关于变分不等式的研究
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摘要
本篇论文我们研究几类半变分不等式解的存在性问题
在第一章我们首先介绍关于半变分不等式的研究背景及一些概念和引理
在第二章我们研究下面p-laplacian方程的Dirichlet边界问题其中Ω∈R~n是有界域，位势函数j(z,·)是局部Lipschitz的．很多学者已经用各种一方法对此问题作了大量的研究，如拓扑度理论，山路引理，环绕定理等．在应用这些理论中，所要解决的首要问题就是要证明非光滑PS紧性条件，而对紧性条件的刻画上，也已经有了更一般的非光滑PS紧性条件。本章将通过用更一般的非光滑PS紧性条件来证明解的存在性．
在第三章我们研究关于p(x)-laplacian方程的Dirichlet边界问题由于p(x)-laplacian相对于p-laplacian具有史加复杂的非线性性质，如它是不同胚的且p(x)-laplacian的特征值的下确界等于0,因此处理起来难度加大，所以通过对位势函数给出合理假设，也对p(x)给出一定的要求，利用非光滑临界点理论中的环绕定理来证明多解的存在性.
在第四章我们研究拟线性半变分不等式径向解与非径向解问题对于半线性情形的径向与非径向解已经解决,本文考虑拟线性情况,利用非光滑情形的喷泉定理和山路引理来证明非平凡解和无穷多径向解与非径向解的存在性.这里主要用到的工具是局部Lipschitz泛函的对称临界原理.
In this paper we deal with solutions of existence theorems for several case of hemivariational inequality.
In the first chapter, we introduce some definitions, lemmas and background of hemivariational inequality.
In the second chapter, we study the problem of the following p-laplacian equation with Dirichlet boundary .WhereΩ∈R~n is a bounded domain , the potential function j(z, ?) is locally Lipschitz. The problem is studied by so many scholars with different methods. Such as topology degree theory, mountain pass lemma, linking theorem and so on. Using the above the-orem, the first thing we need to do is proving nonsmooth PS compactness condition. In the compactness condition, we already had the generalized nonsmooth PS compactness condtion. This chapter we will prove the existence of a theorem through the generalized nonsmooth PS compactness condition.
In the third chapter, we study the problem of the following p(x)-laplacian equation with Dirichlet boundary .The p(x)-laplacian possesses more complicated nonlinearity than the p-laplacian, for example, it is inhomogeneous and in general the infimum of the eigenvalues of p(x)-laplacian equals to 0, so it is more difficult to deal with. Through our assumptions for the nonsmooth potential and also for p(x), using linking theorem of the nonsmooth critical theorem, we prove a multiplicity result.
In the fourth chapter, we concerned with the problem of quasilinear hemivaria-tional inequality, to study radial and unradial solutions.The radial and unradial solutions of seniilinear hemivariational inequality had been obtained. so this chapter, we consider quasilinear. Using the fountain theorem and mountain theorem with nonsmooth condition, we prove the existence of an nontrivial solution, as well as infinitely many radially and unradially solutions. In the proofs we use the principle of symmetric criticality for locally Lipschitz functions.
在第一章我们首先介绍关于半变分不等式的研究背景及一些概念和引理
在第二章我们研究下面p-laplacian方程的Dirichlet边界问题其中Ω∈R~n是有界域，位势函数j(z,·)是局部Lipschitz的．很多学者已经用各种一方法对此问题作了大量的研究，如拓扑度理论，山路引理，环绕定理等．在应用这些理论中，所要解决的首要问题就是要证明非光滑PS紧性条件，而对紧性条件的刻画上，也已经有了更一般的非光滑PS紧性条件。本章将通过用更一般的非光滑PS紧性条件来证明解的存在性．
在第三章我们研究关于p(x)-laplacian方程的Dirichlet边界问题由于p(x)-laplacian相对于p-laplacian具有史加复杂的非线性性质，如它是不同胚的且p(x)-laplacian的特征值的下确界等于0,因此处理起来难度加大，所以通过对位势函数给出合理假设，也对p(x)给出一定的要求，利用非光滑临界点理论中的环绕定理来证明多解的存在性.
