РЕЗЮМЕТА на представените статии от доц. д-р Цанко Дончев Дончев
1. Fuzzy integro–differential equations with compactness type conditions, Hacet.J. Math. Stat.
– to appear (with V. Lupulescu and A. Nosheen)
Abstract.
In the paper fuzzy integro-differential equations with almost continuous right hand sides are studied. The existence of solution is proved under compactness type conditions.
2. Nonlinear Evolution Inclusions with One-sided Perron Right-hand Side Journal of Dynamical and Control Systems – to appear (with O. Carja and V. Postolache)
Abstract.
In a Banach space X with uniformly convex dual, we study the evolution inclusion of the form x
0
( t ) ∈ Ax ( t ) + F ( x ( t )) , where A is an m-dissipative operator and F is an upper hemicontinuous multifunction with nonempty convex and weakly compact values. If X
∗ is uniformly convex and F theorem. Afterward, sufficient conditions for near viability and (strong) invariance of a set K ⊆
D ( A ) are established. As applications, we derive ε − δ lower semicontinuity of the solution set and, consequently, the propagation of continuity of the minimum time function associated with the null controllability problem.
3. One sided Perron differential inclusions Set Valued and Variational Analysis to appear (with A.
Lazu and A. Nosheen)
Abstract.
The main qualitative properties of the solution set of almost lower (upper) semicontinuous one-sided Perron differential inclusion with state constraints in finite dimensional spaces are studied.
Using the technique introduced by Veliov (Nonlinear Anal. 23, 1027–1038 (1994).) we give sufficient conditions for the solution map of the above state constrained differential inclusion to be continuous in the sense of Hausdorff metric. An application on the propagation of the continuity of the state constrained minimum time function associated with the nonautonomous differential inclusion and the target zero is given. Some relaxation theorems are proved, which are used afterward to derive necessary and sufficient conditions for invariance.
4. Numerical Approximations of Impulsive Delay Differential Equations, Num. Funct. Anal. Opt.
– to appear (with Q. Din and D. Kolev)
Abstract.
The paper deals with numerical approximations of impulsive functional differential equations with non-fixed time of jumps which depend on the trajectory. However the right hand side depends also on variable delay and it is assumed to be Lipschitz w.r.t. special type of norm defined in the paper. The right-hand side can be discontinuous at jump and delay times. The jump times of the approximate and exact solutions do not need to coincide. Some numerical examples are provided to illustrate the theory.
5. Filippov–Pliss lemma and m-dissipative differential inclusions, Journal of Global Optimization
53 No 3 (2012) (with Q. Din and D. Kolev)
Abstract.
In this paper we prove a variant of the well known Filippov–Pliss lemma for evolution inclusions given by multivalued perturbations of m-dissipative differential equations in Banach spaces with uniformly convex dual. The perturbations are assumed to be almost upper hemicontinuous with convex weakly compact values and in addition satisfying one-sided Peron condition.
The result is then applied to prove the connectedness of the solution set of evolution inclusions without compactness and afterward the existence of attractor to autonomous evolution inclusion when the perturbations are one-sided Lipschitz with negative constant.

6. Higher Order Runge-Kutta Methods for Impulsive Differential Systems, Applied Mathematic and Computations 218 (2012) 11790–11798 (with R. Baier and Q. Din)
Abstract.
This paper studies higher order approximations of solutions of differential equations with non-fixed times of impulses. We assume that the right-hand side is sufficiently smooth. Using a Runge-Kutta method of higher order and natural assumptions on the impulsive surfaces and the impulses, we calculate good approximations of the jump times, which enables us to extend the classical results for higher order of convergence of Runge-Kutta methods to more complicated systems.
7. Attractors, approximations and fixed sets of evolution systems, International Journ. Pure Appl.
Math.
74 (2012) 183–197 (with Q. Din and D. Kolev).
Abstract.
We study autonomous evolution inclusions with right-hand sides satisfying a one-sided
Lipschitz (OSL) condition in evolution triple. It is known that the solution set is compact on every bounded interval. Using this fact we prove the existence of a flow invariant compact attractor when the OSL constant is negative.
8. On Peano Theorem for Fuzzy Differential Equations, Fuzzy Sets and Systems 177 (2011) 93–94
(with R. Choudary)
Abstract.
Nieto’s theorem Fuzzy Sets and Systems 102 (1999) 259–262 states that every fuzzy differential equation with bounded and continuous right-hand side admits a solution.
