On a Fractional Differential Equation with r-Laplacian Operator and Nonlocal Boundary Conditions

Abstract

We study the existence and multiplicity of positive solutions of a Riemann-Liouville fractional differential equation with r-Laplacian operator and a singular nonnegative nonlinearity dependent on fractional integrals, subject to nonlocal boundary conditions containing various fractional derivatives and Riemann-Stieltjes integrals. We use the Guo–Krasnosel’skii fixed point theorem in the proof of our main results.

1. Introduction

We consider the nonlinear ordinary fractional differential equation with r-Laplacian operator

$$D_{0+}^{\beta }\left({\phi }_r\left(D_{0+}^{\gamma }w\left(\eta \right)\right)\right)=f\left(\eta ,w\left(\eta \right),I_{0+}^{{\gamma }_1}w\left(\eta \right),\dots ,I_{0+}^{{\gamma }_q}w\left(\eta \right)\right),\eta \in (0,1),$$

(1)

supplemented with the nonlocal boundary conditions

$$\left\{\begin{array}{c}{w}^{\left(k\right)}\left(0\right)=0,k=0,\dots ,p-2,D_{0+}^{\gamma }w\left(0\right)=0,\hfill \\ {\displaystyle {\phi }_r\left(D_{0+}^{\gamma }w\left(1\right)\right)={\int }_0^1{\phi }_r\left(D_{0+}^{\gamma }w\left(\tau \right)\right)d{\mathfrak{H}}_0\left(\tau \right),D_{0+}^{{\alpha }_0}w\left(1\right)=\sum _{j=1}^n{\int }_0^1{D}_{0+}^{{\alpha }_j}w\left(\tau \right)d{\mathfrak{H}}_j\left(\tau \right),}\hfill \end{array}\right.$$

(2)

where $\beta \in (1,2]$, $\gamma \in (p-1,p]$, $p\in \mathbb{N}$, $p\ge 3$, $q,n\in \mathbb{N}$, ${\alpha }_j\in \mathbb{R}$, $j=0,\dots ,n$, $0\le {\alpha }_1<{\alpha }_2<\cdots <{\alpha }_n\le {\alpha }_0<\gamma -1$, ${\alpha }_0\ge 1$, $r>1$, ${\phi }_r\left(\tau \right)={\left|\tau \right|}^{r-2}\tau$, ${\phi }_r^{-1}={\phi }_{\varrho }$, $\varrho =\frac{r}{r-1}$, ${\gamma }_j>0$ for all $j=1,\dots ,q$, the function $f:(0,1)\times {\mathbb{R}}_{+}^{q+1}\to {\mathbb{R}}_{+}$ is continuous and it may be singular at $\eta =0$ and/or $\eta =1$, (that is, f is unbounded in the vicinity of $\eta =0$ and/or $\eta =1$), (${\mathbb{R}}_{+}=[0,\infty )$), $I_{0+}^{\theta }$ is the Riemann–Liouville fractional integral of order $\theta$ (for $\theta ={\gamma }_1,\dots ,{\gamma }_q$), $D_{0+}^{\kappa }$ denotes the Riemann–Liouville fractional derivative of order $\kappa$, for $\kappa =\beta ,\gamma ,{\alpha }_j,j=0,\dots ,n$), and the integrals from the boundary conditions (2) are Riemann–Stieltjes integrals with ${\mathfrak{H}}_j:[0,1]\to \mathbb{R},j=0,\dots ,n$ functions of bounded variation. These general boundary conditions cover multi-point boundary conditions (if ${\mathfrak{H}}_j,j=0,\dots ,n$ are step functions), classical integral conditions (if $d{\mathfrak{H}}_j\left(\tau \right)={\tilde{\mathfrak{H}}}_j\left(\tau \right)d\tau ,j=0,\dots ,n$), or combinations of them. The interest in the study of problems with the r-Laplacian operators is based on the turbulent flow problem in a porous medium; see the paper [1] in which Leibenson introduced a p-Laplacian differential equation. Generalizations of this problem with ordinary or fractional derivatives, for differential equations or systems were obtained in the last years by many authors (see [2] and its references).

In this paper, we present various assumptions on the function f such that problem (1), (2) has at least one or two positive solutions $w\left(t\right),t\in [0,1]$ with $w\left(t\right)>0$ for all $t\in (0,1]$. In the proof of our main results we apply the Guo–Krasnosel’skii fixed point theorem (Theorem 1 below). Below, we present some recent results connected to our problem. In [3], the author studied the existence of solutions of the fractional differential equation containing a fractional integral term

$$D_{0+}^{\alpha }w\left(\eta \right)+f(\eta ,w\left(\eta \right),I_{0+}^{p}w\left(\eta \right)))=0,\eta \in (0,1),$$

with the nonlocal boundary conditions

$$w\left(0\right)=w^{\prime }\left(0\right)=\cdots =w^{(n-2)}\left(0\right)=0,D_{0+}^{{\gamma }_0}w\left(1\right)=\sum _{i=1}^m{\int }_0^1{D}_{0+}^{{\gamma }_i}w\left(\tau \right)d{\mathfrak{H}}_i\left(\tau \right),$$

(3)

where $m\in \mathbb{N}$, $\alpha \in \mathbb{R}$, $\alpha \in (n-1,n]$, $n\in \mathbb{N}$, $n\ge 2$, ${\gamma }_i\in \mathbb{R}$ for all $i=0,\dots ,m$, $0\le {\gamma }_1<{\gamma }_2<\cdots <{\gamma }_m<\alpha -1$, ${\gamma }_0\in [0,\alpha -1)$, $p>0$, f is a nonlinear function, and ${\mathfrak{H}}_i$, $i=1,\dots ,m$ are bounded variation functions. In the proof of the main theorems, the author applied the Krasnosel’skii fixed point theorem for the sum of two operators, the Banach contraction mapping principle, the nonlinear alternative of Leray-Schauder type, and the Leray-Schauder alternative theorem. In [4], the authors investigated the existence of positive solutions of the fractional differential equation with a positive parameter

$$D_{0+}^{\alpha }w\left(\eta \right)+\lambda h\left(\eta \right)f(\eta ,w\left(\eta \right))=0,\eta \in (0,1),$$

subject to the nonlocal boundary conditions (3), where $\alpha \in \mathbb{R}$, $\alpha \in (n-1,n]$, $n\in \mathbb{N}$, $n\ge 3$, ${\gamma }_0\ge 1$, the nonnegative function $f(\eta ,w)$ may have singularity at $w=0$, and the nonnegative function $h\left(\eta \right)$ may be singular at $\eta =0$ and/or $\eta =1$, and $\lambda$ is a positive parameter. They present various assumptions for the functions h and f, and give intervals for the parameter $\lambda$ such that the above problem has positive solutions. These intervals for $\lambda$ were expressed with the aid of the principal characteristic value of a particular linear operator. They use the fixed point index theory in the proof of the main theorems. A related semipositone fractional boundary value problem was also studied in [4]. In [5], under some assumptions on the data of the problem, the authors studied the positive solutions of the singular nonlinear fractional differential equation

$$D_{0+}^{\alpha }w\left(\eta \right)+f(\eta ,w\left(\eta \right),D_{0+}^{{\alpha }_1}w\left(\eta \right),\dots ,D_{0+}^{{\alpha }_{n-2}}w\left(\eta \right))=0,\eta \in (0,1),$$

with the nonlocal boundary conditions

$$\left\{\begin{array}{c}w\left(0\right)=D_{0+}^{{\gamma }_1}w\left(0\right)=\cdots =D_{0+}^{{\gamma }_{n-2}}w\left(0\right)=0,\hfill \\ {\displaystyle D_{0+}^{{\beta }_1}w\left(1\right)={\int }_0^{\eta }h\left(s\right)D_{0+}^{{\beta }_2}w\left(s\right)dA\left(s\right)+{\int }_0^{1}a\left(s\right)D_{0+}^{{\beta }_3}w\left(s\right)dA\left(s\right),}\hfill \end{array}\right.$$

where $\alpha \in (n-1,n]$, $n\ge 3$, ${\alpha }_k,{\gamma }_k\in (k-1,k]$, $k=1,\dots ,n-2$, $\alpha -{\gamma }_j\in (n-j-1,n-j]$, $j=1,\dots ,n-2$, $\alpha -{\alpha }_{n-2}-1\in (1,2]$, ${\gamma }_{n-2}\ge {\alpha }_{n-2}$, ${\beta }_1\ge {\beta }_2$, ${\beta }_1\ge {\beta }_3$, $\alpha \ge {\beta }_i+1$, ${\beta }_i\ge {\alpha }_{n-2}+1$, $i=1,2,3$, ${\beta }_1\le n-1$, the function $f:(0,1)\times {\mathbb{R}}_{+}^{n-1}\to {\mathbb{R}}_{+}$ is continuous, $a,h\in C((0,1),{\mathbb{R}}_{+})$, and A is a function of bounded variation. We would also mention the recent papers [6,7,8,9,10,11] which investigate fractional differential equations/systems with $\phi$-Laplacian operators, and some recent monographs dedicated to the study of fractional differential equations and systems, namely [12,13,14,15,16], which contain interesting fractional boundary value problems with many examples and applications. The novelty of our work consists of a combination between the fractional Equation (1) which contains an r-Laplacian operator and varied fractional integral terms, and the general boundary conditions (2) with many fractional derivatives of diverse orders.

The paper is structured in the following way. In Section 2 we study the linear fractional boundary value problem associated with our problem (1), (2), and the properties of the corresponding Green functions. In Section 3, we present and prove the main existence theorems for (1), (2), and in Section 4 we give two examples which illustrate our obtained results. Finally, in Section 5 we present the conclusions for our paper.

2. Auxiliary Results

We consider the fractional differential equation

$$D_{0+}^{\beta }\left({\phi }_r\left(D_{0+}^{\gamma }w\left(\eta \right)\right)\right)=v\left(\eta \right),\eta \in (0,1),$$

(4)

with the boundary conditions (2), where $v\in C(0,1)\cap L^1(0,1)$. We denote by

$${\displaystyle {\mathfrak{a}}_1=1-{\int }_0^1{\vartheta }^{\beta -1}d{\mathfrak{H}}_0\left(\vartheta \right),{\mathfrak{a}}_2=\frac{\Gamma \left(\gamma \right)}{\Gamma (\gamma -{\alpha }_0)}-\sum _{i=1}^n\frac{\Gamma \left(\gamma \right)}{\Gamma (\gamma -{\alpha }_i)}{\int }_0^1{\vartheta }^{\gamma -{\alpha }_i-1}d{\mathfrak{H}}_i\left(\vartheta \right).}$$

(5)

Lemma 1.

