含有广义p-Laplacian算子的双曲型非线性微分方程的单调性方法

Abstract

Abstract:

By using the results on the perturbations of the ranges for pseudo-monotone operators and maximal monotone operators, we present the abstract result for the existence of the unique solution of hyperbolic type nonlinear differential equation involving the generalized p-Laplacian with mixed boundaries. It is an extension of the corresponding studies on nonlinear elliptic equation and nonlinear parabolic equation with generalized p-Laplacian. Some new techniques are employed in the discussion. Moreover, the connections between the unique solution studied in this paper and the zero point of a maximal monotone operator is exemplified.

Key words: Pseudo-monotone operator, p-Laplacian, Subdifferential, Hyperbolic type nonlinear differential equation, Mixed boundaries

CLC Number:

O177.91

Wei Li, Agarwal Ravi P. Monotonicity Method for Hyperbolic Type Nonlinear Differential Equation with Generalized p-Laplacian[J].Acta mathematica scientia,Series A, 2015, 35(5): 867-877.

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References

[1] Barbu V. Nonlinear Semigroups and Differential Equations in Banach Spaces. Leyden:Noordhoff, 1976
[2] Pascali D, Sburlan S. Nonlinear Mappings of Monotone Type. The Netherlands:Sijthoff and Noordhoff, 1978
[3] Lions J L. Quelques Methods de Resolution des Problems aux Limites Nonlineaires. Paris:Dunod Gauthier-Villars, 1969
[4] Calvert B D, Gupta C P. Nonlinear elliptic boundary value problems in L^p-spaces and sums of ranges of accretive operators. Nonlinear Analysis, 1978, 2:1-26
[5] Brezis H. Equations et inequalitions nonlinearies dans les espaces vectoriels an dualite. Ann Inst Fourier Grenoble, 1968, 18(1):115-175
[6] Chen Z C, Luo T. The initial-boundary value problem for quasilinear integro-differential equations. International J Differential Equa Appl, 2002, 6:299-306
[7] Reich S. The range of sums of accretive and monotone operators. J Math Anal Appli, 1979, 68:310-317
[8] Wei L, Zhou H Y. Research on the existence of solution of equation involving p-Laplacian operator. Applied Mathematics A J Chinese Univ(Ser B), 2006, 21:191-202
[9] 魏利. 与广义p-Laplace算子相关的非线性边值问题在L^s(Ω)空间中解的存在性. 数学物理学报, 2010, 30(A):1111-1116
[10] Wei L, He Z. The applications of theories of accretive operators to nonlinear elliptic boundary value problems in L^p-spaces. Nonlinear Analysis, 2001, 46:199-211
[11] Wei L, Zhou H Y. The existence of solutions of nonlinear boundary value problem involving the p-Laplacian operator in L^s-spaces. J Systems Science and Complexity, 2005, 18:511-521
[12] Wei L, Agarwal R P. Existence of solutions to nonlinear Neumann boundary value problem with generalized p-Laplacian operator. Comput Math Appl, 2008, 56:530-541
[13] Wei L, Agarwal R P, Wong P J Y. Existence of solutions to nonlinear parabolic boundary value problems with generalized p-Laplacian operator. Advances in Math Sci Appli, 2010, 20(2):423-445
[14] Kamimura S, Takahashi W. Approximating solutions of maximal monotone operators in Hilbert spaces. J Approx Theory, 2000, 106(2):226-240
[15] Rockafellar R T. Monotone operators and the proximal point algorithm. SIAM J Control Optim, 1976, 14:877-898
[16] Ibaraki T, Takahashi W. Weak and strong convergence theorems for new resolvents of maximal monotone operators in Banach spaces. Adv Math Econ, 2007, 10:51-64

Monotonicity Method for Hyperbolic Type Nonlinear Differential Equation with Generalized p-Laplacian

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