Nonlocal boundary value problems of fractional order at resonance with integral conditions
 HaiE Zhang^{1}Email author
https://doi.org/10.1186/s1366201713798
© The Author(s) 2017
Received: 2 February 2017
Accepted: 26 September 2017
Published: 13 October 2017
Abstract
Keywords
MSC
1 Introduction
In recent years, fractional calculus theory has become a popular area of investigation in view of its widespread applications. Furthermore, fractional differential equation, as a branch of fractional calculus, has been a hot area of research of differential equation with not only numerous theoretical developments, but also countless applications to practical problems. For example, in order to describe certain problems raised in science and engineering, the fractional differential equation is superior to the classical integer one, especially in the fields of biology, physics, mechanics, ecological engineering, finance and other fields which propose the process of memory and genetic properties. For more details about fractional differential equations, one can see [1–4].
For the sake of readers, we will concisely list some necessary symbols now.
The proof of our major results will be shown by employing the coincidence degree theory of Mawhin [21], which plays an extremely important role in investigating the existence of various types of resonant problems. Now, we present it here.
Theorem 1.1
([21])
 (i)
\(Lx\neq\lambda Nx\) for each \((x,\lambda)\in[(\operatorname {dom}(L)\backslash \operatorname {Ker}(L))\cap\partial\Omega]\times(0,1)\);
 (ii)
\(Nx\notin \operatorname {Im}(L)\) for each \(x\in \operatorname {Ker}(L)\cap\partial \Omega\);
 (iii)
\(\operatorname {deg}(JQN_{\operatorname {Ker}(L)} ,\operatorname {Ker}(L) \cap\Omega,\theta)\neq0\), where \(Q:Z \rightarrow Z\) is a projection as above with \(\operatorname {Im}(L) = \operatorname {Ker}(Q)\) and \(J:\operatorname {Im}(Q)\rightarrow \operatorname {Ker}(L)\) is any isomorphism.
Then the operator equation \(Lx=Nx\) has at least one solution in \(\operatorname {dom}(L)\cap\overline{\Omega}\).
The remainder of the thesis is organized as follows. Firstly, we list several necessary definitions and lemmas. Secondly, we obtain the solvability for BVP (1.3). Finally, an example is also given to elucidate the major results.
2 Preliminary lemmas
Lemma 2.1
([20])
Now, we present some conclusions owing to Bai [7], which are basal throughout the paper.
Definition 2.2
([7])
Remark 2.3
([7])
Built upon functional analysis theory, it follows that \(C^{\mu}[0, 1]\) is a Banach space endowed with \(\Vert u(t)\Vert _{C^{\mu}}= \Vert D_{0+}^{\mu}u \Vert _{\infty}+\cdots+\Vert D_{0+}^{\mu(N1)}u\Vert _{\infty }+\Vert u\Vert _{\infty}\) as its norm.
Lemma 2.4
([7])
3 Main results
 (a)
\(2<\alpha\leq3\), \(0<\beta\leq1\) are real numbers;
 (b)
\(\int_{0}^{1}t^{\alpha\beta1}\,dA(t)=1\).
Lemma 3.1
Proof
Let \(u(t)=I_{0^{+}}^{\alpha}y(t)\), then \(u\in \operatorname {dom}(L)\) and \(D_{0+} ^{\alpha}u(t)=y(t)\). Therefore, \(y\in \operatorname {Im}(L)\). □
Lemma 3.2
The mapping \(L : \operatorname {dom}(L)\cap Y \rightarrow Z\) is an index zero Fredholm operator.
Proof
Observing that \(y\in \operatorname {Im}(L)\), we can get \(Qy=\theta\), and then \(y\in \operatorname {Ker}(L)\). Otherwise, if \(y\in \operatorname {Ker}(Q)\), we may get that \(Qy=\theta\), i.e., \(y\in \operatorname {Im}(L)\). So, \(\operatorname {Ker}(Q)=\operatorname {Im}(L)\).
Denote \(y \in Z\) in the way of \(y = (y Qy)+Qy\) so that \(y \in Z\), \(Qy \in \operatorname {Im}(L) = \operatorname {Ker}(Q)\) and \(Qy\in \operatorname {Im}(Q)\). Thereby, \(Z = \operatorname {Im}(L)+\operatorname {Im}(Q)\). In addition, make \(y_{0}\in \operatorname {Im}(L)\cap \operatorname {Im}(Q)\) and suppose that \(y_{0}(s) = c\) is not identically zero on \([0, 1]\). Afterwards, because of \(y_{0} \in \operatorname {Im}(L)\), we get \(Q(y_{0})=Q(c)=cQ(1)=0\) by (3.4) and then derive \(c= 0\), which is contradictory. Whereupon, \(\operatorname {Im}(L)\cap \operatorname {Im}(Q) = {0}\); thus \(Z = \operatorname {Im}(L)\oplus \operatorname {Im}(Q)\). Note that \(\dim \operatorname {Ker}(L) = 1 = \operatorname {co}\dim \operatorname {Im}(L)\), that is, L is a Fredholm operator of index zero. □
Lemma 3.3
Proof
Lemma 3.4
Assume that \(f:[0,1]\times\mathbb{R}^{3}\rightarrow\mathbb{R}\) meeting the Carathéodory conditions, then \(K_{P}(IQ)N : Y \rightarrow Y\) is a completely continuous operator.