In this note we show that the provided proof of this result is not correct.
9. Generic Properties of Multifunctions. Application to Differential Inclusions, Nonlinear Analysis
74 (2011) 2585-2590
Abstract.
We prove that almost all in Baire sense Caratheodory multi-functions in finite dimensional space are Kamke continuous. Further the main properties of differential inclusions with Kamke and one sided Kamke right-hand sides are studied.
As a corollary we prove that for almost all optimal control problems, the relaxation and relaxation stability properties hold.
10. Singularly perturbed Evolution Inclusions, SIAM J. Control and Optimization 48 (2010) 4572–
4590
Abstract.
Singularly perturbed differential inclusions in evolution triple are studied with the help of the averaging method. Upper and lower limit of the slow solution set are estimated when the small parameter tends to zero and upper or the lower limit averaged system exists. Sufficient conditions of the existence of the limit of the (full) solution set are presented. Under these conditions some properties of the limit occupational measure sets are studied. Illustrative example is provided.
11. On the Theorem of Filippov – Pli´ Control and Cybernetics 38 (2009)
1251–1271. (with E. Farkhi)
Abstract.
In the paper some known and new extensions of the famous theorem of Filippov
(1967) and a theorem of Pli? (1965) for differential inclusions are presented. We replace the Lipschitz condition on the set-valued map in the right-hand side by a weaker onesided Lipschitz (OSL), onesided Kamke (OSK) or a continuity-like condition (CLC). We prove new Filippov-type theorems for singularly perturbed and evolution inclusions with OSL right-hand sides. In the CLC case we obtain two extended theorems, one of which implies directly the relaxation theorem. We obtain also a theorem in Banach spaces for OSK multifunctions. Some applications to exponential formulae are surveyed.
12. Discrete Approximation of Impulsive Differential Inclusions, Num. Funct. Anal. Opt.
31
(2010) 653 - 678 (with R. Baier)
Abstract.
The paper deals with the approximation of the solution set and the reachable sets of an impulsive differential inclusion with variable times of impulses. It is strongly connected to the

paper of the second author S. Margenov et al (eds.), Lecture Notes in Comput. Sci.
4818 , pp. 309–
316, Springer Verlag, Berlin, 2008. and is its continuation. We achieve order of convergence 1 for the Euler approximation under Lipschitz assumptions on the set-valued right-hand side and on the functions describing the jump surfaces and jumps themselves. Another criterion prevents the beating phenomena and generalizes available conditions. Several test examples illustrate the conditions and the practical evaluation of the jump conditions.
13. Extensions of Clarke’s proximal characterization for reachable mappings of differential inclusions, J. Math. Anal. Appl.
348 (2008) 454–460 (with A. Dontchev)
Abstract.
In this paper we show that Clarke’s proximal characterization of reachable mappings for Lipschitz continuous differential inclusions is valid for a larger class of continuous and locally oneside Kamke continuous inclusions. Our result is generic, in the sense of Baire category, in the set of all continuous set-valued mappings with compact and convex values, while Clarke’s result is not.
We also give a proximal characterization of reachable mappings for upper semi-continuous differential inclusions.
14.
Discrete approximations, relaxation, and optimization of one-side Lipschitzian differential inclusions in Hilbert spaces, J. Diff. Eqns.
243 (2007) 301-328 (with E. Farkhi and B. Mordukhovich)
Abstract.
We study discrete approximations of nonconvex differential inclusions in Hilbert spaces and dynamic optimization/optimal control problems involving such differential inclusions and their discrete approximations. The underlying feature of the problems under consideration is a modified onesided Lipschitz condition imposed on the right-hand side (i.e., on the velocity sets) of the differential inclusion, which is a significant improvement of the conventional Lipschitz continuity.
Our main attention is paid to establishing efficient conditions that ensure the strong approximation (in the
W
1 ,p
-norm as p ≥ 1 ) of feasible trajectories for the one-sided Lipschitzian differential inclusions under consideration by those for their discrete approximations and also the strong convergence of optimal solutions to the corresponding dynamic optimization problems under discrete approximations. To proceed with the latter issue, we derive a new extension of the Bogolyubov-type relaxation/density theorem to the case of differential inclusions satisfying the modified one-sided Lipschitzian condition.
All the results obtained are new not only in the infinitedimensional Hilbert space framework but also in finite-dimensional spaces.