If ${\mathfrak{a}}_1\ne 0$ and ${\mathfrak{a}}_2\ne 0$, then the unique solution $w\in C[0,1]$ of problem (4), (2) is given by

$${\displaystyle w\left(\eta \right)={\int }_0^1{\mathfrak{G}}_2(\eta ,\zeta ){\phi }_{\varrho }\left({\displaystyle {\int }_0^1{\mathfrak{G}}_1(\zeta ,\vartheta )v\left(\vartheta \right)d\vartheta }\right)d\zeta ,\eta \in [0,1],}$$

(6)

where

$${\displaystyle {\mathfrak{G}}_1(\eta ,\zeta )={\mathfrak{g}}_1(\eta ,\zeta )+\frac{{\eta }^{\beta -1}}{{\mathfrak{a}}_1}{\int }_0^1{\mathfrak{g}}_1(\vartheta ,\zeta )d{\mathfrak{H}}_0\left(\vartheta \right),(\eta ,\zeta )\in [0,1]\times [0,1],}$$

(7)

with

$${\displaystyle {\mathfrak{g}}_1(\eta ,\zeta )=\frac{1}{\Gamma \left(\beta \right)}\left\{\begin{array}{c}{\eta }^{\beta -1}{(1-\zeta )}^{\beta -1}-{(\eta -\zeta )}^{\beta -1},0\le \zeta \le \eta \le 1,\hfill \\ {\eta }^{\beta -1}{(1-\zeta )}^{\beta -1},0\le \eta \le \zeta \le 1,\hfill \end{array}\right.}$$

(8)

and

$${\displaystyle {\mathfrak{G}}_2(\eta ,\zeta )={\mathfrak{g}}_2(\eta ,\zeta )+\frac{{\eta }^{\gamma -1}}{{\mathfrak{a}}_2}\sum _{i=1}^n\left({\displaystyle {\int }_0^1{\mathfrak{g}}_{3i}(\vartheta ,\zeta )d{\mathfrak{H}}_i\left(\vartheta \right)}\right),(\eta ,\zeta )\in [0,1]\times [0,1],}$$

(9)

with

$$\begin{array}{c}{\displaystyle {\mathfrak{g}}_2(\eta ,\zeta )=\frac{1}{\Gamma \left(\gamma \right)}\left\{\begin{array}{c}{\eta }^{\gamma -1}{(1-\zeta )}^{\gamma -{\alpha }_0-1}-{(\eta -\zeta )}^{\gamma -1},0\le \zeta \le \eta \le 1,\hfill \\ {\eta }^{\gamma -1}{(1-\zeta )}^{\gamma -{\alpha }_0-1},0\le \eta \le \zeta \le 1,\hfill \end{array}\right.}\hfill \\ {\displaystyle {\mathfrak{g}}_{3i}(\eta ,\zeta )=\frac{1}{\Gamma (\gamma -{\alpha }_i)}\left\{\begin{array}{c}{\eta }^{\gamma -{\alpha }_i-1}{(1-\zeta )}^{\gamma -{\alpha }_0-1}-{(\eta -\zeta )}^{\gamma -{\alpha }_i-1},0\le \zeta \le \eta \le 1,\hfill \\ {\eta }^{\gamma -{\alpha }_i-1}{(1-\zeta )}^{\gamma -{\alpha }_0-1},0\le \eta \le \zeta \le 1.\hfill \end{array}\right.}\hfill \\ i=1,\dots ,n.\hfill \end{array}$$

(10)

Proof. 

We denote by ${\phi }_r\left(D_{0+}^{\gamma }w\left(\eta \right)\right)={\psi }_1\left(\eta \right),\eta \in (0,1)$. Then problem (4), (2) is equivalent to the following boundary value problems

$$\left(I\right)\left\{\begin{array}{c}{D}_{0+}^{\beta }{\psi }_1\left(\eta \right)=v\left(\eta \right),\eta \in (0,1),\hfill \\ {\displaystyle {\psi }_1\left(0\right)=0,{\psi }_1\left(1\right)={\int }_0^1{\psi }_1\left(s\right)d{\mathfrak{H}}_0\left(s\right),}\hfill \end{array}\right.$$

and

$$\left(\mathit{II}\right)\left\{\begin{array}{c}{D}_{0+}^{\gamma }w\left(\eta \right)={\phi }_{\varrho }\left({\psi }_1\left(\eta \right)\right),\eta \in (0,1),\hfill \\ {\displaystyle w^{\left(k\right)}\left(0\right)=0,k=0,\dots ,p-2,D_{0+}^{{\alpha }_0}w\left(1\right)=\sum _{j=1}^n{\int }_0^1{D}_{0+}^{{\alpha }_j}w\left(\tau \right)d{\mathfrak{H}}_j\left(\tau \right).}\hfill \end{array}\right.$$

Using Lemma 4.1.5 from [15], the unique solution ${\psi }_1\in C[0,1]$ of problem $\left(I\right)$ is

$${\displaystyle {\psi }_1\left(\eta \right)=-{\int }_0^1{\mathfrak{G}}_1(\eta ,\zeta )v\left(\zeta \right)d\zeta ,\eta \in [0,1],}$$

(11)

where ${\mathfrak{G}}_1$ is given by (7). Using Lemma 2.4.2 from [13], the unique solution $w\in C[0,1]$ of problem $\left(II\right)$ is

$${\displaystyle w\left(\eta \right)=-{\int }_0^1{\mathfrak{G}}_2(\eta ,\vartheta ){\phi }_{\varrho }\left({\psi }_1\left(\vartheta \right)\right)d\vartheta ,\eta \in [0,1],}$$

(12)

where ${\mathfrak{G}}_2$ is given by (9). By the relations (11) and (12) we deduce the solution w of problem (4), (2) given by relation (6). □

Lemma 2.

We assume that ${\mathfrak{a}}_1>0$, ${\mathfrak{a}}_2>0$, ${\mathfrak{H}}_i,i=0,\dots ,n$ are nondecreasing functions. Then the functions ${\mathfrak{G}}_1$ and ${\mathfrak{G}}_2$ given by (7) and (9) have the properties

(a) ${\mathfrak{G}}_1,{\mathfrak{G}}_2:[0,1]\times [0,1]\to [0,\infty )$ are continuous functions;

(b) ${\mathfrak{G}}_1(\eta ,\zeta )\le {\mathfrak{J}}_1\left(\zeta \right),\forall (\eta ,\zeta )\in [0,1]\times [0,1]$, where

$${\mathfrak{J}}_1\left(\zeta \right)={\mathfrak{h}}_1\left(\zeta \right)+\frac{1}{{\mathfrak{a}}_1}{\int }_0^1{\mathfrak{g}}_1(\vartheta ,\zeta )d{\mathfrak{H}}_0\left(\vartheta \right),\forall \zeta \in [0,1],$$

with ${\mathfrak{h}}_1\left(\zeta \right)=\frac{1}{\Gamma \left(\beta \right)}{(1-\zeta )}^{\beta -1},\zeta \in [0,1]$;

(c) ${\mathfrak{G}}_2(\eta ,\zeta )\le {\mathfrak{J}}_2\left(\zeta \right),\forall (\eta ,\zeta )\in [0,1]\times [0,1]$, where

$${\displaystyle {\mathfrak{J}}_2\left(\zeta \right)={\mathfrak{h}}_2\left(\zeta \right)+\frac{1}{{\mathfrak{a}}_2}\sum _{i=1}^n{\int }_0^1{\mathfrak{g}}_{3i}(\vartheta ,\zeta )d{\mathfrak{H}}_i\left(\vartheta \right),\forall \zeta \in [0,1],}$$

with ${\mathfrak{h}}_2\left(\zeta \right)=\frac{1}{\Gamma \left(\gamma \right)}{(1-\zeta )}^{\gamma -{\alpha }_0-1}(1-{(1-\zeta )}^{{\alpha }_0}),\zeta \in [0,1]$.

(d) ${\mathfrak{G}}_2(\eta ,\zeta )\ge {\eta }^{\gamma -1}{\mathfrak{J}}_2\left(\zeta \right),\forall (\eta ,\zeta )\in [0,1]\times [0,1]$.

Proof. 

The above properties follow easily from the definition of the Green functions ${\mathfrak{G}}_1$ and ${\mathfrak{G}}_2$ and the properties of the functions ${\mathfrak{g}}_1,{\mathfrak{g}}_2,{\mathfrak{g}}_{3i},i=1,\dots ,n$ (see [13,15]). □

Lemma 3.

We assume that ${\mathfrak{a}}_1>0$, ${\mathfrak{a}}_2>0$, ${\mathfrak{H}}_i,i=0,\dots ,n$ are nondecreasing functions, and $v\in C(0,1)\cap L^1(0,1)$ with $v\left(\eta \right)\ge 0$ for all $\eta \in (0,1)$. Then the solution w of problem (4), (2) satisfies the properties $w\left(\eta \right)\ge 0$ for all $\eta \in [0,1]$ and $w\left(\eta \right)\ge {\eta }^{\gamma -1}w\left(\tau \right)$ for all $\eta ,\tau \in [0,1]$.

Proof. 

By using the properties from Lemma 2, we obtain $w\left(\eta \right)\ge 0$ for all $\eta \in [0,1]$. Besides, we find

$$\begin{array}{c}{\displaystyle w\left(\eta \right)\ge {\int }_0^1{\eta }^{\gamma -1}{\mathfrak{J}}_2\left(\zeta \right){\phi }_{\varrho }\left({\displaystyle {\int }_0^1{\mathfrak{G}}_1(\zeta ,\vartheta )v\left(\vartheta \right)d\vartheta }\right)d\zeta }\hfill \\ {\displaystyle \ge {\eta }^{\gamma -1}{\int }_0^1{\mathfrak{G}}_2(\tau ,\zeta ){\phi }_{\varrho }\left({\displaystyle {\int }_0^1{\mathfrak{G}}_1(\zeta ,\vartheta )v\left(\vartheta \right)d\vartheta }\right)d\zeta ={\eta }^{\gamma -1}w\left(\tau \right),\forall \eta ,\tau \in [0,1].}\hfill \end{array}$$

□

In the proof of our main results we will apply the Guo–Krasnosel’skii fixed point theorem which we present now (see [17]).

Theorem 1.

Let $\mathcal{X}$ be a Banach space and let $\mathcal{C}\subset \mathcal{X}$ be a cone in $\mathcal{X}$. Assume ${\Xi }_1$ and ${\Xi }_2$ are bounded open subsets of $\mathcal{X}$ with $0\in {\Xi }_1\subset \overline{{\Xi }_1}\subset {\Xi }_2$ and let $\mathfrak{A}:\mathcal{C}\cap (\overline{{\Xi }_2}\setminus {\Xi }_1)\to \mathcal{C}$ be a completely continuous operator such that, either

(i) $\parallel \mathfrak{A}u\parallel \le \parallel u\parallel ,u\in \mathcal{C}\cap \partial {\Xi }_1$, and $\parallel \mathfrak{A}u\parallel \ge \parallel u\parallel ,u\in \mathcal{C}\cap \partial {\Xi }_2$, or (ii) $\parallel \mathfrak{A}u\parallel \ge \parallel u\parallel ,u\in \mathcal{C}\cap \partial {\Xi }_1,$ and $\parallel \mathfrak{A}u\parallel \le \parallel u\parallel ,u\in \mathcal{C}\cap \partial {\Xi }_2$.

Then the operator $\mathfrak{A}$ has a fixed point in $\mathcal{C}\cap (\overline{{\Xi }_2}\setminus {\Xi }_1)$.

3. Existence of Positive Solutions

First we introduce the basic assumptions that we will use in this section.

(K1)

$\beta \in (1,2]$, $\gamma \in (p-1,p]$, $p\in \mathbb{N}$, $p\ge 3$, $q,n\in \mathbb{N}$, ${\alpha }_i\in \mathbb{R}$, $i=0,\dots ,n$, $0\le {\alpha }_1<\cdots <{\alpha }_n\le {\alpha }_0<\gamma -1$, ${\alpha }_0\ge 1$, $r>1$, ${\phi }_r\left(\tau \right)={\left|\tau \right|}^{r-2}\tau$, $\varrho =\frac{r}{r-1}$, ${\gamma }_j>0$, $j=1,\dots ,q$, ${\mathfrak{H}}_i:[0,1]\to \mathbb{R}$, $i=0,\dots ,n$ are nondecreasing functions, ${\mathfrak{a}}_1>0$, ${\mathfrak{a}}_2>0$ (given by (5)).

