Existence of solutions of nonlinear systems subject to arbitrary linear non-local boundary conditions

IF 1.4 3区数学 Q1 MATHEMATICS

Journal of Fixed Point Theory and Applications Pub Date : 2023-10-05 DOI:10.1007/s11784-023-01083-7

Alberto Cabada, Lucía López-Somoza, Mouhcine Yousfi

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Abstract

Abstract In this paper, we obtain an explicit expression for the Green’s function of a certain type of systems of differential equations subject to non-local linear boundary conditions. In such boundary conditions, the dependence on certain parameters is considered. The idea of the study is to transform the given system into another first-order differential linear system together with the two-point boundary value conditions. To obtain the explicit expression of the Green’s function of the considered linear system with non-local boundary conditions, it is assumed that the Green’s function of the homogeneous problem, that is, when all the parameters involved in the non-local boundary conditions take the value zero, exists and is unique. In such a case, the homogeneous problem has a unique solution that is characterized by the corresponding Green’s function g . The expression of the Green’s function of the given system is obtained as the sum of the function g and a part that depends on the parameters involved in the boundary conditions and the expression of function g . The novelty of our work is that in the system to be studied, the unknown functions do not appear separated neither in the equations nor in the boundary conditions. The existence of solutions of nonlinear systems with linear non-local boundary conditions is also studied. We illustrate the obtained results in this paper with examples.

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任意线性非局部边界条件下非线性系统解的存在性

摘要本文得到了一类非局部线性边界条件下的微分方程组的格林函数的显式表达式。在这种边界条件下，考虑了对某些参数的依赖。本研究的思想是结合两点边值条件，将给定系统转化为另一个一阶微分线性系统。为了得到所考虑的具有非局部边界条件的线性系统的格林函数的显式表达式，假设齐次问题的格林函数存在且唯一，即当非局部边界条件所涉及的所有参数均取0时。在这种情况下，齐次问题有一个唯一解，其特征为对应的格林函数g。得到给定系统的格林函数表达式为函数g和依赖于边界条件中涉及的参数和函数g表达式的部分的和。我们的工作的新颖之处在于，在待研究的系统中，无论是在方程中还是在边界条件中，未知函数都不会出现分离。研究了具有线性非局部边界条件的非线性系统解的存在性。本文用实例对所得结果进行了说明。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Fixed Point Theory and Applications 数学-数学

CiteScore

3.10

自引率

5.60%

发文量

审稿时长

>12 weeks

期刊介绍： The Journal of Fixed Point Theory and Applications (JFPTA) provides a publication forum for an important research in all disciplines in which the use of tools of fixed point theory plays an essential role. Research topics include but are not limited to: (i) New developments in fixed point theory as well as in related topological methods, in particular: Degree and fixed point index for various types of maps, Algebraic topology methods in the context of the Leray-Schauder theory, Lefschetz and Nielsen theories, Borsuk-Ulam type results, Vietoris fractions and fixed points for set-valued maps. (ii) Ramifications to global analysis, dynamical systems and symplectic topology, in particular: Degree and Conley Index in the study of non-linear phenomena, Lusternik-Schnirelmann and Morse theoretic methods, Floer Homology and Hamiltonian Systems, Elliptic complexes and the Atiyah-Bott fixed point theorem, Symplectic fixed point theorems and results related to the Arnold Conjecture. (iii) Significant applications in nonlinear analysis, mathematical economics and computation theory, in particular: Bifurcation theory and non-linear PDE-s, Convex analysis and variational inequalities, KKM-maps, theory of games and economics, Fixed point algorithms for computing fixed points. (iv) Contributions to important problems in geometry, fluid dynamics and mathematical physics, in particular: Global Riemannian geometry, Nonlinear problems in fluid mechanics.