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1 Introduction

Fractional calculus (FC) has a history of more than 300 years, there are some applications of FC with in various fields of mathematics it self. During the last few decades,FC has obtained vigorous development in the applied sciences and gained considerable popularity. Compared with classical integer-order models, fractional derivatives and integrals are more suitable to describe the memory and hereditary properties of various materials, fractional derivatives are more advantageous in simulating mechanical and electrical properties of real materials and describing rheological properties of rocks and many other fields. Based on the description of their properties in terms of fractional derivatives, fractional differential equations (FDEs) are generated naturally, and how to solve these equations is also very necessary. For example, some new models that involve FDEs have been applied successfully, e.g., in mechanics (theory of viscoelasticity and viscoplasticity [6, 24]), (bio-)chemistry (modelling of polymers and proteins [11]),electrical engineering (transmission of ultrasound waves [3, 28]), medicine (modelling of human tissue under mechanical load [31]). . . Accordingly, more and more researchers and scholars devote themselves to the study of various problem of FDEs. In particular,study of boundary value problems for nonlinear FDEs is particularly concerned among these problems.

The aim of this paper is to investigate the following fractional differential equation:

subjected to boundary condition

where $1<\alpha \le 2,0\le \beta \le \alpha -1\le 1<\gamma <\alpha .$ ${\lambda }_1,{\lambda }_2\ge 0,{\lambda }_1+{\lambda }_2=1,b_i\ge 0$ $(i=1,2,...,m),h_i\in C\left(I_i\right)\cap L^1\left(I_i\right)(i=1,2,...,m))$is nonnegative, $I_i\subset [0,1]$ is measurable, $\textcolor{red}{}m\ge 1$ is an integer. $\textcolor{red}{}g$ is singular at $x=0.D_{0+}^{\alpha }$ is the standard Riemann–Liouville fractional derivative of order $\textcolor{red}{}\alpha$ defined by

where $\text{Γ}$ denotes the Gamma function, and $\left[\alpha \right]$ denotes the integer part of number$\textcolor{red}{}\alpha$,provided that the right side is pointwise defined on$(0,\mathrm{\infty })$; see [14, 25].

When ${\lambda }_2=0,0\le \beta \le 1$, there has been a great deal of literature on the fractional differential equation of such boundary conditions; see [2, 4, 9, 15, 16, 30]. As for$\textcolor{red}{}\beta =0$,for example, in [2],

where $1<\alpha <2,0<\mu \le \alpha -1,f$ satisfies the Carathéodory conditions on $[0,1]\times \text{B},\text{B}=(0,\infty )\times ℝ,f(t,x,y)$ is positive and singular at $x=0$. Based on regularization and sequential techniques, the existence of a positive solution was obtained.

By Schauder fixed point theorem and the Banach contraction principle, Rehman et al.[26] investigated existence and uniqueness of solutions for a class of nonlinear multipointboundary value problems for fractional differential equation

where$1<\alpha \le 2,0<\beta <1,0<{\xi }_i<1(i=1,2,...,m-2),$ ${\zeta }_i\ge 0,\gamma =\sum _{i=1}^{m-2}{\zeta }_i{\xi }_i^{\alpha -\beta -1}<1.$.

In [15], the author studied the nonlinear fractional differential equation

subjected with the boundary conditions$u\left(0\right)=u\left(1\right)=0$. Under Carathéodory conditions,using the Leray–Schauder continuation principle, the existence of at least onesolution was obtained.

KyuNam et al. [16] discussed the existence and uniqueness of solutions for a class ofintegral boundary value problems of nonlinear multiterm fractional differential equation

where$1<\alpha <2,0<{\beta }_1<···<{\beta }_n<1,\alpha -{\beta }_n\ge 1,f:[0,1]\times {ℝ}^{n+1}\to ℝ$, and $g:[0,1]\times ℝ\to ℝ$ are continuous. The existence results are established by the Banachfixed point theorem, and approximate solutions are determined by the Daftardar-Gejji and Jafari iterative method (DJIM) and the Adomian decomposition method (ADM).

In this article, we will deal with singular nonlinear fractional differential equation with a new boundary condition, which is a generalization of many previous researches.

To the best of our knowledge, when ${\lambda }_2\ne 0$, neither ${\lambda }_1=0$ nor ${\lambda }_1\ne 0$ was studied like this type of boundary condition. Furthermore, the nonlinear term contained not only lower derivative of order $\textcolor{red}{}\beta$, but also another lower derivative of order . In comparison with the above literature, our results about the difference include both $\alpha -\beta \ge 1$ and$0<\alpha -\gamma <1$, this has never been seen before.

The stability of differential equations has grown to be one of the considerable areas inthe field of mathematical analysis, and we find many different types of stability, such asexponential [7,17,23], Mittag-Leffler [20,27], Hyers–Ulam (HU) stability and other typesof stability [13,18,19,21,22]. Among these kinds of stability, Hyers–Ulam stability and itsvarious types provide a bridge between the exact and numerical solutions, so researchers devoted their work to the study of different kinds of HU stability for nonlinear fractionaldifferential equation; see [5, 10, 12, 29].

