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Existence results for fractional order differential equation with nonlocal Erdélyi–Kober and generalized Riemann–Liouville type integral boundary conditions at resonance
Advances in Difference Equations volume 2018, Article number: 243 (2018)
Abstract
In this paper, we discuss a nonlinear fractional order boundary value problem with nonlocal Erdélyi–Kober and generalized Riemann–Liouville type integral boundary conditions. By using Mawhin continuation theorem, we investigate the existence of solutions of this boundary value problem at resonance. It is shown that the boundary value problem
has at least one solution under some suitable conditions, where \(\alpha, \beta\in\mathbb{R}\), \(0<\zeta, \xi<T\).
1 Introduction
In this paper, we intend to discuss the following boundary value problem at resonance:
where \({}^{c}D^{q}\) is the Caputo fractional derivative of order \(1< q\leq2\), \(I^{\gamma,\delta}_{\eta}\) is a Erdélyi–Kober type integral of order \(\delta>0\) with \(\eta>0\) and \(\gamma\in \mathbb{R}\), \({}^{\rho}I^{p}\) denotes the generalized Riemann–Liouville type integral of order \(p>0\), \(\rho>0\), and \(\alpha, \beta\in\mathbb{R}\).
Boundary value problems at resonance have aroused people’s interest these days (see [5, 6, 8, 9, 14, 16, 17, 19–21, 25, 29–33, 35, 40, 41, 43]). For instance, in [17], Jiang and Qiu studied the existence of solutions for the following \((k,n-k)\) conjugate boundary value problem at resonance:
where \(1\leq k\leq n-1\), \(0<\xi_{1}<\xi_{2}<\cdots<\xi_{m}<1\). Integral boundary value problems have also gained many people’s attention and have been applied to many fields, such as physics, chemistry, and engineering, see [11, 13, 22, 29–31, 35]. Besides, the subject of fractional differential equations has attracted much attention, see [1–5, 7, 10, 12, 15, 23, 24, 27, 28, 32–34, 36–39, 42]. For example, in [5], Zhang and Bai investigated the existence of solutions for the following m-point boundary value problems:
by using the coincidence degree theory of Mawhin. Very recently, in [2], the authors considered boundary value problem (1) under the nonresonance condition \(v_{1}v_{4}+v_{2}v_{3}\neq0\). They established the existence and uniqueness results of BVP (1) by using the standard fixed point theorems.
Inspired by the work above, in this paper, we intend to discuss the boundary value problem (1) under the resonance condition \(v_{1}v_{4}+v_{2}v_{3}=0\). We shall study resonant BVP (1) in three different cases of \(\operatorname{dim}\operatorname{ker}L=1\). Different from the above results, the boundary conditions we study are nonlocal Erdélyi–Kober type integral and generalized Riemann–Liouville type integral. To the best of our knowledge, it is innovative to study the boundary value problem with the nonlocal Erdélyi–Kober type integral and generalized Riemann–Liouville type integral by using the method of Mawhin continuation theorem.
The organization of this paper is as follows. In Sect. 2, we provide some definitions, lemmas, and Mawhin continuation theorem which will be used to prove the main results. In Sect. 3, we will give our main results and the proof, some lemmas will also be given to prove the solvability of BVP (1).
2 Preliminaries
Firstly, for the convenience of the reader, we recall some definitions and lemmas.
Definition 2.1
The fractional integral of order q with the lower limit zero for a function f is defined by
provided the right-hand side is point-wise defined on \([0,\infty)\), \(\Gamma(\cdot)\) is the gamma function.
Definition 2.2
([2])
The generalized fractional integral of order \(q>0\) and \(\rho>0\) for a function \(f(t)\) is defined by
provided the right-hand side is point-wise defined on \((0,\infty)\).
Definition 2.3
([2])
The Erdélyi–Kober fractional integral of order \(\delta>0\) with \(\eta>0\) and \(\gamma\in\mathbb{R}\) of a continuous function \(f(t)\) is defined as
provided the right-hand side is point-wise defined on \(\mathbb{R}_{+}\).
Definition 2.4
The Riemann–Liouville fractional derivative of order \(q>0\), \(n-1< q< n\), \(n\in\mathbb{N}\) can be written as
where the function \(f(t)\) has absolutely continuous derivative up to order \((n-1)\).
