**3. Conclusions**

As we pointed out, the theory of parabolic differential equations and inclusions written in the abstract operator form is growing rapidly. We refer the reader to [1–3] for the theory of PDE and their investigations as abstract equations. Especially the multivalued evolution equations are comprehensively studied in [4,5,18]. In the book by [5], the authors study differential inclusions in evolution (Gelfand) triple. The authors provide many interesting results and examples. In that case, the compactness assumptions are crucially used. In [17], the author prove relaxation theorem in that case.

In [4], the author restricted the study to Banach spaces with uniformly convex duals and *A* generating a compact semigroup, or he used compactness-type assumptions regarding the Kuratowski (or Hausdorff) measure of noncompactness. In that case, every limit solution is also an integral one. That implies that our existence results extend the existence result there. Notice also [19] where lower semicontinuous perturbations of m-dissipative operators are considered. The existence theorem there is used in the proof of Theorem 6 in this paper. We recall also the book by [18], devoted to nonlocal problems of evolution inclusions with time lag. The main assumptions there are that *A* is completely continuous and generates a compact semigroup. We mention also [22] where functional evolution inclusions are studied.

In [12], the author uses full Perron condition in the case of ordinary differential inclusions in Banach spaces. The author assumes that the multifunction *F* has strongly compact values.

The one-sided Perron condition as used here was introduced in [23]. Using integral representation of the solutions the author defined the so-called weak solutions (which are developed in [8]). Here the integral representation of the solution does not hold when *A* is nonlinear and we use limit solutions. The case of a Banach space with uniformly convex dual was studied in [13] where it was shown that if *F* has compact values, then the solution set of (1) is compact *Rδ* and a relaxation theorem has

been proved. No other compactness conditions were used. The paper [14] was devoted to Lemma of Filippov–Pli´s. The papers [15,16] study the problem (1) in the case when the Banach space has uniformly convex dual.

In the present paper we introduce the so-called limit solutions for the fully nonlinear evolution inclusion (1) and we study their properties. In general, the limit solutions of (1) are not solutions of the relaxed system (6).

	- It appears that the notion of limit solutions is meaningful and it deserves further investigations.

**Author Contributions:** Conceptualization, methodology, investigation, writing–original draft preparation, writing–review and editing, T.D., S.B., O.C., N.J. and A.I.L. All authors contributed equally in writing this article. All authors have read and agreed to the published version of the manuscript.

**Funding:** The work of A. I. Lazu was supported by a gran<sup>t</sup> of Romanian Ministry of Research and Innovation, CNCS - UEFISCDI, project number PN-III-P1-1.1-TE-2016-0868, within PNCDI III. The work of the other authors was supported by the Bulgarian National Science Fund under Project KP-06-N32/7.

**Acknowledgments:** The authors thank the reviewers for their valuable comments and suggestions which improved the paper.

**Conflicts of Interest:** The authors declare no conflict of interest.
