 of Theorem 1.
Indeed, it is easy to see that is continuous:so the assumptions are satisfied with a positive number , , , , and .
If we choose the positive weighted number satisfying , then ; hence, all the hypothesis of Theorem 4 hold; therefore, we have a unique intuitionistic fuzzy solution of equations (72)–(74).
We will use the method of steps proposed in [22, 23] to find an intuitionistic fuzzy solution of equations (72)–(74).
The classical solution of the crisp equation is .
Assume that the parametric forms of corresponding intuitionistic fuzzy number arewhere they verify the conditions of Lemma 1.
The function is defined by
The -cuts of arewhere
Ifthen
Then, we solve the following state-dependent delay partial functional differential equations:where with initial conditions for :and for ,
We get
Then,
Now we denote
It easy to see that and satisfy the conditions of Proposition 1, and by Lemma 1, we can build the intuitionistic fuzzy solution for equations (52)–(54) by the form as follows for every :
Then, is an intuitionistic fuzzy solution which verifies the initial conditions (53)–(54) and can be written as follows:
Figures 4 and 5 represent the membership and nonmembership functions of triangular intuitionistic fuzzy numbers and and the simulation of -cuts of the intuitionistic fuzzy solution at some values of .
In Figure 6, we show the surface of the intuitionistic fuzzy solution with triangular intuitionistic fuzzy numbers and .
5. Conclusion
This research work proposed an integral boundary value problem for an intuitionistic fuzzy partial functional differential equation with state-dependent delay. We have proved the existence-uniqueness of intuitionistic fuzzy solutions for these problems with integral boundary conditions by applying the Banach fixed point theorem. These results are illustrated by numerical examples. Studying local and nonlocal intuitionistic fuzzy delay partial differential equations is the next step that will be considered.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors are responsible for study conception and design, data collection, analysis and interpretation of results, and manuscript preparation.