1. Introduction
Boundary value problems for ODEs, with special initial-boundary conditions, are intensively investigated for their many applications in physics and mathematics [
1,
2] in a wide range of problems from vibrations to the theory of elasticity [
3]. In mathematical terms, these problems are often described by the multipoint boundary value problems [
1]. This theory was mainly described in the original papers of Il’in and Moiseev [
4], with further developments by several authors who contributed with fundamental results based on the Leray–Schauder Continuation Theorem and corresponding nonlinear generalizations, the degree theory, and fixed point theorem (FPT).
Many authors have studied various aspects of boundary value problems with multipoint boundary conditions for the differential equations having broad applications in several branches of physics and applied mathematics [
5,
6,
7,
8,
9,
10,
11].
Differential equations with integral boundary conditions also have many applications in modeling and analyzing of many physical systems as blood flow problems, chemical engineering, thermoelasticity, underground water flow, population dynamics, etc. [
12,
13,
14,
15,
16,
17,
18,
19,
20,
21,
22,
23,
24,
25,
26,
27,
28,
29,
30,
31,
32,
33].
In this paper, an original approach based on the construction of a suitable Green function is proposed for the analysis of the multipoint BPV, so that a problem on a differential system is converted into an equivalent integral equation. Comparing with the results obtained by Multy and Sivasundaram [
34], we do not use the fundamental matrix of the equation. The main advantage of our choice is that we don’t require the existence of the derivative of the equation with respect to the phase coordinates. Then the uniqueness of the solution is studied for the integral equation by means of the Banach contraction mapping principle (BCMP), while the existence is also shown by using Schaefer’s fixed point theorem (SFPT).
The organization of the paper is as follows: in
Section 2 are given some preliminary remarks about this problem, together with some related definitions and known methods.
Section 3 deals with the proof of uniqueness, while in
Section 4 is given a proof of the existence by means of the fixed point theorem. Some applications are given in
Section 5. Conclusion and future perspectives are discussed in
Section 6.
2. Preliminary Remarks
Let us start by considering the following nonlinear differential system
with multipoint boundary conditions
where
are
-order constant matrices with
are some given continuous functions; the points
are arbitrarely chosen in the finite interval
. Let
be the Banach space of all continuous functions from
into
with the norm
.
In general the solution of (1)–(2) is characterized by the following:
Definition 1. A functionis a solution of (1) and (2) if and for eachboundary conditions (2) are fulfilled.
Let us now study the following problem:
We have that
Lemma 1. Forthe solution of the BVP (3) and (4) is unique and it is given bywherewith Proof. For any
the solution
fulfills
where
is an arbitrary constant vector. Let us chose
in such a way that
fulfills Equation (4). There follows
which implies
If we put this value into Equation (5), we get
Since the equality
holds, from Equation (7) we get
From where, by taking
, there follows
Let us write this equation in the equivalent form
where
E is the identity matrix.
Since
takes place, there follows
For
, Equation (8) gives
So that by defining
it is obtained
Continuing this process in a similar way, for the next segment
we get
where
and so on. There follows that Equations (3)–(4) can be expressed by
So that the proof is given. □
3. Uniqueness of the Solution
The uniqueness of the solution of problem (1) and (2) is proven here by taking into account the following:
Lemma 2. Let, thenis solution of the BVP (1)–(2) iffis a solution of the following integral equation Proof. The proof is obtained similarly to Lemma 1 so that by a direct computation, we can see that the solution of Equation (11) fulfills BVP (1) and (2). □
Let us now assume that:
Hypothesis 1 (H1). is a continuous function;
Hypothesis 2 (H2). There exists a constantsuch that the inequalityholds for eachand all Hypothesis 3 (H3). There exists a constantsuch thatfor eachand all
We can show that:
Theorem 1. [Uniqueness]. By assuming that, (H1) and (H2) holds andwhereThen BVP (1), (2) admits a unique solution on the interval.
