Abstract
In this work, we deal with the following class of fractional differential equations with fractional derivative boundary conditions:
where \(n\ge 3\), \(n-1< \alpha < n\), \(1 \le \beta \le n-2\), \(D^{\alpha }\) and \(D^{\beta }\) are the standard Riemann-Liouville fractional derivatives and a is a continuous function on [0, 1]. The associated Green’s function is derived in term of a series of functions by the perturbed approach. Sharp estimates on it are established. We give sufficient conditions for existence results by the means of Schauder’s fixed point theorem. Some examples are given to illustrate our results.
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1 Introduction
In this paper, we study the following problem for the Riemann-Liouville fractional differential equation with fractional derivative boundary conditions
where \(n-1< \alpha < n\), \(n>2\), \(1 \le \beta \le n-2\). \(a: [0,1]\longrightarrow {\mathbb {R}}\) be a continuous function. f and w are appropriate functions to be specified later. \(D^{\alpha }\) is the \(\alpha \)-th left Riemann-Liouville fractional derivative. Fractional calculus has been used to model physical and engineering processes, which are found to be best described by fractional differential equations. It is worth noting that the standard mathematical models of integer-order derivatives, including nonlinear models, do not work adequately in many cases. In the recent years, fractional calculus has been applied in various fields such as mechanics, electricity, chemistry, biology, economics. For more details, see [1, 14, 15].
The Green’s function is a fundamental tool for studying the existence of solutions of BVPs of fractional order. By using the theory of the fixed point on cones, the existence of solutions can be shown based on the construction of the mentioned function and its main properties. More precisely, in the literature, the Green’s functions are derived by considering of a linear equation with one term
subject to certain boundary conditions and convert it to an integral equation, which is the aim of many papers; see for example, [2,3,4,5,6, 8, 12]. However, this method is not applicable in the presence of a perturbed term
and also it is not possible to show the positivity of the associated Green’s function. To overcome this difficulty, Graef et al., [7, 9,10,11] proposed new techniques for the construction of the related Green’s function of (2) with various kinds of boundary conditions. By using spectral theory, they obtained the Green’s function as a series of functions.
In [7], Graef et al. considered the boundary value problem consisting of the fractional differential equation
and subject to boundary conditions
where \(2< \alpha < 3\), \(a \in {\mathcal {C}}([0,1])\), \(w \in {\mathcal {C}}([0,1])\) satisfies \(w(t) \ge 0\) a.e. on [0, 1] and \(f\in {\mathcal {C}}({\mathbb {R}},{\mathbb {R}})\). The authors obtained the Green’s function associated to the problem as a series of functions and showed its positivity. And then by deriving certain property of the series, they established the existence and the uniqueness of solutions of the above problem.
Recently, Zou [18] studied the following problem
where \(2< \alpha < 3\), \(a \in {\mathcal {C}}[0,1]\), and \(f\in {\mathcal {C}}([0,1]\times [0,\infty ) ,{\mathbb {R}})\). The author derived new properties of the associated Green’s function than ones given in [7], to obtain the existence of positive solutions.
On the other hand, in [17], Zhen and Wang considered the following fractional differential equations:
where \(a \in {\mathcal {C}}([0,1])\), \(w\in L^1[0,1]\) with \(w(t) \ne 0\) a.e on [0, 1] and \(f \in {\mathcal {C}}({\mathbb {R}}\times [0,1],{\mathbb {R}})\). The authors used the perturbation approach for deriving the associated Green’s function. And so, the existence of solutions is established by the fixed point theorem.
Motivated by the cited papers, in this work, we first improve main properties on the Green’s function \(G_{0}(t,s)\) associated to the following linear BVP
Next, by applying the spectral theory, we derive the Green’s function associated to problem (1) and we establish some estimates. Moreover, we show its positivity which is the main tool to ensure our main result. Finally, we give sufficient conditions on the nonlinearity to obtain the existence of solutions of problem (1).
The paper is developed as follows: In the next section, we get suitable properties on the Green’s function \(G_{0}(t,s)\). After that, we construct the Green’s function G(t, s) as a series of functions and we prove some new estimates on it. Section 3 is devoted to show the existence of nontrivial and positive solutions of problem (1). In the last section, two examples are given to illustrate our results.
