# Multiplicity Solutions of Fractional Impulsivep-Laplacian Systems: New Result

Authors: Rafik Guefaifia; Mohamed Abdalla; Tahar Bouali; Fares Kamache; Bahri Belkacem Cherif; Ibrahim Mekawy
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, the existence of multiplicity distinct weak solutions is proved for differentiable functionals for perturbed systems of impulsive nonlinear fractional differential equations. Further, examples are given to show the feasibility and efficacy of the key findings. This work is an extension of the previous works to Banach space.


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## Body

## 1. Introduction

This paper explores the perturbed impulsive fractional differential system(1)tDTαiϕp0Dtαiuit=λFuit,u+μGuit,u+hiuit,t∈0,T,ΔtDTαi−1ϕp0cDtαiuittj=Iijuitj,1≤i≤n,j=1,2,⋯,m,ui0=uiT=0,where u=u1,u2,⋯,un, n≥1, 0<αi≤1 for 1≤i≤n, 0Dtαi and tDTαi are the left and right Riemann–Liouville fractional derivatives of order αi, respectively, φps=sp−2s, s≠0, φp0=0, p>1, λ>0, μ>0, T>0, and F,G:0,T×ℝn⟶ℝ are L1-Caratheodory functions, and they satisfy in the following standard summability condition:
(2)sup∣ξ∣≤ς1maxF.,ξ,G.,ξ,Fξi.,ξ,Gξi.,ξ,i=1,⋯,n∈L10,T,for any ς1>0 with ξ=ξ1,ξ2,⋯,ξn and ξ=∑i=1nξipp, and hi:ℝ⟶ℝ is a p−1-Lipschitz continuous function with the Lipschitz constant Li>0, i.e.,
(3)hiξ1−hiξ2≤Liξ1−ξ2p−1,for every ξ1,ξ2∈ℝ, satisfying hi0=0 for 1≤i≤n. The operator Δ is defined as 0<t0<t1<⋯<tm+1=T and
(4)ΔtDTαi−1ϕp0cDtαiuitj=DtTαi−1ϕp0cDtαiuitj+−DtTαi−1ϕp0cDtαiuitj−,where
(5)DtTαi−1ϕp0cDtαiuitj+=limt⟶tj+DtTαi−1ϕp0cDtαiuit,DtTαi−1ϕp0cDtαiuitj−=limt⟶tj−DtTαi−1ϕp0cDtαiuit,and Ij∈Cℝ,ℝ is a p−1-Lipschitz continuous function with the Lipschitz constant Lij>0, i.e.,
(6)Ijξ1−Ijξ2≤Lijξ1−ξ2p−1.Here,Fui and Gui are the partial derivatives of F and G with respect to ui for 1≤i≤n, respectively.In science and engineering, fractional differential equations (FDEs) have recently proved to be useful methods for modeling a broad variety of phenomena. In viscoelasticity, electrochemistry, power, porous media, and electromagnetism, for instance, see [1–33] and the references therein. Many articles have recently investigated the existence of solutions to boundary value problems for FDEs, and we refer the reader to one of them [2, 18–20, 34–46] and the references therein. For example, Kamache et al. [40] investigated the existence of three solutions for a class of fractional p-Laplacian systems using a variational structure and critical point theory.In [36], we investigated the existence of solutions of the periodic boundary value problem for a nonlinear impulsive fractional differential equation with periodic boundary conditions:
(7)D2αut=ft,u,Du,t∈0,1\t0,t1,⋯,tm,0<α<1,(8)limt⟶0+t1−αut=u1,(9)limt⟶0+t1−αDαut=Dαu1,(10)limt⟶tj+t−tj1−αut−utj=Ijut,(11)limt⟶tj+t−tj1−αDαut−Dαutj=I¯jut,where Dαut=D0tαut=1/Γ2−αd/dt∫0tt−τ−αuτdτ is the standard Riemann–Liouville fractional derivative, D2αu=DαDαu is the sequential Riemann–Liouville fractional derivative presented by Miller and Ross on p. 209 of [14], 0<t0<t1<⋯<tm=1, Ij,I¯j∈Cℝ,ℝj=1,⋯,m, and f is continuous at every point t,u,v∈0,1×ℝ×ℝ. By using the method of upper and lower solutions and its associated monotone iterative method, the author studies the existence and uniqueness of the solution of the periodic boundary value problem for the nonlinear impulsive fractional differential equation (7).Upon using variational methods and critical point theory, the presence of one weak solution for the system was also demonstrated in [19] with μ=0 and Iij=0 for i=1,⋯,n and j=1,⋯,m.Impulsive effects are a common phenomenon triggered by short-term perturbations that are negligible in relation to the original operation’s total duration. Such perturbations can be approximated fairly well as instantaneous changes of state or in the form of impulses. Such phenomena governing equations can be interpreted as impulsive differential equations. There has been a surge in interest in the study of impulsive differential equations in recent years, as these equations provide a natural framework for mathematical modeling of many real-world phenomena, especially in control theory, physics, chemistry, population dynamics, biotechnology, economics, and medical fields. Under such boundary conditions, the presence of solutions for impulsive differential equations with variational structures is determined by variational methods. See, for example, [36] as well as the references therein. Many scholars have recently studied fractional differential equations with impulses using variational methods, fixed point theorems, and critical point theory, due to the rapid growth in the theory of fractional calculus and impulsive differential equations, as well as their broad applications in a variety of fields (see, for example, [35, 44] and the references therein for a thorough discussion, as well as the sources therein for more details). For example, Gao et al. provided sufficient conditions for the existence and uniqueness of solutions for a class of impulsive integrodifferential equations with nonlocal conditions involving the Caputo fractional derivative using the Schaefer fixed point theorems (see [45]).The existence of infinitely many solutions for the system (1) was discussed in [46] using variational methods. Some new parameters to guarantee that the system (1), in the case μ=0, has at least two nontrivial and nonnegative solutions were obtained in [30] under appropriate hypotheses and using variational methods.Recently, in Reference [27], perturbed systems of impulsive nonlinear fractional differential equations were studied, including continuous nonlinear Lipschitz terminology where at least three distinct weak solutions were demonstrated based on the modern critical point theory of differentiable functions, but here, we will prove the existence of three distinct weak solutions for differentiable functionals for perturbed systems of impulsive nonlinear fractional differential equations.Most precisely, in this work, we extend the last work [38] to Banach space, where we show that there are at least three weak solutions for the system (1), which involves two parameters λ and μ. Furthermore, we do not need any asymptotic conditions of the nonlinear term at infinity in our new findings. The proof is based on a three-critical point theorem proved by Bonanno and Candito in [32], which we will revisit in the following section (Theorem 1). Theorem 10 is our most important finding. As a result, Theorem 11 can be deduced. Theorem 11 is shown in Example 1. When it comes to a scalar situation (n=1), we obtain Theorems 14 and 15 as special cases of Theorems 10 and 11. Theorem 15 is shown in Example 2. Under appropriate conditions on the nonlinear term at zero and at infinity, we obtain the presence of at least two positive solutions in Theorem 16.The present paper is organized as follows. In Section2, we recall some basic definitions and preliminary results, while Section 3 is devoted to the existence of multiple weak solutions for the eigenvalue system (1).


