THIRD ORDER THREE POINT FUZZY BOUNDARY VALUE PROBLEM UNDER GENERALIZED DIFFERENTIABILITY

Abstract

In this article, we investigate third order three-point fuzzy boundary value problem to using a generalized differentiability concept. We present the new concept of solution of third order three-point fuzzy boundary value problem. Some illustrative examples are provided.

1. Introduction

Fuzzy differential equations is a natural way to model dynamical systems under possibility uncertainty. In [14], Puri and Ralescu introduced the concept of H-derivative of a fuzzy number valued function. Bede [4] proved that the fuzzy two-point boundary value problem is not equivalent to the integral equation expressed by Green’s function under Hukuhara differentiability [16] (generalization of the H-derivative) in the fuzzy differential equation and using fuzzy Aumantype integral in the integral equation. Satio [15] gave a new representation of fuzzy numbers with bounded supports and proved that fuzzy number means a bounded continuous curve in the two-dimensional metric space. Under this new structure and certain conditions, Prakash et.al [13] proved a third order threepoint boundary value problem of fuzzy differential equation is equivalent to a corresponding fuzzy integral equation. Bede [5] defined the generalized differentiability of fuzzy number valued functions. Two point boundary value problem under generalized differentiability is considered in [9]. In [12] the existence and uniqueness of solution for a first-order linear fuzzy differential equation with impulses subject to periodic boundary conditions are obtained. Recently an algorithm for the solution of second order fuzzy initial value problems with fuzzy coefficients, fuzzy initial values and fuzzy forcing functions is given in [2]. Analytical and numerical solution of fuzzy initial value problems under generalized differentiability are considered in [1,3]. However it should be emphasized that most of the works in this direction are mainly concerned with fuzzy initial value problem, periodic boundary value problem and two point boundary value problem there has been no attempts made to study third order three-point fuzzy boundary value problem under generalized differentiability.

 

2. Preliminaries

Let us denote by ℝF the class of fuzzy subsets u : ℝ → [0, 1]; satisfying the following properties:

Then ℝF is called the space of fuzzy numbers. For 0 < r ≤ 1, set [u]r = {s ∈ ℝ|u(s) ≥ r} and [u]0 = cl{s ∈ ℝ|u(s) > 0}. Then the r- level set [u]r is a non-empty compact interval for all 0 ≤ r ≤ 1. The following Theorem gives the parametric form of a fuzzy number.

Theorem 2.1 ([7,8]). The necessary and sufficient conditions for to define the parametric form of a fuzzy number are as follows:

We refer as the lower and upper branches on u, respectively. For u ∈ ℝF ; we define the length of u as: A crisp number α is simply represented by (0 ≤ r ≤ 1) is called singleton. For u, v ∈ ℝF and α ∈ ℝ, the sum u+v and the scalar multiplication αu are defined by

For u, v ∈ ℝF, we say u = v if and only if

The metric structure is given by the Hausdorff distance D : ℝF × ℝF → ℝ+∪{0}, by

Definition 2.2. Let x, y ∈ ℝF. If there exists z ∈ ℝF such that x = y + z then z is called the H-difference of x, y and it is denoted x ⊖ y.

In this paper the sign “⊖” stands always for H-difference and x⊖y≠ x+(−1)y in general. Usually we denote x+(-1)y by x-y, while x ⊖ y stands for the Hdifference. In the sequel, we fix I = [a, c], for a, c ∈ ℝ.

Remark 2.1. A function F is said to be a fuzzy number valued function if its range is a space of fuzzy numbers.

Definition 2.3. Let F : I → ℝF be a fuzzy number valued function. If there exists an element F′(t0) ∈ ℝF such that for all h > 0 sufficiently near to 0, F(t0 + h) ⊖ F(t0), F(t0) ⊖ F(t0 - h) exist and the limits (in the metric D)

exist and equal to F′(t0), then F said to be differentiable at t0 ∈ (a, c). If t0 is the end points of I, then we consider the corresponding one-sided derivative. Here the limits are taken in the metric space (ℝF, D).

In this paper we considered the following third order three-point fuzzy boundary value problem

with boundary conditions

where and f : I × ℝF × ℝF × ℝF → ℝF is continuous fuzzy function.

 

3. Generalized fuzzy derivatives

The definition of the Hukuhara differentiability is a straightforward generalization of the Hukuhara differentiability of a set-valued function. Bede and Gel in [5] showed that if F(t) = c.g(t) where c is a fuzzy number and g : [a, b] → R+ is a function with g′(t) < 0, then F is not Hukuhara differentiable. To avoid this difficulty, they introduced a more general definition of derivative for fuzzy function.