2 Preliminaries
In this section, we firstly present some necessary definitions and properties about fractional calculus theory. We refer the reader to [14, 15] for more details.
Definition 1
[14] The Riemann-Liouville fractional integral of order
\( \alpha >0\ \)for a measurable function \(f:(0,+\infty )\rightarrow {\mathbb {R}}\) is defined as
where \(\Gamma \) is the Euler Gamma function, provided that the right-hand side is pointwise defined on\((0,+\infty ).\)
Definition 2
[14] The Riemann-Liouville fractional derivative of order \( \alpha >0\ \) for a measurable function \(f:(0,+\infty )\rightarrow {\mathbb {R}} \) is defined as
provided that the right-hand side is pointwise defined on \( {\mathbb {R}} ^{+}\).
Here \(n=[\alpha ]+1\), \([\alpha ]\) denotes the integer part of the real number \(\alpha \)
Lemma 3
[14] Let \(\alpha >0\). Let \(u\in {\mathcal {C}} (0,1)\cap L^{1}(0,1).\) Then
-
(i)
\(D^{\alpha }I^{\alpha }u=u.\)
-
(ii)
For \(\delta >\alpha -1,\) \(D^{\alpha }t^{\delta }=\dfrac{\Gamma (\delta +1)}{\Gamma (\delta -\alpha +1)}t^{\delta -\alpha }.\) Moreover, we have \(D^{\alpha }t^{\alpha -i}=0,\ i=1,2,\ldots ,n.\)
-
(iii)
\(D^{\alpha }u(t)=0\) if and only if \(u(t)=c_{1}t^{\alpha -1}+c_{2}t^{\alpha -2}+\cdots +c_{n}t^{\alpha -n},\) \(c_{i}\in {\mathbb {R}},\ \mathrm {i=1,2,\ldots ,n.}\)
-
(iv)
Assume that \(D^{\alpha }u\in {\mathcal {C}} (0,1)\cap L^{1}(0,1),\) then we have
$$\begin{aligned} I^{\alpha }D^{\alpha }u(t)=u(t)+c_{1}t^{\alpha -1}+c_{2}t^{\alpha -2}+\cdots +c_{n}t^{\alpha -n}, c_{i}\in {\mathbb {R}} ,\ i=1,2,\ldots ,n. \end{aligned}$$
Lemma 4
-
(i)
Let \(\sigma , \lambda \in (0, \infty )\), \(c, x \in [0,1]\). Then
$$\begin{aligned} \min ( 1, \frac{\sigma }{\lambda }) (1-c x^{\sigma }) \le (1-c x^{\sigma }) \le \max (1, \frac{\sigma }{\lambda }) (1-c x^{\sigma }). \end{aligned}$$ -
(ii)
For \(x,t \in [0,1]\), we have \( xt\le \min (x,t) \le t\).
3 Estimates on the Green’s function
In this section, we shall construct the explicit expression of the Green’s function associated to the following BVP
First, we give the expression of the Green’s function related to problem (3). The proof of the following lemma is essentially given in [13, Theorem 3.1].
Lemma 5
Let \(h \in {\mathcal {C}}([0,1])\), then the fractional boundary value problem
has a unique solution
where for \(t,s \in [0,1]\),
Here, for \(x \in {\mathbb {R}}\), \(x^{+} = max(x,0)\).
Proof
From Lemma 3, the solution of problem (5) satisfy
Since \(u^{(j)}(0)=0, \ \ 0\le j \le n-2\), it is clear that \(c_2=c_3=\cdots =c_n=0\).
Now, we take the R.L fractional derivative of order \(\beta \) of (8) for \(t>0\) we obtain
Then the condition \([D^{\beta } u(t)]_{t=1}=0\) implies that
and so,
As a consequence, the unique solution u of problem (5) is given by
\(\square \)
Next, we give some properties of the Green’s function \(G_{0}\), stated in [13, Theorem 3.2].
Proposition 6
The function \(G_{0}(t,s)\) has the following properties:
-
(i)
\(G_{0}\) is continuous on \([0,1]\times [0,1]\).
-
(ii)
\(G_{0}(t,s)\ge 0\) for \(t,s \in [0,1]\).
