# On Existence of Multiplicity of Weak Solutions for a New Class of Nonlinear Fractional Boundary Value Systems via Variational Approach

Authors: Fares Kamache; Salah Mahmoud Boulaaras; Rafik Guefaifia; Nguyen Thanh Chung; Bahri Belkacem Cherif; Mohamed Abdalla
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

This paper deals with the existence of solutions for a new class of nonlinear fractional boundary value systems involving the left and right Riemann-Liouville fractional derivatives. More precisely, we establish the existence of at least three weak solutions for the problem using variational methods combined with the critical point theorem due to Bonano and Marano. In addition, some examples inℝ3 and ℝ4 are given to illustrate the theoritical results.


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## Body

## 1. Introduction

Fractional differential equations (FDEs) are a generalization of ordinary differential equations (ODEs), as they contain fractional derivatives whose degree is not necessarily an integer. This is what makes it receive great attention from researchers due to its ability to model some difficult and complex phenomena in many fields, including engineering, science, biology, economics, and physics (for more information, see [1–22]). One of the most investigated issues is the existence of solutions for the fractional initial and boundary value problems by using some fixed point theorems, coincidence degree theory, and monotone interactive method. Among the most important of these are the works mentioned in Oldham and Spanier and Podlubny’s books (see [13, 23]) and the work of Metzler and Klafter (see [24]). Furthermore, the first to use the critical point theorem was Jiao and Zhou in [6] to study the following problem:
(1)DtTαD0tαut=∇Ft,ut,a.et∈0,T,u0=ut=0,where D0Tα and DtTα are the left and right Riemann-Liouville fractional derivatives with 0<α≤1, respectively, and F:0,T×ℝ⟶ℝn is a suitable function satisfying some hypothesis and Ft,x is the gradient of F with respect to x.In [22], the authors have used variational methods to investigate the existence of weak solutions for the following system:
(2)DtTαatD0tαut=λFut,ut,vt,a.et∈0,T,DtTβbtD0tβvt=λFvt,ut,vt,a.et∈0,T,u0=uT=0,v0=vT=0,for D0Tα and DtTα are the left and right Riemann-Liouville fractional derivatives with 0<α≤1 and Fs denotes the partial derivative of F with respect to s. In [?], Zhao et al. obtained the existence of infinitely many solutions for system (2) with perturbed functions hi,i=1,2.Yet, there are a few findings for fractional boundary value problems which were established exploiting this approach due to its difficulty in establishing a suitable space and variational functional for fractional problems.In this work, we shall study the existence of three weak solutions for the following system:(3)DtTαiaitD0tαiuit=λFuit,u1t,u2t,⋯,unt+hiuit,a.et∈0,T,ui0=uiT=0,for 1≤i≤n, where αi∈0;1, D0Tαi and DtTαi are the left and right Riemann–Liouville fractional derivatives of order αi, respectively, ai∈L∞0,T with
(4)ai0=essinf0,Tai>0,for1≤i≤n,λ>0, F:0,T×ℝn⟶ℝ is a measurable function for all x1,⋯,xn∈ℝn and is C1 with respect to x1,⋯,xn∈ℝn for a.e. t∈0,T, Fui denotes the partial derivative of F with respect to ui, respectively, and hi:ℝ⟶ℝ are Lipschitz continuous functions with the Lipschitz constants Li>0, for 1≤i≤n, i.e.,
(5)hix1−hix2≤Lix1−x2,for all x1,x2∈ℝ and hi0=0, for 1≤i≤n. In order to state the main results, we introduce the following conditions:(F0) For all C>0 and any 1≤i≤n(6)supx1,⋯,xn≤CFuit,x1,⋯,xn∈L10,T.(F1) Ft;0,⋯,0=0, for a.e. t∈0;T.In the present study, motivated by the results introduced in [12, 13, 25], using the three critical point theorems due to Ricceri ([26], see Theorem 2.6 in the next section), we ensure the existence of at least three solutions for system (3). For other applications of Ricceri’s result for perturbed boundary value problems, the interested readers are referred to the papers [11–13, 23–25, 27].We divided the paper as follows: in the second section, we put some preliminary facts, while in the third section we presented the main result and its proof. Finally, we proposed two practical examples of our theorem.


