Multiplicity of positive solutions for Sturm-Liouville boundary value problems of fractional differential equations with p-Laplacian

During the past decades, much attention has been focused on the study of equations with p-Laplacian differential operator. The motivation for those works stems from the applications in the modeling of different physical and natural phenomena: non-Newtonian mechanics [1], system of Monge-Kantorovich partial differential equations [2], population biology [3], nonlinear flow laws [4], combustion theory [5]. There exist a very large number of papers devoted to the existence of solutions for the equation with p-Laplacian operator.

subject to various boundary conditions, has been studied by many authors, see [6, 7] and the references therein.

satisfying different boundary conditions have been established by using fixed-point theorems and monotone iterative technique, see [8, 9] and the references therein.

where $\varphi (u^{\prime })=|u^{\prime }{|}^{p-2}{u}^{\prime }$, $p>1$, $f:(r,R)\times (0,\mathrm{\infty })\to \mathbb{R}$ and λ is a positive parameter, f is p-superlinear or p-sublinear at ∞ and maybe singular at $u=0$.

However, few papers can be found in the literature on the existence of multiple positive solutions for the third-order Sturm-Liouville boundary value problem with p-Laplacian.

where ${\varphi }_p(s)=|s{|}^{p-2}s$, $p>1$, ${({\varphi }_p)}^{-1}={\varphi }_q$, $1/p+1/q=1$, $\alpha ,\beta ,\gamma ,\delta \ge 0$, $f\in C([0,1]\times [0,\mathrm{\infty }),[0,\mathrm{\infty }))$. By means of the Leggett-Williams fixed-point theorems, some existence and multiplicity results of positive solutions are obtained. In later work, Yang and Yan [12] also studied the above problem by means of the fixed-point index method.

Recently, fractional differential equations have been of great interest. The motivation for those works stems from both the intensive development of the theory of fractional calculus itself and the applications such as economics, engineering and other fields [13–17]. Much attention has been focused on the study of the existence and multiplicity of solutions or positive solutions for boundary value problems of fractional differential equations by the use of techniques of nonlinear analysis (fixed-point theorems [18–25], upper and lower solutions method [26], fixed-point index theory [27, 28], coincidence theory [29], etc.).

where $1<\alpha \le 2$, $0<\beta \le 1$, $D_{0^{+}}^{\alpha }$, $D_{0^{+}}^{\beta }$ are the standard Caputo fractional derivatives, ${\varphi }_p(s)=|s{|}^{p-2}s$, $p>1$, ${\varphi }_p^{-1}={\varphi }_q$, $1/p+1/q=1$, $\xi ,\eta ,\gamma ,\delta \ge 0$, $\rho :=\xi \gamma +\xi \delta +\eta \gamma >0$, and $f:[0,1]\times [0,+\mathrm{\infty })\to [0,+\mathrm{\infty })$ is continuous. By means of the properties of the Green’s function, Leggett-Williams fixed-point theorems and fixed-point index theory, we establish the existence of at least two or at least three positive solutions for the Sturm-Liouville boundary value problem (1.1). As an application, an example is given to demonstrate the main result.

The rest of this paper is organized as follows. In Section 2, we shall introduce some definitions and lemmas to prove our main results. In Section 3, we state our main results. We prove our main results by Leggett-Williams fixed-point theorems and fixed-point index theory in Section 4. As an application, an example is presented to illustrate our main result in Section 5.

For the convenience of the reader, we give some background materials from fractional calculus theory to facilitate analysis of problem (1.1). These materials can be found in the recent literature, see [14, 17, 18, 31–33].

Definition 2.1 ([17])

provided the right side is pointwise defined on $(0,+\mathrm{\infty })$.

Definition 2.2 ([17])

where n is the smallest integer greater than or equal to α, provided that the right side is pointwise defined on $(0,+\mathrm{\infty })$.

Remark 2.1 ([14])

By Definition 2.2, under natural conditions on the function $f(t)$, as $\alpha \to n$ the Caputo derivative becomes a conventional n th derivative of the function $f(t)$.

Remark 2.2 ([18])

given in particular that $D_{0^{+}}^{\alpha }{t}^{\mu }=0$, $\mu =0,1,\dots ,n-1$, where $D_{0^{+}}^{\alpha }$ is the Caputo fractional derivative, and n is the smallest integer greater than or equal to α.

From the definition of the Caputo derivative and Remark 2.2, we can obtain the following statement.

