\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2015 (2015), No. 138, pp. 1--15.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2015 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2015/138\hfil Variational approach to fractional BVP] {Variational approach to fractional boundary value problems with two control parameters} \author[M. Ferrara, A. Hadjian \hfil EJDE-2015/138\hfilneg] {Massimiliano Ferrara, Armin Hadjian} \address{Massimiliano Ferrara \newline Department of Law and Economics, University Mediterranea of Reggio Calabria, Via dei Bianchi, 2-89127 Reggio Calabria, Italy} \email{massimiliano.ferrara@unirc.it} \address{Armin Hadjian \newline Department of Mathematics, Faculty of Basic Sciences, University of Bojnord, P.O. Box 1339, Bojnord 94531, Iran} \email{a.hadjian@ub.ac.ir} \thanks{Submitted February 18, 2015. Published May 20, 2015.} \subjclass[2010]{58E05, 26A33, 34A08, 34B15, 45J05, 91A80, 91B55} \keywords{Fractional differential equations; Caputo fractional derivatives; \hfill\break\indent variational methods; multiple solutions} \begin{abstract} This article concerns the multiplicity of solutions for a fractional differential equation with Dirichlet boundary conditions and two control parameters. Using variational methods and three critical point theorems, we give some new criteria to guarantee that the fractional problem has at least three solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} Fractional differential equations have been proved to be valuable tools in the modeling of many phenomena in various fields of physic, chemistry, biology, engineering and economics. There has been significant development in fractional differential equations, one can see the monographs of Miller and Ross \cite{MiRo}, Samko et al \cite{SaKiMa}, Podlubny \cite{Pod}, Hilfer \cite{Hil}, Kilbas et al \cite{k1} and the papers \cite{AHM,Bai1,Bai2,BL,BeHaNt,Ko,LaVa,WaZh,WDC,Zh} and references therein. Critical point theory has been very useful in determining the existence of solutions for integer order differential equations with some boundary conditions; see for instance, in the vast literature on the subject, the classical books \cite{MaWi,Rab,Str,Will} and references therein. But until now, there are a few results for fractional boundary value problems (briefly BVP) which were established exploiting this approach, since it is often very difficult to establish a suitable space and variational functional for fractional problems. The aim of this article is to study the nonlinear fractional boundary value problem \begin{equation}\label{e1.1} \begin{gathered} \frac{d}{dt} \Big({}_0D_t^{\alpha-1}({}_0^c D_t^{\alpha} u(t)) -{}_t D_T^{\alpha-1}({}_t^cD_T^{\alpha} u(t))\Big) +\lambda f(t,u(t))+\mu g(t,u(t))=0, \\ \quad\text{ a.e. } t\in [0,T],\\ u(0)=u(T)=0, \end{gathered} \end{equation} where $\alpha \in (1/2,1]$, ${}_0 D_t^{\alpha-1}$ and ${}_t D_T^{\alpha-1}$ are the left and right Riemann-Liouville fractional integrals of order $1-\alpha$ respectively, $_0^c D_t^{\alpha}$ and $_t^c D_T^{\alpha}$ are the left and right Caputo fractional derivatives of order $0<\alpha\leq 1$ respectively, $\lambda$ and $\mu$ are positive real parameters, and $f,g:[0,T]\times\mathbb{R}\to\mathbb{R}$ are continuous functions. In this article, employing two three critical point theorems which we recall in the next section (Theorems \ref{the2.1} and \ref{the2.2}), we establish the exact collections of the parameters $\lambda$ and $\mu$, for which the problem \eqref{e1.1} admits at least three weak solutions; see Theorems \ref{the3.1} and \ref{the3.2}. For more information, we refer the reader to \cite{Bai3,GaMo,Nya} where the existence and multiplicity of solutions for problem \eqref{e1.1}, with $\mu=0$, using the critical point theory is proved; see also \cite{AHR, DagHeiMolica} where analogous variational approaches have been developed on studying nonlinear perturbed