二阶微分包含的边界值问题
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摘要
二阶微分包含作为二阶微分方程与集值分析的交叉学科,在力学,工程学以及优化与控制理论中有着广泛的应用.下面举一个牛顿力学的例子.
考察在一个在外力f的作用下运动的质点m,由牛顿第二运动定律,其位移函数x(t)满足下面的方程这里外力f随时间t,以及t时刻的位移x,速度v=x'的变化而变化.
如果我们把质点m置于一个中心力场(引力场或电磁场)中,则外力f可以表示为一个凸势的梯度,即
在很多情形下,质点m的运动不仅要受到一个中心力场的作用,还会受到其他力的影响,这种力的作用往往来自于很多方向,它的变化是随机的,不连续的,我们称之为扰动.这时该质点的运动状态只能用一个微分包含来描述其中,F(t,x(t),x'(t))表示一个集值扰动.
如果这个扰动是可控的,即F=F(t,x(t),x'(t),u(t)),其中量u(t)被一个集值函数U(t,x(t),x'(t))所控制.这样我们就得到一个反馈控制系统
近三十多年来,作为一个新的数学分支,二阶微分包含越来越受到学者们的关注.罗马尼亚的V.Barbu,N.C.Apreutesei,美国的N.H.Pavel,希腊的N.S.Papageorgiou,意大利的F.Papalini,中国的S.Hu,法国的D.Motreanu以及加拿大的M.Frigon等学者先后研究了带有边界条件的二阶微分包含:这里,A是R~N上的一个单调算子,F是满足一定条件的集值扰动,BC表示一个边界条件,它具有如下几种形式:
在边值问题的研究中,有一个重要的课题,那就是寻找落在给定集合C中的解.这个集合C被称为流不变集.下面是几种特殊的流不变集:
为了解决这类问题,我们通常会采用截断与罚函数技术,得到一个辅助问题,利用逼近技术与不动点原理,可以证明该辅助问题的解是存在的.借助于先验估计,还可以证明这样得到的解一定落在流不变集中,从而是原问题的解.
近年来,许多学者把注意力集中在带p-Laplace(或类p-Laplace)算子的二阶微分系统上.在美国的J.Mawhin,智利的D.Man(?)sevich以及中国的M.Zhang等工作的基础上,一系列具有高质量的文章涌现出来.这些工作集中讨论了p-Laplace算子的Fucik谱,共振及非共振条件,正解(或负解)以及符号变化的解,得到了一批好的结果.
本篇论文主要研究二阶微分包含的边界值问题,根据所使用工具的不同,它大致可以分为两个部分.在第一部分,我们利用不动点原理与Yosida逼近研究带有单调项的集值边界问题.该部分共分三章.
第一章研究下面的模型:这里,a:R~N→R~N是一个类p-Laplace算子,集值扰动F满足Hartman型条件(见第一章H(F)_1(iv)).
第二章,我们在解管道存在的假设下考虑半线性问题
第三章主要是研究非线性数值微分包含在这一章,我们总假设与之相应的上,下解是存在的.为了研究这三个边值问题,我们利用截断函数(见(1.2.1),(1.2.2),(2.1.6),(2.1.7),(3.2.1),(3.2.2)),得到一个定义在函数空间上的变换τ,通过对τ的仔细研究,我们发现了函数x与它的截断函数τx导数之间的内在联系(见(1.2.5),(1.4.7),(2.1.11),(2.3.11),(3.3.8)),这就使得我们能够在逼近方程两边施行对偶积,并得到逼近解的界.利用这一改进的方法,本文改进并推广了Papalini,Papageorgiou,Mawhin以及Frigon等人工作的部分结果.
本文的第二部分也分为三章.在第四章,我们继续研究边值问题这里,g(t,·)是一族下半连续的真凸函数,边界条件是用凸函数的次微分表示的.解决这一问题的难点在于g(t,·)的有效域是随t变化的,逼近解的界无法用上面的方法得到.我们的讨论是围绕函数族g(t,·)展开的.在一些合理的条件之下,我们证明了g(·,)的Nemytskij泛函(?)仍是下半连续的真凸函数,其次微分等同于(?)g(t,·)的Nemytskij算子.这样,再次利用不动点原理,我们解决了一类微分包含解的存在性问题,同时得到了一个Banach空间上极大单调算子的嵌入定理(见第四章,定理4.15),该定理具有很高的理论和应用价值.
