Infinite Existence Solutions of Fractional Systems with Lipschitz Nonlinearity

Guefaifia, Rafik; Boulaaras, Salah Mahmoud; Cherif, Bahri; Radwan, Taha

doi:https://doi.org/10.1155/2020/6679101

Journal of Function Spaces

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Abstract Introduction Preliminaries Data Availability Conflicts of Interest Authors’ Contributions References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 6679101 | https://doi.org/10.1155/2020/6679101

Infinite Existence Solutions of Fractional Systems with Lipschitz Nonlinearity

Rafik Guefaifia,¹Salah Mahmoud Boulaaras,^2,3Bahri Cherif,²and Taha Radwan^2,4

Academic Editor: Maria Alessandra Ragusa

Received02 Nov 2020

Revised06 Dec 2020

Accepted08 Dec 2020

Published17 Dec 2020

Abstract

The paper deals with the existence of infinitely many solutions of a class of perturbed nonlinear fractional -Laplacian differential systems using one control parameter combined with the variational method.

1. Introduction

Fractional differential equations (FDEs) involve fractional derivatives of the form , where is not necessarily an integer. They are generalizations of the ordinary differential equations to a random (noninteger) order. FDEs have attracted considerable interest due to their ability to model complex phenomena in several fields of science, engineering, physics, biology, and economics (see [1–7]). In summary, many improvements have been made in the theory of partial calculus and partial differential equations and partial and ordinary differential equations (see [8–18], [2, 5]). Numerous studies have explored the existence and solutions of different nonlinear elementary and boundary value problems through the use of various nonlinear analysis tools and techniques (see, for example, [7, 19–38]). Some of these ways are the fixed point theorems, critical point theory, monotone iterative methods, coincidence degree theory, and variational methods (see [30]).

Motivated by the above, the interest of this paper is the infinite existence solutions of the following fractional system where is a positive real parameter, , and , and , are the left and right Riemann-Liouville fractional derivatives of order , respectively, , , where is continuous in for any , is a function in , and is the partial derivative of with respect to , and are two Lipschitz continuous functions of order with Lipschitzian constants for , i.e.,

2. Preliminaries

We give some basic lemmas and notations and construct a variational framework in order to apply critical point theory to prove the existence of an infinite number of solutions to the system (1).

Let be a real Banach space, and in addition, let denote the class of all functionals that possess the following property:

If is a sequence in converge weakly to with inf ; thus, has a subsequence converge strongly to .

For offer, if is uniformly convex and is a continuous strictly increasing function, then the functional belongs to.

Definition 1 (see Kilbas et al. [4] chapter 2, p. 87). Let be a function defined on [a, b]. The right and left Riemann-Liouville fractional derivatives of for a function are defined by for all , provided the right-hand sides are pointwise defined on , where and .
Here, is the standard gamma function given by Set the functions space such that with As usual, denotes the mapping set having times continuously differentiable on . In particularly, we have

Definition 2 (see [31]). Let , for the derivative fractional space Thus, for all , we de ne the norm for as follows:

Lemma 3 (see [3]). Let and . For any , we have Also, if and , then Under the result of Lemma 3, we note that for , and for p and .
Under (14), we can see that (11) is equivalent to the following norm: For , . Analogous to the space , we define the fractional derivative space as Then, for any , the norm of is defined by Similar with (14) and (15), we get for , and Moreover, if and , then, based upon (19), the weighted norm is equivalent to (18), for every .
In the following discussion, for any , denote the space of with the norm where and are defined in (16) and (21), respectively.
Clearly, is embedded compactly on

Lemma 4 (see [33]). For , and . The derivative fractional space is a reflexive separable Banach space.

Lemma 5. Assume that and the sequence converge weakly to in , i.e., . Then, converges strongly to in , i.e., , as .

Definition 6 (see [3]). We point out to a weak solution to the system (1), for all such that for all .
We define for all : for every .

Lemma 7. Let satisfy (2) and , defined by (25). Thus, defined by is a Gâteaux function weakly sequentially differentiable over with

Proof. Assume that as . According to Lemma 5 that converges uniformly to on . Then, there exists such that and for any .
Then, for any and . Furthermore, at every , and by the Lebesgue Convergence Theorem Now we prove the Gâteaux differentiability of . Assume that and ; thus, where Thus is a Gâteaux differentiable for all .
Likewise, we have which is a Gâteaux differentiable for all .
Therefore, is a Gâteaux differentiable for all with its derivative For any three elements , , and of , it is easy to see that which implies where Hence, is a compact operator.

Similarly to the proof of Theorem 5.1 of [4], we have

Lemma 8 (see [36]). Let , , and . If is a nontrivial weak solution of problem (1), then is also a nontrivial solution of problem (1).

