On a class of nonlinear Volterra-Fredholm q-integral equations

In this paper we investigate the existence and uniqueness of positive continuous solutions for a q-analogue of Volterra and Fredholm integral equations of first and second kinds. We derive the results by using three fixed point theorems introduced by Bushell in [7, 8]. Bushell derived his theorems by using the Cayley-Hilbert projective metric and Banach fixed point theorem. We also include some uniqueness criteria for the solutions of certain nonlinear q-integral equations provided that the solution exists in certain function spaces.

[1]  Tzong-Mo Tsai,et al.  Applications of Hilbert's projective metric to a class of positive nonlinear operators , 2006 .

[2]  M. Schlosser.  BASIC HYPERGEOMETRIC SERIES , 2007 .

[3]  Anatoly A. Kilbas,et al.  On Solution of Nonlinear Abel–Volterra Integral Equation☆ , 1999 .

[4]  Shyam L. Kalla,et al.  On the Riemann-Liouville Fractional q-Integral Operator Involving a Basic Analogue of Fox H-Function , 2005 .

[5]  R. Nussbaum.  Hilbert's Projective Metric and Iterated Nonlinear Maps , 1988 .

[6]  R. Gorenflo,et al.  Fractional Calculus: Integral and Differential Equations of Fractional Order , 2008, 0805.3823.

[7]  Miomir S. Stanković,et al.  On q–Analogues of Caputo Derivative and Mittag–Leffler Function , 2007 .

[8]  A. Potter.  Hilbert's projective metric applied to a class of positive operators , 1976 .

[9]  P. Bushell,et al.  The Cayley-Hilbert metric and positive operators , 1986 .

[10]  A. Potter,et al.  APPLICATIONS OF HILBERT'S PROJECTIVE METRIC TO CERTAIN CLASSES OF NON-HOMOGENEOUS OPERATORS , 1977 .

[11]  A. Potter.  Existence Theorem for a Non-Linear Integral Equation , 1975 .

[12]  Shui-Nee Chow,et al.  Dynamics of Infinite Dimensional Systems , 1987 .

[13]  F. Mainardi,et al.  Fractals and fractional calculus in continuum mechanics , 1997 .

[14]  P. Bushell.  Hilbert's metric and positive contraction mappings in a Banach space , 1973 .

[15]  G. Birkhoff.  Extensions of Jentzsch’s theorem , 1957 .

[16]  R. Nussbaum.  Iterated nonlinear maps and Hilbert's projective metric: a summary , 1987 .

[17]  R. Nussbaum.  Iterated nonlinear maps and Hilbert’s projective metric. II , 1989 .

[18]  P. Bushell,et al.  On a class of Volterra and Fredholm non-linear integral equations , 1976, Mathematical Proceedings of the Cambridge Philosophical Society.

[19]  V. K. Henner,et al.  Ordinary and Partial Differential Equations , 2013 .

[20]  Mahmoud H. Annaby,et al.  q-Fractional Calculus and Equations , 2012 .

[21]  Waleed A. Al-Salam.  Some Fractional q -Integrals and q -Derivatives , 1966 .

[22]  M. Ismail,et al.  Classical and Quantum Orthogonal Polynomials in One Variable: Bibliography , 2005 .

[23]  Natig M. Atakishiyev,et al.  LETTER TO THE EDITOR: On a one-parameter family of q-exponential functions , 1996 .