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NONLOCAL EVOLUTION SYSTEMS

 

Tzanko Donchev

Department of Mathematics, UACG, 1 Hr. Smirnenski bvd, 1046 Sofia, Bulgaria

E-mail address: tzankodd@gmail.com

 

In this talk we study an evolution inclusion with nonlocal initial condition having the

form:

ẋ ∈ Ax + F (t, x) on I = [t₀ , T ],

f_x(t) ∈ F (t, x(t))

x(t₀) = g(x(·)) ∈ ⊂ X.

 

Here A is m-dissipative operator, generating C₀ semigroup and X is a Banach space.

The existence of solutions is proved in general under the following assumptions:

 

• Compactness type assumptions w.r.t. some measure of noncompactness. In case of linear A (even depending on t) we refer the reader to [3]. In case A is m-dissipative (in general nonlinear) the problem is studied in several papers (cf [4]).

• A generates compact semigroup. This approach is comprehensively studied in [2] (where mainly delay systems are studied), where many examples are provided. Here F should have weakly compact values and A should be of complete continuous type. The state space should be separable. Notice that if A is continuous (even 0) then it generates non compact semigroup.

•  

In this talk we assume that F satisfies some dissipative type conditions, i.e. F (t, ·) is Lipschitz. Under some strong assumptions on the Lipschitz constants of F and g(·) the existence and even relaxation theorem has been proved. In case of uniformly convex dual of the state space this condition can be relaxed to one sided Lipschitz. Notice that OSL multimaps can have negative constant in which case multipoint boundary valued problems can be successfully studied.

In [1] it is assumed that F(t, ·) is one sided Perron and F has strongly compact values. If g(·) is also completely continuous then the existence and compactness of the solution set is proved.

 

Some open problems are also discussed.

 

References

[1] Bilal S., Carja O., Donchev T., Lazu A. I., Nonlocal problem for evolution i nclusions with one-sided Perron nonlinearities, RACSAM https://doi.org/10.1007/s13398-018-0589-6

[2] Burlică M., Necula M., Roşu D., Vrabie I., Delay Differential Evolutions Subjected to Nonlocal Initial Conditions. Monographs and Research Notes in Mathematics. CRC Press, New York (2016)

[3] Cardinali T., Precup R., Rubiconi F., A unified existence theory for evolution equations and systems under nonlocal conditions, J. Math. Anal. Appl. 432 (2015) 10 39–1057.

[4] Zhu L., Huang Q., Li G., Existence and asymptotic properties of solutions of nonlinear multivalued differential inclusions with nonlocal conditions. J. Math. Anal. Appl. 390(2), 523-534 (2012)