Lemma 2.1 ([17])

as the unique solution, where n is the smallest integer greater than or equal to α.

Lemma 2.2 ([17])

for some $c_i\in \mathbb{R}$, $i=0,1,2,\dots ,n-1$, where n is the smallest integer greater than or equal to α.

which completes the proof. □

where $G(t,s)$ is defined as (2.2).

which completes the proof. □

Lemma 2.5 The Green’s function $G(t,s)$ defined by (2.2) is continuous on $[0,1]\times [0,1]$.

The method of proof is similar to Lemma 3.2 in [30], and we omit it here.

Definition 2.3 ([31])

If $P\subset E$ is a cone, we denote the order induced by P on E by ≤. For $u,v\in P$, we write $u\le v$ if $v-u\in P$.

Definition 2.4 ([31])

for all $x,y\in P$ and $0\le t\le 1$.

Definition 2.5 ([31])

Let $0<a<b$ be given and let φ be a nonnegative continuous concave functional on the cone P. Define the convex sets $P_r$, ${\overline{P}}_r$ and $P(\phi ,a,b)$ by $P_r=\{y\in P|\parallel y\parallel <r\}$, ${\overline{P}}_r=\{y\in P|\parallel y\parallel \le r\}$, $P(\phi ,a,b)=\{y\in P|a\le \phi (y),\parallel y\parallel \le b\}$.

Lemma 2.6 (Leggett-Williams [32])

Let $T:{\overline{P}}_c\to {\overline{P}}_c$ be a completely continuous operator and let φ be a nonnegative continuous concave functional on P such that $\phi (y)\le \parallel y\parallel$ for all $y\in {\overline{P}}_c$. Suppose that there exist $0<a<b<d\le c$ such that

(A₁) $\{y\in P(\phi ,b,d)|\phi (y)>b\}\ne \mathrm{\varnothing }$ and $\phi (Ty)>b$ for $y\in P(\phi ,b,d)$;

(A₂) $\parallel Ty\parallel <a$ for $\parallel y\parallel \le a$;

(A₃) $\phi (Ty)>b$ for $y\in P(\phi ,b,c)$ with $\parallel Ty\parallel >d$.

Then T has at least three fixed points $y_1$, $y_2$, and $y_3$ in $\overline{P_c}$ satisfying $\parallel y_1\parallel <a$, $\phi (y_2)>b$, $\parallel y_3\parallel >a$, and $\phi (y_3)<b$.

Lemma 2.7 ([32])

Let $T:{\overline{P}}_c\to P$ be a completely continuous operator and let φ be a nonnegative continuous concave functional on P such that $\phi (y)\le \parallel y\parallel$ for all $y\in {\overline{P}}_c$. Suppose that there exist $0<a<b<c$ such that

(B₁) $\{y\in P(\phi ,b,c)|\phi (y)>b\}\ne \mathrm{\varnothing }$, and $\phi (Ty)>b$ for $y\in P(\phi ,b,c)$;

(B₂) $\parallel Ty\parallel <a$ for $\parallel y\parallel \le a$;

(B₃) $\phi (Ty)>\frac{b}{c}\parallel Ty\parallel$ for $y\in {\overline{P}}_c$ with $\parallel Ty\parallel >c$.

Then T has at least two fixed points $y_1$ and $y_2$ in ${\overline{P}}_c$ satisfying $\parallel y_1\parallel <a$, $\parallel y_2\parallel >a$ and $\phi (y_2)<b$.

Lemma 2.8 ([31])

Let P be a closed convex set in a Banach space E and let Ω be a bounded open set such that ${\mathrm{\Omega }}_p:=\mathrm{\Omega }\cap P\ne \mathrm{\varnothing }$. Let $T:{\overline{\mathrm{\Omega }}}_p\to P$ be a compact map. Suppose that $x\ne Tx$ for all $x\in \partial {\mathrm{\Omega }}_p$.

(C₁) (Existence) If $i(T,{\mathrm{\Omega }}_p,P)\ne 0$, then T has a fixed point in ${\mathrm{\Omega }}_p$.

(C₂) (Normalization) If $u\in {\mathrm{\Omega }}_p$, then $i(\hat{u},{\mathrm{\Omega }}_p,P)=1$, where $\hat{u}(x)=u$ for $x\in {\overline{\mathrm{\Omega }}}_p$.

