Positive solutions of a Hammerstein integral equation with a singular nonlinear term

Abstract

In this paper the existence of a positive measurable solution of the Hammerstein equation of the first kind with a singular nonlinear term at the origin is presented.

A stability result for elliptic equations with singular nonlinearity and its applications to homogenization problems

“…Mathematical study involved has been ongoing, especially since the work by Crandall, Rabinowitz and Tartar [10]. For basics of the singular elliptic problems, we refer to [39,18,26,9,44,24,35] and the references therein.…”

The inapplicability of variational methods in the cohesive crack problem for initially rigid traction‐separation relation and its solution using integral equations

“…A complicated/realistic cohesive law crack problem would give an integral equation in which the integrand exists as a reciprocal. The existence of a solution to such an equation has been studied earlier [56][57][58][59][60]. However, till date numerical method to solve the integral equation for complicated cohesive relation has not been reported though some directions towards these are available [61,62].…”

Coupled systems of Hammerstein integral equations

“…The existence, uniqueness, multiplicity, positivity and location of solutions are the most studied and predominant elements as regards Hammerstein integral equations. Citing just a few examples in the literature, we mention [24], where the authors use fixed point index theory to establish their main result, based on a priori estimates achieved by nonnegative matrices; in [11], Coclite studies the existence of a positive measurable solution of the Hammerstein equation of the first kind with a singular nonlinear term at the origin; in [8], the authors contribute, by monotone iterative methods, combined with the classical fixed point index, proving two results concerning non-decreasing and non-increasing operators in a shell, in the presence of an upper or a lower solution; in [9], Cardinali et al, examine multivalued Hammerstein integral equations defined in a separable reflexive Banach space, obtaining existence results for convex and nonconvex problems; in [13], the researchers study solutions of the nonlinear Hammerstein integral equation with changingsign kernels by using a variational principle of Ricceri and critical points theory techniques (they combine the effects of a sublinear and superlinear nonlinear terms to establish new existence and multiplicity results); in [26], the authors study the existence and the uniqueness of iterative positive solutions for a class of nonlinear singular integral equations in which the nonlinear terms may be singular in both time and space variables. By using the fixed point theorem of mixed monotone operators in cones, they establish the conditions for the existence and uniqueness of positive solutions to the problem.…”