The paper is organized as follows. In Section 2, we will present some useful lemma sand give some valuable preliminary results. In Section 3, we prove the existence and uniqueness of positive solution to problem (1), (2) by using Leray–Schauder nonlinear alternative theorem and Boyd–Wong’s contraction principles. In Section 4, we investigate various kinds of HU stability of solutions.

2 Auxiliary results

To simplify our statements, we introduce the following spaces:

• $C[0,1]$ be the Banach space of all continuous functions from $[0,1]$ to $\textcolor{red}{}ℝ$ equipped with the norm defined by $\left|\right|z|{|}_{\infty }=max\{\left|z\right(x\left)\right|,x\in [0,1]\}.$ .

• $C_{\alpha -\gamma }[0,1]=\{z:(0,1]\to ℝ|x^{1+\gamma -\alpha }z\left(x\right)|.$end owed the norm

• $L_{\alpha -\gamma }^p[0,1]=\{z:(0,1]\to ℝ|x^{1+\gamma -\alpha }z\in L^p[0,1]\}$ end owed the norm

where $0<1/p<\alpha -\gamma$ .

First, we introduce some fundamental facts of the fractional calculus theory, which are used in this paper; see [14, 25].

Property 1.Let $\alpha ,\beta >0$.

1. (i) If$f\in L(0,1)$, then$I_{0+}^{\alpha }{I}_{0+}^{\beta }f\left(t\right)=I_{0+}^{\alpha +\beta }f\left(t\right),D_{0+}^{\alpha }{I}_{0+}^{\alpha }f\left(t\right)=f\left(t\right)$;

2. (ii) If $\textcolor{red}{}n-1<\alpha <n,f\left(x\right)\in L_1(a,b)$ has a summable derivative $D_{\alpha +}^{\alpha }f$, then

where $f_{n-\alpha }\left(x\right)=\left(I_{a+}^{n-\alpha }f\right)\left(x\right)$;

1. (iii) $I_{0+}^{\alpha }{D}_{0+}^{\alpha }f\left(t\right),0<\alpha <1$, where $f\in C[0,1],D_{0+}^{\alpha }f\in C(0,1)\cap L(0,1)$;

2. (iv)$D_{0+}^{\gamma }u\left(t\right)=D^1{D}_{0+}^{\gamma -1}u\left(t\right)(\gamma >1)$, where $D^1=d/dt$ , provided that $D_{0+}^{\gamma }u\left(t\right)$ exists;

3. (v) Let $\alpha >0$ . Assume that $\{f_k{\}}_{k=1}^{\infty }$ is a uniformly convergent sequence of continuous functions on $[a,b]$ and that $D_{\alpha +}^{\alpha }{f}_k$ exists for every $\textcolor{red}{}k$ . Moreover, assume that $\{D_{\alpha +}^{\alpha }{f}_k{\}}_{k=1}^{\infty }$ converges uniformly on $[a+\in ,b]$ for every $\epsilon >0$ . Then for every $x\in (a,b]$ , we have

Next, for the sake of readers’ convenience, we present some necessary lemmas, which will be used in the main results that follow.

Lemma 1. Suppose $\text{ϕ}\in L_{\alpha -\gamma }^p[0,1]$ and $0\le \beta \le \alpha -1\le 1<\gamma <\alpha \le 2$ ,

Then the linear fractional boundary value problem

has a unique solution given by $\text{ϑ}\left(x\right)={\int }_0^{1}G(x,s)\varphi (s)ds$ , where

and

in which

and

Proof. It is very well known that the equation $D_{0+}^{\alpha }\text{ϑ}\left(x\right)+\varphi \left(x\right)=0\mathrm{}$ is equivalent to the following integral equation:

The boundary condition $\text{ϑ}\left(0\right)=0$ implies that$c_2=0$. Using the property of Riemann–Liouville fractional derivative, we know

Combining the boundary condition in (3), it follows that

Therefore the unique solution of problem (3) is given by

Lemma 2. Let $G$ be the Green function related to problem (3), which is given by expression (4). Then for $0\le \beta \le \alpha -1\le 1<\gamma <\alpha \le 2,\varpi >1/T(\alpha -\beta )-1/T(\alpha -\gamma ),G,K,K_0$ have the following properties:

1. (i) $G,K,K_0$ are nonnegative and continuous on$[0,1]\times [0,1)$;

1. (ii) $K(0,s)=K_0(0,s)=0$ for all $s\in [0,1),K(x,0)=K_0(x,0)=0$ for all$x\in [0,1]$, and

1. (iii) For$(x,s)\in [0,1]\times [0,1),$,

1. (iv) Let $M=1+(1/\varpi )\sum _{i=1}^m{b}_i{\int }_{I_i}{h}_i\left(\tau \right){\tau }^{\alpha -1}d\tau$, then for $(x,s)\in [0,1]\times [0,1),$

Proof. From the condition $\text{ϖ}>1T(\alpha -\beta )-1/T(\alpha -\gamma )$ we easily get

In addition, we also have a relation${\lambda }_2{\rho }_1+{\lambda }_1{\rho }_2=1/T\left(\alpha \right)$.

