1. Introduction
Fractional calculus has become increasingly popular lately as a result of some interesting properties of the fractional derivative. For instance, the fractional derivative has a memory property that enables its future state to be determined by the current state and all the previous states. This makes fractional differential equations applicable in various fields of science and engineering [
1,
2,
3].
When the corresponding homogeneous equation of a fractional boundary value problem (FBVP) has a trivial solution then the FBVP is a non-resonance problem and its solution can be obtained using fixed point theorems, see [
4,
5,
6,
7] and the references cited therein. When the homogeneous equation of a FBVP has a non-trivial solution then the problem is a resonance problem and the solution can be obtained using topological degree methods [
8,
9,
10,
11,
12,
13,
14,
15].
In [
16], the authors consider a higher-order fractional boundary value problem involving mixed fractional derivatives:
where
is the left Caputo fractional derivative of order
and
is the right Caputo fractional derivative of order
, where
are integers.
Guezane Lakoud et al. [
17] obtained existence results for a fractional boundary value problem at resonance on the half-line:
where
is the left Caputo fractional derivative of order
, and
is the right Caputo fractional derivative of order
.
Zhang and Liu [
15] considered the following FBVP
where
,
is the Riemann–Liouville derivative of order
,
is a Caratheodory function,
and
are two monotonic sequences with
,
,
.
Imaga et al. [
18] obtained existence results for the following fractional-order boundary value problem at resonance on the half-line with integral boundary conditions:
where
is the left Caputo fractional derivative on the half line and
the right Riemann–Louville fractional derivative on the half-line,
,
,
, with
and
.
is a continuous function.
Chen and Tang [
9] established existence of positive solutions for a FBVP at resonance in an unbounded domain:
where
is Riemann–Liouville fractional derivative,
and
is continuous.
Motivated by the results above, we will use the Mawhin coincidence degree theory [
19] to study the solvability of the following mixed fractional-order m-point boundary value problem at resonance on the half-line:
where
is a continuous function,
is the Caputo fractional derivative,
is the Riemann–Liouville fractional derivative,
,
,
,
,
,
and
. The resonant conditions are
and
.
In
Section 2 of this work the required lemmas, theorem, and definitions will be given, while
Section 3 is dedicated to stating and proving the main existence results. An example will be given in
Section 4.
2. Materials and Methods
In this section, we will give some definitions and lemmas that will be used in this work.
Let
U,
Z be normed spaces,
a Fredholm mapping of zero index and
projectors that are continuous, such that:
Then,
is invertible. The inverse of the mapping
L will be denoted by
while the generalized inverse,
is defined as
.
Definition 1. Let be a Fredholm mapping, E a metric space and a non-linear mapping. N is said to be L-compact on E if and are continuous and compact on E. Additionally, N is L-completely continuous if it is L-compact on every bounded .
Theorem 1. ([
19]).
Let L be a Fredholm map of index zero and let N be L-compact on where is an open and bounded. Assume that the following conditions are satisfied:- (i)
for every ;
- (ii)
for every ;
- (iii)
, where is a projection with .
Then, the abstract equation has at least one solution in .
Definition 2. ([
20]).
Let , the Caputo and Riemann–Liouville fractional integral of a function x on is defined by: Definition 3. ([
20]).
Let , the Caputo and Riemann–Liouville fractional derivative of a function x on is defined by:where .
Lemma 1. ([
21]).
Let . The general solution of the Riemman–Liouville fractional differential equation:is , where while, the general solution of the Caputo fractional differential equation: is , where and is the smallest integer greater than or equal to a.
Lemma 2. ([
21]).
Let and , thenholds almost everywhere on for some . Similarly, holds almost everywhere on for some , .
Lemma 3. ([
21]).
Let , , then:- (i)
;
- (ii)
, for , in particular for , , where N is the smallest integer greater than or equal to a;
- (iii)
, ;
- (iv)
.
Let
with the norm
defined on
U where:
Let
equipped with the norm
. The spaces
and
can be shown to be Banach Spaces. Additionally, define
, with domain
and the non-linear operator
will be defined by
hence, Equations (
3) and (
4) may be written as
Definition 4. The set is said to be relatively compact if are uniformly bounded; equicontinuous on any compact subinterval of and equiconvergent at: .
Definition 5. The set is said to be equiconvergent at if given there exists a , such that: where .
