The geometric lattice intuitionistic fuzzy functions and applications

1.1 Background

In 1983, Atanassov introduced the concept of ”Intuitionistic fuzzy sets - IFSs” [2, 3]. It is a generalization of the fuzzy sets that it could be an important idea when describing a problem with a variable language (fuzzy) and was pretty useful in situations when describing a problem. Because of the flexibility of Intuitionistic fuzzy sets in handling uncertainty, they are a tool for a more human consistent reasoning under the undefined event perfect and vague. [12] used the same terminology "intuitionistic fuzzy set" as Atanassov but different in meaning to build the concept of intuitionistic fuzzy logic and IFSs. In the present time, the IFS theory has been applied to many different fields, for example, in [6] the author discussed intuitionistic fuzzy medical diagnosis consisting of three major steps: symptom identification, formulation of medical knowledge. In [11] the author had discussed an application of intuitionistic fuzzy multiset in medical diagnosis, in [4] the author proposed method of many measurement tools and multi-criteria decisionmaking.

1.2 Approach and objective

The main objective of this study is the intuitionistic fuzzy numbers (IFNs) which is related to the intuitionistic fuzzy sets (IFSs) [2, 3]. In the present approach, we will study them through L-fuzzy sets, which means us describe them as special cases of Goguen’s L-fuzzy sets [8] with elements taking values in a complete lattice L G * = ( L * , ≼ G ) [5], where L * = { x = ( x 1 , x 2 ) ∈ [ 0 , 1 ] 2 | 0 ≤ x 1 + x 2 ≤ 1 } y = ( y 1 , y 2 ) ≼ G x = ( x 1 , x 2 ) ⇔ y 1 ≤ x 1 and y 2 ≥ x 2 .

As we all know, the most real-world problems are studied through differential equations, since the structure of solutions of differential equations can explain many natural phenomena. One specific example is the fuzzy sets were introduced in 1965. However, more than 20 years later, people began to pay attention to its analytic structure, By launched the concept of the addition between two fuzzy sets and the scalar multiplication between a fuzzy set and a non-negative real number, these two operations along with Zadeh’s extended principle have been the basis for the development of semi-linear metric spaces of fuzzy sets, which has led to studies of fuzzy differential equations thrive. In recent years, there have been many studies revolving around the concept of "fuzzy differential equations". As for the intuitionistic fuzzy sets (IFSs), which K.A. Atanassov introduced in his researched since 1983. But until 1999, he began to present the operators on IFSs, so far there have been extensive research articles on these operators. Based on this ideas, Xu and Yager [10] defined the intuitionistic fuzzy numbers which are considered as the basic elements of the intuitionistic fuzzy sets. Most recently, In [1] the authors gave the definition of derivative operations for IFNs and their limited character analysis.

Consequently, from the above viewpoints, we built the basic foundations for IFNs in this paper. Specifically, we built the space for lattice intuitionistic fuzzy numbers (LIFNs) which are the IFNs when this IFNs located in L G * . In addition, a problem that we need a mathematical analysis model to describe how time-varying intuitionistic fuzzy processes and to consider the structure of this model. Therefore, we introduce the concept of geometric lattice intuitionistic fuzzy functions, classify them and examine the derivatives and integrals of them. From the continuous, differentiability and integrable properties of the geometric lattice intuitionistic fuzzy functions, we build the initial value problem of geometric lattice intuitionistic fuzzy differential equation (IVP for GLIFDE). Moreover, to demonstrate the applicability of the results of this paper, we use the tool which is geometric lattice intuitionistic fuzzy differential equation on the L G * space, to produce models for a disease state in a population at a certain time. Whereby, when a disease penetrates into a certain population, there will be two processes occur in this population. The first process is the increase in the number of infected members in the population, the second process is the decrease in the number of non-infected members in the population. To simulate a continuous dynamic system of this two processes in L G * space, we assume that x₁ (t) is process increasing and x₂ (t) is process decreasing over time and x ( t ) = ( x 1 ( t ) , x 2 ( t ) ) ∈ L G * .

1.3 Outline

This paper includes: In section 2, we collect the fundamental notions to use in the next section. In section 3, we proceed to construct two operations: addition operation ( ⊕ G : L G * × L G * → L G * ) and scalar multiplication operation ( ⊙ : ℝ + × L G * → L G * ), such that ( L G * , ⊕ G , ⊙ ) is a semi-linear space. Then we analyze the order in this semi-linear space to form the concept of geometric differences and provide a definition of the differentiable properties of geometric lattice intuitionistic fuzzy functions on L G * . We introduce about the initial value problem of geometric lattice intuitionistic fuzzy differential equation (IVP for GLIFDE) and the medical intuitionistic fuzzy model that describes the developmental process of a disease in a population over time.

