Hybrid fixed point theory in partially ordered normed linear spaces and applications to fractional integral equations. (English) Zbl 1279.45005

Dhage, Bapurao C.

Differ. Equ. Appl. 5, No. 2, 155-184 (2013).

Some basic hybrid fixed point theorems of Banach and Schauder type and some hybrid fixed point theorems of Krasnoselskii type involving the sum of two operators are proved in a partially ordered normed linear space. The author improves his previous results under weaker conditions and applies them to nonlinear Volterra type fractional integral equations for proving the existence of solutions under certain monotonic conditions blending with the existence of either a lower or an upper solution type function.
The nonlinear integral equations of the following types are considered \[ x(t)=h(t)+\frac{1}{\Gamma(q)}\int_{t_0}^{t}(t-s)^{q-1}g(s,x(s))ds \] and \[ x(t)=f(t,x(t))+\frac{1}{\Gamma(q)}\int_{t_0}^{t}(t-s)^{q-1}g(s,x(s))ds, \] where \(0<q<1\), \(t\in[t_0,t_0+a]\), \(a>0\).

Reviewer: Alexander N. Tynda (Penza)

MSC:

45G05	Singular nonlinear integral equations
45M99	Qualitative behavior of solutions to integral equations
47H09	Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10	Fixed-point theorems
26A33	Fractional derivatives and integrals

Keywords:

Hybrid fixed point theorem; partially ordered normed linear space; fractional integral equation; existence theorem; Banach fixed point theorem; Schauder fixed point theorem; Krasnoselskii fixed point theorem; nonlinear Volterra type fractional integral equations