Solvability for a coupled system of fractional differential equations with impulses at resonance

Zhang et al. Boundary Value Problems 2013, 2013:80 http://www.boundaryvalueproblems.com/content/2013/1/80 RESEARCH Open Access Solvability for a coupled system of fractional differential equations with impulses at resonance Xiaozhi Zhang, Chuanxi Zhu* and Zhaoqi Wu * Correspondence: Department of Mathematics, Nanchang University, Nanchang, 330031, P.R. China Abstract In this paper, some Banach spaces are introduced. Based on these spaces and the coincidence degree theory, a 2m-point boundary value problem for a coupled system of impulsive fractional differential equations at resonance is considered, and the new criterion on existence is obtained. Finally, an example is also given to illustrate the availability of our main results. MSC: 34A08; 34B10; 34B37 Keywords: coupled system; impulsive fractional differential equations; at resonance; coincidence degree 1 Introduction Recently, Wang et al. [] presented a counterexample to show an error formula of solutions to the traditional boundary value problem for impulsive differential equations with fractional derivative in [–]. Meanwhile, they introduced the correct formula of solutions for an impulsive Cauchy problem with the Caputo fractional derivative. Shortly afterwards, many works on the better formula of solutions to the Cauchy problem for impulsive fractional differential equations have been reported by Li et al. [], Wang et al. [], Fečkan [], etc. Fractional differential equations have been paid much attention to in recent years due to their wide applications such as nonlinear oscillations of earthquakes, Nutting’s law, charge transport in amorphous semiconductors, fluid dynamic traffic model, non-Markovian diffusion process with memory etc. [–]. For more details, see the monographs of Hilfer [], Miller and Ross [], Podlubny [], Lakshmikantham et al. [], Samko et al. [], and the papers of [, –] and the references therein. In recent years, many researchers paid much attention to the coupled system of fractional differential equations due to its applications in different fields [–]. Zhang et al. [] investigated a three-point boundary value problem at resonance for a coupled system of nonlinear fractional differential equations given by ⎧ β– α ⎪ ⎪ ⎨D+ u(t) = f (t, v(t), D+ v(t)), β D v(t) = g(t, u(t), Dα– + u(t)), ⎪ + ⎪ ⎩ u() = v() = ,  < t < ;  < t < ; u() = σ u(η ), v() = σ v(η ), © 2013 Zhang et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Zhang et al. Boundary Value Problems 2013, 2013:80 http://www.boundaryvalueproblems.com/content/2013/1/80 Page 2 of 23 β– where  < α, β ≤ ,  < η , η < , σ , σ > , σ ηα– = σ η = , Dα+ is the standard Riemann-Liouville fractional derivative and f , g : [, ] × R → R are continuous. And Wang et al. [] considered a m-point boundary value problem (BVP) at resonance for a coupled system as follows: ⎧ β– β– α ⎪ ⎪ ⎪D+ u(t) = f (t, v(t), D+ v(t), D+ v(t)),  < t < ; ⎪ ⎪ ⎨Dβ+ v(t) = g(t, u(t), Dα– α–  < t < ; + u(t), D+ u(t)),    m –α α– α– ⎪ ⎪ I+ u() = , D+ u() = i= ai D+ u(ξi ), u() = m ⎪ i= bi u(ηi ); ⎪ ⎪ m m ⎩ –β β– β– I+ v() = , D+ v() = i= ci D+ v(γi ), v() = i= di v(δi ), where  < α, β ≤ . With the help of the coincidence degree theory, many existence results have been given in the above literatures. It is worth mentioning that the orders of derivative in the nonlinear function on the right-hand of equal signs are all fixed in the above works, but the opposite case is more difficult and complicated, then this work attempts to deal exactly with this case. What is more, this case of arbitrary order derivative included in the nonlinear functions is very important in many aspects [, ]. There are significant developments in the theory of impulses especially in the area of impulsive differential equations with fixed moments, which provided a natural description of observed evolution processes, regarding as important tools for better understanding several real word phenomena in applied sciences [, , –]. In addition, motivated by the better formula of solutions cited by the work of Zhou et al. [, , ], the aim of this work is to discuss a boundary value problem for a coupled system of impulsive fractional differential equation. Exactly, this paper deals with the m-point boundary value problem of the following coupled system of impulsive fractional differential equations at resonance: ⎧ p ⎪ Dα+ u(t) = f (t, v(t), D+ v(t)), ⎪ ⎪ ⎪ ⎪ ⎪ u(t ) = A (v(t ), Dp v(t )), ⎪ ⎪ i i i i + ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ i = , , . . . , k; q v(ti ) = Ci (u(ti ), D+ u(ti )), ⎪ ⎪ ⎪ ⎪ i = , , . . . , k; ⎪ ⎪ ⎪ ⎪ m ⎪ α– ⎪ Dα– ⎪ i= ai D+ u(ξi ), + u() = ⎪ ⎪ ⎪  ⎩Dβ– v() = m c Dβ– v(ζ ), i i= i + + q β D+ v(t) = g(t, u(t), D+ u(t)),  < t < ; q p D+ u(ti ) = Bi (v(ti ), D+ v(ti )), p q D+ v(ti ) = Di (u(ti ), D+ u(ti )), (.)  –α u() = m u(ηi ); i= bi ηi m –β v() = i= di θi v(θi ), where  < α, β < , α – q ≥ , β – p ≥  and  < ξ < ξ < · · · < ξm < ,  < η < η < · · · < ηm < ,  < ζ < ζ < · · · < ζm < ,  < θ < θ < · · · < θm < . f , g : [, ] × R → R satisfy Carathéodory conditions, Ai , Bi , Ci , Di : R × R → R. w(ti ) = w(ti+ ) – w(ti– ), Dr+ w(ti ) = + + r r – D+ w(ti ) – D+ w(ti ), here w ∈ {u, v}, r ∈ {p, q}, w(ti ) and w(ti– ) denote the right and left limits of w(t) at t = ti , respectively, and the fractional derivative is understood in the Riemann-Liouville sense. k, m, ai , bi , ci , di (i = , , . . . , m) are fixed constant satisfying m m m m m m i= ai = i= bi = i= ci = i= di =  and i= bi ηi = i= di θi = . Zhang et al. Boundary Value Problems 2013, 2013:80 http://www.boundaryvalueproblems.com/content/2013/1/80 Page 3 of 23 The coupled system (.) happens to be at resonance in the sense that the associated linear homogeneous coupled system ⎧ β α ⎪ ⎪ ⎨D+ u(t) = , D+ v(t) = , m α– Dα– i= ai D+ u(ξi ), + u() = ⎪ ⎪ m ⎩ β– β– D+ v() = i= ci D+ v(ζi ),  < t < ;  –α u() = m u(ηi ); i= bi ηi m –β v() = i= di θi v(θi ) has (u(t), v(t)) = (h t α– + h t α– , h t β– + h t β– ), ci ∈ R, i = , , ,  as a nontrivial solution. To solve this interesting and important problem and to overcome the difficulties caused by the impulses, we will construct some Banach spaces, then we shall obtain the new solvability results for the coupled system (.) with the help of a coincidence degree continuation theorem. The main contributions of this work are Lemma . and Lemma . in Section  since the calculations are disposed well. The plan of this work is organized as follows. Section  contains some necessary notations, definitions and lemmas that wi (...truncated)