Definition 2.5
The Caputo derivative of order q for a function \(f:[0,\infty )\rightarrow\mathbb{R}\) is defined as
Lemma 2.1
([18])
Given that \(x\in C^{1}[0, 1]\) with a fractional derivative of order q (\(1< q<2\)) that belongs to \(C(0, 1)\cap L(0, 1)\), then
Lemma 2.2
([2])
Let \(\delta,\eta>0\), \(\gamma,q\in\mathbb{R}\), then we can get
Lemma 2.3
([2])
Let \(q, p>0\), then we have
Definition 2.6
([26])
Assume that X and Y are real Banach spaces, \(L: \operatorname{dom}L \subset X \rightarrow Y\) is a Fredholm operator of index zero if the following conditions hold:
-
(1)
ImL is a closed subspace of Y;
-
(2)
\(\operatorname{dim} \operatorname{Ker}L=\operatorname {co}\operatorname{dim} \operatorname{Im}L<\infty\).
Let X, Y be real Banach spaces, \(L: \operatorname{dom}L\subset X\rightarrow Y\) be a Fredholm operator of index zero, and \(N: X\rightarrow Y\) be a nonlinear continuous map. \(P: X\rightarrow X\), \(Q: Y\rightarrow Y\) are continuous projectors such that
It follows that
is invertible, and the inverse of the mapping is denoted by \(K_{P}\) (generalized inverse operator of L). Let Ω be an open bounded subset of X with \(\operatorname{dom} L\cap\Omega\neq\emptyset\), the mapping \(N: X\rightarrow Y\) will be called L-compact on Ω̅ if \(QN(\overline{\Omega})\) is bounded and \(K_{P}(I-Q)N: \overline{\Omega}\rightarrow X\) is compact.
Theorem 2.1
(Mawhin continuation theorem [26])
Let \(L: \operatorname{dom}L \subset X \rightarrow Y\) be a Fredholm operator of index zero and N be L-compact on Ω̅. The equation \(L\varphi=N\varphi\) has at least one solution in \(\operatorname{dom}L \cap\overline{\Omega}\) if the following conditions are satisfied:
-
(1)
\(Lx\neq\lambda Nx\) for every \((x,\lambda) \in[(\operatorname {dom}L\setminus\operatorname{ker}L)\cap\partial\Omega] \times(0,1)\);
-
(2)
\(Nx \notin\operatorname{Im}L\) for every \(x\in\operatorname {ker}L \cap\partial\Omega\);
-
(3)
\(\operatorname{deg} (QN|_{\operatorname{ker}L},\Omega\cap\operatorname {ker}L,0 )\neq0\), where \(Q:Y\rightarrow Y\) is a projection such that \(\operatorname{Im}L =\operatorname{ker}Q\).
Let \(Y=C[0,T]\) with the norm \(\|x\|_{\infty}=\max_{t\in[0,1]}|x(t)|\) and \(X=C^{1}[0,T]\) with the norm \(\|x\|=\max\{\|x\|_{\infty},\|x'\|_{\infty}\}\). Obviously, X and Y are Banach spaces.
An operator L is defined as \(L: Lx(t)={}^{c}D^{q}x(t)\) with
Define the operator \(N:X\rightarrow Y\) as follows:
So problem (1) becomes \(Lx=Nx\).
Let
then we consider the following three resonant conditions:
- \((A1)\) :
-
\(v_{1}=v_{3}=0\), \(v_{2}\neq0\), \(v_{4}\neq0\);
- \((A2)\) :
-
\(v_{2}=v_{4}=0\), \(v_{1}\neq0\), \(v_{3}\neq0\);
- \((A3)\) :
-
\(v_{i}\neq0\) (\(i=1,2,3,4\)), \(v_{1}v_{4}+v_{2}v_{3}=0\).