1. (i) For $0\le x\le s<1,K(x,s)\ge 0$ is obvious. For $\textcolor{red}{}0\le x\le s\le 1$,

In the same way, we get $K_0$ is nonnegative, then we have $G\ge 0$. It is obvious that $G,K,K_0$ are continuous on $[0,1]\times [0,1)$.

1. (ii) The conclusion is obvious, we omit it.

2. (iii) First, we introduce an inequality. For $\lambda ,\mu \in (0,\infty )$ and$a,x\in [0,1]$, we have

When $s\le x$, using inequality (5), we obtain

When $x\le s$,

In addition,

Again, we can get the property of $K_0$, that is,

Therefore,

1. (iv) By substituting the two inequalities (iii) into formula (4) we naturally come to the conclusion.

Consider the problem $\text{ϑ}=\mathrm{G\vartheta }$, where operator $G$ is defined by

where $\mathrm{g\vartheta }\left(s\right)=g(s,\vartheta (s),D_{0+}^{\beta }\text{ϑ}(s),D_{0+}^{\gamma }\text{ϑ}(s\left)\right),G$ is defined in (4). In order to prove that problem (1), (2) has a solution, we just have to show that operator $G$ has a fixed point.

Take the fractional derivative of order $\textcolor{red}{}\beta$ for $\mathrm{G\vartheta }$, we have

where

Similarly, we have

where

Then

By simple deduction we can get the following properties.

Lemma 3. Let $0\le \beta \le \alpha -1\le 1<\gamma <\alpha \le 2,\varpi >1/T(\alpha -\gamma )$ . The functions $K_i,G_i(i=1,2)$ defined in (8), (9) and (10) have the following properties:

1. (i) $K_1,G_1$ are nonnegative and continuous on $[0,1]\times [0,1)$;

2. (ii) For any $(x,s)\in [0,1]\times [0,1)$,

For any $(x,s)\in [0,1]\times [0,1),$,

where

and $M$ is defined in Lemma 2;

1. (iv) $K_2(x,s)>0,0<x\le s<1$, and $K_2(x,s)<0,0<s<x\le 1$

Lemma 4. Integral operators $I_{0+}^{\alpha },I_{0+}^{\alpha -\beta },I_{0+}^{\alpha -\gamma }$ have properties:

1. (i) $I_{0+}^{\alpha },I_{0+}^{\alpha -\beta }:L_{\alpha -\gamma }^p[0,1]\to C[0,1]$ are continuous;

2. (ii) $I_{0+}^{\alpha -\gamma }:L_{\alpha -\gamma }^p[0,1]\to C_{\alpha -\gamma }[0,1]$ is continuous.

Proof. (i) Since $1\le \alpha -\beta <\alpha \le 2$, we only show the continuity of $I_{0+}^{\alpha }$. First, we prove that $I_{0+}^{\alpha }:L_{\alpha -\gamma }^p[0,1]\to C[0,1]$ is well defined, that is, for any $f\in L_{\alpha -\gamma }^p[0,1]$, we have$I_{0+}^{\alpha }f\in C[0,1]$. Notice that $(\alpha -\gamma -1)q+1>0$ because of the condition $1/p<\alpha -\gamma$.

For any $0\le x_1\le x_2\le 1$,

then we naturally get the continuity of $I_{0+}^{\alpha }f\left(x\right)$ as $x_2\to x_1$

Let $\left\{f_n\right\}$ be convergent sequence in the space$L_{\alpha -\gamma }^p[0,1]$, i.e., there exists a function $f_0\in L_{\alpha -\gamma }^p[0,1]$ such that$\left|\right|f_n-f_0|{|}_{L_{\alpha -\gamma }^p}\to 0(n\to \infty )$. In order to show that operator $I_{0+}^{\alpha }$ is continuous, we have to prove$\left|\right|I_{0+}^{\alpha }{f}_n-I_{0+}^{\alpha }{f}_0|{|}_{\infty }\to 0(n\to \infty )$. In fact,

where

According to the inequality above, we have $\left|\right|I_{0+}^{\alpha }{f}_n-I_{0+}^{\alpha }{f}_0|{|}_{\infty }\to 0$ as $n\to \mathrm{\infty }$.

1. (ii) For any$\textcolor{red}{}f\in L_{\alpha -\gamma }^p[0,1]$, denote $F\left(x\right)=x^{1+\gamma -\alpha }{I}_{0+}^{\alpha -\gamma }f\left(x\right),x\in [0,1]$.

where

From this inequality we know $F$ is continuous on $\textcolor{red}{}t=0$ if one supplies the definition of $F$ on$t=0:F\left(0\right)=0$.

For any $\textcolor{red}{}0<x_1\le x_2\le 1,$,

Now, we will evaluate these formulae (13)–(15), respectively.