3.1 The metric semi-linear space of lattice intuitionistic fuzzy numbers

By Atanassov’s concept every intuitionistic fuzzy set (IFS) (see [2, 3]): A = { ( u , μ A ( u ) , ν A ( u ) ) | u ∈ U , 0 ≤ μ A ( u ) + ν A ( u ) ≤ 1 } , if we put x = (x₁, x₂) = (μ _A (u), ν _A (u)) then we have a rectangle R x = { ( v , w ) | 0 ≤ v ≤ x 1 , 0 ≤ w ≤ x 2 } .

Definition 3.1. We denote L * = { x = ( x 1 , x 2 ) ∈ [ 0 , 1 ] × [ 0 , 1 ] | 0 ≤ x 1 + x 2 ≤ 1 } ,(3.1) and classifying ≼ _G order in L^* for ∀x, y ∈ L^*: y = ( y 1 , y 2 ) ≼ G x = ( x 1 , x 2 ) ⇔ y 1 ≤ x 1 and y 2 ≥ x 2 . We say that L G * = ( L * , ≼ G ) is a complete lattice and every x ∈ L G * is called a lattice intuitionistic fuzzy number (LIFN). A simple description for the Definition 3.1 is provided in Fig. 1.

OPEN IN VIEWER

Fig. 1

A geometrical interpretation of LIFNs y ≼ _G x in L G * .

Remark 3.1. In L G * , each lattice intuitionistic fuzzy number x ∈ L G * corresponds to a fuzzy rectangle R x = { ( v , w ) | 0 ≤ v ≤ x 1 , 0 ≤ w ≤ x 2 } .

Definition 3.2. We denote that:

θ = ( 0 , 0 ) ∈ L G * is called the zero element of L G * ;

Remark 3.2. As we all know, before the concept of the intuitionistic fuzzy number appears, in mathematics, there existed a concept which is "interval". Based on the definition of the closed interval, if [ a , b ] ⊂ ℝ , we always have a ≤ b and there is no reverse case. For example, we have [0, 2] while [2, 0] does not satisfy the definition of an interval. Similarly, we have the concept of an open-ended interval. In the general case, if ( a , b ) ⊂ ℝ (or using symbol] a,  b [to represent an open-ended interval), we always have a < b and there is no reverse case. For example, we have (0, 2) while (2, 0) does not satisfy the definition of an interval. However, with a intuitionistic fuzzy number x = (x₁, x₂), then x₁ may be larger than x₂ or vice versa. For example, we have the IFN x = (x₁, x₂) = (0.2, 0.7) and the IFN y = (x₂, x₁) = (0.7, 0.2). Nevertheless, in some special cases, the IFNs are mathematically equivalent to the intervals, for example, we have a closed interval [0,  0.8] (or an open-ended interval (0,  0.8)) and we also have a IFN (0,  0.8) . Therefore, we need to distinguish that the difference of arithmetic operations of these two objects. Specifically, we have some arithmetic operations for intervals by endpoint formulas the following (can see [9]): [a, b] + [c, d] = [a + b, c + d] (addition), [a, b] - [c, d] = [a - d, b - c] (subtraction), and for all k ∈ ℝ we have k . [a, b] = [ka,  kb] if k ≥ 0 or k . [a, b] = [kb,  ka] if k < 0 (scalar multiplication). However, These three operators are not suitable for IFNs, for example, giving two IFNs x = (0.3,  0.6) and y = (0.8,  0.1) then x + y = (1.1, 0.7) ∉ L^*,  x - y = (-0.5,  0.5) ∉ L^*, and k . x = (k . 0.6,  k . 0.3) ∉ L^* if k < 0 . Therefore, for the IFNs, we need to rebuild the appropriate arithmetic operations on L^* .

Remark 3.3. We can give some kinds of the binary addition operators in L G * , for example:

the coordinate addition operation of two elements of x , y ∈ L G * , that denote by z = x+ y = (x₁ + y₁,  x₂ + y₂);

In family of the intuitionistic fuzzy numbers L G * all above kinds of the binary addition operators will only satisfy the commutative property, but without satisfying the distributed property. To have a binary addition operation with both properties, we will extend the average addition operation ⊕ of two elements (without losing the properties of the double addition) by the other concept of close addition operation ⊕ _G in L G * as follows:

Definition 3.3. Let m elements x ( m ) = ( x 1 ( m ) , x 2 ( m ) ) ∈ L G * , we say that there exists an geometric addition operation of this m elements as follows: z = ( x ( 1 ) ⊕ G x ( 2 ) ⊕ G . . . . ⊕ G x ( m ) ) ∈ L G * iff z = ( x 1 ( 1 ) + . . . + x 1 ( m ) 2 m - p - 1 , x 2 ( 1 ) + . . . + x 2 ( m ) 2 m - p - 1 ) , with p is number of component x^(j) = (0, 0) = θ,  j = 1, 2, …, m and satisfy i / ( x ( 1 ) ⊕ G x ( 2 ) ⊕ G . . . ⊕ G x ( n ) ) ⊕ G x ( n + 1 ) ⊕ G . . . . ⊕ G x ( m ) = ( x 1 ( 1 ) + . . . + x 1 ( m ) 2 m - p - 1 , x 2 ( 1 ) + . . . + x 2 ( m ) 2 m - p - 1 ) ; ii / x ( 1 ) ⊕ G x ( 2 ) ⊕ G . . . ⊕ G ( x ( n ) ⊕ G x ( n + 1 ) . . . . ⊕ G x ( m ) ) = ( x 1 ( 1 ) + . . . + x 1 ( m ) 2 m - p - 1 , x 2 ( 1 ) + . . . + x 2 ( m ) 2 m - p - 1 ) , where n ≤ m .

