Nonlinear Fredholm equations in modular function spaces

Abstract

We investigate the existence of solutions in modular function spaces of the Fredholm integral equation $$ \Phi(\theta) = g(\theta) + \int^1_0 f(\theta,\sigma, \Phi(\sigma)) \,d\sigma, $$ where $\Phi(\theta), g(\theta)\in L_{\rho}, \theta\in [0,1], f: [0,1]\times[0,1]\times L_{\rho}\to \mathbb{R}$. An application in the variable exponent Lebesgue spaces is derived under minimal assumptions on the problem data.

Keywords

• Electrorheological fluids
• fixed point
• Fredholm equations
• modular function spaces
• variable exponent spaces