• Title, Summary, Keyword: Leray-Schauder degree

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LERAY-SCHAUDER DEGREE THEORY APPLIED TO THE PERTURBED PARABOLIC PROBLEM

  • Jung, Tacksun;Choi, Q-Heung
    • Korean Journal of Mathematics
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    • v.17 no.2
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    • pp.219-231
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    • 2009
  • We show the existence of at least four solutions for the perturbed parabolic equation with Dirichlet boundary condition and periodic condition when the nonlinear part cross two eigenvalues of the eigenvalue problem of the Laplace operator with boundary condition. We obtain this result by using the Leray-Schauder degree theory, the finite dimensional reduction method and the geometry of the mapping. The main point is that we restrict ourselves to the real Hilbert space instead of the complex space.

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AN APPLICATION OF THE LERAY-SCHAUDER DEGREE THEORY TO THE VARIABLE COEFFICIENT SEMILINEAR BIHARMONIC PROBLEM

  • Choi, Q-Heung;Jung, Tacksun
    • Korean Journal of Mathematics
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    • v.19 no.1
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    • pp.65-75
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    • 2011
  • We obtain multiplicity results for the nonlinear biharmonic problem with variable coefficient. We prove by the Leray-Schauder degree theory that the nonlinear biharmonic problem has multiple solutions for the biharmonic problem with the variable coefficient semilinear term under some conditions.

EXISTENCE OF MULTIPLE SOLUTIONS OF A SEMILINEAR BIHARMONIC PROBLEM WITH VARIABLE COEFFICIENTS

  • Jung, Tacksun;Choi, Q-Heung
    • Journal of the Chungcheong Mathematical Society
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    • v.24 no.1
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    • pp.121-130
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    • 2011
  • We obtain multiplicity results for the biharmonic problem with a variable coefficient semilinear term. We show that there exist at least three solutions for the biharmonic problem with the variable coefficient semilinear term under some conditions. We obtain this multiplicity result by applying the Leray-Schauder degree theory.

BOUNDARY VALUE PROBLEM FOR ONE-DIMENSIONAL ELLIPTIC JUMPING PROBLEM WITH CROSSING n-EIGENVALUES

  • JUNG, TACKSUN;CHOI, Q-HEUNG
    • East Asian mathematical journal
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    • v.35 no.1
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    • pp.41-50
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    • 2019
  • This paper is dealt with one-dimensional elliptic jumping problem with nonlinearities crossing n eigenvalues. We get one theorem which shows multiplicity results for solutions of one-dimensional elliptic boundary value problem with jumping nonlinearities. This theorem is that there exist at least two solutions when nonlinearities crossing odd eigenvalues, at least three solutions when nonlinearities crossing even eigenvalues, exactly one solutions and no solution depending on the source term. We obtain these results by the eigenvalues and the corresponding normalized eigenfunctions of the elliptic eigenvalue problem and Leray-Schauder degree theory.

STUDY ON THE PERTURBED PIECEWISE LINEAR SUSPENSION BRIDGE EQUATION WITH VARIABLE COEFFICIENT

  • Jung, Tacksun;Choi, Q-Heung
    • Korean Journal of Mathematics
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    • v.19 no.2
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    • pp.233-242
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    • 2011
  • We get a theorem that there exist at least two solutions for the piecewise linear suspension bridge equation with variable coefficient jumping nonlinearity and Dirichlet boundary condition when the variable coefficient of the nonlinear term crosses first two successive negative eigenvalues. We obtain this multiplicity result by applying Leray-Schauder degree theory.

TOPOLOGICAL APPROACH FOR THE MULTIPLE SOLUTIONS OF THE NONLINEAR PARABOLIC PROBLEM WITH VARIABLE COEFFICIENT JUMPING NONLINEARITY

  • Jung, Tacksun;Choi, Q-Heung
    • Korean Journal of Mathematics
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    • v.19 no.1
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    • pp.101-109
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    • 2011
  • We get a theorem which shows that there exist at least two or three nontrivial weak solutions for the nonlinear parabolic boundary value problem with the variable coefficient jumping nonlinearity. We prove this theorem by restricting ourselves to the real Hilbert space. We obtain this result by approaching the topological method. We use the Leray-Schauder degree theory on the real Hilbert space.