Definition 3.4. We say that there exists an geometric binary addition operation of two elements of L G * as follows:

x ⊕ G y = ( x 1 + y 1 2 m - p - 1 , x 2 + y 2 2 m - p - 1 ) , if both x and y are different from zero elements in L G * (that means x ≠ θ = (0, 0) and y ≠ θ = (0, 0)), that means m = 2,  p = 0.

The properties of this geometric binary addition operation in Definition 3.4 will be in the following Theorem 3.2. To easily visualize the geometric binary addition operation in Definition 3.4, we will provide some simple examples.

Example 3.1. Now, we consider the geometric binary addition operation with two elements in L G * , for x = ( x 1 , x 2 ) = ( 0.2 , 0.7 ) ∈ L G * and y = ( y 1 , y 2 ) = ( 0.6 , 0.2 ) ∈ L G * , from the Definition 3.4, then we have x ⊕ G y = ( x 1 + y 1 2 m - p - 1 , x 2 + y 2 2 m - p - 1 ) = ( 0.2 + 0.6 2 2 - 0 - 1 , 0.7 + 0.2 2 2 - 0 - 1 ) = ( 0.4 , 0.45 ) = ( z 1 , z 2 ) = z , It is clear that z belongs to L G * because z satisfies 0 ≤ 0.4 + 0.45 ≤ 1. The illustration for the case of the geometric binary addition operation with two elements in L G * is represented in Fig. 2. But assuming that x = θ = ( 0 , 0 ) ∈ L G * and y = ( y 1 , y 2 ) = ( 0.6 , 0.2 ) ∈ L G * then p = 1 and m = 2, so we have x ⊕ G y = ( x 1 + y 1 2 m - p - 1 , x 2 + y 2 2 m - p - 1 ) = ( 0 + 0.6 2 2 - 1 - 1 , 0 + 0.2 2 2 - 1 - 1 ) = ( 0.6 , 0.2 ) = ( z 1 , z 2 ) = z = y .

The next, we consider the geometric binary addition operation with three elements in L G * , for three elements x = (x₁, x₂) = (0.1, 0.9),  y = (y₁, y₂) = (0.8, 0.2) and t = (t₁, t₂) = (0.2, 0.7) clearly all these three elements belong to L G * . All three elements x,  y,  t are different from θ = (0, 0) implies that p = 0 and because the geometric binary addition operation has three elements, we have m = 3. Thus x ⊕ G y ⊕ G t = ( x 1 + y 1 + t 1 2 3 - 0 - 1 , x 2 + y 2 + t 2 2 3 - 0 - 1 ) = ( 0.1 + 0.8 + 0.2 2 2 , 0.9 + 0.2 + 0.7 2 2 ) = ( 0.275 , 0.45 ) = ( z 1 , z 2 ) = z ∈ L G * .

An important point in Definition 3.3 are two conditions i/ and ii/, we will explain in detail why these two conditions must have. Specifically, when we examine the properties in Definition 2.3, then the property (x ⊕ _G y) ⊕ _G z = x ⊕ _G (y ⊕ _G z) is not satisfied, because (x ⊕ _G y) ⊕ _G z ≠ x ⊕ _G (y ⊕ _G z). For example, let x = ( 0.3 , 0.7 ) , y = ( 0.9 , 0.1 ) , z = ( 0.2 , 0.6 ) ∈ L G * then (x ⊕ _G y) ⊕ _G z = ( 0.3 + 0.9 2 2 - 0 - 1 + 0.2 2 2 - 0 - 1 , 0.7 + 0.1 2 2 - 0 - 1 + 0.6 2 2 - 0 - 1 ) = (0.4, 0.5) = v ∈ L G * and x ⊕ _G (y ⊕ _G z) = ( 0.3 + 0.9 + 0.2 2 2 - 0 - 1 2 2 - 0 - 1 , 0.7 + 0.1 + 0.6 2 2 - 0 - 1 2 2 - 0 - 1 ) = (0.425, 0.525) = w ∈ L G * . We see v ≠ w. Therefore, we add to the Definition 3.3 with two conditions i and ii, in order to property (x ⊕ _G y) ⊕ _G z = x ⊕ _G (y ⊕ _G z) can be satisfied. From the example above, by using two conditions i/,  ii/ in the Definition 3.3 with m = 3 and n = 2, we have ( x ⊕ G y ) ⊕ G z = ( 0.3 + 0.9 + 0.2 2 3 - 0 - 1 , 0.7 + 0.1 + 0.6 2 3 - 0 - 1 ) = x ⊕ G ( y ⊕ G z ) .