Proof
It is manifested that \(K_{P}\) is compact by way of Remark 2.3 and Lemma 2.4. Due to the continuity of \(K_{P}\), \(IQ\) and the boundedness of N, the conclusion can be made that this lemma holds. □
Theorem 3.5
 \((H_{1})\) :

There exist four functions a, b, c, r which are continuous on \([0,1]\) such that for all \((x, y, z)\in\mathbb{R}^{3}\),$$ \bigl\vert f(t,x,y,z)\bigr\vert \leq a(t)\vert x\vert + b(t)\vert y \vert + c(t)\vert z\vert + r(t), \quad t\in[0,1]; $$
 \((H_{2})\) :

There exists a constant \(M > 0\) such that for \(u \in \operatorname {dom}(L)\), if \(\vert D_{0+}^{\alpha1}u(t)\vert > M\) for all \(t \in [0, 1]\), then$$ \int_{0}^{1} \biggl( \int_{0}^{1}G(t,s)f\bigl(s,u(s),D_{0+}^{\alpha1}u(s),D _{0+}^{\alpha2}u(s)\bigr)\,ds \biggr) \,dA(t)\neq0; $$
 \((H_{3})\) :

There exists \(M^{*} > 0\) such that for any \(c \in \mathbb{R}\), if \(\vert c\vert > M^{*}\), afterwards eitheror else$$ c \int_{0}^{1} \biggl( \int_{0}^{1}G(t,s)f\bigl(s,cs^{\alpha1},c\Gamma( \alpha)s,c\Gamma(\alpha)\bigr)\,ds \biggr) \,dA(t)< 0; $$$$ c \int_{0}^{1} \biggl( \int_{0}^{1}G(t,s)f\bigl(s,cs^{\alpha1},c\Gamma( \alpha)s,c\Gamma(\alpha)\bigr)\,ds \biggr) \,dA(t)> 0; $$
 \((H_{4})\) :

\(0<\eta\eta_{1}<1\), where \(\eta_{1}=\int_{0}^{1}\vert a(t)\vert \,ds + \int_{0}^{1}\vert b(t)\vert \,ds+ \int_{0}^{1}\vert c(t)\vert \,ds\).
If hypotheses \((H_{1})\)\((H_{4})\) are satisfied, then BVP (1.3) has at least one solution in \(\operatorname {dom}(L)\).
Proof
Denote \(\eta_{2}=\int_{0}^{1}\vert r(s)\vert \,ds\).
Now, the proof will be divided into four steps.
The first step: Deploy \(\Omega_{1}= \{ u\in \operatorname {dom}(L) \operatorname {Ker}(L)\mid Lu= \lambda Nu,\lambda\in[0, 1] \} \) and prove \(\Omega_{1}\) to be a bounded set. Taking \(u\in\Omega_{1}\), then \(u\in \operatorname {dom}(L) \operatorname {Ker}(L)\) and \(Lu=\lambda Nu\), so \(\lambda\neq0\) and \(Nu\in \operatorname {Im}(L)=\operatorname {Ker}(Q)\subset Z\). Accordingly, \(Q(Nu)=\theta\). From \((H_{3})\), we have that \(\vert D_{0+} ^{\alpha1}u(0)\vert \leq M\).
Analogous to the above discussion, we may arrive at \(\Omega^{(2)}_{3}\) is bounded too.
 (i)
\(Lx\neq\lambda Nx\) for every \((x,\lambda)\in[(\operatorname {dom}(L) \operatorname {Ker}(L)) \cap\partial\Omega]\times(0,1)\);
 (ii)
\(Nx\notin \operatorname {Im}(L)\) for every \(x\in \operatorname {Ker}(L)\cap\partial \Omega\).
In sum, all hypotheses of Theorem 1.1 are met. Thereby, BVP (1.3) has at least one solution in \(\operatorname {dom}(L)\cap\overline{\Omega}\). □
4 Example
In this section, an example is given to elucidate the accuracy of the main results.
Declarations
Acknowledgements
The thesis is funded by the Educational Science Research Project of Tangshan University (170322) and the Backbone teachers’ Cultivating Program of Tangshan University. The author would like to thank the anonymous referees very much for helpful comments and suggestions which led to the improvement of presentation and quality of the work.