Definition 3.1. Let F : I → ℝF and fix t0 ∈ (a, c). If there exists an element F′(t0) ∈ ℝF such that for all h > 0 sufficiently near to 0, F(t0+h)⊖F(t0), F(t0)⊖F(t0−h) exist and the limits (in the metric D)

exist and equal to F′(t0), then F said to be (1)-differentiable at t0 and it is denoted by If for all h > 0 sufficiently near to 0, F(t0) ⊖ F(t0 + h), F(t0 − h) ⊖ F(t0) exist and the limits (in the metric D)

exist and equal to F′(t0), then F is said to be (2)-differentiable and it is denoted by If t0 is the end points of I, then we consider the corresponding one-sided derivative.

Theorem 3.2 ([6,10]). Let F : I → ℝF and let F(t) = (f(t, r), g(t, r)) for each r ∈ [0, 1].

Definition 3.3. Let F : I → ℝF and let n,m ∈ {1, 2}. If exists on a neighborhood of t0 as a fuzzy number valued function and it is (m)-differentiable at t0 as a fuzzy number valued function, then F is said to be (n,m)-differentiable at t0 ∈ I and is denoted by

Theorem 3.4 ([10]). Let F : I → ℝF, and let F(t) = (f(t, r), g(t, r)).

Remark 3.1. For each of these four derivatives, we have again two possibilities.

Definition 3.5. Let F : I → ℝF and let n, m, l ∈ {1, 2}. If exist on a neighborhood of t0 as fuzzy number valued functions and is (l)-differentiable at t0 as a fuzzy number valued function, then F is said to be (n, m, l)-differentiable at t0 ∈ I and it is denoted by

Theorem 3.6 ([10]). Let for n,m ∈ {1, 2}and let F(t) = (f(t, r), g(t, r)).

 

4. Three-point fuzzy boundary value problem

In this section, we consider fuzzy boundary value problem (1)-(2) with generalized differentiability and introduce a new class of solutions.

Definition 4.1. Let y : I → ℝF and let n, m, l ∈ {1, 2}. If and exist on I as fuzzy number valued functions, for all t ∈ I and then y is said to be a (n, m, l) solution for the fuzzy boundary value problem (1)-(2) on I,

Definition 4.2. Let n, m, l ∈ {1, 2} and let I1 be an interval such that I1 ⊂ I. If exist on I1 as fuzzy number valued functions and for all t ∈ I1, then y is said to be a (n, m, l) solution for the fuzzy differential equation (1) on I1.

Definition 4.3. Let ni,mi,li ∈ {1, 2} and i ∈ {1, 2, 3, 4}. If there exists a fuzzy number valued function y : I → ℝF such that

where y1 : [a, b] ∪ {c} → ℝF and y2 : [b, c] ∪ {a} → ℝF are the fuzzy number valued functions with and if there exist t1 ∈ (a, b) and t2 ∈ (b, c) such that y1 is a (n1,m1,l1)-solution and a (n2,m2,l2)-solution of the equation (1) on (a, t1) and on (t1, b) respectively and y2 is a (n3,m3,l3)-solution and a (n4,m4,l4) solution of the equation (1) on (b, t2) and on (t2, c) respectively. Then we say that y is a generalized solution of the fuzzy boundary value problem (1)-(2).

By Theorem 3.2, Theorem 3.4 and Theorem 3.6, we can translate the fuzzy boundary value problem (1)-(2) to a system of ordinary boundary value problems hereafter, called corresponding (n,m,l)-system for problem (1)-(2). Therefore, possible system of ordinary boundary value problems for the problem (1)-(2) are as follows:

(1,1,1)-system:

(1,1,2)-system:

with the boundary condition as in (3).

(1,2,1)-system:

with the boundary condition as in (3).

(1,2,2)-system:

with the boundary condition as in (3).

(2,1,1)-system:

with the boundary condition as in (3).

(2,1,2)-system:

with the boundary condition as in (3).

(2,2,1)-system:

with the boundary condition as in (3).

(2,2,2)-system:

with the boundary condition as in (3).

