• Title/Summary/Keyword: periodic boundary value problem

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ON PERIODIC BOUNDARY VALUE PROBLEMS OF HIGHER ORDER NONLINEAR FUNCTIONAL DIFFERENCE EQUATIONS WITH p-LAPLACIAN

  • Liu, Yuji;Liu, Xingyuan
    • Communications of the Korean Mathematical Society
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    • v.24 no.1
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    • pp.29-40
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    • 2009
  • Motivated by [Linear Algebra and its Appl. 420(2007), 218-227] and [Linear Algebra and its Appl. 425(2007), 171-183], we, in this paper, study the solvability of periodic boundary value problems of higher order nonlinear functional difference equations with p-Laplacian. Sufficient conditions for the existence of at least one solution of this problem are established.

MULTIPLE POSITIVE SOLUTIONS OF PERIODIC BOUNDARY VALUE PROBLEMS WITH IMPULSE

  • Song, Xiaohua;Zhao, Zengqin;Wang, Xin
    • Journal of applied mathematics & informatics
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    • v.27 no.3_4
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    • pp.875-883
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    • 2009
  • At least two positive solutions of a first-order periodic boundary value problem with impulse are obtained by establishing a new cone and the theorem of fixed point index. And at the end of this paper we give an example to illustrate the application of our main results.

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TRIPLE SOLUTIONS FOR THREE-ORDER PERIODIC BOUNDARY VALUE PROBLEMS WITH SIGN CHANGING NONLINEARITY

  • Tan, Huixuan;Feng, Hanying;Feng, Xingfang;Du, Yatao
    • Journal of applied mathematics & informatics
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    • v.32 no.1_2
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    • pp.75-82
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    • 2014
  • In this paper, we consider the periodic boundary value problem with sign changing nonlinearity $$u^{{\prime}{\prime}{\prime}}+{\rho}^3u=f(t,u),\;t{\in}[0,2{\pi}]$$, subject to the boundary value conditions: $$u^{(i)}(0)=u^{(i)}(2{\pi}),\;i=0,1,2$$, where ${\rho}{\in}(o,{\frac{1}{\sqrt{3}}})$ is a positive constant and f(t, u) is a continuous function. Using Leggett-Williams fixed point theorem, we provide sufficient conditions for the existence of at least three positive solutions to the above boundary value problem. The interesting point is the nonlinear term f may change sign.

MULTIPLE SOLUTIONS FOR THE NONLINEAR PARABOLIC PROBLEM

  • Jung, Tacksun;Choi, Q-Heung
    • Journal of the Chungcheong Mathematical Society
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    • v.22 no.2
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    • pp.251-259
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    • 2009
  • We investigate the multiple solutions for the nonlinear parabolic boundary value problem with jumping nonlinearity crossing two eigenvalues. We show the existence of at least four nontrivial periodic solutions for the parabolic boundary value problem. We restrict ourselves to the real Hilbert space and obtain this result by the geometry of the mapping.

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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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SOLVABILITY FOR THE PARABOLIC PROBLEM WITH JUMPING NONLINEARITY CROSSING NO EIGENVALUES

  • Jung, Tacksun;Choi, Q-Heung
    • Korean Journal of Mathematics
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    • v.16 no.4
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    • pp.545-551
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    • 2008
  • We investigate the multiple solutions for a parabolic boundary value problem with jumping nonlinearity crossing no eigenvalues. We show the existence of the unique solution of the parabolic problem with Dirichlet boundary condition and periodic condition when jumping nonlinearity does not cross eigenvalues of the Laplace operator $-{\Delta}$. We prove this result by investigating the Lipschitz constant of the inverse compact operator of $D_t-{\Delta}$ and applying the contraction mapping principle.

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Approximating Coupled Solutions of Coupled PBVPs of Non-linear First Order Ordinary Differential Equations

  • Dhage, Bapurao Chandrabhan
    • Kyungpook Mathematical Journal
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    • v.56 no.1
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    • pp.221-233
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    • 2016
  • The present paper proposes a new monotone iteration method for existence as well as approximation of the coupled solutions for a coupled periodic boundary value problem of first order ordinary nonlinear differential equations. A new hybrid coupled fixed point theorem involving the Dhage iteration principle is proved in a partially ordered normed linear space and applied to the coupled periodic boundary value problems for proving the main existence and approximation results of this paper. An algorithm for the coupled solutions is developed and it is shown that the sequences of successive approximations defined in a certain way converge monotonically to the coupled solutions of the related differential equations under some suitable mixed hybrid conditions. A numerical example is also indicated to illustrate the abstract theory developed in the paper.

EXISTENCE OF MULTIPLE PERIODIC SOLUTIONS FOR SEMILINEAR PARABOLIC EQUATIONS WITH SUBLINEAR GROWTH NONLINEARITIES

  • Kim, Wan-Se
    • Journal of the Korean Mathematical Society
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    • v.46 no.4
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    • pp.691-699
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    • 2009
  • In this paper, we establish a multiple existence result of T-periodic solutions for the semilinear parabolic boundary value problem with sublinear growth nonlinearities. We adapt sub-supersolution scheme and topological argument based on variational structure of functionals.