在第四章我们研究拟线性半变分不等式径向解与非径向解问题对于半线性情形的径向与非径向解已经解决,本文考虑拟线性情况,利用非光滑情形的喷泉定理和山路引理来证明非平凡解和无穷多径向解与非径向解的存在性.这里主要用到的工具是局部Lipschitz泛函的对称临界原理.
In this paper we deal with solutions of existence theorems for several case of hemivariational inequality.
In the first chapter, we introduce some definitions, lemmas and background of hemivariational inequality.
In the second chapter, we study the problem of the following p-laplacian equation with Dirichlet boundary .WhereΩ∈R~n is a bounded domain , the potential function j(z, ?) is locally Lipschitz. The problem is studied by so many scholars with different methods. Such as topology degree theory, mountain pass lemma, linking theorem and so on. Using the above the-orem, the first thing we need to do is proving nonsmooth PS compactness condition. In the compactness condition, we already had the generalized nonsmooth PS compactness condtion. This chapter we will prove the existence of a theorem through the generalized nonsmooth PS compactness condition.
In the third chapter, we study the problem of the following p(x)-laplacian equation with Dirichlet boundary .The p(x)-laplacian possesses more complicated nonlinearity than the p-laplacian, for example, it is inhomogeneous and in general the infimum of the eigenvalues of p(x)-laplacian equals to 0, so it is more difficult to deal with. Through our assumptions for the nonsmooth potential and also for p(x), using linking theorem of the nonsmooth critical theorem, we prove a multiplicity result.
In the fourth chapter, we concerned with the problem of quasilinear hemivaria-tional inequality, to study radial and unradial solutions.The radial and unradial solutions of seniilinear hemivariational inequality had been obtained. so this chapter, we consider quasilinear. Using the fountain theorem and mountain theorem with nonsmooth condition, we prove the existence of an nontrivial solution, as well as infinitely many radially and unradially solutions. In the proofs we use the principle of symmetric criticality for locally Lipschitz functions.
引文
[1] Landesman E, Robinson S, Rumbos A. Multiple solutions of semilinear elliptic problems at resonance[J]. Nonlinear Anal.: Theory Methods Appl., 1995, 24:1049-1059.
[2] Mizoguchi N. Asymptotically linear elliptic equations without nonresonance con-ditions[J]. J. Differential Equations, 1994, 113: 150-165.
[3] Hirano N, Nishimura T. Multiplicity results for semilinear elliptic problems at resonance and with jumping nonlinearities[J]. J. Math. Anal. AppL, 1993, 180:566-586.
[4] Motreanu D, Panagiotopoulos PD. A minimax approach to the eigenvalue problem of hemivariationai inequalities and applications[J]. Appl.Anal., 1995, 58: 53-76.
[5] Motreanu D, Panagiotopoulos PD. Minimax Theorems and Qualitative Properties of the Solutions of Hemivariationai Inequalities[M]. Dordrecht: Kluwer Academic Publishers, 1999.
[6] Marano SA, Motreanu D. On a three critical points theorem for non-differentiable functions and applications to nonlinear boundary value problems[J]. Nonlinear Anal.: Theory Methods AppL 2002, 48A: 37-52.
[7] Gasinski L, Papageorgiou NS. An existence theorem for nonlinear hemivariationai inequalities at resonance[J]. Bull. Austral. Math. Soc., 2001, 63: 1-14.
[8] Gasinski L, Papageorgiou NS. Solutions and multiple solutions for quasilinear hemivariationai inequalities at resonance[J]. Proc. Roy. Soc. Edinburgh Math.,2001, 131A: 1091-1111.
[9] Kyritsi ST, Papageorgiou NS. Hemivariationai inequalities with the potential crossing the Orst eigenvalue[J]. Bull. Austral. Math. Soc, 2001, 64: 381-393.
[10] Naniewica Z, Panagiotopoulos PD. Mathematical theory of Hemivariationai Inequalities and Applications[M]. New York: Marcel Dekker, 1994.
[11] Chang KC. Variational methods for nondifferentiable functionals and their applications to partial differential equations[J]. J. Math. Anal. AppL, 1981, 80: 102- 129.
[12] Lindqvist P. On the equation div((?))+(?)=0[J]. Proc. Amer.Math. Soc, 1990,109: 157-164.
[13] Motreanu D, Varga C. Some critical point results for locally lipschitz function-als[J]. Comm. Appl. Nonlinear Anal., 1997, 4: 17-33.