differential equations. A special case of Theorem \ref{the3.1} is the following theorem. \begin{theorem}\label{t1.1} Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function. Put $ F(\xi):=\int_0^\xi f(x)dx$ for each $\xi\in \mathbb{R}$. Assume that $F(d)>0$ for some $d>0$ and $F(\xi)\geq 0$ in $[0,d)$ and $$ \liminf_{\xi\to 0}\frac{F(\xi)}{\xi^2} =\limsup_{|\xi|\to+\infty}\frac{F(\xi)}{\xi^2}=0. $$ Then, there is $\lambda^\star>0$ such that for each $\lambda>\lambda^\star$ and for every continuous function $g:[0,T]\times\mathbb{R}\to \mathbb{R}$ satisfying the asymptotic condition $$ \limsup_{|\xi|\to+\infty}\frac{\sup_{t\in[0,T]} \int_{0}^{\xi}g(t,s)ds}{\xi^2}<+\infty, $$ there exists $\delta^\star_{\lambda,g}>0$ such that, for each $\mu\in[0,\delta^\star_{\lambda,g}[$, the problem \begin{gather*} \frac{d}{dt} \Big({}_0 D_t^{\alpha-1}({}_0^c D_t^{\alpha} u(t)) -{}_t D_T^{\alpha-1}({}_t^c D_T^{\alpha} u(t))\Big) +\lambda f(u(t))+\mu g(t,u(t))=0, \\ \text{ a.e. } t\in [0,T],\\ u(0)=u(T)=0, \end{gather*} admits at least three solutions. \end{theorem} The following result is a consequence of Theorem \ref{the3.2}. \begin{theorem}\label{t1.2} Let $f:\mathbb{R}\to\mathbb{R}$ be a nonnegative continuous function such that $\lim_{t\to 0} f(t)/t=0$ and $$ \int_{0}^{10}f(s)ds<\frac{25}{12}\int_{0}^{1}f(s)ds. $$ Then, for every $$ \lambda\in\Big]\frac{24}{\int_{0}^{1}f(s)ds},\frac{50}{\int_{0}^{10}f(s)ds}\Big[ $$ and for every nonnegative continuous function $g:[0,1]\times\mathbb{R}\to \mathbb{R}$, there exists $\delta^\star>0$ such that, for each $\mu\in[0,\delta^\star[$, the problem \begin{gather*} 2u''(t)+\lambda f(u(t))+\mu g(t,u(t))=0, \quad\text{a.e. } t\in [0,1],\\ u(0)=u(1)=0, \end{gather*} admits at least three solutions. \end{theorem} \section{Preliminaries} The original three critical point theorem is due to Pucci and Serrin \cite{PS1,PS2} and establishes that if $X$ is a real Banach space and a function $f:X\to\mathbb{R}$ is of class $C^1$, satisfies the Palais-Smale condition, and has two local minima, then $f$ has at least three distinct critical points. This result has been extended in the framework of problems depending on a real parameter by Ricceri \cite{Ricceri}, who also established a precise range of the parameter that guarantees the existence of at least three critical points. Our main tools are critical point theorems that we recall here in a convenient form. The first result has been obtained in \cite{BonaMara} and it is a more precise version of \cite[Theorem 3.2]{BonaCand}. The second one has been established in \cite{BonaCand}. \begin{theorem}[{\cite[Theorem 3.6]{BonaMara}}]\label{the2.1} Let $X$ be a reflexive real Banach space; $\Phi:X\to \mathbb{R}$ be a coercive, continuously G\^{a}teaux differentiable and sequentially weakly lower semicontinuous functional whose G\^{a}teaux derivative admits a continuous inverse on $X^\ast;$ $\Psi:X\to \mathbb{R}$ be a continuously G\^{a}teaux differentiable functional whose G\^{a}teaux derivative is compact such that $\Phi(0)=\Psi(0)=0$. Assume that there exist $r>0$ and $\overline{x}\in X$, with $r<\Phi(\overline{x})$, such that \begin{itemize} \item[(A1)] $\frac{\sup_{\Phi(x)\leq r}\Psi(x)}{r}<\frac{\Psi(\overline{x})}{\Phi(\overline{x})};$ \item[(A2)] for each $\lambda\in \Lambda_r:=]\frac{\Phi(\overline{x})}{\Psi(\overline{x})},\frac{r}{\sup_{\Phi(x)\leq r}\Psi(x)}[$ the functional $I_\lambda:=\Phi-\lambda\Psi$ is coercive. \end{itemize} Then, for each $\lambda\in\Lambda_r$ the functional $I_\lambda$ has at least three distinct critical points in $X$. \end{theorem} \begin{theorem}[{\cite[Theorem 3.3]{BonaCand}}]\label{the2.2} Let $X$ be a reflexive real Banach space; $\Phi:X\to\mathbb{R}$ be a convex, coercive and continuously G\^ateaux differentiable functional whose derivative admits a continuous inverse on $X^\ast$; $\Psi:X\to\mathbb{R}$ be a continuously G\^ateaux differentiable functional whose derivative