Lemma 2.4
Assume that \((A1)\) holds. Then there exists \(z\in Y\) such that
Proof
We define two linear functionals \(B_{1}, B_{2}: X\rightarrow \mathbb{R}\) as follows:
Let \(\varphi(t)=1\), \(\psi(t)=t\). It follows from \((A1)\) and Lemmas 2.2 and 2.3 that
So, (2) can be rewritten by
For convenience, set
If there is \(\widetilde{z}\in Y\) such that \(B\widetilde{z}\neq0\) and, as a result, \(z=\frac{\widetilde{z}}{B\widetilde{z}}\in Y\) with \(Bz=1\). Assume the contrary. Then \(B(J^{q}z)=0\) for all \(z\in Y\), and, in particular, for integer \(n\geq2\),
By (3), \(B(1)=B(t)=0\). Therefore, \(B(g)=0\) for every polynomial g. Note that \(B \neq0\) on all of X, there exists \(x_{0}\in X\) such that \(Bx_{0}\neq0\). Thus, there exists a sequence of polynomials \(g_{n}\) such that \(\|x_{0}-g_{n}\|_{X}<\frac{1}{n}\). So, we deduce that
for all integer n, which is a contradiction. Thus, there exists \(z\in Y\) satisfying (2). Thus the lemma holds. □
Similar to the proof of Lemma 2.4, we also get the following lemmas.
Lemma 2.5
Assume that \((A2)\) holds. Then there exists \(z_{1}\in Y\) such that
Lemma 2.6
Assume that \((A3)\) holds. Then there exists \(z_{2}\in Y\) such that
Remark 2.1
The main idea of Lemmas 2.4, 2.5, and 2.6 comes from [16, 19, 20].
3 Main results
Assume that the following conditions hold in this paper:
- (H1):
-
\(f:[0,1]\times\mathbb{R}^{2}\rightarrow\mathbb{R}\) is a continuous function.
- (H2):
-
There exist nonnegative functions \(u, v, w\in C[0,T]\) such that
$$\bigl\vert f (t,x_{1},x_{2} ) \bigr\vert \leq u(t) \vert x_{1} \vert +v(t) \vert x_{2} \vert +w(t),\quad t \in[0,T], x_{1},x_{2}\in \mathbb{R}. $$ - (H3):
-
There exists a constant \(M>0\) such that if \(|x(t)|+|x'(t)|>M\) for all \(t\in[0,T]\), then
$$v_{2}\bigl(\beta{}^{\rho}I^{p}J^{q}Nx( \xi)-J^{q}Nx(T)\bigr)+ \alpha v_{4} I^{\gamma,\delta}_{\eta}J^{q}Nx( \zeta)\neq0. $$ - (H3′):
-
There exists a constant \(M>0\) such that if \(|x'(t)|>M\) for all \(t\in[0,T]\), then
$$\beta{}^{\rho}I^{p}J^{q}Nx(\xi)-J^{q}Nx(T) \neq0. $$ - (H4):
-
There is a constant \(D>0\) such that either
$$ c v_{2}\bigl(\beta{}^{\rho}I^{p}J^{q}N \phi_{1}(\xi)-J^{q}N\phi_{1}(T)\bigr)+ c\alpha v_{4} I^{\gamma,\delta}_{\eta}J^{q}N \phi_{1}(\zeta)>0 $$(5)or
$$ c v_{2}\bigl(\beta{}^{\rho}I^{p}J^{q}N \phi_{1}(\xi)-J^{q}N\phi_{1}(T)\bigr)+ c\alpha v_{4} I^{\gamma,\delta}_{\eta}J^{q}N \phi_{1}(\zeta)< 0 $$(6)holds if \(|c|>D\), where \(\phi_{1}(t)=c\).
- (H4′):
-
There is a constant \(D>0\) such that either
$$c \beta{}^{\rho}I^{p}J^{q}N\phi_{2}( \xi)-cJ^{q}N\phi_{2}(T)>0 $$or
$$c \beta{}^{\rho}I^{p}J^{q}N\phi_{2}( \xi)-cJ^{q}N\phi_{2}(T)< 0 $$holds if \(|c|>D\), where \(\phi_{2}(t)=ct\).
- (H4″):
-
There is a constant \(D>0\) such that either
$$c v_{2}\bigl(\beta{}^{\rho}I^{p}J^{q}N \phi_{3}(\xi)-J^{q}N\phi_{3}(T)\bigr)+ c\alpha v_{4} I^{\gamma,\delta}_{\eta}J^{q}N \phi_{3}(\zeta)>0 $$or
$$c v_{2}\bigl(\beta{}^{\rho}I^{p}J^{q}N \phi_{3}(\xi)-J^{q}N\phi_{3}(T)\bigr)+ c\alpha v_{4} I^{\gamma,\delta}_{\eta}J^{q}N \phi_{3}(\zeta)< 0 $$holds if \(|c|>D\), where \(\phi_{3}(t)=c(1+kt)\), \(k=\frac{v_{1}}{v_{2}}\).