(K2)

The function $f\in C((0,1)\times {\mathbb{R}}_{+}^{q+1},{\mathbb{R}}_{+})$ and there exist the functions $\psi \in C((0,1),{\mathbb{R}}_{+})$ and $\chi \in C([0,1]\times {\mathbb{R}}_{+}^{q+1},{\mathbb{R}}_{+})$ with $\Lambda ={\int }_0^1{(1-\tau )}^{\beta -1}\psi \left(\tau \right)d\tau \in (0,\infty )$ such that

$$f(\eta ,z_1,\dots ,z_{q+1})\le \psi \left(\eta \right)\chi (\eta ,z_1,\dots ,z_{q+1}),\forall \eta \in (0,1),z_i\in {\mathbb{R}}_{+},i=1,\dots ,q+1.$$

We will investigate the existence of at least one or two positive solutions for problem (1), (2). A solution of problem (1), (2) is a function $w\in C[0,1]$ with $D_{0+}^{\gamma }w\in C[0,1]$ and $D_{0+}^{\beta }\left({\phi }_r\left(D_{0+}^{\gamma }w\right)\right)\in C(0,1)$, which satisfies the fractional differential Equation (1) and the boundary conditions (2). By assumption $\left(K2\right)$ the nonnegative function f may be singular at $\eta =0$ and/or $\eta =1$, and it is bounded above by the product between a singular function $\psi$ (satisfying the condition from $\left(K2\right)$) and a nonsingular function $\chi$.

We introduce the Banach space $\mathfrak{X}=C[0,1]$ with supremum norm $\parallel w\parallel ={sup}_{\eta \in [0,1]}$$\left|w\right(\eta \left)\right|$, and the cone $\mathfrak{P}=\{w\in \mathfrak{X},w(\eta )\ge 0,\forall \eta \in [0,1\left]\right\}$.

Lemma 4.

We assume that assumptions $\left(K1\right)$ and $\left(K2\right)$ are satisfied. Then the function $w\in \mathfrak{P}$ is a solution of problem (1), (2) if and only if $w\in \mathfrak{P}$ is a solution of the integral equation

$${\displaystyle w\left(\eta \right)={\int }_0^1{\mathfrak{G}}_2(\eta ,\zeta ){\phi }_{\varrho }\left({\displaystyle {\int }_0^1{\mathfrak{G}}_1(\zeta ,\vartheta )f(\vartheta ,w\left(\vartheta \right),I_{0+}^{{\gamma }_1}w\left(\vartheta \right),\dots ,I_{0+}^{{\gamma }_q}w\left(\vartheta \right))d\vartheta }\right)d\zeta ,\eta \in [0,1].}$$

(13)

Proof. 

We assume that $w\in \mathfrak{P}$ is a solution of problem (1), (2). We denote by $u\left(\eta \right)={\phi }_r\left(D_{0+}^{\gamma }w\left(\eta \right)\right)$. Then the function u satisfies the problem

$$\left\{\begin{array}{c}{D}_{0+}^{\beta }u\left(\eta \right)=f(\eta ,w\left(\eta \right),I_{0+}^{{\gamma }_1}w\left(\eta \right),\dots ,I_{0+}^{{\gamma }_q}w\left(\eta \right)),\hfill \\ u\left(0\right)=0,u\left(1\right)={\int }_0^{1}u\left(s\right)d{\mathfrak{H}}_0\left(s\right).\hfill \end{array}\right.$$

(14)

By Lemma 4.1.5 from [15] the solution u of (14) is

$${\displaystyle u\left(\eta \right)=-{\int }_0^1{\mathfrak{G}}_1(\eta ,\zeta )f(\zeta ,w\left(\zeta \right),I_{0+}^{{\gamma }_1}w\left(\zeta \right),\dots ,I_{0+}^{{\gamma }_q}w\left(\zeta \right))d\zeta ,\eta \in [0,1].}$$

(15)

Besides, the function w satisfies the problem

$$\left\{\begin{array}{c}{D}_{0+}^{\gamma }w\left(\eta \right)={\phi }_{\varrho }\left(u\left(\eta \right)\right),\hfill \\ {\displaystyle w^{\left(k\right)}\left(0\right)=0,k=0,\dots ,p-2,D_{0+}^{{\alpha }_0}w\left(1\right)=\sum _{j=1}^n{\int }_0^1{D}_{0+}^{{\alpha }_j}w\left(\tau \right)d{\mathfrak{H}}_j\left(\tau \right).}\hfill \end{array}\right.$$

(16)

By Lemma 2.4.2 from [13] the solution w of problem (16) is

$${\displaystyle w\left(\eta \right)=-{\int }_0^1{\mathfrak{G}}_2(\eta ,\vartheta ){\phi }_{\varrho }\left(u\left(\vartheta \right)\right)d\vartheta ,\eta \in [0,1].}$$

(17)

Then by relations (15) and (17) we obtain that $w\in \mathfrak{P}$ is solution of the integral Equation (13).

Conversely, we assume that $w\in \mathfrak{P}$ is a solution of the integral Equation (13). We denote by $v(\eta )=f(\eta ,w(\eta ),I_{0+}^{{\gamma }_1}w(\eta ),\dots ,I_{0+}^{{\gamma }_q}w(\eta )),\eta \in (0,1)$ and $u\left(\eta \right)={\int }_0^1{\mathfrak{G}}_1(\eta ,\zeta )v\left(\zeta \right)$$d\zeta ,\eta \in [0,1]$. Then by Lemma 4.1.4 from [15] we have

$$\begin{array}{c}{\displaystyle u\left(\eta \right)=-\frac{1}{\Gamma \left(\beta \right)}{\int }_0^{\eta }{(\eta -s)}^{\beta -1}v\left(s\right)ds+\frac{{\eta }^{\beta -1}}{{\mathfrak{a}}_1\Gamma \left(\beta \right)}\left[{\displaystyle {\int }_0^1{(1-s)}^{\beta -1}v\left(s\right)ds}\right.}\hfill \\ \left.{\displaystyle -{\int }_0^1\left({\displaystyle {\int }_s^1{(\tau -s)}^{\beta -1}d{\mathfrak{H}}_0\left(\tau \right)}\right)v\left(s\right)ds}\right],\eta \in [0,1],\hfill \end{array}$$

(18)

which gives us that $u\in C[0,1]$. Next by Lemma 2.4.1 from [13] we deduce

$$\begin{array}{c}{\displaystyle w\left(\eta \right)={\int }_0^1{\mathfrak{G}}_2(\eta ,\zeta ){\phi }_{\varrho }\left(u\left(\zeta \right)\right)d\zeta }\hfill \\ {\displaystyle =-\frac{1}{\Gamma \left(\gamma \right)}{\int }_0^{\eta }{(\eta -s)}^{\gamma -1}\left({\phi }_{\varrho }\left(u\left(s\right)\right)\right)ds}\hfill \\ {\displaystyle +\frac{{\eta }^{\gamma -1}}{{\mathfrak{a}}_2\Gamma (\gamma -{\alpha }_0)}{\int }_0^1{(1-s)}^{\gamma -{\alpha }_0-1}{\phi }_{\varrho }\left(u\left(s\right)\right)ds}\hfill \\ {\displaystyle -\frac{{\eta }^{\gamma -1}}{{\mathfrak{a}}_2}\sum _{i=1}^n\frac{1}{\Gamma (\gamma -{\alpha }_i)}{\int }_0^1\left({\displaystyle {\int }_0^s{(s-\tau )}^{\gamma -{\alpha }_i-1}{\phi }_{\varrho }\left(u\left(\tau \right)\right)d\tau }\right)d{\mathfrak{H}}_i\left(s\right),\eta \in [0,1].}\hfill \end{array}$$

Hence we obtain $D_{0+}^{\gamma }w\left(\eta \right)=-{\phi }_{\varrho }\left(u\left(\eta \right)\right)$, and then we find that $D_{0+}^{\gamma }w\in C[0,1]$. From this last information and the relation (18) we obtain

$$D_{0+}^{\beta }\left({\phi }_r\left(D_{0+}^{\gamma }w\left(\eta \right)\right)\right)=D_{0+}^{\beta }({\phi }_r(-{\phi }_{\varrho }\left(u\left(\eta \right)\right))=D_{0+}^{\beta }\left(-\left({\displaystyle {\int }_0^1{\mathfrak{G}}_1(\eta ,\vartheta )v\left(\vartheta \right)d\vartheta }\right)\right).$$

Hence $D_{0+}^{\beta }\left({\phi }_r\left(D_{0+}^{\gamma }w\left(\eta \right)\right)\right)=v\left(\eta \right)$. This gives us $D_{0+}^{\beta }\left({\phi }_r\left(D_{0+}^{\gamma }w\right)\right))\in C(0,1)$. Because the boundary value problems

$$\left(\tilde{I}\right)\left\{\begin{array}{c}{D}_{0+}^{\beta }u\left(\eta \right)=-v\left(\eta \right),\hfill \\ u\left(0\right)=0,u\left(1\right)={\int }_0^{1}u\left(s\right)d{\mathfrak{H}}_0\left(s\right),\hfill \end{array}\right.$$

and

$$\left(\widetilde{II}\right)\left\{\begin{array}{c}{D}_{0+}^{\gamma }w\left(\eta \right)=-{\phi }_{\varrho }\left(u\left(\eta \right)\right),\hfill \\ w^{\left(k\right)}\left(0\right)=0,k=0,\dots ,p-2,D_{0+}^{{\alpha }_0}w\left(1\right)=\sum _{j=1}^n{\int }_0^1{D}_{0+}^{{\alpha }_j}w\left(\tau \right)d{\mathfrak{H}}_j\left(\tau \right),\hfill \end{array}\right.$$

have unique solutions (by Lemma 4.1.5 from [15], and Lemma 2.4.2 from [13]), we deduce that the function w satisfies Equation (1) and the conditions (2). □

We define the operator $\mathfrak{A}:\mathfrak{X}\to \mathfrak{X}$ by

$${\displaystyle \mathfrak{A}w\left(\eta \right)={\int }_0^1{\mathfrak{G}}_2(\eta ,\zeta ){\phi }_{\varrho }\left({\displaystyle {\int }_0^1{\mathfrak{G}}_1(\zeta ,\vartheta )f(\vartheta ,w\left(\vartheta \right),I_{0+}^{{\gamma }_1}w\left(\vartheta \right),\dots ,I_{0+}^{{\gamma }_q}w\left(\vartheta \right))d\vartheta }\right)d\zeta ,}$$

for $w\in \mathfrak{X}$ and $\eta \in [0,1]$. We see that w is solution of the integral Equation (13) if and only w is a fixed point of operator $\mathfrak{A}$.

Lemma 5.

We assume that assumptions $\left(K1\right)$ and $\left(K2\right)$ are satisfied. Then operator $\mathfrak{A}:\mathfrak{P}\to \mathfrak{P}$ is a completely continuous operator.

Proof. 

