Abstract： In this work， the existence of a unique solution of Volterra-Hammerstein integral equation of the second kind （V-HIESK） is discussed. The Volterra integral term （VIT） is considered in time with a continuous kernel， while the Fredholm integral term （FIT） is considered in position with a generalized singular kernel. Using a numerical technique， V-HIESK is reduced to a nonlinear system of Fredholm integral equations （SFIEs）. Using Toeplitz matrix method we have a nonlinear algebraic system of equations. Also， many important theorems related to the existence and uniqueness of the produced algebraic system are derived. Finally， some numerical examples when the kernel takes the logarithmic， Carleman， and Cauchy forms， are considered.
　　Keywords： Singular integral equation； Nonlinear Volterra –Fredholm integral equation； Toeplitz matrix； Cauchy kernel； Carleman kernel.
　　Al-Bugami A. M. （2013）. Toeplitz Matrix Method and Volterra-Hammerstien Integral Equation With a Generalized Singular Kernel. Progress in Applied Mathematics， 6（2）， 16-42. Available from http：//www.cscanada.net/index.php/ pam/article/view/j.pam.1925252820130602.2593 DOI：10.3968/j.pam.1925 252820130602. 2593
　　 1. INTRODUCTION
　　Many authors have interested in solving the Volterra-Fredholm integral equation， Abdou and Salama， in [1]， obtained the solution in one， two and three dimensional for the V –FIT of the first kind using spectral relationships. In [2]， EL-Borai et al. studied the existence and uniqueness of solution of nonlinear integral equation of the second kind of type V - FIE. Maleknejad and Sohrabi， in[3]， solved the nonlinear V –F –Hammerstein integral equations in terms of Legendre polynomials. In [4]， Ezzati and Najafalizadeh， used Chebyshev polynomials to solve linear and nonlinear Volterra-Fredholm integral equations. Shazad， in [5]， solved Volterra-Fredholm integral equation by using least squares technique. Shali et al.， in [6]， studied the numerical solvability of a class of nonlinear Volterra-Fredholm integral equations.
　　In this work， we consider the V –HIESKK
　　Applying H？lder inequality to Hammerstein integral term， and taking in account（5）， the above inequality becomes
　　Since ？？？1， then the inequality （13） is true only if ？？（x ，t ）？？？（x ，t ） ； that is， the solution of Eq. （1） is unique.
　　2.2 The existence of a unique solution using Banach fixed point theorem：
　　 2.3 The existence of at least one solution： 　　If the condition （d-2） of theorem （1） is not verified， then the existence of at least one solution of Eq. （1） can be
　　established by virtue of the following theorem.
　　Theorem 2. Consider Eq. （1） with the same conditions for the functions k （ g （x ）？？g （y ） ）， （ ， ）F t？ and f （x ，t ） as
　　Applying H？lder inequality to Hammerstein integral term， then using conditions （a）， （i） of theorem （1） and （2）， respectively， and Eq.（22）， the above inequality becomes
　　In this section， a numerical method is used， in the mixed integral Eq. （1） to obtain a system of nonlinear integral equations with a generalized singular kernel in position. For this aim we divide the interval
　　 4. THE TOEPLITZ MATRIX METHOD
　　Here， we will discuss the solution of Eq. （1） numerically using Toeplitz matrix method， and ？？[？b，b] . For this， write Eq. （1） in the form
　　represent a matrices of order （2N ？1） whose elements are zeros except the first and the last rows （columns）.
　　The error term R is determined from the following formula
　　The existence of a unique solution of the algebraic system （42）， will be proved according to the Banach fixed point theorem. For this aim， we consider the following assumption
　　The above inequality， after using condition （60）， holds for each integer ，m hence from condition （62）， we find
　　 6. CONCLUSION
　　（1） When the values of λ and υ are increasing and the values of the time T kept fixed， the error is increasing， where the atomic bond between the particles of the material is increasing.
　　（2） When the values of time T are increasing and the values of λ， υ and N kept fixed， the error is increasing.
　　（3） The Toeplitz matrix method is the efficient numerical method， for solving the V - HIESK with singular kernels， compared to the other methods.
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