An application of the Krasnoselskii theorem to systems of algebraic equations

Wang, Haiyan; Wang, Melinda; Wang, Emily

doi:10.1007/s12190-011-0498-8

Cited by 8 publications

(11 citation statements)

References 14 publications

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“…Thus, all conditions of Theorem 12 hold. In [11,23], it is assumed that is a positive or nonnegative matrix, where is independent on the variable . In the above section, the coefficient matrix is dependent on the variable .…”

Section: More General Resultsmentioning

confidence: 99%

“…Generally, there are no good methods for solving such systems, even in the simple case of only two equations of the form: 1 ( 1 , 2 ) = 0 and 2 ( 1 , 2 ) = 0; see van der Laan et al [14]. Thus, some existing theorems of zero points or fixed points had been extensively established by a number of authors; see [1,2,4,[10][11][12][13][15][16][17][18][19][20][21][22][23] and so forth.…”

Section: Introductionmentioning

confidence: 99%

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Existence of Positive Solutions for a Class of Nonlinear Algebraic Systems

Zhang

Feng

2016

Mathematical Problems in Engineering

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Based on Guo-Krasnoselskii’s fixed point theorem, the existence of positive solutions for a class of nonlinear algebraic systems of the formx=GFxis studied firstly, whereGis a positiven×nsquare matrix,x=col⁡(x1,x2,…,xn), andF(x)=col⁡(f(x1),f(x2),…,f(xn)), where,F(x)is not required to be satisfied sublinear or superlinear at zero point and infinite point. In addition, a new cone is constructed inRn. Secondly, the obtained results can be extended to some more general nonlinear algebraic systems, where the coefficient matrixGand the nonlinear term are depended on the variablex. Corresponding examples are given to illustrate these results.

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Section: More General Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Existence of Positive Solutions for a Class of Nonlinear Algebraic Systems

Zhang

Feng

2016

Mathematical Problems in Engineering

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“…The CTE method is developed to find interaction solutions between solitons and any other types of solitary waves [17]. The method has been valid for lots of nonlinear integrable systems [20][21][22][23][24]. According to the CTE method [20][21][22][23][24][25], the expansion solution has the form…”

Section: Cte Methods For Bsmkdv-b Systemmentioning

confidence: 99%

“…The method has been valid for lots of nonlinear integrable systems [20][21][22][23][24]. According to the CTE method [20][21][22][23][24][25], the expansion solution has the form…”

Section: Cte Methods For Bsmkdv-b Systemmentioning

confidence: 99%

Interaction solutions for supersymmetric mKdV-B equation

Ren¹

2016

Chinese Journal of Physics

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The N = 1 supersymmetric mKdV-B system is transformed to a system of coupled bosonic equations by using the bosonization approach. The bosonized supersymmetric mKdV-B (BSmKdV-B) equation includes the usual mKdV equation and a linear partial differential equation. The bosonization approach can thus effectively avoid difficulties caused by anticommutative fermionic fields of the supersymmetric systems. The consistent tanh expansion (CTE) method is applied to the BSmKdV-B equation. A nonauto-Bäcklund (BT) theorem is obtained by using CTE method. The interaction solutions among solitons and other complicated waves including Painlevé II waves and periodic cnoidal waves are given through a nonauto-BT theorem. The features of the soliton-cnoidal interaction solutions are investigated both in analytical and graphical ways by combining the mapping and deformation method.

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“…f : R → R is continuous, and G = (g ij ) n×n is an n × n square matrix. The existence of positive solutions for system (1) has been extensively studied in the literature; see [2,3,5,6,10,11,13,15,17,20,21,24,[27][28][29] and the references therein. However, to the best of our knowledge, almost all obtained results require that the coefficient matrix G ≥ 0 or G > 0, where G ≥ 0 if g ij ≥ 0 and G > 0 if g ij > 0 for (i, j) ∈ [1, n] An n × n square matrix G is called a sign-changing coefficient matrix if its elements change the sign.…”

Section: Introductionmentioning

confidence: 99%

Positive solutions of a nonlinear algebraic system with sign-changing coefficient matrix

et al. 2020

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Existence of positive solutions for the nonlinear algebraic system $x=\lambda GF ( x ) $ x = λ G F ( x ) has been extensively studied when the $n\times n$ n × n coefficient matrix G is positive or nonnegative. However, to the best of our knowledge, few results have been obtained when the coefficient matrix changes sign. In this case, some commonly applied analysis methods such as the cone theory, the Krein–Rutman theorem, the monotone iterative techniques, and so on cannot be directly applied. In this note, we prove the existence of positive solutions for the above nonlinear algebraic system with sign-changing coefficient matrix taking the advantages of the classical Brouwer fixed point theorem combined with a decomposition condition on the coefficient matrix. We provide an example in solving a second-order difference equation with periodic boundary conditions to illustrate the applications of the results.

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An application of the Krasnoselskii theorem to systems of algebraic equations

Cited by 8 publications

References 14 publications

Existence of Positive Solutions for a Class of Nonlinear Algebraic Systems

Existence of Positive Solutions for a Class of Nonlinear Algebraic Systems

Interaction solutions for supersymmetric mKdV-B equation

Positive solutions of a nonlinear algebraic system with sign-changing coefficient matrix

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