Existence and exponential decay for a nonlinear wave equation with nonlocal boundary conditions of 2 N ‐point type

This paper is devoted to the study of a nonlinear wave equation with initial conditions and nonlocal boundary conditions of 2 N ‐point type, which connect the values of an unknown function u ( x , t )   at x  = 1,x  = 0,   x  =  η i ( t ) , and x  =  θ i ( t ), where 0 < η 1 ( t ) < η 2 ( t ) < … < ηN − 1( t ) < 1 ,   0 < θ 1 ( t ) < θ 2 ( t ) < … < θN − 1( t ) < 1 ,   for all t  ≥ 0.   First, we prove local existence of a unique weak solution by using density arguments and applying the Banach's contraction principle. Next, under the suitable conditions, we show that the problem considered has a unique global solution u ( t )   with energy decaying exponentially as t  → + ∞ . Finally, we present numerical results.