is compact, such that $$ \inf_{X}\Phi=\Phi(0)=\Psi(0)=0. $$ Assume that there are two positive constants $r_1,r_2$ and $\overline{x}\in X$, with $2r_1<\Phi(\overline{x})< r_2/2$, such that \begin{itemize} \item[(A3)] $\frac{\sup_{\Phi(x)0$ for a function $u$ are defined by \begin{gather*} {}_a D_t^{-\alpha}u(t):=\frac{1}{\Gamma(\alpha)} \int_a^t (t-s)^{\alpha-1}u(s) ds,\\ {}_t D_b^{-\alpha}u(t) := \frac{1}{\Gamma(\alpha)} \int_t^b (s-t)^{\alpha-1} u(s) ds, \end{gather*} for every $t \in [a,b]$, provided the right-hand sides are pointwise defined on $[a,b]$, where $\Gamma(\alpha)$ is the standard gamma function given by $$ \Gamma(\alpha):=\int_0^{+\infty}z^{\alpha-1}e^{-z}dz. $$ \end{definition} Set $AC^n([a,b],{\mathbb{R}})$ as the space of functions $u:[a,b]\to {\mathbb{R}}$ such that $u$ belongs to $C^{n-1}([a,b], {\mathbb{R}})$ and $u^{(n-1)}$ belongs to $AC([a,b],{\mathbb{R}})$. Here, as usual, $C^{n-1}([a,b],{\mathbb{R}})$ denotes the set of mappings having $(n-1)$ times continuously differentiable on $[a,b]$. In particular we denote $AC([a,b],{\mathbb{R}}):=AC^1([a,b],{\mathbb{R}})$. \begin{definition}\label{def2.2}\rm Let $\gamma \ge 0$ and $n \in \mathbb{N}$. (i) If $\gamma \in (n-1, n)$ and $u\in AC^n([a, b], {\mathbb{R}})$, then the left and right Caputo fractional derivatives of order $\gamma$ for function $u$ denoted by $_a^c D_t^{\gamma}u(t)$ and $_t^c D_b^{\gamma}u(t)$, respectively, exist almost everywhere on $[a, b]$, $_a^c D_t^{\gamma}u(t)$ and $_t^c D_b^{\gamma}u(t)$ are represented by \begin{gather*} {}_a^c D_t^{\gamma}u(t) = \frac{1}{\Gamma(n-\gamma)} \int_a^t (t-s)^{n-\gamma-1} u^{(n)}(s) ds, \\ {}_t^c D_b^{\gamma}u(t) = \frac{(-1)^n}{\Gamma(n-\gamma)} \int_t^b (s-t)^{n-\gamma-1} u^{(n)}(s) ds, \end{gather*} for every $t \in [a, b]$, respectively. (ii) If $\gamma = n - 1$ and $u \in AC^{n-1}([a, b], {\mathbb{R}})$, then $_a^c D_t^{n-1}u(t)$ and $_t^c D_b^{n-1}u(t)$ are represented by \[ {}_a^c D_t^{n-1}u(t) = u^{(n-1)}(t), \quad\text{and}\quad _t^c D_b^{n-1}u(t) = (-1)^{(n-1)} u^{(n-1)}(t), \] for every $t\in [a,b]$. \end{definition} With these definitions, we have the rule for fractional integration by parts, and the composition of the Riemann-Liouville fractional integration operator with the Caputo fractional differentiation operator, which were proved in \cite{k1} and \cite{SaKiMa}. \begin{proposition}\label{prop2.1} We have the following property of fractional integration \begin{equation} \int_a^b [_a D_t^{-\gamma}u(t)] v(t) dt = \int_a^b [_t D_b^{-\gamma}v(t)] u(t) dt, \quad \gamma > 0,\label{e2.1} \end{equation} provided that $u \in L^p([a, b], {\mathbb{R}})$, $v \in L^q([a, b], {\mathbb{R}})$ and $p \ge 1$, $q \ge 1$, $1/p + 1/q \le 1 + \gamma$ or $p \neq 1$, $q \neq 1$, $1/p + 1/q = 1 + \gamma$. \end{proposition} \begin{proposition}\label{prop2.2} Let $n \in \mathbb{N}$ and $n-1 < \gamma \le n$. If $u \in AC^n([a, b], {\mathbb{R}})$ or $u \in C^n([a, b], {\mathbb{R}})$, then \begin{gather*} {}_a D_t^{-\gamma}({}_a^c D_t^{\gamma} u(t)) = u(t) - \sum _{j=0}^{n-1} \frac{u^{(j)}(a)}{j!}(t-a)^j, \\ {}_t D_b^{-\gamma}({}_t^c D_b^{\gamma} u(t)) = u(t) - \sum _{j=0}^{n-1} \frac{(-1)^ju^{(j)}(b)}{j!}(b-t)^j, \end{gather*} for every $t \in [a, b]$. In particular, if $0<\gamma\le 1$ and $u \in AC([a, b], {\mathbb{R}})$ or $u \in C^1([a,b],{\mathbb{R}})$, then \begin{equation}\label{e2.2} {}_a D_t^{-\gamma}({}_a^c D_t^{\gamma} f(t))=f(t)-f(a),\quad \text{and}\quad _t D_b^{-\gamma}({}_t^c D_b^{\gamma} f(t))=f(t)-f(b). \end{equation} \end{proposition} \begin{remark}\label{rmk2.1}\rm By \eqref{e2.1} and Definition \ref{def2.2}, it is obvious that $u \in AC([0, T],\mathbb{R})$ is a solution of problem \eqref{e1.1} if and only if $u$ is a solution of the boundary value problem \begin{equation}\label{e2.3} \begin{gathered} \frac{d}{dt} \left({}_0 D_t^{-\beta}(u'(t)) + {}_t D_T^{-\beta}(u'(t))\right) +\lambda f(t,u(t)) = 0 \quad \text{ a.e. } t\in [0,T]\\ u(0) = u(T) = 0, \end{gathered} \end{equation} where $\beta := 2(1-\alpha) \in [0, 1)$. Recall that a function $u\in AC([0,T],\mathbb{R})$ is called a solution of BVP \eqref{e2.3} if: \begin{itemize} \item[(i)] the map $t\mapsto {}_0 D_t^{-\beta}(u'(t)) + {}_t D_T^{-\beta}(u'(t))$ is differentiable for almost every $t\in [0,T]$, and \item[(ii)] the function $u$ satisfies \eqref{e2.3}. \end{itemize} \end{remark} To establish a variational structure for the main problem, it is necessary to construct appropriate function spaces. Following \cite{JZ}, we denote by $C_0^{\infty}([0,T],\mathbb{R})$ the set of all functions $g\in C^{\infty}([0,T],\mathbb{R})$ with $g(0)=g(T)=0$. \begin{definition}\label{def2.3}\rm Let $0 < \alpha \le 1$. The fractional derivative space $E_0^{\alpha}$ is defined by the closure of $C_0^{\infty}([0,T],\mathbb{R})$ with respect to the norm \[ \|u\|:=\Big(\int_0^T |_0^c D_t^{\alpha} u(t)|^2 dt+\int_0^T |u(t)|^2dt\Big)^{1/2}, \] for every $u \in E^{\alpha}_0$. \end{definition} \begin{remark} \label{rmk2.2} \rm It is obvious that the fractional derivative space $E_0^{\alpha}$ is the space of functions $u \in L^2([0,T],\mathbb{R})$ having an $\alpha$-order Caputo fractional derivative $_0^c D_t^{\alpha} u \in L^2([0, T],\mathbb{R})$ and $u(0)=u(T)=0$. \end{remark} \begin{proposition}\label{prop2.3} Let $\alpha \in (0, 1]$. The fractional derivative space $E_0^{\alpha}$ is reflexive and separable Banach space. \end{proposition} For $u\in E^\alpha_0$, set \begin{gather*} \|u\|_{L^s}:=\Big(\int_0^T |u(t)|^s dt \Big)^{1/s},\quad(s\geq 1),\\ \|u\|_\infty:=\max_{t\in [0,T]}|u(t)|. \end{gather*} One has the following two Lemmas. \begin{lemma}\label{lem2.1} Let $\alpha \in (1/2, 1]$. For all $u\in E_0^{\alpha}$, we have \begin{gather} \|u\|_{L^2} \le \frac{T^{\alpha}}{\Gamma(\alpha+1)} \|\empty_0^c D_t^{\alpha}u\|_{L^2}, \label{e2.4} \\ \|u\|_{\infty} \le \frac{T^{\alpha - 1/2}}{\Gamma(\alpha)\sqrt{2\alpha-1}} \|_0^c D_t^{\alpha} u\|_{L^2}. \label{e2.5} \end{gather} \end{lemma} Hence, we can consider $E_0^{\alpha}$ with respect to the (equivalent) norm \begin{equation} \|u\|_{\alpha}:=\Big(\int_0^T |_0^c D_t^{\alpha} u(t)|^2 dt\Big)^{1/2}=\|_0^c D_t^{\alpha} u\|_{L^2},\quad \forall u \in E_0^{\alpha} \label{e2.6} \end{equation} in the following analysis. \begin{lemma}[\cite{JZ}] \label{ineq} Let $\alpha \in (1/2, 1]$, then for every $u \in E_0^{\alpha}$, we have \begin{equation} |\cos (\pi \alpha)| \|u\|_{\alpha}^2 \le - \int_0^T {}_0^c D_t^{\alpha} u(t) \cdot {}_t^c D_T^{\alpha} u(t) dt \le \frac{1}{|\cos (\pi \alpha)|} \|u\|_{\alpha}^2. \label{e2.7} \end{equation} \end{lemma} In the rest of this article, $f,g:[0,T]\times\mathbb{R}\to\mathbb{R}$ are continuous functions, and $\lambda,\mu>0$ are real parameters. Put $$ F(t,\xi):=\int_0^\xi f(t,s)\,ds, \quad G(t,\xi):=\int_0^\xi g(t,s)\,ds, $$ for all $(t,\xi)\in[0,T]\times\mathbb{R}$. Set $ G^c:=\int_0^T\max_{|\xi|\leq c}G(t,\xi)\,dt$ for all $c>0$ and $G_d:=\inf_{[0,T]\times[0,d]}G$ for all $d>0$. Clearly, $G^c\geq 0$ and $G_d\leq 0$. We consider the functional $I_\lambda:E_0^\alpha\to\mathbb{R}$, defined by \begin{equation}\label{e2.8} I_\lambda(u):=\Phi(u)-\lambda\Psi(u),\quad u\in E_0^\alpha, \end{equation} where \begin{gather}\label{e2.9} \Phi(u):= -\int_0^T {}_0^c D_t^{\alpha}u(t)\cdot{}_t^c D_T^{\alpha} u(t)\,dt, \\ \label{e2.10} \Psi(u):=\int_0^T[F(t,u(t))+\frac{\mu}{\lambda}G(t,u(t))]dt. \end{gather} Clearly, $\Phi$ and $\Psi$ are G\^{a}teaux differentiable functionals whose derivatives at the point $u \in E_0^{\alpha}$ are \begin{align*} \Phi'(u)(v)&=-\int_0^T({}_0^c D_t^{\alpha}u(t)\cdot{}_t^c D_T^{\alpha}v(t)+{}_t^c D_T^{\alpha}u(t)\cdot{}_0^c D_t^{\alpha}v(t))\,dt,\\ \Psi'(u)(v)&=\int_0^T \big[f(t,u(t))+\frac{\mu}{\lambda}g(t,u(t))\big]v(t)\,dt\\ &=-\int_0^T \int_0^t \big[f(s,u(s))+\frac{\mu}{\lambda}g(s,u(s))\big]\,ds\cdot v'(t)\,dt, \end{align*} for every $v \in E_0^{\alpha}$. By Definition \ref{def2.2} and \eqref{e2.2}, we have \[ \Phi'(u)(v)=\int_0^T({}_0 D_t^{\alpha-1}({}_0^c D_t^{\alpha} u(t))-{}_t D_T^{\alpha-1}({}_t^c D_T^{\alpha}u(t)))\cdot v'(t)\,dt. \] Hence, $I_{\lambda}=\Phi-\lambda\Psi\in C^1(E_0^{\alpha},\mathbb{R})$. Moreover, a critical point of the functional $I_\lambda$ is a solution of \eqref{e1.1}. Indeed, if $u_\star\in E_0^{\alpha}$ is a critical point of $I_{\lambda}$, then \begin{equation}\label{e2.11} \begin{aligned} 0=I_{\lambda}'(u_\star)(v) &=\int_0^T \Big({}_0 D_t^{\alpha-1}({}_0^c D_t^{\alpha}u_\star(t)) -{}_t D_T^{\alpha-1}({}_t^c D_T^{\alpha}u_\star(t))\\ &\quad+\lambda\int_0^t f(s,u_\star(s))\,ds+\mu\int_0^t g(s,u_\star(s))\,ds\Big)\cdot v'(t)\,dt, \end{aligned} \end{equation} for every $v\in E_0^{\alpha}$. We can choose $v\in E_0^{\alpha}$ such that \[ v(t)=\sin\frac{2k\pi t}{T} \quad \text{or} \quad v(t)=1-\cos \frac{2k\pi t}{T}, \quad (k=1, 2, \dots). \] The theory of Fourier series and \eqref{e2.11} imply \begin{equation}\label{e2.12} \begin{aligned} &{}_0 D_t^{\alpha-1}({}_0^c D_t^{\alpha} u_\star(t))-{}_t D_T^{\alpha-1}({}_t^c D_T^{\alpha} u_\star(t))\\ &+\lambda\int_0^t f(s,u_\star(s))\,ds+\mu\int_0^t g(s,u_\star(s))\,ds=\kappa \end{aligned} \end{equation} a.e. on $[0,T]$ for some $\kappa\in\mathbb{R}$. By \eqref{e2.12}, it is easy to show that $u_\star\in E_0^{\alpha}$ is a solution of \eqref{e1.1}. To conclude this section, we cite a recent monograph by Krist\'aly, R\u adulescu and Varga \cite{KrisRadVar} as a general reference on variational methods adopted here. \section{Main results} In this section we establish our main abstract results. We put \begin{gather*} \Omega:=\frac{T^{\alpha-\frac{1}{2}}}{\Gamma(\alpha)\sqrt{2\alpha-1}},\\ \begin{aligned} C(T,\alpha) &:= \int_0^{T/4}t^{2-2\alpha}\,dt+\int_{T/4}^{3T/4}\big[t^{1-\alpha} -\big(t-\frac{T}{4}\big)^{1-\alpha}\big]^2\,dt\\ &\quad +\int_{3T/4}^T\big[t^{1-\alpha}-\big(t-\frac{T}{4}\big)^{1-\alpha} +\big(t-\frac{3T}{4}\big)^{1-\alpha}\big]^2\,dt, \end{aligned} \\ \omega_{\alpha,d}:=\frac{16d^2}{T^2\Gamma^2(2-\alpha) |\cos(\pi\alpha)|}C(T,\alpha). \end{gather*} Fixing $c,d>0$ such that $$ \frac{\omega_{\alpha,d}}{\int_{T/4}^{{3T}/4} F(t,d)\,dt}<\frac{c^2|\cos(\pi\alpha)|}{\Omega^2\int_0^T\max_{|\xi|\leq c}F(t,\xi)\,dt} $$ and selecting \begin{equation}\label{e3.1} \lambda\in\Lambda:=\Big]\frac{\omega_{\alpha,d}}{\int_{T/4}^{{3T}/4} F(t,d)\,dt},\frac{c^2|\cos(\pi\alpha)|}{\Omega^2\int_0^T\max_{|\xi|\leq c}F(t,\xi)\,dt}\Big[, \end{equation} put \begin{equation}\label{e3.2} \delta:=\min\Big\{\frac{ c^2|\cos(\pi\alpha)|-\lambda\Omega^2\int_0^T\max_{|\xi|\leq c}F(t,\xi)\,dt}{\Omega^2G^c},\frac{\omega_{\alpha,d}-\lambda \int_{T/4}^{{3T}/4}F(t,d)\,dt}{TG_d}\Big\} \end{equation} and \begin{equation}\label{e3.3} \overline{\delta}:=\min\Big\{\delta,\frac{1} {\max\big\{0,\frac{2T\Omega^2}{|\cos(\pi\alpha)|} \limsup_{|\xi|\to+\infty}\frac{\sup_{t\in[0,T]}G(t,\xi)}{\xi^2}\big\}}\Big\}, \end{equation} where we read $\frac{r}{0}=+\infty$ whenever this case occurs. With the above notation we are able to prove the following result. \begin{theorem}\label{the3.1} Assume that there exist positive constants $c,d$, with \begin{equation}\label{e3.4} c<\Big(\frac{4\Omega d}{T\Gamma(2-\alpha)}\Big)\sqrt{C(T,\alpha)}, \end{equation} such that \begin{itemize} \item[(A6)] $F(t,\xi)\geq 0$, for each $(t,\xi)\in([0,\frac{T}{4}]\cup[\frac{3T}{4},T])\times[0,d]$; \item[(A7)] \[ \frac{\int_0^T\max_{|\xi|\leq c}F(t,\xi)\,dt}{c^2}<\frac{|\cos(\pi\alpha)|}{\Omega^2}\frac{\int_{T/4}^{{3T}/4} F(t,d)\,dt}{\omega_{\alpha,d}}; \] \item[(A8)] \[ \limsup_{|\xi|\to +\infty}\frac{\sup_{t\in[0,T]}F(t,\xi)}{\xi^2} <\frac{\int_0^T\max_{|\xi|\leq c}F(t,\xi)\,dt}{2c^2T}. \] \end{itemize} Then, for every $\lambda\in\Lambda$, where $\Lambda$ is given by \eqref{e3.1}, and for every continuous function $g:[0,T]\times\mathbb{R}\to\mathbb{R}$ such that $$ \limsup_{|\xi|\to +\infty} \frac{\sup_{t\in[0,T]}G(t,\xi)}{\xi^2}<+\infty, $$ there exists $\overline{\delta}>0$ given by \eqref{e3.3} such that, for each $\mu\in [0,\overline{\delta}[$, problem \eqref{e1.1} admits at least three solutions. \end{theorem} \begin{proof} Fix $\lambda$, $g$ and $\mu$ as in the conclusion. It suffices to show the functional $I_\lambda$ defined in \eqref{e2.8} has at least three critical points in $E_0^\alpha$. We prove this by verifying the conditions given in Theorem \ref{the2.1}. Note that $\Phi$ defined in \eqref{e2.9} is a nonnegative G\^{a}teaux differentiable and sequentially weakly lower semicontinuous functional, and its G\^{a}teaux derivative admits a continuous inverse on $(E_0^\alpha)^\ast$. Further, from Lemma \ref{ineq}, the functional $\Phi$ is coercive. Indeed, one has $$ \Phi(u)\geq |\cos(\pi\alpha)|\|u\|_\alpha^2\to +\infty, $$ as $\|u\|_\alpha\to +\infty$. Moreover, $\Psi$ defined in \eqref{e2.10} is a continuously G\^{a}teaux differentiable functional whose G\^{a}teaux derivative is compact. We will verify (A1) and (A2) of Theorem \ref{the2.1}. Let $w$ be the function defined by \begin{equation}\label{e3.4'} w(t):=\begin{cases} \frac{4d}{T}t, & t\in [0,T/4),\\ d, & t\in [T/4,3T/4],\\ \frac{4d}{T}(T-t), & t\in (3T/4,T], \end{cases} \end{equation} and put $$ r:=\frac{|\cos(\pi\alpha)|}{\Omega^2}c^2. $$ It is easy to check that $w(0)=w(T)=0$ and $w\in L^2([0,T])$. Moreover, $w$ is Lipschitz continuous on $[0,T]$, and hence $w$ is absolutely continuous on $[0,T]$. By calculations, we have \begin{equation*} {}_0^cD_t^\alpha w(t)=\begin{cases} \frac{4d}{T\Gamma(2-\alpha)}t^{1-\alpha}, & t\in [0,T/4),\\ \frac{4d}{T\Gamma(2-\alpha)}\big[t^{1-\alpha} -\big(t-\frac{T}{4}\big)^{1-\alpha}\big], & t\in [T/4,3T/4],\\ \frac{4d}{T\Gamma(2-\alpha)}\big[t^{1-\alpha} -\big(t-\frac{T}{4}\big)^{1-\alpha}+\big(t-\frac{3T}{4}\big)^{1-\alpha}\big], & t\in (3T/4,T]. \end{cases} \end{equation*} Obviously, ${}_0^cD_t^\alpha w$ is continuous on $[0,T]$ and \begin{align*} &\int_0^T|{}_0^cD_t^\alpha w(t)|^2\,dt\\ &= \frac{16d^2}{T^2\Gamma^2(2-\alpha)} \Big\{\int_0^{T/4}t^{2-2\alpha}\,dt +\int_{T/4}^{3T/4}\big[t^{1-\alpha} -\big(t-\frac{T}{4}\big)^{1-\alpha}\big]^2\,dt\\ &\quad +\int_{3T/4}^T\big[t^{1-\alpha}-\big(t-\frac{T}{4}\big)^{1-\alpha} +\big(t-\frac{3T}{4}\big)^{1-\alpha}\big]^2\,dt\Big\}\\ &:= \frac{16d^2}{T^2\Gamma^2(2-\alpha)}C(T,\alpha). \end{align*} Therefore, from inequality \eqref{e3.4} one has $$ \Phi(w)\geq|\cos(\pi\alpha)|\|w\|^2_\alpha =\frac{16d^2|\cos(\pi\alpha)|}{T^2\Gamma^2(2-\alpha)}C(T,\alpha)>r. $$ Also, by using condition (A6), since $0\leq w(t)\leq d$ for each $t\in[0,T]$, we infer \begin{align*} \Psi(w) &= \int_0^T[F(t,w(t))+\frac{\mu}{\lambda}G(t,w(t))]dt\\ &\geq \int_{T/4}^{{3T}/4}F(t,d)\,dt+\frac{\mu}{\lambda}\int_0^TG(t,w(t))\,dt\\ &\geq \int_{T/4}^{{3T}/4}F(t,d)\,dt+\frac{\mu}{\lambda}TG_d. \end{align*} For all $u\in E_0^\alpha$ with $\Phi(u)\leq r$, by Lemma \ref{ineq}, we have $$ |\cos(\pi\alpha)|\|u\|_\alpha^2\leq\Phi(u)\leq r, $$ which implies $$ \|u\|_\alpha^2\leq\frac{1}{|\cos(\pi\alpha)|}r. $$ On the other hand, by Lemma \ref{lem2.1}, when $\alpha>1/2$, for each $u\in E_0^{\alpha}$ we have \begin{equation}\label{e3.5} \|u\|_{\infty}\le \Omega\Big(\int_0^T |_0^c D_t^{\alpha} u(t)|^2\,dt \Big)^{1/2}=\Omega\|u\|_{\alpha}. \end{equation} Thus, we obtain $$ |u(t)|\leq\Omega\sqrt{\frac{r}{|\cos(\pi\alpha)|}}=c,\quad\forall t\in[0,T]. $$ Therefore, \begin{align*} \frac{\sup_{\Phi(u)\leq r}\Psi(u)}{r} &\leq \frac{\int_0^T\max_{|\xi|\leq c}F(t,\xi)\,dt+\frac{\mu}{\lambda}\int_0^T\max_{|\xi|\leq c}G(t,\xi)\,dt} {\frac{|\cos(\pi\alpha)|}{\Omega^2}c^2}\\ &= \frac{\Omega^2}{c^2|\cos(\pi\alpha)|}\Big(\int_0^T\max_{|\xi|\leq c}F(t,\xi)\,dt+\frac{\mu}{\lambda}G^c\Big). \end{align*} From this, if $G^c=0$, it is clear that \begin{equation}\label{e3.6} \frac{\sup_{\Phi(u)\leq r}\Psi(u)}{r}<\frac{1}{\lambda}, \end{equation} while, if $G^c>0$, it turns out to be true bearing in mind that $$ \mu<\frac{ c^2|\cos(\pi\alpha)|-\lambda\Omega^2\int_0^T\max_{|\xi|\leq c}F(t,\xi)\,dt}{\Omega^2G^c}. $$ On the other hand, taking into account that $$ \Phi(w)\leq\frac{1}{|\cos(\pi\alpha)|}\|w\|_\alpha^2=\omega_{\alpha,d}, $$ we have \[ \frac{\Psi(w)}{\Phi(w)} \geq\frac{\int_{T/4}^{{3T}/4}F(t,d)\,dt +\frac{\mu}{\lambda}G_d}{\omega_{\alpha,d}}. \] Hence, if $G_d= 0$, we find \begin{equation}\label{e3.7} \frac{\Psi(w)}{\Phi(w)}>\frac{1}{\lambda}, \end{equation} while, if $G_d<0$, the same relation holds since $$ \mu<\frac{\omega_{\alpha,d}-\lambda\int_{T/4}^{{3T}/4} F(t,d)\,dt}{TG_d}. $$ Therefore, from \eqref{e3.6} and \eqref{e3.7}, condition (A1) of Theorem \ref{the2.1} is verified. Now, to prove the coercivity of the functional $I_\lambda$, first we assume that $$ \limsup_{|\xi|\to +\infty}\frac{\sup_{t\in[0,T]}F(t,\xi)}{\xi^2}>0. $$ Therefore, fixing $$ \limsup_{|\xi|\to +\infty}\frac{\sup_{t\in[0,T]}F(t,\xi)}{\xi^2}<\varepsilon <\frac{\int_0^T\max_{|\xi|\leq c}F(t,\xi)\,dt}{2c^2T}, $$ from (A8) there is a function $h_\varepsilon\in L^1([0,T])$ such that $F(t,\xi)\leq\varepsilon\xi^2+h_\varepsilon(t)$, for each $t\in[0,T]$ and $\xi\in\mathbb{R}$. Taking \eqref{e3.5} into account and since \[ \lambda<\frac{ c^2|\cos(\pi\alpha)|}{\Omega^2\int_0^T\max_{|\xi|\leq c}F(t,\xi)\,dt}, \] it follows that \begin{equation} \label{e3.8} \begin{aligned} \lambda\int_0^T F(t,u(t))\,dt &\leq \lambda\Big(\varepsilon\int_0^T(u(t))^2dt+\int_0^T h_\varepsilon(t)\,dt\Big) \\ &<\frac{c^2|\cos(\pi\alpha)|}{\Omega^2\int_0^T\max_{|\xi|\leq c}F(t,\xi)\,dt}\Big(\varepsilon\Omega^2T\|u\|_\alpha^2 +\|h_\varepsilon\|_{L^1([0,T])}\Big), \end{aligned} \end{equation} for each $u\in E_0^\alpha$. Since $\mu<\overline{\delta}$, we obtain $$ \limsup_{|\xi|\to+\infty}\frac{\sup_{t\in[0,T]}G(t,\xi)}{\xi^2} <\frac{|\cos(\pi\alpha)|}{2T\mu \Omega^2}; $$ then, there is a function $h_\mu\in L^1([0,T])$ such that $$ G(t,\xi)\leq\frac{|\cos(\pi\alpha)|}{2T\mu\Omega^2}\xi^2+h_\mu(t), $$ for each $t\in[0,T]$ and $\xi\in\mathbb{R}$. Thus, taking again \eqref{e3.5} into account, it follows that \begin{equation} \label{e3.9} \begin{aligned} \int_0^TG(t,u(t))\,dt &\leq \frac{|\cos(\pi\alpha)|}{2T\mu\Omega^2}\int_0^T(u(t))^2dt +\int_0^Th_\mu(t)\,dt\\ &\leq \frac{|\cos(\pi\alpha)|}{2\mu}\|u\|_\alpha^2+\|h_\mu\|_{L^1([0,T])}, \end{aligned} \end{equation} for each $u\in E_0^\alpha$. Finally, putting together \eqref{e3.8} and \eqref{e3.9}, we have \begin{align*} I_\lambda(u) &= \Phi(u)-\lambda\Psi(u)\\ &\geq |\cos(\pi\alpha)|\|u\|_\alpha^2-\frac{c^2|\cos(\pi\alpha)|} {\Omega^2\int_0^T\max_{|\xi|\leq c}F(t,\xi)\,dt}\Big(\varepsilon\Omega^2T\|u\|_\alpha^2 +\|h_\varepsilon\|_{L^1([0,T])}\Big)\\ &\quad -\frac{|\cos(\pi\alpha)|}{2}\|u\|_\alpha^2-\mu\|h_\mu\|_{L^1([0,T])}\\ &= |\cos(\pi\alpha)|\Big(\frac{1}{2}-\frac{c^2T}{\int_0^T\max_{|\xi|\leq c}F(t,\xi)\,dt}\varepsilon\Big)\|u\|_\alpha^2\\ &\quad -\frac{c^2|\cos(\pi\alpha)|\|h_\varepsilon\|_{L^1([0,T])}} {\Omega^2\int_0^T\max_{|\xi|\leq c}F(t,\xi)\,dt}-\mu \|h_\mu\|_{L^1([0,T])}. \end{align*} On the other hand, if $$ \limsup_{|\xi|\to +\infty}\frac{\sup_{t\in[0,T]}F(t,\xi)}{\xi^2}\leq 0, $$ then there exists a function $h_\varepsilon\in L^1([0,T])$ such that $F(t,\xi)\leq h_\varepsilon(t)$ for each $t\in[0,T]$ and $\xi\in\mathbb{R}$. Arguing as before we obtain $$ I_\lambda(u)\geq\frac{|\cos(\pi\alpha)|}{2}\|u\|_\alpha^2 -\frac{c^2|\cos(\pi\alpha)|\|h_\varepsilon\|_{L^1([0,T])}} {\Omega^2\int_0^T\max_{|\xi|\leq c}F(t,\xi)\,dt}-\mu \|h_\mu\|_{L^1([0,T])}\,. $$ Both cases lead to the coercivity of $I_\lambda$ and condition (A2) of Theorem \ref{the2.1} is verified. Since, from \eqref{e3.6} and \eqref{e3.7}, $$ \lambda\in\Lambda\subseteq\Big]\frac{\Phi(w)}{\Psi(w)}, \frac{r}{\sup_{\Phi(u)\leq r}\Psi(u)}\Big[, $$ Theorem \ref{the2.1} ensures the existence of at least three critical points for the functional $I_\lambda$ and the proof is complete. \end{proof} Now, we state a variant