Corollary 3.1. For three elements x , y , z ∈ L G * and x,  y,  z ≠ θ then the extended geometric addition operation in Definition 3.3 satisfies:

( x ⊕ G y ) ⊕ G z = ( x 1 + y 1 + z 1 4 , x 2 + y 2 + z 2 4 ) = x ⊕ G y ⊕ G z ;

OPEN IN VIEWER

Fig. 2

Geometric binary addition operation of the LIFNs z = x ⊕ G y ∈ L G * .

Proof. (x ⊕ _G y) ⊕ _G z = x ⊕ _G (y ⊕ _G z) ? Particular, using the extended geometric addition operation by Definition 3.3 in the case m = 3,  n = 2,  p = 0 then we have: ( x ⊕ G y ) ⊕ G z = ( x 1 + y 1 + z 1 4 , x 2 + y 2 + z 2 4 ) , x ⊕ G ( y ⊕ G z ) = ( x 1 + y 1 + z 1 4 , x 2 + y 2 + z 2 4 ) .

Theorem 3.2. The geometric binary addition operation in Definition 3.4 with the extended geometric addition operation in Definition 3.3 of the lattice intuitionistic fuzzy numbers satisfies:

x⊕ _G y = y ⊕ _G x;

Proof. We prove this properties of geometric binary addition operation of the lattice intuitionistic fuzzy numbers by Definition 3.4: x ⊕ _G y = y ⊕ _G x ? Particular, because x _i ,  y _i ∈ [0, 1],  i = 1, 2,  m = 2,  p = 0 then we have: x ⊕ G y = ( x 1 + y 1 2 , x 2 + y 2 2 ) = ( y 1 + x 1 2 , y 2 + x 2 2 ) = y ⊕ G x . 2/ (x ⊕ _G y) ⊕ _G z = x ⊕ _G (y ⊕ _G z) ? Particular, because 0 ≤ x₁ + x₂ ≤ 1 and θ = ( 0 , 0 ) ∈ L G * , using the extended geometric addition operation of the lattice intuitionistic fuzzy numbers by Definition 3.3 and Corollary 3.1 the case m = 3,  n = 2, p = 0 . x ⊕ G θ = x ∈ L G * ? Particular, because 0 ≤ x₁ + x₂ ≤ 1 and θ = ( 0 , 0 ) ∈ L G * , using the geometric binary addition operation of the lattice intuitionistic fuzzy numbers by Definition 3.4 in the case m = 2,  p = 1 then we have: x ⊕ G θ = ( x 1 + 0 2 2 - 1 - 1 , x 2 + 0 2 2 - 1 - 1 ) = ( x 1 2 0 , x 2 2 0 ) = ( x 1 , x 2 ) = x . □

Remark 3.4.By the Theorem 3.2 the new closed binary addition operation ⊕ _G in L G * satisfies that is a binary addition of two elements of x , y ∈ L G * . This mathematical operation ⊕ _G in L G * is a commutative and distributive extended binary addition operator.

Definition 3.5. Let x = ( x 1 , x 2 ) ∈ L G * , λ ∈ ℝ + , we say that there exists a nonegative scalar multiplication operation λ ⊙ x = ( λ . x 1 , λ . x 2 ) ∈ L G * if 0 ≤ λ . x₁ + λ . x₂ ≤ 1, and denote y = λ ⊙ x ∈ L G * .

Theorem 3.3. Assuming that the scalar multiplication operation in Definition 3.5 is exists then it’s satisfies:

λ⊙ (x ⊕ _G y) = λ ⊙ x ⊕ _G λ ⊙ y;

Proof. We have to prove this properties of the scalar multiplication operation in Definition 3.5:

1/ λ ⊙ (x ⊕ _G y) = λ ⊙ x ⊕ _G λ ⊙ y ? Particular, because x _i ,  y _i ∈ [0, 1],  i = 1, 2,  m = 2,  p = 0 then we have: λ ⊙ ( x ⊕ G y ) = ( λ . x 1 + y 1 2 , λ . x 2 + y 2 2 ) = ( λ . x 1 + λ . y 1 2 , λ . x 2 + λ . y 2 2 ) = ( λ ⊙ x ) ⊕ G ( λ ⊙ y ) . 2/ (λ + β) ⊙ x = (λ ⊙ x) + (β ⊙ x) ? Particular, because x i , y i ∈ [ 0 , 1 ] , i = 1 , 2 , ( λ + β ) ∈ ℝ + then we have: ( λ + β ) ⊙ x = ( ( λ + β ) x 1 , ( λ + β ) x 2 ) = ( λ x 1 + β x 1 , λ x 2 + β x 2 ) = λ ⊙ x + β ⊙ x . 3/ (λ . β) ⊙ x = λ ⊙ (β ⊙ x) ? Particular, because x₁, x₂ ∈ [0, 1] and θ = ( 0 , 0 ) ∈ L G * then we have: ( λ . β ) ⊙ x = ( ( λ . β ) x 1 , ( λ . β ) x 2 ) = ( λ . ( β . x 1 ) , λ . ( β . x 2 ) ) = λ ⊙ ( β ⊙ x ) . 4/ x ⊙ 1 = x ∈ L G * ? Particular, because x _i ∈ [0, 1], i = 1, 2 then we have: x ⊙ 1 = ( x 1 . 1 , x 2 . 1 ) = ( x 1 , x 2 ) = x . 5/ x ⊙ 0 = θ ∈ L G * ? Particular, because x _i ∈ [0, 1], i = 1, 2 then we have: x ⊙ 0 = ( x 1 . 0 , x 2 . 0 ) = ( 0 , 0 ) = θ ∈ L G * .

