Existence and Uniqueness Theorems of Ordered Contractive Map in Banach Lattices

and Applied Analysis 3 In fact, for n 1, using the fact that P is normal, we have ∥ ∥ ∥A u0 −Au0 ∥ ∥ ∥ ≤ N‖L Au0 − u0 ‖. 2.3 Suppose that 2.2 is true when n k then when n k 1, we obtain ∥ ∥ ∥A 2 u0 −An 1 u0 ∥ ∥ ∥ ∥ ∥ ∥A ( A 1 u0 ) −A A u0 ∥ ∥ ∥ ≤ N ∥ ∥ ∥L ( A 1 u0 −An u0 )∥ ∥ ∥ ≤ N ∥ ∥ ∥L 1 Au0 − u0 ∥ ∥ ∥. 2.4 For any m,n ∈ N,m > n, since P is normal cone, we have ‖A u0 −An u0 ‖ ∥ ∥ ∥ ( A u0 −Am−1 u0 ) · · · ( A 1 u0 −An u0 )∥ ∥ ∥ ≤ N ∥ ∥ ∥ ( Lm−1 Lm−2 · · · L ) Au0 − u0 ∥ ∥ ∥ ≤ Nr (( Lm−1 Lm−2 · · · L )) ‖Au0 − u0‖ ≤ N ( r ( Lm−1 ) r ( Lm−2 ) · · · r L ) ‖Au0 − u0‖. 2.5 Here N is the normal constant. Given a α such that r L < α < 1, since limn→ ∞‖L‖ r L < α < 1, there exists a n0 ∈ N such that ‖L‖ < α, n ≥ n0. 2.6 For any m,n ∈ N,m > n ≥ n0, since P is normal cone, we have ‖A u0 −An u0 ‖ ≤ N ( r ( Lm−1 ) r ( Lm−2 ) · · · r L ) ‖Au0 − u0‖ ≤ N ( αm−1 αm−2 · · · α ) ‖Au0 − u0‖ ≤ N ( α − α 1 − α ) ‖Au0 − u0‖ ≤ N ( α 1 − α ) ‖Au0 − u0‖. 2.7 This implies that {An u0 } is a Cauchy sequence in E. The complete character of E implies the existence of x∗ ∈ P such that lim n→ ∞ A u0 x∗. 2.8 Next, we prove that x∗ is a fixed point of A in E. Since A is decreasing and u0 ≤ Au0, we can get Au0 ≤ Au0. 4 Abstract and Applied Analysis So Au0 −A2 u0 ≤ L Au0 − u0 , 2.9 then Au0 − u0 Au0 − u0 − ( Au0 −A2 u0 ) ≥ I − L Au0 − u0 ≥ θ. 2.10 It is easy to know that A2 is increasing and A2 u0 ≤ A4 u0 , A3 u0 ≤ A u0 . 2.11 By induction, we obtain that u0 ≤ A2 u0 ≤ · · · ≤ A2n u0 ≤ · · · ≤ A2n 1 u0 ≤ · · · ≤ A3 u0 ≤ Au0. 2.12 Hence, the sequence {An u0 } has an increasing Cauchy subsequence {A2n u0 } and a decreasing Cauchy subsequence {A2n 1 u0 } such that lim n→ ∞ A2n u0 u∗, lim n→ ∞ A2n 1 u0 v∗. 2.13 Thus Lemma 1.1 implies that A2n u0 ≤ u∗, v∗ ≤ A2n 1 u0 . Since {An u0 } is a Cauchy sequence, we can get that u∗ v∗ x∗. Moreover ‖Ax∗ − x∗‖ ≤ ∥ ∥ ∥Ax∗ −A ( A2n u0 )∥ ∥ ∥ ∥ ∥ ∥A n 1 u0 − x∗ ∥ ∥ ∥ ≤ N ∥ ∥ ∥L ( x∗ −A2n u0 )∥ ∥ ∥ ∥ ∥ ∥A n 1 u0 − x∗ ∥ ∥ ∥ ≤ Nα ∥ ∥ ∥x∗ −A2n u0 ∥ ∥ ∥ ∥ ∥ ∥A n 1 u0 − x∗ ∥ ∥ ∥. 2.14 Thus ‖Ax∗ − x∗‖ 0. That is Ax∗ x∗. Hence x∗ is a fixed point of A in E. Case (II). On the contrary, suppose that u0 is not comparable to Au0. Now, since E is a Banach lattice, there exists v0 such that inf{Au0, u0} v0. That is v0 ≤ Au0 and v0 ≤ u0. Since A is a decreasing operator, we have Au0 ≤ Av0, Au0 ≤ Av0. 