Assume that p : [a, b] → ℝ+, that 1 p ∈ L 1 [ a, b ], and that the mappings f : L l o c 1 ( a, b ] → A R [ a, b ] and c, d : L l o c 1 ( a, b ] → ℝ are increasing and order-bounded. Then, the IVP (5.1) has the smallest and greatest solutions in Y, and they are increasing with respect to f, c and d.
Thus A and B satisfy all the conditions of Theorem 1.3 and so the inclusion (uin A u) +B u) ) has a solution in Y. Therefore the inclusion problem (1.1 - 1.2 1.1 - 1.2lution in Y and thasproof is completed.
Then the problem (1.1) has at least one solution in Y. Proof Let Ω be a bounded open set of Y such that ⋃ i = 1 3 Ω ¯ i ⊂ Ω.
Then, in view of Theorem 1.6, the equation (0,0) = T u,v) + partialPhi u,v) (2.1) has a solution in Y, which is denoted by ((u,v)).
We say that x* is the least (respectively the greatest) solution of (1) in Y if x* ≤ x (respectively x* ≥ x) for any other solution x ∈ Y. Notice that the least solution in a subset Y is a minimal solution in Y, but the converse is false in general, and an analgous remark is true for maximal and greatest solutions.
Thus by Theorem 2.1 we can find a unique common fixed point of (f_{1}), (f_{2}), (f_{3}), (f_{4}), (f_{5}), and (f_{6}) in Y, that is, the system (2.1) of Urysohn integral equations has a unique common solution in Y. □.
Suppose (H1 - H3) hold, then the problem (1.1) and (1.2) has at least on solution in Y. Let Ω to be a bounded open subset of Y, such that (bigcup_{i=1}^{3}{overline{Omega}_{i}}subsetOmega).
Theorem 3.1 Suppose (H1 - H3) hold, then problem (1.1) has at least one solution in Y. Proof Let Ω be a bounded open set of Y such that ⋃ i = 1 3 Ω ¯ i ⊂ Ω.
Then, by Theorem 2.1, L ( u, v ) = N ( u, v ) has at least one solution in dom ( L ) ∩ Ω ¯, i.e., the problem (1.1) has at least one solution in Y, which completes the proof.
Then system (1.1) has a unique solution in Y. Next, the local stability of the nonlinear inverse problem stated in Theorem 1.1 will be proved following the ideas in [22].
