Abstract
Higher order fractional differential equations are important tools to deal with precise models of materials with hereditary and memory effects. Moreover, fractional differential inequalities are useful to establish the properties of solutions of different problems in biomathematics and flow phenomena. In the present work, we are concerned with the nonexistence of global solutions to a higher order fractional differential inequality with a nonlinearity involving Caputo fractional derivative. Namely, using nonlinear capacity estimates, we obtain sufficient conditions for which we have no global solutions. The a priori estimates of the structure of solutions are obtained by a precise analysis of the integral form of the inequality with appropriate choice of test function.
Highlights
Differential equations of fractional order are a interesting model in different areas of engineering sciences such as modeling of materials with memory and hereditary effects
Higher order fractional differential equations are important tools to deal with precise models of materials with hereditary and memory effects
We are concerned with the nonexistence of global solutions to a higher order fractional differential inequality with a nonlinearity involving Caputo fractional derivative
Summary
Differential equations of fractional order are a interesting model in different areas of engineering sciences such as modeling of materials with memory and hereditary effects (see, for example, the books [1,2]). The investigation of global solutions to different classes of fractional differential equations and inequalities has been a point of interest for many researchers. In [5], useful properties of Mittag–Leffler functions, together with fixed-point methods, lead to the existence and uniqueness of global solutions in the case of delay fractional differential equations. In the case where y1 > 0, the authors proved that for all q > 1, problem (2) admits no global solution. In the case of y1 = 0, the authors established that, if γ ≤ min{α, β}, for all q > 1, the only global solution to problem (2) is y ≡ y0, while, if γ > min{α, β}, for all. Before presenting our main results, let us mention what we mean by a global solution to problem (1).
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