Chandrasekhar quadratic and cubic integral equations via Volterra-Stieltjes quadratic integral equation

1 Introduction

Quadratic integral equations arise in many applications in the real world. For example, problems in the theory of radioactive transfer, in the theory of neutron transport and the kinetic theory of gasses lead to quadratic integral equations (see [1,2,3, 4,5]).

The existence of solutions of the integral equations of Volterra-Stiltjes type has been studied by J. Banaś in [6,7, 8,9,10, 11,12].

Consider the delay quadratic integral equation of Volterra-Stieltjes type

Here we study the existence of solutions of the quadratic integral equation (1) in the class of continuous functions. The continuous dependence of the unique solution of the quadratic integral equation (1) on the functions g 1 , g 2 and φ will be studied. As an application, we use the concept and properties of the fractional-order integral (see, for example, [13,14] and [15,16,17, 18,19,20, 21,22]) to obtain the solutions of the fractional-order quadratic integral equation

the Chandrasekhar quadratic integral equation

and the Chandrasekhar cubic integral equation

will be considered.

2 Existence of at least one solution

Now we consider the delay quadratic integral equation of Volterra-Stieltjes type (1) under the following assumptions:

1. φ : [ 0 , 1 ] → [ 0 , 1 ] is continuous and increasing such that φ ( t ) ≤ t .

2. a ∈ C [ 0 , 1 ] .

3. f i : [ 0 , 1 ] × [ 0 , 1 ] × R → R are continuous such that

   ∣ f 1 ( t , s , x ) ∣ ≤ b 1 ( t , s ) + k 1 ( t , s ) ∣ x ∣ p , p ≥ 1

   and

   ∣ f 2 ( t , s , x ) ∣ ≤ b 2 ( t , s ) + k 2 ( t , s ) ∣ x ∣ ,

   where b i , k i : [ 0 , 1 ] × [ 0 , 1 ] → R + are continuous and

   b = sup t { b i ( t , s ) : t , s ∈ [ 0 , 1 ] , i = 1 , 2 } ,

   k = sup t { k i ( t , s ) : t , s ∈ [ 0 , 1 ] , i = 1 , 2 } .

4. (1) The function g 1 is continuous on Λ , where

   Λ = ( t , s ) : 0 ≤ s ≤ t ≤ 1 .

5. (2) The function g 2 : [ 0 , 1 ] × R → R is continuous with

   μ = max { sup { ∣ g 2 ( t , 1 ) ∣ : t ∈ [ 0 , 1 ] } , sup { ∣ g 2 ( t , 0 ) ∣ : t ∈ [ 0 , 1 ] } } .

6. (1) For each ε > 0 there exists δ > 0 for all t 1 , t 2 ∈ I such that t 1 < t 2 and t 2 − t 1 < δ the following inequality holds:

   ⋁ 0 t 1 [ g 1 ( t 2 , s ) − g 1 ( t 1 , s ) ] ≤ ε .

7. (2) For all t 1 , t 2 ∈ I such that t 1 < t 2 the function s → g 2 ( t 2 , s ) − g 2 ( t 1 , s ) is nondecreasing on [ 0 , 1 ] .

8. (1) g 1 ( t , 0 ) = 0 for any t ∈ [ 0 , 1 ] .

9. (2) g 2 ( 0 , s ) = 0 for any s ∈ [ 0 , 1 ] .

10. The function s → g 1 ( t , s ) is of bounded variation on [0, t] for each fixed t ∈ I .

11. There exists a positive root r of the algebraic equation

    (2) k 2 K μ r p + 1 + b k K μ r p + ( b k μ K − 1 ) r + ( a + b 2 K μ ) = 0 .

    If p = 1 , then the algebraic equation (2) has the form

    k 2 K μ r 2 + ( 2 b k K μ − 1 ) r + ( a + b 2 K μ ) = 0 .

    If p = 2 , the algebraic equation (2) will be of the form

    k 2 K μ r 3 + b k K μ r 2 + ( b k μ K − 1 ) r + ( a + b 2 K μ ) = 0 .

For the existence of at least one solution of the quadratic integral equation (1), we have the following theorem.

3 Uniqueness of the solution

To study the uniqueness of the solution of the functional integral equation (1), we replace the assumption (iii) by:

1. f i : [ 0 , 1 ] × [ 0 , 1 ] × R → R , i = 1 , 2 are continuous and satisfy the Lipschitz condition

   ∣ f 1 ( t , s , x ) − f 1 ( t , s , y ) ∣ ≤ k 1 ∣ x − y ∣ p ≤ 2 r p − 1 k 1 ∣ x − y ∣ , x , y ∈ Q ,

   ∣ f 2 ( t , s , x ) − f 2 ( t , s , y ) ∣ ≤ k 2 ∣ x − y ∣ , x , y ∈ Q

   and k = max { 2 r p − 1 k 1 , k 2 } .

From assumption ( i i i ) ∗ we have consecutively

Hence,

Similarly,

For the uniqueness of the solution of the functional integral equation (1) we have the following theorem.