Choose a constant $\textcolor{red}{}\eta$ satisfying$0<\eta <min\{1+\gamma -\alpha ,\alpha -\gamma -1/p\}$, then

where

In the end, as for (15), let $x_2\to x_1$, then

Taking all the conclusions above into (13)–(15), we can get $\left|F\right(x_2)-F(x_1\left)\right|\to 0$ as$x_2\to x_1$, which implies that $F\left(x\right)$ is continuous on $[0,1]$ and$I_{0+}^{\alpha -\gamma }f\in C_{\alpha -\gamma }[0,1]$.

Combining (12), we easily infer the continuity of integral operator $I_{0+}^{\alpha -\gamma }$ .

We define a normed vector space

equipped with the norm$||\vartheta ||=max\{||\vartheta |{|}_{\infty ,}\left|\right|D_{0+}^{\beta }\text{ϑ}\left|\right|\infty ,\left|\right|D_{0+}^{\gamma }\text{ϑ}\left|{|}_{*}\right\}$.

Lemma 5. $(X,||·|\left|\right)$ is Banach space.

Proof. Let $\left\{{\text{ϑ}}_n\right\}$ be Cauchy sequence in$(X,||·|\left|\right)$. Clearly, $\left\{{\text{ϑ}}_n\right\},\left\{D_{0+}^{\beta }{\text{ϑ}}_n\right\}$ are also Cauchy sequence in the space $C[0,1]$. Therefore, $\left\{{\text{ϑ}}_n\right\},\left\{D_{0+}^{\beta }{\text{ϑ}}_n\right\}$ converge to some $\text{ϑ},u\in C[0,1]$. By similar work of Su and Liu [30], we have $u=D_{0+}^{\beta }\text{ϑ}$.

Let $U_n\left(t\right)=t^{1+\gamma -\alpha }{D}_{0+}^{\gamma }{\text{ϑ}}_n\left(t\right)$. Evidently, $\left\{U_n\right(t\left)\right\}$ is Cauchy sequence in $C[0,1]$,then there exists $\textcolor{red}{}\mu \in C[0,1]$ such that$U_n\left(t\right)\to \mu \left(t\right)$ in $C[0,1]$. That is to say, for any $\in >0$, there is a positive integer $\textcolor{red}{}N$,

Choose arbitrary positive number $\textcolor{red}{}a<1$, we will prove $\textcolor{red}{}\left\{D_{0+}^{\gamma }{\text{ϑ}}_n\right\}$ uniformly converges to $t^{\alpha -\gamma -1}\mu \left(t\right)$on$\textcolor{red}{}[a,1]$. In fact, for all$\textcolor{red}{}n>N,t\in [a,1]$, we have

By property of fractional calculus, for every$t\in (0,1]$, we have$\left(\underset{n\to \mathrm{\infty }}{\lim }{D}_{0+}^{\gamma }{\text{ϑ}}_n\right)\left(t\right)=\left(D_{0+}^{\gamma }\underset{n\to {\mathrm{\infty }}^{\textcolor{red}{}}}{\lim }{\text{ϑ}}_n\right)\left(t\right)$, Hence, $t^{\alpha -\gamma -1}\mu \left(t\right)=D_{0+}^{\gamma }\text{ϑ}\left(t\right)$ for all$t\in (0,1]$. So, $D_{0+}^{\gamma }\text{ϑ}\left(t\right)\in C(0,1]$ and$\underset{t\to 0+}{\lim }{t}^{1+\gamma -\alpha }{D}_{0+}^{\gamma }\text{ϑ}\left(t\right)=\underset{t\to 0+}{\lim }\mu \left(t\right)=\mu \left(0\right)$, i.e.,$D_{0+}^{\gamma }\text{ϑ}\left(t\right)\in C_{\alpha -\gamma }[0,1]$.

The proof is completed.

Let$K=\{\text{ϑ}\in X|\vartheta (x)\ge 0,D_{0+}^{\beta }\text{ϑ}(x)\ge 0,x\in [0,1\left]\right\}$. Apparently, $\textcolor{red}{}K$ is a cone of$\textcolor{red}{}X$.

3 Existence and uniqueness results

This section deals with existence and uniqueness of solutions for problem (1), (2). Thenonlinear term $\textcolor{red}{}g$ satisfies the following assumptions:

1. C1) $g:(0,1]\times {ℝ}^{+\times }{ℝ}^{+}\times ℝ\to {ℝ}^{+}$ is continuous, $\textcolor{red}{}g(x,0,0,0)$ does not vanish on any compact interval of$(0,1]$. Furthermore, there exist nonnegative functions ${\sigma }_i\in L_{\alpha -\gamma }^p[0,1](i=1,2,3)$ and continuous and nondecreasing functions ${\text{ϑ}}_i:{ℝ}^{+}\to {ℝ}^{+}(i=1,2,3)$ such that for any$x\in (0,1],u,v\in {ℝ}^{+},\omega \in ℝ$,

1. (C2) There exists a positive number $R$ such that

where $\overline{{\mathrm{\sigma }}_i}=\varrho \left|\right|{\sigma }_i|{|}_{L_{\alpha -\gamma }^p(i=1,2,3)}$.

