Nonlinear mixed Volterra-Fredholm integrodifferential equation with nonlocal condition.
The aim of this paper is to study the global existence of solutions of the equation \[ \frac {\text{d}}{\text{d}t}[x(t)-u(t,x(t))]=Ax(t)+f\left (t,x(t),\int ^{t}_{0}k(t,s,x(s))\text{d}s,\int ^{b}_{0}h(t,s,x(s))\text{d}s\right ) \] satisfying the initial data \(x(0)=x_0+g(x)\). In order to study the global existence of mild solutions to the above problem, the authors use a theorem whose proof is based on an application of the Leray-Schauder alternative. This theorem ensures simultaneously the global existence of solutions and the maximal interval of existence, a fact which is somewhat remarkable. The preceeding results of those two authors were gradually obtained and the result of this paper represents a further work of the previous authors' research. The existence and uniqueness of mild and strong solutions of a nonlinear Volterra integro-differential equation with nonlocal condition (the analysis is based on semigroups theory and the Banach fixed point theorem, and inequalities were established by T. H. Gronwall and B. G. Pachpatte) were proved in [Demonstr. Math. 43, No. 3, 643--652 (2010; Zbl 1228.45009)]. The existence and uniqueness of mild and strong solutions of a nonlinear mixed Volterra-Fredholm integro-differential equation with nonlocal condition (the results was obtained by using the Schauder fixed point theorem and the theory of semigroups) were proved in [the authors, Tamkang J. Math. 41, No. 4, 361--373 (2010; Zbl 1256.45001)]. The paper gives a theoretical method, which makes it possible to obtain some results on the global existence of mild solutions of some nonlinear integro-differential equations (with nonlocal condition).
Volterra-Fredholm integro-differential equation
scientific article; zbMATH DE number 6121193
scientific article