Sufficient Conditions for Existence of Positive Solutions for a Caputo Fractional Singular Boundary Value Problem

TL;DR

This paper establishes sufficient conditions for positive solutions to a class of fractional singular boundary value problems involving Caputo derivatives, addressing singularities in both the independent and dependent variables.

Contribution

It provides new sufficient conditions for existence of positive solutions in fractional singular boundary value problems with Caputo derivatives, considering singularities in variables.

Findings

Derived sufficient conditions for solution existence

Addressed singularities in variables

Extended theory to fractional Caputo derivatives

Abstract

We present sufficient conditions for the existence of positive solutions for a class of fractional singular boundary value problems in presence of Caputo fractional derivative. Further, the nonlinearity involved has singularity with respect to independent variable as well as with respect to dependent variable.

Equations

$$\begin{array}{rl}{}^C{D}_{0^{+}}^{\mu }x(t)+f(t,x(t))& =\omega x(t),t\in (0,1),1<\mu \le 2,\omega >0,\\ x^{\prime }(0)=0,x(1)& =0,\end{array}$$

$${}^C{D}_{0^{+}}^{\mu }x(t)=\frac{1}{\Gamma (n-\mu )}{\int }_0^t\frac{x^{(n)}(\tau )}{(t-\tau {)}^{\mu -n+1}}d\tau .$$

$$\begin{array}{rl}\mathcal{L}\{({}^C{D}_{0^{+}}^{\mu }x)(t)\}(s)& =s^{\mu }\mathcal{L}\{x(t)\}(s)-\sum _{k=0}^{n-1}{s}^{\mu -k-1}{x}^{(k)}(0),n-1<\mu \le n,\\ \mathcal{L}\{t^{\nu -1}{E}_{\mu ,\nu }(\omega t^{\mu })\}(s)& =\frac{s^{\mu -\nu }}{s^{\mu }-\omega },\mu >0,\nu >0,\left|\frac{\omega }{s^{\mu }}\right|<1,\end{array}$$

$$\begin{array}{rl}{}^C{D}_{0^{+}}^{\mu }x(t)+y(t)& =\omega x(t),t\in (0,1),1<\mu \le 2,\omega >0,\\ x^{\prime }(0)=0,x(1)& =0,\end{array}$$

$$x(t)={\int }_0^{1}G(t,\tau )y(\tau )d\tau ,t\in [[0,1]],$$

$$G(t,\tau )=\{\begin{array}{ll}\frac{E_{\mu ,1}(\omega t^{\mu })}{E_{\mu ,1}(\omega )}(1-\tau {)}^{\mu -1}{E}_{\mu ,\mu }(\omega (1-\tau {)}^{\mu })-(t-\tau {)}^{\mu -1}{E}_{\mu ,\mu }(\omega (t-\tau {)}^{\mu }),& 0\le \tau \le t\le 1,\\ \frac{E_{\mu ,1}(\omega t^{\mu })}{E_{\mu ,1}(\omega )}(1-\tau {)}^{\mu -1}{E}_{\mu ,\mu }(\omega (1-\tau {)}^{\mu }),& 0\le t\le \tau \le 1.\end{array}$$

$${}^C{D}_{0^{+}}^{\mu }x(t)+y_{\ast }(t)=\omega x(t),t>0,1<\mu \le 2,\omega >0,$$

$$y_{\ast }(t)=\{\begin{array}{ll}y(t),& 0<t<1,\\ 0,& t\ge 1.\end{array}$$

$$\mathcal{L}\{{}^C{D}_{0^{+}}^{\mu }x(t)\}(s)+\mathcal{L}\{y_{\ast }(t)\}(s)=\omega \mathcal{L}\{x(t)\}(s)$$

$$\mathcal{L}\{x(t)\}(s)=\frac{s^{\mu -1}}{s^{\mu }-\omega }x(0)+\frac{s^{\mu -2}}{s^{\mu }-\omega }{x}^{\prime }(0)-\frac{1}{s^{\mu }-\omega }\mathcal{L}\{y_{\ast }(t)\}(s)$$

$$x(t)=E_{\mu ,1}(\omega t^{\mu })x(0)+t{E}_{\mu ,2}(\omega t^{\mu })x^{\prime }(0)-{\int }_0^t(t-\tau {)}^{\mu -1}{E}_{\mu ,\mu }(\omega (t-\tau {)}^{\mu })y_{\ast }(\tau )d\tau ,t\ge 0$$