Then we can present the following theorem.
Theorem 3.1
Suppose that \((A1)\) and (H1)–(H4) are satisfied, then there must be at least one solution of problem (1) in X provided that \(2T^{q}\|u\|_{\infty}+2T^{q-1}\|v\| _{\infty}<\Gamma(q)\).
To prove the theorem, we need the following lemmas.
Lemma 3.1
Assume that \((A1)\) holds, then \(L:\operatorname{dom}L \subset X\rightarrow Y\) is a Fredholm operator with index zero. And a linear continuous projector \(P:X\rightarrow X \) can be defined by
Furthermore, define the linear operator \(K_{p}:\operatorname {Im}L\rightarrow\operatorname{dom}L\cap\operatorname{ker}P\) as follows:
such that \(K_{p}=(L|_{\operatorname{dom}L\cap\operatorname{ker}P})^{-1}\).
Proof
Let \(\varphi(t)=1\), \(\psi(t)=t\). From \((A1)\) and Lemma 2.4, we can easily get
Moreover, we can obtain that
On the one hand, suppose \(y\in\operatorname{Im}L\), then there exists \(x \in\operatorname{dom}L\) such that
Then we have
where \(c_{0}, c_{1}\in\mathbb{R}\). Furthermore, for \(x \in\operatorname{dom}L\),
and
The above two equalities imply that
Using (3) and (7), we get the system
From this together with the second boundary value condition of (1), we can get
By using the eliminated element method, equalities (8) and (9) are changed into the equality
So we obtain that
On the other hand, if \(y\in Y\) satisfies \(v_{2}(\beta{}^{\rho }I^{p}J^{q}y(\xi)-J^{q}y(T))+ \alpha v_{4} I^{\gamma,\delta}_{\eta}J^{q}y(\zeta)=0\), we let
Then we conclude that
and
Besides,
therefore
That is, \(x \in\operatorname{dom}L\), then \(y\in\operatorname{Im}L\). In conclusion,
We define the linear operator \(P:X\rightarrow X\) as
It is obvious that \(P^{2}x=Px\) and \(\operatorname{Im}P=\operatorname {ker}L\). For any \(x\in X\), together with \(x=(x-Px)+Px\), we have \(X=\operatorname {ker}P+\operatorname{ker}L\). It is easy to obtain that \(\operatorname{ker}L\cap\operatorname{ker}P=\emptyset\), which implies
Next the operator \(Q:Y\rightarrow Y\) is defined as follows:
where B is given by (4) and \(z\in Y\) satisfying \(B(J^{q}z)=1\).
Obviously, Q is a projection operator such that \(\operatorname {ker}Q=\operatorname{Im}L\) and \(\operatorname{Im} L=\{cz(t): c\in \mathbb{R}\}\). For any \(y\in Y\), because \(y=(y-Qy)+Qy\), we have \(Y=\operatorname{Im}L+ \operatorname{Im}Q\). Moreover, by a simple calculation, we can get \(\operatorname{Im}Q\cap\operatorname {Im}L=\emptyset\). Above all, \(Y=\operatorname{Im}L\oplus\operatorname{Im}Q\).
To sum up, we can get that ImL is a closed subspace of Y; \(\operatorname{dim}\operatorname{ker}L=\operatorname{co} \operatorname{dim}\operatorname{Im}L<+\infty\); that is, L is a Fredholm operator of index zero.
We now define the operator \(K_{p}y:Y\rightarrow X\) as follows:
For any \(y\in\operatorname{Im}L\), we have
then \((K_{p}y)(t)\in\operatorname{dom}L\). In addition, \((K_{p}y)(0)=0\), which means \(K_{p}y\in \operatorname{ker}P\). Therefore
Next we will prove that \(K_{p}\) is the inverse of \(L|_{\operatorname {dom}L\cap\operatorname{ker}P}\). It is clear that
By Lemma 2.1, for each \(x\in\operatorname{dom}L\cap\operatorname {ker}P\), we have \(x(0)=0\) and
This implies that \(K_{p}Lx=x\). So \(K_{p}=(L|_{\operatorname{dom}L\cap \operatorname{ker}P})^{-1}\). Thus the lemma holds. □
Lemma 3.2
N is L-compact on Ω̅ if \(\operatorname{dom}L\cap \overline{\Omega}\neq\emptyset\), where Ω is a bounded open subset of X.
