Existence of solutions of Dirichlet problems for one dimensional fractional equations

1.   Introduction

In this paper, we are eager to investigate the nonlinear fractional boundary value problem (BVP) of the form

where $\lambda > 0$ is a real parameter, $\alpha \in (1/2, 1]$, $_0^c D_t^{\alpha}$ and $_t^c D_T^{\alpha}$ are the left and right Caputo fractional derivatives of order $\alpha$, respectively, ${}_0 D_t^{\alpha-1}$ and ${}_t D_T^{\alpha-1}$ are the left and right Riemann-Liouville fractional integrals of order $1-\alpha$, respectively, and $f:[0, T]\times\mathbb{R}\to\mathbb{R}$ is a continuous function.

Fractional differential equations (i.e. $\alpha \in (1/2, 1)$) arise in real applications in many fields of sciences such as conservation laws, minimal surfaces, water waves and ultra-relativistic limits of quantum mechanics. Due to these applications, non-local fractional problems are extensively investigated. There has been remarkable progress in fractional differential equations; the interested reader may see the books ^{[16,18,24,26,29]}.

Critical point theory has been very effective in specifying the existence and multiplicity of solutions for integer order differential equations provided that the equation has a variational construction on some appropriate Sobolev spaces, e.g., we refer to ^{[12,20,21,23,27,30]} and the references therein for detailed discussions. But until now, there are little consequences on the existence of solutions to fractional BVPs which were proved by the variational methods, since it is frequently hard to establish a proper space and variational functional for fractional differential equations. For instance, in ^{[1,2,3,4,5,13,14,17]}, variational methods are applied to study the existence and multiplicity of solutions for fractional BVPs.

An attractive physical case is considered in ^[17] where Jiao and Zhou, by applying the critical point theory, proved the existence and multiplicity of solutions for the problem

where $\beta\in[0, 1)$, $F:[0, T]\times{\Bbb R}^N\rightarrow \Bbb R$ (with $N\geq 1$) is a suitable given function and $\nabla F(t, x)$ is the gradient of $F$ at $x$ (see Remark 2.9 below for the relation of this problem with problem (1.1)).

Also, Bai in ^[5], applying a critical point result for differentiable functionals proved by Bonanno ^[7], discussed the existence of at least one non-zero solution for the problem

The authors in ^[13] obtained, by using three critical point theorems, for the following BVP for fractional order differential equations

the existence of at least three solutions, where $\lambda, \mu > 0$ are two parameters.

More recently, Galewski and Molica Bisci in ^[14] studied the problem (1.2), in the case $\lambda = 1$. With an asymptotic behaviour of $f$ at zero and using a critical point result ^[8], they proved the existence of one non-zero solution for the problem.

In this paper, we will prove the existence of infinitely many solutions for problem (1.1) with rather various hypotheses on the function $f$. We need that the potential $F$ of $f$ assures an appropriate oscillatory behavior either at infinity or at the origin.

Our results are based on the variational principle due to Ricceri ^[28]. We address the eager reader to the book ^[19] as a comprehensive reference on critical point theory adopted here.

Finally, we note that an interesting and careful analysis of fractional BVPs was extended in the nice and recent works ^{[6,9,10,11,15,22,25,32,33,34]} and the references therein.

This paper is organized as follows. In Section 2, we state some preliminary definitions and properties of the fractional calculus that will be required in the paper. In Section 3, our principal result, Theorem 3.1, and some significant conclusions (see Corollaries 3.3, 3.4 and 3.6) are presented. Then Example 3.8 is given as an application of Corollary 3.3.