[14] Hu S, Papageorgiou NS. Handbook of Multivalued Analysis Volume II: Applications[M]. Dordrecht: Kluwer Academic Publishers, 2000.
[15] Hu S, Papageorgiou NS. Handbook of Multivalued Analysis Volume I: The-ory[M]. Dordrecht: Kluwer Academic Publishers, 1997.
[16] Clarke FH. Optimization and Nonsmooth Analysis[M]. New York: Wiley, 1983.
[17] Adams R. Sobolev Spaces[M]. New York: Academic Press, 1975.
[18] Z. Denkowski, L. Gasi(?)ski, N.S. Papageorgiou, Existence and multiplicity of solutions for semilinear hemivariational inequalities at resonance[J]. Nonlinear Anal., 2007, 66: 1329-1340.
[19] X.L. Fan, Solutions for p(x)-Laplacian Dirichlet problem with singular coeffi-cients[J]. J. Math. Anal. Appl., 2005, 312: 464-477.
[20] X.L. Fan, On the sub-supersolution method for p(x)-Laplacian equations[J]. J.Math. Anal. Appl., 2007, 330: 665-682.
[21] M.E. Filippakis, L. Gasi(?)ski, N.S. Papageorgiou, Quasilinear hemivariational inequalities with strong resonance at infinity[J]. Nonlinear. Anal.. 2004. 56 : 331-345.
[22] X.L. Fan, Q.H. Zhang, Existence of solutions for p(x)-Laplacian Dirichlet prob-lem[J]. Nonlinear Anal., 2003, 52: 1843-1852.
[23] X.L. Fan,D. Zhao, On the generalized Orlicz-Sobolev space (?)(Ω)[J], J.GansuEduc. College, 1998, 12(1): 1 - 6.
[24] X.L. Fan, D. Zhao, On the spaces L~(p(x))(Ω) and (?)(Ω)[J]. J. Math. Anal.Appl., 2001,262: 749-760.
[25] X.L. Fan, Q.H. Zhang, D. Zhao, Eigenvalues of p(x)-Laplacian Dirichlet problem[J]. J. Math. Anal. Appl., 2005, 302: 306-317.
[26] X.L. Fan, Y.Z. Zhao, D. Zhao, Compact imbedding theorem with symmetry of Strauss-Lions type for the space (?)(Ω)[J]. J. Math. Anal. Appl. 2001, 255:333-348.
[27] A.B. Zang, p(x)-Laplacian equations satisfying Cerami condition[J]. J. Math.Anal. Appl, 2008, 337: 547-555.
[28] D. Goeleven, D. Motreanu, P.D. Panagiotopoulos, Eigenvalue problem for varitional hemivariational inequalities at resonance[J]. Nonlinear Anal., 1981, 33:161-180.
[29] L. Gasinski, N.S. Papageorgiou, Multiple solutions for semilinear hemivariational inequalies at resonance[J]. Publ. Math. Debrecen, 2001, 59: 121-146.
[30] L. Gasinski, N.S. Papageorgiou, Nonlinear Hemivariational inequalities at reso-nance[J]. Bull. Austral. Math. Soc., 1999, 60: 353-364.
[31] L. Gasi(?)ski, N.S. Papageorgiou, Existence of solutions and of multiple solutions for eigenvalue problems in hemivariational inequalities[J]. Adv. Math. Sci. Appl.,2001, 11:437-464.
[32] L. Gasinski, N.S. Papageorgiou, Nonsmooth critical point theory and nonlinear boundary value problems[M]. Boca Raton: Chapman and Hall, CRC Press, 2005.
[33] N.S.Papageorgiou,et al., Three nontrivial solutions for the p-Laplacian with a non-smooth potential[J]. Nonlinear Anal., 2008, 68: 3812-3827
[34] S.T. Kyritsi, N.S. Papageorgiou, Pairs of positive solutions for nonlinear elliptic equations with the p-Laplacian and a nonsmooth pothtial[J]. Ann. Mat. Pura Appl., 2005, 184: 449-472.
[35] S.T. Kyritsi, N.S. Papageorgiou, Nonsmooth critical point theory on closed convex sets and nonlinear hemivariational inequalities[J]. Nonlinear Anal. 2005, 61: 373-403.