-
(iii)
\(\max _{t \in [0,1]} G_{0}(t,s)=G_{0}(1,s)\) for \(s \in [0,1].\)
Proof
It is obvious to see that \(G_{0}(t,s)\) is continuous on \([0,1]\times [0,1]\). Hence, (i) holds.
Let’s prove (iii). If \(0 \le t \le s \le 1\), then we obtain
Thus, \(G_0(t,s)\) is increasing with respect to t. As a direct consequence, we deduce that,
Now, if \(0 \le s \le t \le 1\), then we get
So, to show that \(\frac{\partial G_0(t,s)}{\partial t} \ge 0\), we must prove that
Using the fact that \(\alpha -\beta -1 < \alpha -2 \), we have
Which implies that \( \frac{\partial G_0(t,s)}{\partial t} \ge 0\). As a result, we get that is increasing with respect
Hence, it follows that
To show (ii), let fix \( s \in [0,1]\), we have that \(G_0(0,s)=0.\) And, from (iii), \(G_0(t,s)\) is increasing in t for each \(s \in [0,1]\), we get that \(G_0(t,s) \ge 0\) for each \((t,s) \in [0,1]\times [0,1]\). Therefore, (ii) holds and the proof is complete. \(\square \)
Now, we state new sharp estimates on \(G_{0}\) that will be used later.
Lemma 7
-
(i)
Let \(n\in {\mathbb {N}}\), \(n-1< \alpha <n\) and \(1\le \beta \le n-2\). Define the function H(t, s) on \([0,1]\times [0,1]\) by
$$\begin{aligned} H(t,s)= \frac{1}{\Gamma (\alpha )} t ^{\alpha -2} (1-s)^{\alpha -\beta -1} \min (t,s). \end{aligned}$$(10)Then \(G_{0}\) has the following property
$$\begin{aligned} H(t,s) \le G_{0}(t,s) \le (\alpha -1)\beta H(t,s). \end{aligned}$$(11) -
(ii)
For \(t,s\in [0,1]\), we have
$$\begin{aligned} t^{\alpha -1}\frac{s(1-s)^{\alpha -\beta -1}}{\Gamma (\alpha )} \le G_{0}(t,s) \le (\alpha -1)\beta t^{\alpha -2} \frac{s(1-s)^{\alpha -\beta -1}}{\Gamma (\alpha )} . \end{aligned}$$(12)
Proof
-
(i)
For \(t\in (0,1]\) and \(s \in [0,1)\), we have
$$\begin{aligned} \Gamma (\alpha ) G_{0}(t,s)= t^{\alpha -1}(1-s)^{\alpha -\beta -1} [1- \frac{((t-s)^+)^{\alpha -1}}{t^{\alpha -1}(1-s)^{\alpha -\beta -1}}] \end{aligned}$$From Lemma 4, for \(\lambda =1\), \(\sigma =\alpha -1\), \(c=(1-s)^{\alpha }\) and \(x=\frac{(t-s)^+}{t(1-s)}\), we obtain
$$\begin{aligned}&t^{\alpha -1}(1-s)^{\alpha -\beta -1} (1-(1-s)^{\beta } \frac{(t-s)^+}{t(1-s)}) \\&\quad \le \Gamma (\alpha ) G_{0}(t,s) \le (\alpha -1) t^{\alpha -1}(1-s)^{\alpha -\beta -1} (1-(1-s)^{\beta } \frac{(t-s)^+}{t(1-s)}). \end{aligned}$$Applying again Lemma 4, for \(\lambda =1\), \(\sigma =\beta \ge 1\), \(x =(1-s)\) and \(c=\frac{(t-s)^+}{t(1-s)}\), we get
$$\begin{aligned}&t^{\alpha -1}(1-s)^{\alpha -\beta -1}(1-\frac{(t-s)^+}{t}) \le \Gamma (\alpha ) G_{0}(t,s) \\&\quad \le (\alpha -1) t^{\alpha -1}(1-s)^{\alpha -\beta -1} \beta (1-\frac{(t-s)^+}{t}). \end{aligned}$$By using the fact that \((1-\frac{(t-s)^+}{t})=\frac{1}{t} \min (t,s)\), we conclude that
$$\begin{aligned} H(t,s) \le G_{0}(t,s) \le (\alpha -1) \beta H(t,s). \end{aligned}$$ - (ii)
The proof is completed. \(\square \)
In the next lemma, we derive an important property of Green’s function \(G_{0}\).
