A positive fixed point theorem with applications to systems of Hammerstein integral equations

We present new criteria on the existence of fixed points that combine some monotonicity assumptions with the classical fixed point index theory. As an illustrative application, we use our theoretical results to prove the existence of positive solutions for systems of nonlinear Hammerstein integral equations. An example is also presented to show the applicability of our results.


Introduction
In this manuscript we pursue the line of research developed in the recent papers [4,5,8,12,23] in order to deal with fixed point theorems on cones that mix monotonicity assumptions and conditions in one boundary, instead of imposing conditions on two boundaries as in the celebrated cone compression/expansion fixed point theorem of Krasnosel'skiȋ. In order to do this we employ the well-known monotone iterative method, combined with the classical fixed point index. In Section 2 we prove two results concerning non-decreasing and non-increasing operators in a shell, in presence of an upper or of a lower solution; in Remark 2.4 we present a comparison with previous results in this direction.
(with the obvious meaning when f (s) = 0), then the BVP (1.1) has at least a positive solution.
Note that the above theorem is valid for a specific Green's function. On the other hand the existence of nonnegative solutions for systems of Hammerstein integral equations has been widely studied, see for example [1,6,7,9,10,11,13,14,16,17,18,19,20,21,22,24,25] and references therein. In Section 3 we give an extension of Theorem 1.1 to the context of systems of Hammerstein integral equations of the type providing, under suitable assumptions on the kernels and the nonlinearities, the existence of a positive solution.
In order to show the applicability of our results, we discuss the following system of secondorder ODEs, subject to local and nonlocal boundary conditions, that generates two different kernels, computing all the constants that occur in our theory.

Two fixed point theorems in cones
A subset K of a real Banach space X is a cone if it is closed, K + K ⊂ K, λK ⊂ K for all λ ≥ 0 and K ∩ (−K) = {θ}. A cone K defines the partial ordering in X given by x y if and only if y − x ∈ K.
We reserve the symbol "≤" for the usual order on the real line. For x, y ∈ X, with x y, we define the ordered interval [x, y] = {z ∈ X : x z y}.
The cone K is normal if there exists d > 0 such that for all x, y ∈ X with 0 x y then x ≤ d y .
We denote the closed ball of center x 0 ∈ X and radius r > 0 as and the intersection of the cone with the open ball centered at the origin and radius r > 0 as We recall a well known result of the fixed point theory, known as the monotone iterative method (see, for example, [28,Theorem 7.A] or [2]). In the next Proposition we recall the main properties of the fixed point index of a completely continuous operator relative to a cone, for more details see [3,15]. In the sequel the adherence and the boundary of subsets of K are understood to be relative to K.
Then the fixed point index i K (T, D K ) has the following properties: We state our first result on the existence of non-trivial fixed points.
the map T is non-decreasing in the set Then the map T has at least one non-zero fixed point x 1 in K, that either belongs to P or belongs to Suppose first that we can choose α ∈ K with α = R and T α α. Since α β and due to the normality of the cone K we have that [α, β] ⊂ P which implies that T is nondecreasing on [α, β]. Then we can apply the Theorem 2.1 to ensure the existence of a fixed point of T on [α, β], which, in particular, is a non-trivial fixed point.
Now suppose that such α does not exist. Thus T x x for all x ∈ K with x = R, which by Proposition 2.2, (iii) implies that i K (T, K R ) = 1. Since, by assumption, i K (T, V ) = 0 we get the existence of a non-trivial fixed point Remark 2.4. We note that we can use either Proposition 2.2, (i) or Proposition 2.2, (ii) in order to check the assumption (3) in Theorem 2.3. We also stress that P is contained in the set {x ∈ K : R d ≤ x ≤ d β }. Therefore Theorem 2.3 is a genuine generalization of the previous fixed point theorems obtained in [4,5,8,12]. Moreover, we show in the applications that in many cases is useful to apply Theorem 2.3 with a set V different from K r .
We observe that, following some ideas introduced in [5, Theorem 2.1], it is possible to modify the assumptions of Theorem 2.3 in order to deal with non-increasing operators. The next result describes precisely this situation.
Theorem 2.5. Let X be a real Banach space, K a cone with nonempty interior (i.e. solid) and T : K → K a completely continuous operator. Assume that Then the map T has at least one non-zero fixed point such that either belongs to P or belongs to Proof. Let x ∈ K be such that x = R. Then by (1') we have that x α and since x, α ∈ P it follows from (2') that T x T α α x. Now, if for some x ∈ ∂K R is the case that T x x then we are done. If not, T x x for all x ∈ ∂K R which by Proposition 2.2 implies that i K (T, K R ) = 0. This result together with (3') give the existence of a non-zero fixed point with the desired localization property.

An application to a system of Hammerstein integral equations
We now apply the results of the previous Section in order to prove the existence of positive solutions of the system of integral equations where we assume the following assumptions: Set c = min{c 1 , c 2 } and let us definẽ and consider the cone K in C[a, b] × C[a, b] defined by which is a normal cone with d = 1.
Under our assumptions it is routine to check that the integral operator leaves K invariant and is completely continuous. Now we present our main result concerning the existence of positive solutions for the system (3.1).
Theorem 3.1. Assume that the assumptions (H 1 ) − (H 5 ) hold and moreover: Then the system (3.1) has at least one positive solution in K provided that Proof. Due to (3.2) we can fix β i ∈ (0, B i ), i = 1, 2, such that On the other hand, for M > max 1 λ 1 γ 1, * , 1 λ 2 γ 2, * let ρ = ρ(M) > 0 as in (H 7 ) and fix Let us check that assumptions of Theorem 2.3 are satisfied with Since β is constant and R < min Moreover, since β i − T i β ∞ ≤ β i , i = 1, 2, and taking into account (3.3) we have for t ∈ [a i , b i ] and i = 1, 2, As a consequence, we have T β β, and the claim is proven.
we have for all t ∈ [a, b] and i = 1, 2, Moreover, for all t ∈ [a i , b i ], r ∈ [0, 1] and i = 1, 2, T v, and since P ⊂ {x ∈ K : x β}, T is also non-decreasing on P.
Thus, we obtain ρ = min Therefore by Proposition 2.2 we have that i K (T, V ) = 0 and the proof is finished.
Remark 3.2. The following condition, similar to the one given in [6], implies (H 7 ) and it is easier to check.