[36] S.T. Kyritsi, N.S. Papageorgiou, Multiple solutions of constant sign for nonlinear nonsmooth eigenvalue problems near resonance[J]. Calc. Var. Partial Differential Equations, 2004, 20: 1-24.
[37] V.V. Jikov, S.M. Kozlov, O.A. Oleinik, Homogenization of Differential Operators and Integral Functionals[M]. Berlin: Springer-Verlag, 1994.
[38] D.Kandilakis, N.Kourogenis, N.S.Papageorgiou, Two nontrivial critical points for nonsmooth functional via local linking and applications[J]. J. Global Optim.,2006,34: 317-337.
[39] Z. Denkowski, S. Migorski, N.S. Papageorgiou, An Introduction to Nonlinear Analysis: Theory[M]. New York: Kluwer Academic/Plenum, 2003.
[40] J. Musielak, Orlicz Spaces and Modular spaces[M]. Berlin: Springer, 1983.
[41] D. Motreanu, V. Radulescu, Variational and non-variational Methods in Nonlinear Analysis and Boundary Value Problems[M]. Boston: Kluwer Academic Publishers, 2003.
[42] P.D. Panagiotopoulos, Hemivariational Inequalities. Applications in Mechanics and Engineering[M]. Berlin: Springer-Verlag, 1993.
[43] F. Gazzola, V. Radulescu, A nonsmooth critical point approach to some nonlinear elliptic equations in (?)~N[J]. Differential Integral Equations, 2000, 13: 47-60.
[44] A. Krist(?)ly, Infinitely many radial and non-radial solutions for a class of hemivariational inequalities[J]. Rocky Mountain Journal of Mathematics, 2005, 35:1173-1190.
[45] T. Bartsch, Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on (?)~N[J]. Comm. Partial Differential Equations, 1995, 20: 1725-1741.
[46] T. Bartsch, M. Willem, Infinitely many non-radial solutions of an Euclidean scalar field equation[J]. J. Funct. Anal., 1993, 117: 447-460.
[47] T. Bartsch, M. Willem, Infinitely many radial solutions of a semilinear elliptic problem in (?)~N[J]. Arch. Rational Mech. Anal., 1993, 124: 261-276.
[48] W. Krawcexicz, W. Marzantowicz, Some remarks on the Lusternik-Schnirelman method for nondifferentiable functionals invariant with respect to a finite group action[J]. Rocky Mountain Journal of Mathematics, 1990, 20: 1041-1049.
[49] N. C. Kourogenis, N. S. Papageorgiou, Nonsmooth critical point theory and nonlinear elliptic equations at resonance[J]. Kodai Math. J., 2000, 23: 108-135.
[50] D. Motreanu, C. Varga, A nonsmooth equivariant minimax principle[J]. Comm.Appl. Anal. 1999,3: 115-130.
[51] T. Bartsch, Infinitely many solutions of a symmetric dirichlet problem[J]. Nonlinear Anal. 1993,20: 1205-1216.
[52] G. Arioli, F. Gazzola, Existence and multiplicity results for quasilinear elliptic defferential systems[J]. Comm. Partial Differential Equatiions, 2000, 25: 125-153.
[2] Mizoguchi N. Asymptotically linear elliptic equations without nonresonance con-ditions[J]. J. Differential Equations, 1994, 113: 150-165.
[3] Hirano N, Nishimura T. Multiplicity results for semilinear elliptic problems at resonance and with jumping nonlinearities[J]. J. Math. Anal. AppL, 1993, 180:566-586.
[4] Motreanu D, Panagiotopoulos PD. A minimax approach to the eigenvalue problem of hemivariationai inequalities and applications[J]. Appl.Anal., 1995, 58: 53-76.
[5] Motreanu D, Panagiotopoulos PD. Minimax Theorems and Qualitative Properties of the Solutions of Hemivariationai Inequalities[M]. Dordrecht: Kluwer Academic Publishers, 1999.
[6] Marano SA, Motreanu D. On a three critical points theorem for non-differentiable functions and applications to nonlinear boundary value problems[J]. Nonlinear Anal.: Theory Methods AppL 2002, 48A: 37-52.
[7] Gasinski L, Papageorgiou NS. An existence theorem for nonlinear hemivariationai inequalities at resonance[J]. Bull. Austral. Math. Soc., 2001, 63: 1-14.
