Existence in the large for nonlinear delay evolution inclusions with nonlocal initial conditions

This paper deals with a nonlinear Volterra delay evolution inclusion subjected to a nonlocal implicit initial condition. The evolution inclusion involves an $m$-dissipative operator (possibly multivalued and/or nonlinear) and a noncompact interval. We first consider the evolution inclusion subjected to a local initial condition and prove an existence result for bounded $C^0$-solutions. Then, using a fixed point theorem for upper semicontinuous multifunctions with contractible values, we obtain a global solvability result for the original problem. Finally, we present an example to illustrate the abstract result.

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Existence of solutions for some nonlinear delay integrodifferential equations with nonlocal initial conditions

Demonstratio Mathematica ◽

10.1515/dema-2013-0365 ◽

2012 ◽

Vol 45 (1) ◽

Author(s):

Zuomao Yan

Keyword(s):

Fixed Point ◽

Fixed Point Theorem ◽

Existence Of Solutions ◽

Initial Conditions ◽

Mild Solutions ◽

Integrodifferential Equations ◽

Resolvent Operators ◽

First Order ◽

Nonlocal Initial Conditions ◽

Nonlinear Delay

AbstractThe main purpose of this paper is the existence of mild solutions for a class of first-order nonlinear delay integrodifferential equations with nonlocal initial conditions in Banach spaces. We show that the solutions are given by the application of the theory of resolvent operators and the Sadovskii’s fixed point theorem. An example is presented in the end to show the applications of the obtained results.

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Almost periodic solutions for nonlinear delay evolutions with nonlocal initial conditions

Journal of Evolution Equations ◽

10.1007/s00028-013-0198-y ◽

2013 ◽

Vol 13 (3) ◽

pp. 693-714 ◽

Cited By ~ 7

Author(s):

Ioan I. Vrabie

Keyword(s):

Periodic Solutions ◽

Initial Conditions ◽

Almost Periodic Solutions ◽

Almost Periodic ◽

Nonlocal Initial Conditions ◽

Nonlinear Delay

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One Sided Lipschitz Evolution Inclusions in Banach Spaces

Mathematics ◽

10.3390/math9243265 ◽

2021 ◽

Vol 9 (24) ◽

pp. 3265

Author(s):

Ali N. A. Koam ◽

Tzanko Donchev ◽

Alina I. Lazu ◽

Muhammad Rafaqat ◽

Ali Ahmad

Keyword(s):

Banach Spaces ◽

Lipschitz Condition ◽

Differential Inclusions ◽

Initial Conditions ◽

Lipschitz Constant ◽

Evolution Inclusions ◽

Nonlocal Initial Conditions ◽

Limit Solution ◽

The Right ◽

Dissipative Differential Inclusions

Using the notion of limit solution, we study multivalued perturbations of m-dissipative differential inclusions with nonlocal initial conditions. These solutions enable us to work in general Banach spaces, in particular L1. The commonly used Lipschitz condition on the right-hand side is weakened to a one-sided Lipschitz one. No compactness assumptions are required. We consider the cases of an arbitrary one-sided Lipschitz condition and the case of a negative one-sided Lipschitz constant. Illustrative examples, which can be modifications of real models, are provided.

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