ELLIPTIC PROBLEM WITH A VARIABLE COEFFICIENT AND A JUMPING SEMILINEAR TERM

  • Choi, Q-Heung;Jung, Tacksun
    • Korean Journal of Mathematics
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    • v.20 no.1
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    • pp.125-135
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    • 2012
  • We obtain the multiple solutions for the fourth order elliptic problem with a variable coefficient and a jumping semilinear term. We have a result that there exist at least two solutions if the variable coefficient of the semilinear term crosses some number of the eigenvalues of the biharmonic eigenvalue problem. We obtain this multiplicity result by applying the Leray-Schauder degree theory.

INVARIANCE OF DOMAIN THEOREM FOR DEMICONTINUOUS MAPPINGS OF TYPE ( $S_+$)

  • Park, Jong-An
    • Bulletin of the Korean Mathematical Society
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    • v.29 no.1
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    • pp.81-87
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    • 1992
  • Wellknown invariance of domain theorems are Brower's invariance of domain theorem for continuous mappings defined on a finite dimensional space and Schauder-Leray's invariance of domain theorem for the class of mappings I+C defined on a infinite dimensional Banach space with I the identity and C compact. The two classical invariance of domain theorems were proved by applying the homotopy invariance of Brower's degree and Leray-Schauder's degree respectively. Degree theory for some class of mappings is a useful tool for mapping theorems. And mapping theorems (or surjectivity theorems of mappings) are closely related with invariance of domain theorems for mappings. In[4, 5], Browder and Petryshyn constructed a multi-valued degree theory for A-proper mappings. From this degree Petryshyn [9] obtained some invariance of domain theorems for locally A-proper mappings. Recently Browder [6] has developed a degree theory for demicontinuous mapings of type ( $S_{+}$) from a reflexive Banach space X to its dual $X^{*}$. By applying this degree we obtain some invariance of domain theorems for demicontinuous mappings of type ( $S_{+}$). ( $S_{+}$).

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ONE-DIMENSIONAL JUMPING PROBLEM INVOLVING p-LAPLACIAN

  • Jung, Tacksun;Choi, Q-Heing
    • Korean Journal of Mathematics
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    • v.26 no.4
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    • pp.683-700
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    • 2018
  • We get one theorem which shows existence of solutions for one-dimensional jumping problem involving p-Laplacian and Dirichlet boundary condition. This theorem is that there exists at least one solution when nonlinearities crossing finite number of eigenvalues, exactly one solutions and no solution depending on the source term. We obtain these results by the eigenvalues and the corresponding normalized eigenfunctions of the p-Laplacian eigenvalue problem when 1 < p < ${\infty}$, variational reduction method and Leray-Schauder degree theory when $2{\leq}$ p < ${\infty}$.

EXISTENCE OF POSITIVE SOLUTIONS FOR SINGULAR IMPULSIVE DIFFERENTIAL EQUATIONS WITH INTEGRAL BOUNDARY CONDITIONS

  • Miao, Chunmei;Ge, Weigao;Zhang, Zhaojun
    • The Pure and Applied Mathematics
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    • v.21 no.3
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    • pp.147-163
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    • 2014
  • In this paper, we study the existence of positive solutions for singular impulsive differential equations with integral boundary conditions $$\{u^{{\prime}{\prime}}(t)+q(t)f(t,u(t),u^{\prime}(t))=0,\;t{\in}\mathbb{J}^{\prime},\\{\Delta}u(t_k)=I_k(u(t_k),u^{\prime}(t_k)),\;k=1,2,{\cdots},p,\\{\Delta}u^{\prime}(t_k)=-L_k(u(t_k),u^{\prime}(t_k)),\;k=1,2,{\cdots},p,\\u=(0)={\int}_{0}^{1}g(t)u(t)dt,\;u^{\prime}=0,$$) where the nonlinearity f(t, u, v) may be singular at v = 0. The proof is based on the theory of Leray-Schauder degree, together with a truncation technique. Some recent results in the literature are generalized and improved.