最后两章研究的是具有周期边界的二阶微分系统.通过在函数空间上引入合适的能量泛函,我们把解的存在性问题转化为一个求能量泛函临界点的问题.
这里有两个模型.第五章处理的是第一个模型对于这一模型,我们定义泛函Φ=φ+ψ,这里φ是从函数族g(t,·)导出的凸泛函,ψ是一个与f(t,·)相关的局部Lipschitz泛函.利用广义PS-条件以及最小作用原理,我们证明了下面问题的解是存在的,它仍被称为泛函Φ的临界点.这一模型在边值问题中有更广泛的应用.第五章的结果还可以抽象出来,作为临界点理论的一个补充(见第五章,定理5.10).
最后一章讨论的模型二也是一个数值微分包含,我们的讨论集中在弱AR条件上(见H(f)(iv)),运用非光滑C-条件以及非光滑形式的山路引理,我们证明了模型二的正解的存在性.这是第一次在周期问题中讨论弱AR条件.在第六章的最后,多解及同宿解的存在性也一并加以讨论.
综上所述,在这篇文章中,我们研究了二阶微分包含的边界值问题.以集值分析,凸分析与非光滑分析以及临界点理论为理论根据,本文研究了几类模型的解的存在性及多样性,引入并讨论了两个新的模型(第四、五章).这项工作推广了Papageorgiou,Papalini以及Frigon等学者的部分研究成果,其试用的方法可供这一领域的其他研究者借鉴.
As a intersection of second order differential equation theory and set-valued analysis, the new subject second order differential inclusions has wide application in mechanics, engineering, optimization and control theory. Here is an illustrative example.
Consider a particle m moving under the rule of an external force f.By Newton's second law of motion, we know that the displacement x(t) of m conforms the following equationwhere the force f imposed on m varies about the time t,the displacement x and the velocity v = x'.
If we place the particle m into a central (gravitational or electro-magnetic) field, then the force f can be represented by the gradient of a convex potential, namely,
Sometimes, the external force can be decomposed into two factors, one comes from a central field, which changes smoothly, while the other one is a perturbation, denoted by F, which comes from many directions and sources, and changed discontinuously and uncertainly. To describe the motion of m precisely in this case, we can only appeal to the differential inclusion
If the perturbation is controllable with the form F(t, x(t),x'(t),u(t)), where the data u(t) can be controlled by a set-valued function U(t,x(t),x'(t)),then we get a feed back control system
In the last three decades, the new subject has been received an increasing interest. A lot of scholars, such as V. Barbu (Romania), N. H. Pavel (U.S.A.), N. C. Apreutesei (Romania), N. S. Papageorgiou (Greece), S. Hu (P.R.C.), F. Papalini (Italy), D. Motreanu (France) and M. Frigon (Canada) etc studied the second order differential inclusions with various boundary conditionswhere A is a maximal monotone map on R~N(especially, it is equal to the gradient or subdifferential operator of a convex function), F is a multivalued perturbation satisfying some conditions, and BC denotes a boundary condition having the following forms:
In dealing with the boundary value problems, there is an important task, that is to seek for a solution lying in a given set C, which contains the following forms:
To complete this task, we always use the method of truncations and penalization and obtain an auxiliary problem, which solutions can be proved existing by means of fixed point theory and approximation. And from an priori estimate, we can find that all the solutions lie in the given set C,and then solve the original problem automatically.
Recently, many authors paid their attentions to the boundary value problems driven by the p-Laplacian (or p-Laplacian-like) operators. Based on the works of J. Mawhin (U.S.A), D. Man(?)sevich (Chile) and M. Zhang (P.R.C), a series of papers with high quality appeared. In these works, authors' discussions were concentrated on the Fucik spectrum, the resonance and unresonance conditions, and the constant sign and nodal solutions, etc.
This thesis is devoted to study the boundary value problems for second order inclusions, which contents can be divided into two parts.
In the first part, we investigate the multivalued boundary problems with monotone terms. Our approach relies on the theory of fixed points of set-valued maps and Yosida approximation of maxiaml monotone operators. This part contains three chapters.
In Chapter 1,we study the following modelwhere a :R~N→R~N is a classical p-Laplacian-like operator, and the multivalued perturbation satisfies the generalized Hartman's condition (see Chapter 1H(F)_(iv))
In Chapter 2, we consider the semilinear problemwith a pair of solution tube existing.(see Section 1, Chapter 2)
And Chapter 3 is devoted to the scalar nonlinear problemcoupled with a pair of upper-lower solutions.
