1. Introduction
In the present paper, we investigate the Cauchy problem for the generalized Kawahara–KdV system:
where
Hypothesis 1. , on for , , , on , , ,,on , , , , p, , Kondo and Pes [
1] proved the local well-posedness of this system in analytic Gevrey spaces
with
,
.
The range of the type of equations that this model encompasses is obviously broad and can represent many physical phenomena. As examples, we can consider the nonlinear case
When
and
, taking
and taking the same choices to
with
and
, we have a coupled system of modified KdV equations (see [
2,
3]):
Considering in (
2) the case when
and
, since
, we obtain a more general system, treated in [
4] as
In order to find a more general and more complicated systems, we can consider
and
; then, we notice that the term nonlinear is more general:
If we change
z by
w, and consider again all identical null polynomials, except
, we obtain the Kawahara system [
5]
The study of nonlinear partial differential equations (PDEs) has garnered significant attention in recent years due to their wide-ranging applications in various fields such as fluid dynamics, plasma physics, and optical communications [
6,
7,
8]. In particular, fractional-order PDEs, which generalize classical PDEs by incorporating nonlocal effects, have been the subject of extensive research, including the analysis of the Kaup–Kupershmidt equation and Korteweg–De Vries (KdV)-type equations within different operators [
6,
7]. Additionally, the investigation of nonlinear wave phenomena in plasma and fluid systems has led to the development of analytical solutions for various nonlinear PDEs, such as the nonlinear Schrodinger equation with a detuning term [
8].
Shah et al. [
6] conducted a comparative analysis of the fractional-order Kaup–Kupershmidt equation using different operators, offering valuable insights into the behavior of the equation and its solutions. Similarly, Shah et al. [
7] explored the analytical investigation of fractional-order KdV-type equations under the Atangana–Baleanu–Caputo operator, focusing on the modeling of nonlinear waves in plasma and fluid systems. Furthermore, Shah et al. [
8] analyzed optical solitons for the nonlinear Schrodinger equation with a detuning term using the iterative transform method, which has important implications for the understanding and control of optical communication systems.
Building on these foundational studies, our research aims to further advance the understanding of nonlinear PDEs by applying a novel topological approach to the generalized Kawahara–KdV system. We seek to demonstrate the existence of classical and non-negative solutions, thus contributing to the broader knowledge of nonlinear PDEs and their applications in various scientific and engineering contexts.
Theorem 1. We suppose that Hypothesis 1 holds. Then, the initial value problem (1) has at least one solution Theorem 2. We suppose that Hypothesis 1 holds. Then, the initial value problem (1) has at least two non-negative solutions We organized the paper as follows. In the second section, we introduce and state some auxiliary results related the to our system and its symmetrical problem. In the next
Section 3, we prove Theorem 1 for the existence of at least one solution. In
Section 4, we show the existence of at least two non-negative solutions in in Theorem 2. In
Section 5, we introduce an example illustrating the main results.
2. Preliminary Results
In order to prove the existence of the solution, we shall use the following fixed-point Theorem.
Theorem 3. Let , , be a Banach space and Let also , , be a continuous function, reside in a compact subset of , andThen, there exists such that Proof. Then,
is compact and continuous. Thus, owing to the Schauder fixed-point theorem, it follows that there exists
such that
Assume that
. Thus,
and
Then,
contradicts (
3). Thus,
and
or
or
which completes our proof. □
Let be a real Banach space.
Definition 1. A mapping is said to be completely continuous if it is continuous and maps bounded sets into relatively compact sets.
The definition of l-set contraction is related to the Kuratowski measure of noncompactness, which we recall for completeness.
Definition 2. Let be the class of all bounded sets of . The Kuratowski measure of noncompactnessis defined bywhereis the diameter of , . We refer the reader to [
9] for the main symmetrical properties of the measure of noncompactness.
Definition 3. A mapping is said to be an l-set contraction if it is continuous, bounded, and there exists a constant such thatfor any bounded set . The mapping is said to be a strict set contraction if . If
is a completely continuous mapping, then
is 0-set contraction (see [
10] (p. 264)).
Definition 4. Let and be real Banach spaces. A mapping is said to be expansive if there exists a constant such that Definition 5. A closed, convex set ϖ in ϱ is said to be cone if
- 1.
for any and for any ;
- 2.
implies .
Let us denote .
Lemma 1. Let ϱ be a convex closed subset of a Banach space and be a bounded open subset where For small enough values of , let be a strict k-set contraction that satisfies Thus,
Proof. Let the homotopic deformation be
defined by
For each the operator is continuous and uniformly continuous in t, where is a strict set contraction for each . Notice that has no fixed point on . On the contrary,
If , such that , contradicting
If , such that ; then, with contradicting the assumption. From the invariance under homotopy and the normalization symmetrical properties of the index, we deduce
We have as for
Let us define the convex deformation
by
For all
x,
F is continuous, and uniformly continuous in
t. The mapping
is a strict set contraction
. We mention that
has no fixed point on
. We have
, and thus we have
According to the invariance properties, the homotopy of the index ensures the claim. □
3. Proof of Theorem 1
Let
be a space endowed with
provided it exists. Define
with
We define for
Then, the IVP (
1) can be rewritten as
Lemma 2. Suppose . If and , then Proof. We have
,
. Since
, we have
for any
,
. Then,
on
,
, and then
on
. As above,
on
. The proof is now completed. □
For
, we define the operators
.
Lemma 3. Suppose . If satisfiesthen is a solution to (1). Proof. We have
, where we differentiate with respect to
t to have (
5). Let
in (
6). We thus obtain
Thus,
is a solution to (
1). The proof is now completed. □
Lemma 4. Suppose . If and ; then, Proof. As above,
which completes the proof. □
Let
Hypothesis 2. There exists a function , on , , , and such that. We will give some examples for
g and
that satisfy Hypothesis 2. For
, define the operators
Lemma 5. Suppose Hypothesis 1 and Hypothesis 2. If satisfiesthen is a solution to (1). Proof. Differentiating the Equation (5) two times in
t and
times in
x, we have
Since
and
are continuous functions on
, we have
Using Lemma 3, we obtain the main result. □
Lemma 6. Suppose Hypothesis 1 and Hypothesis 2. If , , then Proof. The inequality
,
will be used. We have
and
and
and so on. As above,
. Thus,
which completes our proof. □
Suppose
Hypothesis 3. Let , , and satisfy and .
Let
denote the set of all equi-continuous in
with respect to the norm
. Also let
be the closure of
, where
and
Note that
is a compact set in
. For
, we define
Owing to the Lemma 5, we have f
Thus,
is continuous, and
resides in a compact subset of
. One can suppose that
such that
and
or
or
for
. Then,
,
This is a contradiction. By Theorem 3, we see that
has a fixed point
. Then,
, whereupon
Owing to the Lemma 5, we have
as a solution to (
1), which completes the proof.
5. Example
Let
and
,
,
. Then,
i.e., (Hypothesis 3) holds. Next,
i.e., (Hypothesis 4) holds. Take
Hence, there exists
such that
. Note that acccording to
and [
12] (p. 707, Integral 79), we have
Then,
i.e., (Hypothesis 3) holds. Therefore, for the IVP
all conditions of Theorems 1 and 2 are fulfilled.