Existence of solutions for a class of IBVP for nonlinear hyperbolic equations

Abstract

We study a class of initial boundary value problems of hyperbolic type. A new topological approach is applied to prove the existence of non-negative classical solutions. The arguments are based upon a recent theoretical result. Keywords Hyperbolic equations Á Positive solution Á Fixed point Á Cone Á Sum of operators Mathematics Subject Classification 47H10 Á 58J20 Á 35L15 This article is part of the section ''Theory of PDEs'' edited by Eduardo Teixeira.

Existence of Solutions of Boundary Value Problem for Nonlinear One-Dimensional Wave Equations by Fixed Point Method

“…A transformation which allows certain additional conditions used to derive several existence results by restoring the theory of a new fixed point theory indexed in cones to strict the sect of contraction mappings. Some of these results have been improved in several directions, and they have been applied to obtain existence results of initial and boundary value problems subject to ordinary and partial differential equations ( 20)- (23). e existence of solutions (7) when the spatial variable x ranges over the finite interval of the real line has supplemented with boundary conditions to specify how the solution u(x, t)is constrained to be at the two end pointsx � 0 and x � L. is is linked with the two most commonly arising types of boundary conditions: Dirichlet and Neuman boundary conditions [16].…”

“…Moreover, the solutions for a class of IBVP for nonlinear one-dimensional wave equations were obtained and reported by several different authors and also very applicable in different mathematical problems and physics [20]. In general, there is a vast literature concerning to existence of nonnegative solutions for a class of IBVP for nonlinear wave equations which is mentioned in [20,22,23,34]. However, they did not apply the fixed point method to determine solutions of the initial boundary value problem for nonlinear one-dimensional wave equations by using the specific method.…”

Two nonnegative solutions for two-dimensional nonlinear wave equations

“…To prove our main result we use a new topological approach. This approach can be used for investigations for existence of at least one and at least two classical solutions for initial value problems, boundary value problems and initial boundary value problems for some classes ordinary differential equations, partial differential equations and fractional differential equations (see [2,3,4,7,10,12,13,15,16] and references therein). So far, for the authors they are not known investigations for existence of multiple solutions for the IVP (1.1).…”