Abstract
In this article, the variational method together with two control parameters is used for introducing the proof for the existence of infinitely many solutions for a new class of perturbed nonlinear system having -Laplacian fractional-order differentiation.
1. Introduction
One of the main applications of fractional calculus science is the fractional-order differential equations (FDEs). Various natural phenomena are modeled mathematically through the FDEs, and this is evident in numerous areas of physics, engineering, chemistry, and other fields. The fractional-order partial differential equations have several applications in many fields such as engineering, biophysics, physics, mechanics, chemistry, and biology (see [1–7]). More and more efforts have been made in the fractional calculus field especially in FDEs (see, for instance, [2, 5, 8–14, 27–39]). Solution existence for a lot of boundary value problems and several nonlinear elementary problems is studied via a huge number of techniques and nonlinear mathematical tools (see [7, 15–23]): the theory of critical point, fixed-point theory, technique of monochromatic iterative, theory degree of coincidence, and the change methods. Motivated by multiple works involved in this domain, we concentrate in this paper on the existence of several infinite solutions to the following fractional-order differentiation system: for , , and are the left and right fractional-order derivatives of the Riemann-Liouville operator, respectively,
, are positive and nonnegative real parameters, respectively, are continuous functions according to for any and are with respect to for a.e. and for it is to say . Also, and indicates partial derivatives of and according to , respectively, and are -order of Lipschitz continuous functions with constants of Lipschizian, i.e.,
In the last few months, we treated the same area of this study, in [18], by using variational methods together with a critical point theory due to Bonano and Marano. We got at least three weak solutions for the following nonlinear dual-Laplace systems with respect to two parameters:
In addition, in [24], by using the variational method and Ricceri’s critical point theorems, the existence of three weak solutions has been used to investigate the following class of perturbed nonlinear fractional -Laplacian differential systems: where some necessary conditions on the primitive function of nonlinear terms and have been considered. Then, in [25], the same last methods have been used for problem (5), the existence of multiplicity of weak solutions for the following perturbed nonlinear fractional differential systems: where Lipschitz nonlinearity order of has been used.
Most recently, in [26], the authors proved the existence of infinitely multiple solutions of the following perturbed nonlinear fractional -Laplacian differential systems: where one control parameter with the variational method has been used.
Motivated by recently mentioned papers, the main contribution of this article is to use two control parameters and variational method to study a class of a nonlinear perturbed fractional-order -Laplacian differential system which is defined in (6), where we can prove that the studied system admits sequences of weak different solutions, strongly converge to zero.
2. Preliminaries
In this section, we introduce some notations, lemmas that are required for the subsequential. Then, a variational framework is constructed; then, the critical point theory is applied to explore the existence of infinite solutions for the system given in (6).
We denote the class of all functionals where is real Banach space which has the following properties.
If is a sequence in converge weakly to and thus has a subsequence that strongly converge to .
As an example, suppose a uniformly convex with is an increasing, continuous strictly function, then the functional .
Definition 1 (see [4]). Let be a defined function on . The left and right Riemann-Liouville fractional derivatives of order , respectively, are given as where the right-hand sides are pointwise defined over , , and
The gamma function, , is given by
Setting the space of functions where
As familiar, we denote the mappings set indicates -times continuously differentiable on
Actually, we imply
Definition 2 (see [22]). Let (, ), we introduce the space of the fractional-order derivative as follows: then, the norm of can be defined , as the following
Lemma 3 (see [3]). Let For all , we have
Also, if and then
Hence, the operator according to the norm can be considered
for , that is equivalent to (13).
Definition 4. Suppose a Cartesian product of spaces ; that is to say, provided with the norm where is defined in (16).
Clearly, is embedded compact in
Lemma 5 (see [23]). For and The space of the fractional-order derivative is a reflexive separable Banach space.
Lemma 6 (see [16]). Assume that and be the sequence that weakly converges to in , i.e., Then, have a strong convergence to in i.e., , as
Definition 7 (see [3]). We indicate a weak solution for system (6), any such that
We define
Lemma 8. Suppose that satisfy (3) and defined through (20). Then, be the functional that is described by is a weakly continuous sequentially Gâteaux differentiable functional on together with a compact derivative for every
Proof. Assume that in as It follows from Lemma 8 that converges uniformly to on Therefore, there exist constants , such that .
Then,
for any , & . Moreover, and via the convergence theorem in the Lebesgue sense, we have
In the following, a Gâteaux differentiability of will be implemented. Let and , then we claim where , . Hence, is a Gâteaux differentiable at any .
Analogously, we have that at any is a differentiable in Gâteaux sense.
Therefore, to each with derivative is differentiable with a Gâteaux description
For any three elements , and of it is easy to see that
which implies
where
Consequently, the operator is compact.
Below, we recall Theorem 2.5 of [23] which is an essential tool in our paper.