2.   Preliminaries

In the present section, first we present several needful definitions and properties of the fractional calculus which are required further in this paper. Let $AC([a, b], {\mathbb{R}}) = AC^1([a, b], {\mathbb{R}})$ be the space of absolutely continuous functions on the interval $[a, b]$, where $a < b$ are real numbers (see ^[18]). Set $AC^n([a, b], {\mathbb{R}})$ the space of functions $u:[a, b]\rightarrow {\mathbb{R}}$ such that $u\in C^{n-1}([a, b], {\mathbb{R}})$ and $u^{(n-1)}\in AC([a, b], {\mathbb{R}})$. Here, $C^{n-1}([a, b], {\mathbb{R}})$ signifies the set of mappings that are $(n-1)$ times continuously differentiable on $[a, b]$.

Definition 2.1 (^[18,26]). Let $u\in L^1([a, b], {\mathbb{R}})$. We denote by ${}_a D_t^{-\gamma}u(t)$ and ${}_t D_b^{-\gamma}u(t)$ the left and right Riemann-Liouville fractional integrals of order $\gamma > 0$ for function $u$, respectively, that are defined by

and

for every $t \in [a, b]$, while the right-hand sides are pointwise defined on $[a, b]$, where $\Gamma > 0$ is the standard gamma function given by

We note that ${}_a D_t^{-\gamma}$ and ${}_t D_b^{-\gamma}$ are linear and bounded operators from $L^1([a, b], {\mathbb{R}})$ into $L^1([a, b], {\mathbb{R}})$; see also Lemma 2.6 below.

Definition 2.2 (^[18,26]). Let $u\in AC^n([a, b], {\mathbb{R}})$. We denote by ${}_a D_t^{\gamma}u(t)$ and ${}_t D_b^{\gamma}u(t)$ the left and right Riemann-Liouville fractional derivatives of order $\gamma$ $(n-1\leq\gamma < n$ and $n\in\Bbb N)$ for function $u$, respectively, that are defined by

and

where $t\in[a, b]$. Specially, if $0\leq\gamma < 1$, then

and

Definition 2.3 (^[18]). Suppose that $\gamma \ge 0$ and $n \in \mathbb{N}$.

$(\rm i)$ Let $\gamma \in (n-1, n)$ and $u\in AC^n([a, b], {\mathbb{R}})$. We denote by $_a^c D_t^{\gamma}u(t)$ and $_t^c D_b^{\gamma}u(t)$ the left and right Caputo fractional derivatives of order $\gamma$ for function $u$, respectively. These derivatives exist almost everywhere on $[a, b]$. $_a^c D_t^{\gamma}u(t)$ and $_t^c D_b^{\gamma}u(t)$ are illustrated by

and

for every $t \in [a, b]$, respectively. Specially, if $0 < \gamma < 1$, then

and

$(\rm ii)$ 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 illustrated by

for every $t\in [a, b]$. Specially, $_a^c D_t^{0}u(t) = _t^c D_b^{0}u(t) = u(t), \, \, t\in[a, b]$.

If $u\in AC^n([a, b], {\mathbb{R}})$, then the relation between the Riemann-Liouville fractional derivative and the Caputo fractional derivative is expressed by the following

Proposition 2.4 (^[18]). The left and right Riemann-Liouville fractional integral operators have the feature of a semigroup, that is

at every point $t\in[a, b]$ for a continuous function $u$, and for a.e. point in $[a, b]$ if $u\in L^1([a, b], {\mathbb{R}})$.

Proposition 2.5 (^[29]). We have

with the condition that $u \in L^p([a, b], {\mathbb{R}})$, $v \in L^q([a, b], {\mathbb{R}})$ and $p \ge 1$, $q \ge 1$ and $1/p + 1/q \le 1 + \gamma$, or $p \neq 1$, $q \neq 1$ and $1/p + 1/q = 1 + \gamma$.

For any $u \in L^2([0, T], \mathbb{R})$ and for every fixed $t\in[0, T]$, set

Lemma 2.6 ([17,Lemma 3.1]). Let $\alpha \in (0, 1]$. For any $u \in L^2([0, T], \mathbb{R})$, we have

Let $C_0^{\infty}([0, T], \mathbb{R})$ be the collection of all functions $g\in C^{\infty}([0, T], \mathbb{R})$ with compact support contained in $(0, T)$. Then any function $g\in C_0^{\infty}([0, T], \mathbb{R})$ satisfies $g(0) = g(T) = 0$.

