Existence and Multiple of Positive Solution for Nonlinear Fractional Difference Equations with Parameter ()
1. Introduction
There have been of great interest recently on fractional difference equations. It is caused by the development of the theory of fractional calculus and discrete fractional calculus, also by its applications, see [1]-[7]. We noted that most papers on discrete fractional difference equation are devoted to solvability of linear initial fractional difference equations [8] [9]. Recently, there are some papers dealing with the existence of solutions of nonlinear boundary value problems, we also refer the readers to [10] [11]. However, there are few papers consider parameter intervals of fractional difference boundary value problems. In the present work, our purpose is to the parameter intervals of the following fractional difference boundary value problem
(1.1)
(1.2)
where
,
is an integer,
is continuous,
for
and
. For
, define
.
F. M. Atici and P. W. E. [10] studied fractional difference boundary value problem
(1.3)
with the boundary value condition (1.2). By using Krasnosel’skii fixed point theorem under condition
(H1)
,
;
(H2)
, where
is a positive function,
is a non-negative function and ![]()
(H3)
, where
is a positive function,
is a non-negative function and ![]()
They get the following.
Theorem 1.1[10] Assume that conditions (H1) and (H2) are satisfied, then problem (1.1) and (1.2) has at least one solution. Assume that conditions (H1) and (H3) are satisfied, then problem (1.1) and (1.2) has at least one solution.
The following conditions will be used in the paper
(A1)
, where
is a positive function,
is continuous, and there exist
such that
;
(A2)
.
2. Preliminaries
Recall the factorial polynomial
where
denotes the special Gamma function and if
for some
, we assume the product is zero. We shall employ the convention that division at a pole yields zero. For arbitrary
, define
We also appeal to the convention that ![]()
is a pole of the Gamma function and
is not a pole, then
. Let
,
and
defined on
, Miller and Ross [12] have defined the
-th fractional sum of
by
(2.1)
where
, also define the
-th fractional difference
![]()
where
and
with
,
.
Lemma 2.1 [10] Let
,
, the unique solution problem
(2.2)
is
where
(2.3)
Lemma 2.2 [10] The Green’s function
in Lemma 2.1 satisfies the following conditions:
(i)
for
and
;
(ii)
for
;
(iii) There exists a positive number
such that for ![]()
(2.4)
where
(2.5)
In the rest of the paper, we will use the fixed point index theory in cones to deal with (1.1) and (1.2).
Lemma2.3 [12] Let
be a Banach space,
be a cone, and suppose that
are bounded open balls of
centered at the origin with
. Suppose further that
is a completely continuous operator such that either
(i)
,
and
,
, or
(ii)
,
and
, ![]()
holds, then
has a fixed point in
.
We will need the following notations. Let
![]()
Then
is a Banach space with the norm ![]()
So,
is a solution of (1.1) and (1.2) if, and only if
is a fixed point of the operator
defined by
![]()
Note
, let
be defined by (2.5) and define cones
in
by
For some
,
Since
is finite dimensional, we have the
is compact. Obviously,
.
Lemma 2.4 Suppose that conditions (A1) hold, and there exist two different positive numbers
and
such that
where
.
Then, problem (1.1), (1.2) has at least one positive solution
such that
.
Proof. We can suppose that
. For
,
, there is
, then
![]()
these mains that for
, there is
. For
,
, there is
, then
![]()
these mains that for
, there is
. By using Lemma 2.3, there exist
such that
. This means that,
is a solution of problems (1.1), (1.2) and
. Also, because
,
so
for
, taking into account that conditions(A1) and (A2) hold and
, we have that
for
, i.e.
is a positive solution of (1.1), (1.2).
3. Main Results
For some
, denote ![]()
By using Lemma 2.4, we get
Theorem 3.1 Assume that (A1) hold, and
and
, then, there exist
, for every
, problem (1.1) and (1.2) has at least two positive solutions.
Theorem 3.2 Assume that (A1) hold, and
or
, then, for every
, problem (1.1) and (1.2) has at least one positive solutions.
Theorem 3.3 Assume that (A1) hold, and
and
, then, for every
, problem (1.1) and (1.2) has at least two positive solutions.
Theorem 3.4 Assume that (A1) and (A2) hold, and
or
, then, for every
, problem (1.1) and (1.2) has at least one positive solutions.
Acknowledgements
Author was supported by the NSF of Gansu Province (No. 2013GS08288).