1. (C3) There exist $L_i\in C_{\alpha -\gamma }[0,1](i=1,2,3)$ and $x_i:{ℝ}^{+}\to {ℝ}^{+}(i=1,2,3)$are upper semicontinuous from the right and nondecreasing such that for any$x\in (0,1],u_i,v_i\in {ℝ}^{+},{\omega }_i\in ℝ(i=1,2)$, we have

1. (C4) Denote

It satisfies $\text{ϕ}\left(x\right)<x$ for all $\textcolor{red}{}x>0$, where$\overline{L_i}=\varrho \left|\right|L_i\left|{|}_{L_{\alpha -\gamma }^p}\right(i=1,2,3)$.

1. (C4´) Denote

it satisfies $\text{ϕ}\left(x\right)<x$ for all $x>0$.

Lemma 6. Suppose that (C1) holds, and $0\le \beta \le \alpha -1\le 1<\gamma <\alpha \le 2,\varpi >1/T(\alpha -\beta )-1/T(\alpha -\gamma )$ . Then $G:K\to K$ is completely continuous.

Proof. Let us first show that the operator $\textcolor{red}{}G$ is well defined. For any $\text{ϑ}\in K$, from (C1),since${\mathrm{\sigma }}_i\in L_{\alpha -\gamma }^p[0,1](i=1,2,3)$, we know$g_{\text{ϑ}}\in L_{\alpha -\gamma }^p[0,1]$. Where upon, according to Hölder inequality and Lemma 2, we have ${\int }_0^1(1-s{)}^{\alpha -\beta -1}{g}_{\text{ϑ}}\left(s\right)ds,{\int }_0^1(1-s{)}^{\alpha -\gamma -1}\times g_{\text{ϑ}}\left(s\right)ds<\infty$ and${\int }_0^1(\sum _{i=1}^m{b}_i{\int }_{I_i}{K}_0(\tau ,s)h_i\left(\tau \right)d\tau )g_{\text{ϑ}}\left(s\right)ds<\infty$, and we deduce that $I_{0+}^{\alpha }{g}_{\text{ϑ}}\left(x\right)\in C[0,1]$ from Lemma 4. Then by (6) we get $\mathrm{G\vartheta }\in C[0,1]$. Similarly, using Lemma 4 and formulae (7), (11), we get the conclusions:$D_{0+}^{\beta }\mathrm{G\vartheta }\in C[0,1],D_{0+}^{\gamma }\mathrm{G\vartheta }\in C_{\alpha -\gamma }[0,1]$. Furthermore, we naturally get $\mathrm{G\vartheta }\left(x\right),D_{0+}^{\beta }\mathrm{G\vartheta }\left(x\right)\ge 0,x\in [0,1]$ from thefact that $\textcolor{red}{}g$ and Green’s functions $G,G_1$ are nonnegative. So,$\mathrm{G\vartheta }\in K$.

Suppose that ${\text{ϑ}}_n\to {\text{ϑ}}_0(n\to \infty )$ in cone $\textcolor{red}{}K$, then there exists a constant $M>0$ such that$\left|\right|{\text{ϑ}}_n\left|\right|\le M(n=0,1,...)$. In order to get the conclusion that operator $\textcolor{red}{}G$ is continuous, let us start with the fact that ${\mathrm{g\vartheta }}_n\to {\mathrm{g\vartheta }}_0(n\to \infty )$ in $L_{\alpha -\gamma }^p[0,1]$, where$g_{\text{ϑ}n}\left(x\right)=g(x,{\text{ϑ}}_n(x),D_{0+}^{\beta }{\text{ϑ}}_n(x),D_{0+}^{\gamma }{\text{ϑ}}_n(x\left)\right)(n=0,1,...)$. By condition (C1) it follows that

Moreover, on the basis of the continuity of$\textcolor{red}{}g$, we deduce that $g_{\text{ϑ}n}\left(x\right)\to g_{\text{ϑ}0}\left(x\right),n\to \infty$,for all $x\in (0,1]$. Taking advantage of Lebesgue dominated convergence theorem, we know${\int }_0^1|x^{1+\gamma -\alpha }(g_{\text{ϑ}n}\left(x\right)-g_{\text{ϑ}n}\left(x\right)){|}^{p}dx\to 0,n\to \infty$. Thereupon, by Lemma 4, we have $I_{0+}^{\alpha }{\mathrm{g\vartheta }}_n$ convergence to $I_{0+}^{\alpha }{\mathrm{g\vartheta }}_0$ in $C[0,1]$. In addition, by Hölder inequality, we canalso get and${\int }_0^1(\sum _{i=1}^m{b}_i{\int }_{I_i}{K}_0(\tau ,s)h_i\left(\tau \right)d\tau \left)\right|g_{\text{ϑ}n}\left(s\right)-g_{\text{ϑ}0}\left(s\right)ds\to 0$as $\textcolor{red}{}n\to \infty$. Synthesizingthe above conclusions, from the expression of $G{\text{ϑ}}_{\textcolor{red}{}}$ (6), we have $G{\text{ϑ}}_n\to G{\text{ϑ}}_0$ in $C[0,1]$. Analogously, we can deduce $D_{0+}^{\beta }G{\text{ϑ}}_n\to D_{0+}^{\beta }G{\text{ϑ}}_0$ in $C[0,1]$ and $D_{0+}^{\gamma }G{\text{ϑ}}_n\to D_{0+}^{\gamma }G{\text{ϑ}}_0$ in$C_{\alpha -\gamma }[0,1]$. To wit, $G{\text{ϑ}}_n\to G{\text{ϑ}}_0(n\to \infty )$ in$\textcolor{red}{}K$.