15. Discrete Approximations and Fixed Set Iterations in Banach Spaces, SIAM Journal Optimization 18 (2007) 895-906 (with E. Farkhi and S. Reich)
Abstract.
We study autonomous differential inclusions with right-hand sides satisfying a onesided
Lipschitz (OSL) condition in Banach spaces with uniformly convex duals. We first show that the solution set is closed and obtain estimates for Euler-type discrete approximations. We then use these results to derive an analogue of the exponential formula for the reachable set, as well as results regarding the existence and approximation of a strongly invariant attractor in the case of a negative OSL constant.
As a by-product, conditions for controllability of the reverse-time system are obtained.
16. Strong invariance for discontinuous differential inclusions in a Hilbert space, An. Stiint. Univ.
"A. Cuza" Iasi, Tomul LI, S. I-a, Matematica , (2005), f.2, 265-279 (with V. Rios and P. Wolenski)
Abstract.
This paper presents two characterizations, i.e. necessary and sufficient conditions, of strong invariance with Hamilton-Jacobi inequalities for a differential inclusion in a Hilbert space.
We study two cases when the right-hand side is almost upper semicontinuous with convex compact values o when it is almost upper hemicontinuous with convex weakly compact values. In both cases we essentially use the so called one sided Lipschitz condition.
17. Averaging of functional differential inclusions in Banach spaces, J. Math. Anal. Appl.
311
(2005) 402-415 (with G. Grammel)

Abstract.
Averaging schemes for functional differential inclusions in Banach spaces with slow and fast time variables are studied. Under mild suppositions on the regularity, the periodic case and the case of nonexistence of an average are investigated. The accuracy of the averaging technique is considered as well. In particular, for periodic systems, the usual linear approximation is achieved.
Under stronger regularity conditions, approximation orders for Krylov?Bogoliubov?Mitropolskii type right-hand sides are derived.
18. Strong Invariance and one-sided Lipschitz multifunctions, Nonlinear Analysis TMA 60 (2005)
849-862 (with V. Rios and P. Wolenski)
Abstract.
This paper studies the strong invariance property of a differential inclusion in finite dimensions under the assumption of a locally one-sided Lipschitz condition.
19. Averaging of Perturbed One Sided Lipschitz Differential Inclusions, ZAA 23 (2004) 1-10 (with
M. Kamenskii and M. Quincampoix)
Abstract.
We consider one sided Lipschitz differential inclusions perturbed with multimap, satisfying compactness type conditions. The state space is Banach with uniformly convex dual. Averaging result on a finite interval is proved. The averaging of functional differential inclusions is also studied
20. Surjectivity and Fixed Points of Relaxed Dissipative Multifunctions. Differential Inclusions
Approach, J. Math. Anal. Appl.
299 (2004) 525-533
Abstract.
One from the most important properties of accretive and monotone operators is the existence of zeros and surjectivity.
In the paper we introduce relaxed variants of dissipative, accretive and monotone operators. Using essentially the properties of the solution set of appropriate differential inclusions we study the existence of zeros of such operators. As corollaries the existence of fixed points of relaxed contractive and relaxed nonexpansive multifunctions are obtained.
21. Stability for the Solutions of Parabolic Equations with "Maxima", Pan-Amer. Math. J.
20
(2010) 1–19 (with N. Kitanov and D. Kolev)
Abstract.
In this paper we study a class of reaction-diffusion equations under initial and boundary conditions and with nonlinear reaction terms containing ?maxima?. By assuming that the initial density as well the boundary data are H?older continuous, and that the reaction function has a certain rate we give two stability criteria. We extend the existence and uniqueness result for the parabolic equation with delay to the case with ?maxima?. The uniqueness and asymptotic behavior of the solutions are discussed as well. The above mentioned equations are used for mathematical simulation in theoretical physics, thermodynamics, chemistry, mechanics of materials, biology, ecology, optimal control, etc.
22. Singular perturbations in infinite dimensional control systems, SIAM J. Control and Optim.
42 (2003) 1795 - 1812 (with A. Dontchev)
Abstract.
In this paper we consider a singularly perturbed control system involving differential inclusions in Banach spaces with slow and fast solutions. Using the averaging approach, we obtain sufficient conditions for the Hausdorff convergence of the set of slow solutions in the supremum norm.
We present applications of the theorem to prove convergence of the fast solutions in terms of invariant measures and convergence of equi-Lipschitz solutions.