Authors’ contributions
The main idea of this paper was proposed by HEZ. She prepared the manuscript initially and performed all the steps of the proofs in this research. The author read and approved the final manuscript.
Competing interests
The author declares that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. NorthHolland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006) View ArticleMATHGoogle Scholar
 Miller, KS, Ross, B: An Introduction to the Fractional Calculus and Fractional Differential Equation. Wiley, New York (1993) MATHGoogle Scholar
 Podlubny, I: Fractional Differential Equations, Mathematics in Science and Engineering. Academic Press, New York (1999) MATHGoogle Scholar
 Nonnenmacher, TF, Metzler, R: On the RiemannLiouville fractional calculus and some recent applications. Fractals 3, 557566 (1995) MathSciNetView ArticleMATHGoogle Scholar
 Bai, ZB, Zhang, YH: Solvability of fractional threepoint boundary value problems with nonlinear growth. Appl. Math. Comput. 218(5), 17191725 (2011) MathSciNetMATHGoogle Scholar
 Bai, ZB: On solutions of some fractional mpoint boundary value problems at resonance. Electron. J. Qual. Theory Differ. Equ. 2010, 37 (2010) MathSciNetMATHGoogle Scholar
 Bai, ZB: Solvability for a class of fractional mpoint boundary value problem at resonance. Comput. Math. Appl. 62, 12921302 (2011) MathSciNetView ArticleMATHGoogle Scholar
 Infante, G, Zima, M: Positive solutions of multipoint boundary value problems at resonance. Nonlinear Anal. 69, 24582465 (2008) MathSciNetView ArticleMATHGoogle Scholar
 Jiang, W: The existence of solutions to boundary value problems of fractional differential equations at resonances. Nonlinear Anal. 74, 19871994 (2011) MathSciNetView ArticleMATHGoogle Scholar
 Kosmatov, N: A boundary value problem of fractional order at resonance. Electron. J. Differ. Equ. 135, 1 (2010) MathSciNetMATHGoogle Scholar
 Liang, SQ, Mu, L: Multiplicity of positive solutions for singular threepoint boundary value problem at resonance. Nonlinear Anal. 71, 24972505 (2009) MathSciNetView ArticleMATHGoogle Scholar
 Zhang, HE, Sun, JP: Positive solutions of thirdorder nonlocal boundary value problems at resonance. Bound. Value Probl. 2012, 102 (2012) MathSciNetView ArticleMATHGoogle Scholar
 Zhang, Y, Bai, ZB, Feng, T: Existence results for a coupled system of nonlinear fractional threepoint boundary value problems at resonance. Comput. Math. Appl. 61, 10321047 (2011) MathSciNetView ArticleMATHGoogle Scholar
 Bashir, A, Ntouyas, SK: Nonlocal fractional boundary value problems with slitstrips boundary conditions. Fract. Calc. Appl. Anal. 18, 261280 (2015) MathSciNetMATHGoogle Scholar
 Garout, D, Ahmad, B, Alsaedi, A: Existence theorems for semilinear Caputo fractional differential equations with nonlocal discrete and integral boundary conditions. Fract. Calc. Appl. Anal. 19, 463479 (2016) MathSciNetMATHGoogle Scholar
 Agarwal, RP, Ahmad, B, Garout, D, Alsaedi, A: Existence results for coupled nonlinear fractional differential equations equipped with nonlocal coupled flux and multipoint boundary conditions. Chaos Solitons Fractals (2017). doi:10.1016/j.chaos.2017.03.025 MathSciNetGoogle Scholar
 Webb, JRL, Infante, G: Positive solutions of nonlocal boundary value problems: a unified approach. J. Lond. Math. Soc. (2) 74, 673693 (2006) MathSciNetView ArticleMATHGoogle Scholar
 Webb, JRL, Infante, G: Nonlocal boundary value problems of arbitrary order. J. Lond. Math. Soc. (2) 79, 238258 (2009) MathSciNetView ArticleMATHGoogle Scholar
 Webb, JRL: Positive solutions of some higher order nonlocal boundary value problems. Electron. J. Qual. Theory Differ. Equ. 29, 1 (2009) MathSciNetView ArticleMATHGoogle Scholar
 Zhang, HE: Iterative solutions for fractional nonlocal boundary value problems involving integral conditions. Bound. Value Probl. 2016, 3 (2016) MathSciNetView ArticleMATHGoogle Scholar
 Mawhin, J: Topological Degree Methods in Nonlinear Boundary Value Problems. NSFCBMS Regional Conference Series in Mathematics. Am. Math. Soc., Providence (1979) View ArticleMATHGoogle Scholar