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CONTENTSIntroduction...........................................................................................................51. Preliminaries on convex and set-valued analysis..............................................6 1.1. Convexity of sets...........................................................................................6 1.2. Convexity of set-valued mappings.................................................................9 1.3. Closed convex processes and invex set-valued mappings..........................122....

We consider differential inclusions with state constraints in a Banach space and study the properties of their solution sets. We prove a relaxation theorem and we apply it to prove the well-posedness of an optimal control problem.

The connectivity of the efficient point set and of some proper efficient point sets in locally convex spaces is investigated.

Let X, Y be two separable F-spaces. Let (Ω,Σ,μ) be a measure space with μ complete, non-atomic and σ-finite. Let $N_F$ be the Nemytskiĭ set-valued operator induced by a sup-measurable set-valued function $F:\Omega \times X\to 2^Y$. It is shown that if $N_F$ maps a modular space $(N\left(L(\Omega ,\Sigma ,\mu ;X)\right),{\varrho }_{N,\mu })$ into subsets of a modular space $(M\left(L(\Omega ,\Sigma ,\mu ;Y)\right),{\varrho }_{M,\mu })$, then $N_F$ is automatically modular bounded, i.e. for each set K ⊂ N(L(Ω,Σ,μ;X)) such that $r_K=sup{\varrho }_{N,\mu }\left(x\right):x\in K<\infty$ we have $sup{\varrho }_{M,\mu }\left(y\right):y\in N_F\left(K\right)<\infty$.

A strong edge-coloring of a graph is a proper edge-coloring where each color class induces a matching. We denote by 𝜒's(G) the strong chromatic index of G which is the smallest integer k such that G can be strongly edge-colored with k colors. It is known that every planar graph G has a strong edge-coloring with at most 4 Δ(G) + 4 colors [R.J. Faudree, A. Gyárfás, R.H. Schelp and Zs. Tuza, The strong chromatic index of graphs, Ars Combin. 29B (1990) 205–211]. In this paper, we show that if G is...