Open Access

Existence of integro-differential solutions for a class of abstract partial impulsive differential equations

Journal of Inequalities and Applications20112011:135

DOI: 10.1186/1029-242X-2011-135

Received: 1 March 2011

Accepted: 7 December 2011

Published: 7 December 2011

Abstract

In this study, we investigate the existence of integro-differential solutions for a class of abstract partial impulsive differential equations.

Keywords

integro-differential equations neutral differential equations analytic semigroup of compact operators; non-autonomous operators family of evolution operators mild solutions

1. Introduction

In this study, we established two existence results of solutions for a class of impulsive functional differential equations which can be described in the following form
d d t D ( t , u t ) = A ( t ) D ( t , u t ) + f ( t , u t , 0 t e ( t , s , u s ) d s ) , t [ 0 , b ] , t t i , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equ1_HTML.gif
(1.1)
u 0 = φ , φ , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equ2_HTML.gif
(1.2)
Δ u ( t i ) = I i ( u t i ) , i { 1 , , m } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equ3_HTML.gif
(1.3)

where A ( t ) : D ( A ( t ) ) X X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq1_HTML.gif is a family of unbounded linear closed operators such that for each t [0, b], A(t) is the infinitesimal generator of analytic semigroup of linear bounded operators (S t (s))s ≥ 0on a Banach space X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq2_HTML.gif, endowed with the norm X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq3_HTML.gif; the history u t : ( - , 0 ] X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq4_HTML.gif is defined as u t (θ) = u(t + θ), θ ≤ 0; https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq5_HTML.gif is a set of measurable functions φ : (-∞, 0] → X endowed with appropriate seminorm; the operator D(t, ϕ) is defined as D(t, ϕ) = ϕ(0) + g(t, ϕ), where the functions g : [ 0 , b ] × X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq6_HTML.gif, f : [ 0 , b ] × × X X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq7_HTML.gif and I i : X X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq8_HTML.gif, i are appropriate functions for all i {1, ..., m}; 0 < t1 < < t m < b is a sequence of fixed real numbers and the symbol Δξ(t) represents the jump of the function ξ at the moment t, this means that Δξ (t) = ξ (t+) - ξ(t-), where the notation ξ(t+) and ξ(t-) represent, respectively, the right and the left-hand side limits of the function ξ at t.

There are many physical phenomena that are described by means of impulsive differential equations, for instance, biological systems, electrical engineering, chemical reactions, among others can be modeled by impulsive differential equations, a good survey on impulsive differential equations can be found in [1] see also [2, 3]. However, impulsive actions can influence the behavior of solutions making the analysis more difficult. Motivated by this facts, the studies of such systems have drawn the attention of many researchers during last years.

Recently, Park et al. [4] have investigated the problem
d d t x ( t ) - g ( t , x t , 0 t a ( t , s , x s ) d s ) = A x ( t ) + f ( t , x t , 0 t e ( t , s , x s ) d s ) , t [ 0 , b ] , t t i , i = 1 , 2 , , m , Δ x ( t i ) = I i ( x ( t i - ) ) , i = 1 , 2 , , m , x 0 = φ h . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equ4_HTML.gif
(1.4)
In this model, the operator A : D ( A ) X X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq9_HTML.gif is the infinitesimal generator of a compact analytic semigroup of bounded linear operators (T(t))t ≥ 0on a Banach space X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq2_HTML.gif such that 0 ρ(A), where ρ(A) is the resolvent set of the operator A ; a , e : [ 0 , b ] 2 × h X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq10_HTML.gif, f , g : [ 0 , b ] × h × X X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq11_HTML.gif and I i : X X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq12_HTML.gif, i = 1, 2,..., m are given functions that satisfy suitable conditions. Using the theory of fractional powers and priori estimates for compact operators, the authors established some existence result for the problem (1.4). Lately, Balachandran and Annapoorani [5] investigated the following class of abstract problem (1.5)
d d t x ( t ) - g ( t , x t ) = A ( t ) x ( t ) + f ( t , x t ) + 0 t e ( t , s , x s ) d s , t I = [ 0 , b ] , t t i , i = 1 , 2 , , m , Δ x ( t i ) = I i ( x ( t i - ) ) , i = 1 , 2 , , m , x 0 = φ h . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equa_HTML.gif
In the system (1.5), it was assumed that for each t [0, b] the operator A(t) is the infinitesimal generator of compact analytic semigroup of bounded linear operators on a Banach space X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq2_HTML.gif. Moreover, the domain, D(A(t)), of the operators A(t) is assumed to be independent of t [0, b] and dense in X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq2_HTML.gif, i.e., A ( t ) : D X X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq13_HTML.gif with D ¯ = X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq14_HTML.gif. To get their results, the authors used the conditions of Acquistapace and Terreni, see [6], to guarantee the existence of an evolution family of operators associated with the non-autonomous abstract Cauchy problem
u ( t ) = A ( t ) u ( t ) , t [ 0 , b ] , u ( 0 ) = x X . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equb_HTML.gif

Then, using fractional powers and operators theory the authors get some existence result based on a priori bounded estimates for compact operators.