2.15 This shows that v0 ≤ Av0. Similarly as the proof of case I , we can get that A has a fixed point x∗ in E. Finally, we prove that A has a unique fixed point x∗ in E. In fact, let u∗ and v∗ be two fixed points of A in E. Abstract and Applied Analysis 5 1 If u∗ is comparable to v∗, A u∗ u∗ is comparable to A v∗ v∗ for every n 0, 1, 2, . . ., and ‖u∗ − v∗‖ ‖Anu∗ −Anv∗‖ ≤ Nαn‖u∗ − v∗‖, 2.16 which implies u∗ v∗. 2 If u∗ is not comparable to v∗, there exists either an upper or a lower bound of u∗ and v∗ because E is a Banach lattice, that is, there exists z∗ ∈ E such that z∗ ≤ u∗, z∗ ≤ v∗ or u∗ ≤ z∗, u∗ ≤ z∗. Monotonicity implies that A z∗ is comparable to A u∗ and A v∗ , for all n 0, 1, 2, . . ., and ‖u∗ − v∗‖ ‖A u∗ −An v∗ ‖ ≤ ‖A z∗ −An u∗ ‖ ‖A z∗ −An v∗ ‖ ≤ Nαn‖u∗ − z∗‖ Nαn‖z∗ − v∗‖. 2.17and Applied Analysis 5 1 If u∗ is comparable to v∗, A u∗ u∗ is comparable to A v∗ v∗ for every n 0, 1, 2, . . ., and ‖u∗ − v∗‖ ‖Anu∗ −Anv∗‖ ≤ Nαn‖u∗ − v∗‖, 2.16 which implies u∗ v∗. 2 If u∗ is not comparable to v∗, there exists either an upper or a lower bound of u∗ and v∗ because E is a Banach lattice, that is, there exists z∗ ∈ E such that z∗ ≤ u∗, z∗ ≤ v∗ or u∗ ≤ z∗, u∗ ≤ z∗. Monotonicity implies that A z∗ is comparable to A u∗ and A v∗ , for all n 0, 1, 2, . . ., and ‖u∗ − v∗‖ ‖A u∗ −An v∗ ‖ ≤ ‖A z∗ −An u∗ ‖ ‖A z∗ −An v∗ ‖ ≤ Nαn‖u∗ − z∗‖ Nαn‖z∗ − v∗‖. 2.17 This shows that ‖u∗ − v∗‖ → 0 when n → ∞. HenceA has a unique fixed point x∗ in E. Theorem 2.2. LetE be a real Banach lattice, and let P ⊂ E be a normal cone. Suppose thatA : P → P is a completely continuous and increasing operator such that there exists a linear operator L : E → E with spectral radius r L < 1 and Au −Av ≤ L u − v , for u, v ∈ P with v ≤ u. 2.18 Then the operator A has a unique fixed point u∗ in P . Proof. For any r > 0, let Ω {x ∈ P : ‖x‖ ≤ r}. Now we suppose the following two cases. Case (I). Firstly, suppose that there exists u0 ∈ ∂Ω such that u0 ≤ Au0. If Au0 u0, then the proof is finished. Suppose Au0 / u0. Since u0 ≤ Au0 and A is nondecreasing, we obtain by induction that u0 ≤ Au0 ≤ A2 u0 ≤ A3 u0 ≤ · · · ≤ A u0 ≤ A 1 u0 ≤ · · · . 2.19 Similarly as the proof of Theorem 2.1, we can get that {An u0 } is a Cauchy sequence in E. Since E is complete, by Lemma 1.1, there exists u∗ ∈ E,A u0 ≤ u∗ such that lim n→ ∞ A u0 u∗. 2.20 Next, we prove that u∗ is a fixed point of A, that is, Au∗ u∗. In fact ‖Au∗ − u∗‖ ≤ ‖Au∗ −A A u0 ‖ ∥ ∥ ∥A 1 u0 − u∗ ∥ ∥ ∥ ≤ N‖L u∗ −An u0 ‖ ∥ ∥ ∥A 1 u0 − u∗ ∥ ∥ ∥ ≤ Nα‖u∗ −An u0 ‖ ∥ ∥ ∥A 1 u0 − u∗ ∥ ∥ ∥. 