Approximating positive solutions of nonlinear first order ordinary quadratic differential equations

Abstract: In this paper, the authors prove the existence as well as approximations of the positive solutions for an initial value problem of first-order ordinary nonlinear quadratic differential equations. An algorithm for the solutions is developed and it is shown that the sequence of successive approximations converges monotonically to the positive solution of related quadratic differential equations under some suitable mixed hybrid conditions. We base our results on the Dhage iteration method embodied in a recent hybrid fixed-point theorem of Dhage (2014) in partially ordered normed linear spaces. An example is also provided to illustrate the abstract theory developed in the paper.


Introduction
Given a closed and bounded interval J = [t 0 , t 0 + a], of the real line ℝ for some t 0 , a ∈ ℝ with t 0 ≥ 0, a > 0, consider the initial value problem (in short IVP) of first-order ordinary nonlinear quadratic differential equation, (in short HDE) f (t,x(t)) = g(t, x(t)), t ∈ J, x(t 0 ) = x 0 ∈ ℝ,

ABOUT THE AUTHORS
The key research project of the authors of the paper is to prove existence and find the algorithms for different nonlinear equations that arise in mathematical analysis and allied areas of mathematics via newly developed Dhage iteration method. The quadratic differential equations form an important class in the theory of differential equations. In the present paper, it is shown that the new method is also applicable to such type of nonlinear quadratic differential equations for proving the existence as well as approximations of the solutions under mixed monotonic and geometric conditions.

PUBLIC INTEREST STATEMENT
It is known that many of the natural, physical, biological, and social processes or phenomena are governed by mathematical models of nonlinear differential equations. So if a person is engaged in the study of such complex universal phenomena and not having the knowledge of sophisticated nonlinear analysis of this paper, then one may convinced the use of the results of this paper, in particular when one comes across a certain dynamic process which is based on a mathematical model of quadratic differential equations. In such situations, the application of the results of the present paper yields numerical concrete solutions under some suitable natural conditions thereby which it is possible to improve the situation for better desired goals.
By a solution of the QDE (1.1), we mean a function x ∈ C 1 (J, ℝ) that satisfies is a continuously differentiable function for each x ∈ ℝ, and (ii) x satisfies the equations in (1.1) on J,where C(J, ℝ) is the space of continuously differentiable real-valued functions defined on J.
The QDE (1.1) with = 0 is well known in the literature and is a hybrid differential equation with a quadratic perturbation of second type. Such differential equations can be tackled with the use of hybrid fixed-point theory (cf. Dhage 1999;2013;2014a). The special cases of QDE (1.1) have been discussed at length for existence as well as other aspects of the solutions under some strong Lipschitz and compactness-type conditions which do not yield any algorithm to determine the numerical solutions. See Dhage and Regan (2000), Dhage and Lakshmikantham (2010) and the references therein. Very recently, the study of approximation of the solutions for the hybrid differential equations is initiated in Dhage, Dhage, and Ntouyas (2014) via hybrid fixed-point theory. Therefore, it is of interest and new to discuss the approximations of solutions for the QDE (1.1) along the similar lines. This is the main motivation of the present paper and it is proved that the existence of the solutions may be proved via an algorithm based on successive approximations under weaker partial continuity and partial compactness-type conditions.

Auxiliary results
Unless otherwise mentioned, throughout this paper that follows, let E denotes a partially ordered real-normed linear space with an order relation ⪯ and the norm ‖ ⋅ ‖. It is known that E is regular if {x n } n∈ℕ is a nondecreasing (resp. nonincreasing) sequence in E such that x n → x * as n → ∞, then x n ⪯ x * (resp. x n ⪰ x * ) for all n ∈ ℕ. Clearly, the partially ordered Banach space, C(J, ℝ) is regular and the conditions guaranteeing the regularity of any partially ordered normed linear space E may be found in Nieto and Lopez (2005) and Heikkilä and Lakshmikantham (1994) and the references therein.
We need the following definitions in the sequel.
is a relatively compact subset of E for all totally ordered sets or chains C in E.  is called uniformly partially compact if  (C) is a uniformly partially bounded and partially compact on E.  is called partially totally bounded if for any totally ordered and bounded subset C of E,  (C) is a relatively compact subset of E. If  is partially continuous and partially totally bounded, then it is called partially completely continuous on E.
Definition 2.5 (Dhage, 2009) The order relation ⪯ and the metric d on a nonempty set E are said to be compatible if {x n } n∈ℕ is a monotone, that is, monotone nondecreasing or monotone nonincreasing sequence in E and if a subsequence {x n k } n∈ℕ of {x n } n∈ℕ converges to x * implies that the whole sequence {x n } n∈ℕ converges to x * . Similarly, given a partially ordered normed linear space (E, ⪯, ‖ ⋅ ‖), the order relation ⪯ and the norm ‖ ⋅ ‖ are said to be compatible if ⪯ and the metric d defined through the norm ‖ ⋅ ‖ are compatible.
Clearly, the set ℝ of real numbers with usual order relation ≤ and the norm defined by the absolute value function | ⋅ | has this property. Similarly, the finite-dimensional Euclidean space ℝ n with usual componentwise order relation and the standard norm possesses the compatibility property.
Definition 2.6 (Dhage, 2010) An upper semi-continuous and nondecreasing function : such that for all comparable elements x, y ∈ E. If (r) = k r, k > 0, then  is called a partially Lipschitz with a Lipschitz constant k.
Let (E, ⪯, ‖ ⋅ ‖) be a partially ordered normed linear algebra. Denote and The elements of the set  are called the positive vectors in E. The following lemma follows immediately from the definition of the set , which is oftentimes used in the hybrid fixed-point theory of Banach algebras and applications to nonlinear differential and integral equations.
Definition 2.7 An operator  : E → E is said to be positive if the range R( ) of  is such that R ( ) ⊆ . The Dhage iteration principle or method (in short DIP or DIM) developed in Dhage (2010;2013; 2014a) may be rephrased as "monotonic convergence of the sequence of successive approximations to the solutions of a nonlinear equation beginning with a lower or an upper solution of the equation as its initial or first approximation" and which forms a useful tool in the subject of existence theory of nonlinear analysis. The Dhage iteration method is different from other iterations methods and embodied in the following applicable hybrid fixed-point theorem of Dhage (2014b), which is the key tool for our work contained in the present paper. A few other hybrid fixed-point theorems containing the Dhage iteration principle appear in Dhage (2010;2013;2014a;2014b  (2.1) Then the operator equation has a positive solution x * in E and the sequence {x n } of successive iterations defined by x n+1 = x n x n , n = 0, 1, … ; converges monotonically to x * .