Definition 2.7. Suppose that $0 < \alpha \le 1$. We define the fractional derivative space $E_0^{\alpha}$ by the closure of $C_0^{\infty}([0, T], \mathbb{R})$ with respect to the norm

for every $u \in E^{\alpha}_0$.

Remark 2.8. $(\rm i)$ For any $u\in E_0^{\alpha}$, with the fact that $u(0) = 0$, one has $_0^c D_t^{\alpha}u(t) = _0 D_t^{\alpha}u(t), \, t\in[0, T]$ according to the equality (10) of ^[17].

$(\rm ii)$ According to Lemma 2.6, for every $u\in C_0^{\infty}([0, T], \mathbb{R})$, one has $u \in L^2([0, T], \mathbb{R})$ and $_0^c D_t^{\alpha}u \in L^2([0, T], \mathbb{R})$. Thus, it is obvious that $E_0^{\alpha}$ 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 $u(0) = u(T) = 0$.

Remark 2.9. In view of Definition 2.3, for every $u \in AC([0, T], \mathbb{R})$, BVP (1.1) transforms to

where $\beta : = 2(1-\alpha) \in [0, 1)$.

Furthermore by Proposition 2.4, it is clear that $u \in AC([0, T], \mathbb{R})$ is a solution of BVP (2.1) if and only if $u$ is a solution of the problem

For completeness we recall that a function $u\in AC([0, T], \mathbb{R})$ is named a solution of BVP (2.2) if:

$({\rm{j}})$ The map

is differentiable for a.e. $t\in [0, T]$, and

$({\rm{jj}})$ The function $u$ assures (2.2).

Proposition 2.10 ([31,Lemma 4.2]). Suppose that $\alpha \in (0, 1]$. The space $E_0^{\alpha}$ is a separable and reflexive Banach space.

Lemma 2.11 ([17,Proposition 3.2]). Suppose that $\alpha \in (1/2, 1]$. For all $u\in E_0^{\alpha}$, one has

Hence, we can consider $E_0^{\alpha}$ equipped with the equivalent norm

Lemma 2.12 ([17,Proposition 4.1]). Suppose that $\alpha \in (1/2, 1]$. For every $u \in E_0^{\alpha}$, one has

We refer our main results for $\alpha \in (1/2, 1]$ rather than $\alpha \in (0, 1/2]$, since by Lemmas 2.11 and 2.12, for $\alpha \in (1/2, 1]$, the space $E_0^{\alpha}$ is compactly embedded in $C([0, T], \Bbb R)$ and the functional $-\int_0^T {}_0^c D_t^{\alpha}u(t)\cdot{}_t^c D_T^{\alpha} u(t)\, dt$ is coercive.

We formulate the following version of Ricceri's variational principle [28,Theorem 2.5], that is our principal tool for establishing the principal result of this paper.

Theorem 2.13. Suppose that $X$ is a reflexive real Banach space, and let $\Phi, \Psi:X\rightarrow \mathbb{R}$ be two Gâteaux differentiable functionals such that $\Phi$ is sequentially weakly lower semicontinuous, strongly continuous and coercive, and $\Psi$ is sequentially weakly upper semicontinuous. For any $r > \inf_X\Phi,$ let

Put

Then, the following properties hold$:$

$\rm(a)$ If $\gamma < +\infty,$ then for each $\lambda\in (0, 1/\gamma)$, the following alternative holds: Either

$\rm(a_1)$ $I_\lambda$ possesses a global minimum, or

$\rm(a_2)$ there is a sequence $\{u_n\}$ of critical points $($local minima$)$ of $I_\lambda$ such that

$\rm(b)$ If $\delta < +\infty,$ then for each $\lambda\in (0, 1/\delta)$, the following alternative holds: Either

$\rm(b_1)$ there is a global minimum of $\Phi$ which is a local minimum of $I_\lambda$, or

$\rm(b_2)$ there is a sequence $\{u_n\}$ of pairwise distinct critical points $($local minima$)$ of $I_\lambda$ that converges weakly to a global minimum of $\Phi$, with $\lim\limits_{n\rightarrow +\infty}\Phi(u_n) = \inf\limits_{u\in X}\Phi(u)$.