We denote by $M_0={\int }_0^1{\mathfrak{J}}_1\left(\tau \right)\psi \left(\tau \right)d\tau$. By using $\left(K2\right)$ and Lemma 2, we deduce that $M_0>0$. In addition we find

$$\begin{array}{c}{\displaystyle M_0={\int }_0^1{\mathfrak{J}}_1\left(\tau \right)\psi \left(\tau \right)d\tau ={\int }_0^1\left({\displaystyle {\mathfrak{h}}_1\left(\tau \right)+\frac{1}{{\mathfrak{a}}_1}{\int }_0^1{\mathfrak{g}}_1(\zeta ,\tau )d{\mathfrak{H}}_0\left(\zeta \right)}\right)\psi \left(\tau \right)d\tau }\hfill \\ {\displaystyle \le \frac{1}{\Gamma \left(\beta \right)}{\int }_0^1{(1-\tau )}^{\beta -1}\psi \left(\tau \right)d\tau +\frac{1}{{\mathfrak{a}}_1}{\int }_0^1\left({\displaystyle {\int }_0^1\frac{1}{\Gamma \left(\beta \right)}{\zeta }^{\beta -1}{(1-\tau )}^{\beta -1}d{\mathfrak{H}}_0\left(\zeta \right)}\right)\psi \left(\tau \right)d\tau }\hfill \\ {\displaystyle =\frac{1}{\Gamma \left(\beta \right)}{\int }_0^1{(1-\tau )}^{\beta -1}\psi \left(\tau \right)d\tau \left[{\displaystyle 1+\frac{1}{{\mathfrak{a}}_1}\left({\displaystyle {\int }_0^1{\zeta }^{\beta -1}d{\mathfrak{H}}_0\left(\zeta \right)}\right)}\right]<\infty .}\hfill \end{array}$$

Using again Lemma 2 we obtain that $\mathfrak{A}$ maps $\mathfrak{P}$ into $\mathfrak{P}$.

We will show that $\mathfrak{A}$ maps bounded sets into relatively compact sets. Let $\mathfrak{D}\subset \mathfrak{P}$ be an arbitrary bounded set. Then there exists ${\Xi }_1>0$ such that $\parallel w\parallel \le {\Xi }_1$ for all $w\in \mathfrak{D}$. By the continuity of $\chi$ we deduce that there exists ${\Xi }_2>0$ such that ${\Xi }_2=sup\{\chi (\eta ,z_1,\dots ,z_{q+1});\eta \in [0,1],z_i\in [0,\omega ],i=1,\dots ,q+1\}>0$, where $\omega ={\Xi }_{1}max\{1,\frac{1}{\Gamma ({\gamma }_1+1)},\cdots ,\frac{1}{\Gamma ({\gamma }_q+1)}\}$. We denote $f_w(\tau )=f(\tau ,w(\tau ),I_{0+}^{{\gamma }_1}w(\tau ),\dots ,I_{0+}^{{\gamma }_q}w(\tau ))$ and ${\chi }_w\left(\tau \right)=\chi (\tau ,w\left(\tau \right),I_{0+}^{{\gamma }_1}w\left(\tau \right),\dots ,I_{0+}^{{\gamma }_q}w\left(\tau \right))$. Based on the inequalities $|I_{0+}^{{\gamma }_i}w\left(\eta \right)|\le \frac{{\Xi }_1}{\Gamma ({\gamma }_i+1)},i=1,\dots ,q$, and by Lemma 2, we find for any $w\in \mathfrak{D}$ and $\eta \in [0,1]$

$$\begin{array}{c}{\displaystyle \mathfrak{A}w\left(\eta \right)\le {\int }_0^1{\mathfrak{J}}_2\left(\zeta \right){\phi }_{\varrho }\left({\displaystyle {\int }_0^1{\mathfrak{J}}_1\left(\tau \right)\psi \left(\tau \right){\chi }_w\left(\tau \right)d\tau }\right)d\zeta }\hfill \\ {\displaystyle \le {\Xi }_2^{\varrho -1}{\phi }_{\varrho }\left({\displaystyle {\int }_0^1{\mathfrak{J}}_1\left(\tau \right)\psi \left(\tau \right)d\tau }\right){\int }_0^1{\mathfrak{J}}_2\left(\zeta \right)d\zeta =M_0^{\varrho -1}{\Xi }_2^{\varrho -1}{M}_1,}\hfill \end{array}$$

where $M_1={\int }_0^1{\mathfrak{J}}_2\left(\zeta \right)d\zeta$. Then $\parallel \mathfrak{A}w\parallel \le M_0^{\varrho -1}{\Xi }_2^{\varrho -1}{M}_1$ for all $w\in \mathfrak{D}$, and so $\mathfrak{A}\left(\mathfrak{D}\right)$ is bounded.

We will prove next that $\mathfrak{A}\left(\mathfrak{D}\right)$ is equicontinuous. By using Lemma 1, for $w\in \mathfrak{D}$ and $\eta \in [0,1]$ we have

$$\begin{array}{c}{\displaystyle \mathfrak{A}w\left(\eta \right)={\int }_0^1\left({\displaystyle {\mathfrak{g}}_2(\eta ,\zeta )+\frac{{\eta }^{\gamma -1}}{{\mathfrak{a}}_2}\sum _{i=1}^n\left({\displaystyle {\int }_0^1{\mathfrak{g}}_{3i}(\tau ,\zeta )d{\mathfrak{H}}_i\left(\tau \right)}\right)}\right){\phi }_{\varrho }\left({\displaystyle {\int }_0^1{\mathfrak{G}}_1(\zeta ,\vartheta )f_w\left(\vartheta \right)d\vartheta }\right)d\zeta }\hfill \\ {\displaystyle ={\int }_0^{\eta }\frac{1}{\Gamma \left(\gamma \right)}[{\eta }^{\gamma -1}{(1-\zeta )}^{\gamma -{\alpha }_0-1}-{(\eta -\zeta )}^{\gamma -1}]{\phi }_{\varrho }\left({\displaystyle {\int }_0^1{\mathfrak{G}}_1(\zeta ,\vartheta )f_w\left(\vartheta \right)d\vartheta }\right)d\zeta }\hfill \\ {\displaystyle +{\int }_{\eta }^1\frac{1}{\Gamma \left(\gamma \right)}{\eta }^{\gamma -1}{(1-\zeta )}^{\gamma -{\alpha }_0-1}{\phi }_{\varrho }\left({\displaystyle {\int }_0^1{\mathfrak{G}}_1(\zeta ,\vartheta )f_w\left(\vartheta \right)d\vartheta }\right)d\zeta }\hfill \\ {\displaystyle +\frac{{\eta }^{\gamma -1}}{{\mathfrak{a}}_2}{\int }_0^1\sum _{i=1}^n\left({\displaystyle {\int }_0^1{\mathfrak{g}}_{3i}(\tau ,\zeta )d{\mathfrak{H}}_i\left(\tau \right)}\right){\phi }_{\varrho }\left({\displaystyle {\int }_0^1{\mathfrak{G}}_1(\zeta ,\vartheta )f_w\left(\vartheta \right)d\vartheta }\right)d\zeta .}\hfill \end{array}$$

Then for any $\eta \in (0,1)$ we obtain

$$\begin{array}{c}{\displaystyle {\left(\mathfrak{A}w\right)}^{\prime }\left(\eta \right)={\int }_0^{\eta }\frac{1}{\Gamma \left(\gamma \right)}[(\gamma -1){\eta }^{\gamma -2}{(1-\zeta )}^{\gamma -{\alpha }_0-1}-(\gamma -1){(\eta -\zeta )}^{\gamma -2}]}\hfill \\ \times {\phi }_{\varrho }\left({\displaystyle {\int }_0^1{\mathfrak{G}}_1(\zeta ,\vartheta )f_w\left(\vartheta \right)d\vartheta }\right)d\zeta \hfill \\ {\displaystyle +{\int }_{\eta }^1\frac{1}{\Gamma \left(\gamma \right)}(\gamma -1){\eta }^{\gamma -2}{(1-\zeta )}^{\gamma -{\alpha }_0-1}{\phi }_{\varrho }\left({\displaystyle {\int }_0^1{\mathfrak{G}}_1(\zeta ,\vartheta )f_w\left(\vartheta \right)d\vartheta }\right)d\zeta }\hfill \\ {\displaystyle +\frac{(\gamma -1){\eta }^{\gamma -2}}{{\mathfrak{a}}_2}{\int }_0^1\sum _{i=1}^n\left({\displaystyle {\int }_0^1{\mathfrak{g}}_{3i}(\tau ,\zeta )d{\mathfrak{H}}_i\left(\tau \right)}\right){\phi }_{\varrho }\left({\displaystyle {\int }_0^1{\mathfrak{G}}_1(\zeta ,\vartheta )f_w\left(\vartheta \right)d\vartheta }\right)d\zeta .}\hfill \end{array}$$

So for any $\eta \in (0,1)$ we deduce

$$\begin{array}{c}{\displaystyle {|\left(\mathfrak{A}w\right)}^{\prime }\left(\eta \right)|\le \frac{1}{\Gamma (\gamma -1)}{\int }_0^{\eta }[{\eta }^{\gamma -2}{(1-\zeta )}^{\gamma -{\alpha }_0-1}+{(\eta -\zeta )}^{\gamma -2}]}\hfill \\ \times {\phi }_{\varrho }\left({\displaystyle {\int }_0^1{\mathfrak{J}}_1\left(\vartheta \right)\psi \left(\vartheta \right){\chi }_w\left(\vartheta \right)d\vartheta }\right)d\zeta \hfill \\ {\displaystyle +\frac{1}{\Gamma (\gamma -1)}{\int }_{\eta }^1{\eta }^{\gamma -2}{(1-\zeta )}^{\gamma -{\alpha }_0-1}{\phi }_{\varrho }\left({\displaystyle {\int }_0^1{\mathfrak{J}}_1\left(\vartheta \right)\psi \left(\vartheta \right){\chi }_w\left(\vartheta \right)d\vartheta }\right)d\zeta }\hfill \\ {\displaystyle +\frac{(\gamma -1){\eta }^{\gamma -2}}{{\mathfrak{a}}_2}{\int }_0^1\sum _{i=1}^n\left({\displaystyle {\int }_0^1{\mathfrak{g}}_{3i}(\tau ,\zeta )d{\mathfrak{H}}_i\left(\tau \right)}\right){\phi }_{\varrho }\left({\displaystyle {\int }_0^1{\mathfrak{J}}_1\left(\vartheta \right)\psi \left(\vartheta \right){\chi }_w\left(\vartheta \right)d\vartheta }\right)d\zeta }\hfill \\ \le {\Xi }_2^{\varrho -1}{M}_0^{\varrho -1}\left\{{\displaystyle \frac{1}{\Gamma (\gamma -1)}{\int }_0^{\eta }[{\eta }^{\gamma -2}{(1-\zeta )}^{\gamma -{\alpha }_0-1}+{(\eta -\zeta )}^{\gamma -2}]d\zeta }\right.\hfill \\ +\left.{\displaystyle \frac{1}{\Gamma (\gamma -1)}{\int }_{\eta }^1{\eta }^{\gamma -2}{(1-\zeta )}^{\gamma -{\alpha }_0-1}d\zeta +\frac{(\gamma -1){\eta }^{\gamma -2}}{{\mathfrak{a}}_2}{\int }_0^1\sum _{i=1}^n\left({\displaystyle {\int }_0^1{g}_{3i}(\tau ,\zeta )d{\mathfrak{H}}_i\left(\tau \right)}\right)d\zeta }\right\}.\hfill \end{array}$$

