Nontrivial solutions for a higher fractional differential equation with fractional multi-point boundary conditions

Jia Min, Zhang Xinguang, Gu, Gu Xuemai

Description

This paper investigates the existence and uniqueness of nontrivial solutions to a class of fractional nonlocal multi-point boundary value problems of higher order fractional differential equation, this kind of problems arise from viscoelasticity, electrochemistry control, porous media, electromagnetic and signal processing of wireless communication system. Some sufficient conditions for the existence and uniqueness of nontrivial solutions are established under certain suitable growth conditions, our proof is based on Leray-Schauder nonlinear alternative and Schauder fixed point theorem. MSC: 34B15, 34B25.

Sujets

• Fractional calculus
• Green function

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 3
• Langue: English

Extrait

Jia et al.Boundary Value Problems2012,2012:70 http://www.boundaryvalueproblems.com/content/2012/1/70

R E S E A R C HOpen Access Nontrivial solutions for a higher fractional differential equation with fractional multi-point boundary conditions

1* 2*1 Min Jia, Xinguang Zhangand Xuemai Gu

* Correspondence: [e-mail]; [e-mail] 1 Communication Research Center, Harbin Institute of Technology, Harbin 150080, China 2 School of Mathematical and Informational Sciences, Yantai University, Yantai 264005, China

Abstract This paper investigates the existence and uniqueness of nontrivial solutions to a class of fractional nonlocal multi-point boundary value problems of higher order fractional differential equation, this kind of problems arise from viscoelasticity, electrochemistry control, porous media, electromagnetic and signal processing of wireless communication system. Some sufficient conditions for the existence and uniqueness of nontrivial solutions are established under certain suitable growth conditions, our proof is based on Leray-Schauder nonlinear alternative and Schauder fixed point theorem. MSC:34B15; 34B25 Keywords:fractional differential equation; nontrivial solution; Green function; Leray-Schauder nonlinear alternative

1 Introduction The purpose of this paper is to establish the existence and uniqueness of nontrivial solu- tions to the following higher fractional differential equation:    α µµ µ  n– –Dx(t) =f t,x(t),Dx(t),Dx(t), . . . ,Dx(t) , <t< ,   p–  (.) µ µµ i  x() = ,Dx() = ,Dx() =a j Dx(ξ j ), ≤i≤n– ,  j=

wheren≥,n∈N,n– <α≤n,n–l– <α–µ l <n–l, forl= , , . . . ,n– , andµ–µ n– >  p– α–µ– α ,α–µ ≤,α–µ> ,a∈[, +∞),  <ξ<ξ<∙ ∙ ∙<ξ< ,ξ n–j p– j= a j j = ,Dis n the standard Riemann-Liouville derivative, andf: [, ]×R→Ris continuous. Differential equations of fractional order occur more frequently in different research ar- eas such as engineering, physics, chemistry, economics, etc. Indeed, we can find numerous applications in viscoelasticity, electrochemistry control, porous media, electromagnetic and signal processing of wireless communication system, etc. [–]. For an extensive collection of results about this type of equations, we refer the reader to the monograph by Kilbas et al. [], Miller and Ross [], Podlubny [], the papers [–] and the references therein. Recently, Salem [] has investigated the existence of Pseudo solutions for the nonlin- earm-point boundary value problem of a fractional type. In particular, he considered the

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