$$x(t)=\frac{E_{\mu ,1}(\omega t^{\mu })}{E_{\mu ,1}(\omega )}{\int }_0^1(1-\tau {)}^{\mu -1}{E}_{\mu ,\mu }(\omega (1-\tau {)}^{\mu })y(\tau )d\tau -{\int }_0^t(t-\tau {)}^{\mu -1}{E}_{\mu ,\mu }(\omega (t-\tau {)}^{\mu })y(\tau )d\tau ,t\in [[0,1]],$$

$$G(t,\tau )\le E_{\mu ,\mu }(\omega ),\text{ for }(t,\tau )\in [[0,1]]\times [[0,1]].$$

$${\int }_0^{1}G(t,\tau )d\tau$$

$$=\frac{E_{\mu ,1}(\omega t^{\mu })}{E_{\mu ,1}(\omega )}{\int }_0^1{\tau }^{\mu -1}{E}_{\mu ,\mu }(\omega {\tau }^{\mu })d\tau -{\int }_0^t{\tau }^{\mu -1}{E}_{\mu ,\mu }(\omega {\tau }^{\mu })d\tau$$

$$=\frac{E_{\mu ,1}(\omega t^{\mu })E_{\mu ,\mu +1}(\omega )-t^{\mu }{E}_{\mu ,1}(\omega )E_{\mu ,\mu +1}(\omega t^{\mu })}{\omega E_{\mu ,1}(\omega )}=\frac{{\sigma }_{\mu ,\omega }(t)}{\omega E_{\mu ,1}(\omega )}$$

$$|f(t,x)|\le q(t)(u(x)+v(x)),t\in (0,1),x\in (0,\infty ),$$

$${\int }_0^{1}q(t)dt<\infty ,\text{ and }{\int }_0^{1}q(t)u\left(c{\sigma }_{\mu ,\omega }(t)\right)dt<\infty \text{ for }c>0.$$

$$\frac{R}{E_{\mu ,\mu }(\omega ){\chi }_{{}_R}\left(1+\frac{q(R)}{p(R)}\right)}>1$$

$${\chi }_r={\int }_0^{1}q(t)u\left(\frac{{\gamma }_{{}_r}{\sigma }_{\mu ,\omega }(t)}{\omega E_{\mu ,1}(\omega )}\right)dt.$$

$$\frac{R-\epsilon }{E_{\mu ,\mu }(\omega ){\chi }_{{}_{R+\epsilon }}\left(1+\frac{q(R+\epsilon )}{p(R+\epsilon )}\right)}\ge 1$$

$$\begin{array}{rl}{}^C{D}_{0^{+}}^{\mu }x(t)+f\left(t,\min \{\max \{x(t)+\frac{1}{m},\frac{1}{m}\},R\}\right)& =\omega x(t),t\in (0,1),1<\mu \le 2,\omega >0,\\ x^{\prime }(0)=0,x(1)& =0,\end{array}$$

$$x(t)={\int }_0^{1}G(t,\tau )f\left(\tau ,\min \{\max \{x(\tau )+\frac{1}{m},\frac{1}{m}\},R\}\right)d\tau ,t\in [[0,1]].$$

$$T_{m}x(t)={\int }_0^{1}G(t,\tau )f\left(\tau ,\min \{\max \{x(\tau )+\frac{1}{m},\frac{1}{m}\},R\}\right)d\tau ,t\in [[0,1]].$$

$$x_m(t)={\int }_0^{1}G(t,\tau )f\left(\tau ,\min \{\max \{x(\tau )+\frac{1}{m},\frac{1}{m}\},R\}\right)d\tau ,t\in [[0,1]]$$

$$x_m(t)\ge {\int }_0^{1}G(t,\tau ){\gamma }_{{}_R}d\tau \ge \frac{{\gamma }_{{}_{R+\epsilon }}{\sigma }_{\mu ,\omega }(t)}{\omega E_{\mu ,1}(\omega )}$$

$$\begin{array}{rl}{x}_m(t)& \le E_{\mu ,\mu }(\omega ){\int }_0^{1}q(\tau )u(\min \{\max \{x_m(\tau )+\frac{1}{m},\frac{1}{m}\},R\})\left(1+\frac{v(\min \{\max \{x_m(\tau )+\frac{1}{m},\frac{1}{m}\},R\})}{u(\min \{\max \{x_m(\tau )+\frac{1}{m},\frac{1}{m}\},R\})}\right)d\tau \\ & \le E_{\mu ,\mu }(\omega ){\int }_0^{1}q(\tau )u\left(\frac{{\gamma }_{{}_{R+\epsilon }}{\sigma }_{\mu ,\omega }(\tau )}{\omega E_{\mu ,1}(\omega )}\right)\left(1+\frac{v(R+\epsilon )}{u(R+\epsilon )}\right)d\tau \\ & =E_{\mu ,\mu }(\omega ){\chi }_{{}_{R+\epsilon }}\left(1+\frac{v(R+\epsilon )}{u(R+\epsilon )}\right)\le R-\epsilon \end{array}$$

