Abstract
In this paper, we study the existence of the hybrid fractional pantograph equation {D?0+[x(t)/f(t,x(t),x(?t))= g(t,x(t), x(?t)), 0 < t < 1, x(0) = 0, where ?,?,? ?((0,1) and D?0+ denotes the Riemann-Liouville fractional derivative. The results are obtained using the technique of measures of noncompactness in the Banach algebras and a fixed point theorem for the product of two operators verifying a Darbo type condition. Some examples are provided to illustrate our results.
Highlights
Fractional differential equations are a very important tool in modelling many phenomena of physics and, they deserve an independent study of their theories parallel to the well-known theory of differential equations, [10, 12, 15, 17]
A great number of papers about differential and integral equations with a modified argument have appeared in the literature recently. Such equations arise in a wide variety of applications such as the modelling of problems from the natural and social sciences, for example, physics, biology and economics
A special class of these equations is the differential equation with affine modification of the argument which can be delay differential equations or differential equations
Summary
Fractional differential equations are a very important tool in modelling many phenomena of physics and, they deserve an independent study of their theories parallel to the well-known theory of differential equations, [10, 12, 15, 17]. A great number of papers about differential and integral equations with a modified argument have appeared in the literature recently. Such equations arise in a wide variety of applications such as the modelling of problems from the natural and social sciences, for example, physics, biology and economics. Results concerning with such kind of equations appear in the papers [5, 6, 7, 8, 11, 14, 16, 18, 20], for example. The main tool in our study is a fixed point theorem for the product of two operators satisfying a condition of Darbo with respect to a measure of noncompactness. We give some applications and examples where our results may be applied and we compare these results with others appearing in the literature
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