2.21 6 Abstract and Applied Analysis Now, by the convergence of {An u0 } to u∗, we can get ‖Au∗ − u∗‖ 0. This proves that u∗ is a fixed point of A. Case (II). On the contrary, suppose that x Ax for all x ∈ ∂Ω. Thus Lemma 1.2 implies the existence of a fixed point in this case also. Finally, similarly as the proof of Theorem 2.1, we can get that A has a unique fixed point x∗ in P . Theorem 2.3. LetE be a real Banach lattice, and let P ⊂ E be a normal cone. Suppose thatA : P → P is a completely continuous and increasing operator which satisfies the following assumptions: i there exists a linear operator L : E → E with spectral radius r L < 1 and Au −Av ≤ L u − v , for u, v ∈ P with v ≤ u; 2.22 ii S {x ∈ P : Ax ≤ x} is bounded. Then the operator A has a unique nonzero fixed point u∗ in P . Proof. Firstly, for any r > 0, let Ω {x ∈ P : ‖x‖ ≤ r}. Now we suppose the following two cases. Case (I). Suppose that there exists u0 ∈ ∂Ω such that u0 ≤ Au0. Similarly as proof of Theorem 2.1, we get that A has a nonzero fixed point u∗ in P . Case (II). On the contrary, suppose that x Ax for all x ∈ ∂Ω. Now, since S is bounded there exists R > r such that Ax x for all x ∈ P with ‖x‖ R. Thus Lemma 1.3 implies the existence of a nonzero fixed point in this case. Finally, similarly as the proof of Theorem 2.1, we can get thatA has a unique non-zero fixed point u∗ in P . 3. Applications In this section, we use Theorem 2.1 to show the existence of unique solution for the first-order initial value problem u′ t f t, u t , t ∈ I 0, T , u 0 u0, 3.1 where T > 0 and f : I × R → R is a continuous function. Theorem 3.1. Let f : I × R → R be continuous, and suppose that there exists 0 < μ < λ, such that −μ(y − x) ≤ f(t, y) λy − [f t, x λx] ≤ 0, ∀y ≥ x. 3.2 Then 3.1 has a unique solution u∗. Abstract and Applied Analysis 7 Proof. It is easy to know that E C I is a Banach space with maximum norm ‖ · ‖, and it is also a Banach lattice with maximum norm ‖ · ‖. Let P {u ∈ E|u t ≥ 0, for all t ∈ I}, and P is a normal cone in Banach lattice E. Equation 3.1 can be written asand Applied Analysis 7 Proof. It is easy to know that E C I is a Banach space with maximum norm ‖ · ‖, and it is also a Banach lattice with maximum norm ‖ · ‖. Let P {u ∈ E|u t ≥ 0, for all t ∈ I}, and P is a normal cone in Banach lattice E. Equation 3.1 can be written as u′ t λu t f t, u t λu t , t ∈ I 0, T , u 0 u0. 3.3 This problem is equivalent to the integral equation