一类非线性项包含导数的p-Laplacian边值问题对称解的存在性
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摘要
研究了一维p-Laplacian动力方程{(φ_p(u′(t))′+h(t)f(t,u(t),u′(t))=0,u(0)=u(1)=ω,u′(0)=-u′(1),t∈[0,1]两点边值问题对称正解的存在性.利用锥压缩和锥拉伸不动点定理,得到了该边值问题一个对称正解的存在性定理.
In this paper,we study the following dynamic equation for the two-point BVPs with p-Laplacian in the form of {(φ_p(u′(t))′+h(t)f(t,u(t),u′(t))=0, u(0)=u(1)=ω,u′(0)=-u′(1),t∈[0,1].The existence of at least one positive symmetric solution is obtained by using the fixed point theorem of cone compression and expansion.
In this paper,we study the following dynamic equation for the two-point BVPs with p-Laplacian in the form of {(φ_p(u′(t))′+h(t)f(t,u(t),u′(t))=0, u(0)=u(1)=ω,u′(0)=-u′(1),t∈[0,1].The existence of at least one positive symmetric solution is obtained by using the fixed point theorem of cone compression and expansion.
引文
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[4]孙伟平,葛渭高.一类非线性边值问题正解的存在性[J].数学学报,2001,44(4):577-580.doi:10.3321/j.issn:0583-1431.2001.04.001SUN Weiping,GE Weigao.The Existence of Positive Solutions for a Class of Nonlinear Boundary Value Problems[J].Acta Mathematica Sinica,2001,44(4):577-580.
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[7]ZHU Y,ZHU J.Existence of Multiple Positive Solutions for Nth-order p-Laplacian M-Point Singular Boundary Value Problems[J].J Appl Math Comput,2010,34:393-405.doi:10.1007/s12190-009-0329-3
[8]AVERY R I,CHYAN C J,HENDERSIN J.Twin Positive Solutions of Boundary Value Problem for Ordinary Differential Equations and Finite Difference Equations[J].Comput Math Appl,2001,42:695-704.doi:10.1016/S0898-1221(01)00188-2
[9]MA R,ZHU B.Existence of Positive Solutions for a Semipositone Boundary Value Problem on the Half-line[J].Comput Math Appl,2009,58:1672-1686.doi:10.1016/j.camwa.2009.07.005
[10]ZHAO J,MIAO C,GE W,et al.Multiple Symmetric Positive Solutions to a New Kind of Four Point Boundary Value Problem[J].Nonlinear Anal,2009,71:9-18.doi:10.1016/j.na.2008.10.026
[11]陈艳丽.二阶非线性积分-微分方程边值问题的正解[J].河北师范大学学报(自然科学版),2014,38(5):445-450.doi:10.11826/j.issn.1000-5854.2014.05.003CHEN Yanli.Positive Solutions of Nonlinear Second-order Boundary Value Problems for Integro-differential Equations[J].Journal of Hebei Normal University(Natural Science Edition),2014,38(5):445-450.
[12]袁晓红,周德高,许方,等.非线性项带导数的p-Laplacian边值问题解的存在性[J].徐州工程学院学报(自然科学版),2010,25(1):1-5.doi:10.3969/j.issn.1674-358X.2010.01.001YUAN Xiaohong,ZHOU Degao,XU Fang,et al.Existence of Solution of BVPs for p-Laplacian Dynamic Equations Involving Derivative[J].Journal of Xuzhou Institute of Technology(Natural Science Edition),2010,25(1):1-5.
[13]薛益民,苏莹,胡飞.非线性项包含导数的边值问题的一个对称解[J].合肥工业大学学报(自然科学版),2016,39(5):716-720.doi:10.3969/j.issn.1003-5060.2016.05.027XUE Yimin,SU Ying,HU Fei.Existence of Symmetric Solutions to Boundary Value Problem with Nonlinear Term Involving Derivative[J].Journal of Hefei University of Technology(Natural Science Edition),2016,39(5):716-720.
[14]ZHAO X,HU W M,JIANG D Q.Existence Theory for Single and Multiple Positive Solutions to Singular Boundary Value Problems for Second-order Differential Systems p-Laplacian[J].Chin Quart J Math,2013,28(3):345-354.doi:10.1016/S0898-1221(00)00158-9
[15]LIU Y J.Picard Boundary Value Problems of Second Order p-Laplacian Differential Equations[J].Chin Quart J Math,2011,26(1):77-84.doi:10.1007/s11424-011-8045-0
[16]薛益民,苏莹.一类p-Laplacian边值问题多个对称解的存在性[J].安徽大学学报(自然科学版),2016,40(4):30-36.doi:10.3969/j.issn.1000-2162.2016.04.006XUE Yimin,SU Ying.Existence of Multiple Symmetric Solutions to a p-Laplacian Boundary Value Problem with Nonlinear Term Involving Derivative[J].Journal of Anhui University(Natural Science Edition),2016,40(4):30-36.
