Existence theory for nonlinear volterra integral and differential equations

Implicit type Volterra integrodifferential equation

“…From the hypotheses, we have Remark 3. We note that our approach to the study of IVP (1.5)−(1.6) is different from those in [3,5,7,10] and we believe that the results given here are of independent interest. We also note that the idea employed here can be extended to the study of higher order integrodifferential equation of the form y (n) (t) = f t, y (t) , .…”

“…x ′ (t) = F t, x (t) , t 0 K (t, s, x (s)) ds , x (0) = 0, (1.1) by using the topological transversality argument and a certain integral inequality with explicit estimate on the unknown function (see also [7,10]). In (author?)…”

“…Integrating both sides of equation (1.4) from 0 to t ∈ R + , one may arrive at a slight variant of equation of the form (1.2). The vast literature exists dealing with the special and even more general versions of equations (1.1), (1.2) and (1.4) by using different techniques (see [1,5,6,9,10]). Owing to the importance of equations of the forms (1.1), (1.2) and (1.4) arising in many physical problems, the simple, unified and concise treatment of these equations is desired.…”

Existence of Solutions for Volterra Integro-Differential Equations with Implicit Derivative

“…The integro-differential equations have been studied in various papers with the help of several tools of functional analysis, topology and fixed point theory. For instance, we can refer to [1], [2], [4], [5], [6], [7]. So, the crucial key of our approach in order to find solutions of equation (1) consists in the use of a very useful fixed point theorem for multivalued, compact, uppersemicontinuous maps, with acyclic values in a Banach space.…”

Measure of noncompactness in the study of solutions for a system of integral equations

“…In recent years, measure of noncompactness which was introduced by Kuratowski [16] in 1930 and has provided powerful tools for obtaining the solutions of a large variety of integral equations and systems, see Aghajani et al [2], [4], [3], Banas [7], Banas and Rzepka [11], Mursaleen and Mohiuddine. [17], Araba et al [6], Deepmala and Pathak [14], Shaochun and Gan [18], Sikorska [19] and many others.…”