Lemma 8
For \(t,\tau ,s \in (0,1)\), we have
Proof
By Lemma 7 (i), for \(t,\tau ,s \in (0,1)\), we obtain
If \(t\le s\). Then
On the other hand, if \(s\le t\) we have
which completes the proof. \(\square \)
Now, let \(G: [0,1]\times [0,1] \longrightarrow {\mathbb {R}}\) be defined by
where \(G_{0}\) is given by (7) and \(G_{k}: [0,1]\times [0,1] \longrightarrow {\mathbb {R}}\),
In order to express the Green’s function associated with the linear problem (4), we shall use the spectral theory in Banach spaces. To this end, we require the following lemma.
Lemma 9
[16] Let X be a Banach space and \( {\mathcal {A}}:X \longrightarrow X\) be a linear operator with the operator norm \(\Vert {\mathcal {A}}\Vert \) and spectral radius \(r( {\mathcal {A}})\) of \( {\mathcal {A}}\). Then
-
(i)
\(r( {\mathcal {A}}) \le \Vert {\mathcal {A}} \Vert \);
-
(ii)
if \( r( {\mathcal {A}}) < 1\), then \(( {\mathcal {I}}- {\mathcal {A}})^{-1}\) exists and \(( {\mathcal {I}}- {\mathcal {A}})^{-1}=\sum \nolimits _{n=0}^{\infty } {\mathcal {A}}^n\), where \( {\mathcal {I}}\) stands for the identity operator.
Let X denotes the Banach space \(({\mathcal {C}}([0,1]), \Vert . \Vert )\) where \(\left\| u \right\| =\max \nolimits _{0\le t\le 1}\left| u(t)\right| \), and let
The next theorem, which is our main contribution in this paper, presents a careful analysis of Green’s function which allows us to deduce the existence results of our problem (1).
Theorem 10
Let \({\overline{a}}=\max _{t\in [0,1]} \vert a(t) \vert < \frac{1}{\sigma }\). Then, G, defined by (14) as a series of functions, is uniformly convergent for \((t,s) \in [0,1]\times [0,1]\) and continuous on \([0,1]\times [0,1]\). Furthermore, G is the Green’s function for the problem
Moreover, we get
Proof
Let \(y \in X\), assume that u is the solution of problem
Then, u satisfies
which implies
Define \( {\mathcal {A}}\) and \( {\mathcal {B}}: X \longrightarrow X\) by
Then, equation (20) becomes
We divide the proof into several steps:
Step 1: Let us verify that \(\Vert {\mathcal {B}} \Vert <1\). For any \(u \in X\) with \(\Vert u \Vert =1\) and \(t\in [0,1]\), by (12), we have
Thus, since \({\overline{a}} < \frac{1}{\sigma }\), we have \(\Vert {\mathcal {B}} \Vert <1.\) And by Lemma 9, we deduce that
Step 2: We shall prove that
It is clear that, for \(k=0\), (25) holds. Assume that (25) holds for \(k=m-1\). Then, by (15), (21), (22), (25) and Fubini’s Theorem, we have
Thus, (25) holds for \(k=m\). And so, (25) holds for any \(k=0,1,2,...\)
Step 3: We show that for \(k=0,1,2,...\)
For \(k=0\), (26) holds. Assume that (26) holds for \(k=m \ge 0\). Then, for \(t,s\in [0,1]\), we have
So, (26) holds for \(k=m+1\). Then, by induction, for any \(k=0,1,2,...\), we have that (26) holds.
Since \(\sigma {\overline{a}}<1\), then, for \((t,s) \in [0,1]\times [0,1]\), we obtain
Thus, G is uniformly convergent on \([0,1]\times [0,1]\), and G satisfies (18).
In addition, from (14), (24) and (25), we get
Step 4: We shall verify that u, defined by (27), is a solution of problem (19). From (14), (21) and (22), we deduce that u satisfies (24). Moreover, by (21) and (22), u satisfies (20).