Hence for any $\eta \in (0,1)$ we find

$$\begin{array}{c}{|\left(\mathfrak{A}w\right)}^{\prime }\left(\eta \right)|\le {\Xi }_2^{\varrho -1}{M}_0^{\varrho -1}\left[{\displaystyle \frac{1}{\Gamma (\gamma -1)}\left({\displaystyle \frac{{\eta }^{\gamma -2}}{\gamma -{\alpha }_0}+\frac{{\eta }^{\gamma -1}}{\gamma -1}}\right)}\right.\hfill \\ +\left.{\displaystyle \frac{(\gamma -1){\eta }^{\gamma -2}}{{\mathfrak{a}}_2}{\int }_0^1\sum _{i=1}^n\left({\displaystyle {\int }_0^1\frac{1}{\Gamma (\gamma -{\alpha }_i)}{(1-\zeta )}^{\gamma -{\alpha }_0-1}d\zeta }\right){\tau }^{\gamma -{\alpha }_i-1}d{\mathfrak{H}}_i\left(\tau \right)}\right]\hfill \\ ={\Xi }_2^{\varrho -1}{M}_0^{\varrho -1}\left[{\displaystyle \frac{1}{\Gamma (\gamma -1)}\left({\displaystyle \frac{{\eta }^{\gamma -2}}{\gamma -{\alpha }_0}+\frac{{\eta }^{\gamma -1}}{\gamma -1}}\right)}\right.\hfill \\ \left.{\displaystyle +\frac{(\gamma -1){\eta }^{\gamma -2}}{{\mathfrak{a}}_2(\gamma -{\alpha }_0)}\sum _{i=1}^n\frac{1}{\Gamma (\gamma -{\alpha }_i)}{\int }_0^1{\tau }^{\gamma -{\alpha }_i-1}d{\mathfrak{H}}_i\left(\tau \right)}\right].\hfill \end{array}$$

(19)

We denote by

$$\begin{array}{c}{\displaystyle \xi \left(\eta \right)=\frac{1}{\Gamma (\gamma -1)}\left({\displaystyle \frac{{\eta }^{\gamma -2}}{\gamma -{\alpha }_0}+\frac{{\eta }^{\gamma -1}}{\gamma -1}}\right)}\hfill \\ {\displaystyle +\frac{(\gamma -1){\eta }^{\gamma -2}}{{\mathfrak{a}}_2(\gamma -{\alpha }_0)}\sum _{i=1}^n\frac{1}{\Gamma (\gamma -{\alpha }_i)}{\int }_0^1{\tau }^{\gamma -{\alpha }_i-1}d{\mathfrak{H}}_i\left(\tau \right),\eta \in (0,1).}\hfill \end{array}$$

This function $\xi \in L^1(0,1)$ because

$${\displaystyle {\int }_0^1\xi \left(\eta \right)d\eta =\frac{1}{\Gamma \left(\gamma \right)}\left({\displaystyle \frac{1}{\gamma -{\alpha }_0}+\frac{1}{\gamma }}\right)+\frac{1}{{\mathfrak{a}}_2(\gamma -{\alpha }_0)}\sum _{i=1}^n\frac{1}{\Gamma (\gamma -{\alpha }_i)}{\int }_0^1{\tau }^{\gamma -{\alpha }_i-1}d{\mathfrak{H}}_i\left(\tau \right)<\infty .}$$

(20)

So for any ${\eta }_1,{\eta }_2\in [0,1]$, with ${\eta }_1<{\eta }_2$ and $w\in \mathfrak{D}$, by (19) and (20) we obtain

$${\displaystyle |\left(\mathfrak{A}w\right)\left({\eta }_1\right)-\left(\mathfrak{A}w\right)\left({\eta }_2\right)|=\left|{\displaystyle {\int }_{{\eta }_1}^{{\eta }_2}{\left(\mathfrak{A}w\right)}^{\prime }\left(\tau \right)d\tau }\right|\le {\Xi }_2^{\varrho -1}{M}_0^{\varrho -1}{\int }_{{\eta }_1}^{{\eta }_2}\xi \left(\tau \right)d\tau .}$$

(21)

By (20), (21) and the property of the absolute continuity of the integral function, we deduce that $\mathfrak{A}\left(\mathfrak{D}\right)$ is equicontinuous. Using the Arzela-Ascoli theorem, we conclude that $\mathfrak{A}\left(\mathfrak{D}\right)$ is relatively compact. In addition, with standard arguments, we show that $\mathfrak{A}$ is continuous on $\mathfrak{P}$. Therefore $\mathfrak{A}$ is a completely continuous operator on $\mathfrak{P}$. □

Next we define the cone

$${\mathfrak{P}}_0=\left\{w\in \mathfrak{P},w\left(\eta \right)\ge {\eta }^{\gamma -1}\parallel w\parallel ,\forall \eta \in [0,1]\right\}.$$

Under the assumptions $\left(K1\right)$ and $\left(K2\right)$, by Lemma 3, we obtain that $\mathfrak{A}\left(\mathfrak{P}\right)\subset {\mathfrak{P}}_0$, and so ${\mathfrak{A}|}_{{\mathfrak{P}}_0}:{\mathfrak{P}}_0\to {\mathfrak{P}}_0$ (denoted again by $\mathfrak{A}$) is also a completely continuous operator. For $\kappa >0$ we denote by $B_{\kappa }$ the open ball with the center at zero of radius r and by ${\overline{B}}_{\kappa }$ and $\partial B_{\kappa }$ its closure and its boundary, respectively. We also denote by ${\gamma }_0=0$, $M_0={\int }_0^1{\mathfrak{J}}_1\left(\tau \right)\psi \left(\tau \right)d\tau$, $M_1={\int }_0^1{\mathfrak{J}}_2\left(\tau \right)d\tau$, and for ${\theta }_1,{\theta }_2\in (0,1)$, ${\theta }_1<{\theta }_2$, by $M_2={\int }_{{\theta }_1}^{{\theta }_2}{\mathfrak{J}}_2\left(\zeta \right){\left({\int }_{{\theta }_1}^{\zeta }{\mathfrak{G}}_1(\zeta ,\tau )d\tau \right)}^{\varrho -1}d\zeta$.

Theorem 2.

We suppose that assumptions $\left(K1\right)$, $\left(K2\right)$,

(K3)

There exist $b_1,\dots ,b_{q+1}\ge 0$ with ${\sum }_{i=1}^{q+1}{b}_i>0$, and $\sigma \ge 1$ such that

$${\displaystyle {\chi }_0=\underset{{\sum }_{i=1}^{q+1}{b}_i{z}_i\to 0}{lim\; sup}\underset{\eta \in [0,1]}{max}\frac{\chi (\eta ,z_1,\dots ,z_{q+1})}{{\phi }_r\left({(b_1{z}_1+\cdots +b_{q+1}{z}_{q+1})}^{\sigma }\right)}<l_1,}$$

where $l_1={(M_0{M}_1^{r-1}{\rho }_1^{\sigma (r-1)})}^{-1}$, with ${\rho }_1={\sum }_{i=1}^{q+1}\frac{b_i}{\Gamma ({\gamma }_{i-1}+1)}$;

(K4)

There exist $c_1,\dots ,c_{q+1}\ge 0$ with ${\sum }_{i=1}^{q+1}{c}_i>0$, ${\theta }_1,{\theta }_2\in (0,1)$, ${\theta }_1<{\theta }_2$, and $\lambda >1$ such that

$${\displaystyle f_{\infty }=\underset{{\sum }_{i=1}^{q+1}{c}_i{z}_i\to \infty }{lim\; inf}\underset{\eta \in [{\theta }_1,{\theta }_2]}{min}\frac{f(\eta ,z_1,\dots ,z_{q+1})}{{\phi }_r(c_1{z}_1+\cdots +c_{q+1}{z}_{q+1})}>l_2,}$$

where $l_2=\lambda (M_2^{r-1}{\rho }_2^{r-1}{\theta }_1^{(\gamma -1)(r-1)})-1$, with ${\rho }_2={\sum }_{i=1}^{q+1}\frac{c_i{\theta }_1^{{\gamma }_{i-1}+\gamma -1}\Gamma \left(\gamma \right)}{\Gamma (\gamma +{\gamma }_{i-1})}$,

hold. Then there exists a positive solution $w\left(\eta \right),\eta \in [0,1]$ of problem (1), (2).

Proof. 

By $\left(K3\right)$ there exists $R\in (0,1)$ such that

$$\chi (\eta ,z_1,\dots ,z_{q+1})\le l_1{\phi }_r\left({(b_1{z}_1+\cdots +b_{q+1}{z}_{q+1})}^{\sigma }\right),$$

(22)

for all $\eta \in [0,1]$, $z_i\ge 0,i=1,\dots ,q+1$ with ${\sum }_{i=1}^{q+1}{b}_i{z}_i\le R$. We define $R_1\le min\{R/{\rho }_1,R\}$. For any $w\in \mathfrak{X}$ and $\eta \in [0,1]$ we have $I_{0+}^{{\gamma }_i}w\left(\eta \right)\le \frac{\parallel w\parallel }{\Gamma ({\gamma }_i+1)}$, for all $i=1,\dots ,q$. Then for any $w\in {\overline{B}}_{R_1}\cap \mathfrak{P}$ and $\zeta \in [0,1]$, we find

$$\begin{array}{c}{b}_{1}w\left(\zeta \right)+b_2{I}_{0+}^{{\gamma }_1}w\left(\zeta \right)+\cdots +b_{q+1}{I}_{0+}^{{\gamma }_q}w\left(\zeta \right)\hfill \\ \le \left({\displaystyle b_1+\frac{b_2}{\Gamma ({\gamma }_1+1)}+\cdots +\frac{b_{q+1}}{\Gamma ({\gamma }_q+1)}}\right)\parallel w\parallel ={\rho }_1\parallel w\parallel \le {\rho }_1{R}_1\le R.\hfill \end{array}$$

Hence by (22) and Lemma 2, for any $w\in \partial B_{R_1}\cap {\mathfrak{P}}_0$ and $\eta \in [0,1]$ we obtain

$$\begin{array}{c}{\displaystyle \left(\mathfrak{A}w\right)\left(\eta \right)\le {\int }_0^1{\mathfrak{J}}_2\left(\zeta \right){\phi }_{\varrho }\left({\mathfrak{J}}_1\left(\vartheta \right)f_w\left(\vartheta \right)d\vartheta \right)d\zeta }\hfill \\ =M_1{\phi }_{\varrho }\left({\displaystyle {\int }_0^1{\mathfrak{J}}_1\left(\vartheta \right)f_w\left(\vartheta \right)d\vartheta }\right)\le M_1{\phi }_{\varrho }\left({\displaystyle {\int }_0^1{\mathfrak{J}}_1\left(\vartheta \right)\psi \left(\vartheta \right){\chi }_w\left(\vartheta \right)d\vartheta }\right)\hfill \\ \le M_1{\phi }_{\varrho }\left({\displaystyle {\int }_0^1{\mathfrak{J}}_1\left(\vartheta \right)\psi \left(\vartheta \right)l_1{\phi }_r\left({\left(b_{1}w\left(\vartheta \right)+b_2{I}_{0+}^{{\gamma }_1}w\left(\vartheta \right)+\cdots +b_{q+1}{I}_{0+}^{{\gamma }_q}w\left(\vartheta \right)\right)}^{\sigma }\right)d\vartheta }\right)\hfill \\ \le M_1{\phi }_{\varrho }\left({\phi }_r\left(({\rho }_1{\parallel w\parallel )}^{\sigma }\right)\right){\phi }_{\varrho }\left(l_1\right){\phi }_{\varrho }\left(M_0\right)=M_1{M}_0^{\varrho -1}{l}_1^{\varrho -1}{\rho }_1^{\sigma }{\parallel w\parallel }^{\sigma }\hfill \\ \le M_1{M}_0^{\varrho -1}{l}_1^{\varrho -1}{\rho }_1^{\sigma }\parallel w\parallel =\parallel w\parallel .\hfill \end{array}$$

