Positive solutions of an infinite boundary value problem for \(n\)th-order nonlinear impulsive singular integro-differential equations in Banach spaces
impulsive singular integro-differential equation
In an ordered Banach space \(E\), the existence of a positive (in the sense of the ordering) solution of the integro-differential equation NEWLINE\[NEWLINEx^{(n)}(t)=-f(t,x(t),x'(t),\ldots,x^{(n-1)},\int_0^tK(t,s)x(s)ds, \int_0^\infty H(t,s)x(s)ds)NEWLINE\]NEWLINE with jump conditions (for \(i=1,\ldots,n-1\) and \(k=1,2,\ldots\)) NEWLINE\[NEWLINEx^{(i)}(t_k^+)-x^{(i)}(t_k^-)=I_{ik}(x(t_k),\ldots,x^{(n-1)}(t_k))NEWLINE\]NEWLINE and boundary conditions with given \(x^{(i)}(0)\) \((i=1,\ldots,n-2)\) and \(x^{n-1}(\infty)\) is shown where the function \(f\) can have a singularity at \(t=0\). Also, it is shown that a certain method of successive approximation converges to the solution. The main tool for the existence result is an application of Mönch's fixed point theorem.
scientific article; zbMATH DE number 6047339
scientific article