Periodic and almost periodic solutions for differential equations with reflection of the argument

Nontrivial solutions of Hammerstein integral equations with reflections

“…In a recent paper Cabada and Tojo [6] studied, by means of methods and results present in [4,5], the first order operator u ′ (t) + ω u(−t) coupled with periodic boundary value conditions, describing the eigenvalues of the operator and providing the expression of the associated Green's function in the non-resonant case. One motivation for studying this particular problem is that differential equations with reflection of the argument have seen growing interest along the years, see for example the papers [1,3,6,10,11,21,[23][24][25][26]31] and references therein. In [6] the authors provide the range of values of the real parameter ω for which the Green's function has constant sign and apply these results to prove the existence of constant sign solutions for the nonlinear periodic problem with reflection of the argument u ′ (t) = h(t, u(t), u(−t)), t ∈ [−T, T ], u(−T ) = u(T ).…”

Nonlinear perturbed integral equations related to nonlocal boundary value problems

“…On the other hand, BVPs where nonlocal terms occur in the differential equation have been studied by a number of authors. For example, the case of equations with reflection of the argument has been investigated by Andrade and Ma [3], Cabada and co-authors [6], Piao [45,46], Piao and Xin [47], Wiener and Aftabizadeh [61], the case of equations with deviated arguments has be en studied by Jankowski [34][35][36], Figueroa and Pouso [14] and Szatanik [51,52] and the case of equations that involve the average of the solution has been considered by Andrade and Ma [3], Chipot and Rodrigues [8] and Infante [27].…”

Nonzero solutions of perturbed Hammerstein integral equations with deviated arguments and applications

“…We point out that when σ(t) = a + b − t this type of perturbed Hammerstein integral equation is well-suited to treat problems with reflections. Differential equations with reflection of the argument have been subject to a growing interest along the years, see for example the papers [1,3,6,7,8,9,22,23,45,52,53,55,56,57,71] and references therein. We apply our theory to prove the existence of nontrivial solutions of the first order functional periodic boundary value problem…”