In this article, we investigate third order threepoint fuzzy boundary value problem to using a generalized differentiability concept. We present the new concept of solution of third order threepoint fuzzy boundary value problem. Some illustrative examples are provided.
AMS Mathematics Subject Classification : 47H10, 54H25.
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 Hderivative of a fuzzy number valued function. Bede
[4]
proved that the fuzzy twopoint boundary value problem is not equivalent to the integral equation expressed by Green’s function under Hukuhara differentiability
[16]
(generalization of the Hderivative) 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 twodimensional 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 firstorder 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 threepoint fuzzy boundary value problem under generalized differentiability.
2. Preliminaries
Let us denote by ℝ
_{F}
the class of fuzzy subsets
u
: ℝ → [0, 1]; satisfying the following properties:

(1)uis normal, that is, there existx0∈ ℝ withu(x0) = 1.

(2)uis convex fuzzy set, that is,

(3)uis upper semicontinuous on ℝ.

(4)is compact, whereĀdenotes the closure ofA.
Then ℝ
_{F}
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 nonempty 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:

(1)is a bounded monotonic increasing (nondecreasing) leftcontinuous function∀r∈ (0, 1]and rightcontinuous for r= 0.

(2) ū(r)is a bounded monotonic decreasing (nonincreasing) leftcontinuous function∀r∈ (0, 1]and rightcontinuous for r= 0.

(3)0 ≤r≤ 1.
We refer
as the lower and upper branches on
u
, respectively. For
u
∈ ℝ
_{F}
; we define the length of
u
as:
A crisp number
α
is simply represented by
(0 ≤
r
≤ 1) is called singleton. For
u, v
∈ ℝ
_{F}
and
α
∈ ℝ, the sum
u
+
v
and the scalar multiplication
αu
are defined by
For
u, v
∈ ℝ
_{F}
, we say
u
=
v
if and only if
The metric structure is given by the Hausdorff distance
D
: ℝ
_{F}
× ℝ
_{F}
→ ℝ
_{+}
∪{0}, by
Definition 2.2.
Let
x, y
∈ ℝ
_{F}
. If there exists
z
∈ ℝ
_{F}
such that
x
=
y
+
z
then
z
is called the Hdifference of
x, y
and it is denoted
x
⊖
y
.
In this paper the sign “⊖” stands always for Hdifference and
x
⊖
y
≠
x
+(−1)
y
in general. Usually we denote x+(1)y by xy, 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
→ ℝ
_{F}
be a fuzzy number valued function. If there exists an element
F′
(
t
_{0}
) ∈ ℝ
_{F}
such that for all
h
> 0 sufficiently near to 0,
F
(
t
_{0}
+
h
) ⊖
F
(
t
_{0}
),
F
(
t
_{0}
) ⊖
F
(
t
_{0}

h
) exist and the limits (in the metric D)
exist and equal to
F′
(
t
_{0}
), then
F
said to be differentiable at
t
_{0}
∈ (
a, c
). If
t
_{0}
is the end points of
I
, then we consider the corresponding onesided derivative. Here the limits are taken in the metric space (ℝ
_{F}
,
D
).
In this paper we considered the following third order threepoint fuzzy boundary value problem
with boundary conditions
where
and
f
:
I
× ℝ
_{F}
× ℝ
_{F}
× ℝ
_{F}
→ ℝ
_{F}
is continuous fuzzy function.
3. Generalized fuzzy derivatives
The definition of the Hukuhara differentiability is a straightforward generalization of the Hukuhara differentiability of a setvalued 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
→ ℝ
_{F}
and fix
t
_{0}
∈ (
a, c
). If there exists an element
F′
(
t
_{0}
) ∈ ℝ
_{F}
such that for all
h
> 0 sufficiently near to 0,
F
(
t
_{0}
+
h
)⊖
F
(
t
_{0}
),
F
(
t
_{0}
)⊖
F
(
t
_{0}
−
h
) exist and the limits (in the metric D)
exist and equal to
F′
(
t
_{0}
), then
F
said to be (1)differentiable at
t
_{0}
and it is denoted by
If for all
h
> 0 sufficiently near to 0,
F
(
t
_{0}
) ⊖
F
(
t
_{0}
+
h
),
F
(
t
_{0}
−
h
) ⊖
F
(
t
_{0}
) exist and the limits (in the metric D)
exist and equal to
F′
(
t
_{0}
), then
F
is said to be (2)differentiable and it is denoted by
If
t
_{0}
is the end points of
I
, then we consider the corresponding onesided derivative.
Theorem 3.2
(
[6
,
10]
).
Let F
:
I
→ ℝ
_{F} and let F
(
t
) = (
f
(
t
,
r
),
g
(
t
,
r
))
for each r
∈ [0, 1].