Proof
It follows from the continuity of f in condition (H1) and \(z\in Y\) that \((I-Q)N(\overline{\Omega})\) is bounded. In addition,
and
are equi-continuous and uniformly bounded. By Ascoli–Arzela theorem, we get \(K_{p}(I-Q)N:\overline{\Omega}\rightarrow X\) is compact. Thus, N is L-compact. The proof is completed. □
Lemma 3.3
The set \(\Omega_{1}=\{x \in\operatorname{dom}L \setminus \operatorname{ker} L:Lx=\lambda Nx, \lambda\in[0,1]\}\) is bounded if (H1)–(H3) are satisfied.
Proof
Take \(x \in\Omega_{1}\), then \(x\notin\operatorname{ker}L\), so \(\lambda\neq0\) and \(Nx\in\operatorname{Im}L\). Thus we have
where \(z\in Y\) satisfying \(B(J^{q}z)=1\). So we get
According to (H3), there exists at least a point \(t_{0}\in[0,T]\) such that
Using the Newton–Leibnitz formula, we have
In addition, for \(Lx=\lambda Nx\) and \(x\in\operatorname{dom}L\), we have
and
Take \(t=t_{0}\) in (12), we get
This together with \(|x'(t_{0})|\leq M\) and (11) implies that
Then we conclude that
Therefore, we can obtain that
Combining this with (11), we have
Then \(\Omega_{1}\) is bounded. The proof of the lemma is completed. □
Lemma 3.4
The set \(\Omega_{2}=\{x:x\in\operatorname{ker}L, Nx\in\operatorname{Im}L\}\) is bounded if (H1), (H4) hold.
Proof
Let \(x\in\Omega_{2}\), then \(x(t)\equiv c\) and \(Nx\in\operatorname{Im}L\), so we can get
According to (H4), we have \(|c|\leq D\), that is to say, \(\Omega_{2}\) is bounded. We complete the proof. □
Lemma 3.5
The set \(\Omega_{3}=\{x\in\operatorname {ker}L:\lambda x+\alpha(1-\lambda)JQNx=0, \lambda\in[0,1]\}\) is bounded if conditions (H1), (H4) are satisfied, where \(J :\operatorname{Im}Q\rightarrow\operatorname{ker}L\) is a linear isomorphism defined by
and
where \(z_{1}\) is introduced in Lemma 2.4.
Proof
Suppose that \(x\in\Omega_{3}\), we have \(x(t)=c\) and
thus we have
If \(\lambda=0\), by condition (H4) we have \(|c|\leq D\). If \(\lambda=1\), then \(c=0\). If \(\lambda\in(0,1)\), we suppose \(|c|> D\), then
which contradicts with \(\lambda c^{2}>0\), so \(|c|\leq D\). Then the lemma holds. □
Theorem 3.1 can be proved now.
Proof of Theorem 3.1
Suppose that \(\Omega\supset \bigcup_{i=1}^{3}\overline{\Omega_{i}}\cup\{0\}\) is a bounded open subset of X, from Lemma 3.2 we know that N is L-compact on Ω̅. In view of Lemmas 3.3 and 3.4, we can get:
-
(i)
\(Lx\neq\lambda Nx\) for every \((x, \lambda) \in [(\operatorname{dom}L \setminus\operatorname{ker}L) \cap\partial \Omega] \times(0,1)\);
-
(ii)
\(Nx \notin\operatorname{Im}L\) for every \(x\in \operatorname{ker}L \cap \partial\Omega\).
Set \(H(x,\lambda)=\lambda Jx+\alpha(1-\lambda)QNx\). It follows from Lemma 3.5 that we have \(H(x,\lambda)\neq0\) for any \(x\in \partial\Omega\cap\operatorname{ker}L\). So, by the homotopic property of degree, we have
All the conditions of Theorem 2.1 are satisfied. So there must be at least one solution of problem (1) in X. The proof of Theorem 3.1 is completed. □
Theorem 3.2
Suppose that \((A2)\) and (H1), (H2), (H3′), (H4′) are satisfied, then there must be at least one solution of problem (1) in X provided that \(2T^{q}\|u\|_{\infty}+2T^{q-1}\|v\| _{\infty}<\Gamma(q)\).