Our strategy of solving (1)-(2) is based on the selection of derivative type in the fuzzy boundary value problem. We first choose the type of solution and translate problem (1)-(2) to the corresponding system of boundary value problems. Then, we solve the obtained boundary value problems system. Finally we find such a domain in which the solution and its derivatives have valid level sets according to the type of differentiability and using the Representation theorem [11] we can construct the solution of the fuzzy boundary problem (1)-(2).

Remark 4.1. If y is the (n, m, l)-solution of (1) on I1 ⊆ I for m, n, l ∈ {1, 2}, then y is (n, m, l)-differentiable on I1 and y(t) is not (n, m, l)-differentiable in t0 ∈ (I\I1):

 

5. Examples

Example 5.1. Consider the following third order three-point fuzzy boundary value problem:

If y is a (1,1,1)-solution of (4)-(5), then and satisfies the (1,1,1)-system associated with (4). Similarly for other system. On the other hand, by direct calculation, the corresponding solution of the (1,1,1), (1,2,2), (2,1,2), and (2,2,1) systems has necessarily the following expression

By the Representation Theorem [11] and Theorem 2.1, we see represents a valued fuzzy number when t3−3t2+2t ≥ 0. Hence (6) represents fuzzy number for t ∈ [0,1] or t = 2. Now we find the range of (1,1,1),(1,2,2),(2,1,2) and (2,2,1)-solutions of the differential equation (4) separately.

(1, 1, 1)-solution

The (1)-derivative of (6) in that case is given by:

and it is a fuzzy number when Then it is again (1)-differentiable

and it is a fuzzy number when t = 2. By the Definition 3.5, does not exist. Hence y in (6) is not a (1,1,1)-solution of the fuzzy differential equation (4).

(1, 2, 2)-solution

is a fuzzy number when or t = 2. y′′(t) = ((2−r)(t−1), r(t−1)) and y′′′(t) = (r, 2−r) are fuzzy numbers when Hence y(t), are valid fuzzy numbers for and y in (6) is a (1,2,2)-solution of the fuzzy differential equation (4) on

(2, 1, 2)-solution

are fuzzy numbers when Hence are valid fuzzy numbers for and y in (6) is a (2,1,2)-solution of the fuzzy differential equation (4) on

(2, 2, 1)-solution

is a fuzzy numbers when y′′(t) = (r(t − 1), (2 − r)(t − 1)) is a fuzzy number when t = 1. By the Definition 3.5, does not exist. Hence y in (6) is not a (2,2,1)-solution of the fuzzy differential equation (4).

The solution of the remaining four systems (1,1,2), (1,2,1), (2,1,1), and (2,2,2) has the following form

By the Representation Theorem [11] and Theorem 2.1, we see represents a valued fuzzy number t3 − 3t2 + 2t ≤ 0. Hence (7) represents fuzzy real number for t = 0 or t ∈ [1, 2]. Now we find the range of (1,1,2), (1,2,1), (2,1,1), and (2,2,2)-solutions of the differential equation (4) separately.

(1, 1, 2)-solution

is a fuzzy number when y′′(t) = ((2−r)(t−1), r(t−1)) is a fuzzy number when t = 1. By the Definition 3.5, does not exist. Hence y in (7) is not a (1,1,2)-solution of the fuzzy differential equation (4).

(1, 2, 1)-solution

y′′(t) = (r(t − 1), (2 − r)(t − 1)) and y′′′(t) = (r, 2 − r) are fuzzy numbers when Hence are valid fuzzy numbers for and y in (7) is a (1,2,1)-solution of the fuzzy differential equation (4) on

(2, 1, 1)-solution

is a fuzzy number when t = 0 or y′′(t) = (r(t−1), (2−r)(t−1)) and y′′′(t) = (r, 2−r) are fuzzy numbers when Hence are valid fuzzy numbers for and y in (7) is a (2,1,1)-solution of the fuzzy differential equation (4) on

(2, 2, 2)-solution

is a fuzzy number when t = 0 or y′′(t) = ((2 − r)(t − 1), r(t − 1)) is a fuzzy number when t = 0. By the Definition 3.5, does not exist. Hence y in (7) is not a (2,2,2)-solution of the fuzzy differential equation (4).

There exists a fuzzy number valued function y : [0, 2] → ℝF such that

where for all t ∈ [0, 1] ∪ {2} and for all t ∈ [1, 2] ∪ {0} are fuzzy number valued functions and there exist such that y1 is a (1, 2, 2) solution and (2, 1, 2) solution of the equation (4) on respectively, y2 is a (1, 2, 1) solution and (2, 1, 1) solution of the equation (4) on respectively and y1 y2 satisfy the boundary conditions (5). Therefore y in (8) is a generalized solution of the fuzzy boundary value problem (4)-(5). y1 and y2 are shown in Figure 1 and Figure 2 respectively for different values of t. From these figures we see that y1 and y2 are fuzzy number valued functions. In Figure 3 and Figure 4, lower and upper branch of the generalized solution y are shown respectively for different values of r.