(1)If F is (1)differentiable then f(t,r)and g(t,r)are differentiable functions and

(2)If F is (2)differentiable then f(t,r)and g(t,r)are differentiable functions and
Definition 3.3.
Let
F
:
I
→ ℝ
_{F}
and let
n,m
∈ {1, 2}. If
exists on a neighborhood of
t
_{0}
as a fuzzy number valued function and it is (
m
)differentiable at
t
_{0}
as a fuzzy number valued function, then
F
is said to be (
n,m
)differentiable at
t
_{0}
∈
I
and is denoted by
Theorem 3.4
(
[10]
).
Let F
:
I
→ ℝ
_{F}
,
and let F
(
t
) = (
f
(
t
,
r
),
g
(
t
,
r
)).

(1)Ifis (1)differentiable, then f′(t,r)and g′(t,r)are differentiable functions and

(2)Ifis (2)differentiable, then f′(t,r)and g′(t,r)are differentiable functions and

(3)Ifis (1)differentiable, then f′(t,r)and g′(t,r)are differentiable functions and

(4)Ifis (2)differentiable, then f′(t,r)and g′(t,r)are differentiable functions and
Remark 3.1.
For each of these four derivatives, we have again two possibilities.
Definition 3.5.
Let
F
:
I
→ ℝ
_{F}
and let
n, m, l
∈ {1, 2}. If
exist on a neighborhood of
t
_{0}
as fuzzy number valued functions and
is (
l
)differentiable at
t
_{0}
as a fuzzy number valued function, then
F
is said to be (
n, m, l
)differentiable at
t
_{0}
∈
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
)).

(1)Ifis (1)differentiable, then f′′(t,r)and g′′(t,r)are differentiable functions and

(2)Ifis (2)differentiable, then f′′(t,r)and g′′(t,r)are differentiable functions and

(3)Ifis (1)differentiable, then f′′(t,r)and g′′(t,r)are differentiable functions and

(4)Ifis (2)differentiable, then f′′(t,r)and g′′(t,r)are differentiable functions and

(5)Ifis (1)differentiable, then f′′(t,r)and g′′(t,r)are differentiable functions and

(6)Ifis (2)differentiable, then f′′(t,r)and g′′(t,r)are differentiable functions and

(7)Ifis (1)differentiable, then f′′(t,r)and g′′(t,r)are differentiable functions and