Lemma 4. and
where and .
Proof. Consider
for
, then by Lemma 1
Applying the boundary condition
, gives
. Thus,
. Next, consider
for
and
, then
From
we obtain
. Therefore,
By boundary condition
and the conditions
,
, (5) gives
Similarly,
by boundary condition
and resonant conditions
and
, (6) gives
□
Let
. Let the operator
be defined as
where
and
is the algebraic cofactor of
.
Lemma 5. The following holds:
- (i)
is a Fredholm operator of index zero;
- (ii)
the generalized inverse may be written as
Proof. (i) For , it is easily be seen that , , , and . Hence, , thus is a projector.
We now prove that . Let , since then . Conversely, if , then by , . Therefore, .
Let
, then
and
, hence,
. Assuming
, then since
, then from equation
gives
, since
. Therefore
and
. Thus
implying
L is a Fredholm mapping of index zero.
(ii) Let
a continuous projector be defined as:
Similarly, for
, we have
Since
,
and
, then
Therefore,
. Furthermore,
and
Proof of Lemma 5 is complete. □
Lemma 6. The operator N is L-compact on , where is open and bounded with .
Proof. Therefore,
is bounded. In addition,
. In the following steps, we show that
is compact. Let
and
, then:
and
From (8), (11)–(14), we see that
is bounded. Next, the equi-continuity of
will be proved. For
,
with
and
, then:
and
Thus, (15)–(17) shows that
is equi-continuous on the compact set
. Finally, we show equi-convergence at
. Let
be a constant such that
In addition, since
, then for same
, there exist
, such that for
, we have
and
Hence, is equi-convergent at . Therefore, by Definition 1, is compact, therefore, the non-linear operator N is L-compact on . This concludes proof of Lemma 6. □
3. Results and Discussion
Here, the conditions for the existence of solutions to problem (1.1) subject to (1.2) is proved.
Theorem 2. Let f be a continuous function. If () and () holds, then, the following conditions also hold:
- ()
There exists functions , such that for all and , - ()
There exist constants , such that for if for , then either - ()
There exists a constant , such that if or , then either where satisfying .
Then, the boundary value problem (3) and (4) has at least one solution provided: Proof. The proof will be completed in four stages.
Stage 1. We will establish that
is bounded. Let
then
. This means that
and
, hence,
. By Lemma 5, we have
Since
, then
. Additionally, by (
) there exists
, such that
, therefore
Therefore, from (25), we have
and from (24) and (26), we have
Thus, is bounded.
Step 2. Let . For , then . and . Thus, from (), we have and . Hence, is bounded.
Step 3. For
,
, the isomorphism
is as
Let
, then
. Since
, then
When
, we obtain
. When
,
, which contradicts (22) and (23). Hence, from (
), we obtain
, and
. For
, if
or
by (22) and (28), we have
which is a contradiction. Hence,
is bounded.
Similarly, if (23) holds and , can be shown to be bounded by similar argument.
Step 4. Let
. Finally, we will show that a solution of (
3) and (
4) exists in
. We have shown in Steps 1 and 2 that (i) and (ii) of Theorem 1 hold. Finally, we show that (iii) also holds. Let
, then following the arguments of Step 3, it follows that for every
,
. Therefore, by the homotopy property of degree
Therefore, by Theorem 1 at least one solution of (
3) and (
4) exists in
U. □
4. Conclusions
This work considered a mixed fractional-order boundary value problem at resonance on the half-line. The Mawhin’s coincidence degree theory was used to establish existence of solutions when the dimension of the kernel of the linear fractional differential operator is two. The result obtained is new and an example was used to demonstrate the result obtained.
5. Example
Example 1. Consider the following boundary value problem: Here , , , , , , , , ,
. , , , .
, ,
. Then,
. Then,
. Hence,
Finally, conditions () - () can also be shown to hold. Therefore (29) and (30) has at least one solution.
Author Contributions
Conceptualization O.F.I., methodology O.F.I., S.A.I. and P.O.O., manuscript preparation O.F.I. and P.O.O. All authors have read and agreed to the published version of the manuscript.
Funding
This research received no external funding.
Data Availability Statement
Not applicable.
Acknowledgments
The authors are grateful to Covenant University for its support.
Conflicts of Interest
The authors declare no conflict of interest.
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