[8] Gasinski L, Papageorgiou NS. Solutions and multiple solutions for quasilinear hemivariationai inequalities at resonance[J]. Proc. Roy. Soc. Edinburgh Math.,2001, 131A: 1091-1111.
[9] Kyritsi ST, Papageorgiou NS. Hemivariationai inequalities with the potential crossing the Orst eigenvalue[J]. Bull. Austral. Math. Soc, 2001, 64: 381-393.
[10] Naniewica Z, Panagiotopoulos PD. Mathematical theory of Hemivariationai Inequalities and Applications[M]. New York: Marcel Dekker, 1994.
[11] Chang KC. Variational methods for nondifferentiable functionals and their applications to partial differential equations[J]. J. Math. Anal. AppL, 1981, 80: 102- 129.
[12] Lindqvist P. On the equation div((?))+(?)=0[J]. Proc. Amer.Math. Soc, 1990,109: 157-164.
[13] Motreanu D, Varga C. Some critical point results for locally lipschitz function-als[J]. Comm. Appl. Nonlinear Anal., 1997, 4: 17-33.
[14] Hu S, Papageorgiou NS. Handbook of Multivalued Analysis Volume II: Applications[M]. Dordrecht: Kluwer Academic Publishers, 2000.
[15] Hu S, Papageorgiou NS. Handbook of Multivalued Analysis Volume I: The-ory[M]. Dordrecht: Kluwer Academic Publishers, 1997.
[16] Clarke FH. Optimization and Nonsmooth Analysis[M]. New York: Wiley, 1983.
[17] Adams R. Sobolev Spaces[M]. New York: Academic Press, 1975.
[18] Z. Denkowski, L. Gasi(?)ski, N.S. Papageorgiou, Existence and multiplicity of solutions for semilinear hemivariational inequalities at resonance[J]. Nonlinear Anal., 2007, 66: 1329-1340.
[19] X.L. Fan, Solutions for p(x)-Laplacian Dirichlet problem with singular coeffi-cients[J]. J. Math. Anal. Appl., 2005, 312: 464-477.
[20] X.L. Fan, On the sub-supersolution method for p(x)-Laplacian equations[J]. J.Math. Anal. Appl., 2007, 330: 665-682.
[21] M.E. Filippakis, L. Gasi(?)ski, N.S. Papageorgiou, Quasilinear hemivariational inequalities with strong resonance at infinity[J]. Nonlinear. Anal.. 2004. 56 : 331-345.
[22] X.L. Fan, Q.H. Zhang, Existence of solutions for p(x)-Laplacian Dirichlet prob-lem[J]. Nonlinear Anal., 2003, 52: 1843-1852.
[23] X.L. Fan,D. Zhao, On the generalized Orlicz-Sobolev space (?)(Ω)[J], J.GansuEduc. College, 1998, 12(1): 1 - 6.
[24] X.L. Fan, D. Zhao, On the spaces L~(p(x))(Ω) and (?)(Ω)[J]. J. Math. Anal.Appl., 2001,262: 749-760.
[25] X.L. Fan, Q.H. Zhang, D. Zhao, Eigenvalues of p(x)-Laplacian Dirichlet problem[J]. J. Math. Anal. Appl., 2005, 302: 306-317.
[26] X.L. Fan, Y.Z. Zhao, D. Zhao, Compact imbedding theorem with symmetry of Strauss-Lions type for the space (?)(Ω)[J]. J. Math. Anal. Appl. 2001, 255:333-348.
[27] A.B. Zang, p(x)-Laplacian equations satisfying Cerami condition[J]. J. Math.Anal. Appl, 2008, 337: 547-555.
[28] D. Goeleven, D. Motreanu, P.D. Panagiotopoulos, Eigenvalue problem for varitional hemivariational inequalities at resonance[J]. Nonlinear Anal., 1981, 33:161-180.
[29] L. Gasinski, N.S. Papageorgiou, Multiple solutions for semilinear hemivariational inequalies at resonance[J]. Publ. Math. Debrecen, 2001, 59: 121-146.
[30] L. Gasinski, N.S. Papageorgiou, Nonlinear Hemivariational inequalities at reso-nance[J]. Bull. Austral. Math. Soc., 1999, 60: 353-364.