FIGURE 1.for different t.

FIGURE 2.for different t ∈ [1, 2].

FIGURE 3.Lower branch of generalized solution different r.

FIGURE 4.Upper branch of generalized solution different r.

Example 5.2. Consider the following third order three-point boundary value problem:

By direct calculation, the corresponding solution of the (1,1,1), (1,2,2), (2,1,2), and (2,2,1) systems has necessarily the following expression

By the Representation Theorem [11] and Theorem 2.1, we see represents a valued fuzzy number 2t3 − 3t2 + t ≥ 0. Hence (11) represents fuzzy number for or t = 1. Now we find the range of (1,1,1),(1,2,2),(2,1,2) and (2,2,1)-solutions of the fuzzy differential equation (9) separately.

(1,1,1)-solution

The (1)-derivative of (9) in that case is given by:

and it a fuzzy number when or t = 1. Then it is again (1)-differentiable

and it is a fuzzy number when t = 1. By the Definition 3.5, does not exist. Hence y in (11) is not a (1,1,1)-solution of the fuzzy differential equation (9).

(1, 2, 2)-solution

is a fuzzy number when and y′′′(t) = (r−1, 1−r) are fuzzy numbers when Hence are valid fuzzy numbers for and y (11) is (1,2,2)-solution of the fuzzy differential equation (9) on

(2, 1, 2)-solution

and y′′′(t) = (r − 1, 1 − r) are fuzzy numbers when Hence are valid fuzzy numbers for and y (11) is (2,1,2)-solution of the fuzzy differential equation (9) on

(2, 2, 1)-solution

is a fuzzy number when is a fuzzy number when By the Definition 3.5, does not exist. Hence y in (11) is not a (2,2,1)-solution of the fuzzy differential equation (11).

The solution of the remaining four systems (1,1,2), (1,2,1), (2,1,1), and (2,2,2) has the following form

By the Representation Theorem [11] and Theorem 2.1, we see represents a valued fuzzy number 2t3 −3t2 + t ≤ 0. Hence (12) represents fuzzy real number for Now we find the range of (1,1,2), (1,2,1), (2,1,1), and (2,2,2)-solutions of the fuzzy boundary value problem separately.

(1, 1, 2)-solution

is a fuzzy number when is a fuzzy number when By the Definition 3.5, does not exist. Hence y in (12) is not a (1,1,2)-solution of the fuzzy differential equation (9).

(1, 2, 1)-solution

and y′′′(t) = (r − 1, 1 − r) are fuzzy numbers when Hence are valid fuzzy numbers for and y (12) is (1,2,1)-solution of the fuzzy differential equation (9) on

(2, 1, 1)-solution

is a fuzzy number when t = 0 or and y′′′(t) = (r−1, 1−r) are fuzzy numbers when Hence are valid fuzzy numbers for and y (12) is (2,1,1)-solution of the fuzzy differential equation (9) on

(2, 2, 2)-solution

is a fuzzy number when t = 0 or is a fuzzy number when t = 0. By the Definition 3.5, does not exist. Hence y in (12) is not a (2,2,2)-solution of the fuzzy differential equation (9).

There exists a fuzzy number valued function y : [0, 1] → ℝF such that

where for all and for all are fuzzy number valued function and there exist such that y1 is a (1, 2, 2)-solution and (2, 1, 2)-solution of the equation (9) on respectively, y2 is a (1, 2, 1)-solution and (2, 1, 1)-solution of the equation (9) on respectively and y1 y2 satisfy the boundary conditions (10). Therefore y in (13) is a generalized solution of the fuzzy boundary value problem (9)-(10). y1 and y2 are shown inFigure 5 and Figure 6 respectively for different values of t. From these figures we see that y1 and y2 are fuzzy number valued functions. In Figure 7 and Figure 8, lower and upper branch of the generalized solution y are shown respectively for different values of r.

FIGURE 5.for different t.

FIGURE 6.for different t.

FIGURE 7.Lower branch of generalized solution for different r.

FIGURE 8.Upper branch of generalized solution for different r.