Infinite Existence Solutions of Fractional Systems with Lipschitz Nonlinearity

Laboratory of Mathematics, Informatics and System (LAMIS), Larbi Tebessi University, Tebessa, Algeria Department of Mathematics, College of Sciences and Arts, Qassim University, Ar Rass, Saudi Arabia Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO), University of Oran 1 Ahmed Ben Bella, Algeria Department of Mathematics and Statistics, Faculty of Management Technology and Information Systems, Port Said University, Port Said, Egypt


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 X be a real Banach space, and in addition, let Y X denote the class of all functionals that possess the following property: If fw n g is a sequence in X converge weakly to w ∈ X with lim n→∞ inf ϕðw n Þ ≤ ϕðwÞ; thus, fw n g has a subsequence converge strongly to w.
For offer, if X is uniformly convex and S : ½0,+∞Þ ⟶ ℝ is a continuous strictly increasing function, then the functional w ⟶ SðkwkÞ belongs toY X .
for all t ∈ ½a, b, provided the right-hand sides are pointwise defined on ½a, b, where n − 1 ≤ α < n and n ∈ ℕ.
Here, ΓðαÞ is the standard gamma function given by with As usual, C n−1 ð½a, b, ℝÞ denotes the mapping set having ðn − 1Þ times continuously differentiable on ½a, b. In particularly, we have Definition 2 (see [31]). Let 0 < α ≤ 1, for 1 < p < ∞ the derivative fractional space Thus, for all u ∈ E p α , we de ne the norm for E p α as follows: Lemma 3 (see [3]). Let 0 < α ≤ 1 and 1 < p < ∞. For any u ∈ E p α , we have Also, if α > p and 1/p Under the result of Lemma 3, we note that for 0 < α ≤ 1, and for α > p and 1/p + 1/q = 1.
Under (14), we can see that (11) is equivalent to the following norm: Then, for any v ∈ E p β , the norm of E p β is defined by
Definition 6 (see [3]). We point out to a weak solution to the system (1), for all ðu, vÞ ∈ X such that for all ðx, yÞ ∈ X.
We define for all x ∈ ℝ: for every t ∈ ½0, T. (2) and H i ðxÞ, Θ i ðxÞ, i = 1, 2, defined by (25). Thus, Θðu, vÞ: X ⟶ ℝ defined by is a Gâteaux function weakly sequentially differentiable over X with Proof. Assume that as n → +∞. According to Lemma 5 that ðu n , v n Þ converges uniformly to ðu, vÞ on ½0, T. Then, there exists c 1 , c 2 > 0 such that ku n k ∞ ≤ c 1 and kv n k ∞ ≤ c 2 for any n ∈ ℕ. Then, for any n ∈ ℕ and t ∈ ½0, T. Furthermore, H 1 ðu n ðtÞÞ ⟶ H 1 ðuðtÞÞ and H 2 ðv n ðtÞÞ ⟶ H 2 ðvðtÞÞ at every t ∈ ½0, T, and by the Lebesgue Convergence Theorem Now we prove the Gâteaux differentiability of Θ. Assume 3 Journal of Function Spaces that u, x ∈ E p α and s ≠ 06; thus, where Thus is a Gâteaux differentiable for all u ∈ E p α . Likewise, we have which is a Gâteaux differentiable for all v ∈ E p α . Therefore, is a Gâteaux differentiable for all ðu, vÞ ∈ X with its derivative For any three elements ðu 1 , v 1 Þ, ðu 2 , v 2 Þ, and ðx, yÞ of X, it is easy to see that which implies where Hence, Θ ′ : X ⟶ X * is a compact operator.
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 X be a reflexive real Banach space. Let ϕ, Ψ : X ⟶ ℝ be two Gâteaux differentiable functionals such that ϕ is sequentially weakly lower semicontinuous, strongly continuous, and coercive and Ψ is sequentially weakly upper semicontinuous. For every r > inf X ϕ, put Then, (1) If γ < +∞ and λ ∈ 0, 1/γ½, the following alternative holds: either the functional ϕ − λΨ has a global minimum or there exists a sequence fu n g of local minima ϕ − λΨ such that lim n→+∞ ϕðu n Þ = +∞ (2) If γ < +∞ and λ ∈ 0, 1/δ½, the following alternative holds: either there exists a global minimum of ϕ or the following alternative holds: either there exists a global minimum of ϕ − λΨ or there exists a sequence {u n } of pairwise distinct local minima of ϕ − λΨ, with lim n→+1 ϕðu n Þ = inf X ϕ, which weakly converges to a global minimum of ϕ
Proof. Our goal is to apply a portion (1) of Lemma 9 to problem (1). First, by taking endowed with kðu, vÞk X similar to what is considered in (22). We define the following functional for all ðu, vÞ ∈ X, where Since X is embedded compact in it is well known that is a well-defined Gâteaux differentiable functional whose Gâteaux derivative at the point ðu, vÞ ∈ X is the functional Ψ ′ ðu, vÞ ∈ X * , given by for every ðx, yÞ ∈ X. We claim that the functional Ψ is a sequentially weakly upper semicontinuous functional on X. Indeed, for fixed ðu, vÞ ∈ X, suppose that fðu n , v n Þg ⊂ X, ðu n , v n Þ ⇀ ðu, vÞ in X as n ⟶ +∞. Then, ðu n , v n Þ converges uniformly to ðu, vÞ on ½0, T. Hence, which implies that it is sequentially weakly upper semicontinuous. Hence, the claim is true.
Concerning the functional ϕ, we can show that what is defined by (56) is a sequentially weakly lower semicontinuous, strongly continuous, and coercive functional on X. In fact since (2) holds for every x 1 , x 2 ∈ R and h 1 ð0Þ = h 2 ð0Þ = 0, one has jh i ðxÞj ≤ L i jxj p−1 , i = 1, 2, for all x ∈ ℝ. It follows from 5 Journal of Function Spaces (14), (20), and Lemma 5 that for all ðu, vÞ ∈ X and similarly for all ðu, vÞ ∈ X. So ϕ is coercive. Moreover, ϕ + Θ is a continuous functional on X, and Θ, from Lemma 5, is Gâteaux differentiable sequentially weakly continuous and therefore continuous on X, then ϕ is a continuous functional on X. It is not difficult to verify that the functional is a Gâteaux differentiable functional with the differential Furthermore, ϕ is also sequentially weakly lower semicontinuous on X since Θ is sequentially weakly lower semicontinuous, and if ðu n , v n Þ ⇀ ðu, vÞ in X then It is easy to show that the critical points of the functional I λ 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 ðu, vÞ ∈ X. Hence, So, for every r > 0, from the definition of and by using (61), one has Set Note that ϕð0, 0Þ = 0, and from the condition (H1), Ψð0 , 0Þ ≥ 0. Hence, for every r > 0, and it follows from (68) that where Ω Mr k = u, v ð Þ∈ X : Journal of Function Spaces Let fξ n g be a sequence of positive numbers such that ξ n ⟶ +∞ and lim ξ→+∞ inf Ð T 0 sup x j j+ y j j≤ξ F t, x, y ð Þdt Put r n = ðk/p2 p MÞξ p n for all n ∈ ℕ. Let ðu, vÞ ∈ ϕ 1 ð1−∞, r n Þ, by (68) one has Hence, Λ⊑0, 1/γ½: For λ ∈ Λ, we shall show that the functional I λ is unbounded from below.
Indeed, since B ∞ /ρ Δ > 1/λ, we can choose a sequence fη n g of positive numbers and ε > 0 such that η n ⟶ +∞ and