L 1-solutions of the initial value problems for implicit differential equations with Hadamard fractional derivative

In this paper, we study the existence of integrable solutions for initial value problems for fractional order implicit differential equations with Hadamard fractional derivative. Our results are based on Schauder’s fixed point theorem and the Banach contraction principle fixed point theorem.

Nonlocal Initial Value Problem for Hybrid Generalized Hilfer-type Fractional Implicit Differential Equations

In this paper, we prove some existence results of solutions for a class of nonlocal initial value problem for nonlinear fractional hybrid implicit differential equations under generalized Hilfer fractional derivative. The result is based on a fixed point theorem on Banach algebras. Further, examples are provided to illustrate our results.

On the Cauchy problem for implicit differential equations of higher orders

The article is devoted to the study of implicit differential equations based on statements about covering mappings of products of metric spaces. First, we consider the system of equations Φ_i (x_i,x_1,x_2,…,x_n )=y_i, i=(1,n,) ̅ where 〖 Φ〗_i: X_i×X_1×… ×X_n→Y_i, y_i∈Y_i, X_i and Y_i are metric spaces, i=(1,n) ̅. It is assumed that the mapping 〖 Φ〗_i is covering in the first argument and Lipschitz in each of the other arguments starting from the second one. Conditions for the solvability of this system and estimates for the distance from an arbitrary given element x_0∈X to the set of solutions are obtained. Next, we obtain an assertion about the action of the Nemytskii operator in spaces of summable functions and establish the relationship between the covering properties of the Nemytskii operator and the covering of the function that generates it. The listed results are applied to the study of a system of implicit differential equations, for which a statement about the local solvability of the Cauchy problem with constraints on the derivative of a solution is proved. Such problems arise, in particular, in models of controlled systems. In the final part of the article, a differential equation of the n-th order not resolved with respect to the highest derivative is studied by similar methods. Conditions for the existence of a solution to the Cauchy problem are obtained.

On Chaplygin’s theorem for an implicit differential equation of order n

We consider the Cauchy problem for the implicit differential equation of order n g(t,x,(x,) ̇…,x^((n)) )=0,t ∈ [0; T],x(0)= A. It is assumed that A=(A_0,…,A_(n-1) )∈R^n, the function g:[0,T] × R^(n+1)→ R is measurable with respect to the first argument t∈[0,T], and for a fixed t, the function g(t,∙)×R^(n+1)→ R is right continuous and monotone in each of the first n arguments, and is continuous in the last n+1-th argument. It is also assumed that for some sufficiently smooth functions η,ν there hold the inequalities ν^((i) ) (0)≥ A_i ≥ η^((i) ) (0),i= (0,n-1,) ̅ ν^((n) ) (t)≥ η^((n) ) (t),t∈[0; T]; g(t; ν(t),ν ̇(t),...,ν^((n) ) (t) )≥ 0,g(t,η(t),η ̇(t),…,η^((n)) (t))≤ 0,t∈[0; T]. Sufficient conditions for the solvability of the Cauchy problem are derived as well as estimates of its solutions. Moreover, it is shown that under the listed conditions, the set of solutions satisfying the inequalities η^((n) ) (t)≤x^((n) ) (t)≤ν^((n) ) (t), is not empty and contains solutions with the largest and the smallest n -th derivative. This statement is similar to the classical Chaplygin theorem on differential inequality. The proof method uses results on the solvability of equations in partially ordered spaces. Examples of applying the results obtained to the study of second-order implicit differential equations are given.