(C₃) (Homotopy) Let $\nu :[0,1]\times {\overline{\mathrm{\Omega }}}_p\to P$ be a compact map such that $x\ne \nu (t,x)$ for $x\in \partial {\mathrm{\Omega }}_p$ and $t\in [0,1]$. Then $i(\nu (0,\cdot ),{\mathrm{\Omega }}_p,P)=i(\nu (1,\cdot ),{\mathrm{\Omega }}_p,P)$.

(C₄) (Additivity) If $U_1$, $U_2$ are disjoint relatively open subsets of ${\mathrm{\Omega }}_p$ such that $x\ne Tx$ for $x\in {\overline{\mathrm{\Omega }}}_p\setminus (U_1\cup U_2)$, then $i(T,{\mathrm{\Omega }}_p,P)=i(T,U_1,P)+i(T,U_2,P)$, where $i(T,U_j,P)=i(T{|}_{{\overline{U}}_j},U_j,P)$ ($j=1,2$).

Lemma 2.9 ([33])

Let P be a cone in a Banach space E. For $q>0$, define ${\mathrm{\Omega }}_q=\{x\in p|\parallel x\parallel <q\}$. Assume that $T:{\overline{\mathrm{\Omega }}}_q\to P$ is a compact map such that $x\ne Tx$ for $x\in \partial {\mathrm{\Omega }}_q$. Thus, one has the following conclusions:

(D₁) if $\parallel x\parallel \le \parallel Tx\parallel$ for $x\in \partial {\mathrm{\Omega }}_q$, then $i(T,{\mathrm{\Omega }}_q,P)=0$;

(D₂) if $\parallel x\parallel \ge \parallel Tx\parallel$ for $x\in \partial {\mathrm{\Omega }}_q$, then $i(T,{\mathrm{\Omega }}_q,P)=1$.

where λ is given as in Lemma 2.5.

Then $T:P\to P$ is completely continuous.

Thus, $T(P)\subset P$. In view of non-negativity and continuity of $G(t,s)$, $\frac{{(s-\tau )}^{\beta -1}}{\mathrm{\Gamma }(\beta )}$ and $f(t,u(t))$, we find that $T:P\to P$ is continuous.

That is to say, $T(\mathrm{\Omega })$ is equicontinuous. By the Arzela-Ascoli theorem, we see that $T:P\to P$ is completely continuous. The proof is completed. □

We are now ready to prove our main results.

Theorem 3.1 Let $f(t,u)$ be nonnegative continuous on $[0,1]\times [0,+\mathrm{\infty })$. Assume that there exist constants a, b with $b>a>0$ such that

(H₁) $f(t,u)\ge {\varphi }_p(Nb)$, for $(t,u)\in [\frac{1}{4},\frac{3}{4}]\times [b,\frac{b}{{\lambda }^2}]$;

(H₂) $f(t,u)<{\varphi }_p(Ma)$, for $(t,u)\in [0,1]\times [0,a]$.

Then the boundary value problem (1.1) has at least two positive solutions $u_1$ and $u_2$ satisfying $\parallel u_1\parallel <a$, ${min}_{\frac{1}{4}\le t\le \frac{3}{4}}{u}_2(t)<b$ and $\parallel u_2\parallel >a$, where λ is given as in Lemma  2.5.

Evidently, for each $u\in P$, we have $\theta (u)\le \parallel u\parallel$.

So condition (B₃) of Lemma 2.7 is satisfied. Thus using Lemma 2.7, T has at least two fixed points. Consequently, the boundary value problem (1.1) has at least two positive solutions $u_1$ and $u_2$ in ${\overline{P}}_{\frac{b}{{\lambda }^2}}$ satisfying $\parallel u_1\parallel <a$, ${min}_{\frac{1}{4}\le t\le \frac{3}{4}}{u}_2(t)<b$ and $\parallel u_2\parallel >a$. The proof is completed. □

Theorem 3.2 Let $f(t,u)$ be nonnegative continuous on $[0,1]\times [0,+\mathrm{\infty })$. Assume that there exist constants a, b, c with ${\lambda }^{2}c>b>a>0$ such that

(H₃) $f(t,u)<{\varphi }_p(Ma)$, for $(t,u)\in [0,1]\times [0,a]$;

(H₄) $f(t,u)\ge {\varphi }_p(Nb)$, for $(t,u)\in [\frac{1}{4},\frac{3}{4}]\times [b,\frac{b}{{\lambda }^2}]$;

(H₅) $f(t,u)\le {\varphi }_p(Mc)$, for $(t,u)\in [0,1]\times [0,c]$.