Existence of at least one solution of singular Volterra-Hammerstein integral equation and its numerical solution

domain of integration and T is the time. The kernel of Hammerstein integral term has a singularity, while the kernel of Volterra is continuous in time. Using a quadratic numerical method with respect to time, we have a system of Hammerstein integral equations (SHIEs) in position. The existence of at least one solution for the SHIEs is considered and discussed. Moreover, using Toeplitz matrix method (TMM), the SHIEs are transformed into a nonlinear algebraic system (NAS). Many theorems related to the existence of at least one solution for this system are proved. Finally, numerical results and the estimate error of it are calculated and computed using Mable 12.


INTRODUCTION
Linear and nonlinear singular integral equations have received considerable interest the mathematical applications in different areas of sciences.The different numerical methods play an important role in solving the nonlinear integral equations (NIE).Kummer and Sloan (2003) used a new collection type method to discuss the solution of HIE with continuous kernel.Kummer (1988) used a discrete collection-type method to discuss the solution of HIE with continuous kernel.Hacia (1993) used projection-iteration methods, and an approximate method to discuss the solution of HIE with continuous kernel.Moreover, the super convergence of some numerical methods for HIE with continuous kernels is observed and developed through the work of many authors (Zhang, 2008;Diago and Lima, 2008;Kaneko and Xu, 1996).When the kernel of HIE has a singular term, new different numerical methods were used (Lardy, 1981;Abdou et al., 2005;Abdou et al., 2009;Vainikko, 2011).More Email: rmatoog_777@ yahoo.comMSC: 45B05, 45E10, 65R10 Author(s) agree that this article remain permanently open access under the terms of the Creative Commons Attribution License 4.0 International License information for solving the NIE using different methods can be found in the work of Abdou (2003), Abdou and Al-Bigamy (2013a), Bazm and Babolian (2012) and Abdou et al. (2013b).
Consider a general formula of V-HIE of the second kind 0 ( , ) ( , ) ( ) ( , ) ( , , ( , ))  defines the kind of IE; while λ has a physical meaning and  is the domain of integration with respect to position.To discuss the existence of at least one solution of Equation ( 1) in ( ) [0, ], 1, 1   , we write IE (1) in the integral operator form     0 ( , ) ( , ) ( ) , , ( , ) , ( , 0, ,0 1). (1.3) Then, Let  S be the set of functions  in   , is a constant, and assume the following necessary conditions: i) The kernel of position iii) The given function ( , ) f x t with its partial derivatives with respect to position x and time t are continuous in the space

BASIC THEOREMS AND DEFINITIONS
We state the famous theorems used in proving the principal theorem as follows: Theorem 1 (without proof) Let S be a closed and convex set in a Hilbert space, and K is a continuous mapping of S into itself.Suppose that the set S is compact, and then K has at least one fixed point in S (Abdou et al., 2005).

Theorem 2 (Modified Schauder fixed point; (without proof))
Let S be a closed set, convex set, in a Hilbert space, and K is a continuous mapping of S into itself.Suppose that () KS is compact, then K has at least one fixed point in S (Abdou et al., 2011).
In the remainder part of this paper, the existence of at least one solution for V-HIE (1), under the necessary conditions, in the space ( ) [0, ], 1, 1,  will be proved.In addition, using a quadratic numerical method with respect to time, we obtain SHIEs, where the existence of at least one solution of the system can be proved.
Moreover, using TMM, we represent the SHIEs in the form of NAS.Many different theorems are derived to prove the existence of at least one solution of the NAS.Finally, some examples, when the kernel of position takes a logarithmic form, Carleman function and Cauchy kernel are calculated numerically and the error estimate, in each case, is computed.

The principal theorem of at least one solution
Here, we state several lemmas that lead to prove the following principal theorem:
Lemma 2: Under the conditions (i-iii) the integral operator W of Equation (2) maps the set S  into itself.
Proof: In the light of Equations ( 2) and (3), we get Applying Cauchy-Schwarz inequality to Hammerstein integral term, and then using the conditions (i-iii), the above inequality can be adapted in the form The above inequality shows that, the operator W maps the set  S into itself.
, then in the light of condition (iv) the above inequality reduces to This implies the continuity of n Ft be two sequences of continuous functions satisfy the conditions Then, after neglecting very small constants, and for positive integer Now, in view of Lemma 4, we define the sequence of operator   n W as: .
Lemma 5: The integral operator (8) maps the set S  continuously into itself.Moreover, the integral operator (8) is continuous.
The proof of Lemma ( 5) can be obtained directly after using Lemmas (2) and (3).
Lemma 6: Under the same conditions (i-iv) and Lemma 4, the set () WS  of Equation ( 8) is compact.