## 2. Preliminaries

LetX be a nonempty set and Φ,Ψ:X⟶ℝ be two functions. For all r1,r2,r3>infXΦ, r2>r1,r3>0, we define
(12)φr≔infu∈Φ−1−∞,rsupv∈Φ−1−∞,rΨv−Ψur−Φu,βr1,r2≔infu∈Φ−1−∞,rsupv∈Φ−1r1,r2Ψv−ΨuΦv−Φu,γr2,r3≔supu∈Φ−1−∞,r2+r3r3,αr1,r2,r3≔maxφr1,φr2,γr2,r3.Theorem 1 ([32], Theorem 3.3).
LetX be a reflexive real Banach space; let Φ:X⟶ℝ be a coercive and continuously Gateaux differentiable and sequentially weakly lower semicontinuous functional whose Gateaux derivative admits a continuous inverse on X∗, where X∗ is the dual space of X, and let Ψ:X⟶ℝ be a continuously Gateaux differentiable functional whose Gateaux derivative is compact, such that
a1: Φ is convex, and infXΦ=Φ0=Ψ0=0.
a2: for every u1,u2∈X such that Ψu1≥0 and Ψu2≥0, one has
(13)infs∈0,1Ψsu1+1−su2≥0.
Assume that there are three positive constantsr1, r2, and r3 with r1<r2, such that
(14)a3φr1<βr1,r2,a4φr2<βr1,r2,a5γr2,r3<βr1,r2.
Then, for eachλ∈1/βr1,r2,1/αr1,r2,r3, the functional Φ−λΨ admits three distinct critical points u1, u2, and u3 such that u1∈Φ−1−∞,r1, u2∈Φ−1r1,r2, and u3∈Φ−1−∞,r2+r3.
Now, we introduce some important fractional calculus concepts and properties that will be used in this paper.
LetC0∞0,T,ℝn be the set of all functions x∈C0∞0,T,ℝn with x0=xT=0 and the norm
(15)x∞=maxt∈0,Txt
Denote the norm of the spaceLp0,T,ℝn for 1≤p<∞ by
(16)xLp=∫0Txspds.
The following lemma yields the boundedness of the Riemann–Liouville fractional integral operators from the spaceLp0,T,ℝn to the space Lp0,T,ℝn, where 1≤p<∞.Definition 2 [35].
The left and right Riemann–Liouville fractional derivatives of orderαi for the function u are defined in the following forms, respectively,
(17)0Dtαiut=1Γ1−αiddt∫0tt−s−αiusds,t>0,tDtαiut=1Γ1−αiddt∫tTs−t−αiusds,t<T,where u is a function defined on 0,T and αi>0 for 1≤i≤n, and Γαi is the standard gamma function given by
(18)Γz=∫0+∞zαi−1e−zdz.Definition 3 (see [40]).
Letαi≥0 for 1≤i≤n and n∈ℕ.
(i)
Ifαi∈n−1,n and u∈ACn0,T,ℝn, then the left and right Caputo fractional derivatives of order αi for function u denoted by D0ctαiu and DtcTαiu, respectively, exist almost everywhere on 0,T and for 1≤i≤n, where D0ctαiu and DtcTαiu are represented by
(19)D0ctαiu=1Γn−αi∫0tt−sn−αi−1unsds,t∈0,T,DtcTαiu=−1nΓn−αi∫tTt−sn−αi−1unsds,t∈0,T,respectively.(ii)
Ifαi=n−1 and u∈ACn0,T,ℝn, then D0ctn−1ut and DtcTn−1ut are represented by D0ctn−1ut=un−1t and DtcTn−1ut=−1n−1un−1tLemma 4.
Let0<αi≤1 for 1≤i≤n, 1≤p<∞, and u∈Lp0,T,ℝn. Then
(20)D0ξ−αtuLp0,t≤tαiΓαi+1uLp0,t,forξ∈0,t,t∈0,T.Proposition 5 (see [40]).
From fractional integration, we have(21)∫0TD0t−αutvtdt=∫0TDtT−αvtutdt,α>0,provided that u∈Lp0,T,ℝ, v∈Lq0,T,ℝ, and p≥1,1/p+1/q≤1+α or p≠1,q≠1,1/p+1/q=1+α.Definition 6 (see [40]).