Definition 3.6. The geometric distance between intuitionistic fuzzy numbers x = ( x 1 , x 2 ) , y = ( y 1 , y 2 ) ∈ L G * denote by H L G * ( x , y ) = sup { | x 1 - y 1 | , | x 2 - y 2 | } .

Theorem 3.4. The geometric binary addition operation in Definition 3.4, the scalar multiplication operation in Definition 3.5 of the lattice intuitionistic fuzzy number on L G * and the geometric distance between two lattice intuitionistic fuzzy numbers in Definition 3.6 make that L G * become to the metric semi-linear space of the lattice intuitionistic fuzzy numbers (LIFNs).

Proof. From Definition 2.3, using Theorems 3.2 – 3.3, we have L G * is a semi-linear space and Definition 3.6 we get the metric semi-linear space of the lattice intuitionistic fuzzy numbers.□

Definition 3.7. Let x : [ t 0 , T ] → L G * , with x ( t ) = ( x 1 ( t ) , x 2 ( t ) ) ∈ L G * and t, t + h ∈ [t₀, T],   ∀ h > 0 . We say that

the geometric lattice intuitionistic fuzzy function x (t) is strict monotone increasing by t iff x (t) ≼ _G1x (t + h) in L G * with x₁ (t)≤ x₁ (t + h),  x₂ (t + h) ≤ x₂ (t),   ∀ t ∈ [t₀, T];

Definition 3.8. Let x : [ t 0 , T ] → L G * , with x ( t ) = ( x 1 ( t ) , x 2 ( t ) ) ∈ L G * and t,  t + h ∈ [t₀, T],   ∀ h > 0 . We say that

the geometric lattice intuitionistic fuzzy function x (t) is monotone decreasing by t iff x (t + h) ≼ _G2x (t) in L G * , with x₁ (t+ h) ≤ x₁ (t),  x₂ (t) ≤ x₂ (t + h),   ∀ t ∈ [t₀, T];

If exist x ( t ) ⊖ G 1 y ( t ) = z ( t ) or x ( t ) ⊖ G 2 y ( t ) = z ( t ) , then we say that exists x ( t ) ⊖ G y ( t ) and is called geometric difference (with symbol ⊖ G ) .

Theorem 3.5. Assume that x ( t ) , y ( t ) , z ( t ) ∈ L G * , with geometric difference ⊖ G , then there are exist the following properties:

If x (t) ⊖ G 1 y (t), x (t) ⊖ G 1 z (t) exist, then H L G * [x (t) ⊖ G 1 y (t), x (t) ⊖ G 1 z (t)] = H L G * [y (t), z (t)];

The same result when replacing ⊖ G 1 by ⊖ G 2 .

Proof. (a/) Suppose that x ( t ) ⊖ G 1 y ( t ) , x ( t ) ⊖ G 1 z ( t ) exist, from Definition 3.6, we have x ( t ) ⊖ G 1 y ( t ) = ( x 1 ( t ) - y 1 ( t ) , y 2 ( t ) - x 2 ( t ) ) , x ( t ) ⊖ G 1 z ( t ) = ( x 1 ( t ) - z 1 ( t ) , z 2 ( t ) - x 2 ( t ) ) and H L G * [ y ( t ) , z ( t ) ] = sup { | y 1 - z 1 | , | y 2 - z 2 | } , so H L G * [ x ( t ) ⊖ G 1 y ( t ) , x ( t ) ⊖ G 1 z ( t ) ] = sup { | z 1 - y 1 | , | y 2 - z 2 | } = sup { | y 1 - z 1 | , | y 2 - z 2 | } =

H L G * [ y ( t ) , z ( t ) ] . (b/) Suppose that x (t) ⊖ G 1 y (t), x (t) ⊖ G 1 z (t), z (t) ⊖ G 1 y ( t ) exist, this implies that x (t) ⊖ G 1 y (t) ∈ L G * , x (t) ⊖ G 1 z (t) ∈ L^* and z (t) ⊖ G 1 y (t) ∈ L G * . Thus [ x ( t ) ⊖ G 1 y ( t ) ) ] ⊖ G 1 [x (t) ⊖ G 1 z (t)] ∈ L G * and

[ x ( t ) ⊖ G 1 y ( t ) ] ⊖ G 1 [ x ( t ) ⊖ G 1 z ( t ) ] = ( x 1 ( t ) - y 1 ( t ) , y 2 ( t ) - x 2 ( t ) ) ⊖ G ( x 1 ( t ) - z 1 ( t ) , z 2 ( t ) - x 2 ( t ) ) = ( x 1 ( t ) - y 1 ( t ) - x 1 ( t ) + z 1 ( t ) , y 2 ( t ) - x 2 ( t ) - z 2 ( t ) + x 2 ( t ) ) = ( z 1 ( t ) - y 1 ( t ) , y 2 ( t ) - z 2 ( t ) ) = z ( t ) ⊖ G 1 y ( t ) .

