Nontrivial solutions of systems of perturbed Hammerstein integral equations with functional terms

We discuss the solvability of a fairly general class of systems of perturbed Hammerstein integral equations with functional terms that depend on several parameters. The nonlinearities and the functionals are allowed to depend on the components of the system and their derivatives. The results are applicable to systems of nonlocal second order ordinary differential equations subject to functional boundary conditions, this is illustrated in an example. Our approach is based on the classical fixed point index.


Introduction
Nonlocal differential equations have seen recently growing attention by researchers, both in the context of ODEs and PDEs. One motivation for studying this class of equations is that nonlocal terms often occur in physical models, we refer the reader to the paper by Stanćzy [14] for nonlocalities involving averaging processes, and to the review by Ma [13] for Kirchhoff-type problems.
Here we proceed in a different way; rather than studying a specific boundary value problem (BVP), we provide new results regarding the existence and non-existence of non-zero solutions of the following class of systems of integral equations with functional terms, namely where i = 1, 2, . . . , n, u = (u 1 , . . . , u n ), u ′ = (u ′ 1 , . . . , u ′ n ), f i are continuous, γ ij are continuously differentiable, h ij and w i are suitable functionals, λ i and η ij are positive parameters.
When dealing with systems of second order BVPs, the functional terms w i occurring in (1.4) can be used to incorporate the nonlocalities that appear in the differential equations, while the functionals h ij originate directly from the BCs. In the context of positive solutions, the idea of incorporating the nonlocal terms of differential equations within the nolinearities has been exploited in the case of equations by Fija lkowski and Przeradzki [4] and Enguiça and Sanchez [3], while the case of systems of second order elliptic operators has been considered by the author [8,9]. We seek solutions of the system (1.4) in a product of cones of a kind that differs from (1.3); in particular we work on products of cones in the space C 1 [0, 1] where the functions are positive on a subinterval of [0, 1] and are allowed to change sign elsewhere, this follows the line of research initiated by the author and Webb in [11]. We stress that ours is a larger cone than the one used by the author and Minhós [10], where some additional constrains on the growth of the derivatives are embedded within the cone, a setting not applicable to the present class of systems due to the assumptions on the kernels. As in the case of elliptic equations [9], our approach can cover different kinds of nonlocalities in the differential equations and several types of BCs: local, nonlocal, linear and nonlinear. There exists a wide literature on nonlocal/nonlinear BCs, we refer the reader to the papers [2,5] and references therein.
The proof of the existence result relies on the classical fixed point index, while for the non-existence we use an elementary argument. We conclude by illustrating, in an example, how our theoretical results can be applied to a system of nonlocal second order ODEs that presents coupling between the components of the system in the nonlocal terms occurring in the equations and in the BCs. 2

Existence and nonexistence of nontrivial solutions
We discuss the solvability of the system of perturbed integral equations of the type We make the following assumptions on the terms that occur in (2.1).
We work in the product space We define the operator T as With the assumptions above, it routine to show that T maps K to K and is compact.
The next result summarizes the main properties of the classical fixed point index for compact maps, for more details we refer the reader to [1,7]. In what follows the closure and the boundary of subsets of a cone K are understood to be relative to K. (i) If there exists e ∈ C \ {0} such that x = Tx + λe for all x ∈ ∂D C and all λ > 0, then For ρ ∈ (0, ∞), we define the set  .
In the next Lemma, we restrict the growth of the nonlinearities in only one of the components.
Proof. We show that u = T u + σ1 for every u ∈ ∂K ρ and every σ > 0. If not, there exists u ∈ ∂K ρ and σ > 0 such that u = T u + σ1. Note that u i 0 ∞ ≤ u = ρ, therefore for every a contradiction, since σ > 0. Therefore we obtain i K (T, K ρ ) = 0.
With these ingredients we can state the following existence and localization result.
Theorem 2.5. Assume that either of the following conditions holds.
Proof. We prove the result under the assumption (S), the other case is similar. If T has fixed point either on ∂K ρ 1 or on ∂K ρ 2 we are done. If this is not the case, by Lemma 2.3 we have i K (T, K ρ 1 ) = 0 and by Lemma 2.2 we obtain i K (T, K ρ 2 ) = 1. Therefore we have i K (T, K ρ 2 \ K ρ 1 ) = 1, which proves the result.
We now provide a non-existence result that allows different growths in the components of the system.
for every u ∈ K ρ and j = 1, 2, Then the system (2.1) has at most the zero solution in K ρ .
If i 0 ∈ J , then the assumptions in (N J ) imply that, for every t ∈ [a i , b i ], we havẽ We conclude with the following example, which illustrates the applicability of the above results.