Let0<αi≤1 for 1≤i≤n. The fractional derivative space H0αi0,T (denoted by H0αi for short) is defined by the closure C0∞0,T,ℝ, that is,
(22)H0αi=C0∞0,T,ℝ¯,with respect to the weighted norm
(23)uiαip=∫0TD0tαiuitpdt+∫0Tuitpdt,for every ui∈H0ai and for 1≤i≤n.Remark 7.
It is obvious that the fractional derivative spaceH0αi is the space of functions ui∈Lp0,T,ℝ having an αi order Riemann–Liouville fractional derivative 0Dtαiui∈Lp0,T,ℝ and ui0=uiT=0 for 1≤i≤n. From [12] (Proposition 3.1), we know that for 0<αi≤1, the space H0αi is a reflexive and separable Banach space.Lemma 8 (see [40]).
Let0<αi≤1 for 1≤i≤n, and 1<p<∞. For any ui∈H0αi, we have
(24)uiLp≤TαiΓαi+1D0tαiuitLp.
Moreover, ifαi>1/p, then
(25)ui∞≤Tαi−1/pΓαiαi−1q+11/qD0tαiuitLp,where 1/p+1/q=1. Upon using (23), we observe that
(26)uiαi=D0tαiuitLp=∫0TD0tαiuitpdt1/p,∀ui∈H0αi,for 1≤i≤n, which is equivalent to (15). Then, we have
(27)∑i=1nuiLpp≤S∑i=1nuiαip,and if αi>1/p, then
(28)∑i=1nui∞p≤M∑i=1nuiαip,with
(29)S=maxTpαiΓαi+1p,1≤i≤n,M=maxTpαi−1Γαi+1pαi−1q+1p/q,1≤i≤n.
Now, we letX be the Cartesian product of n Sobolev spaces H0αi,⋯,H0αi, i.e., X=H0αi×⋯×H0αn, which is a reflexive Banach space endowed with the norm
(30)u1,⋯,un=∑i=1nuiαi.
Obviously,X is compactly embedded in C00,Tn.Definition 9.
We mean by a (weak) solution of the system (1) any function u=u1,⋯,un∈X such that
(31)∑i=1n∫0TD0tαiuitp−2D0tαiuit⋅D0tαivitdt−∑i=1n∫0Thiuitvitdt+∑i=1n∑j=1maitjIijuitjvitj−λ∑i=1n∫0TFuit,uvitdt−μ∑i=1n∫0TGuit,uvitdt=0,for every v=v1,⋯,vn∈X.
Put(32)Hix=∫0xhiξdξ,forallx∈ℝ,for 1≤i≤n.
We need the following conditions:
H1: 1/p<αi≤1 for 1≤i≤n.
H2: Iij0=0, and there exists a constant Lij>0 such that
(33)Iijs1−Iijs2≤Lijs1−s2p−1,foranys1,s2∈ℝi=1,⋯,n,j=1,⋯,m.
H3: ∑i=1nLiTpαi/Γαi+1pai¯+MCma¯∞<1, where C=maxi∈1,⋯,n,ji∈1,⋯,mLij and a¯=maxait,t∈0,T,1≤i≤n.
Put(34)σ=min1−LiTpαiΓαi+1p,1≤i≤n,ρ=max1+LiTpαiΓαi+1pai¯,1≤i≤n,ϱ1=σ−MCma¯∞,ϱ2=σ+MCma¯∞.


## 3. Main Results

In this section, we present our key findings regarding the existence of at least three weak system solutions (1). For any ς>0, we denote by Qς the set xi,⋯,xn∈ℝn:1/p∑i=1nxip≤ς. For positive constants θ and η, set
(35)Gθ≔∫0Tmaxx1,⋯,xn∈QθGt,x1,⋯,xndt,Gη≔inf0,T×0,Γ2−α1η×⋯×0,Γ2−αnηGt,x1,⋯,xndt.For the rest of this article, positive constants will be used (θ and η), and let Θ and η be the vectors in ℝn defined by
(36)Θ=θp,⋯,θp,η¯=Γ2−α1η,⋯,Γ2−αnη,respectively.Set(37)Ciαi,γ=1pγTp∫0γTtp1−αidt+∫γT1−γTt1−αi−t−γT1−αipdt+∫1−γTTt1−αi−t−γT1−αi−1−1−γT1−αip,for 0<γ<1/p, and
(38)K1=maxCiαi,γ,1≤i≤n,K2=minCiαi,γ,1≤i≤n.Fixing four positive constantsθ1, θ2, θ3, and η, put
(39)δλ,G≔min1pMminϱ1θ1p−pMλ∫0TFt,Θ1dtGθ1,ϱ1θ2p−pMλ∫0TFt,Θ2dtGθ2,ϱ1θ3p−θ2p−pMλ∫0TFt,Θ3dtGθ3K1nϱ2ηp−λ∫γT1−γTFt,n¯dt−∫0TFt,Θ1dtTGη−Gθ1,for 0<γ<1/p.Theorem 10.
LetF:0,T×ℝn⟶ℝ be nonnegative. Assume that there exist positive constants γ<1/p, θ1, θ2, θ3, and η with θ1<pMK2n1/pη and pMK1nϱ2/ϱ11/pη<θ2<θ3 such that
A1:
(40)max∫0TFt,Θ1dtθ1p,∫0TFt,Θ2dtθ2p,∫0TFt,Θ3dtθ3p−θ2p<ϱ1pM∫γT1−γTFt,η¯dt−∫0TFt,Θ1dtK1nϱ2ηp.
