Existence and Uniqueness Results for a Smooth Model of Periodic Infectious Diseases

and Applied Analysis 3 Moreover g is a nonnegative function of class C1 on [0, +∞) and g󸀠(x) = −x exp(−x/2) for x > 0, lim x→0 + g (x)


Introduction
By modelling some infectious diseases with periodic contact rate that varies seasonally, Cooke and Kaplan [1] came up with the nonlinear integral equation where () represents the proportion of infections in the population at time ,  : R×[0, ∞) → [0, ∞); (, )  → (, ) is a (nonnegative) continuous function which is -periodic in the variable ; and  is a positive real number corresponding to the length of time an individual remains infectious.This has attracted many mathematicians such as Leggett and Williams [2], Nussbaum [3], and Agarwal and O'Regan [4] who have considered many variants of this model and used cone theoretic arguments to establish their existence results.
In this paper, we consider  as a positive real parameter and prove under suitable conditions (5) the existence of a unique curve of periodic positive solutions when  is of separable variables; say (, ) ≡ ()() with  : R → [0, +∞) continuous and -periodic, and  : [0,+∞) → [0, +∞) is of class C 1 .Furthermore we show a uniqueness result for bounded solutions of (1) when (, 0) ≡ 0,  is continuous and continuously differentiable with respect to its second variable , and  > 0 is sufficiently small.

The Results
In the sequel  denotes a positive constant real number, C  (R) denotes the real Banach space of -periodic continuous functions from R to R equipped with the supremum norm C 1  (R) denotes the space of -periodic continuously differentiable functions from R to R, and   (R) denotes the real Banach space of bounded continuous functions from R to R equipped with the supremum norm Given a function of two variables  : (, )  → (, ), we shall set Theorem 1.Let  : R → [0, +∞) be a (nonnegative) continuous -periodic function that is not identically equal to zero and  : [0, +∞) → [0, +∞) be a nonnegative continuous function of class C 1 .Suppose, moreover, that there exists a real number  0 > 0 such that where  = (1/) ∫  0 () (the mean value of ).Then there exists  ∈ (0, ) and a unique curve of nontrivial nonnegative -periodic solutions  ∈ C 1 (( − ,  + ); and for each  ∈ ( − ,  + ), that is,   solves (1) with (, ) ≡ ()().
Remarks 2. (i) For  sufficiently closed but not equal to , the solution   provided by Theorem 1 is not constant (since it can be seen in the proof that (/)(,  0 ) ̸ ≡ 0).(ii) The assumptions of this theorem are satisfied (due to the intermediate value theorem) when  : R → [0, +∞) is a nonnegative continuous -periodic function that is not identically equal to zero and  : [0, +∞) → [0, +∞) is a nonnegative continuous function of class C 1 such that lim sup (iii) The conclusion of Theorem 1 still holds, according to its proof, when  : R → [0, +∞) is a nonnegative continuous -periodic function that is not identically equal to zero, for some real number  1 > 0,  is continuously differentiable from [0,  1 ] into [0, +∞), and there exists a real number  0 ∈ (0,  1 ) that satisfies the conditions (5).
(iv) Note that if  : R → [0, +∞) is a nonnegative continuous -periodic function that is not identically equal to zero and  : [0, +∞) → [0, +∞) is a nonnegative continuous function of class C 1 which is superlinear or for which there exists a positive number  * such that then (1)  the zero function is an isolated solution of ( 1).
Example 4. The assumptions of this theorem are satisfied in each of the next two cases followed by an illustration of part (iii) of Remarks 2: (i) Let () =  − for every  ≥ 0 and () = (1/2)(1 + sin(2)) for all  ∈ R and  = 1.
Clearly  is a 2-periodic nonnegative function with Moreover  is a nonnegative function of class C 1 on [0, +∞) and   () = − exp(− 2 /2) for  > 0, Then we can conclude according to part (ii) of Remarks 2.
The result follows from part (iii) of Remarks 2.
Proof of Theorem 1. Suppose that the assumptions of Theorem 1 are satisfied.
Step 1.Let g be a real-valued C 1 -extension of  to R; for instance, which may change sign; in other words g is defined from we shall need just a positive real number  1 >  0 such that for the sake of generality (see Remarks 2(iii)).Hence Now set Clearly Ω is open in C  (R) and contains the constant function  0 .Moreover consider the mapping Then  is well-defined by the -periodicity of  and the continuity of both  and .Also for every (, ) ∈ (0, +∞)×Ω fixed, we have Thus for (, ) ∈ (0, +∞) × Ω, (, ) = 0 if and only if  is a positive solution of (1) with (, ) ≡ ()().