On the nonlinear Hammerstein integral equations in Banach spaces and application to the boundary value problem of fractional order

Abstract

In this paper, we study the existence of solutions of the operator equations $p+\lambda Gfx=x$ in the Banach space $C[I,E]$. It is assumed the vector-valued function $f$ is nonlinear Pettis-integrable. Some additional assumptions imposed on $f$ are expressed in terms of a weak measure of noncompactness. To encompass the full scope of the paper, we investigate the existence of pseudo-solutions for the nonlinear boundary value problem of fractional type

$$-\frac{d^{\alpha }}{d{t}^{\alpha }}x\left(t\right)=\lambda f(t,x\left(t\right))\text{,}\text{a.e. on }[0,1]\text{,}x\left(0\right)=x\left(1\right)=0,\alpha \in (1,2]\text{,}$$

under the Pettis integrability assumption imposed on $f$.