[31] L. Gasi(?)ski, N.S. Papageorgiou, Existence of solutions and of multiple solutions for eigenvalue problems in hemivariational inequalities[J]. Adv. Math. Sci. Appl.,2001, 11:437-464.
[32] L. Gasinski, N.S. Papageorgiou, Nonsmooth critical point theory and nonlinear boundary value problems[M]. Boca Raton: Chapman and Hall, CRC Press, 2005.
[33] N.S.Papageorgiou,et al., Three nontrivial solutions for the p-Laplacian with a non-smooth potential[J]. Nonlinear Anal., 2008, 68: 3812-3827
[34] S.T. Kyritsi, N.S. Papageorgiou, Pairs of positive solutions for nonlinear elliptic equations with the p-Laplacian and a nonsmooth pothtial[J]. Ann. Mat. Pura Appl., 2005, 184: 449-472.
[35] S.T. Kyritsi, N.S. Papageorgiou, Nonsmooth critical point theory on closed convex sets and nonlinear hemivariational inequalities[J]. Nonlinear Anal. 2005, 61: 373-403.
[36] S.T. Kyritsi, N.S. Papageorgiou, Multiple solutions of constant sign for nonlinear nonsmooth eigenvalue problems near resonance[J]. Calc. Var. Partial Differential Equations, 2004, 20: 1-24.
[37] V.V. Jikov, S.M. Kozlov, O.A. Oleinik, Homogenization of Differential Operators and Integral Functionals[M]. Berlin: Springer-Verlag, 1994.
[38] D.Kandilakis, N.Kourogenis, N.S.Papageorgiou, Two nontrivial critical points for nonsmooth functional via local linking and applications[J]. J. Global Optim.,2006,34: 317-337.
[39] Z. Denkowski, S. Migorski, N.S. Papageorgiou, An Introduction to Nonlinear Analysis: Theory[M]. New York: Kluwer Academic/Plenum, 2003.
[40] J. Musielak, Orlicz Spaces and Modular spaces[M]. Berlin: Springer, 1983.
[41] D. Motreanu, V. Radulescu, Variational and non-variational Methods in Nonlinear Analysis and Boundary Value Problems[M]. Boston: Kluwer Academic Publishers, 2003.
[42] P.D. Panagiotopoulos, Hemivariational Inequalities. Applications in Mechanics and Engineering[M]. Berlin: Springer-Verlag, 1993.
[43] F. Gazzola, V. Radulescu, A nonsmooth critical point approach to some nonlinear elliptic equations in (?)~N[J]. Differential Integral Equations, 2000, 13: 47-60.
[44] A. Krist(?)ly, Infinitely many radial and non-radial solutions for a class of hemivariational inequalities[J]. Rocky Mountain Journal of Mathematics, 2005, 35:1173-1190.
[45] T. Bartsch, Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on (?)~N[J]. Comm. Partial Differential Equations, 1995, 20: 1725-1741.
[46] T. Bartsch, M. Willem, Infinitely many non-radial solutions of an Euclidean scalar field equation[J]. J. Funct. Anal., 1993, 117: 447-460.
[47] T. Bartsch, M. Willem, Infinitely many radial solutions of a semilinear elliptic problem in (?)~N[J]. Arch. Rational Mech. Anal., 1993, 124: 261-276.
[48] W. Krawcexicz, W. Marzantowicz, Some remarks on the Lusternik-Schnirelman method for nondifferentiable functionals invariant with respect to a finite group action[J]. Rocky Mountain Journal of Mathematics, 1990, 20: 1041-1049.
[49] N. C. Kourogenis, N. S. Papageorgiou, Nonsmooth critical point theory and nonlinear elliptic equations at resonance[J]. Kodai Math. J., 2000, 23: 108-135.
[50] D. Motreanu, C. Varga, A nonsmooth equivariant minimax principle[J]. Comm.Appl. Anal. 1999,3: 115-130.
[51] T. Bartsch, Infinitely many solutions of a symmetric dirichlet problem[J]. Nonlinear Anal. 1993,20: 1205-1216.
[52] G. Arioli, F. Gazzola, Existence and multiplicity results for quasilinear elliptic defferential systems[J]. Comm. Partial Differential Equatiions, 2000, 25: 125-153.