Nontrivial solutions of Hammerstein integral equations with reflections

Using the theory of fixed point index, we establish new results for the existence of nonzero solutions of Hammerstein integral equations with reflections. We apply our results to a first order periodic boundary value problem with reflections.


Introduction
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 ). (1.1) The methodology, analogous to the one utilized by Torres [27] in the case of ordinary differential equations, is to rewrite the problem (1.1) as an Hammerstein integral equation with reflections of the type
In this paper we continue the study of [6] and we prove new results regarding the existence of nontrivial solutions of Hammerstein integral equations with reflections of the form where the kernel k is allowed to be not of constant sign. In order to do this, we extend the results of [15], valid for Hammerstein integral equations without reflections, to the new context. We make use of a cone of functions that are allowed to change sign combined with the classical fixed point index for compact maps (we refer to [2] or [9] for further information). As an application of our theory we prove the existence of nontrivial solutions of the periodic problem with reflections (1.1).

The case of kernels that change sign
We begin with the case of kernels that are allowed to change sign. We impose the following conditions on k, f , g that occur in the integral equation
We recall the following definition.
Definition 2.1. Let X be a Banach Space. A cone on X is a closed, convex subset of X such that λ x ∈ K for x ∈ K and λ ≥ 0 and K ∩ (−K) = {0}.
Here we work in the space C[−T, T ], endowed with the usual supremum norm, and we use the cone The cone K has been essentially introduced by Infante and Webb in [15] and later used in [7,8,[12][13][14][16][17][18]. K is similar to a type of cone of non-negative functions first used by Krasnosel'skiȋ, see e.g. [19], and D. Guo, see e.g. [9]. Note that functions in K are positive on the subset [a, b] but are allowed to change sign in [−T, T ].
We require some knowledge of the classical fixed point index for compact maps, see for example [2] or [9] for further information. If Ω is a bounded open subset of K (in the relative topology) we denote by Ω and ∂ Ω the closure and the boundary relative to K. When D is an open bounded subset of X we write D K = D ∩ K, an open subset of K.
Next Lemma is a direct consequence of classical results from degree theory. (1) If there exists e ∈ K \ {0} such that x = F x + λe for all x ∈ ∂ D K and all λ > 0, then i K (F, D K ) = 0.
(2) If µx = F x for all x ∈ ∂ D K and for every µ ≥ 1, then i K (F, D K ) = 1.

Definition 2.3.
We use the following sets: The set V ρ was introduced in [18] and is equal to the set called Ω ρ/c in [16]. The notation V ρ makes it clear that choosing c as large as possible yields a weaker condition to be satisfied by f in Lemma 2.7. A key feature of these sets is that they can be nested, that is Theorem 2.4. Assume that hypotheses (C 1 )-(C 4 ) hold for some r > 0. Then F maps K r into K and is compact. When these hypotheses hold for each r > 0, F is compact and maps K into K.
Proof. For u ∈ K r and t ∈ [−T, T ] we have, Therefore we have that F u ∈ K for every u ∈ K r .
The compactness of F follows from the fact that the Hammerstein integral operator that occurs in (2.1) is compact (this a consequence of Proposition 3.1 of Chapter 5 of [22]).
In the sequel, we give a condition that ensures that, for a suitable ρ > 0, the index is 1 on K ρ .

Lemma 2.5. Assume that
Then the fixed point index, i K (F, K ρ ), is equal to 1.
Proof. We show that µu = F u for every u ∈ ∂ K ρ and for every µ ≥ 1. In fact, if this does not happen, there exist µ ≥ 1 and u ∈ ∂ K ρ such that µu = F u, that is   This contradicts the fact that µ ≥ 1 and proves the result.
Remark 2.6. We point out that, as in [29], a stronger (but easier to check) condition than (I 1 ρ ) is given by the following.
Let us see now a condition that guarantees the index is equal to zero on V ρ for some appropriate ρ > 0.
In fact, if not, there exist u ∈ ∂ V ρ and λ ≥ 0 such that u = F u + λe. Then we have Thus we get, for t ∈ [a, b], Taking the minimum over [a, b] gives ρ > ρ a contradiction.
Proof. We prove that In fact, if not, there exist u ∈ ∂ V ρ such that u = F u. Then we have Thus we get, for t ∈ [a, b], Taking the minimum over [a, b] gives ρ > ρ a contradiction.

Remark 2.8.
We point out that, as in [29], a stronger (but easier to check) condition than (I 0 ρ ) is given by the following.
The above Lemmas can be combined to prove the following Theorem. Here we deal with the existence of at least one, two or three solutions. We stress that, by expanding the lists in conditions (S 5 ), (S 6 ) below, it is possible to state results for four or more positive solutions, see for example the paper by Lan [20] for the type of results that might be stated. We omit the proof which follows directly from the properties of the fixed point index stated in Lemma 2.2, (3). Theorem 2.9. The integral equation (2.1) has at least one non-zero solution in K if either of the following conditions hold.
The integral equation (2.1) has at least two non-zero solutions in K if one of the following conditions hold.
The integral equation (2.1) has at least three non-zero solutions in K if one of the following conditions hold.