Therefore, u is a solution of problem (19) and so G is the Green’s function of problem (4). \(\square \)
The following result ensures the positivity of the related Green’s function.
Proposition 11
Let \(\gamma =\frac{((\alpha -1)\beta )^2}{\Gamma (\alpha )}\int _{0}^{1} \tau ^{\alpha -1}(1-\tau )^{\alpha -\beta -1}d\tau .\) Assume that \(\gamma {\overline{a}} < \frac{1}{2}\). Then for \((t,s)\in [0,1]\times [0,1]\), we have
Proof
Since \(\gamma {\overline{a}} < \frac{1}{2}\). Then by Lemma 8, we obtain
And, from the expression of G given by (14), we get
This implies that
So, it follows that \(0 \le G(t,s) \le G_{0}(t,s)\). This completes the proof. \(\square \)
An immediate consequence of Proposition 11 and Lemma 7 (ii) is the following result.
Corollary 12
For \((t,s)\in [0,1]\times [0,1]\), we have
4 Existence results
In this section, using the estimates of G(t, s) derived above, sufficient conditions on the non-linearity f are discussed to guarantee the existence of solutions of problem (1).
Hereinafter, we suppose the following assumption:
-
(H)
\(w: (0,1) \rightarrow {\mathbb {R}}\) such that \(w(t) \not \equiv 0\) a.e on (0, 1) and \(0<\sigma _{w}=\int _{0}^{1} s (1-s)^{\alpha -\beta -1} w(s) ds < \infty .\)
Theorem 13
Assume that (H) holds. In addition if the following conditions hold:
- (C\(_{1}\)):
-
\(f: [0,1] \times {\mathbb {R}} \longrightarrow {\mathbb {R}}\) is a continuous function.
- (C\(_{2}\)):
-
\(f(t,0) \not \equiv 0\) on [0, 1] and
$$\begin{aligned} \lim \limits _{\vert u \vert \rightarrow \infty } \max \limits _{t \in [0,1]} \frac{\vert f(t, u )\vert }{\vert u \vert }=0. \end{aligned}$$
Then Problem (1) has at least one nontrivial solution.
Proof
From (C\(_{2}\)), for \(\varepsilon = \left( \frac{(\alpha -1)\beta }{\Gamma (\alpha )} \sigma _{w} \right) ^{-1} >0\), there exists \(B>0\) such that for each \(t \in [0,1]\) and \(\vert u \vert \ge B\), we have \(\vert f(t, u )\vert \le \varepsilon \vert u \vert .\)
Moreover, by (C\(_{1}\)), there exists \(M>0\) such that
Let \(R= \max \lbrace B, \frac{M}{\varepsilon } \rbrace \). Then
Let
It is clear that \(\Omega \) is a non-empty, convex and closed set.
Define the operator \(T: \Omega \longrightarrow X\) by
where G(t, s) is defined by (14). Obviously, u is a solution of problem (1) if and only if u is a fixed point of T.
Step 1: Let us prove that \(T(\Omega ) \subset \Omega \).
By (C\(_{1}\)) and Theorem 10, we get that \(T: \Omega \longrightarrow X\) is continuous.
Let \(u \in \Omega \), then by Corollary 12 and (31), we have for \(t \in [0,1]\),
Thus, \(\Vert T u \Vert \le R\). And so, \(T(\Omega ) \subset \Omega \).
Step 2: We shall show that T is uniformly bounded.
Let S be bounded set of \( \Omega \), then there exists a positive constant N such that \(\Vert u \Vert = N\), for all \(u \in S\).
Let \(N_{1}=1+ \max \nolimits _{t\in [0,1], u \in [-N,N]} \vert f(t,u)\vert \). Then, by (H), (C\(_{1}\)) and Corollary 12, we obtain for all \(u \in S\) and \(t\in [0,1]\)
Hence, T(S) is uniformly bounded.
Step 3: Let us prove that T(S) is equicontinuous on [0, 1].
