Infinite Systems of Differential and Integral Equations: Current State and Some Open Problems Article Swipe
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· 2025
· Open Access
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· DOI: https://doi.org/10.3390/sym17040575
· OA: W4409335036
This paper presents some topics of the theory of infinite systems of differential and integral equations. Our considerations focus on showing the symmetries that can be encountered in the theory of nonlinear differential and integral equations from the viewpoint of initial conditions, such as the symmetry of the behaviour of solutions of differential equations with respect to initial conditions, the symmetry of the behaviour of solutions in +∞ and −∞ and some other essential properties of solutions of differential and integral equations. First of all, we describe the fundamental facts connected with the theory of infinite systems of both differential and integral equations. Particular attention is paid to the location of infinite systems of the mentioned equations in a suitable Banach space. Indeed, we define the spaces in question and describe the basic properties of those spaces. Next, we discuss conditions imposed on terms of equations of the considered infinite systems that guarantee the existence of solutions of those systems and allow us to obtain some essential information on those solutions. Moreover, after the description of the current state of investigations concerning the theory of infinite systems of differential and integral equations, we formulate a few open problems concerning the mentioned systems of equations.