After using Lemma 4 and conditions (i-iv), and then applying Hölder inequality to HI term, the above inequality yields Therefore, .
n n m n  Then from (11) for large j , we get the Cauchy sequence .
After the above discussion, the principle theorem is proved:

SYSTEM OF HAMMERSTEIN INTEGRAL EQUATIONS
Here, quadratic numerical method is used, in Equation ( 1) to obtain SHIEs in position.For this aim, we divide the interval   Where, ). 2 The values of j u and ; p p k  are depending on the number of derivatives of ( , ) Ft with respect to time, see Atkinson (2011).
Using (10) in (1), and neglecting   Where, we used the following notations
The formula ( 11 Following the same way of Lemmas 2, 3 and 6 of the principal theorem 3, we can directly proof the following lemmas and principal theorem of SHIEs. Lemma 7 (without proof): Under the conditions (i) -(iva), the operator Lemma 8 (without proof): Under that, the conditions (i), (ii) and (iv-b) V is a compact operator in the space

Principal Theorem 4 of SHIEs:
According to Lemmas 7 and 8, the SHIEs of the second kind ( 11) have at least one solution.

THE TOEPLITZ MATRIX METHOD
Here, we will discuss the solution of Equation (1) numerically, using TMM in one dimensional, and [ , ] bb    .For this, write the integral term Equation (11) in the form (Abdou et al., 2011) ( 1) 1 ))} ; ( ; where, ( ) ( ) B x for all 0 j i   are arbitrary functions to be determined, and In the light of TMM (Abdou et al., 2011), the integral formula ( 11) yields where The error term for each value of j , is determined from the following formula The existence of at least one solution of the NAS The existence of at least one solution of the NAS (18) in the space  , will be proved according to Schauder fixed point theorem.For this, we write it in the operator form: Let  be the set of the two families ,  are constants, and then consider the following conditions: For the known set Now we state and prove some lemmas that lead directly to the principal theorem of NAS.
The above inequality proves that, the operator T maps the set  into itself.In addition, the inequality (24) define the boundedness of the operator T and T .
Lemma 10: Under the conditions ( 21) and ( 23), T is continuous in the set  .
Lemma 13: Under the same conditions of lemma 12, the set Proof: From the formula (31), we have


Hence, with the aid of ( 26) and ( 27), there exists a positive integer Inequality (32) shows that, .
Hence, the sequence The previous Lemmas 9-13 show that, T is a continuous operator maps the set  , which is evidently closed and convex set into itself, and ( ) T  is a compact set.Therefore, we can state the following by theorem:

Principal Theorem 5 of the NAS
The NAS (18) has at least one solution in set  under the condition .

  
Now, it is suitable to consider the following theorem which proves the convergence of one sequence of approximate solutions to some solution of Equation ( 18 The above inequality, after using the two conditions ( 30) and ( 31), holds for each integer m , hence  34) with logarithmic kernel are solved numerically, in different times, using TMM.The error, in each case, is computed.We see that as t increases the error increases.Also, the error, using Maple 12, in the linear case is less than the error in the nonlinear case. (where Since as j  , so that

NUMERICAL EXAMPLES AND DISCUSSION
For the integral equation  2).The importance of Carleman kernel comes from the work of Arytiunian (1959) that has shown that the plane contact problem in the nonlinear theory of plasticity, in its first approximation can be reduced to Fredholm integral equation of the first kind with Carleman kernel.5) The third case when the kernel takes the Cauchy kernel The results are computing, at, 0.01, 0.1, 0.4 t  and 0.8 t  and 30 N  , (Table 3).In Table 3, the mixed integral equation in linear and nonlinear case (34) with Cauchy are solved numerically in different times, using TMM.The error increases with increase in the time and further increases the linearity of the equation.
The importance of the above kernel is found in the work of Abdou and Salama (2004).
6) The Toeplitz matrix method is considered one of the best methods for solving the singular integral equations with discontinuous kernel, where the singular part disappears and the solution is obtained directly.  , we have the potential kernel; see Abdou (2002).For1/ 2 1   , we have the generalized potential function (Abdou et al., 2013b) VI term in time.The constant 

.
) represents SHIEs and its solution depends on the given function The formula (11) can be written in the integral operator form If the conditions (19) and (20) are verified, and the sequence of functions By virtue of the formula (31), we get

7)
From Tables (1) -(3), we note that as N increases the error decreases while at t increases the error increases.

Table 3 .
In Table2the linear and nonlinear mixed integral Equation (34) with Carleman function are solved numerically, in different times, using TMM.The error, in each case, is computed.As increases the error are increases.In addition as the time increases, the error increases Principal example: Cauchy kernel