Some Existence Results on Positive Solutions for an Iterative Second-order Boundary-value Problem with Integral Boundary Conditions

abstract: The present paper is devoted to study a nonlinear second order differential equation with iterative source term and integral boundary conditions. The utilization of some suitable fixed point theorems ultimately led us to establish some sufficient conditions that guarantee the existence, uniqueness and continuous dependence of positive bounded solutions. The obtained results are illustrated by an example.


Introduction
Iterative differential equations constitute a particular type of the so-called delay differential equations where the delays which depend on both the time t and the state variable x are defined implicitly by the iterates. To the best of our knowledge, the concept of iterative equations appeared for the first time in 1815 by Babbage [2] in his essay towards the calculus of function in which he was interested in finding a function equalling its n-th iterate. But the first step of the theory for iterative differential equations is said to have begun in the late twentieth century see [14].
Recently they have been attracting great interest. Many researchers have concentrated on studying first order iterative differential equations by different approaches such as fixed point theory, Picard's successive approximation and the technique of nonexpansive operators. But the literature related to the equations of higher order is limited since the presence of the iterates increases the difficulty of studying them. This motivates us to investigate the following second order iterative differential equation: [1] (t) , x [2] (t) , ..., x [n] (t) = 0, 0 < t < b, (1.1) where x [0] (t) = t, x [1] (t) = x (t) , ..., x [n] (t) = x [n−1] (x (t)) and f : [0, b] × R n → [0, +∞) is a continuous function with respect to its arguments. Equation (1.1) describes diffusion phenomena with a source or a reaction term. For instance, in thermal conduction, it can be interpreted as the one-dimensional heat conduction equation which models the steady-states of a heated bar of length b with a controller at x = b that adds or removes heat according to a sensor, while the left endpoint is maintained at 0 • C and f is the distributed temperature source function depending on delayed temperatures. We refer the interested reader to [9,10,13] and the references therein for more details. We would like to mention some recent results on this type of equations with integral boundary conditions which arises in many areas of applied mathematics such as physics, electrodynamics, infectious diseases, population dynamics and many others. In [12], G. Infante used the theory of fixed point index for treating the following equation: In [8], by virtue of Krasnoselskii's fixed point theorem, A. Boucherif investigated the existence of positive solutions of the following problem: In [4], M. Benchohra et al. used nonlinear alternative Leray Schauder type and Banach contraction principle to establish the existence of solutions of the following second-order boundary value problem: −y ′′ (t) = f (t, y (t)) , a.e 0 < t < 1, where f : [0, 1] × R −→ R is a given function and g : [0, 1] −→ R is an integrable function. In [11], by using Schauder's fixed point theorem, Juan Galvis, Edixon M. Rojas, Alexander V. Sinitsyn proved the existence of positive solutions of the problem −u ′′ (t) + a (t) f (u (t)) = 0, 0 < t < γ, This work is a continuation of the obove montioned works, some other works on positivity and boundary value problems [1,3] and our recent papers on iterative problems (see [5,6]). Our main contribution to this important area is to show that the fixed point theory can be applied successfully to iterative problems with integral boundary conditions. We briefly outline the structure of the paper. In the beginning of it, we provide some basic concepts which are useful in the sequel. In the third section we prove our main results by means of Banach and Schauder's fixed point theorems. Finally, an illustrative example is given in the last section.

Preliminaries
Before attempting to prove our desired results, we start by defining a subset of C ([0, b] , R) that it will be able to contain the iterates and the solutions if they exist. To this end, we define a subset CB Int of C ([0, b] , R) as follows: It's obvious that CB Int , equipped with the supremum norm then CB Int (L, M ) is a closed convex and bounded subset of CB Int . Throughout this paper we assume that the function f (t, x 1 , x 2 , ..., x n ) is globally Lipschitz in x 1 , ..., x n . i.e., there exist n positive constants c 1 , c 2 , ..., c n such that and we introduce the following constants has a unique solution given by

From Lemma 2.3, we can also obtain
which proves the continuity of the operator A. ✷ Lemma 3.3. Suppose that condition (2.1) holds. If

Uniqueness
In this section, we present our uniqueness result. Proof: Let ϕ, ψ be two distinct solutions of (1.1)-(1.2). Using the same technique as in the proof of 3.3 we can prove Using (3.4) leads us to Since we have a contradiction, the fixed point must be unique.

Continuous dependence
In this section, we show that the unique solution proved in the last theorem depends continuously on the function f.  We have   This implies that Therefore, This completes the proof. ✷