3.   Main results

In the present section, we state and prove our principal result. Let

and

where $F$ is the potential of $f$ defined by

We suppose that the following condition holds:

$(\rm f_1)$ $F(t, \xi)\geq 0$ for any $(t, \xi)\in ([0, \frac{T}{4}]\cup[\frac{3T}{4}, T])\times\mathbb{R}$.

Our principal result reads as follows.

Theorem 3.1. Suppose that $f:[0, T]\times\mathbb{R}\to\mathbb{R}$ is a continuous function whose potential satisfies $(\rm f_1)$. Assume that there exist real sequences $\{a_n\}$ and $\{b_n\}$ in $(0, +\infty)$, with $\lim\limits_{n\rightarrow +\infty}b_n = +\infty$, such that$:$

$(\rm h_1)$ For some $n_0\in\Bbb N$ we have $a_n < \frac{T|\cos(\pi\alpha)|\Gamma(2-\alpha)} {4\kappa \sqrt{C(T, \alpha)}}b_n$ for each $n\geq n_0;$

$(\rm h_2)$ $\mathcal{A}_\infty: = \lim\limits_{n\rightarrow +\infty} \frac{ \int_0^T \max_{|\xi|\leq b_n}F(t, \xi)\, dt-\int_{T/4}^{3T/4}F(t, a_n)\, dt} { T^2|\cos(\pi\alpha)|^2\Gamma^2(2-\alpha)b_n^2-16\kappa^2a_n^2C(T, \alpha)} < \frac{B^\infty}{16\kappa^2C(T, \alpha)}$.

Then, for each

problem $(1.1)$ admits an unbounded sequence of solutions in $E_0^\alpha$.

Proof. We want to apply Theorem 2.13 to problem (1.1). For this, we define the functionals $\Phi, \Psi:X\rightarrow \mathbb{R}$ by

and put

for every $u\in X: = E_0^\alpha$.

Obviously, $\Phi$ and $\Psi$ are Gâteaux differentiable functionals whose derivatives at $u \in E_0^{\alpha}$ are

for every $v \in E_0^{\alpha}$. By Definition 2.3 and (2.1), we have

Hence, $I_{\lambda} = \Phi-\lambda\Psi\in C^1(E_0^{\alpha}, \mathbb{R})$ and $\Phi$ and $\Psi$ are sequentially weakly lower and upper semicontinuous, respectively.

Also by applying Lemma 2.12, we deduce that the functional $\Phi$ is coercive. Indeed, we have

as $\|u\|_\alpha\rightarrow +\infty$.

Further, we prove that a critical point of $I_\lambda$ is a solution of (1.1). For this, if $u_*\in E_0^{\alpha}$ is a critical point of $I_{\lambda}$, then

for every $v\in E_0^{\alpha}$. The Du Bois-Reymond Lemma and (3.1) imply

a.e. on $[0, T]$ for some $m\in\mathbb{R}$. By (3.2), it is obvious to prove that $u_*\in E_0^{\alpha}$ is a solution of (1.1).

By Lemma 2.11, when $\alpha > 1/2$, for each $u\in E_0^{\alpha}$ one has

First we establish that $\lambda < 1/\gamma$, for any fixed $\lambda$ as in the conclusion. For this, put

Then, for all $u\in E_0^\alpha$ with $\Phi(u) < r_n$, by applying Lemma 2.12, we see that

which implies

Thus, by (3.3)–(3.5) we obtain

for any $u\in E_0^\alpha$ with the condition $\Phi(u) < r_n$. Then, for every $n\in\Bbb N$, we get that

Let $w_n$ be defined by

for each $n\in\Bbb N$.