Using Theorem 10, we obtain that G is uniformly continuous on \([0,1]\times [0,1]\). Then for \(t_{1}, t_{2} \in [0, 1]\) such that \(t_{1}\le t_{2}\) and for each \(s \in [0, 1]\), we obtain \(\vert G(t_{2},s)-G(t_{1},s)\vert \rightarrow 0\) as \(t_{2} \longrightarrow t_{1}\) and
The Lebesgue control convergence guarantee that T(S) is equicontinuous. Consequently by Ascoli’s theorem, we conclude that T(S) is relatively compact. Therefore, by Schauder fixed point theorem, T has at least one fixed point in \(\Omega \). Hence, problem (1) has a solution u in \(\Omega \). It is clear that \(u(t) \equiv 0\) is not a solution of (1). \(\square \)
Now, we will be concerned with the existence of positive solution to the problem (1) under the following conditions:
- (C\(_{1}^{\prime }\)):
-
\(f: [0,1] \times [0,\infty ) \longrightarrow {\mathbb {R}}\) is a continuous function.
- (C\(_{3}\)):
-
There exist \(d_{2}>d_{1}>0\) such that
$$\begin{aligned} \inf _{u \in P} \int _{0}^{1}s(1-s)^{\alpha -\beta -1} w(s)f(s,u(s)) ds \ge d_{1} \Gamma (\alpha ) \left( 1-\frac{\gamma {\overline{a}}}{1-\gamma {\overline{a}}} \right) ^{-1} \end{aligned}$$(33)and
$$\begin{aligned} \sup _{u\in P} \int _{0}^{1}s(1-s)^{\alpha -\beta -1} w(s)f(s,u(s)) ds \le d_{2} \frac{\Gamma (\alpha )}{(\alpha -1)\beta }, \end{aligned}$$(34)where
$$\begin{aligned} P= \lbrace u \in X: t^{\alpha -1} d_{1} \le u \le d_{2} \rbrace . \end{aligned}$$(35)
Theorem 14
Under assumptions (H), (C\(_{1}^{\ \prime }\)) and (C\(_{3}\)) Problem (1) has at least one positive solution in P.
Proof
Define the operator \(T: P\rightarrow X\) by
It is clear that u is a fixed point of T if and only if u is a solution of problem (1).
Let us prove that \(T(P) \subset P\). Let \(t \in [0,1]\) and \( u \in P\). Then, by Corollary 12, (C\(_{1}^{\ \prime }\)) and (C\(_{3}\)), we have
and
Thus, \(T(P) \subset P\). Using standard arguments, we conclude, by Schauder fixed point theorem, that T has at least one fixed point \(u \in P\). Since \(u(t)>0\), \(t\in [0,1]\), this implies the existence of a positive solution of problem (1) in P. \(\square \)
As a consequence of Theorem 14, we deduce the following corollaries.
Corollary 15
Assume that (H) holds. Moreover, if there exist \(d_{2}>d_{1}>0\) such that for \(t\in [0,1]\), f(t, .) is non-decreasing on \([0,d_{2}]\), satisfying
and
Then problem (1) has at least one positive solution in P.
Corollary 16
Assume that (H) holds. In addition, if there exist \(d_{2}>d_{1}>0\) such that for \(t\in [0,1]\), f(t, .) is non-increasing on \([0,d_{2}]\), satisfying
and
Then problem (1) has at least one positive solution in P.
5 Examples
In this section, we give two examples to illustrate the applicability of the obtained results.
Example 17
Consider the following problem
Set \(a(t)= \frac{-e^{t}}{3}\), \(w(t)=\frac{1}{1-t}\) and \(f(t,u)=\ln (t+2) (u+1) e^{-u}\). It is not difficult to verify that (C\(_{1}\)) and (C\(_{2}\)) are satisfied. By direct calculation, we obtain \({\overline{a}} \simeq 0.906093\), \(\sigma \simeq 0.10746118\), \(\sigma _{w}= \frac{1}{2}\) and \(\gamma \simeq 0.265947\).
Hence, \(\gamma {\overline{a}} \simeq 0.240972 < \frac{1}{2}\) and \({\overline{a}} \sigma \simeq 0.0973698 <1\). From Theorem 13, problem (36) has at least one nontrivial solution.