In the end, Ascoli–Arzela theorem guarantees operator $G:K\to K$ is compact. That is to say, we can deduce that $G\left(B\right)$ is bounded and equicontinuous for any bounded subset$B\subseteq K$. The proof can be obtained by the conventional procedure, so we omit this step.

From the above we conclude that the operator $\textcolor{red}{}G$ is completely continuous.

Theorem 1. (See [1].) Let E be a Banach space with $\textcolor{red}{}C\subseteq E$ closed and convex. Assume that $\textcolor{red}{}U$ is relatively open subset of $\textcolor{red}{}C$ with $\textcolor{red}{}0\in U$ and $A:\bar{U}\to C$ is a continuous compactmap. Then either

1. (i) A has a fixed point in $\textcolor{red}{}\bar{U}$ or

2. ii) There exists $\textcolor{red}{}u\in \mathrm{\delta }U$ and $\textcolor{red}{}\lambda \in (0,1)$ with$\textcolor{red}{}u=\lambda Au$.

Theorem 2. Assume that (C1), (C2) hold, and $\textcolor{red}{}0\le \beta \le \alpha -1\le 1<\gamma <\alpha \le 2,\varpi >1/T(\alpha -\beta )-1/T(\alpha -\gamma )$ . Then BVP (1), (2) has at least one positive solution.

Proof. By applying nonlinear alternative of Leray–Schauder-type fixed point theorem (Theorem 1), we will prove that $\textcolor{red}{}G$ has a fixed point. Let $B_R=\{u\in K|\left|u\right||<R\},R$ is given in condition (C2). Consider the following integral equation:

where $\lambda \in (0,1)$. We claim that any solution of (16) for any $\textcolor{red}{}\mathrm{\lambda }\in (0,1)$ must satisfies $||\vartheta ||\ne R$. Otherwise, assume that $\textcolor{red}{}\text{ϑ}$ is a solution of (16) for some $\lambda \in (0,1)$ such that$||\vartheta ||\ne R$. Hence, from condition (C1) and Lemma 2(iv), for any $\textcolor{red}{}x\in [0,1]$, we have

so,

Similarly, in view of Lemma 3 and (7), we have

then

According to (11), we have

then

Therefore,

This is a contradiction and the claim is proved. Leray–Schauder nonlinear alternative theorem guarantees that operator $\textcolor{red}{}G$ has a fixed point$\textcolor{red}{}\text{ϑ}\in \overline{Br}$. Since $\textcolor{red}{}g(x,0,0,0)$ does not vanish on any compact interval of$\textcolor{red}{}(0,1)$, we know $\textcolor{red}{}\text{ϑ}$ must be positive.

Next, our uniqueness result for problem (1), (2) relies on Boyd–Wong’s contraction principle [8].

Theorem 3. Let $\textcolor{red}{}X$ be a complete metric space and suppose $T:X\to X$ satisfies

where $\textcolor{red}{}\text{ϕ}:[0,\infty )\to [0,\infty )$ is upper semicontinuous function from the right (i.e., $r_j\downarrow r\ge 0\Rightarrow \underset{j\to \infty }{\lim sup}\text{ϕ}\left(r_j\right)\le \varphi \left(r\right)$ , and for $\textcolor{red}{}x>0,0\le \varphi \left(x\right)<x$ for $\textcolor{red}{}x>0)$ .Then $\textcolor{red}{}T$ has a unique fixed point $x\in X$ .

Theorem 4. Assume that (C3), (C4) hold, and $\textcolor{red}{}0\le \beta \le \alpha -1\le 1<\gamma <\alpha \le 2,\varpi >1/T(\alpha -\beta )-1/T(\alpha -\gamma )$ . Then BVP (1), (2) has a unique positive solution.

Proof. For any $\textcolor{red}{}\text{ϑ},y\in K$ and $\textcolor{red}{}x\in [0,1]$, by using Lemma 2(iv) and condition (C3), we get

so

Similarly,

Synthesizing the above three inequalities and combining with condition (C4), we get

Then Boyd–Wong’s contraction principle can be applied and $\textcolor{red}{}G$ has a unique fixed pointwhich is the unique solution of problem (1), (2).

