ANALYTICAL RESULTS FOR QUADRATIC INTEGRAL EQUATIONS WITH PHASE–LAG TERM

Abstract In the present paper, we are concerning with a quadratic integral equation with phase–lag term. In the following pages, sufficient conditions are given for the existence of positive continuous solution to quadratic integral equations. The method used here depends on both Tychonoff fixed point principle and Arzelà–Ascoli theorem. A concrete example illustrating the mentioned applicability is also included.


Introduction
Phase lag has a very important role in our applied science and there are currently one, dual and three phases and each phase has a different applications. For example the three-phase-lag model incorporates the microstructural interaction effect in the fast-transient process of heat transport. It describes the finite time required for the various microstructural interactions to take place, including the phonon-electron interaction in metals, the phonon scattering in dielectric crystals, insulators, and semiconductors, and the activation of molecules at extremely low temperature, by the resulting phase lag (time delay) in the process of heat transport see [9,11,13,23]. Integral equations with phase lag term are the mathematical model of many evolutionary in problems chemistry, engineering, quantum mechanics, biology, optimal control systems, mathematical physics and so on. For example, integral equations for the dual lag model of heat transfer.
Integral equations create a very important and significant part of mathematical analysis and their applications to real-world problems. On the other hand, normality and continuity are very useful tools in the wide area of functional analysis such as the metric fixed-point theory and the theory of operator equations in Banach spaces. They are also used in the studies of functional equations, ordinary and partial differential equations, fractional partial differential equations, integral and integro-differential equations, optimal control theory, etc., see [1-8, 14, 17-22]. In our investigations, we apply the Tychonoff fixed point principle [12] to prove the existence and uniqueness of solution of the quadratic integral equation with phase-lag term.
In this paper, the existence of at least one continuous solution for the quadratic integral equation of phase-lag term (in short QIEPLT), will be proved, where δt is the phase-lag constant, the function ψ(t) is unknown in the Banach space and continuous with their derivative with respect to time. The kernel k(t, τ ) is positive and continuous, the functions a(t), g(τ, ψ) are continuous its derivatives with respect to time. Let I = [0, 1], denote by E = C(I) the space of continuous functions defined on I with norm ∥ψ∥ = max t∈I |ψ(t)|. Using Taylor expansion after neglecting the second derivative in Eq. (1.1), we get The quadratic integro-differential equation (QIDE) is a kind of functional equation that has associate integral and derivatives of an unknown function. These equations were named after the leading mathematicians who have first studied them, such as quadratic Fredholm, quadratic Volterra. quadratic Fredholm and Volterra equations are the most encountered types. By comparing the expressions with the same powers of parameter δt, we receive the relations and Integrating Eq. (1.5) twice and using initial condition (1.3), we get (1.6) integration by parts and applying the Leibniz's rule, we obtain The contribution of this work can be summarized in the following four points: • Introducing the preliminaries and auxiliary results about the fixed point theorem needed in the following points of the paper.
• The existence and uniqueness of the solution of a quadratic integral equation of Volterra type (1.4), under certain conditions, will be discussed and proved using Banach's fixed point method in the space E = C(I).
• We study existence of solution of quadratic integral equation (1.7) by using Tychonoff fixed point theorem.
• An example is given to show the applications of our results.

Preliminaries
In this section, the existence results will be based on the following fixed-point theorems and definitions that are used in the paper. Notice that a normed vector space is a locally convex topological vector space so this theorem extends the Schauder fixed point theorem.

Theorem 2.3 (Arzelà-Ascoli Theorem [15]). Let E be a compact metric space and C(E) the Banach space of real or complex valued continuous functions normed by
such that is uniformly bounded and equicontinuous, then the closure of F is compact.

Existence of positive continuous solution
In this section, we study the existence of at least one solution of the integral equation (1.1), to achieve this, the existence and uniqueness of the Eqs. (1.4) and (1.7) were discussed in the following subsections:

The existence and uniqueness of solution of Eq. (1.4)
Here, we prove the existence of positive continuous solution for Eq. (1.4). To facilitate our discussion, let us first state the following assumptions: To prove the existence and the uniqueness solution of Eq. (1.4), we use the continuity of the integral operator, with the help of Banach fixed point. For this the integral equation (1.4) can be written in the integral operator form: (3.1)

Theorem 3.1. If the conditions (i) -(iv) are satisfied and the integral operator (3.1) is a continuous, then equation (1.4) has an unique solution ψ(t) in the Banach
Proof. For the continuity, we assume the two functions ψ 1 (t) and ψ 2 (t) in the space C([0, 1]) satisfy the integral operator then, Using the properties of the norm and the conditions (i)-(iv), we get

The existence and uniqueness of solution of Eq. (1.7)
Equation (1.7) can be written in the following integral operator from: (KG)(t) = 1 0 k(t, s)g(s, ψ(s))ds, Assume that g is a real function defined on the set I × R + , we consider the superposition operator (Gψ)(t) = g(t, ψ(t)) under the some following assumptions.
(a) g is continuous on the set I × R + .