The case of non-negative kernels
We now assume the functions k, f , g that occur in (2.1) satisfy the conditions (C 1 ) − (C 4 ) in the previous section, where (C 2 ) and (C 4 ) are replaced with the following.
These hypotheses enable us to work in the cone of non-negative functions that is smaller than the cone (2.2). It is possible to show that F is compact and leaves the cone K ′ invariant. The conditions on the index are given by the following Lemmas, the proofs are omitted as they are similar to the ones in the previous section.

Lemma 3.2. Assume that
A result equivalent to Theorem 2.9 is clearly valid in this case, with nontrivial solutions belonging to the cone (3.1).

The case of kernels with extra positivity
We now assume the the functions k, f , g that occur in (2.1) satisfy the conditions ( These hypotheses enable us to work in the cone Once gain F is compact and leaves the cone K ′′ invariant. The assumptions on the index are as follows.

Lemma 4.3. Assume that
A result similar to Theorem 2.9 holds in this case.
and secondlyf Remark 4.5. Note that results similar to Sections 2, 3 and 4 hold when the kernel k is negative on a strip, negative and strictly negative. This gives nontrivial solutions that are negative on an interval, negative and strictly negative respectively.

An application
We now turn our attention to the first order functional periodic boundary value problem We apply the shift argument of [6] (a similar idea has been used in [27,30]), by fixing ω ∈ \ {0} and considering the equivalent expression Following the ideas developed in [6], we can verify that the functional boundary value problem (5.3)-(5.4) can be rewritten into a Hammerstein integral equation of the type Also, k(t, s) can be expressed in the following way (see [6] for details): The results that follow are meant to prove that we are under the hypothesis of Theorem 2.4.
The sign properties of the kernel (5.6) can be summarized as follows: The following lemma relates the sign of k for ζ positive and negative.
Now we have the following result.
Proof. By Lemma 5.2, it is enough to prove that k is strictly positive in S for ζ ∈ ( π 4 , π 2 ). We do here the proof for the connected component 1 − π 4ζ , π 4ζ × [−1, 1] of S. For the other one the proof is analogous.
We study now the different cases for the value of y.
We know give a technical lemma that will be used afterwards.  If g(a) < g(b) < p or p < g(b) < g(a) then f (g(a)) < f (g(b)), ) if and only if g( a+b 2 ) > p.  (1) and (2) are straightforward. Also, realize that, since g is affine, we have that g a+b Let us prove (3) as (4) is analogous: Proof. We know by Lemma 5.3 that k is positive in so, differentiating and doing the proper substitutions we get that Therefore, ∂ 2 k ∂ t 2 < 0 in S r , which means that any minimum of k with respect to t has to be in the boundary of the differentiable regions of S r . Thus, it is clear that, in S r , z∈[a,b] k(z, y). Also, realize that the arguments of the cosine in (5.7) are affine functions and that the cosine function is strictly decreasing in [0, π] and symmetric with respect to zero. We can apply Lemma 5.5 to get Finally, we have to compare the cases (5.17b) with (5.17c) for y ∈ [a, b) and (5.17d) with (5.17c) for y ∈ [−b, −a). Using again Lemma 5.5, we obtain the following inequality.
To compare (5.17d) with (5.17c) for y ∈ [−b, b) realize that k is continuous in the diagonal z = − y (see [6]). Hence, since the expressions of (5.17d) and (5.17c) are already locally minimyzing (in their differentiable components) for the variable z, it is clear that (5.17d)≥(5.17c) for y ∈ [−b, −a). Therefore, It is easy to check that the following order holds: Thus, we get the following expression To find the infimum of this function we will go through several steps in which we discard different cases. First, it is easy to check the inequalities (5.19g)≥(5.19h)= (5.19a) and (5.19d)≥(5.19e), so we need not to think about (5.19d), (5.19g) and (5.19h) anymore. Now, realize that |ζ(1 − b) − π 4 | ≤ |ζb + π 4 | ≤ π. Since the cosine is decreasing in [0, π] and symmetric with respect to zero this implies that (5.19e)≥(5.19a).
Note that (5.19c) can be written as Its derivative is which only vanishes at y = 1 2 for y ∈ [a, b].
Therefore y = 1 2 is a maximum of the function. Since g 1 is symmetric with respect to 1 2 and a is the symmetric point of b with respect to 1 2 , g(a) = g(b) is the infimum of (5.19c) which is contemplated in (5.19b) for y = b.
The integral equation (2.1) has at least two non-zero solutions in K if one of the following conditions hold.
The integral equation (2.1) has at least three non-zero solutions in K if one of the following conditions hold.