## 2. Preliminaries

In this section, introducing some necessary definitions and preliminary facts.Definition 1 [28].
Letu be a function defined on 0,T and αi>0 for 1≤i≤n. The left and right Riemann–Liouville fractional integrals of order αi for the function u are defined by
(7)D0t−αiut=1Γαi∫0tt−sαi−1usds,t∈0,T,D0t−αiut=1Γαi∫tTs−tαi−1usds,t∈0,T,for 1≤i≤n, provided the RHS are pointwise given on 0,T, where Γαi is the standard gamma function defined by
(8)Γz=∫0+∞zαi−1e−zdz.Definition 2 [25].
Let0<αi≤1 for 1≤i≤n. The fractional derivative space H0αi is given by the closure C0∞0,T,ℝ, that is
(9)H0αi=C0∞0,T,ℝ¯,with the norm
(10)uiαi=∫0TaitD0tαiuit2dt+∫0Tuit2dt1/2,for every ui∈H0αi and for 1≤i≤n.We point out thatH0αi (0<αi≤1) is a reflexive and separable Banach space (see [22], Proposition 3.1) for details.For everyui∈H0αi, set
(11)uiLs≔∫0Tuitsdt1/s,s≥1ui∞=maxt∈0,Tuit.Definition 3 [27].
We mean by a weak solution of system (3), any u=u1,u2,⋯,un∈X such that for all v=v1,v2,⋯,vn∈X,
(12)∫0T∑i=1nait0Dtαiuit0Dtαivitdt−λ∫0T∑i=1nFuit,u1t,u2t,⋯,untvitdt−∫0T∑i=1nhiuivitdt=0.Lemma 4 [27].
Let0<αi≤1, for 1≤i≤n. ∀ui∈H0αi, we have
(13)uiL2≤TαiΓαi+1D0tαiuiL2.Moreover,(14)ui∞≤TαiΓαi2αi−1D0tαiuiL2.From Lemma4, we easily observe that
(15)uiL2≤TαiΓαi+1αi0∫0TaitD0tαiuit2dt1/2.for 0<αi≤1, and
(16)ui∞≤Tαi−1/2Γαiai02αi−1∫0TaitD0tαiuit2dt1/2.By using (15), the norm of (10) is equivalent to
(17)uiαi∫0TaitD0tαiuit2dt1/2,∀ui∈H0αi.Throughout this paper, letX be the Cartesian product of the n spaces H0αi for 1≤i≤n, i.e., X=H0α1×H0α2×⋯×H0αn; we equip X with the norm defined by
(18)u=∑i=1nuiH0αi,u=u1,u2,⋯,un,where uiH0αi is given in (17). We have X compactly embedded in C0,T,ℝn.Theorem 5 [25].
LetX be a reflexive real Banach space and Φ:X⟶ℝ be a coercive, continuously Gâteaux differentiable sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on X∗, bounded on bounded subsets of X,Ψ:X⟶ℝ a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact such that
(19)Φ0=Ψ0=0.
Suppose that∃r>0 and x¯∈X, with r<Φx¯, satisfying
(a1) supΦu≤rΨu/r<Φx¯/Ψx¯.
(a2) For each λ∈Λλ;=Φx¯/Ψx¯,r/supΦu≤rΨu, the functional Φ−λΨ is coercive.
Hence,∀λ∈Λλ, the functional Φ−λΨ has at least three critical points in the space X.