Therefore

$$\parallel \mathfrak{A}w\parallel \le \parallel w\parallel ,\forall w\in \partial B_{R_1}\cap {\mathfrak{P}}_0.$$

(23)

From $\left(K4\right)$ there exists $C_1>0$ such that

$$f(\eta ,z_1,\dots ,z_{q+1})\ge l_2{\phi }_r(c_1{z}_1+\dots +c_{q+1}{z}_{q+1})-C_1,$$

(24)

for all $\eta \in [{\theta }_1,{\theta }_2]$ and $z_i\ge 0,i=1,\dots ,q+1$. By definition of $I_{0+}^{\alpha }$, for any $w\in {\mathfrak{P}}_0$, $\eta \in [0,1]$ and $i=1,\dots ,q$ we find

$$\begin{array}{c}{\displaystyle I_{0+}^{{\gamma }_i}w\left(\eta \right)=\frac{1}{\Gamma \left({\gamma }_i\right)}{\int }_0^{\eta }{(\eta -\zeta )}^{{\gamma }_i-1}w\left(\zeta \right)d\zeta \ge \frac{1}{\Gamma \left({\gamma }_i\right)}{\int }_0^{\eta }{(\eta -\zeta )}^{{\gamma }_i-1}{\zeta }^{\gamma -1}\parallel w\parallel d\zeta }\hfill \\ {\displaystyle \stackrel{\zeta =\eta y}{=}\frac{\parallel w\parallel }{\Gamma \left({\gamma }_i\right)}{\int }_0^1{(\eta -\eta y)}^{{\gamma }_i-1}{\eta }^{\gamma -1}{y}^{\gamma -1}\eta dy=\frac{\parallel w\parallel }{\Gamma \left({\gamma }_i\right)}{\eta }^{{\gamma }_i+\gamma -1}{\int }_0^1{y}^{\gamma -1}{(1-y)}^{{\gamma }_i-1}dy}\hfill \\ {\displaystyle =\frac{\parallel w\parallel }{\Gamma \left({\gamma }_i\right)}{\eta }^{{\gamma }_i+\gamma -1}B(\gamma ,{\gamma }_i)=\frac{\parallel w\parallel {\eta }^{{\gamma }_i+\gamma -1}\Gamma \left(\gamma \right)}{\Gamma (\gamma +{\gamma }_i)},}\hfill \end{array}$$

(25)

where $B(p,q)$ is the first Euler function. Then by using (24) and (25), for any $w\in {\mathfrak{P}}_0$ and $\eta \in [{\theta }_1,{\theta }_2]$ we obtain

$$\begin{array}{c}{\displaystyle \left(\mathfrak{A}w\right)\left(\eta \right)\ge {\int }_{{\theta }_1}^{{\theta }_2}{\mathfrak{G}}_2(\eta ,\zeta ){\phi }_{\varrho }\left({\displaystyle {\int }_{{\theta }_1}^{\zeta }{\mathfrak{G}}_1(\zeta ,\vartheta )f_w\left(\vartheta \right)d\vartheta }\right)d\zeta }\hfill \\ {\displaystyle \ge {\theta }_1^{\gamma -1}{\int }_{{\theta }_1}^{{\theta }_2}{\mathfrak{J}}_2\left(\zeta \right)\left({\displaystyle {\int }_{{\theta }_1}^{\zeta }{\mathfrak{G}}_1(\zeta ,\vartheta )\left[l_2{\left(c_{1}w\left(\vartheta \right)+c_2{I}_{0+}^{{\gamma }_1}w\left(\vartheta \right)+\cdots +c_{q+1}{I}_{0+}^{{\gamma }_q}w\left(\vartheta \right)\right)}^{r-1}\right.}\right.}\hfill \\ {\left.\left.-C_1\right]d\vartheta \right)}^{\varrho -1}d\zeta \hfill \\ {\displaystyle \ge {\theta }_1^{\gamma -1}{\int }_{{\theta }_1}^{{\theta }_2}{\mathfrak{J}}_2\left(\vartheta \right)\left({\displaystyle {\int }_{{\theta }_1}^{\zeta }{\mathfrak{G}}_1(\zeta ,\vartheta )\left[l_2\left({\displaystyle c_1{\theta }_1^{\gamma -1}\parallel w\parallel +c_2\frac{{\theta }_1^{{\gamma }_1+\gamma -1}\Gamma \left(\gamma \right)}{\Gamma (\gamma +{\gamma }_1)}\parallel w\parallel +\cdots }\right.\right.}\right.}\hfill \\ {\left.\left.{\left.{\displaystyle +c_{q+1}\frac{{\theta }_1^{{\gamma }_q+\gamma -1}\Gamma \left(\gamma \right)}{\Gamma (\gamma +{\gamma }_q)}\parallel w\parallel }\right)}^{r-1}-C_1\right]d\vartheta \right)}^{\varrho -1}d\zeta \hfill \\ =M_2{\theta }_1^{\gamma -1}{\left[l_2({\rho }_2{\parallel w\parallel )}^{r-1}-C_1\right]}^{\varrho -1}\hfill \\ ={\left(M_2^{r-1}{\theta }_1^{(\gamma -1)(r-1)}{l}_2{\rho }_2^{r-1}{\parallel w\parallel }^{r-1}-M_2^{r-1}{\theta }_1^{(\gamma -1)(r-1)}{C}_1\right)}^{\varrho -1}\hfill \\ ={\left({\lambda \parallel w\parallel }^{r-1}-C_2\right)}^{\varrho -1},C_2=M_2^{r-1}{\theta }_1^{(\gamma -1)(r-1)}{C}_1.\hfill \end{array}$$

Then we deduce

$$\parallel \mathfrak{A}w\parallel \ge {\left({\lambda \parallel w\parallel }^{r-1}-C_2\right)}^{\varrho -1},\forall w\in {\mathfrak{P}}_0.$$

We choose $R_2\ge max\{1,C_2^{\varrho -1}/{(\lambda -1)}^{\varrho -1}\}$, and we conclude

$$\parallel \mathfrak{A}w\parallel \ge \parallel w\parallel ,\forall w\in \partial B_{R_2}\cap {\mathfrak{P}}_0.$$

(26)

By Lemma 5, (23), (26) and Theorem 1 (i), we deduce that $\mathfrak{A}$ has a fixed point $w\in ({\overline{B}}_{R_2}\setminus B_{R_1})\cap {\mathfrak{P}}_0$, so $R_1\le \parallel w\parallel \le R_2$, and $w\left(\eta \right)\ge {\eta }^{\gamma -1}\parallel w\parallel$ for all $\eta \in [0,1]$, (so $w\left(\eta \right)>0$ for all $\eta \in (0,1]$). Hence $w\left(\eta \right),\eta \in [0,1]$ is a positive solution of problem (1), (2). □

Theorem 3.

We suppose that assumptions $\left(K1\right)$, $\left(K2\right)$,

(K5)

There exist $d_1,\dots ,d_{q+1}\ge 0$ with ${\sum }_{i=1}^{q+1}{d}_i>0$, and $\mu <1$ such that

$${\displaystyle {\chi }_{\infty }=\underset{{\sum }_{i=1}^{q+1}{d}_i{z}_i\to \infty }{lim\; sup}\underset{\eta \in [0,1]}{max}\frac{\chi (\eta ,z_1,\dots ,z_{q+1})}{{\phi }_r(d_1{z}_1+\cdots +d_{q+1}{z}_{q+1})}<l_3,}$$

where $l_3=\mu {\left(M_0{M}_1^{r-1}{\rho }_3^{r-1}\right)}^{-1}$, with ${\rho }_3={\sum }_{i=1}^{q+1}\frac{d_i}{\Gamma ({\gamma }_{i-1}+1)}$;

(K6)

There exist $e_1,\dots ,e_{q+1}\ge 0$ with ${\sum }_{i=1}^{q+1}{e}_i>0$, ${\theta }_1,{\theta }_2\in (0,1)$, ${\theta }_1<{\theta }_2$, and $\nu \in (0,1]$ such that

$${\displaystyle f_0=\underset{{\sum }_{i=1}^{q+1}{e}_i{z}_i\to 0}{lim\; inf}\underset{\eta \in [{\theta }_1,{\theta }_2]}{min}\frac{f(\eta ,z_1,\dots ,z_{q+1})}{{\phi }_r\left({(e_1{z}_1+\cdots +e_{q+1}{z}_{q+1})}^{\nu }\right)}>l_4,}$$

where $l_4=(M_2^{r-1}{\rho }_4^{\nu (r-1)}{\theta }_1^{(\gamma -1)(r-1)})-1$, with ${\rho }_4={\sum }_{i=1}^{q+1}\frac{e_i{\theta }_1^{{\gamma }_{i-1}+\gamma -1}\Gamma \left(\gamma \right)}{\Gamma (\gamma +{\gamma }_{i-1})}$,

hold. Then there exists a positive solution $w\left(\eta \right),\eta \in [0,1]$ of problem (1), (2).

Proof. 

From $\left(K5\right)$ there exists $C_3>0$ such that

$$\chi (\eta ,z_1,\dots ,z_{q+1})\le l_3{\phi }_r(d_1{z}_1+\cdots +d_{q+1}{z}_{q+1})+C_3,$$

(27)

for any $\eta \in [0,1]$ and $z_i\ge 0,i=1,\dots ,q+1$. By using $\left(K2\right)$ and (27), for any $w\in {\mathfrak{P}}_0$ we obtain

$$\begin{array}{c}{\displaystyle \left(\mathfrak{A}w\right)\left(\eta \right)\le {\int }_0^1{\mathfrak{J}}_2\left(\zeta \right){\phi }_{\varrho }\left({\displaystyle {\int }_0^1{\mathfrak{J}}_1\left(\vartheta \right)f_w\left(\vartheta \right)d\vartheta }\right)d\zeta }\hfill \\ \le M_1{\phi }_{\varrho }\left({\displaystyle {\int }_0^1{\mathfrak{J}}_1\left(\vartheta \right)\psi \left(\vartheta \right){\chi }_w\left(\vartheta \right)d\vartheta }\right)\hfill \\ \le M_1{\phi }_{\varrho }\left({\displaystyle {\int }_0^1{\mathfrak{J}}_1\left(\vartheta \right)\psi \left(\vartheta \right)\left[l_3{\phi }_r\left(d_{1}w\left(\vartheta \right)+d_2{I}_{0+}^{{\gamma }_1}w\left(\vartheta \right)+\cdots +d_{q+1}{I}_{0+}^{{\gamma }_q+1}w\left(\vartheta \right)\right)+C_3\right]d\vartheta }\right)\hfill \\ \le M_1{\phi }_{\varrho }\left({\displaystyle {\int }_0^1{\mathfrak{J}}_1\left(\vartheta \right)\psi \left(\vartheta \right)\left[l_3{\left({\displaystyle d_1\parallel w\parallel +d_2\frac{\parallel w\parallel }{\Gamma ({\gamma }_1+1)}+\cdots +d_{q+1}\frac{\parallel w\parallel }{\Gamma ({\gamma }_i+1)}}\right)}^{r-1}+C_3\right]d\vartheta }\right)\hfill \\ =M_1{\phi }_{\varrho }\left[l_3{\left({\displaystyle d_1+\frac{d_2}{\Gamma ({\gamma }_1+1)}+\cdots +\frac{d_{q+1}}{\Gamma ({\gamma }_i+1)}}\right)}^{r-1}{\parallel w\parallel }^{r-1}+C_3\right]{\left({\displaystyle {\int }_0^1{\mathfrak{J}}_1\left(\vartheta \right)\psi \left(\vartheta \right)d\vartheta }\right)}^{\varrho -1}\hfill \\ =M_1{M}_0^{\varrho -1}{\left(l_3{\rho }_3^{r-1}{\parallel w\parallel }^{r-1}+C_3\right)}^{\varrho -1}.\hfill \end{array}$$