(8)Ifis (2)differentiable, then f′′(t,r)and g′′(t,r)are differentiable functions and
4. Threepoint 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
→ ℝ
_{F}
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
I
_{1}
be an interval such that
I
_{1}
⊂
I
. If
exist on
I
_{1}
as fuzzy number valued functions and
for all
t
∈
I
_{1}
, then
y
is said to be a (
n, m, l
) solution for the fuzzy differential equation (1) on
I
_{1}
.
Definition 4.3.
Let
n_{i}
,
m_{i}
,
l_{i}
∈ {1, 2} and
i
∈ {1, 2, 3, 4}. If there exists a fuzzy number valued function
y
:
I
→ ℝ
_{F}
such that
where
y
_{1}
: [
a, b
] ∪ {
c
} → ℝ
_{F}
and
y
_{2}
: [
b, c
] ∪ {
a
} → ℝ
_{F}
are the fuzzy number valued functions with
and if there exist
t
_{1}
∈ (
a, b
) and
t
_{2}
∈ (
b, c
) such that
y
_{1}
is a (
n
_{1}
,
m
_{1}
,
l
_{1}
)solution and a (
n
_{2}
,
m
_{2}
,
l
_{2}
)solution of the equation (1) on (
a, t
_{1}
) and on (
t
_{1}
,
b
) respectively and
y
_{2}
is a (
n
_{3}
,
m
_{3}
,
l
_{3}
)solution and a (
n
_{4}
,
m
_{4}
,
l
_{4}
) solution of the equation (1) on (
b
,
t
_{2}
) and on (
t
_{2}
,
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
I
_{1}
⊆
I
for
m, n, l
∈ {1, 2}, then
y
is (
n, m, l
)differentiable on
I
_{1}
and
y
(
t
) is not
(n, m, l
)differentiable in
t
_{0}
∈ (
I
\
I
_{1}
):
5. Examples
Example 5.1.
Consider the following third order threepoint 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
t
^{3}
−3
t
^{2}
+2
t
≥ 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
t
^{3}
− 3
t
^{2}
+ 2
t
≤ 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] → ℝ
_{F}
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
y
_{1}
is a (1, 2, 2) solution and (2, 1, 2) solution of the equation (4) on
respectively,
y
_{2}
is a (1, 2, 1) solution and (2, 1, 1) solution of the equation (4) on
respectively and
y
_{1}
y
_{2}
satisfy the boundary conditions (5). Therefore
y
in (8) is a generalized solution of the fuzzy boundary value problem (4)(5).
y
_{1}
and
y
_{2}
are shown in
Figure 1
and
Figure 2
respectively for different values of
t
. From these figures we see that
y
_{1}
and
y
_{2}
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
.
for different t.
for different t ∈ [1, 2].
Lower branch of generalized solution different r.
Upper branch of generalized solution different r.
Example 5.2.
Consider the following third order threepoint 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 2
t
^{3}
− 3
t
^{2}
+
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 2
t
^{3}
−3
t
^{2}
+
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] → ℝ
_{F}
such that
where
for all
and
for all
are fuzzy number valued function and there exist
such that
y
_{1}
is a (1, 2, 2)solution and (2, 1, 2)solution of the equation (9) on
respectively,
y
_{2}
is a (1, 2, 1)solution and (2, 1, 1)solution of the equation (9) on
respectively and
y
_{1}
y
_{2}
satisfy the boundary conditions (10). Therefore y in (13) is a generalized solution of the fuzzy boundary value problem (9)(10).
y
_{1}
and
y
_{2}
are shown in
Figure 5
and
Figure 6
respectively for different values of
t
. From these figures we see that
y
_{1}
and
y
_{2}
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
.
for different t.
for different t.
Lower branch of generalized solution for different r.
Upper branch of generalized solution for different r.
Acknowledgements
We wish to thank the reviewers for their helpful comments and suggestions to revise and enhance portion of this paper.
BIO
P. Prakash received M.Phil. and Ph.D from Bharathiar University, Coimbatore. His research interests include fuzzy differential equation, partial differential equation and numerical analysis.
Department of Mathematics, Periyar University, Salem636 001, India.
email:pprakashmaths@gmail.com
N. Uthirasamy received M.Sc. from Erode Arts and Science College, Erode and M.Phil from Alagappa University. His research interests include fuzzy differential equation and numerical analysis.
Department of Mathematics, K.S.Rangasamy College of Technology, Tiruchengode  637 215, India.
email:samyres@rediffmail.com
G. Sudha Priya received M.Sc. from Sri Sarada College for Women, Salem and M.Phil from Periyar University. Her research interests include analysis, differential equation and numerical analysis.
Department of Mathematics, Periyar University, Salem636 001, India.
email:priyasudha1985@gmail.com
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