Coupled lower and upper solution approach for the existence of Solutions of nonlinear coupled system with nonlinear coupled boundary conditions

Talib, Imran; Asif, Naseer Ahmad; Tunç, Cemil

doi:10.4067/s0716-09172016000100007

Cited by 5 publications

(3 citation statements)

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“…The methodology of upper and lower solutions has found extensive applications in the analysis of boundary value problems associated with nonlinear differential equations [3][4][5][6][7][8][9]. In addition, some scholars have also focused on how to apply upper and lower solution methods to solve coupled differential systems [10][11][12][13], but currently, there are few achievements in this type of research.…”

Section: Introductionmentioning

confidence: 99%

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Upper and lower solutions method for a class of second-order coupled systems

Yu,

Bai,

Shang

2024

Bound Value Probl

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This paper provides a class of upper and lower solution definitions for second-order coupled systems by transforming the fourth-order differential equation into a second-order differential system. Then, by constructing a homotopy parameter and utilizing the maximum principle, we propose an upper and lower solutions method for studying a class of second-order coupled systems with Dirichlet boundary conditions and obtain an existence result.

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Section: Introductionmentioning

confidence: 99%

“…In 2016, by applying the upper and lower solutions method combined with Schauder's fixed point theorem, Talib, Asif, and Tunc [11] studied the coupled second-order system…”

Section: Introductionmentioning

confidence: 99%

Upper and lower solutions method for a class of second-order coupled systems

Yu,

Bai,

Shang

2024

Bound Value Probl

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“…The approach was further extended in [34] to second-order nonlinear coupled ODSs with generalized NBCs. One of the interests in this paper was the unification of the existence criteria of certain NBVPs which have been previously treated on a case-by-case basis.…”

Section: Introductionmentioning

confidence: 99%

New results and applications on the existence results for nonlinear coupled systems

et al. 2021

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In this manuscript, we study a certain classical second-order fully nonlinear coupled system with generalized nonlinear coupled boundary conditions satisfying the monotone assumptions. Our new results unify the existence criteria of certain linear and nonlinear boundary value problems (BVPs) that have been previously studied on a case-by-case basis; for example, Dirichlet and Neumann are special cases. The common feature is that the solution of each BVPs lies in a sector defined by well-ordered coupled lower and upper solutions. The tools we use are the coupled lower and upper solutions approach along with some results of fixed point theory. By means of the coupled lower and upper solutions approach, the considered BVPs are logically modified to new problems, known as modified BVPs. The solution of the modified BVPs leads to the solution of the original BVPs. In our case, we only require the Nagumo condition to get a priori bound on the derivatives of the solution function. Further, we extend the results presented in (Franco et al. in Extr. Math. 18(2):153–160, 2003; Franco et al. in Appl. Math. Comput. 153:793–802, 2004; Franco and O’Regan in Arch. Inequal. Appl. 1:423–430, 2003; Asif et al. in Bound. Value Probl. 2015:134, 2015). Finally, as an application, we consider the fully nonlinear coupled mass-spring model.

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Second-order strongly nonlinear impulsive coupled systems

Minhós,

Rodrigues

2024

J. Fixed Point Theory Appl.

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This paper presents sufficient conditions for the solvability of a second-order coupled system, composed of two differential equations involving different laplacians applied to fully discontinuous nonlinearities, two-point boundary conditions, and generalized impulsive effects. Applying the lower and upper solutions technique and Schauder’s fixed point theorem, it is obtained an existence and localization theorem, based on local monotone properties on the nonlinearities and the impulsive functions.

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Coupled lower and upper solution approach for the existence of Solutions of nonlinear coupled system with nonlinear coupled boundary conditions

Cited by 5 publications

References 20 publications

Upper and lower solutions method for a class of second-order coupled systems

Upper and lower solutions method for a class of second-order coupled systems

New results and applications on the existence results for nonlinear coupled systems

Second-order strongly nonlinear impulsive coupled systems

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