An Investigation of an Integral Equation Involving Convex–Concave Nonlinearities

: We investigate the existence and uniqueness of positive solutions to an integral equation involving convex or concave nonlinearities. A numerical algorithm based on Picard iterations is provided to obtain an approximation of the unique solution. The main tools used in this work are based on partial-ordering methods and ﬁxed-point theory. Our results are supported by examples.

Equation (2) was proposed in [1] as a mathematical model to analyze the observed periodic outbreaks of certain infectious diseases. Namely, for a given population, ϑ(σ), , and λ(σ, ϑ(σ)) represent, respectively, the proportion of infectious individuals at time σ, the length of time for which an individual is infective, and the proportion of new infective individuals per unit of time. Several investigations of Equation (2) have been carried out. In [1], sufficient conditions ensuring the existence of nontrivial periodic solutions to (2), as well as sufficient conditions for which all solutions to (2) approach zero as σ → ∞, were provided. In [2,3], using Krasnosel'skii-type fixed point theorems, the existence of at least one nontrivial periodic solution to (2) was proved under certain conditions on λ. The same question was investigated in [4] using fixed-point index theory. In [5,6], the question of points of bifurcations of positive periodic solutions to (2) was studied. For other contributions related to the study of (2), see, e.g., [7][8][9][10] and the references therein.
Various interesting contributions dealing with generalized variants of (2) have been performed by many authors. In [11], the existence of positive almost periodic solutions to integral equations of the form λ(s, ϑ(s)) ds, σ ∈ R, has been studied. In [12], the neutral integral equation has been considered. In [13], the existence of multiple periodic solutions to integral equations of the form has been investigated using various fixed-point theorems. For more contributions related to generalized variants of (2), see, e.g., [14][15][16][17][18][19][20][21][22] and the references therein.
In [23], sufficient conditions for the existence of a principal solution to a nonlinear Volterra integral equation of the second kind on the half-line and on a finite interval have been derived. Furthermore, a method for computing the boundary of an interval outside of which the solution can blow up has been proposed (see also [24]). In [25], the local solvability and blow-up of solutions to an abstract nonlinear Volterra integral equation have been investigated. Recently, in [26], the authors proposed a new method and a tool to validate the numerical results of Volterra integral equations with discontinuous kernels in linear and nonlinear forms obtained from the Adomian decomposition method.
In this paper, Equation (1) is investigated. Namely, using partial-ordering methods and a fixed-point theorem for monotone and convex/concave operators defined in a normal solid cone, we derive sufficient conditions, ensuring the existence and uniqueness of positive solutions. Moreover, in order to approximate the solution, a numerical algorithm based on Picard iterations is provided.
The main tools of partial-ordering methods and fixed-point theory that will be used in this paper are presented in Section 2. The main results, as well as their proofs, are presented in Section 3. Finally, some examples are studied in Section 4.

Preliminaries
Let B be a Banach space over R with respect to a certain norm · B . We denote, by 0 B , the zero vector of B. Let C ⊂ B (C = {0 B }) be nonempty, closed, and convex. We say Here, for α ∈ R, αC denotes the subset of B defined by Let C be a cone in B. Then C induces a partial-order C in B defined by for all x, y ∈ B. We use the notation x ≺ C y to indicate that x C y and The notation x C y indicates that y − x ∈C, whereC is the interior of C. IfC = ∅, We say that C is a solid cone. We say that C is normal, if there exists ρ ≥ 1 such that for all x, y ∈ B. Let S : A ⊂ B → B be a given operator. Then, Lemma 1 (see [27]). Suppose that C is a normal solid cone and S : [x, y] → B is increasing, where x, y ∈ B and x ≺ C y. Assume that one of the following conditions is satisfied: (i) S is concave, Sx C x and Sy C y.
(ii) S is convex, Sx C x and Sy C y. Then, (II) There exist γ > 0 and 0 < θ < 1, such that for all z 0 ∈ [x, y], the sequence {z n } n≥0 defined by z n+1 = Sz n , for all n converges to z and satisfies z n − z B ≤ γθ n , for all n.
Proof. Let us introduce the set Then, C is a normal solid cone in B, and its interior is given bẙ The partial order induced by C is defined by We shall prove that S([x, y]) ⊂ B.
Finally, applying Lemma 1 and observing that any fixed point of S is a solution to (1), the conclusion of Theorem 1 follows.
Proof. From the proof of Theorem 1, the mapping S : [x, y] → B is nondecreasing. Using the convexity of ξ(σ, ·) and λ(σ, ·) (see (iv)), we deduce that S is convex. Moreover, by (10) and (vi), we have Then Sx C x and Sy C y. Finally, using Lemma 1, the conclusion of Theorem 2 follows.

Some Examples
Consider the nonlinear integral equation where > 0 is a constant.
Proof. Notice that (12) is a special case of (1) with Moreover, we have µ ∈ C(R × R, [0, ∞)) and This shows that conditions (i) and (ii) of Theorem 1 are satisfied with m = 0 and M = .
Consider now the integral equation where > 0 is a constant.

Conflicts of Interest:
The authors declare no conflict of interest.