$$x_m(t)={\int }_0^{1}G(t,\tau )f\left(\tau ,x_m(\tau )+\frac{1}{m}\right)d\tau ,t\in [[0,1]]$$

$$\frac{{\gamma }_{{}_{R+\epsilon }}{\sigma }_{\mu ,\omega }(t)}{\omega E_{\mu ,1}(\omega )}\le x_m(t)<R,t\in [[0,1]]$$

$$x_{n_k}(t)={\int }_0^{1}G(t,\tau )f\left(\tau ,x_{n_k}(\tau )+\frac{1}{n_k}\right)d\tau$$

$$x(t)={\int }_0^{1}G(t,\tau )f(\tau ,x(\tau ))d\tau$$

Full text

1. introduction

Mathematical models involving fractional order derivatives offer better description of physical phenomena such as in mechanics, in control systems, fluid flow in porous media, signal and image processing, aerodynamics, electromagnetics, viscoelasticity [2, 3, 8, 10]. Recently, the area of fractional order boundary value problems (BVPs) has achieved a great progress in respect of both theoretically and physical applications [4, 5, 6, 7, 9, 13, 14, 15]. Since most of the nonlinear fractional differential equations do not have exact analytic solution, therefore, results to establish existence of solutions have attracted attention of many researchers [16, 21, 24, 25, 26, 27]. However, few articles in literature have studied the existence of solution for singular BVPs of fractional order, see [1, 12, 17, 19, 20, 23].

In this article, we establish criteria for positive existence of the following fractional order BVP

[TABLE]

where ${}^{C}D_{0^{+}}^{\,\mu}$ Caputo fractional left derivative of order $\mu$, $f:(0,1)\times(0,\infty)\rightarrow\mathbb{R}$ is continuous and singular at $t=0$, $t=1$ and $x=0$. We prove positive existence for BVP (1.1) in the space $X:=\{x:x\in C\llbracket 0,1\rrbracket,\,{}^{C}D_{0^{+}}^{\,\mu}\,x\in C(0,1)\}$. By positive solution $x$ of BVP (1.1) we mean $x\in X$ satisfies BVP (1.1) and $x(t)>0$ for $t\in\llbracket 0,1)$.

The rest of the paper is organized as follows. In Section 2, the definition of fractional derivative and some preliminaries lemmas are presented. In Section 3, by the use of fixed-point theorem and results of functional analysis, the existence of positive solution has established. An example is presented to illustrate the main theorem.

2. preliminaries

⟦** 1****.**

[18, 22] The Caputo fractional left derivative of a function $x\in AC^{n}\llbracket 0,\infty)$ of order $\mu\in(n-1,n\rrbracket$, $n\in\mathbb{N}$, is

[TABLE]

Further, the following Laplace transforms are essential for our work

[TABLE]

where $E_{\mu,\nu}(t):=\sum_{k=0}^{\infty}\frac{t^{k}}{\Gamma(\mu k+\nu)}$ is the modified Mittag-Leffler function.

⟦** 1****.**

Let $y\in L\llbracket 0,1\rrbracket$, then the BVP

[TABLE]

has integral representation

[TABLE]

where

[TABLE]

Proof.

Consider the extended differential equation

[TABLE]

where $y_{*}:(0,\infty)\rightarrow\mathbb{R}$ is defined as

[TABLE]

Taking Laplace transform of (2.5), we have

[TABLE]

which in view of (2.1), leads to

[TABLE]

Taking inverse Laplace transform we have

[TABLE]

Now employing BCs (2.2), we have

[TABLE]

which is equivalent to (2.3). ∎

⟦** 2****.**

The Green’s function (2.4) satisfies

• •

⟦(1).⟧* $G:\llbracket 0,1\rrbracket\times\llbracket 0,1\rrbracket\rightarrow\llbracket 0,\infty)$ is continuous and is positive on $\llbracket 0,1)\times\llbracket 0,1)$;*

• •

⟦(2).⟧* $G(t,\tau)\leq E_{\mu,\mu}(\omega)$, for $(t,\tau)\in\llbracket 0,1\rrbracket\times\llbracket 0,1\rrbracket$; and*

• •

⟦(3).⟧* $\int_{0}^{1}G(t,\tau)d\tau=\frac{\sigma_{\mu,\omega}(t)}{\omega E_{\mu,1}(\omega)}$, for $t\in\llbracket 0,1\rrbracket$, $\sigma_{\mu,\omega}(t):=E_{\mu,1}(\omega t^{\mu})E_{\mu,\mu+1}(\omega)-t^{\mu}E_{\mu,1}(\omega)E_{\mu,\mu+1}(\omega t^{\mu})$.*

Proof.