of Theorem \ref{the3.1} in which no asymptotic condition on $g$ is requested. In such a case, the functions $f$ and $g$ are supposed to be nonnegative. Fixing positive constants $c_1,c_2$ and $d$ such that $$ \frac{3}{2}\frac{\omega_{\alpha,d}}{\int_{T/4}^{{3T}/4}F(t,d)\,dt}< \frac{|\cos(\pi\alpha)|}{\Omega^2} \min\Big\{\frac{c_1^2}{\int_0^T F(t,c_1)\,dt}, \frac{c_2^2}{ 2\int_0^T F(t,c_2)\,dt}\Big\}, $$ and selecting \begin{equation}\label{e3.10} \lambda\in\Lambda' :=\Big]\frac{3}{2}\frac{\omega_{\alpha,d}}{\int_{T/4}^{{3T}/4}F(t,d)\,dt}, \frac{|\cos(\pi\alpha)|}{\Omega^2}\min\Big\{\frac{c_1^2}{\int_0^T F(t,c_1)\,dt}, \frac{c_2^2}{ 2\int_0^T F(t,c_2)\,dt}\Big\}\Big[, \end{equation} we put \begin{equation}\label{e3.11} \delta^\star:=\min\Big\{\frac{c_1^2|\cos(\pi\alpha)|-\lambda\Omega^2 \int_0^T F(t,c_1)\,dt}{\Omega^2G^{c_1}}, \frac{c_2^2|\cos(\pi\alpha)|-2\lambda\Omega^2 \int_0^T F(t,c_2)\,dt}{2\Omega^2G^{c_2}}\Big\}. \end{equation} With the above notation we have the following multiplicity result. \begin{theorem}\label{the3.2} Assume that there exist three positive constants $c_1,c_2$ and $d$ with \begin{equation}\label{e3.12} c_1<\frac{2d\Omega}{T\Gamma(2-\alpha)}\sqrt{2C(T,\alpha)} <\frac{c_2|\cos(\pi\alpha)|}{2}, \end{equation} such that \begin{itemize} \item[(A9)] $f(t,\xi)\geq 0$ for all $(t,\xi)\in[0,T]\times[0,c_2]$; \item[(A10)] $$ \max\Big\{\frac{\int_0^T F(t,c_1)\,dt}{ c_1^2},\frac{2\int_0^T F(t,c_2)\,dt}{ c_2^2}\Big\} <\frac{2}{3}\frac{|\cos(\pi\alpha)|\int_{T/4}^{{3T}/4} F(t,d)\,dt}{\omega_{\alpha,d}\Omega^2}. $$ \end{itemize} Then, for each $\lambda\in\Lambda'$, where $\Lambda'$ is given by \eqref{e3.10}, and for every nonnegative continuous function $g:[0,T]\times\mathbb{R}\to \mathbb{R}$, there exists $\delta^\star>0$ given by \eqref{e3.11} such that, for each $\mu\in[0,\delta^\star[$, problem \eqref{e1.1} admits at least three distinct solutions $u_i$, $i=1,2,3$, such that $$ 0\leq u_i(t)\lambda^\star:=\frac{2\omega_{\alpha,d}}{TF(d)}$ for some $d>0$ such that $F(d)>0$. Recalling that $$ \liminf_{\xi\to 0}\frac{F(\xi)}{\xi^2}=0, $$ there is a sequence $\{c_n\}\subset]0,+\infty[$ such that $\lim_{n\to +\infty}c_{n}=0$ and $$ \lim _{n\to +\infty}\frac{\max_{|\xi| \leq c_{n}}F(\xi)}{c_n^2}=0. $$ Indeed, one has $$ \lim_{n\to +\infty}\frac{\max_{|\xi| \leq c_n}F(\xi)}{c_n^2}=\lim_{n\to +\infty}\frac{F(\xi_{c_n})}{\xi_{c_n}^2} \frac{\xi_{c_n}^2}{c_n^2}=0, $$ where $ F(\xi_{c_n})=\max_{|\xi| \leq c_{n}}F(\xi).$ Therefore, there exists $\overline{c}>0$ such that $$ \frac{\max_{|\xi| \leq\overline{c}}F(\xi)}{\overline{c}^2}<\frac{|\cos(\pi\alpha)|}{\Omega^2} \min\Big\{\frac{F(d)}{2\omega_{\alpha,d}},\frac{1}{T\lambda}\Big\} $$ and $\overline{c}<\frac{4\Omega d}{T\Gamma(2-\alpha)}\sqrt{C(T,\alpha)}$. Hence, the conclusion follows from Theorem \ref{the3.1} \end{proof} \begin{proof}[Proof of Theorem \ref{t1.2}] Our aim is to apply Theorem \ref{the3.2} by choosing $c_2=10$ and $d=1$. Therefore, taking into account that $\alpha=T=1$, one has \begin{gather*} \frac{3}{2}\frac{\omega_{\alpha,d}}{\int_{T/4}^{{3T}/4} F(t,d)dt}=\frac{24}{\int_{0}^{1}f(s)ds}, \\ \frac{|\cos(\pi\alpha)|}{\Omega^2}\frac{c_2^2}{2\int_0^T F(t,c_{2})dt}=\frac{50}{\int_{0}^{10}f(s)ds}. \end{gather*} Since $\lim_{t\to 0} f(t)/t=0$, one has $$ \lim_{t\to 0}\frac{\int_{0}^{t}f(s)ds}{t^{2}}=0. $$ Then, there exists a positive constant $c_1<2$ such that \begin{gather*} \frac{\int_{0}^{c_1}f(s)ds}{c_1^{2}}<\frac{1}{24}\int_{0}^{1}f(s)ds, \\ \frac{c_1^{2}}{\int_{0}^{c_1}f(s)ds}>\frac{50}{\int_{0}^{10}f(s)ds}. \end{gather*} Hence, a simple computation shows that all assumptions of Theorem \ref{the3.2} are satisfied, and the conclusion follows. \end{proof} \begin{thebibliography}{00} \bibitem{AHM} G. A. Afrouzi, A. Hadjian, G. Molica Bisci; {Some results for one dimensional fractional problems}, submitted. \bibitem{AHR} G. A. Afrouzi, A. Hadjian, V. 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