Example.

Let$$\begin{gathered} \alpha =7/4,\beta =1/2,\gamma =3/2,I_1=[0,1/4],I_2=[1/4,3/4],I_3=[3/4,1],I_4= \\ [0,1],b_1=0.1,b_2=0.05,b_3=0.1,b_4=0.02,h_1\left(s\right)=s^{-1/2},h_2\left(s\right)=s^{-1}, \\ {h}_3\left(s\right)=s^{-1/4},h_4\left(s\right)=s^{1/4},{\lambda }_1=0.9,{\lambda }_2=0.1 \end{gathered}$$. A simple calculation yields

and$1/T(\alpha -\beta )-1/T(\alpha -\gamma )\approx 0.82741$, then $\textcolor{red}{}\text{ϖ}>1/T(\alpha -\beta )-1/T(\alpha -\gamma )$.

Let $g(x,u,\upsilon ,w)=x^{-1/2}(1+u^{1/2})+x^{-1/4}{\upsilon }^{1/4}+x^{-1/4}{\omega }^{2/3}$, then $g:(0,1]\times {ℝ}^{+}\times {ℝ}^{+}\times ℝ\to {ℝ}^{+}$ is continuous, and $g(x,0,0,0)$ does not vanish on any compact interval of $(0,1]$. Let$$\begin{gathered} {\mathrm{\sigma }}_1\left(x\right)=x^{-1/2},{\mathrm{\sigma }}_2\left(x\right)=x^{-1/4},{\mathrm{\sigma }}_3\left(x\right)=x^{-3/4},{\text{ϑ}}_1\left(u\right)=1+u^{1/2, \\ } \\ {\text{ϑ}}_2\left(\upsilon \right)=1+{\upsilon }^{1/4},{\text{ϑ}}_3\left(\omega \right)={\omega }^{2/3} \end{gathered}$$, then for $\textcolor{red}{}x\in (0,1],u,\upsilon \in {ℝ}^{+},\omega \in ℝ$.

Furthermore,$\overline{{\mathrm{\sigma }}_1}=8.5634,\overline{{\mathrm{\sigma }}_2}=8.0337,\overline{{\mathrm{\sigma }}_3}=9.8240$,

Take $R=2^1=262144$,

Hence, conditions (C1), (C2) in Theorem 2 hold, then problem (17) has a positive solution.

4 Stability analysis

In this section, we consider the Banach space $$\begin{gathered} X=\{u\in C[0,1\left]\right|D_{0+}^{\beta }u\in C[0,1],D^{\gamma }u\in \\ {C}_{\alpha +\gamma }[0,1]\} \end{gathered}$$equipped with the norm$\left|\right|u\left|\right|=\left|\right|u|{|}_{\mathrm{\infty }}+|\left|D_{0+}^{\beta }u\right|{|}_{\infty }+\left|\right|D_{0+}^{\gamma }u|{|}_{*}$. Let us introduce some definitions related to Ulam stability.

Suppose that function $H\in C_{\alpha -\gamma }[0,1]$ is nonnegative and$\textcolor{red}{}\in >0$. Consider thein equalities given below:

Definition 1. We say that $\overline{\text{ϑ}}\in X$ is a solution of inequality (18): if there is $N_{\overline{\text{ϑ}}}\in C(0,1]$ which depends on $\textcolor{red}{}\overline{\text{ϑ}}$, such that $\left|N_{\overline{\text{ϑ}}}\right(x\left)\right|\le ∊$ and$D_{0+}^{\alpha }\overline{\text{ϑ}}\left(x\right)+g_{\overline{\text{ϑ}}}\left(x\right)=N_{\overline{\text{ϑ}}}\left(x\right)$, meanwhile $\textcolor{red}{}\overline{\text{ϑ}}\left(x\right)$ satisfies boundary condition (2), where

Remark 1. The solution of inequalities (19), (20) can be defined as well.

Definition 2. BVP (1), (2) is Hyers–Ulam stable: if there is a constant $\textcolor{red}{}C>0$ such thatfor any $\textcolor{red}{}\epsilon >0$ and for each solution $\overline{\text{ϑ}}\in X$ of inequality (18), there exists a unique solution $\textcolor{red}{}\text{ϑ}\in X$ of BVP (1), (2) satisfying

Definition 3. BVP (1), (2) is generalized Hyers–Ulam stable: if there is a function $\textcolor{red}{}\mathrm{\Psi }\in C\left(\right[0,1\left]{ℝ}^{+}\right)$ with $\textcolor{red}{}\mathrm{\Psi }\left(0\right)=0$ such that for any $\textcolor{red}{}∊>0$ and for each solution $\textcolor{red}{}\overline{\text{ϑ}}\in X$ of inequality (18), there exists a unique solution $\textcolor{red}{}\text{ϑ}\in X$ of BVP (1), (2) with

