On a set-valued functional integral equation of Volterra-Stieltjes type

Abstract

In this paper, we study the existence of continuous solutions of a set-valued functional integral equation for the Volterra-Stieltjes type. The asymptotic stability of the solutions will be studied. The continuous dependence of the solution on the set of selections of the set-valued function will be proven. As an application, we study the existence of solutions of an initial value problem of arbitrary (fractional) order differential inclusion.

An outlook on stability of implicit θ -Caputo fractional differential equation with application involving the Riemann-Stieltjes type

“…with integral bounday conditions (4.2)-(4.3). Recently, the problem of such a type was intensively investigated in [19].…”

“…Fractional calculus is a potent tool in applied mathematics, offering a way to analyze a wide range of problems in various scientific and technical fields. Fractional derivatives have yielded significant results in [5,[18][19][20][21][22][23]. The study of partial fractional differential equations, as well as ordinary differential equations, has made substantial progress in recent years.…”

A Nonlocal (Two-point with Parameters) Boundary Value Problem of Differential Inclusion

“…where H(A, B) is the Hausdorff metric between the two subsets A, B ∈ I × E (see [13]) Remark 3 From this assumptions we can deduce that there exists a function…”

“…The models of the arbitrary (fractional-orders) differential and integral equations have many applications ( see [1], [3]- [13] and [15]- [16]). Here, we are concerning with the nonlinear differential inclusion dx dt ∈ F 1 (t, x(t), I γ f 2 (t, x(t))), γ ∈ (0, 1) t ∈ (0, T )…”

Results on solvability of nonlinear quadratic integral equations of fractional orders in Banach algebra

“…Further, the continuous dependence of a unique solution on delay functions will be considered. Many authors use fixed point theorems to prove the existence of the solution to nonlinear integral equations, see [9][10][11]13].…”