So we find

$$\begin{array}{c}\parallel \mathfrak{A}w\parallel \le M_1{M}_0^{\varrho -1}{\left(l_3{\rho }_3^{r-1}{\parallel w\parallel }^{r-1}+C_3\right)}^{\varrho -1}\hfill \\ ={\left(M_1^{r-1}{M}_0{l}_3{\rho }_3^{r-1}{\parallel w\parallel }^{r-1}+M_1^{r-1}{M}_0{C}_3\right)}^{\varrho -1}\hfill \\ ={\left({\mu \parallel w\parallel }^{r-1}+C_4\right)}^{\varrho -1},\forall w\in {\mathfrak{P}}_0,C_4=M_0{M}_1^{r-1}{C}_3.\hfill \end{array}$$

We choose $R_3>max\left\{1,{\left(C_4/(1-\mu )\right)}^{\varrho -1}\right\}$. Then we deduce

$$\parallel \mathfrak{A}w\parallel \le \parallel w\parallel ,\forall w\in \partial B_{R_3}\cap {\mathfrak{P}}_0.$$

(28)

By $\left(K6\right)$ there exists ${\tilde{R}}_4\in (0,1]$ such that

$$f(\eta ,z_1,\dots ,z_{q+1})\ge l_4{\phi }_r\left({(e_1{z}_1+\cdots +e_{q+1}{z}_{q+1})}^{\nu }\right),$$

(29)

for all $\eta \in [{\theta }_1,{\theta }_2],$$z_i\ge 0,i=1,\dots ,q+1$, ${\sum }_{i=1}^{q+1}{e}_i{z}_i\le {\tilde{R}}_4.$ We take $R_4\le min\{{\tilde{R}}_4/{\tilde{\rho }}_4,{\tilde{R}}_4\}$, where ${\tilde{\rho }}_4={\sum }_{i=1}^{q+1}\frac{e_i}{\Gamma ({\gamma }_{i-1}+1)}$. Then for any $w\in {\overline{B}}_{R_4}\cap \mathfrak{P}$ and $\eta \in [0,1]$ we have

$$\begin{array}{c}{e}_{1}w\left(\eta \right)+e_2{I}_{0+}^{{\gamma }_1}w\left(\eta \right)+\cdots +e_{q+1}{I}_{0+}^{{\gamma }_q}w\left(\eta \right)\le e_1\parallel w\parallel +e_2\frac{\parallel w\parallel }{\Gamma ({\gamma }_1+1)}+\cdots +e_{q+1}\frac{\parallel w\parallel }{\Gamma ({\gamma }_q+1)}\hfill \\ ={\tilde{\rho }}_4\parallel w\parallel \le {\tilde{\rho }}_4{R}_4\le {\tilde{R}}_4.\hfill \end{array}$$

Therefore, by using (29), we find for any $w\in \partial B_{R_4}\cap {\mathfrak{P}}_0$ and $\eta \in [{\theta }_1,{\theta }_2]$

$$\begin{array}{c}{\displaystyle \left(\mathfrak{A}w\right)\left(\eta \right)\ge {\int }_{{\theta }_1}^{{\theta }_2}{\mathfrak{G}}_2(\eta ,\zeta ){\phi }_{\varrho }\left({\displaystyle {\int }_{{\theta }_1}^{\zeta }{\mathfrak{G}}_1(\zeta ,\vartheta )f_w\left(\vartheta \right)d\vartheta }\right)d\zeta }\hfill \\ {\displaystyle \ge {\theta }_1^{\gamma -1}{\int }_{{\theta }_1}^{{\theta }_2}{\mathfrak{J}}_2\left(\zeta \right){\phi }_{\varrho }\left({\displaystyle {\int }_{{\theta }_1}^{\zeta }{\mathfrak{G}}_1(\zeta ,\vartheta )\left[l_4{\phi }_r\left(e_{1}w\left(\vartheta \right)+e_2{I}_{0+}^{{\gamma }_1}w\left(\vartheta \right)+\cdots \right.\right.}\right.}\hfill \\ \left.\left.{\left.+e_{q+1}{I}_{0+}^{{\gamma }_q}w\left(\vartheta \right)\right)}^{\nu }\right]d\vartheta \right)d\zeta \hfill \\ {\displaystyle \ge {\theta }_1^{\gamma -1}{\int }_{{\theta }_1}^{{\theta }_2}{\mathfrak{J}}_2\left(\zeta \right){\phi }_{\varrho }\left({\displaystyle {\int }_{{\theta }_1}^{\zeta }{\mathfrak{G}}_1(\zeta ,\vartheta )\left[l_4\left(e_1{\theta }_1^{\gamma -1}\parallel w\parallel +e_2\frac{{\theta }_1^{{\gamma }_1+\gamma -1}\Gamma \left(\gamma \right)}{\Gamma (\gamma +{\gamma }_1)}\parallel w\parallel +\cdots \right.\right.}\right.}\hfill \\ \left.\left.{\left.+e_{q+1}\frac{{\theta }_1^{{\gamma }_{q+1}+\gamma -1}\Gamma \left(\gamma \right)}{\Gamma (\gamma +{\gamma }_{q+1})}\parallel w\parallel \right)}^{\nu (r-1)}\right]d\vartheta \right)d\zeta \hfill \\ =M_2{\theta }_1^{\gamma -1}{l}_4^{\varrho -1}{\rho }_4^{\nu }{\parallel w\parallel }^{\nu }\ge M_2{\theta }_1^{\gamma -1}{l}_4^{\varrho -1}{\rho }_4^{\nu }\parallel w\parallel =\parallel w\parallel .\hfill \end{array}$$

Then we obtain

$$\parallel \mathfrak{A}w\parallel \ge \parallel w\parallel ,\forall w\in \partial B_{R_4}\cap {\mathfrak{P}}_0.$$

(30)

By Lemma 5, (28), (30) and Theorem 1 (ii), we conclude that $\mathfrak{A}$ has a fixed point $w\in ({\overline{B}}_{R_3}\setminus B_{R_4})\cap {\mathfrak{P}}_0$, so $R_4\le \parallel w\parallel \le R_3$, and $w\left(\eta \right)\ge {\eta }^{\gamma -1}\parallel w\parallel$ for all $\eta \in [0,1]$, which is a positive solution of problem (1), (2). □

Theorem 4.

We suppose that assumptions $\left(K1\right)$, $\left(K2\right)$, $\left(K4\right)$ and $\left(K6\right)$ hold. In addition the functions ψ and χ satisfy the condition

(K7)

$M_1{\Xi }_0^{\varrho -1}{M}_0^{\varrho -1}<1$, where ${\Xi }_0=max\{\chi (\eta ,z_1,\dots ,z_{q+1}),\eta \in [0,1],z_i\in [0,{\omega }_0],i=1,\dots ,q+1,\}$ and ${\omega }_0=max\left\{1,\frac{1}{\Gamma ({\gamma }_1+1)},\cdots ,\frac{1}{\Gamma ({\gamma }_q+1)}\right\}$.

Then there exist two positive solutions $w_1\left(\eta \right),w_2\left(\eta \right),\eta \in [0,1]$ of problem (1), (2).

Proof. 

If $\left(K1\right)$, $\left(K2\right)$ and $\left(K4\right)$ are satisfied, then by the proof of Theorem 2, we find that there exists $R_2>1$ (we can consider $R_2>1$) such that

$$\parallel \mathfrak{A}w\parallel \ge \parallel w\parallel ,\forall w\in \partial B_{R_2}\cap {\mathfrak{P}}_0.$$

(31)

If $\left(K1\right)$, $\left(K2\right)$ and $\left(K6\right)$ are satisfied, then by the proof of Theorem 3 we deduce that there exists $R_4<1$ (we can consider $R_4<1$) such that

$$\parallel \mathfrak{A}w\parallel \ge \parallel w\parallel ,\forall w\in \partial B_{R_4}\cap {\mathfrak{P}}_0.$$

(32)

We consider now the set $B_1=\{w\in \mathfrak{X},\parallel w\parallel <1\}$. By $\left(K7\right)$, for any $w\in \partial B_1\cap {\mathfrak{P}}_0$ and $\eta \in [0,1]$ we obtain

$$\begin{array}{c}{\displaystyle \left(\mathfrak{A}w\right)\left(\eta \right)\le {\int }_0^1{\mathfrak{J}}_2\left(\zeta \right){\phi }_{\varrho }\left({\displaystyle {\int }_0^1{\mathfrak{J}}_1\left(\vartheta \right)f_w\left(\vartheta \right)d\vartheta }\right)d\zeta }\hfill \\ ={\phi }_{\varrho }\left({\displaystyle {\int }_0^1{\mathfrak{J}}_1\left(\vartheta \right)f_w\left(\vartheta \right)d\vartheta }\right)\left({\displaystyle {\int }_0^1{\mathfrak{J}}_2\left(\zeta \right)d\zeta }\right)\hfill \\ \le M_1{\phi }_{\varrho }\left({\displaystyle {\int }_0^1{\mathfrak{J}}_1\left(\vartheta \right)\psi \left(\vartheta \right){\chi }_w\left(\vartheta \right)d\vartheta }\right)\le M_1{\Xi }_0^{\varrho -1}{M}_0^{\varrho -1}<1=\parallel w\parallel .\hfill \end{array}$$

So we conclude

$$\parallel \mathfrak{A}w\parallel <\parallel w\parallel ,\forall w\in \partial B_1\cap {\mathfrak{P}}_0.$$

(33)

Then by (31), (33) and Theorem 1 (i), we find that problem (1), (2) has one positive solution $w_1\in {\mathfrak{P}}_0$ with $1<\parallel w_1\parallel \le R_2$. By (32), (33) and Theorem 1 (ii), we deduce that problem (1), (2) has another positive solution $w_2\in {\mathfrak{P}}_0$ with $R_4\le \parallel w_2\parallel <1$. So problem (1), (2) has at least two positive solutions $w_1\left(\eta \right),w_2\left(\eta \right),\eta \in [0,1]$. □

4. Examples

Let $\beta =3/2$, $\gamma =10/3$, $p=4$, $r=11/7$, $\varrho =11/4$, $q=2$, ${\gamma }_1=25/8$, ${\gamma }_2=21/5$, $n=2$, ${\alpha }_0=5/4$, ${\alpha }_1=5/6$, ${\alpha }_2=8/7$, $H_0\left(\eta \right)=\{1,\eta \in [0,1/2);3/2,\eta \in [1/2,1]\}$, $H_1\left(\eta \right)=\eta /3,\eta \in [0,1]$, $H_2\left(\eta \right)=\{1/3,\eta \in [0,2/3);13/12,\eta \in [2/3,1]\}$.