Then, for every(41)λ∈K1nϱ2ηp∫γT1−γTFt,η¯dt−∫0TFt,Θ1dt,ϱ1pMminθ1p∫0TFt,Θ1dt,θ2p∫0TFt,Θ2dt,θ3p−θ2p∫0TFt,Θ3dt,,and every nonnegative function G:0,T×ℝn⟶ℝ satisfying Gθ≥0, there exists δλ,G>0 given by (39) such that, for each μ∈0,δλ,G, the system (1) has at least three solutions u1, u2, and u3 such that maxt∈0,Tu1t<θ1, maxt∈0,Tu2t<θ2, and maxt∈0,Tu3t<θ3.Proof.
Our aim is to apply Theorem1 to the system (1). We take X=H0α1×⋯×H0αn and introduce the functionals Φ and Ψ for u=u1,u2,⋯,u3∈X, as follows:
(42)Φu=1p∑i=1nuiαip−∑i=1n∫0THiuitdt+∑i=1n∑j=1maitj∫0uitjIijsds,(43)Ψu=∫0TFt,utdt+μλ∫0TGt,utdt,and we put
(44)Iλu=Φu−λΨu.
Clearly,Φ and Ψ are continuously Gateaux differentiable functionals whose Gateaux derivatives at the point u∈X are given by
(45)Ψ′uv=∫0T∑i=1nFuit,utvitdt+μλ∫0T∑i=1nGuit,itvitdt,Φ′uv=∑i=1n∫0Tait0Dtαiuitp−20Dtαiuit⋅Dtαivitdt−∫0T∑i=1nhiuitvitdt+∑j=1m∑i=1naitjIijuitjvitj,for every v=v1,v2,⋯,vn∈X. Clearly, Φ′,Ψ′∈X∗, and we easily observe that infXΦ=Φ0=Ψ0=0.
We can show by (42) that Φ is sequentially weakly lower semicontinuous. Indeed, taking the sequentially weakly lower semicontinuity property of the norm into account and since Hi is continuous for i=1,⋯,n, it is enough to prove that
(46)∑j=1m∑i=1naitj∫0uitjIijsds,is weakly continuous in X. In fact, for uk=u1k,⋯,unk⊂X, if uk converges to u in X, then there exists S1>0 such that uk∞≤S1. Therefore, we have
(47)∑j=1m∑i=1naitj∫0uiktjIijsds−∑j=1m∑i=1naitj∫0uitjIijsds≤∑j=1m∑i=1naitj∫uitjuiktjIijsds≤S2mna¯∞uk−u∞⟶0,where S2=maxi∈1,⋯,n,j∈,⋯,m,∣s∣≤S1Iijs. So, we have ∣Φuk−Φu∣⟶0; thus, Φ is weakly continuous. Hence, Φ is sequentially weakly lower semicontinuous in X. We show what is required. Since hi0=0, one has hixi≤Lixip−1 for i=1,⋯,n; from (43) and the condition H2, we see that
(48)ϱ1p∑i=1nuiαip≤σp∑i=1nuiαip−Mp∑i=1n∑j=1mLijai∞uiαip≤1p∑i=1nuiαip−∑i=1nLiTpαipΓαi+1pai¯uiαip+∑i=1n∑j=1majtj∫0uitjIijsds≤Φu≤1p∑i=1nuiαip+∑i=1nLiTpαipΓαi+1pai¯uiαip+∑i=1n∑j=1maitj∫0uitjIijsds≤ρp∑i=1nuiαip+Mp∑i=1n∑j=1mLijai∞uiαip≤ϱ2p∑i=1nuiαip,and bearing the condition H3 in mind, it follows limu⟶∞Φu=+∞; namely, Φ is coercive and convex.
For0<γ<1/p, define ω=ω1,ω2,⋯,ωn by
(49)ωit=Γ2−αiηγTt,t∈0,γT,Γ2−αiη,t∈γT,1−γT,Γ2−αiηγTT−1,t∈1−γT,T,for 1≤i≤n. Clearly, ωi0=ωiT=0 and ωi∈Lp0,T for 1≤i≤n. A direct calculation shows that
(50)0Dtαiωit=ηγTt1−αi,t∈0,γT,ηγTt1−αi−t−γT1−αi,t∈γT,1−γT,ηγTt1−αi−t−γT1−αi−t−1−γT1−αi,t∈1−γT,T,for 1≤i≤n. Furthermore,
(51)∫0T0Dtαiωitpdt=ηγTp∫0γTt1−aipdt+∫hT1−γTt1−αi−t−γT1−αipdt+∫1−hTTt1−αi−t−γT1−αi−t−1−γT1−αipdt=pηpCiαi,γ,for 1≤i≤n. Thus, ω∈X, and
(52)ωiαip=pηpCαi,γ,for 1≤i≤n. By using (50) and (52), we have
(53)K2nϱ1ηp≤Φω≤K1nϱ1ηp.
Chooser1=ϱ1/pMθ1p, r2=ϱ1/pMθ2p, and r3=ϱ1/pMθ3p−θ2p. From the conditions θ3>θ2, θ1<pMK2n1/pη, and pMK1nϱ2/ϱ11/pη<θ2, we achieve r3>0 and r1<Φω<r2. From the definition of Φ and considering Equations (24), (27), and (50), one has
(54)Φ−1−∞;r1=u∈X:Φu≤r1⊆u∈X:∑i=1nuiαip≤pr1ϱ1=u∈X:1M∑i=1nui∞p≤pr1ϱ1=u∈X:∑i=1nuiαip≤pMr1ϱ1=u∈X:∑i=1nui∞p≤θ1p.
Hence, sinceF is nonnegative, one has
(55)supu∈Φ−1−∞;r1∫0TFt,ut≤∫0Tmaxx1,x2,⋯,xn∈Qθ1Ft,x1,x2,⋯,xndt≤∫0TFt,Θ1dt.
In a similar way, we have(56)supu∈Φ−1−∞;r2∫0TFt,ut≤∫0TFt,Θ2dt,supu∈Φ−1−∞;r2+r3∫0TFt,ut≤∫0TFt,Θ3dt.
