NONLOCAL PROBLEMS FOR QUASILINEAR FUNCTIONAL PARTIAL DIFFERENTIAL EQUATIONS OF FIRST ORDER

Existence and uniqueness of almost everywhere solutions of nonlocal problems to functional partial differential systems in diagonal form are investigated. The proof is based on the characteristics and fixed point methods.

System (1) contains as particular cases the system of differential equations with a retarded argument, differential-integral systems and differential-functional equations with operators of the Volterra type (see Section 4).
In this paper, we consider the local existence and uniqueness of generalized solutions of nonlocal problem (1), (2).The method used in the paper is based on characteristics theory and the fixed point theorem.

Assumptions and Lemma
For a ∈ (0, a 0 ] we denote by C ϕ,a [p, ω, q] the set of all functions for (x, y), (x, ȳ) ∈ I a and v satisfies condition (3) on I.
We consider a Carathéodory solution of (1), (2).More precisely, a function z is called a Carathéodory solution of problem ( 1), (2) provided the following conditions hold: (ii) z satisfies (1) almost everywhere in I a and (2) everywhere in R n .
For z ∈ C ϕ,a [p, ω, q] we consider the following problem ( 5) Note that ( 5) is an ordinary differential equation.If Assumption H 1 is satisfied then for every z ∈ C ϕ,a [p, ω, q] the right hand side of (5) satisfies Carathéodory assumptions and the following Lipschitz condition holds, where r a = p+q+ a −τ ω(s) ds.Thus, the existence and uniqueness of the solution g i [z](•; x, y) : [0, a] → R of (5) follows from classical theorems.

Lemma 1. If Assumption
where || • || a denotes the supremum norm in the space C(I a , R n ).
Proof: We will consider the case where x ≤ x.We have, by Assumption H 1 , Hence,by the above inequality and by Gronwall's inequality we get (6).We consider the case x > x analogously to the case x ≤ x.
It follows, from Assumption H 1 , that Utilizing the Gronwall's inequality, we obtain (7).The proof of Lemma 1 is complete.
It remains to prove that the fixed point z of the operator T is the Carathéodory solution of (1), (2).By taking y = g i (x; 0, η) in the equation z i (x, y) = (T z) i (x, y), we get (11) since the solution g i of (5) satisfies the following group property g i (t ; t, g i (t; x, y)) = g i (t ; x, y).By differentiation of (11) with respect to x, using the chain rule differentiation Lemma (4.ii) of [7], and by putting again y = g i (x; 0, η), we obtain that z satisfies (1) almost everywhere in I a .It follows immediately that z satisfies (2).The proof of Theorem 1 is complete.
The functions α and β depend on the functional argument.Therefore, we cannot apply existence theorems from [13], [14] to the above system.