Proof. To show this, let us transform (1) and (2) into a fixed point problem. Let
be an operator, defined as
whose fixed points are solutions of Equations (1) and (2).
Setting
and taking
we show that
, where
For
, by using (H2), we get
Let us show now that
is a contraction map for any
. Thus we write
or
It shows that according to (12),
is a contraction map and therefore (1) and (2) admits a unique solution. □
4. Existence of the Solution
Theorem 2. [Existence] Let us assume that (H1)–(H3) hold. Then there exists at least one solution of (1), (2) on [0, T].
Proof. By taking into account its definition (13), we can use the SFPT to show that there exists a fixed point for . The multistep proof is as follows:
Step 1:
is a continuous operator. In order to show this, let
be a sequence such that
in
. There follows that, for
From this we get
as
, which implies that
is a continuous operator.
Step 2: The operator
maps bounded sets into bounded sets in
. In order to show this, it is enough to prove that for any
there exists a positive constant
such that for each
it is
So that for each
we get
From where there follows
Step 3: Let us show now that maps bounded sets into equicontinuous sets in . Let be a bounded set in as shown in Step 2, and let .
Case 1.
Then,
Case 2.
Then
The r.h.s. of this equation tends to zero for . As a consequence of Steps 1–3 and by taking into account the Ascoli–Arzela theorem, there follows that is continuous.
Step 4: Let us show now that the set
is bounded. Let
, then,
for some
so that, for each
we have
Therefore, is bounded and we can conclude that the operator admits at least one fixed point. As a consequence, there exists at least one solution for the problem (1) and (2) on the interval □
Some more problems for the two-point and the three-point boundary value conditions are studied in [
5,
30,
31,
32,
33,
34].
5. Example: Analysis of the Vibrations of a Non-Homogeneous String
The existence and uniqueness of the solution for a nonlinear first-order equation with multipoint boundary conditions are given for a concrete example.
Example. Let
be a given differential system with the following three-point and integral boundary conditions:
Then for
we obtain
and for
So .
Thus . From here, we can easily see that the given system (14)–(15) has a unique solution.
The solution of system (14)–(15) involves elliptic functions; therefore, the exact solution is a quite impossible problem. However, in some special cases, it is possible to obtain the exact form of the two functions
which also fulfill the boundary-integral conditions
In fact, let us compute the solution of system (14)–(15) after linearization of the trigonometric functions of (14), that is
Moreover, if we also assume that the initial conditions are as follows
the two functions
also fulfill the three-point boundary-integral conditions
. So that, at least in the linearized case, we have explicitly computed the solution of the three-point boundary-integral problem.
At the second-order approximation
the solution can be obtained by solving the equation
which, however, is expressed in terms of Weierstrass elliptic functions.
6. Conclusions
In this paper, a proof of the existence and uniqueness is proved for the solution for a class of nonlinear differential equations with some special boundary conditions. These theorems might be useful in the analysis of several physical problems arising in applied fields, such as problems with impulsive conditions, or wave propagations in non-homogeneous media. So, when the hypotheses of the theorems are fulfilled, then the solution exists and is unique.
The approach given here may be applied to the special cases, for instance, if a physical process is described in terms of a multipoint boundary and is subjected to an impulsive effect at certain points, then it can be studied by the following problem:
under multipoint and integral boundary conditions
with impulsive conditions
Here
is a given function;
are given matrices;
is a given vector, and
Author Contributions
Conceptualization, M.J.M. and Y.S.G.; Formal analysis, M.J.M., Y.A.S., Y.S.G., and C.C.; Investigation, M.J.M., Y.A.S., Y.S.G., and C.C.; Writing—original draft, M.J.M., Y.A.S., Y.S.G., and C.C. All authors have read and agreed to the published version of the manuscript.
Funding
This research received no external funding.
Data Availability Statement
Data is contained within the article.
Conflicts of Interest
The authors declare no conflict of interest.
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