Therefore, since0∈Φ−1−∞;r1 and Φ0=Ψ0=0, one has
(57)φr1≔infu∈Φ−1−∞,r1supu∈Φ−1−∞,r1Ψu−Ψur1−Φu≤supu∈Φ−1−∞,r1Ψur1=supu∈Φ−1−∞,r1∫0TFt,ut+μ/λGt,utdtr1≤pMϱ1⋅∫0TFt,Θ1dt+μ/λGθ1θ1p,(58)φr2≤supu∈Φ−1−∞,r2Ψur2=supu∈Φ−1−∞,r2∫0TFt,ut+μ/λGt,utdtr2≤pMϱ1⋅∫0TFt,Θ2dt+μ/λGθ2θ2p,(59)γr2,r3=supu∈Φ−1−∞,r2+r3Ψur3=supu∈Φ−1−∞,r2+r3∫0TFt,ut+λ/μGt,utdtr3≤pMϱ1⋅∫0TFt,Θ3dt+μ/λGθ3θ3p−θ2p.
On the other hand, for eachu∈Φ−1−∞,r1, one has
(60)βr1,r2≥∫γT1−γTFt,η¯dt−∫0TFt,Θ1dt+μ/λTGη−Gθ1Φω−Φu≥∫γT1−γTFt,η¯dt−∫0TFt,Θ1dt+μ/λTGη−Gθ1K1nϱ2ηp.
Sinceμ<δλ,G, one has
(61)μ<1pMϱ1θ1p−pMλ∫0TFt,Θ1dtGθ1.
This means(62)∫0TFt,Θ1dt+μ/λGθ1ϱ1/pMθ1<1λ.
Furthermore,(63)μ<K1nϱ2ηp−λ∫γT1−γTFt,η¯dt−∫0TFt,Θ1dtTGη−Gθ1.
This means(64)∫γT1−γTFt,η¯dt−∫0TFt,Θ1dt+μ/λTGη−Gθ1K1nϱ2ηp>1λ.
Then,(65)∫0TFt,Θ1dt+μ/λGθ1ϱ1/pMθ1p<1λ<∫γT1−γTFt,η¯dt−∫0TFt,Θ1dt+μ/λTGη−Gθ1K1nϱ2ηp.
In a similar way, we have(66)∫0TFt,Θ2dt+μ/λGθ2ϱ1/pMθ2p<1λ<∫γT1−γTFt,η¯dt−∫0TFt,Θ1dt+μ/λTGη−Gθ1K1nϱ2ηp,(67)∫0TFt,Θ3dt+μ/λGθ3ϱ1/pMθ3p−θ2p<1λ<∫γT1−γTFt,η¯dt−∫0TFt,Θ1dt+μ/λTGη−Gθ1K1nϱ2ηp.
Hence, from (57)–(67), we get
(68)αr1,r2,r3<βr1,r2.
Therefore,a1 and a2 of Theorem 1 are verified.
Now, we show that the functionalIλ satisfies the assumption a2 of Theorem 1. Let u∗=u1∗,u2∗,⋯,un∗ and u∗∗=u1∗∗,u2∗∗,⋯,un∗∗ be two local minima for Iλ. Then, u∗ and u∗∗ are critical points for Iλ; they are weak solutions for the system (1). Since we assumed F is nonnegative and since G is nonnegative, for fixed λ>0 and μ≥0, we have Ft,su∗+1−su∗∗+μ/λGt,su∗+1−su∗∗≥0, and consequently, Ψt,su∗+1−su∗∗≥0 for all s∈0,1. Hence, Theorem 1 implies that for every
(69)λ∈K1nϱ2ηp∫γT1−γTFt,η¯dt−∫0TFt,Θ1dt,ϱ1pMminθ1p∫0TFt,Θ1dt,θ2p∫0TFt,Θ2dt,θ3p−θ2p∫0TFt,Θ3dt,and μ∈0,δλ,G, the functional Iλ has three critical points ui, i=1,2,3, in X such that Φu1<r1, Φu2<r2, and Φu3<r2+r3, that is, maxt∈0,Tu1t<θ1, maxt∈0,Tu2t<θ2, and maxt∈0,Tu3t<θ3. Then, taking into account the fact that the weak solutions of the system (1) are exactly critical points of the functional Iλ, we have the desired conclusion.☐For positive constantsθ1, θ4, and η, set
(70)δλ,G′≔min1pMminϱ1θ1p−pMλ∫0TFt,Θ1dtGθ1,ϱ1θ4p−2pMλ∫0TFt,θ4/2p,θ4/2p,⋯,θ4/2pdtGθ4/2p,ϱ1θ4−2pMλ∫0TFt,Θ4dtGθ4,K1nϱ2ηp−λ∫γT1−γTFt,η¯dt−∫0TFt,Θ1dtTGη−Gθ1,where 0<γ<1/p.Theorem 11.
LetF:0,T×ℝn⟶ℝ satisfy the condition Ft,x1,x2,⋯,xn≥0 for all t,x1,x2,⋯,xn∈0,T×ℝn. Assume that there exist positive constants γ<1/p, θ1, θ4, and η with θ1<minη,pMK2n1/pη and 2pMK1nϱ2/σ−MCma~∞1/pη<θ4 such that
A2:
(71)max∫0TFt,Θ1dtθ1p,2∫0TFt,Θ4dtθ4p<ϱ1ϱ1+pMK1nϱ2∫γT1−γTFt,η¯dtηp.