## 3. Main Results

In this section, by applying Theorem5, we examine the existence of multiple solutions for system (3). For any σ>0, let us define
(20)πσ=x1,⋯,xn∈ℝn:12∑i=1nxi2≤σ.This set will be used in some of our hypotheses with appropriate choices ofσ. For u=u1,u2,⋯,un∈X, we define
(21)Yu≔∑i=1nYiuiwhere Yix=∫0THixsds and Hix=∫0xhizdz1≤i≤n, ∀t∈0;T and x∈ℝ.Furthermore, let(22)k≔max1≤i≤nT2αi−1Γαi2ai02αi−1,M≔min1≤i≤n1−LiT2αiΓαi+12ai0,k~≔max1≤i≤n1+LiT2αiΓαi+12ai0.Theorem 6.
Let1/2<αi≤1, for 1≤i≤n, and suppose that M>0 and the conditions (F0) and (F1) are satisfied. Furthermore, assume that ∃r>0 and a function ω=ω1,ω2,⋯,ωn∈X satisfying
(23)i∑i=1nωiαi22>rM,ii2r∫0TFt,ω1,ω2,⋯,ωndt∑i=1nωiαi2−2Yω1,ω2,⋯,ωn−∫0Tmaxx1,⋯,xn∈πkrM,Ft,x1,⋯,xndt>0,iiilimx1,⋯,xn⟶+∞,⋯,+∞supsupt∈0,TFt,x1,⋯,xn∑i=1nxi2/2≤0.Then, setting(24)Λ=∑i=1nωiαi2/2−Yω1,ω2,⋯,ωn∫0TFt,ω1t,ω2t,⋯,ωntdt,r∫0Tmaxx1,⋯,xn∈πM/krFt,x1,⋯,xndt.∀λ∈Λ system (3) admits at least 3 weak solutions in X.Proof.
For eachu=u1,u2,⋯,un∈X, we introduce the functionals Φ,Ψ:X⟶ℝ as
(25)Φu=∑i=1nuiαi22−Yu,(26)Ψu=∫0TFt,u1t,u2t,⋯,untdt.It is clear thatΦ and Ψ are continuously Gâteaux differentiable functionals whose Gâteaux derivatives at the point u∈X are defined by
(27)Φ′uv=∫0T∑i=1nait0Dtαiuit0Dtαivitdt−∫0T∑i=1nhiuitvitdtΨ′uv=∫0T∑i=1nFuit,u1t,u2t,⋯,untvitdt,for everyv=v1,v2,⋯,vn∈X.We haveΦ′u,Ψ′u∈X∗, where X∗ is the dual space of X. And the functional Φ is sequentially weakly lower semicontinuous and its Gâteaux derivative admits a continuous inverse on X∗; also limuX⟶+∞Φu=+∞ it is coercive. Now, we show that the functional Ψ is sequentially weakly upper semicontinuous and its derivative Ψ′:X⟶X∗ is a compact operator. Let um⇀u in X, where umt=um,1t,um,2t,⋯,um,nt; then certainly um converges uniformly to u on the interval 0,T. Then,
(28)limsupm⟶+∞Ψum≤∫0Tlimsupm⟶+∞Ft,um,1t,um,2t,⋯,um,ntdt=∫0TFt,u1t,u2t,⋯,untdt=Ψu,which gets thatΨ is sequentially weakly upper semicontinuous.Moreover, we have(29)limm⟶+∞Ft,um,1t,um,2t,⋯,um,nt=Ft,u1t,u2t,⋯,unt,forallt∈0,T.Note thatFt,·,⋯,·∈C1ℝn. The Lebesgue control convergence theorem implies that Ψm′u⟶Ψ′u strongly, hence yielding that Ψ′ is strongly continuous on X. Then, Ψ′:X⟶X∗ is a compact operator.We show that required hypothesisΦx¯>r follows from (i) and the definition of Φ by taking x¯=ω. Indeed, as (5) holds for all x1;x2∈ℝ and h10=⋯=hn0=0; one has hix≤Lix,1≤i≤n, for any x∈ℝ. It follows from (15) that
(30)Φω≥∑i=1nωiαi22−∫0T∑i=1nHiωitdt≥∑i=1nωiαi22−∑i=1nLi2∫0Tωit2dt≥∑i=1n12−LiT2αiΓαi+12ai0ωiαi2≥M2∑i=1nωiαi2.From (16), for every ui∈H0αi, we have
(31)maxt∈0,Tuit2≤kuiαi,for1≤i≤n. Hence,
(32)maxt∈0,T∑i=1nuit2≤k∑i=1nuiαi.Assume thatu0t=0,⋯,0 and the supposition (i) deduces that 0<r<Φω and they hold Φu0t=Ψu0t=0 from definitions (25) and (26), which are required assumptions in Theorem 5. Applying relations (16), (17), and (22) gives the following relation:
(33)Φ−1−∞;r=u=u1,u2,⋯,un∈X:Φu≤r=u=u1,u2,⋯,un∈X:∑i=1nuiαi22≤rM⊆u=u1,u2,⋯,un∈X:∑i=1nΓαi2a102αi−12T2αi−1ui∞2≤rM⊆u=u1,u2,⋯,un∈X:12∑i=1nuit2≤krM,which implies that