Remark 2.1
The compatibility of the order relation ⪯ and the norm ‖ ⋅ ‖ in every compact chain of E is held if every partially compact subset S of E possesses the compatibility property with respect to ⪯ and ‖ ⋅ ‖. This simple fact is used to prove the desired characterization of the positive solution of the QDE (1.1) defined on J.

Main results
The QDE (1.1) is considered in the function space C(J, ℝ) of continuous real-valued functions defined on J. We define a norm ‖ ⋅ ‖ and the order relation ≤ in C(J, ℝ) by and for all t ∈ J, respectively. Clearly, C(J, ℝ) is a Banach algebra with respect to above supremum norm and is also partially ordered w.r.t. the above partially order relation ≤. It is known that the partially ordered Banach algebra C(J, ℝ) has some nice properties w.r.t. the above order relation in it. The following lemma follows by an application of Arzelá-Ascoli theorem.
Proof The proof of the lemma is given in Dhage and Dhage (in press). Since it is not well known, we give the details of proof for the sake of completeness. Let S be a partially compact subset of C(J, ℝ) and let {x n } be a monotone nondecreasing sequence of points in S. Then, we have for each t ∈ J.
Suppose that a subsequence {x n k } of {x n } is convergent and converges to a point x in S. Then the subsequence {x n k (t)} of the monotone real sequence {x n (t)} is convergent. By monotone characterization, the whole sequence {x n (t)} is convergent and converges to a point x(t) in ℝ for each t ∈ J. This shows that the sequence {x n (t)} converges pointwise in S. To show the convergence is uniform, it is enough to show that the sequence {x n (t)} is equicontinuous. Since S is partially compact, every chain or totally ordered set and consequently {x n } is an equicontinuous sequence by Arzelá-Ascoli theorem. Hence {x n } is convergent and converges uniformly to x. As a result, ‖ ⋅ ‖ and ≤ are compatible in S. This completes the proof.
We need the following definition in what follows.
Definition 3.1 A function u ∈ C 1 (J, ℝ) is said to be a lower solution of the QDE (1.1) if the function is continuously differentiable and satisfies for all t ∈ J. Similarly, a function v ∈ C 1 (J, ℝ) is said to be an upper solution of the QDE (1.1) if it satisfies the above property and inequalities with reverse sign.
We consider the following set of assumptions in what follows: is injection for each t ∈ J.
(A 3 ) There exists a -function , such that for all t ∈ J and x, y ∈ ℝ, x ≥ y.
(B 2 ) There exists a constant M g > 0 such that g(t, x) ≤ M g for all t ∈ J and x ∈ ℝ.
The QDE (1.1) has a lower solution u ∈ C 1 (J, ℝ). Proof Set E = C(J, ℝ) Then, by Lemma 3.1, every compact chain in E possesses the compatibility property with respect to the norm ‖ ⋅ ‖ and the order relation ≤ in E.
Define two operators  and  on E by d dt e − (t−s) g(s, x n (s)) ds and From the continuity of the integral, it follows that  and  define the maps ,  : E → E. Now by Lemma 3.2, the QDE (1.1) is equivalent to the operator equation We shall show that the operators  and  satisfy all the conditions of Theorem 2.1. This is achieved in the series of following steps.
Step I: and  are nondecreasing on E.
Let x, y ∈ E be such that x ≥ y. Then by hypothesis (A 3 ), we obtain for all t ∈ J. This shows that  is nondecreasing operator on E into E. Similarly using hypothesis (B 3 ), it is shown that the operator  is also nondecreasing on E into itself. Thus,  and  are nondecreasing positive operators on E into itself.
Step II:  is partially bounded and partially -Lipschitz on E.
Let x ∈ E be arbitrary. Then by (A 2 ), for all t ∈ J. Taking supremum over t, we obtain ‖x‖ ≤ M f and so,  is bounded. This further implies that  is partially bounded on E.
Next, let x, y ∈ E be such that x ≥ y. Then, for all t ∈ J. Taking supremum over t, we obtain ‖x − y‖ ≤ (‖x − y‖) for all x, y ∈ E, x ≥ y. Hence,  is a partial nonlinear D-LIpschitz on E which further implies that  is a partially continuous on E.
Step III:  is partially continuous on E.
Let {x n } n∈ℕ be a sequence in a chain C of E such that x n → x for all n ∈ ℕ. Then, by dominated convergence theorem, we have (3.6) x(t) = f (t, x(t)), t ∈ J, e − (t−s) g(s, x(s)) ds, t ∈ J.