Then, for every(72)λ∈Λ′≔ϱ1+pMK1nϱ2ηppM∫γT1−γTFt,η¯dt,ϱ1pMminθ1p∫0TFt,Θ1dt,θ4p2∫0TFt,Θ4dt,and every nonnegative function G:0,T×ℝn⟶ℝ satisfying Gθ≥0, there exists δλ,G′ given by (70) such that, for each μ∈0,δλ,G, the system (1) has at least three solutions u1, u2, and u3 such that maxt∈0,T∣u1t∣<θ1, maxt∈0,T∣u2t∣<θ4/2p, and maxt∈0,T∣u3t∣<θ4.☐Proof.
Chooseθ2=θ4/2p and θ3=θ4. So, by using A2, one has
(73)∫0TFt,Θ2dtθ2p=2∫0TFt,θ4/2p,θ4/2p,⋯,θ4/2pdtθ2p≤2∫0Tt,Θ4dtθ4p<ϱ1ϱ1+pMK1nϱ2∫γT1−γFt,η¯dtηp,(74)∫0TFt,Θ3dtθ3p−θ2p=2∫0TFt,Θ4dtθ4p<ϱ1ϱ1+pMK1nϱ2∫γT1−γTFt,η¯dtηp.
Moreover, taking into account thatθ1<η, by using A2, we have
(75)ϱ1pMK1nϱ2∫γT1−γTFt,η¯dt−∫0TFt,Θ1dtηp>ϱ1pMK1nϱ2∫γT1−γTFt,η¯dtηp−ϱ1pMK1nϱ2∫0TFt,Θ1dtθ1p>ϱ1pMK1nϱ2∫γT1−γTFt,η¯dtηp−ϱ12pMK1nϱ2ϱ1+pMK1nϱ2∫γT1−γTFt,η¯dtηp=ϱ1ϱ1+pMK1nϱ2∫γT1−γTFt,η¯dtηp.
Hence, fromA2, (73), and (74), it is easy to see that the assumption A1 of Theorem 10 is satisfied, and since the critical points of the functional Φ−λΨ are the weak solutions of the system (1), we have the conclusion.☐Now, we present the following example in which the hypotheses of Theorem11 are satisfied.Example 1.
Consider the following system:(76)tD10,65ϕ30Dt0,65u1t=λFu1u1,u2+μGu1u1,u2+h1u1,t∈0,1,t≠t1,tD10,7ϕ30Dt0,7u2t=λFu2u1,u2+μGu2u1,u2+h2u2,t∈0,1,t≠t2,ΔtDT−0,35ϕ30cDt0,65u1ttj=I1ju1tj,j=1,2,ΔtDT−0,3ϕ30cDt0,7u2ttj=I2ju2tj,j=1,2,u10=u11=0,u20=u21=0,where
(77)Fx1,x2=e−1/x1+e−1/x2ifx1x2≠0,e−1/x2ifx1=0,x2≠0,e−1/x1ifx1≠0,x2=0,0ifx1=0,x2=0.
h1x1=1/102sin2x1 and h2x2=1/1031−cos2x2 for every x1,x2∈ℝ, t1=1/3, t2=1/2, and Iijξ=1/102ξ2 for every ξ∈ℝ and for i,j=1,2. By expressions of h1 and h2, we have H1x1=1/2.102x1−1/2sin2x1 and H2x2=1/2.1032x1+1/2sin2x1 for every x1,x2∈ℝ. Choosing γ=1/4, θ1=10−4, θ4=108, and η=1, we clearly observe that all assumptions of Theorem 11 are satisfied. Hence, for every
(78)λ∈0.5Γ0.6521−1/106Γ1.652−1/5×105Γ0.652e−1/Γ1.35+e−1/Γ1.3+31.83041+1/106Γ1.652+1/5×105Γ0.652e−1/Γ1.35+e−1/Γ1.3,1−1/106Γ1.652−1/5×105Γ0.652Γ0.65210244e−1/104,and every nonnegative function G:ℝn⟶ℝ satisfying Gθ≥0, there exists δλ,G′′>0 such that, for each μ∈[0,δλ,G[, the system (74) has at least three solutions u1, u2, and u3 such that maxt∈0,T∣u1t∣<10−4, maxt∈0,Tu2t<108/23, and maxt∈0,T∣u3t∣<108.☐Remark 12.
WhenF does not depend on t, in Theorem 10, the assumption A1 can be written as
(79)maxFΘ1θ1p,FΘ2θ2p,FΘ3θ3p−θ2p<ϱ1pM1−2γTFη¯−TFΘ1K1nϱ2ηp,as well as
(80)Λ≔K1nϱ2ηp1−2γTFη¯−TFΘ1,ϱ1pTMminθ1pFΘ1,θ2pFΘ2,θ3p−θ2pFΘ3,δλ,G≔min1pMminϱ1θ1p−pMλFΘ1Gθ1,ϱ1θ2p−pMλFΘ2Gθ2,ϱ1θ3p−θ2p−pMλFΘ3Gθ3,K1nϱ2ηp−λ1−2γTFη¯−TFΘ1TGη−Gθ1.
In this case, in Theorem11, the assumption A2 follows the form
(81)maxFΘ1θ1p,2FΘ4θ2p<ϱ1ϱ1+pMK1nϱ21−2γTFη¯ηp,as well as
(82)Λ′≔ϱ1+pMK1nϱ2ηppM1−2γTFη¯,ϱ1pTMminθ1pFΘ1,θ4p2FΘ4,δλ,G′≔min1pMminϱ1θ1p−pMTλFΘ1Gθ1,ϱ1θ4p−2pMTλFt,θ4/2p,θ4/2p,⋯,θ4/2pGθ2,ϱ1θ4p−2pTMλFΘ4Gθ3,K1nϱ2ηp−λ1−2γTFη¯−TFt,Θ1TGη−Gθ1.Remark 13.
We observe that, in our results, no asymptotic conditions onF and G are needed and only algebraic conditions on F are imposed to guarantee the existence of solutions. Moreover, in the conclusions of the above results, one of the three solutions may be trivial since the values of Fxit,0,0,⋯,0 and Gxit,0,0,⋯,0 for every t∈0,T, 1≤I≤n, are not determined.