(34)supu∈Φ−1−∞;rΨu=supu∈Φ−1−∞;r∫0TFt,u1t,u2t,⋯,untdt≤∫0Tmaxx1,⋯,xn∈πkrMFt,x1,⋯,xndt.Hence, under the condition (ii), we get the following inequality
(35)supu∈Φ−1−∞;rΨur≤∫0Tmaxx1,⋯,xn∈πkrMFt,x1,⋯,xndt<2r∫0TFt,ω1,ω2,⋯,ωndt∑i=1nωiαi2−2Yω1,ω2,⋯,ωn=r∫0TFt,ω1,ω2,⋯,ωndt∑i=1nωiαi2/2−Yω1,ω2,⋯,ωn=ΨωΦω.Thus, the hypothesis (a1) of Theorem 5 holds.On the other hand, fix0<ε<1/2Tkλ. From (iii) into account, there exist constants τε∈ℝ such that
(36)Ft,x1,⋯,xn≤ε∑i=1nxi2+τε,for any t∈0,T and x1,⋯,xn∈ℝn, by using (36) and (15) yields, it follows that, for each u∈X,
(37)Φu−λΨu=12∑i=1nuiαi2−λ∫0TFt,u1t,u2t,⋯,untdt≥12∑i=1nuiαi2−Tλkε∑i=1nuiαi2−λτε≥12−Tλkε∑i=1nuiαi2−λτε.And from him,(38)limuX⟶+∞Φu−λΨu=+∞.Moreover, analogous to the case ofτε>0, we imply that Φu−λΨu⟶+∞ as uX⟶+∞ with τε≤0. Then, the hypotheses of Theorem 5 hold, which means that system (3) admits at least 3 weak solutions in X, which completes the proof.Now, we present some notations, before the corollary of Theorem6.Put(39)Aiαi=16T2∫0Taitt21−αidt+∫T/4Taitt−T421−αidt+∫3T/4Taitt−3T421−αidt−2∫T/4Taitt2−T4t1−αidt−2∫3T/4Taitt2−3T4t1−αidt+2∫3T/4Taitt2−Tt+3T216t1−αidt,Δ1=min1≤i≤nAiαi:for1≤i≤n,Δ2=max1≤i≤nAiαi:for1≤i≤n.Corollary 7.
Let1/2<αi≤1,1≤i≤n and supposition (iii) in Theorem 6 holds. Suppose that ∃τ>0 and d such that τ/Δ1kMn<d2, and also
(40)i′Ft,x1,⋯,xn≥0,fort,x1,⋯,xn∈0,T4∪3T4,T×0,+∞×⋯×0,+∞,ii′∫0Tmaxx1,⋯,xn∈πτFt,x1,⋯,xndtτM<∫T/43T/4Ft,Γ2−α1d,Γ2−α2d,⋯,Γ2−αnddtnkk~Δ2d2,iii′limx1,⋯,xn⟶+∞,⋯,+∞supsupt∈0,TFt,x1,⋯,xn∑i=1nxi2/2≤0.
Then, setting(41)λ∈Λ′=nk~Δ2d2∫T/43T/4Ft,Γ2−α1d,Γ2−α2d,⋯,Γ2−αnddt,τk∫0Tmaxx1,⋯,xn∈πτFt,x1,⋯,xndtThus, system (3) admits at least three weak solutions in X.Proof.
Choose(42)ωit=4Γ2−αidTt,t∈0,T4,Γ2−αid,t∈T4,3T4,4Γ2−αidhTT−t,t∈3T4,T.
We derive(43)D0tαiωit=4dTt1−αi,t∈0,T4,4dTt1−αi−t−T41−αi,t∈T4,3T4,4dTt1−αi−t−T41−αi−t−3T41−αi,t∈3T4,T.Moreover,(44)∫0TaitD0tαiωit2dt=∫0T/4+∫T/43T/4+∫3T/4TaitD0tαiωitpdt=2Aiαid2.Then,ωi0=ωiT=0,ωit,0Dtαiωit∈L20,T, i=1,2,⋯,n; hence, ω=ω1,ω2,⋯,ωn∈X, and we have
(45)ωiαi2=∫0TaitD0tαiωit2dt=2Aiαid2.By (25), for 1≤i≤n, imply that
(46)Φω=Φω1,ω2,⋯,ωn=∑i=1nωiαi22−Yω≥M2∑i=1nωiαi2=Md2∑i=1nAαi≥nMΔ1d2.Similar to (30) and (46), we have Φω≤nk~Δ2d2.Letr=τM/k. From τ/Δ1kMn<d2, we have
(47).∑i=1nωiαi22≥Φω≥nMΔ1d2>nMΔ1×τΔ1kMn=rM.Thus, the assumption (ii) of Theorem 6 holds.(i′) implies that
(48)Ψω=∫0TFt,ω1,ω2,⋯,ωndt=∫0T/4Ft,ω1,ω2,⋯,ωndt+∫T/43T/4Ft,ω1,ω2,⋯,ωndt+∫3T/4TFt,ω1,ω2,⋯,ωndt≥∫T/43T/4Ft,ω1,ω2,⋯,ωndt.Moreover, by condition (ii′), we have
(49)∫0Tmaxx1,⋯,xn∈πkr/MFt,x1,⋯,xndtr=k∫0Tmaxx1,⋯,xn∈πkr/MFt,x1,⋯,xndtτM<∫T/43T/4Ft,Γ2−α1d,Γ2−α2d,⋯,Γ2−αnddtnk~Δ2d2≤∫T/43T/4Ft,Γ2−α1d,Γ2−α2d,⋯,Γ2−αnddtΦω≤2∫0TFt,ω1,ω2,⋯,ωndt∑i=1nωiαi2−2Yω1,ω,⋯,ωn.Hence, the supposition (ii) of Theorem 6 is verified.Moreover, the supposition (iii) of Theorem 6 holds under (iii′) from Λ′⊆Λ. Theorem 6 is successfully employed to ensure the existence of at least 3 weak solutions for system (3). This completes of the proof.


