Nondecreasing solutions of a quadratic Abel equation with supremum in the kernel

We prove an existence theorem for a quadratic Abel integral equation of the second kind with supremum in the kernel. The quadratic integral equation studied below contains as a special case numerous integral equations encountered in the theory of radiative transfer and in the kinetic theory of gases. We show that the singular quadratic integral equations with supremum has a monotonic solution in C[0,1]. The concept of measure of noncompactness and a fixed point theorem due to Darbo are the main tools in carrying out our proof.

[1]  K. Deimling.  Nonlinear functional analysis , 1985 .

[2]  Józef Banaś,et al.  Measures of noncompactness related to monotonicity , 2001 .

[3]  D. Bainov,et al.  Oscillation Theory for Neutral Differential Equations with Delay , 1991 .

[4]  J. Caballero,et al.  On monotonic solutions of an integral equation of Volterra type with supremum , 2005 .

[5]  Józef Banaś,et al.  Monotonic solutions of a quadratic integral equation of Volterra type , 2004 .

[6]  Mohamed Abdalla Darwish,et al.  On quadratic integral equation of fractional orders , 2005 .

[7]  Snezhana Hristova,et al.  Monotone-iterative techniques of V. Lakshmikantham for a boundary value problem for systems of impulsive differential equations with «Supremum» , 1993 .

[8]  Józef Banaś,et al.  Existence Theorems for Some Quadratic Integral Equations , 1998 .

[9]  Józef Banaś,et al.  Monotonic solutions of a quadratic integral equation of fractional order , 2007 .

[10]  A. Salam,et al.  THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS ON SOLVABILITY OF SOME QUADRATRIC FUNCTIONAL-INTEGRAL EQUATION IN BANACH ALGEBRA , 2008 .

[11]  Prof. Dr. Mohamed Abdalla Darwish.  Dynamic Systems and Applications 17 ( 2008 ) 539-550 ON MONOTONIC SOLUTIONS OF A SINGULAR QUADRATIC INTEGRAL EQUATION WITH SUPREMUM , 2008 .

[12]  Shouchuan Hu,et al.  Integral equations arising in the kinetic theory of gases , 1989 .

[13]  J. Caballero,et al.  Existence of Nondecreasing and Continuous Solutions for a Nonlinear Integral Equation with Supremum in the Kernel , 2007 .

[14]  Prof. Dr. Mohamed Abdalla Darwish.  On global attractivity of solutions of a functional-integral equation , 2007 .

[15]  S. Ntouyas,et al.  Monotonic solutions of a perturbed quadratic fractional integral equation , 2009 .

[16]  Mohamed Abdalla Darwish,et al.  On a perturbed quadratic fractional integral equation of Abel type , 2011, Comput. Math. Appl..

[17]  S. Ntouyas,et al.  On a quadratic fractional Hammerstein–Volterra integral equation with linear modification of the argument , 2011 .