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A study of solutions of some nonlinear integral equations in the space of functions of bounded second variation in the sense of Shiba. (English) Zbl 1495.45002

The authors study the existence and uniqueness of solutions for nonlinear Hammerstein equations of the form \[ x(t)=g(t)+\lambda\int_{I}K(t,s)f(x(s))ds,\quad \lambda\in I,\ t\in I=[0,b],\ b>0, \tag{1} \] Volterra-Hammerstein equations of the form \[ x(t)=g(t)+\int_{0}^{t}K(t,s)f(x(s))ds,\quad t\in[0,b],\tag{2} \] and Volterra equations of the form \[ x(t)=g(t)+\int^{t}_{a}K(t,s)f(x(s))ds,\quad t\in[a,b],\ a<b, \tag{3} \] in the space of functions of bounded second variation in the sense of Shiba, viz, \(\bigwedge^{2}_{p}BV([a,b])\). For the proofs of the theorems for Equations (1) and (2), the authors use the Banach contraction principle, while for equations of type (3) the Leray-Schauder alternative theorem is used. A section is devoted to applications, where a nonlinear Hammerstein-Volterra integral equation is solved by means of numerical methods.

MSC:

45G10 Other nonlinear integral equations
45D05 Volterra integral equations
47N20 Applications of operator theory to differential and integral equations
26A45 Functions of bounded variation, generalizations
65R20 Numerical methods for integral equations
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