□

OPEN IN VIEWER

Fig. 3

Illustration of the real functions x₁ (t),  x₂ (t) belong to C (R, [0, 1]), with x₁ (t) + x₂ (t) =1 .

In order to have an intuitive view of the continuous intuitionistic fuzzy functions in L G * , we recognize that each of the functions x ( t ) ∈ L G * is represented by the form x (t) = (x₁ (t), x₂ (t)), where x _i (t),  i = 1, 2 are the continuous functions on the [0, 1] . So we have a relationship between the continuous real functions x i ∈ C ( ℝ , [ 0 , 1 ] ) , i = 1 , 2 and the continuous geometric lattice intuitionistic fuzzy functions in the metric semi-linear geometric lattice space L G * of LIFNs, we can assume the functions x _i (t) with the graph shown in Fig. 3, at each t _k ,  k = 1, 2 we have the point M _k = (x₁ (t _k ), x₂ (t _k )). The Fig. 4 below shown geometrical interpretation of geometric lattice intuitionistic fuzzy functions in L G * .

OPEN IN VIEWER

Fig. 4

Illustration of the geometric lattice intuitionistic fuzzy functions x (t) = (x₁ (t), x₂ (t)) belong to L G * in the case x₁ (t) + x₂ (t) =1 .

Definition 3.9. Let x : [ t 0 , T ] → L G * . We say that a geometric lattice intuitionistic fuzzy function x (t) is geometric differentiable at t ∈ (t₀, T), if there are exist the geometric differences x ( t + h ) ⊖ G 1 x ( t ) , x ( t ) ⊖ G 1 x ( t - h ) and D G x ( t ) ∈ L G * , such that lim h → 0 + x ( t + h ) ⊖ G 1 x ( t ) h = lim h → 0 + x ( t ) ⊖ G 1 x ( t - h ) h = D G x ( t ) ; or if there exists geometric differences x ( t + h ) ⊖ G 2 x ( t ) , x ( t ) ⊖ G 2 x ( t - h ) and D G x ( t ) ∈ L G * , such that lim h → 0 + x ( t + h ) ⊖ G 2 x ( t ) h = lim h → 0 + x ( t ) ⊖ G 2 x ( t - h ) h = D G x ( t ) , where denote D _G x (t) is geometric derivatives in L G * .

Theorem 3.6. The geometric derivative of the geometric lattice intuitionistic fuzzy function x ( t ) ∈ L G * will have the following form:

D G x ( t ) = ( dx 1 dt , - dx 2 dt ) if the geometric lattice intuitionistic fuzzy function x (t) is strict monotone increasing by t;

Proof. a/ D G x ( t ) = ( dx 1 dt , - dx 2 dt ) . Because the geometric lattice intuitionistic fuzzy function x (t) is strict monotone increasing by t then exists x ( t + h ) ⊖ G 1 x ( t ) ∈ L G * and implies that x (t) ≼ _G1x (t + h) or x₁ (t) ≤ x₁ (t + h), x₂ (t + h) ≤ x₂ (t). By Definition 3.7, we have x ( t + h ) ⊖ G 1 x ( t ) = ( x 1 ( t + h ) - x 1 ( t ) , x 2 ( t ) - x 2 ( t + h ) ) and

lim h → 0 + x ( t + h ) ⊖ G 1 x ( t ) h = ( lim h → 0 + x 1 ( t + h ) - x 1 ( t ) h , lim h → 0 + x 2 ( t ) - x 2 ( t + h ) h ) = ( lim h → 0 + x 1 ( t + h ) - x 1 ( t ) h , - lim h → 0 + x 2 ( t + h ) - x 2 ( t ) h ) = ( dx 1 dt , - dx 2 dt ) = D G x ( t ) . b/ D G x ( t ) = ( - dx 1 dt , dx 2 dt ) . Because the geometric lattice intuitionistic fuzzy function x (t) is monotone decreasing by t then exists x ( t + h ) ⊖ G 2 x ( t ) ∈ L G * and implies that x (t) ≼ _G2x (t + h) or x₁ (t + h) ≤ x₁ (t), x₂ (t) ≤ x₂ (t + h) . By Definition 3.8, we have x ( t + h ) ⊖ G 2 x ( t ) = ( x 1 ( t ) - x 1 ( t + h ) , x 2 ( t + h ) - x 2 ( t ) ) and lim h → 0 + x ( t + h ) ⊖ G 2 x ( t ) h = ( lim h → 0 + x 1 ( t ) - x 1 ( t + h ) h , lim h → 0 + x 2 ( t + h ) - x 2 ( t ) h ) = ( - lim h → 0 + x 1 ( t + h ) - x 1 ( t ) h , lim h → 0 + x 2 ( t + h ) - x 2 ( t ) h ) = ( - dx 1 dt , dx 2 dt ) = D G x ( t ) . □