As an application of Theorem10, we consider the following problem:
(83)tDTαϕp0Dtαut=λft,u+μgt,u+hu,t∈0,T,t≠tj,ΔtDTα−1ϕp0cDtαu1ttj=Ijutj,j=1,2,u0=uT=0,where 1/p<α<1, λ>0, μ≥0, T>0, 0Dtα and tDTαi denote the left and right Riemann–Liouville fractional derivatives of order α, respectively, a0=essinft∈0,Tat, f,g:0,T×ℝ⟶ℝ are L1-Caratheodory functions, and h:ℝ⟶0,+∞ is a p−1-Lipschitz continuous function with the Lipschitz constant L>0, i.e.,
(84)hx1−hx2≤Lx1−x2p−1,for every x1,x2∈ℝ, satisfying h0=0, Ij∈Cℝ,ℝ for j=1,2,⋯,m, such that Ij0=0, 0<t1<t2<⋯<tm<tm+1=T, and Ij is a p−1-Lipschitz continuous function with the Lipschitz constant Lj>0 such that
(85)Ijs1−Ijs2≤Ljs1−s2p−1,for any s1,s1∈ℝ, satisfying Ij0=0, for j=1,2,⋯,m. Put
(86)Ft,x=∫0xft,ξdt,foreveryt,x∈0,T×ℝ,Gt,x=∫0xgt,ξdt,foreveryt,x∈0,T×ℝ,and Hx=∫0xhξdt for every x∈ℝ.
Set(87)σ¯=1−LTpαΓα+1p,ρ¯=1+LTpαΓα+1pa¯,M¯=Tpα−1α0Γαpα−1q+1p/q,ϱ¯1=σ−MCma¯∞,ϱ¯2=σ+MCma¯∞,Cα,γ=1pγTp∫0γTtp1−αdt+∫γT1−γTt1−α−t−γT1−αpdt+∫1−γTTt1−α−t−γT1−α−1−1−γT1−αp,for 0<γ<1/p. We suppose that
(88)K¯=LiTpαΓα+1pα¯i+MC¯ma¯∞<1,where C¯=maxj∈1,2,⋯,mLj.
For positive constantsθ and η, set
(89)Gθ≔∫0Tmaxx≤θpGt,xdt,Gη≔inf0,T×0,Γ2−αG.
Obviously, ifg changes sign on 0,T, then clearly Gθ≥0.☐Now, we give the following straightforward consequences of Theorems10 and 11, respectively.Theorem 14.
Letf:0,T×ℝ⟶ℝ be a nonnegative L1-Caratheodory function. Assume that there exist positive constants γ<1/p, θ1, θ2, θ3, and η with θ3>θ2, θ1<pM¯Cα,γ1/pη, and pM¯Cα,γϱ¯2/ϱ¯11/pη<θ2 such that
(90)max∫0TFt,θ1pdtθ1p,∫0TFt,θ2pdtθ2p,∫0TFt,θ3pdtθ3p−θ2p<ϱ¯1pM¯Cα,γϱ¯2∫γT1−γTFt,Γ2−αηdt−∫0TFt,θ1pdtηp.
Then, for every(91)λ∈Λ″≔Cα,γϱ¯2ηp∫γT1−γTFt,Γ2−αηdt−∫0TFt,θ1pdt,ϱ¯1pM¯minθ1p∫0TFt,θ1pdt,θ2p∫0TFt,θ2pdt,θ3p−θ2p∫0TFt,θ3pdt,and every nonnegative L1-Caratheodory function g:0,T×ℝ⟶ℝ, there existsδλ,g∗>0 given by
(92)δλ,g∗≔min1pM¯minϱ¯1θ1p−pM¯λ∫0TFt,θ1pdtGθ1,ϱ1θ2p−pMλ∫0TFt,θ2pdtGθ2,ϱ¯1θ3p−θ2p−pM¯λ∫0TFt,θ3pdtGθ3,Cα,γϱ¯2ηp−λ∫γT1−γTFt,Γ2−αη¯dt−∫0TFt,θ1pdtTGη−Gθ1,such that, for each μ∈0,δλ,G∗, the system (83) has at least three solutions u1, u2, and u3 such that maxt∈0,T∣u1t∣<θ1, maxt∈0,T∣u2t∣<θ2, and maxt∈0,T∣u3t∣<θ3.☐Proof.
By a similar argument as given in the proof of Theorem10, we ensure the existence of the weak solutions u1, u2, and u3 such that maxt∈0,T∣u1t∣<θ1, maxt∈0,T∣u2t∣<θ2, and maxt∈0,T∣u3t∣<θ3. Now, we show that the weak solutions u1, u2, and u3 are nonnegative. To this end, let u0 be a nontrivial weak solution of the problem (83). Arguing by a contradiction, assume that the set A=t∈0,T:u0t<0 is nonempty and of positive measure. Put v¯t=min0,u0t for all t∈0,T. Clearly, v¯∈Hα, and one has
(93)∫0Tat0Dtαu0tp−20Dtαu0t⋅0Dtαv¯tdt−∫0Thu0tv¯t+∑j=1matjIiju0tjv¯tj−λ∫0Tft,u0v¯tdt−μ∫0Tgt,u0v¯tdt=0.
Thus, from our sign assumptions on the data, we have(94)0≤1−K¯∫Aat0Dtαu0tpdt≤∫Aat0Dtαu0tpdt−∫Ahu0tu0tdt+∑j=1matjIiju0tju0t≤0.
Hence, sinceK¯<1, u0=0 in A and we arrive at a contradiction.☐Theorem 15.