Example 18
Consider the following problem
where \(a(t)=-\frac{t^2}{2}\), \(w(t)=\frac{1}{t}\) and \(f(t,u)=\ln (\frac{1}{2}+t)\cos (\sqrt{u})\). A simple calculation yields to \({\overline{a}} = \frac{1}{2}\), \(\sigma _{w}=\frac{1}{2}\), \(\sigma \simeq 0.188063\) and \(\gamma \simeq 0.2686619\). So, assumptions (H) and (C\(_{1}^{\ \prime }\)) are satisfied. In addition, \({\overline{a}} < \frac{1}{\sigma } \simeq 5.31735\) and \(\gamma {\overline{a}} \simeq 0.134331 < \frac{1}{2}\).
Thus, for \(d_{1}\) small enough and \(d_{2}\) large enough, hypothesis (C\(_{3}\)) holds and so, by Theorem 14, the problem (37) has at least one positive solution.
References
Baleanu, D., Machado, J.A.T., Luo, A.C.J.: Fractional Dynamics and Control. Springer, New York (2012)
Bai, Z., Lu, H.: Positive solutions for boundary value problem of nonlinear fractional differential equations. J. Math. Anal. Appl. 311, 495–505 (2005)
Benmezai, A., Chentout, S., Henderson, J.: Strongly positive-like operators and eigenvalue criteria for existence and nonexistence of positive solutions for a-order fractional boundary value problems. J. Nonlinear Funct. Anal. (2019), Article ID 24
Belmekki, M., Nieto, J.J., Rodriguez-Lopez, R.: Existence of periodic solution for a nonlinear fractional differential equation. Boundary Value Problems, pp 1–18, (2009)
Bourguiba, R., Toumi, F.: Existence Results of a Singular Fractional Differential Equation with Perturbed Term, Memoirs on Differential Equations and Mathematical Physics, pp. 29–44, (2018)
Toumi, F., Wanassi, O.K.: Existence of positive solutions for a semipositone fractional differential equation with perturbed term, Ricerche di Matematica, (2019), pp. 1–20
Graef, J.R., Kong, L., Kong, Q., Wang, M.: On a fractional boundary value problem with a perturbation term. J. Appl. Anal. Comput. 7(1), 57–66 (2017)
Graef, J.R., Kong, L., Yang, B.: Positive solutions for a fractional boundary value problem. Appl. Math. Lett. 56, 49–55 (2016)
Graef, J.R., Kong, L., Kong, Q., Wang, M.: A fractional boundary value problem with dirichlet boundary condition. Commun. Appl. Anal. 19, 497–504 (2015)
Graef, J.R., Kong, L., Kong, Q., Wang, M.: Fractional boundary value problems with integral boundary conditions. Appl. Anal. Int. J. 92(10), 2008–2020 (2013)
Graef, J.R., Kong, L., Kong, Q., Wang, M.: Existence and uniqueness of solutions for a fractional boundary value problem with Dirichlet boundary condition. Electron. J. Qual. Theor. Differ. Equ. 55, (2013)
Graef, J.R., Kong, L., Yang, B.: Positive solutions for a semipositone fractional boundary value problem with a forcing term. Fract. Calc. Appl. Anal. 15(1), 8–24 (2012)
Goodrich, C.S.: Existence of a positive solution to a class of fractional differential equations. Appl. Math. Lett. 23, 1050–1055 (2010)
Kilbas, A., Srivastava, H., Trujillo, J.: Theory and Applications of Fractional Differential Equation. In: North-Holland Mathematics studies, Vol. 204, Elsevier, Amsterdam (2006)
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Yverdon (1993)
Zeidler, E.: Nonlinear Functional Analysis and Its Applications. I. Fixed-Point Theorems, Translated from the German by Peter R. Wadsack, Springer-Verlag, New York, (1986)
Zhen, G., Wang, M.: Existence of solutions for a nonlocal fractional boundary value problem. Rocky Mountain J. Math. 48, 831–843 (2018)
Zou, Y.: Positive Solutions for a Fractional Boundary Value Problem with a Perturbation Term. Journal of Function Spaces, pp. 1–5. (2018)
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Wanassi, O.K., Toumi, F. Existence results for perturbed boundary value problem with fractional order. Ricerche mat 73, 1367–1383 (2024). https://doi.org/10.1007/s11587-021-00677-y
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DOI: https://doi.org/10.1007/s11587-021-00677-y