To prove the theorem, we need the following lemmas.
Lemma 3.6
Assume that \((A2)\) holds, then \(L:\operatorname{dom}L \subset X\rightarrow Y\) is a Fredholm operator with index zero. And the linear continuous projector \(P:X\rightarrow X \) can be defined by
Furthermore, define the linear operator \(K_{p}:\operatorname {Im}L\rightarrow\operatorname{dom}L\cap\operatorname{ker}P\) as follows:
such that \(K_{p}=(L|_{\operatorname{dom}L\cap\operatorname{ker}P})^{-1}\).
Proof
Let \(\varphi(t)=1\), \(\psi(t)=t\). In view of \((A2)\) we know
and we get
and
Besides, operators \(P:X\rightarrow X\), \(Q:Y\rightarrow Y\) can be defined as follows:
and
where \(z_{1}\in Y\) satisfying \(J^{q}z_{1}(T)-\beta{}^{\rho }I^{p}J^{q}z_{1}(\xi)=1\). In addition, for each \(x\in\operatorname{dom}L\cap\operatorname {ker}P\), we have \(x(0)=0\), \(x(T)=0\), then we get the generalized inverse operator of L as follows:
The detailed proof of Lemma 3.6 is similar to that of Lemma 3.1, so we omit it. □
Proof of Theorem 3.2
The proof of Theorem 3.2 is similar to that of Theorem 3.1, we omit it. □
Theorem 3.3
Suppose that \((A3)\) and (H1), (H2), (H3), and (H4″) are satisfied, then there must be at least one solution of problem (1) in X provided that \(2T^{q}\|u\|_{\infty}+2T^{q-1}\|v\| _{\infty}<\Gamma(q)\).
To prove the theorem, we need the following lemmas.
Lemma 3.7
Assume that \((A3)\) holds, then \(L:\operatorname{dom}L \subset X\rightarrow Y\) is a Fredholm operator with index zero. And a linear continuous projector \(P:X\rightarrow X \) can be defined by
where \(k=\frac{v_{1}}{v_{2}}=-\frac{v_{3}}{v_{4}}\). Furthermore, define the linear operator \(K_{p}y:\operatorname{Im}L\rightarrow\operatorname {dom}L\cap\operatorname{ker}P\) as follows:
such that \(K_{p}=(L|_{\operatorname{dom}L\cap\operatorname{ker}P})^{-1}\).
Proof
Let \(\varphi_{1}(t)=1+kt\). In view of \((A3)\) we know
we can easily get
Moreover, we can obtain that
We define the linear operator \(P:X\rightarrow X\) as
and the operator \(Q:Y\rightarrow Y\) as
where \(z_{2}\in Y\) satisfying \(v_{2}(\beta{}^{\rho}I^{p}J^{q}z_{2}(\xi )-J^{q}z_{2}(T))+\alpha v_{4}I^{\gamma,\delta}_{\eta}J^{q}z_{2}(\zeta)=1\).
In addition, for each \(x\in\operatorname{dom}L\cap\operatorname {ker}P\), we have \(x(0)=0\), then we get the generalized inverse operator of L as follows:
The detailed proof of Lemma 3.7 is similar to that of Lemma 3.1, so we omit it. □
Proof of Theorem 3.3
The proof of Theorem 3.3 is similar to that of Theorem 3.1, we omit it. □
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Acknowledgements
We would like to thank the referees very much for their valuable suggestions to improve this paper.
Funding
This work was partially supported by the Natural Science Foundation of China (11371221, 11571207, 51774197), the Shandong Natural Science Foundation (ZR2018MA011), and the Tai’shan Scholar Engineering Construction Fund of Shandong Province of China.
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Sun, Q., Meng, S. & Cui, Y. Existence results for fractional order differential equation with nonlocal Erdélyi–Kober and generalized Riemann–Liouville type integral boundary conditions at resonance. Adv Differ Equ 2018, 243 (2018). https://doi.org/10.1186/s13662-018-1668-x
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DOI: https://doi.org/10.1186/s13662-018-1668-x
Keywords
- Boundary value problem
- Resonance
- Integral conditions