To solve the above three problems, we employ truncated functions (see (1.2.1), (1.2.2), (2.1.6), (2.1.7), (3.2.1), (3.2.2)) and get a transformationτdefined on a function space. By a careful analysis on r, we find the relations between the differentials of x and (τx)' (see (1.2.5), (1.4.7), (2.1.11), (2.3.11), (3.3.8)), which make us perform dual products (or inner products) on the approximate equations and get the boundedness of the approximate solutions. Based on the reformed method, we improve and extend some results of those in the works of Papalini, Papageorgiou, Mawhin and Frigon etc.
The second part also has three chapters. In Chapter 4, we continue to discuss the multivalued boundary problemwhere g(t,·) is a family of lower semicotinuous and convex proper functions, and the boundary conditions arc described by the subdifferentials of two convex functions respectively. This model has not been studied ever before, since the domain Dg(t,·) varies about t,and thus the approximate equations could not be used any longer. To dear with the model, our discussion is concentrated on the family g(t,·).Under some reasonable conditions, we prove that the functional (?)derived from g(t,·)is also lsc,convex and proper, which subdifferential is equal to the Nemytskij operator of (?)g(t,·).Thus, using fixed point theory again, a class of problems have been proved correspondingly. As a by-product, an abstract result about the embedding of a maximal monotone map from a Banach space into one of its dense subspaces has been got simultaneously (see theorem 4.15, Chapter 4). This result has important value in theory and application.
In the last two chapters, we turn to investigate the periodic differential systems.
By introducing a suitable functional defined on some function space, the task to search for a solution for the boundary problem turns to be one to seek for a critical point of the functional.
There are two models we deal with, both of them are concerning periodic solutions.
In Chapter 5, there is model 1, For this model, we introduce the functionalΦ=φ+ψ,whereφis a convex function derived from g(t,·) andψis a locally Lipschitz one associated with f(t.·).
Using the generalized nonsmooth PS-condition and the least action principle, we prove the existence of solutions of the equationwhich are also called the critical points ofΦ.This model has wide application in boundary problems, and its abstract form can be viewed as a useful supplement of critical point theory (see Theorem 5.10, Chapter 5).
In the last chapter, we discuss model 2,Our discussion focusses on the weak AR condition (see H(f)(iv)). By means of nonsmooth C-condition and the Mountain Pass Lemma of nonsmooth type, we prove the existence of positive solutions of the model. This is the first time to investigate the periodic problem using the weak A.R. condition.
At the end of Chapter 6, the multiplicity of solutions and the homoclinic solutions are all taken into account.
To sum up, in this thesis, we study several types of second order differential inclusions with boundary conditions. Employing the tools of set-valued analysis, convex and nonsmooth analysis, and critical point theory, we discuss the existence and multiplicity of solutions for these models. This work improves and extends some results in the works followed by us, discussed two new models (see Chapter 4 and Chapter 5) and provides a useful method for other authors in this field.
考察在一个在外力f的作用下运动的质点m,由牛顿第二运动定律,其位移函数x(t)满足下面的方程这里外力f随时间t,以及t时刻的位移x,速度v=x'的变化而变化.
如果我们把质点m置于一个中心力场(引力场或电磁场)中,则外力f可以表示为一个凸势的梯度,即
在很多情形下,质点m的运动不仅要受到一个中心力场的作用,还会受到其他力的影响,这种力的作用往往来自于很多方向,它的变化是随机的,不连续的,我们称之为扰动.这时该质点的运动状态只能用一个微分包含来描述其中,F(t,x(t),x'(t))表示一个集值扰动.
如果这个扰动是可控的,即F=F(t,x(t),x'(t),u(t)),其中量u(t)被一个集值函数U(t,x(t),x'(t))所控制.这样我们就得到一个反馈控制系统
近三十多年来,作为一个新的数学分支,二阶微分包含越来越受到学者们的关注.罗马尼亚的V.Barbu,N.C.Apreutesei,美国的N.H.Pavel,希腊的N.S.Papageorgiou,意大利的F.Papalini,中国的S.Hu,法国的D.Motreanu以及加拿大的M.Frigon等学者先后研究了带有边界条件的二阶微分包含:这里,A是R~N上的一个单调算子,F是满足一定条件的集值扰动,BC表示一个边界条件,它具有如下几种形式:
在边值问题的研究中,有一个重要的课题,那就是寻找落在给定集合C中的解.这个集合C被称为流不变集.下面是几种特殊的流不变集:
为了解决这类问题,我们通常会采用截断与罚函数技术,得到一个辅助问题,利用逼近技术与不动点原理,可以证明该辅助问题的解是存在的.借助于先验估计,还可以证明这样得到的解一定落在流不变集中,从而是原问题的解.