For every x ( t ) = ( x 1 ( t ) , x 2 ( t ) ) ∈ L G * , we have the initial value problem for geometric lattice intuitionistic fuzzy differential equation (IVP for GLIFDE) under form: { D G x ( t ) = f ( t , x ( t ) ) , t ∈ [ t 0 , T ] , x ( t 0 ) = ( x 1 ( t 0 ) , x 2 ( t 0 ) ) = ( a 1 , a 2 ) ∈ L G * ,(3.2) where f : [ t 0 , T ] × L G * → L G * that means f ( t , x ( t ) ) ∈ L G * and D _G x (t) is geometric derivative of x (t) . A special point of the problem (3.2) is that it can be rewritten into a system of real equations. Because x ( t ) , f ( t , x ( t ) ) ∈ L G * so x ( t ) = ( x 1 ( t ) , x 2 ( t ) ) , f ( t , x ( t ) ) = ( f 1 ( t , x 1 ( t ) , x 2 ( t ) ) , f 2 ( t , x 1 ( t ) , x 2 ( t ) ) ) , with x₁ (t),  x₂ (t),  f₁ (.),  f₂ (.),  x₁ (t₀),  x₂ (t₀) ∈ [0, 1] and 0 ≤ x 1 ( t ) + x 2 ( t ) ≤ 1 , 0 ≤ f 1 ( . ) + f 2 ( . ) ≤ 1 , 0 ≤ x 1 ( t 0 ) + x 2 ( t 0 ) ≤ 1 . Assume that the geometric derivative of geometric lattice intuitionistic fuzzy function x (t) is D _G x (t) exists then the problem (3.2) can be written as follows: In the case, when the geometric lattice intuitionistic fuzzy function x (t) is strict monotone increasing by t . Then the problem 3.2 be written as: { d x 1 ( t ) dt = f 1 ( t , x 1 ( t ) , x 2 ( t ) ) , - d x 2 ( t ) dt = f 2 ( t , x 1 ( t ) , x 2 ( t ) ) , x 1 ( t 0 ) = a 1 , x 2 ( t 0 ) = a 2 , 0 ≤ a 1 + a 2 ≤ 1 t ∈ [ t 0 , T ] . In the case, when the geometric lattice intuitionistic fuzzy function x (t) is monotone decreasing by t . Then the problem 3.2 be written as: { - dx 1 ( t ) dt = f 1 ( t , x 1 ( t ) , x 2 ( t ) ) , dx 2 ( t ) dt = f 2 ( t , x 1 ( t ) , x 2 ( t ) ) , x 1 ( t 0 ) = a 1 , x 2 ( t 0 ) = a 2 , 0 ≤ a 1 + a 2 ≤ 1 t ∈ [ t 0 , T ] .

Example 3.7. Let consider the initial value problem of geometric lattice intuitionistic fuzzy differential equation (IVP for GLIFDE) under the form: { D G x ( t ) = ( 1 - x 1 ( t ) , exp ( - t ) ) , t ∈ [ 0 , 4 ] , x ( 0 ) = ( x 1 ( 0 ) , x 2 ( 0 ) ) = ( 0 , 1 ) ∈ L G * ,(3.3) with x (t) is a strict monotone increasing function by t .

We find the solutions of (IVP for GLIFDE) (3.3). Because x (t) is a strict monotone increasing function, thus (IVP for GLIFDE) (3.3) be written as follows: { dx 1 ( t ) dt = 1 - x 1 ( t ) , - dx 2 ( t ) dt = exp ( - t ) , x 1 ( t 0 ) = 0 , x 2 ( t 0 ) = 1 .(3.4) Solving (3.4), we have one of the solutions of (IVP for GLIFDE) (3.3) that is

( x 1 ( t ) , x 2 ( t ) ) = ( 1 - exp ( - t ) , exp ( - t ) ) , t ∈ [ 0 , 4 ] .

This solution is shown in Figs. 5 and 6.

OPEN IN VIEWER

Fig. 5

Illustration of the real functions x₁ (t), x₂ (t) belong to C ([0, 4], [0, 1]) .

OPEN IN VIEWER

Fig. 6

Solution of (IVP for GLIFDE) (3.3) in L G * for Example 3.7.

Example 3.8. Let consider the initial value problem of geometric lattice intuitionistic fuzzy differential equation (IVP for GLIFDE) under the form: { D G x ( t ) = ( sin ( t ) , 1 - x 1 ( t ) ) , t ∈ [ 0 , π 2 ] , x ( 0 ) = ( x 1 ( 0 ) , x 2 ( 0 ) ) = ( 1 , 0 ) ∈ L G * ,(3.5) with x (t) is a monotone decreasing function by t .

