Density of regular functions in variable exponent Sobolev spaces

In the first chapter of this work, significance of densitiy of smooth functions in variable exponent Sobolev spaces was indicated. Obstacles that cause non-density were explained In the second chapter the historical development, applications and a summary of studies that were done by researcher on variable exponent Lebesgue and variable exponent Sobolev spaces are stated. In the third chapter basic functional analysis, real analysis and concepts related to denseness are given. Sobolev spaces which are classical case of variable exponent Sobolev spaces are mentioned. Definitions and properties of variable exponent Lebesgue and variable exponent Sobolev spaces are given. Researcher works of density in variable exponent Sobolev spaces are given with their proofs. Convolution and mollifier method were explained In the fourth chapter which is the last chapter of this work by using convolution and other tecknique with condition C¹(Rⁿ) to be dense in W^(1,p(.) ) (R^n ) we have shown density of C_0^∞ (R^n ) in W^(1,p(.) ) (R^n ). Besides, a theorem which show pointwise convergence implies weak convergence was proved and an open problem was put.