Lemma 9 (see [23], Theorem 2.5). Let a real reflexive Banach space . Also, suppose two Gâteaux differentiable functionals such that is sequentially weakly lower semicontinuous, strongly continuous, and coercive where sequentially weakly upper semicontinuous achieved for . put
Then, (a)For every & , the functional constraint of to allows a global minimum, which is a critical point (local minimum) for in the space (b)If & , the subsequent alternative holds:
(b1)The functional has a global minimum or
(b2) a sequence of critical points (local minima) for such that (c)If & , the next alternative exists:
(c1) a global minimum of that is a local minimum of or
(c2) a sequence of pairwise disjoint critical points (local minima) for that one converges weakly to a global minimum of together with
3. Principle Results
This section deals with stating and proving our main results. For assistance, suggest
For a given constant set
For any , we set
Theorem 10. Let for . Suppose that there exists such that
(h1) for each
(h2)
Then, for each where
for each nonnegative function achieving the constrain
and for every where
system (1) has an unbounded sequence of weak solutions in space .
Proof. The main aim here is applying Lemma 6 (see [16]) over system (1). For this purpose, fix , and let be a function that satisfies our hypotheses. Since , we claim
Now, fix . Set
for every and . We construct the mappings as follows:
and determine
Let us prove that & satisfy the required constrains. Since is compactly embedded in it is well known that is well-defined Gateaux differentiable functional whose Gateaux derivative at is the functional given by
. Moreover, is sequentially weakly continuous.
The functional is a Gateaux differentiable functional with the differential at ,
for every . Moreover, is sequentially weakly lower semicontinuous, strongly continuous, and coercive functional on .
Obviously, the weak solutions of system (1) are precisely the critical points of the functional . Furthermore, since (3) holds for every and , one has . It obtained from (14) and (15) the following:
, and as a result for this, is coercive.
Now, allow us to verify that . Assume is a positive number sequence such that as and
Put . Since for all and , we have for each . So, from (45) and (47), we have
Consequently, taking into the description that , for all big enough, one has
Moreover, from assumption (h2) and (37), one has which implies
Therefore,
The assumption immediately yields
The succeeding step is to confirm that for a fixed , the functional has no global minimum. Let us verify that is unbounded from below. Since consider is a real sequence and is a positive constant such that as , and
huge adequate. , and define by setting for . Clearly, and for . A direct calculation shows that for . Furthermore, for . Thus, , and in particular,
On the other hand, similar to (45), we have
Bearing in mind assumption () and since is nonnegative, then using definition, we conclude that
So, according to (54), (55), and (60),
for every large enough. Hence, is unbounded from below and so has no global minimum. Therefore, applying Lemma (b), we deduce that there is a sequence of critical points of such that
Here, the outcome is produced.
Remark 11. Under the conditions from Theorem 10, we claimed and system (1) admits infinitely many weak solutions in the space . Also, if , then the result holds and
Here, we point out the following conclusion of Theorem 10 beside
Corollary 12. Suppose such that hypothesis (h1) holds. Assume
(B1)
(B2)
Then, the system
for , has an unbounded sequence of weak solutions in
Follow the same steps of Theorem 10 proving but alternatively of (b) of Lemma 6 applying conclusion (c), the following result will be obtained.
Theorem 13. Suppose that all of Theorem 10 assumptions hold except for hypothesis (h2). Assume thatassumptions hold except for hypothesis (h2). Assume that Then, for each where for every nonnegative function satisfying the condition and for every where a sequence of weak solutions for system (system (1) exists, and it strongly converges to zero in the space .
Proof. Fix and let be a function satisfying (67). Since , one has
Fix and put for every and . We take , and as Theorem 10 proof. We verify that . For this, let be a sequence of positive number such that as and
Through the evidence in addition to definition, we claim . Then, like in display (52) of Theorem 10 proof, can be proven, and hence,
Let be fixed. We conclude that at zero, there is no local minimum for the functional . For this purpose, a sequence of positive numbers is supposed such that when and choosing such that for every large enough . Assume a sequence in the space together with given in (55). Remark that . Hence, like in (62) show, the following can be obtained: large enough . Where , this means at the point zero there is no local minimum for the functional .
Therefore, part (c) of Lemma ensures that there exists a sequence in the space of critical points for that is convergent weakly to the point zero. According to the established truth is compact, we conclude that the critical points strongly converge to zero, and the proof is performed.
Remark 14. According to the conditions ensures that and system admits infinitely many weak solutions in the sapce . Moreover, if , the conclusion exists and
Now, the following example will be presented for illustrating the above result.
Example 15. Consider the system where and . Moreover, for all , let be defined as where are nonnegative continuous functions. Let . We observe that
Now, let : be a function defined by
By definition, and
All hypotheses of Remark 14 are satisfied. Then, for all , system (75) admits a sequence of weak solutions which strongly converges to in
Data Availability
No data were used to support the study.
Conflicts of Interest
This work does not have any conflicts of interest.
Acknowledgments
The fourth author extends their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through a research group program under grant (R.G.P-2/1/42).