[2]KRASNOSEL′SKII M.Positive Solutions of Operator Equations[D].Noordhoff:University of Groningen,1964.doi:10.2307/2316070
[3]WANG J.The Existence of Positive Solutions for the One-dimensional p-Laplacian[J].Proc Amer Math Soc,1997,125:2275-2283.doi:10.1090/S0002-9939-97-04148-8
[4]孙伟平,葛渭高.一类非线性边值问题正解的存在性[J].数学学报,2001,44(4):577-580.doi:10.3321/j.issn:0583-1431.2001.04.001SUN Weiping,GE Weigao.The Existence of Positive Solutions for a Class of Nonlinear Boundary Value Problems[J].Acta Mathematica Sinica,2001,44(4):577-580.
[5]MA R,MA H.Positive Solutions for Nonlinear Discrete Periodic Boundary Value Problems[J].Comput Math Appl,2010,59:136-141.doi:10.1016/j.camwa.2009.07.071
[6]BALANOV Z,KRAWCEWICZ W,NGUYEN M L.Multiple Solutions to Symmetric Boundary Value Problems for Second Order Odes:Equivariant Degree Approach[J].Nonlinear Anal,2014,94:45-64.doi:10.1016/j.na.2013.07.030
[7]ZHU Y,ZHU J.Existence of Multiple Positive Solutions for Nth-order p-Laplacian M-Point Singular Boundary Value Problems[J].J Appl Math Comput,2010,34:393-405.doi:10.1007/s12190-009-0329-3
[8]AVERY R I,CHYAN C J,HENDERSIN J.Twin Positive Solutions of Boundary Value Problem for Ordinary Differential Equations and Finite Difference Equations[J].Comput Math Appl,2001,42:695-704.doi:10.1016/S0898-1221(01)00188-2
[9]MA R,ZHU B.Existence of Positive Solutions for a Semipositone Boundary Value Problem on the Half-line[J].Comput Math Appl,2009,58:1672-1686.doi:10.1016/j.camwa.2009.07.005
[10]ZHAO J,MIAO C,GE W,et al.Multiple Symmetric Positive Solutions to a New Kind of Four Point Boundary Value Problem[J].Nonlinear Anal,2009,71:9-18.doi:10.1016/j.na.2008.10.026
[11]陈艳丽.二阶非线性积分-微分方程边值问题的正解[J].河北师范大学学报(自然科学版),2014,38(5):445-450.doi:10.11826/j.issn.1000-5854.2014.05.003CHEN Yanli.Positive Solutions of Nonlinear Second-order Boundary Value Problems for Integro-differential Equations[J].Journal of Hebei Normal University(Natural Science Edition),2014,38(5):445-450.
[12]袁晓红,周德高,许方,等.非线性项带导数的p-Laplacian边值问题解的存在性[J].徐州工程学院学报(自然科学版),2010,25(1):1-5.doi:10.3969/j.issn.1674-358X.2010.01.001YUAN Xiaohong,ZHOU Degao,XU Fang,et al.Existence of Solution of BVPs for p-Laplacian Dynamic Equations Involving Derivative[J].Journal of Xuzhou Institute of Technology(Natural Science Edition),2010,25(1):1-5.
[13]薛益民,苏莹,胡飞.非线性项包含导数的边值问题的一个对称解[J].合肥工业大学学报(自然科学版),2016,39(5):716-720.doi:10.3969/j.issn.1003-5060.2016.05.027XUE Yimin,SU Ying,HU Fei.Existence of Symmetric Solutions to Boundary Value Problem with Nonlinear Term Involving Derivative[J].Journal of Hefei University of Technology(Natural Science Edition),2016,39(5):716-720.
[14]ZHAO X,HU W M,JIANG D Q.Existence Theory for Single and Multiple Positive Solutions to Singular Boundary Value Problems for Second-order Differential Systems p-Laplacian[J].Chin Quart J Math,2013,28(3):345-354.doi:10.1016/S0898-1221(00)00158-9
[15]LIU Y J.Picard Boundary Value Problems of Second Order p-Laplacian Differential Equations[J].Chin Quart J Math,2011,26(1):77-84.doi:10.1007/s11424-011-8045-0
[16]薛益民,苏莹.一类p-Laplacian边值问题多个对称解的存在性[J].安徽大学学报(自然科学版),2016,40(4):30-36.doi:10.3969/j.issn.1000-2162.2016.04.006XUE Yimin,SU Ying.Existence of Multiple Symmetric Solutions to a p-Laplacian Boundary Value Problem with Nonlinear Term Involving Derivative[J].Journal of Anhui University(Natural Science Edition),2016,40(4):30-36.