Abstract
We investigate the existence of solutions for a class of second‐order q‐difference inclusions with nonseparated boundary conditions. By using suitable fixed‐point theorems, we study the cases when the right‐hand side of the inclusions has convex as well as nonconvex values.
Highlights
The discretization of the ordinary differential equations is an important and necessary step towards finding their numerical solutions
We are concerned with the existence of solutions for the Problems 1.1 - 1.2 when the right-hand side has convex as well as nonconvex values
We assume that F is a compact and convex valued multivalued map
Summary
The discretization of the ordinary differential equations is an important and necessary step towards finding their numerical solutions. Instead of the standard discretization based on the arithmetic progression, one can use an efficient q-discretization related to geometric progression. This alternative method leads to q-difference equations, which in the limit q → 1 correspond to the classical differential equations. We study the existence of solutions for second-order q-difference inclusions with nonseparated boundary conditions given by. We will combine the nonlinear alternative of Leray-Schauder type for single-valued maps with a selection theorem due to Bressan and Colombo for lower semicontinuous multivalued maps with nonempty closed and decomposable values, while in the third result, we will use the fixed-point theorem for generalized contraction multivalued maps due to Wegrzyk.
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