Clearly, we can investigate that $w_n(0) = w_n(T) = 0$ and $w_n\in L^2([0, T])$. Moreover, $w_n$ is Lipschitz continuous on $[0, T]$, and therefore $w_n$ is absolutely continuous on $[0, T]$. We have

Obviously, the function ${}_0^cD_t^\alpha w_n$ is continuous in $[0, T]$, and

Therefore,

Hence, by $\rm(h_1)$, one has $\Phi(w_n) < r_n$ for all $n\geq n_0$. Moreover, by $\rm(f_1)$, we also have

for each $n\in\Bbb N$.

Then, it follows that

for every $n\geq n_0$.

Hence, by the hypothesis $\rm(h_2)$, we get that

By the above relation, since

we also have $\lambda < 1/\gamma$.

Now, we claim that $I_\lambda$ is unbounded from below. By the relation

taking into account the definition of $B^\infty$, there exist a sequence $\{\eta_n\}$ of positive numbers and $\tau > 0$ with the conditions $\lim\limits_{n\rightarrow +\infty}\eta_n = +\infty$ and

for every $n\in\mathbb{N}$ large enough.

For every $n\in\mathbb{N}$, suppose that $s_n\in X$ is defined by

Thus, we obtain

for all $n\in\mathbb{N}$ large enough. By the relation $\lambda\tau > 1$ and $\lim\limits_{n\rightarrow +\infty}\eta_n = +\infty$, we have

Therefore, $I_\lambda$ is unbounded from below, and so, we get that $I_\lambda$ has no global minimum. Then, by applying Theorem 2.13, part $(\rm b)$, there exists a sequence $\{u_n\}$ of critical points of $I_\lambda$ with

So by Lemma 2.12, we have

The proof is complete.

Put

Discussing as in the proof of Theorem 3.1 and exploiting part $\rm (c)$ of Theorem 2.13, we arrive the following.

Theorem 3.2. Suppose that $f:[0, T]\times\mathbb{R}\to\mathbb{R}$ is a continuous function whose potential satisfies $\rm(f_1)$. Assume that there exist real sequences $\{c_n\}$ and $\{d_n\}$ in $(0, +\infty)$, with $\lim\limits_{n\rightarrow +\infty}d_n = 0$, such that$:$

$\rm(h_3)$ For some $n_0\in\Bbb N$ we have $c_n < \frac{T|\cos(\pi\alpha)| \Gamma(2-\alpha)}{4\kappa \sqrt{C(T, \alpha)}}d_n$ for each $n\geq n_0;$

$\rm(h_4)$ $\mathcal{A}_0: = \lim\limits_{n\rightarrow +\infty} \frac{ \int_0^T \max_{|\xi|\leq d_n}F(t, \xi)\, dt-\int_{T/4}^{3T/4}F(t, c_n)\, dt} { T^2|\cos(\pi\alpha)|^2\Gamma^2(2-\alpha)d_n^2-16\kappa^2c_n^2C(T, \alpha)} < \frac{B^0}{16\kappa^2C(T, \alpha)}$.

Then, for each

problem $(1.1)$ has a sequence of non-zero solutions which strongly converges to zero in $E_0^\alpha$.

At the present, we state some remarkable consequences of Theorem 3.1. Let

Corollary 3.3. Suppose that $f:[0, T]\times\mathbb{R}\to\mathbb{R}$ is a continuous function whose potential satisfies $\rm(f_1)$. Assume that

$(\rm h_5)$ $A_\infty < \frac{T^2|\cos(\pi\alpha)|^2\Gamma^2(2-\alpha)}{16\kappa^2C(T, \alpha)}B^\infty$.

Then, for each

problem $(1.1)$ has an unbounded sequence of solutions in $E_0^\alpha$.