We find the solutions of (IVP for GLIFDE) (3.5). Because x (t) is a monotone decreasing function, thus (IVP for GLIFDE) (3.5) be written as follows: { - dx 1 ( t ) dt = sin ( t ) , dx 2 ( t ) dt = 1 - x 1 ( t ) , x 1 ( t 0 ) = 1 , x 2 ( t 0 ) = 0 .(3.6)

Solving (3.6), we have one of the solutions of (IVP for GLIFDE) (3.5) that is ( x 1 ( t ) , x 2 ( t ) ) = ( cos ( t ) , t - sin ( t ) ) , t ∈ [ 0 , π 2 ] , and it’s graph illustrates in Figs. 7 and 8.

OPEN IN VIEWER

Fig. 7

Illustration of the real functions x₁ (t), x₂ (t) belong to C ( [ 0 , π 2 ] , [ 0 , 1 ] ) .

OPEN IN VIEWER

Fig. 8

Solution of (IVP for GLIFDE) (3.5) in L G * for Example 3.8.

In this paper, we use the tool that is geometric lattice intuitionistic fuzzy differential equation on L G * = ( L * , ≼ G ) space, to produce models for a disease state in a population at a certain time. According to our observation, when a disease penetrates into a certain population, there will be two processes occur in this population. The first process is the increase in the number of infected members in the population, the second process is the decrease in the number of non-infected members in the population. To simulate a continuous dynamic system of this two processes in L G * = ( L * , ≼ G ) space, we assume that x₁ (t) is process increasing and x₂ (t) is process decreasing over time and x (t) = (x₁ (t), x₂ (t)) ∈ (L^*, ≼ _G ). We introduce a model describing the developmental process of a disease in a population over time in the following form: { D G x ( t ) = u ( t ) ⊙ f ( t , x ( t ) ) , t ∈ [ t 0 , T ] , x ( t 0 ) = ( 0 , 1 ) ∈ L G * ,(3.7) where x (t) = (x₁ (t), x₂ (t)), f (t, x (t)) = (f₁ (.), f₂ (.)) ∈ L G * , “⊙” is the symbol of the scalar multiplication operation in the definition 3.5, D _G x (t) is geometric derivative of x (t) and u (t) ∈ [0, 1] is control real function (pills and other preventions). In addition, x₁ (t) =1 - x₂ (t) and at the time of the original x₁ (t₀) =0,  x₂ (t₀) =1, this implies that at the initial time the number of infected members is 0% and the number of non-infected members is 100% in the population. In fact, when a disease invades a population, it will encounter some obstacles, for example, vaccines, pills and other preventions, thus we add to the u (t) control real function to the above model, to express the effect of drugs on this disease. Because x₁ (t) is increases process and x₂ (t) is process decreasing over time, thus x ( t ) ∈ L G * is strict monotone increasing function in L G * and (3.7) will be the dynamic system of the form: { dx 1 ( t ) dt = u ( t ) . ( 1 - x 1 ( t ) ) , - dx 2 ( t ) dt = u ( t ) . x 2 ( t ) , x 1 ( t 0 ) = 0 , x 2 ( t 0 ) = 1 t ∈ [ t 0 , T ] .(3.8) Solving of system (3.8), we have solutions x₁ (t),  x₂ (t) are real positive processes in [0, 1] with condition 0 ≤ x₁ (t) + x₂ (t) ≤1, thus x ( t ) ∈ L G * is positive solutions of system (3.8). Specifically, we have

x 1 ( t ) = 1 - 1 exp ( ∫ t 0 t u ( s ) ds ) , x 2 ( t ) = exp ( - ∫ t 0 t u ( s ) ds ) and x ( t ) = ( 1 - 1 exp ( ∫ t 0 t u ( s ) ds ) , exp ( - ∫ t 0 t u ( s ) ds ) ) .

Example 3.9. Let consider one model describing the developmental process of a disease in a population over time with x₁ (t) is increases and x₂ (t) is decreases then the model that describes the developmental process of a disease in a population for relationship between x₁ (t),  x₂ (t) ∈ [0, 1] and the control real functions u (t) (pills and other preventions) will be:

{ D G x ( t ) = u ( t ) ⊙ ( 1 - x 1 ( t ) , x 2 ( t ) ) , t ∈ [ 0 , π ] x ( 0 ) = ( x 1 ( 0 ) , x 2 ( 0 ) ) = ( 0 , 1 ) ∈ L G * .(3.9) Because x₁ (t) is process increasing and x₂ (t) is process decreasing over time, thus x ( t ) = ( x 1 ( t ) , x 2 ( t ) ) ∈ L G * is strict monotone increasing function and (3.9) be written as under type: { d x 1 ( t ) dt = u ( t ) ( 1 - x 1 ( t ) ) , - d x 2 ( t ) dt = u ( t ) x 2 ( t ) , x 1 ( 0 ) = 0 , x 2 ( 0 ) ) = 1 , t ∈ [ 0 , π ] .(3.10) Assume that u (t) = sin(t).