近年来,许多学者把注意力集中在带p-Laplace(或类p-Laplace)算子的二阶微分系统上.在美国的J.Mawhin,智利的D.Man(?)sevich以及中国的M.Zhang等工作的基础上,一系列具有高质量的文章涌现出来.这些工作集中讨论了p-Laplace算子的Fucik谱,共振及非共振条件,正解(或负解)以及符号变化的解,得到了一批好的结果.
本篇论文主要研究二阶微分包含的边界值问题,根据所使用工具的不同,它大致可以分为两个部分.在第一部分,我们利用不动点原理与Yosida逼近研究带有单调项的集值边界问题.该部分共分三章.
第一章研究下面的模型:这里,a:R~N→R~N是一个类p-Laplace算子,集值扰动F满足Hartman型条件(见第一章H(F)_1(iv)).
第二章,我们在解管道存在的假设下考虑半线性问题
第三章主要是研究非线性数值微分包含在这一章,我们总假设与之相应的上,下解是存在的.为了研究这三个边值问题,我们利用截断函数(见(1.2.1),(1.2.2),(2.1.6),(2.1.7),(3.2.1),(3.2.2)),得到一个定义在函数空间上的变换τ,通过对τ的仔细研究,我们发现了函数x与它的截断函数τx导数之间的内在联系(见(1.2.5),(1.4.7),(2.1.11),(2.3.11),(3.3.8)),这就使得我们能够在逼近方程两边施行对偶积,并得到逼近解的界.利用这一改进的方法,本文改进并推广了Papalini,Papageorgiou,Mawhin以及Frigon等人工作的部分结果.
本文的第二部分也分为三章.在第四章,我们继续研究边值问题这里,g(t,·)是一族下半连续的真凸函数,边界条件是用凸函数的次微分表示的.解决这一问题的难点在于g(t,·)的有效域是随t变化的,逼近解的界无法用上面的方法得到.我们的讨论是围绕函数族g(t,·)展开的.在一些合理的条件之下,我们证明了g(·,)的Nemytskij泛函(?)仍是下半连续的真凸函数,其次微分等同于(?)g(t,·)的Nemytskij算子.这样,再次利用不动点原理,我们解决了一类微分包含解的存在性问题,同时得到了一个Banach空间上极大单调算子的嵌入定理(见第四章,定理4.15),该定理具有很高的理论和应用价值.
最后两章研究的是具有周期边界的二阶微分系统.通过在函数空间上引入合适的能量泛函,我们把解的存在性问题转化为一个求能量泛函临界点的问题.
这里有两个模型.第五章处理的是第一个模型对于这一模型,我们定义泛函Φ=φ+ψ,这里φ是从函数族g(t,·)导出的凸泛函,ψ是一个与f(t,·)相关的局部Lipschitz泛函.利用广义PS-条件以及最小作用原理,我们证明了下面问题的解是存在的,它仍被称为泛函Φ的临界点.这一模型在边值问题中有更广泛的应用.第五章的结果还可以抽象出来,作为临界点理论的一个补充(见第五章,定理5.10).
最后一章讨论的模型二也是一个数值微分包含,我们的讨论集中在弱AR条件上(见H(f)(iv)),运用非光滑C-条件以及非光滑形式的山路引理,我们证明了模型二的正解的存在性.这是第一次在周期问题中讨论弱AR条件.在第六章的最后,多解及同宿解的存在性也一并加以讨论.
综上所述,在这篇文章中,我们研究了二阶微分包含的边界值问题.以集值分析,凸分析与非光滑分析以及临界点理论为理论根据,本文研究了几类模型的解的存在性及多样性,引入并讨论了两个新的模型(第四、五章).这项工作推广了Papageorgiou,Papalini以及Frigon等学者的部分研究成果,其试用的方法可供这一领域的其他研究者借鉴.