Proof. Assume that $\{b_n\}$ be a sequence of positive numbers, with $\lim_{n\rightarrow +\infty}b_n = +\infty$, such that

Taking $a_n = 0$ for all $n\geq n_0$, by applying Theorem 3.1, we have the outcome.

A specific case of Corollary 3.3 is the following.

Corollary 3.4. Assume that $f:[0, T]\times\mathbb{R}\to\mathbb{R}$ be a continuous function whose potential satisfies $\rm(f_1)$. Assume that

$\rm (h_6)$ $A_\infty < \frac{|\cos(\pi\alpha)|}{\kappa^2}$ and $B^\infty > \frac{16C(T, \alpha)}{T^2\Gamma^2(2-\alpha)|\cos(\pi\alpha)|}$.

Then, the problem

has an unbounded sequence of solutions in $E_0^\alpha$.

Remark 3.5. We point out that when $f$ is a nonnegative function, hypothesis $\rm (f_1)$ preserves and assumption $\rm (h_5)$ becomes

$\rm(h_5')$ $A_\infty': = \liminf\limits_{\xi\rightarrow +\infty} \frac{ \int_0^T F(t, \xi)\, dt}{\xi^2}< \frac{T^2|\cos(\pi\alpha)|^2\Gamma^2(2-\alpha)}{16\kappa^2C(T, \alpha)}B^\infty$.

In this occasion, $\rm(h_5')$ ensures that for all

problem (1.1) has an unbounded sequence of solutions in $E_0^\alpha$.

Corollary 3.6. Assume that $f:[0, T]\times\mathbb{R}\to\mathbb{R}$ be a continuous function whose potential satisfies $\rm(f_1)$. Assume that there exist real sequences $\{a_n\}$ and $\{b_n\}$ in $(0, +\infty)$, with $\lim\limits_{n\rightarrow +\infty}b_n = +\infty$, such that $\rm(h_1)$ holds and

$\rm (h_7)$ $\int_{T/4}^{3T/4}F(t, a_n)\, dt = \int_0^T\max_{|\xi|\leq b_n}F(t, \xi)\, dt$ for all $n\in \Bbb N$.

If $B^\infty > 0$, then, for all

problem $(1.1)$ has an unbounded sequence of solutions in $E_0^\alpha$.

Proof. By $\rm(h_7)$ we get $\mathcal{A}_\infty = 0$. Therefore, since $B^\infty > 0$, condition $\rm(h_2)$ of Theorem 3.1 holds and the result is obtained.

Remark 3.7. By Theorem 3.2 we get the identical conclusions of Theorem 3.1. Namely, substituting $\xi\rightarrow +\infty$ with $\xi\rightarrow 0^+$, assertions such as Corollaries 3.3, 3.4 and 3.6 can be established. We omit the details.

To conclude, we give an example of application of the main results.

Example 3.8. Consider the problem

where $f:\mathbb{R}\to\mathbb{R}$ is the continuous function defined by

A direct calculation shows

So, $F$ satisfies $\rm(f_1)$ and we have

The above calculations are done using MAPLE. Hence, using Corollary 3.3, for each $\lambda\in]4.5384, +\infty[$, problem (3.6) has an unbounded sequence of solutions in $E_0^{0.7}$.

4.   Conclusions

Taking advantage of a critical point theorem obtained by Ricceri ^[28], the existence of infinitely many solutions for a nonlinear fractional BVP with a parameter is established. More precisely, a concrete interval of positive parameters, for which the treated problem admits infinitely many solutions, is determined without any symmetry or monotonicity assumptions on the nonlinear data. Our goal was achieved by requiring an appropriate oscillatory behavior of the nonlinear term either at infinity or at zero, without any additional conditions.

Acknowledgments

The authors would like to thank the reviewers for their good comments and suggestions which helped them to improve the presentation of this paper.

The work of the second author has been partially supported by the Agencia Estatal de Investigación (AEI) of Spain under Grants MTM2016-75140-P and PID2020-113275GB-I00, and by Xunta de Galicia, grant ED431C 2019/02.

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.