## 4. Examples

In this section, we propose two practical examples of Theorem6.Example 1.
Letα1=0.7,α2=0.65,α3=0.6,a1t=1+t2,a2t=0.5+t,a3t=1+t,T=1. Then, system (3) gets the following form:
(50)Dt10.71+t2D0t0.7u1t=λFu1t,u1t,u2t,u3t+h1u1,t∈0,1,Dt10.650.5+tD0t0.65u2t=λFu2t,u1t,u2t,u3t+h2u2,t∈0,1,Dt10.61+tD0t0.6u3t=λFu3t,u1t,u2t,u3t+h3u3,t∈0,1,u10=u11=0,u20=u21=0,u30=u31=0,where h1u1=1/4sinu1,h2u2=u2/2, and h3u3=1/20arctanu3.
Furthermore,∀t;x1,x2,x3∈0;1×R3; put
(51)Ft,x1t,x2t,x3t=1+t2Gx1,x2,x3,where
(52)Gx1,x2,x3=x12+x22+x322,x12+x22+x32≤1,10x12+x22+x3212−9x12+x22+x321/3,x12+x22+x32>1.
Obviouslyh1,h1,h3⟶ℝ are three Lipschitz continuous functions with Lipschitz constants L1=1/4,L2=1/2,L3=1/20 and h10=h20=h30=0. Clearly, Ft,0,0,0=0,∀t∈0,1, by the direct calculation, we have a10=1, a20=1, and a30=0.5(53)k=max1Γ0.722×0.7−1,1Γ0.6522×0.65−1,1Γ0.62×0.52×0.6−1≈4.509191,M=min1−L1Γ0.7+12,1−L2Γ0.65+12,1−L3Γ0.6+12×0.5≈0.912084
Taking(54)ω1t=Γ1.3t1−t,ω2t=Γ1.35t1−t,ω3t=Γ1.4t1−tD0t0.7ω1t=t0.3−2Γ1.3Γ2.3t1.3,D0t0.65ω2t=t0.35−2Γ1.35Γ2.35t1.35,D0t0.6ω3t=t0.4−2Γ1.4Γ2.4t1.4.By a simple calculation, we obtain(55)ω1t0.72≈0.130566,ω2t0.652≈0.078559,ω3t0.62≈0.102638.Selectr=1×10−3, we find
(56)ω1t0.72+ω2t0.652+ω3t0.62≈0.311763>2rM≈0.002192.We deduce that the supposition (i) holds, and
(57)∫01maxx1,x2,x3∈πkr/MFt,x1,x2,x3dtr=16k2r3M2≈0.130355<2∫01Ft,ω1,ω2,ω3dtω1t0.72+ω2t0.652+ω3t0.62−Yω1,ω2,ω3≈0.365517,limx1,x2,x3⟶+∞,+∞,+∞supsupt∈0,TFt,x1,x2,x3x12/2+x22/2+x32/2=0.Then, suppositions (ii) and (iii) are verified. Hence, in view of Theorem 6 for every λ∈2.7359,7.6714, system (50) has at least 3 weak solutions in the space X=H00.7×H00.65×H00.6.Example 2.
Letα1=0.65,α2=0.75,α3=0.85,α4=0.95, a1t=1+t3,a2t=1+t2, a3t=0.5+t,a4t=1+t,T=1.
Hence, system (3) gives
(58)Dt10.651+t3D0t0.65u1t=λFu1t,u1t,u2t,u3t,u4t+h1u1,t∈0,1,Dt10.751+t2D0t0.75u2t=λFu2t,u1t,u2t,u3t,u4t+h2u2,t∈0,1,Dt10.850.5+tD0t0.85u3t=λFu3t,u1t,u2t,u3t,u4t+h3u3,t∈0,1,Dt10.951+tD0t0.95u4t=λFu4t,u1t,u2t,u3t,u4t+h4u4,t∈0,1,u10=u11=0,u20=u21=0,u30=u31=0,u40=u41=0.
Taking(59)ω1t=Γ1.35t1−t,ω2t=Γ1.25t1−t,ω3t=Γ1.15t1−t,ω4t=Γ1.05t1−t.
Moreover, for allt;x1,x2,x3,x4∈0;1×R4, put
(60)Ft,x1t,x2t,x3t,x4t=1+t2Gx1,x2,x3,x4,where