23. On non-emptiness of viability kernels for infinite dimensional differential inclusions, Appl.
Math. Lett.
16 (2003) 1195-1199 (with M. Quincampoix)
Abstract.
We investigate in?nite-dimensional differential inclusions.
Sufficient conditions for nonemptiness of the viability kernels of such systems are obtained.

24. Fixed Set Iterations for Relaxed Lipschitz Multi-maps, Nonlinear Analysis 53 (2003) 997-1015.
(with E. Farkhi and S. Reich)
Abstract.
A dynamical system described by an autonomous differential inclusion with a righthand side satisfying a relaxed Lipschitz condition, as well as its Euler approximations, are studied.
We investigate the asymptotic properties of the solutions and of the attainable sets. It is shown that the system has a strongly flow invariant set, or a “fixed set”, that is, a set such that each trajectory starting from it does not leave it. This set is also an attractor, i.e., it attracts the continuous and the discrete Euler trajectories as the time tends to infinity. We give estimates of the rate of attraction. An algorithm for approximating the fixed set by the attainable sets of the discrete system is also presented.
25. Relaxed Sub-monotone Mappings, Abstract and Applied Analysis , v.
2003 (2003) 19-31. (with
P. Georgiev)
Abstract.
The notions of "relaxed submonotone" and "relaxed monotone" mappings in Banach spaces are introduced and many their properties are investigated. For example, the Clarke subdifferential of a locally Lipschitz function in a separable Banach space is relaxed submonotone on a residual subset. By example it is shown that this property need not be valid on the whole space. In the last section we prove, under certain hypotheses, the surjectivity of the relaxed monotone mappings.
26. Properties of the reachable set of control systems, System & Control Letters 46 (2002) 379-386.
Abstract.
We investigate some properties of the reachable set of a control system. Representing the system as a differential inclusion and using proximal Hamilton – Jacobi equation we describe its graph. We work in infinitely dimensional Hilbert space and use one sided Lipschitz approach. The funnel equation is considered in the last section. That equation describes the reachable set in arbitrary
Banach space. We consider also the autonomous case and prove the existence of a limit of the reachable set.
27. Multi-valued Perturbations of m-Dissipative Differential Inclusions in Uniformly Convex Spaces,
New Zeland Journal of Mathematics 31 (2002) 19-32
Abstract.
Abstract. The paper deals with multivalued perturbations of m-dissipative differential equations in Banach spaces with uniformly convex dual. The perturbations are assumed to be almost upper hemicontinuous (almost lower semicontinuous) and one sided Lipschitz multifunctions. Some qualitative properties of the solution set are investigated. In the last section we give some directions for further investigations.
28. Properties of One-sided Lipschitz Multi-valued Maps, Nonlinear Analysis 49 (2002) 13-20
Abstract.
In the paper we show that the differential inclusions with one sided Lipschitz righthand side admit a solution on [0 , ∞ ) . Furthermore we prove that the one sided Lipschitz multivalued maps with negative constant are surjective. As a corollary the existence of a fixed point of one sided contractive multifunctions is obtained.
29. Mixed Type Semicontinuous Differential Inclusions in Banach Spaces, Ann. Polon. Math.
LXXVII.3
(2001) 245-259
Abstract.
We consider a class of differential inclusions in (nonseparable) Banach spaces satisfying mixed type semicontinuous hypotheses and prove the existence of solutions for the problem with state constrains. The case of dissipative type conditions and with time lag is also studied. These results are then applied to control systems.
30. Approximation of Lower Semicontinuous Differential Inclusions, Numer. Funct. Anal. Opt.
22 (2001) 55-67
Abstract.
We study the approximation of the solution set of lower semicontinuous differential

inclusions having the form:
˙ ( t ) ∈ F ( t, x ( t )) , by the solution set of the discretized one. When F ( t, · ) is one sided Lipschitz in Hilbert space we obtain
O ( h
1 / 2
) order of approximation using Euler scheme. In some cases a better order of convergence is obtained.
31. Generic Properties of Differential Inclusions and Control Problems, Z. Li et al. (Eds.) NAA
2004, LNCS 3401 (2005) 266-271, Springer, Berlin
Abstract.
We prove that almost all in Baire sence Caratheodory multifunctions in finite dimensional space are Kamke continuous. We obtain as a corollary that almost every differential inclusion with Caratheodory right-hand side satisfies the relaxation property.
Possible applications in Bolza problem, given for differential inclusions are pointed out.