Existence of Solutions for Nonlinear Mixed Type Integrodifferential Functional Evolution Equations with Nonlocal Conditions

and Applied Analysis 3 The evolution system R t, s is said to be equicontinuous if for all bounded set Q ⊂ X, {s → R t, s x : x ∈ Q} is equicontinuous for t > 0. x ∈ C −q, b , X is said to be a mild solution of the nonlocal problem 1.1 , if x t φ t g x t for t ∈ −q, 0 , and, for t ∈ J , it satisfies the following integral equation: x t R t, 0 [ φ 0 g x 0 ] ∫ t 0 R t, s f ( s, xs, ∫ s 0 K s, r, xr dr, ∫b 0 H s, r, xr dr ) ds. 2.1 The following lemma is obvious. Lemma 2.2. Let the evolution system R t, s be equicontinuous. If there exists a ρ ∈ L1 J,R such that ‖x t ‖ ≤ ρ t for a.e. t ∈ J , then the set {∫ t 0 R t, s x s ds} is equicontinuous. Lemma 2.3 see 21 . Let V {xn} ⊂ L1 a, b , X . If there exists σ ∈ L1 a, b ,R such that ‖xn t ‖ ≤ σ t for any xn ∈ V and a.e. t ∈ a, b , then α V t ∈ L1 a, b ,R and α ({∫ t 0 xn s ds : n ∈ N }) ≤ 2 ∫ t 0 α V s ds, t ∈ a, b . 2.2 Lemma 2.4 see 22 . Let V ⊂ C a, b , X be an equicontinuous bounded subset. Then α V t ∈ C a, b ,R R 0,∞ , α V maxt∈ a,b α V t . Lemma 2.5 see 23 . LetX be a Banach space,Ω a closed convex subset inX, and y0 ∈ Ω. Suppose that the operator F : Ω → Ω is continuous and has the following property: V ⊂ Ω countable, V ⊂ co({y0} ∪ F V ) ⇒ V is relatively compact. 2.3 Then F has a fixed point in Ω. Let V t {x t : x ∈ C −q, b , X } ⊂ X t ∈ J , Vt {xt : x ∈ C −q, b , X } ⊂ C −q, 0 , X , α · and αC · denote the Kuratowski measure of noncompactness in X and C −q, b , X , respectively. For details on properties of noncompact measure, see 22 . 3. Existence Result We make the following hypotheses for convenience. H1 g : C 0, b , X → C −q, 0 , X is continuous, compact and there exists a constant N such that ‖g x ‖ −q,0 ≤ N. H2 1 f : J × C −q, 0 , X × X × X → X satisfies the Carathodory conditions, that is, f ·, x, y, z is measurable for each x ∈ C −q, b , X , y, z ∈ X, f t, ·, ·, · is continuous for a.e. t ∈ J . 2 There is a bounded measure function p : J → R such that ∥∥f(t, x, y, z)∥∥ ≤ p t (‖x‖ −q,0 ∥∥y∥∥ ‖z‖), a.e. t ∈ J, x ∈ C([−q, 0], X), y, z ∈ X. 3.1 4 Abstract and Applied Analysis H3 1 For each x ∈ C −q, 0 , X , K ·, ·, x ,H ·, ·, x : J × J → X are measurable and K t, s, · ,H t, s, · : C −q, 0 , X → X is continuous for a.e. t, s ∈ J . 2 For each t ∈ 0, b , there are nonnegative measure functions k t, · , h t, · on 0, b such that ‖K t, s, x ‖ ≤ k t, s ‖x‖ −q,0 , t, s ∈ Δ, x ∈ C ([−q, 0], X), ‖H t, s, x ‖ ≤ h t, s ‖x‖ −q,0 , t, s ∈ J, x ∈ C ([−q, 0], X), 3.2 and ∫ t 0 k t, s ds, ∫b 0 h t, s ds are bounded on 0, b . H4 For any bounded set V1 ⊂ C −q, 0 , X , V2, V3 ⊂ X, there is bounded measure function li ∈ C J,R i 1, 2, 3 such that α ( f t, V1, V2, V3 ≤ l1 t sup −q≤θ≤0 α V1 θ l2 t α V2 l3 t α V3 , a.e. t ∈ J, α (∫ t 0 K t, s, V1 ds ) ≤ k t, s sup −q≤θ≤0 α V1 θ , t ∈ J, α (∫b 0 H t, s, V1 ds ) ≤ h t, s sup −q≤θ≤0 α V1 θ , t ∈ J. 3.3 H5 The resolvent operator R t, s is equicontinuous and there are positive numbers M ≥ 1 and w max { Mp0 1 k0 h0 1, 2M ( l0 1 2l 0 2k0 2l 0 3h0 ) 1 } , 3.4 such that ‖R t, s ‖ ≤ Me−w t−s , 0 ≤ s ≤ t ≤ b, where k0 sup t,s ∈Δ ∫ t 0 k t, s ds, h0 supt,s∈J ∫b 0 h t, s ds, p0 supt∈Jp t , l 0 i supt∈J li t i 1, 2, 3 . Theorem 3.1. Let conditions H1 – H5 be satisfied. Then the nonlocal problem 1.1 has at least one mild solution. Proof. Define an operator F : C −q, b , X → C −q, b , X by Fx t ⎪⎪⎨ ⎪⎪⎩ φ t g x t , t∈[−q, 0], R t, 0 [ φ 0 g x 0 ] ∫ t 0 R t, s f ( s, xs, ∫ s 0 K s, r, xr dr, ∫b 0 H s, r, xr dr ) ds, t ∈ 0, b . 3.5 Abstract and Applied Analysis 5 We have by H1 – H3 and H5 ,and Applied Analysis 5 We have by H1 – H3 and H5 , ‖ Fx t ‖ ≤ ∥∥φ∥∥ −q,0 N ≤ M(∥∥φ∥∥ −q,0 N) : L, t ∈ [−q, 0], ‖ Fx t ‖ ≤ L M ∫ t 0 e s−t ∥∥∥∥f ( s, xs, ∫ s 0 K s, r, xr dr, ∫b 0 H s, r, xr dr ∥∥∥∥ds ≤ L M ∫ t 0 e s−t p s ( ‖xs‖ −q,0 ∫s 0 ‖K s, r, xr ‖dr ∫b 0 ‖H s, r, xr ‖dr ) ds ≤ L Mp0 ∫ t 0 e s−t ( ‖xs‖ −q,0 ∫ s 0 k s, r ‖xr‖ −q,0 dr ∫b 0 h s, r ‖xr‖ −q,0 dr ) ds ≤ L Mp0 1 k0 h0 w−1‖x‖ −q,b , t ∈ 0, b . 3.6