Definition 4. BVP (1), (2) is Hyers–Ulam–Rassias (HUR) stable w.r.t. nonnegative function $\textcolor{red}{}H\in C_{\alpha -\gamma }[0,1]:$ if there is a constant $\textcolor{red}{}C>0$ such that for any $\textcolor{red}{}\epsilon >0$ and foreach solution $\textcolor{red}{}\overline{\text{ϑ}}\in X$ of inequality (19), there is a unique solution $\textcolor{red}{}\text{ϑ}\in X$ of BVP (1), (2) satisfying

Definition 5. BVP (1), (2) is generalized Hyers–Ulam–Rassias stable w.r.t. $H\in C\left(\right(0,1],{ℝ}^{+}):$ if there is a constant $\textcolor{red}{}C>0$ such that for each solution $\textcolor{red}{}\overline{\text{ϑ}}\in X$ of inequality (20),there is a unique solution $\textcolor{red}{}\text{ϑ}\in X$ of BVP (1), (2) satisfying

Theorem 5. Assume that (C3), (C40) hold, and $\textcolor{red}{}0\le \beta \le \alpha -1\le 1<\gamma <\alpha \le 2,\varpi >1/T(\alpha -\beta )-1/T(\alpha -\gamma )$ Let $\Delta =M/T\left(\alpha \right)+M/T(\alpha -\beta )+M+1/T(\alpha -\gamma )$ . If $\textcolor{red}{}\Delta <1$ , then BVP (1), (2) is HU stable.

Proof. For any $\textcolor{red}{}\epsilon >0$, suppose $\textcolor{red}{}\overline{\text{ϑ}}\in X$ be the solution of inequality (18), then$\textcolor{red}{}\overline{\text{ϑ}}\left(x\right)={\int }_0^{1}G(x,s)(g_{\overline{\text{ϑ}}}\left(s\right)-N_{\overline{\text{ϑ}}}\left(s\right))ds$, where$|N_{\overline{\text{ϑ}}}\left(s\right)|\le \in$. Just like the proof method in Theorem 4, we also know that BVP (1), (2) has a unique solution $\textcolor{red}{}\text{ϑ}\in X$ under the new normand $\textcolor{red}{}\text{ϑ}$ can be expressed by $\textcolor{red}{}\text{ϑ}\left(x\right)={\int }_0^{1}G(x,s)g_{\text{ϑ}}(s)ds$ by Lemma 1.

On the basis of Lemma 2, we have

Similarly, we can get

Hence,$\left|\right|\overline{\text{ϑ}}-\overline{\text{ϑ}}\left|\right|\le \Delta \left|\right|\overline{\text{ϑ}}-\overline{\text{ϑ}}\left|\right|+(\epsilon /(\alpha -\gamma \left)\right)\Delta$, then

Let $C=\Delta /\left(\right(\alpha -\gamma \left)\right(1-\Delta \left)\right)$, clearly,

In other words, conclusion (21) in Definition 2 is satisfied, i.e., BVP (1), (2) is Hyers–Ulam stable.

Theorem 6. Suppose that (C3), (C40) hold, and $0\le \beta \le \alpha -1\le 1<\gamma <\alpha \le 2,\varpi >1/T(\alpha -\beta )-1/T(\alpha -\gamma )$ Let $\Delta =M/T\left(\alpha \right)+M/T(\alpha -\beta )+M+1/T(\alpha -\gamma )$ . If $\Delta <1$ , then BVP (1), (2) is generalized HU stable.

Proof. For any $\epsilon >0$, for convenience, let us assume $0<\epsilon <1$. Suppose $\overline{\text{ϑ}}\in X$ be the solution of inequality (18), $\text{ϑ}\in X$ be a unique solution of BVP (1), (2). Following the method of Theorem 5, we know that (22) holds. Thereupon, for any $x\in [0,1]$,

It is obvious that $\mathrm{\Psi }\left(0\right)=0$. Then by Definition 3, BVP (1), (2) is generalized HUstable.

Theorem 7. Let the conditions of Theorem 6 be satisfied, moreover, there exists a nonnegative function $H\in C_{\alpha -\gamma }[0,1]:$ satisfying $x^{\alpha -1}\le H\left(x\right)$ for all $x\in (0,1]$ . Then BVP (1), (2) is HUR stable.

Proof. For any $\epsilon >0$, suppose $\overline{\text{ϑ}}\in X$ be the solution of inequality (19), then$\overline{\text{ϑ}}\left(x\right)={\int }_0^{1}G(x,s)(g_{\overline{\text{ϑ}}}\left(s\right)-N_{\overline{\text{ϑ}}}\left(s\right))ds$, where $|N_{\overline{\text{ϑ}}}\left(s\right)|\le H(x)∊,\vartheta \in X$ is a unique solution of BVP (1), (2). On the similar way of Theorem 5, one can prove

So, combining with these three inequalities, we have

For any $x\in (0,1]$, from (23), we get

Let

then

By Definition 4, BVP (1), (2) is HUR stable.

Remark 2. Under the condition of Theorem 7, imitating the process, we can prove that BVP (1), (2) is generalized HUR stable.