As a intersection of second order differential equation theory and set-valued analysis, the new subject second order differential inclusions has wide application in mechanics, engineering, optimization and control theory. Here is an illustrative example.
Consider a particle m moving under the rule of an external force f.By Newton's second law of motion, we know that the displacement x(t) of m conforms the following equationwhere the force f imposed on m varies about the time t,the displacement x and the velocity v = x'.
If we place the particle m into a central (gravitational or electro-magnetic) field, then the force f can be represented by the gradient of a convex potential, namely,
Sometimes, the external force can be decomposed into two factors, one comes from a central field, which changes smoothly, while the other one is a perturbation, denoted by F, which comes from many directions and sources, and changed discontinuously and uncertainly. To describe the motion of m precisely in this case, we can only appeal to the differential inclusion
If the perturbation is controllable with the form F(t, x(t),x'(t),u(t)), where the data u(t) can be controlled by a set-valued function U(t,x(t),x'(t)),then we get a feed back control system
In the last three decades, the new subject has been received an increasing interest. A lot of scholars, such as V. Barbu (Romania), N. H. Pavel (U.S.A.), N. C. Apreutesei (Romania), N. S. Papageorgiou (Greece), S. Hu (P.R.C.), F. Papalini (Italy), D. Motreanu (France) and M. Frigon (Canada) etc studied the second order differential inclusions with various boundary conditionswhere A is a maximal monotone map on R~N(especially, it is equal to the gradient or subdifferential operator of a convex function), F is a multivalued perturbation satisfying some conditions, and BC denotes a boundary condition having the following forms:
In dealing with the boundary value problems, there is an important task, that is to seek for a solution lying in a given set C, which contains the following forms:
To complete this task, we always use the method of truncations and penalization and obtain an auxiliary problem, which solutions can be proved existing by means of fixed point theory and approximation. And from an priori estimate, we can find that all the solutions lie in the given set C,and then solve the original problem automatically.
Recently, many authors paid their attentions to the boundary value problems driven by the p-Laplacian (or p-Laplacian-like) operators. Based on the works of J. Mawhin (U.S.A), D. Man(?)sevich (Chile) and M. Zhang (P.R.C), a series of papers with high quality appeared. In these works, authors' discussions were concentrated on the Fucik spectrum, the resonance and unresonance conditions, and the constant sign and nodal solutions, etc.
This thesis is devoted to study the boundary value problems for second order inclusions, which contents can be divided into two parts.
In the first part, we investigate the multivalued boundary problems with monotone terms. Our approach relies on the theory of fixed points of set-valued maps and Yosida approximation of maxiaml monotone operators. This part contains three chapters.
In Chapter 1,we study the following modelwhere a :R~N→R~N is a classical p-Laplacian-like operator, and the multivalued perturbation satisfies the generalized Hartman's condition (see Chapter 1H(F)_(iv))
In Chapter 2, we consider the semilinear problemwith a pair of solution tube existing.(see Section 1, Chapter 2)
And Chapter 3 is devoted to the scalar nonlinear problemcoupled with a pair of upper-lower solutions.
To solve the above three problems, we employ truncated functions (see (1.2.1), (1.2.2), (2.1.6), (2.1.7), (3.2.1), (3.2.2)) and get a transformationτdefined on a function space. By a careful analysis on r, we find the relations between the differentials of x and (τx)' (see (1.2.5), (1.4.7), (2.1.11), (2.3.11), (3.3.8)), which make us perform dual products (or inner products) on the approximate equations and get the boundedness of the approximate solutions. Based on the reformed method, we improve and extend some results of those in the works of Papalini, Papageorgiou, Mawhin and Frigon etc.
The second part also has three chapters. In Chapter 4, we continue to discuss the multivalued boundary problemwhere g(t,·) is a family of lower semicotinuous and convex proper functions, and the boundary conditions arc described by the subdifferentials of two convex functions respectively. This model has not been studied ever before, since the domain Dg(t,·) varies about t,and thus the approximate equations could not be used any longer. To dear with the model, our discussion is concentrated on the family g(t,·).Under some reasonable conditions, we prove that the functional (?)derived from g(t,·)is also lsc,convex and proper, which subdifferential is equal to the Nemytskij operator of (?)g(t,·).Thus, using fixed point theory again, a class of problems have been proved correspondingly. As a by-product, an abstract result about the embedding of a maximal monotone map from a Banach space into one of its dense subspaces has been got simultaneously (see theorem 4.15, Chapter 4). This result has important value in theory and application.