Letf:0,T×ℝ⟶ℝ be a nonnegative L1-Caratheodory function. Assume that there exist positive constants γ<1/p, θ1, θ4, and η with θ1<minη,pM¯Cα,γ1/pη and 2pM¯Cα,γϱ¯2/ϱ¯11/pη<θ4 such that
(95)max∫0TFt,θ1pdtθ1p,2∫0TFt,θ4pdtθ4p<ϱ¯1ϱ¯1+pM¯Cα,γϱ¯2∫γT1−γTFt,Γ2−αηdtηp.
Then, for every(96)λ∈ϱ¯1+pM¯Cα,γϱ¯2ηppM¯∫γT1−γTFt,Γ2−αηdt,ϱ¯1pM¯minθ1p∫0TFt,θ1pdt,θ4p2∫0TFt,θ4pdt,and every nonnegative function g:0,T×ℝ⟶ℝ satisfying Gθ≥0, there exists δλ,g∗∗>0 given by
(97)δλ,g∗∗≔min1pM¯ϱ¯1θ1p−pM¯λ∫0TFt,θ1pdtGθ1,ϱ¯1θ4p−2pM¯λ∫0TFt,θ4/2pdt2Gθ4/2p,ϱ¯1θ4p−2PM¯λ∫0TFt,θ4dt2Gθ4,Cα,γϱ¯2ηp−λ∫γT1−γTFt,Γ2−αη−∫0TFt,θ1pdtTGη−Gθ1,such that, for each μ∈0,δλ,g∗∗, the system (83) has at least three solutions u1, u2, and u3 such that maxt∈0,T∣u1t∣<θ1, maxt∈0,T∣u2t∣<θ4/2p, and maxt∈0,T∣u3t∣<θ4.☐Here, in order to illustrate Theorem15, we present the following example.Example 2.
(98)tD10,8ϕ30Dt0,8ut=λfu+μgu+hu,t∈0,1,t≠tj,ΔtD1−0,2ϕ30cDt0,8u1ttj=Ijutj,j=1,2,u0=u1=0,where
(99)fx=6x5,x≤1,6x,x>1.
hx=1/108sin2x, for every x∈ℝ, t1=1/4, t2=1/5, and Ijξ=1/108ξ2, for every ξ∈ℝ and for j=1,2. By expression, we have
(100)Fx=x6,x≤1,6lnx+1,x>1.
Takingγ=1/4, θ1=10−4, θ4=104, and η=1, we clearly observe that all assumptions of Theorem 15 are satisfied.
Then, for each(101)λ∈0.6Γ0.821−1/108Γ1.82−1/3×107Γ1.82Γ1.22+8.92821+1/108Γ1.82+1/3×107Γ1.82Γ1.22,1−1/102Γ1.82−1/3×107Γ1.821.2Γ0.8610126ln102+1,and every nonnegative continuous function g:ℝ⟶ℝ, there exists δλ,g>0 such that, for each μ∈[0,δλ,g[, the system (83) has at least three nonnegative weak solutions u1, u2, and u3 such that maxt∈0,T∣u1t∣<10−4, maxt∈0,T∣u2t∣<104/23, and maxt∈0,T∣u3t∣<104.☐Now, we list some consequences of Theorem15 as follows.Theorem 16.
Letf be a nonnegative continuous and nonzero function such that
(102)limx⟶0+fxxp−1=limx⟶+∞fxxp−1=0,for every λ>λ∗, where
(103)λ∗=infϱ1¯+pM¯Cα,γϱ2¯ηppM¯TFΓ2−αη:η>0,FΓ2−αη>0.
Then, there exists(104)μ¯λ,g=min1pM¯minϱ¯1θ1p−pM¯λTFθ1pGθ1,ϱ¯1θ4p−2pM¯λTFθ4/2p2Gθ4/2p,ϱ¯1θ4p−2pM¯λTFθ4p2Gθ4,Cα,γϱ¯2ηp−λ1−2γTFΓ2−α−TFθ1pTGη−Gθ1,where θ1, θ4, and γ are positive constants with γ<1/p, such that for each μ∈0,μ¯λ,g, the problem
(105)tDTαϕp0Dtαut=λFu+μgu+hu,t∈0,T,t≠tj,ΔtDTα−1ϕp0cDtαu1ttj=Ijutj,j=1,2,u0=uT=0,where g:ℝ⟶ℝ is a nonnegative continuous and nonzero function, has at least two distinct positive weak solutions.☐Proof.
Fixλ>λ∗, put Fx=∫0xfξdξ for all x∈ℝ, and let η>0 such that FΓ2−αη>0 and
(106)λ>ϱ¯1+pM¯Cα,γϱ¯2ηppM¯FΓ2−αη.
From (102), there is θ1>0 such that θ1<minη,pM¯Cα,γ1/Pη and Fθ1p/θ1<ϱ¯1/pM¯Tλ and θ4>0 such that 2pM¯Cα,λϱ¯2/ϱ¯11/pη<θ4 and Fθ4p/θ4<ϱ¯1/2pM¯Tλ. Therefore, Theorem 11 ensures the conclusion.☐Finally, by way of example, we point out the following simple consequence of Theorem16 when μ=0.Theorem 17.
Letf:ℝ⟶ℝ be a continuous function such that xfx>0 for all x≠0 and
(107)limx⟶0+fxxp−1=limx⟶+∞fxxp−1=0.
Then, for everyλ>λ¯, where
(108)λ¯=ϱ¯1+2pM¯Cα,γϱ¯2pM¯T×maxinfη>0ηpFΓ2−αη,infη<0−ηpFΓ2−αη,the problem (105), in the case μ=0, has at least four distinct nontrivial weak solutions.Proof.
Setting(109)f1x=0,ifx<0,fx,ifx≥0,f2x=0,ifx<0,−f−x,ifx≥0,and applying Theorem 16 to f1 and f2, we have the result.☐



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