• Title/Summary/Keyword: Upper and lower solution method

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UPPER AND LOWER SOLUTION METHOD FOR FRACTIONAL EVOLUTION EQUATIONS WITH ORDER 1 < α < 2

  • Shu, Xiao-Bao;Xu, Fei
    • Journal of the Korean Mathematical Society
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    • v.51 no.6
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    • pp.1123-1139
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    • 2014
  • In this work, we investigate the existence of the extremal solutions for a class of fractional partial differential equations with order 1 < ${\alpha}$ < 2 by upper and lower solution method. Using the theory of Hausdorff measure of noncompactness, a series of results about the solutions to such differential equations is obtained.

Upper and lower solutions for a singular p-Laplacian system

  • Kim, Chan-Gyun;Lee, Eun-Kyoung
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.11 no.4
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    • pp.89-99
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    • 2007
  • In this paper, we define the upper and lower solutions for a p-Laplacian system with singular nonlinearity at the boundaries. And we prove the theorem for the upper and power solutions method.

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EXISTENCE OF THE THIRD POSITIVE RADIAL SOLUTION OF A SEMILINEAR ELLIPTIC PROBLEM ON AN UNBOUNDED DOMAIN

  • Ko, Bong-Soo;Lee, Yong-Hoon
    • Journal of the Korean Mathematical Society
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    • v.39 no.3
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    • pp.439-460
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    • 2002
  • We prove the multiplicity of ordered positive radial solutions for a semilinear elliptic problem defined on an exterior domain. The key argument is to prove the existence of the third solution in presence of two known solutions. For this, we obtain some partial results related to three solutions theorem for certain singular boundary value problems. Proof are mainly based on the upper and lower solutions method and degree theory.

THE METHOD OF LOWER AND UPPER SOLUTIONS FOR IMPULSIVE FRACTIONAL EVOLUTION EQUATIONS IN BANACH SPACES

  • Gou, Haide;Li, Yongxiang
    • Journal of the Korean Mathematical Society
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    • v.57 no.1
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    • pp.61-88
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    • 2020
  • In this paper, we investigate the existence of mild solutions for a class of fractional impulsive evolution equation with periodic boundary condition by means of the method of upper and lower solutions and monotone iterative method. Using the theory of Kuratowski measure of noncompactness, a series of results about mild solutions are obtained. Finally, two examples are given to illustrate our results.

Sensitivity Analysis on the Non-tree Solution of the Minimum Cost Flow Problem (최소비용문제의 비정점 최적해에 대한 감도분석)

  • 정호연;박순달
    • Journal of the Korean Operations Research and Management Science Society
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    • v.20 no.1
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    • pp.1-10
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    • 1995
  • The purpose of this paper is to develop a method of the sensitivity analysis that can be applied to a non-tree solution of the minimum cost flow problem. First, we introduce two types of sensitivity analysis. A sensitivity analysis of Type 1a is the well known method applicable to a tree solution. However this method can not be applied to a non-tree solution. So we propose a sensitivity analysis of Type 2 that keeps solutions of upper bounds at upper bounds, those of lower bounds at lower bounds, and those of intermediate values at intermediate values. For the cost coefficient we present a method that the sensitivity analysis of Type 2 is solved by finding the shortest path. Besides we also show that the results of Type 2 and Type 1 are the same in a spanning tree solution. For the right-hand side constant or the capacity, the sensitivity analysis of Type 2 is solved by a simple calculation using arcs with intermediate values.

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EXISTENCE AND UNIQUENESS OF A SOLUTION FOR FIRST ORDER NONLINEAR LIOUVILLE-CAPUTO FRACTIONAL DIFFERENTIAL EQUATIONS

  • Nanware, J.A.;Gadsing, Madhuri N.
    • Nonlinear Functional Analysis and Applications
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    • v.26 no.5
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    • pp.1011-1020
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    • 2021
  • In this paper, first order nonlinear Liouville-Caputo fractional differential equations is studied. The existence and uniqueness of a solution are investigated by using Krasnoselskii and Banach fixed point theorems and the method of lower and upper solutions. Finally, an example is given to illustrate our results.

Sensitivity Analysis on the Degenerate Tree Solution of the Minimum Cost Flow Problem (최소비용문제의 퇴화 정점 최적해에 대한 감도분석)

  • Chung, Ho-Yeon;Park, Soon-Dal
    • IE interfaces
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    • v.7 no.3
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    • pp.193-199
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    • 1994
  • The purpose of this paper is to develop a method of the sensitivity analysis that can be applicable to a degenerate tree solution of the minimum cost flow problem. First, we introduce two types of sensitivity analysis. A sensitivity analysis of Type 1 is the well known method applicable to a spanning tree solution. However, this method have some difficulties in case of being applied to a degenerate tree solution. So we propose a sensitivity analysis of Type 2 that keeps solutions of upper bounds remaining at upper bounds, those of lower bounds at lower bounds, and those of intermediate values at intermediate values. For the cost coefficient, we present a method that the sensitivity analysis of Type 2 is solved by using the method of a sensitivity analysis of Type 1. Besides we also show that the results of sensitivity analysis of Type 2 are union set of those of Type 1 sensitivity analysis. For the right-hand side constant or the capacity, we present a simple method for the sensitivity analysis of Type 2 which uses arcs with intermediate values.

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THE ITERATION METHOD OF SOLVING A TYPE OF THREE-POINT BOUNDARY VALUE PROBLEM

  • Liu, Xiping;Jia, Mei
    • Journal of applied mathematics & informatics
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    • v.27 no.3_4
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    • pp.475-487
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    • 2009
  • This paper studies the iteration method of solving a type of second-order three-point boundary value problem with non-linear term f, which depends on the first order derivative. By using the upper and lower method, we obtain the sufficient conditions of the existence and uniqueness of solutions. Furthermore, the monotone iterative sequences generated by the method contribute to the minimum solution and the maximum solution. And the error estimate formula is also given under the condition of unique solution. We apply the solving process to a special boundary value problem, and the result is interesting.

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