(61)Gx1,x2,x3,x4=x12+x22+x32+x422,x12+x22+x32+x42≤1,10x12+x22+x32+x4212−9x12+x22+x32+x4213,x12+x22+x32+x42>1.
Obviouslyh1,h1,h3,h4⟶ℝ are three Lipschitz continuous functions, h1u1=1/4sinu1,h2u2=u2/20 and h3u3=1/100arctanu3,h4u4=1/10lnu4+1 for all u1,u2,u3,u4∈ℝ with Lipschitz constants L1=1/4,L2=1/20,L3=1/100,L4=1/10 and h10=h20=h30=h40=0. Clearly, Ft,0,0,0,0=0 for any t∈0,1, a10=1, a20=0.5, a30=1, and a40=1The direct calculation, gives(62)k=max1Γ0.6522×0.65−1,1Γ0.752×0.52×0.75−1,1Γ0.8522×0.85−1,1Γ0.9522×0.95−1≈2.663742,M=min1−L1Γ0.65+12,1−L2Γ0.75+12×0.5,1−L3Γ0.85+12,1−L4Γ0.95+12≈0.956042,D0t0.65ω1t=t0.35−2Γ1.35Γ2.35t1.35,D0t0.75ω2t=t0.25−2Γ1.25Γ2.25t1.25,D0t0.85ω3t=t0.15−2Γ1.15Γ2.15t1.15,D0t0.95ω3t=t0.05−2Γ1.05Γ2.05t1.05.So that(63)ω1t0.652≈0.104555,ω2t0.752≈0.158153,ω3t0.852≈0.170894,ω4t0.952≈0.397611.Selectr=1×10−3; we find
(64)ω1t0.652+ω2t0.752+ω3t0.852+ω4t0.952≈0.831213>2rM≈0.002092.We deduce that the supposition (i) holds, and
(65)∫01maxx1,x2,x3,x4∈πkr/MFt,x1,x2,x3,x4dtr=16k2r3M2≈0.041403<2∫01Ft,ω1,ω2,ω3,ω4dtω1t0.652+ω2t0.752+ω3t0.852+ω4t0.952−Yω1,ω2,ω3,ω4≈0.135277,limx1,x2,x3,x4⟶+∞,+∞,+∞,+∞supsupt∈0,TFt,x1,x2,x3,x4x12/2+x22/2+x32/2+x42/2=0.Then, suppositions (ii) and (iii) are verified. Hence, in view of Theorem 6 for every λ∈7.3922,24.1528, system (58) has at least 3 weak solutions in the space X=H00.65×H00.75×H00.85×H00.95.


## 5. Conclusion

In this work, at least 3 weak solutions were obtained for a new class of nonlinear fractional BVPs using a critical three-point theorem due to Bonano and Marano. Some appropriate function spaces and variational frameworks were successfully created for system (3). Finally, we suggested two practical examples of Theorem 6 with a special case discussion ℝ3. As for case ℝ4, it was discussed. This makes our results prominent and distinct than previous ones. In the next work, we extend our recent work to the coupled system for this important problem. Also some numerical examples will be given in order to ensure the theory study by using some famous algorithms which are presented in ([28, 29]).



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*Source: 5544740-2021-03-13.xml*