In the last two chapters, we turn to investigate the periodic differential systems.
By introducing a suitable functional defined on some function space, the task to search for a solution for the boundary problem turns to be one to seek for a critical point of the functional.
There are two models we deal with, both of them are concerning periodic solutions.
In Chapter 5, there is model 1, For this model, we introduce the functionalΦ=φ+ψ,whereφis a convex function derived from g(t,·) andψis a locally Lipschitz one associated with f(t.·).
Using the generalized nonsmooth PS-condition and the least action principle, we prove the existence of solutions of the equationwhich are also called the critical points ofΦ.This model has wide application in boundary problems, and its abstract form can be viewed as a useful supplement of critical point theory (see Theorem 5.10, Chapter 5).
In the last chapter, we discuss model 2,Our discussion focusses on the weak AR condition (see H(f)(iv)). By means of nonsmooth C-condition and the Mountain Pass Lemma of nonsmooth type, we prove the existence of positive solutions of the model. This is the first time to investigate the periodic problem using the weak A.R. condition.
At the end of Chapter 6, the multiplicity of solutions and the homoclinic solutions are all taken into account.
To sum up, in this thesis, we study several types of second order differential inclusions with boundary conditions. Employing the tools of set-valued analysis, convex and nonsmooth analysis, and critical point theory, we discuss the existence and multiplicity of solutions for these models. This work improves and extends some results in the works followed by us, discussed two new models (see Chapter 4 and Chapter 5) and provides a useful method for other authors in this field.
引文
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[38] L. Gasi(?)ski, D. Motreanu, N. S. Papageorgiou, Multiplicity of nontrivial solutions for elliptic equations with nonsmooth potential and resonance at higher eigenvalues, Proc. Indian. Acad. Sci. (Math. Sci),116, (2006), 233-255.
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[41] D. Guo,"Nonlinear functional analysis", The Science and Technology Press of Shandong, 2001.
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[44] N. Halidias and N. S. Papageorgiou, Existence and relaxation results for nonlinear second order multivalued boundary value problems in R~N,J.Differential Equations, 147 (1998), 123-154.
[45] N. Halidias and N. S. Papageorgiou, Existence of solutions for qusilinear second order differential inclusions with nonlinear boundary conditions,J.Comput.Appl. Math, 1113 (2000), 51-64.
[46] N. Halidias and N. S. Papageorgiou, Qusilinear elliptic problems with multivalued terms, Czechoslovak Mathematical Journal, 50 (125) (2000), 803-823
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[49] S. Hu and N. S. Papageorgiou, Positive periodic and homoclinic solutions for nonlinear differential equations with nonsmooth potential, Positivity, 10(2006), 343-363.
[50] F. Hiai and H. Umegaki, Integrals, conditional expectations and martingales of multivalued functions, J. Multivariate Anal, 7 (1977), 149-182.
[51] P. Jebelean, J. Mawhin, Periodic solutions of singular nonlinear perturbations of the ordinary p-Laplacian, adv. Nonlinear Stud., 2 (2002), 299-312.
[52] S. Kyritsi, N. Matzakos, and N. S. Papageorgiou, Periodic problems for strongly nonlinear second-order differential inclusions, J. Differential Equations, 183 (2002), 279-302.
[53] S. Kyritsi, N. Matzakos, N. S. Papageorgiou, Nonlinear boundary value problems for second order differential inclusions, Czechoslovak Math. Jour., 55(2005), 545-579.
[54] D. A. Kandilakis and N. S. Papageorgiou, Existence theorems for nonlinear boundary value problems for second order differential inclusions,J.Differential Equations,132 (1996), 107-125.
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[56] D. Kravvaritis and N. S. Papageorgiou, Boundary value problems for Noncovex differential inclusions, J. Math. Anal,19 (1993), 146-160.
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[61] J. Mawhin, The periodic Ambrosetti-Prodi problem for nonlinear perturbations of the p-Laplacian, J. Eur. Math. Soc, 8 (2006), 375-388.
[62] R. Man(?)sevich and J.Mawhin, Periodic solutions for nonlinear systems with p-Laplacian-like operators, J. Differential Equations,145 (1998), 367-393.
[63] R. Man(?)sevich and J.Mawhin, Boundary value problems for nonlinear differential equations of vector p-Laplacian-like operators, J. Korean Math.Soc.,37 (2000), 665-685.
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