We consider the fractional differential equation

$$D_{0+}^{3/2}\left({\phi }_{11/7}\left(D_{0+}^{10/3}w\left(\eta \right)\right)\right)=f\left(\eta ,w\left(\eta \right),I_{0+}^{25/8}w\left(\eta \right),I_{0+}^{21/5}w\left(\eta \right)\right),\eta \in (0,1),$$

(34)

with the boundary conditions

$$\left\{\begin{array}{c}{\displaystyle w\left(0\right)=w^{\prime }\left(0\right)=w^{\prime \prime }\left(0\right)=0,D_{0+}^{10/3}w\left(0\right)=0,D_{0+}^{10/3}w\left(1\right)=\frac{1}{2^{7/4}}{D}_{0+}^{10/3}w\left({\displaystyle \frac{1}{2}}\right),}\hfill \\ {\displaystyle D_{0+}^{5/4}w\left(1\right)=\frac{1}{3}{\int }_0^1{D}_{0+}^{5/6}w\left(\tau \right)d\tau +\frac{3}{4}{D}_{0+}^{8/7}w\left(\frac{2}{3}\right).}\hfill \end{array}\right.$$

(35)

We obtain here ${\mathfrak{a}}_1\approx 0.64644661$ and ${\mathfrak{a}}_2\approx 1.2243698$. Then assumption $\left(K1\right)$ is satisfied. We also find

$$\begin{gathered} \begin{array}{c}{\displaystyle {\mathfrak{g}}_1(\eta ,\zeta )=\frac{1}{\Gamma (3/2)}\left\{\begin{array}{c}{\eta }^{1/2}{(1-\zeta )}^{1/2}-{(\eta -\zeta )}^{1/2},0\le \zeta \le \eta \le 1,\hfill \\ {\eta }^{1/2}{(1-\zeta )}^{1/2},0\le \eta \le \zeta \le 1,\hfill \end{array}\right.}\hfill \end{array} \\ {\mathfrak{g}}_2(\eta ,\zeta )=\frac{1}{\Gamma (10/3)}\left\{\begin{array}{c}{\eta }^{7/3}{(1-\zeta )}^{13/12}-{(\eta -\zeta )}^{7/3},0\le \zeta \le \eta \le 1,\hfill \\ {\eta }^{7/3}{(1-\zeta )}^{13/12},0\le \eta \le \zeta \le 1,\hfill \end{array}\right. \\ {\mathfrak{g}}_{31}(\eta ,\zeta )=\frac{1}{\Gamma (5/2)}\left\{\begin{array}{c}{\eta }^{3/2}{(1-\zeta )}^{13/12}-{(\eta -\zeta )}^{3/2},0\le \zeta \le \eta \le 1,\hfill \\ {\eta }^{3/2}{(1-\zeta )}^{13/12},0\le \eta \le \zeta \le 1,\hfill \end{array}\right. \\ {\mathfrak{g}}_{32}(\eta ,\zeta )=\frac{1}{\Gamma (46/21)}\left\{\begin{array}{c}{\eta }^{25/21}{(1-\zeta )}^{13/12}-{(\eta -\zeta )}^{25/21},0\le \zeta \le \eta \le 1,\hfill \\ {\eta }^{25/21}{(1-\zeta )}^{13/12},0\le \eta \le \zeta \le 1,\hfill \end{array}\right. \\ {\mathfrak{G}}_1(\eta ,\zeta )=g_1(\eta ,\zeta )+\frac{{\eta }^{1/2}}{2_{a_1}}{g}_1(\frac{1}{2},\zeta ),(\eta ,\zeta )\in [0,1] \times [0,1], \\ {\mathfrak{G}}_2(\eta ,\zeta )=g_2(\eta ,\zeta )+\frac{{\eta }^{7/3}}{a_2}[\frac{1}{3}{\int }_0^1\mathfrak{g}31(\vartheta ,\zeta )d\vartheta +\frac{3}{4}{\mathfrak{g}}_{32}(\frac{2}{3},\zeta \left)\right],(\eta ,\zeta )\in [0,1] \times [0,1], \\ {\mathfrak{h}}_1\left(\zeta \right)=\frac{1}{\Gamma (3/2)}{(1-\zeta )}^{1/2},\zeta \in [0,1], \\ {\mathfrak{h}}_2\left(\zeta \right)=\frac{1}{\Gamma (10/3)}[{(1-\zeta )}^{13/12}-{(1-\zeta )}^{7/3}],\zeta \in [0,1], \end{gathered}$$

In addition we obtain

$$\begin{gathered} \begin{array}{c}{\mathfrak{J}}_1\left(\zeta \right)=\left\{\begin{array}{c}{\mathfrak{h}}_1\left(\zeta \right)+\frac{1}{2{\mathfrak{a}}_1\Gamma (3/2)}\left[{\left(\frac{1}{2}\right)}^{1/2}{(1-\zeta )}^{1/2}-{\left(\frac{1}{2}-\zeta \right)}^{1/2}\right],0\le \zeta \le \frac{1}{2},\hfill \\ {\mathfrak{h}}_1\left(\zeta \right)+\frac{1}{2{\mathfrak{a}}_1\Gamma (3/2)}{\left(\frac{1}{2}\right)}^{1/2}{(1-\zeta )}^{1/2},\frac{1}{2}<\zeta \le 1,\hfill \end{array}\right.\hfill \end{array} \\ {\mathfrak{J}}_2\left(\zeta \right)=\left\{\begin{array}{c}{\mathfrak{h}}_2\left(\zeta \right)+\frac{1}{{\mathfrak{a}}_2}\left\{\frac{2}{15\Gamma (5/2)}{(1-\zeta )}^{13/12}-\frac{2}{15\Gamma (5/2)}{(1-\zeta )}^{5/2}\right.\hfill \\ +\left.\frac{3}{4\Gamma (46/21)}\left[{\left(\frac{2}{3}\right)}^{25/21}{(1-\zeta )}^{13/12}-{\left(\frac{2}{3}-\zeta \right)}^{25/21}\right]\right\},0\le \zeta \le \frac{2}{3},\hfill \\ {\mathfrak{h}}_2\left(\zeta \right)+\frac{1}{{\mathfrak{a}}_2}\left[\frac{2}{15\Gamma (5/2)}{(1-\zeta )}^{13/12}-\frac{2}{15\Gamma (5/2)}{(1-\zeta )}^{5/2}\right.\hfill \\ +\left.\frac{3}{4\Gamma (46/21)}{\left(\frac{2}{3}\right)}^{25/21}{(1-\zeta )}^{13/12}\right],\frac{2}{3}<\zeta \le 1.\hfill \end{array}\right. \end{gathered}$$

Example 1.

We consider the function

$${\displaystyle f(\eta ,z_1,z_2,z_3)=\frac{{(z_1+3z_2+2z_3)}^{4a/7}}{{\eta }^{{\varsigma }_1}{(1-\eta )}^{{\varsigma }_2}},\eta \in (0,1),z_i\ge 0,i=1,2,3,}$$

(36)

where $a>1$, ${\varsigma }_1\in (0,1)$, ${\varsigma }_2\in (0,3/2)$. Here $f(\eta ,z_1,z_2,z_3)=\psi \left(\eta \right)\chi (\eta ,z_1,z_2,z_3)$, $\eta \in (0,1)$, $z_i\ge 0$, $i=1,2,3$, where $\psi \left(\eta \right)=\frac{1}{{\eta }^{{\varsigma }_1}{(1-\eta )}^{{\varsigma }_2}}$, $\eta \in (0,1)$, $\chi (\eta ,z_1,z_2,z_3)={(z_1+3z_2+2z_3)}^{4a/7}$, $\eta \in [0,1]$, $z_i\ge 0$, $i=1,2,3$. We also find

$${\displaystyle \Lambda ={\int }_0^1{(1-\tau )}^{1/2}\psi \left(\tau \right)d\tau ={\int }_0^1{\tau }^{-{\varsigma }_1}{(1-\tau )}^{1/2-{\varsigma }_2}d\tau =B\left(1-{\varsigma }_1,\frac{3}{2}-{\varsigma }_2\right)\in (0,\infty ).}$$

So assumption $\left(K2\right)$ is also satisfied.

Besides, in $\left(K3\right)$, for $b_1=1$, $b_2=3$, $b_3=2$ and $\sigma =1$, we obtain ${\chi }_0=0$, and in $\left(K4\right)$ for $[{\theta }_1,{\theta }_2]\subset (0,1)$, $c_1=1$, $c_2=3$, $c_3=2$, we have $f_{\infty }=\infty$. Then by Theorem 3.1 we deduce that there exists a positive solution $w\left(\eta \right),\eta \in [0,1]$ of problem (34), (35) with the function f given by (36).

Example 2.

We consider the function

$${\displaystyle f(\eta ,z_1,z_2,z_3)=\frac{l_0(\eta +2)}{({\eta }^2+9)\sqrt[7]{{\eta }^3}}\left[{(z_1+4z_2+7z_3)}^{k_1}+{(z_1+4z_2+7z_3)}^{k_2}\right],}$$

(37)

for $\eta \in (0,1]$, $z_1,z_2,z_3\ge 0$, where $l_0>0$, $k_1>\frac{4}{7}$, $k_2\in \left(0,\frac{4}{7}\right)$. Here we have $\psi \left(\eta \right)=\frac{1}{\sqrt[7]{{\eta }^3}}$, $\eta \in (0,1]$, $\chi (\eta ,z_1,z_2,z_3)=\frac{l_0(\eta +2)}{{\eta }^2+9}\left[{(z_1+4z_2+7z_3)}^{k_1}+{(z_1+4z_2+7z_3)}^{k_2}\right]$, $\eta \in [0,1]$, $z_i\ge 0$, $i=1,2,3$. We find here $\Lambda =B\left(\frac{4}{7},\frac{3}{2}\right)\in (0,\infty )$. Then assumption $\left(K2\right)$ is satisfied.

For $[{\theta }_1,{\theta }_2]\subset (0,1)$, $c_1=1$, $c_2=4$ and $c_3=7$ we obtain $f_{\infty }=\infty$, and for $e_1=1$, $e_2=4$, $e_3=7$ and $\nu \in (7k_2/4,1]$ we find $f_0=\infty$. Then assumptions $\left(K4\right)$ and $\left(K6\right)$ are satisfied. We also deduce $M_0={\int }_0^1{\mathfrak{J}}_1\left(\tau \right)\psi \left(\tau \right)d\tau \approx 1.77981899$, $M_1={\int }_0^1{\mathfrak{J}}_2\left(\tau \right)d\tau \approx 0.14128375$, ${\omega }_0=1$, and ${\Xi }_0=\frac{3l_0}{10}({12}^{k_1}+{12}^{k_2})$. If $l_0<\frac{10}{3({12}^{k_1}+{12}^{k_2})M_0{M}_1^{4/7}}$, then the inequality $M_1{\Xi }_0^{7/4}{M}_0^{7/4}<1$ is satisfied, (that is assumption $\left(K7\right)$ is satisfied). For example, if $k_1=1$, $k_2=1/2$ and $l_0<0.3705$, then the above inequality is satisfied. Therefore by applying Theorem 4 we conclude that there exist two positive solutions $w_1\left(\eta \right),w_2\left(\eta \right),\eta \in [0,1]$ of problem (34), (35) with the nonlinearity f given by (37).

5. Conclusions

In this paper, we proved the existence of at least one or two positive solutions for a Riemann–Liouville fractional differential equation with r-Laplacian operator and a general nonlinearity which is dependent of various fractional order integrals and it is singular at $\eta =0$ and $\eta =1$, supplemented with boundary conditions containing Riemann–Stieltjes integrals and fractional derivatives. We presented the associated Green functions with their properties, and we gave two examples illustrating our main theorems.