Abstract
In this paper we deal with generalized coupled systems of integral equations of Hammerstein type with nonlinearities depending on several derivatives of both variables and we underline that both equations and both variables can have a different regularity. This detail is very important as it allows for the application to, for example, boundary value problems with coupled systems composed of differential equations of different orders and distinct boundary conditions. This issue will open a new field of applications to phenomena modeled by coupled systems requiring different types of regularity for the unknown functions.The arguments follow Guo–Krasnosel’skiĭ compression–expansion theory on cones and the kernel functions are nonnegative and verifying adequate sign and growth assumptions.The dependence of the derivatives is overcome by the construction of suitable cones taking into account certain conditions of sublinearity/superlinearity at the origin and at +∞.
Highlights
1 Introduction In this paper we deal with generalized coupled systems of integral equations of Hammerstein type
By using the fixed point theorem of mixed monotone operators in cones, they establish the conditions for the existence and uniqueness of positive solutions to the problem
To the best of our knowledge, it is the first time where coupled systems contain integral equations with nonlinearities depending on several derivatives of both variables and, the derivatives can be of different order on each variable and each equation. Both equations and both variables can have a different regularity. This detail is very important as it allows the application of our results, for example, to boundary value problems with coupled systems composed of differential equations of different orders and distinct boundary conditions on each unknown function
Summary
1 Introduction In this paper we deal with generalized coupled systems of integral equations of Hammerstein type, The theory of integral equations has been and continues to be a field of many research and applications. The existence, uniqueness, multiplicity, positivity and location of solutions are the most studied and predominant elements as regards Hammerstein integral equations.
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