(b)
The function t → g(t, ψ) is nondecreasing on I for any fixed ψ ∈ R + .
(c) For any fixed t ∈ I the function ψ → g(t, ψ) is nondecreasing on R + .

(d)
The function g = g(t, ψ) satisfies the Lipschitz condition with respect to the variable ψ, i.e. there exists a constant l > 0 such that for any t ∈ I and for ψ 1 , ψ 2 ∈ R + the following inequality holds Then the following result is implied.

Theorem 3.2. Assume that the hypotheses (a)-(d) are satisfied and ψ ∈ Ψ I ⊆ C(I). Then d(Gψ) ≤ ld(ψ).
The above theorem follows that when the function g satisfies the Lipschitz condition with a constant l < 1 (cf. the assumption (d)) the superposition operator G generated by the function g improves the degree of monotonicity of any subset Ψ of Ψ I with the coefficient l. Corollary 3.1. Suppose the function g(t, ψ) = g : I × R + → R + satisfies the assumptions (a), (b). Moreover, we assume that g has partial derivative g ψ which is nonnegative and bounded on the set I × R + . Then g satisfies the assumptions (c) and (d) with the Lipschitz constant l defined as follows In order that discuss the existence and uniqueness solution of Eq. (1.7), we assume the following assumptions:

(i) k : I × I → R + is continuous and the functions s → k(t, s) and t → k(t, s)
are nondecreasing on R + for fixed t ∈ I and s ∈ I, respectively such that (v) The inequality where k * = max{k(t, s) : t, s ∈ I}.
Now we can formulate the main existence theorem:

Theorem 3.3. Let the assumptions (i)-(v) be satisfied. Then the quadratic functional integral equation (1.7) has at least one solution ψ = ψ(t) in the space C(I).
Proof. Let S r be the subset of the space C([0, 1]) defined as follows: The space C([0, 1]) is a complete locally convex linear space that has been proved in [12], it is clear that the set S r is nonempty, bounded and closed, but we will prove that the set S r convex. Let ψ 1 , ψ 2 ∈ S r and λ ∈ [0, 1] then we have ≤ λr + r − λr = r.
Then λψ 1 + (1 − λ)ψ 2 ∈ S r which means that is convex set. Let us consider the operator V is defined on the space C(I) by the formula: To show the operator V transforms the space S r into itself. For that let ψ ∈ S r , then Using the properties of the norm and the conditions (i)-(v), we get

s)g(s, ψ(s))ds
using the properties of the norm and the conditions (i)-(v), we obtain Hence, keeping in mind our assumptions and the above-established facts, we arrive at the following relation: This means that the function V S r is equi-continuous on I. By using Arzelà-Ascoli theorem [13], we can say that is V S r compact. Now, Tychonoff fixed point theorem is satisfied all its conditions, then Eq. (1.7) has at least one solution ψ ∈ C(I). This completes the proof.

Example
In this section, we will discuss the following example and applying theories 3.1 and 3.3, then check the results.   (iv) The function g(τ, ψ(τ )) = ln(1 + τ |ψ(τ )|) satisfy the Lipschitz condition with Lipschitz constants Under the following inequality: Using the standard methods we can verify that the function ρ(r) = r(7/e 2 + 1/2e 2 ) attains its maximum at the point r = 0.985 and ρ(0.985) = 0.985(7/e 2 +1/2e 2 ) ≤ 1. So, the number r = 0.985 is a positive solution of the inequality (4.4), hence the theorem 3.3 is true. Finally, taking into account all the above-established facts and theories 3.1 and 3.3 we conclude that the equation (1.1) has at least one solution ψ = ψ(t) defined and continuous on the interval I. Moreover, |ψ| ≤ r = 0.985.

Conclusion
In this work, from the above results and discussion, the following may be conclude, the quadratic integral equation with phase-lag term (1.1) possesses at least one solution ψ(t) in the space C ([0, 1]), under all the assumptions of theories 3.1 and 3.3. Fixed point theories are one of the best methods to prove the existence and uniqueness of any equation.