Generalized Hammerstein Equations and Applications

Graef, John R.; Kong, Lingju; Minhós, Feliz

doi:10.1007/s00025-016-0615-y

Cited by 10 publications

(7 citation statements)

References 19 publications

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“…Motivated by this work we consider the coupled integral system (1). Our results are based on [20] and [15], extending the results to systems of coupled Hammerstein-type integral equations with nonlinearities in both unknown functions and their first derivatives.…”

Section: Introductionmentioning

confidence: 60%

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On coupled systems of Hammerstein integral equations

Sousa¹,

Minhós²

2019

Bound Value Probl

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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'skiȋ 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 +∞.

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Section: Introductionmentioning

confidence: 60%

“…The first derivative of the Green's functions are positive, with ≤ t ≤ s ≤ 1.Therefore (A1) holds and, to show that (A2) is verified, we follow the arguments in[15] (Lemmas 4.1-4.4), for G 1 and any (t, s) ∈ [0, 1] × [0, 1], to obtain 0 ≤ G 1 (t, s) ≤ 10s(1s) := φ…”

mentioning

confidence: 87%

On coupled systems of Hammerstein integral equations

Sousa¹,

Minhós²

2019

Bound Value Probl

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“…We also highlight recent works, not necessarily in real line or half-line, on Hammerstein-type integral equations, with several approaches and applications in References [13,[15][16][17][18][19][20][21][22] and the references therein.…”

Section: Introductionmentioning

confidence: 94%

Solvability of Coupled Systems of Generalized Hammerstein-Type Integral Equations in the Real Line

Minhós

Sousa

2020

Mathematics

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In this work, we consider a generalized coupled system of integral equations of Hammerstein-type with, eventually, discontinuous nonlinearities. The main existence tool is Schauder's fixed point theorem in the space of bounded and continuous functions with bounded and continuous derivatives on R, combined with the equiconvergence at ±∞ to recover the compactness of the correspondent operators. To the best of our knowledge, it is the first time where coupled Hammerstein-type integral equations in real line are considered with nonlinearities depending on several derivatives of both variables and, moreover, the derivatives can be of different order on each variable and each equation. On the other hand, we emphasize that the kernel functions can change sign and their derivatives in order to the first variable may be discontinuous. The last section contains an application to a model to study the deflection of a coupled system of infinite beams.

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“…This type of third order three-point boundary value problems can be seen as a particular case of multipoint problems (as in [12]), nonlocal problems (see [9]), functional problems (as in [3]), or integral equations (see [4]). Therefore all the applications for the above type of problems hold for our problem.…”

Section: Introductionmentioning

confidence: 99%

Existence, non-existence and multiplicity results for a third order eigenvalue three-point boundary value problem

Cabada¹,

López-Somoza²,

Minhós³

2017

J. Nonlinear Sci. Appl.

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This paper provides sufficient conditions to guarantee the existence, non-existence and multiplicity of solutions for a third order eigenvalue fully differential equation, coupled with three point boundary value conditions. Although the change of sign, some bounds for the second derivative of the Green's function are obtained, which allow to define a different kind of cone that, as far as we know, has not been previously used in the literature. The main arguments are based on the fixed point index theory for bounded and unbounded sets. Some examples are presented in order to show that the different existence theorems proved are not comparable.

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Generalized Hammerstein Equations and Applications

Cited by 10 publications

References 19 publications

On coupled systems of Hammerstein integral equations

On coupled systems of Hammerstein integral equations

Solvability of Coupled Systems of Generalized Hammerstein-Type Integral Equations in the Real Line

Existence, non-existence and multiplicity results for a third order eigenvalue three-point boundary value problem

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