Existence results for nonlinear \(q\)-difference equations with nonlocal boundary conditions (Q2839824)

This paper discusses the existence of solutions of the nonlinear \(q\)-difference equations with nonlocal boundary conditions NEWLINE\[NEWLINE D_q^2x(t)=f(t,x(t)),\quad t\in I_q, NEWLINE\]NEWLINE NEWLINE\[NEWLINE \alpha_1x(0)-\beta_1D_qx(0)=\gamma_1x(\eta_1),NEWLINE\]NEWLINE NEWLINE\[NEWLINE\alpha_2x(1)+\beta_2D_qx(1)=\gamma_2x(\eta_2), NEWLINE\]NEWLINE where \(f\in C(I_q\times \mathbb{R}, \mathbb{R})\), \(I_q=\{q^n:n\in\mathbb{N}\}\cup\{0,1\}\), \(q\in(0,1)\), and \(\eta_1, \eta_2 \in \{q^n:n\in\mathbb{N}\}\) with \(\eta_1<\eta_2\).NEWLINENEWLINEIn the second section, the authors first obtain a lemma that provides the unique solution of the boundary value problem for the special case when \(f(t, x(t))=g(t)\) with \(g(t)\in C(I_q,\mathbb{R})\). Then they introduce an operator \(F:C(I_q,\mathbb{R})\rightarrow C(I_q,\mathbb{R})\) by which the existence of a solution of the original boundary value problem is equivalent to the fixed point problem of the equation \(F x=x\).NEWLINENEWLINEThe main results are given in the third section. The first theorem provides a set of conditions under which the boundary value problem has a unique solution using Banach's contraction principle. Their second existence theorem, that gives another set of conditions so that the boundary value problem has at least one solution, is based on Krasnoselskii's fixed point theorem. At last, based on the Leray-Schauder nonlinear alternative, they obtain the third result that the boundary value problem also has at least one solution. Illustrative examples are given. Moreover, some special cases are discussed. This paper will be of interest to anyone who is studying \(q\)-difference equations.