• •

⟦(1).⟧ Clearly, Green’s function $G(t,\tau)$ is continuous for $(t,\tau)\in\llbracket 0,1\rrbracket\times\llbracket 0,1\rrbracket$. Moreover, $G(t,\tau)>0$ for $(t,\tau)\in\llbracket 0,1)\times\llbracket 0,1)$.

• •

⟦(2).⟧ For $\tau\in\llbracket 0,1\rrbracket$ we have $(1-\tau)^{\mu-1}\leq 1$. Consequently, from (2.4), we have

[TABLE]

• •

⟦(3).⟧ Integrating (2.4) with respect to $\tau$ from [math] to $1$, we have

[TABLE]

∎

3. main result

Assume that

• •

⟦(A1).⟧ There exist $q\in C(0,1)$, $u\in C(0,\infty)$ decreasing, and $v\in C\llbracket 0,\infty)$ increasing such that

[TABLE]

[TABLE]

• •

⟦(A2).⟧ There exist a constant $R>\frac{\gamma_{{}_{R}}\,E_{\mu,\mu+1}(\omega)}{\omega E_{\mu,1}(\omega)}$ such that, for $t\in(0,1)$ and $x\in(0,R\rrbracket$, $f(t,x)\geq\gamma_{{}_{R}}$, where the parameter $\gamma_{r}$ is positive and decreasing for $r>0$. Moreover,

[TABLE]

where

[TABLE]

In view of $(A2)$, choose $\varepsilon\in(0,R-\frac{\gamma_{{}_{R}}\,E_{\mu,\mu+1}(\omega)}{\omega E_{\mu,1}(\omega)}\rrbracket$ such that

[TABLE]

For $m\in\mathbb{N}$ with $\frac{1}{m}<\varepsilon$, consider the modified BVP

[TABLE]

which in view of Lemma 1, has integral representation

[TABLE]

Define $T_{m}:C\llbracket 0,1\rrbracket\rightarrow C\llbracket 0,1\rrbracket$ by

[TABLE]

Clearly, fixed points of $T_{m}$ are solutions of BVP (3.2). {thm} Assume that $(A1)$ and $(A2)$ hold. Then the BVP (1.1) has a positive solution.

Proof.

In view of $(A1)$ and Schauder’s fixed point theorem the map $T_{m}$ defined by (3.3) has a fixed point $x_{m}\in C\llbracket 0,1\rrbracket$. Thus

[TABLE]

which in view of $(A2)$ and Lemma 2, leads to

[TABLE]

Also (3.4) in view of Lemma 2, $(A1)$, (3.5) and (3.1), leads to

[TABLE]

Consequently, from (3.5) and (3.6), solution $x_{m}$ of BVP (3.2) satisfies

[TABLE]

and

[TABLE]

which shows that the sequence $\{x_{n}\}_{n=m}^{\infty}$ is uniformly bounded on $\llbracket 0,1\rrbracket$. Moreover, since $G(t,\tau)$ is uniformly continuous on $\llbracket 0,1\rrbracket\times\llbracket 0,1\rrbracket$, by Lebesgue dominated convergence theorem, the sequence $\{x_{n}\}_{n=m}^{\infty}$ equicontinuous on $\llbracket 0,1\rrbracket$. Thus by Arzela Ascoli Theorem the sequence $\{x_{n}\}_{n=m}^{\infty}$ is relatively compact and consequently there exist a subsequence $\{x_{n_{k}}\}_{k=1}^{\infty}$ converging uniformly to $x\in C\llbracket 0,1\rrbracket$. Moreover, in view of (3.7), we have

[TABLE]

as $k\rightarrow\infty$, we obtain

[TABLE]

which in view of Lemma 1, leads to

[TABLE]

Also, ${}^{C}D_{0^{+}}^{\,\mu}\,x\in C(0,1)$. Further, from (3.8) in view of $(A2)$ and Lemma 2, we have

[TABLE]

which shows that $x(t)>0$ for $t\in\llbracket 0,1)$. Hence $x\in X$ is a positive solution of BVP (1.1). ∎

⟦** 1****.**

[TABLE]

where

[TABLE]

Here

[TABLE]

Choose

[TABLE]

Then,

[TABLE]

Moreover,

[TABLE]

[TABLE]

Further,

[TABLE]

where

[TABLE]

Clearly, the assumptions $(A1)$ and $(A2)$ of Theorem 3 are satisfied, therefore, the BVP (3.9) has a positive solution $x\in X$.