Three-Point Boundary Value Problems of Nonlinear Second-Order q-Difference Equations Involving Different Numbers of q

We study a new class of three-point boundary value problems of nonlinear second-order q-difference equations. Our problems contain different numbers of q in derivatives and integrals. By using a variety of fixed point theorems (such as Banach’s contraction principle, Boyd and Wong fixed point theorem for nonlinear contractions, Krasnoselskii’s fixed point theorem, and Leray-Schauder nonlinear alternative) and Leray-Schauder degree theory, some new existence and uniqueness results are obtained. Illustrative examples are also presented.

[1]  A. A. Potapenko,et al.  Method of Successive Approximations , 1964, Encyclopedia of Evolutionary Psychological Science.

[2]  Moustafa El-Shahed,et al.  Positive solutions of q-difference equation , 2009 .

[3]  F. H. Jackson.  q-Difference Equations , 1910 .

[4]  On nonlocal boundary value problems of nonlinear q-difference equations , 2012 .

[5]  J. S. W. Wong,et al.  On nonlinear contractions , 1969 .

[6]  C. R. Adams,et al.  On the Linear Ordinary q-Difference Equation , 1928 .

[7]  L. Vinet,et al.  Automorphisms of the q-oscillator algebra and basic orthogonal polynomials , 1993 .

[8]  Martin Bohner,et al.  The H-Laplace and Q-Laplace Transforms , 2010 .

[9]  Anatol Odzijewicz,et al.  Second order q -difference equations solvable by factorization method , 2003 .

[10]  Mizan Rahman,et al.  Some systems of multivariable orthogonal q-Racah polynomials , 2004, math/0410250.

[11]  Jiang Zeng,et al.  On a q-sequence that generalizes the median Genocchi numbers , 1999 .

[12]  T. E. Mason.  On Properties of the Solutions of Linear q-Difference Equations with Entire Function Coefficients , 1915 .

[13]  L. Vinet,et al.  Symmetries of theq-difference heat equation , 1994 .

[14]  Gaspard Bangerezako,et al.  Variational q-calculus , 2004 .

[15]  Mahmoud H. Annaby,et al.  q -Difference Equations , 2012 .

[16]  R. D. Carmichael,et al.  The General Theory of Linear q-Difference Equations , 1912 .

[17]  The spectral problem for the q-Knizhnik-Zamolodchikov equation , 1993, hep-th/9305091.

[18]  Plamen Simeonov,et al.  Q-difference Operators for Orthogonal Polynomials , 2009, J. Comput. Appl. Math..

[19]  Ahmed Alsaedi,et al.  A study of second-order q-difference equations with boundary conditions , 2012, Advances in Difference Equations.

[20]  Jessada Tariboon,et al.  Sequential Derivatives of Nonlinear q-Difference Equations with Three-Point q-Integral Boundary Conditions , 2013, J. Appl. Math..

[21]  S. Ntouyas,et al.  Boundary value problems for q-difference inclusions , 2011 .

[22]  Bashir Ahmad,et al.  Boundary Value Problems for -Difference Inclusions , 2011 .

[23]  q -Gamma and q -beta functions in quantum algebra representation theory , 1996 .

[24]  Existence of solutions for nonlinear second-order q-difference equations with first-order q-derivatives , 2013 .

[25]  Wen-Xue Zhou,et al.  Existence solutions for boundary value problem of nonlinear fractional q-difference equations , 2013 .