Our analysis is mainly based on the following critical points theorem of Bonanno and Molica Bisci [36], which is a more precise result of Ricceri ([37], Theorem 2.5).

Lemma 9 (see [[36], Theorem 2.1]). Let be a reflexive real Banach space. Let be two Gâteaux differentiable functionals such that is sequentially weakly lower semicontinuous, strongly continuous, and coercive and is sequentially weakly upper semicontinuous. For every , put

Then, (1)If and , the following alternative holds: either the functional has a global minimum or there exists a sequence of local minima such that (2)If and , the following alternative holds: either there exists a global minimum of or the following alternative holds: either there exists a global minimum of or there exists a sequence {u_n} of pairwise distinct local minima of , with , which weakly converges to a global minimum of

3. Main Results

Here, we prove our main results.

Setting

For a given constant , set

For any , we denote by the set

Theorem 10. Suppose that and (H0) hold. In addition,
(H1)
(H2) for any
(H3) there exists where if we set one has where and is given in (45).
Then, for every (1) has an unbounded sequence in (weak solutions).

Proof. Our goal is to apply a portion (1) of Lemma 9 to problem (1). First, by taking endowed with similar to what is considered in (22). We define the following functional for all , where Since is embedded compact in it is well known that is a well-defined Gâteaux differentiable functional whose Gâteaux derivative at the point is the functional , given by for every .
We claim that the functional is a sequentially weakly upper semicontinuous functional on . Indeed, for fixed , suppose that , in as . Then, converges uniformly to on . Hence, which implies that it is sequentially weakly upper semicontinuous. Hence, the claim is true.
Concerning the functional , we can show that what is defined by (56) is a sequentially weakly lower semicontinuous, strongly continuous, and coercive functional on . In fact since (2) holds for every and , one has , , for all . It follows from (14), (20), and Lemma 5 that for all and similarly for all . So is coercive.
Moreover, is a continuous functional on , and , from Lemma 5, is Gâteaux differentiable sequentially weakly continuous and therefore continuous on , then is a continuous functional on . It is not difficult to verify that the functional is a Gâteaux differentiable functional with the differential Furthermore, is also sequentially weakly lower semicontinuous on since is sequentially weakly lower semicontinuous, and if in then It is easy to show that the critical points of the functional and the weak solutions of the problem (1) are the same, and by Lemma 9, we prove our result. According to taking (13) and (20) into account, one has for every .
Hence, So, for every , from the definition of and by using (61), one has Set Note that , and from the condition (H1), . Hence, for every , and it follows from (68) that where Let be a sequence of positive numbers such that and Put for all . Let , by (68) one has which implies Hence, for large enough Thus, for all , Let Then, Hence,
For , we shall show that the functional is unbounded from below.
Indeed, since , we can choose a sequence of positive numbers and such that and for large enough.
For all , and define by setting Clearly and for . A direct calculation shows that Furthermore, Thus, , and This and (61) imply that From (H2), we have According to (80), (88), and (89), we have for large enough. Taking into account the choice of , the above inequality shows that which implies that the functional is unbounded from below and the claim follows.
By using the case (1) of Lemma 9, the functional has a sequence of critical points such that From (22) and (61), we get which implies and the proof of Theorem 10 is complete.

Theorem 11. Assume that and (H0) and (H2) hold. Furthermore, (H4) for all .
(H5) There exists such that, if we put one has where and is given in (45).
Then, for every (1) has a sequence of weak solutions such that .

Proof. Our goal is to apply part (2) of Lemma 9 to and defined in (48) and (51), respectively.
As it has been pointed out before, the functionals and satisfy the assumption regularity required in Lemma 9.
Since for all , then Let be a sequence of positive numbers such that and Setting for all , and working as in the proof of Theorem 10, we can show that and so .
Now fix as in the conclusion, then and there exist a sequence of positive numbers and a constant such that and and in addition For all , and define by setting Clearly for converges strongly to in .
By the same argument as in Theorem 10, we have for large enough. This together with the fact that shows that has no local minimum at zero, and the claim follows.
The alternative of Lemma 9 case (2) ensures the existence of sequence of pairwise distinct local minima of which weakly converges to . This completes the proof of Theorem 11.

Finally, we present an example to illustrate our main results.

Example 12. Consider the following fractional differential system: where . Moreover, for all put where where and for all .

Clearly, are two Lipschitz continuous functions of order 2 with Lipschitzian constants for all . With the aid of direct computation we have that

Let , then we have

Then, . Thus, all conditions of Theorem 10 are satisfied.

In fact, the conditions (H0), (H1), and (H2) hold. For all .

Restriction of on attains its maximum in and

In addition, and so which implies that the condition (H3) holds. Hence, owing to Theorem 10, for each , the coupled system (107) has an unbounded sequence of weak solutions.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

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Copyright © 2020 Rafik Guefaifia et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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