Other authors have studied problems involving impulsive act, for retarded and neutral functional differential equations we cite [717], for applications of impulsive differential equations on biology and neural networks we cite [1821]. On the other hand impulsive fractional differential equations is a topic treated in [22, 23].

In this article, the study of a class of neutral impulsive integro-differential equations is proposed. To get our results, we used the technique involving the fixed point theory of compact and condensing operators. We pointed out that the problem studied in this article has not been considered in the literature, once that the approach used in this study is totally different from those studies mentioned above. Actually, the main difference is that in our study we need to use an assumptions of compactness on the nonlinear equation, and in applications, these assumptions make all differences, because, even in infinite dimensional Hilbert space it is not straightforward handedly with compact sets. However, in our applications, we overcome this difficult using a well-known criterion of compactness in L p (Ω) space [[24], Kolmogorov-Riesz-Weil theorem]. This is the principal motivation of this study.

We now turn to a summary of this study. The second section provides tools which are necessary to establish the main results that are the Theorems 2.3 and 2.4. In third section, we apply our abstract results in concrete examples.

2. Preliminaries

In this study, the symbols ( X , X ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq15_HTML.gif and ( Y , Y ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq16_HTML.gif stand for Banach spaces with their, respectively, norms and we denote by ( Y , X ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq17_HTML.gif the Banach space of bounded linear operators from Y https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq18_HTML.gif into X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq2_HTML.gif endowed with the uniform operator topology; particularly, we denote ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq19_HTML.gif when Y = X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq20_HTML.gif. we start defining the evolution operator associated with the family A(t), t [0. b].

Definition 2.1. A family of operators U(t, s), ts, t, s I is said to be an evolution family associated to the problem (2.1) if the following conditions hold:
  1. (a)

    U(t, s)U(s, r) = U(t, r) for all rst.

     
  2. (b)

    For each x X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq21_HTML.gif, the function (t, s) → U(t, s)x is continuous from {(t, s), ts, t, s I} into X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq2_HTML.gif.

     
  3. (c)

    For each t > s, the function tU(t, s) is continuous differentiable with respect to t and t U ( t , s ) = A ( t ) U ( t , s ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq22_HTML.gif.

     

The family of evolution system U(t, s) is called exponential stable if there are positive constants M ̃ https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq23_HTML.gif and α such that U ( t , s ) ( X ) M ̃ e - α ( t - s ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq24_HTML.gif, for every t, s [0, b].

Throughout this study, A ( t ) : D ( A ( t ) ) X X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq25_HTML.gif denotes a family of unbounded closed linear operators defined in a common domain D, which is independent of t and dense in X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq2_HTML.gif. Moreover, we assume that the system
u ( t ) = A ( t ) u ( t ) , t s , t , s I , u ( s ) = x X , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equ5_HTML.gif
(2.1)

has an associated evolution family of operators U(t, s), ts, t, s I. For additional details and more properties about the family U(t, s), we refer the reader to [6, 25, 26].

To study the problem (2.1), we consider the space of normalized piecewise continuous functions P C ( [ 0 , τ ] , X ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq26_HTML.gif, this means that, a function u : [ 0 , τ ] X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq27_HTML.gif belongs to P C ( [ 0 , τ ] , X ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq26_HTML.gif if u is continuous at tt i , u ( t i - ) = u ( t i ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq28_HTML.gif and u ( t i + ) < https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq29_HTML.gif, for all i = 1,..., m. It is well known that if it is equipped with the norm u P C = sup s [ 0 , a ] u ( s ) X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq30_HTML.gif, then P C ( [ 0 , τ ] , X ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq26_HTML.gif became a Banach space.

The technique used in this study is based on the compactness criterion. For this reason, we will make the following assumptions.

Put t0 = 0, tn+1= τ and for u P C ( [ 0 , τ ] , X ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq31_HTML.gif we denote by u i ̃ P ( [ t i , t i + 1 ] ; X ) , i = 0 , , n https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq32_HTML.gif, the function given by
ũ i ( t ) = u ( t ) , for t ( t i , t i + 1 ] , u ( t i + ) , for t = t i . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equc_HTML.gif

In particular, B ̃ i https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq33_HTML.gif stands the set defined by B ̃ i = { u i ̃ , u B } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq34_HTML.gif, where B P C ( [ 0 , τ ] , X ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq35_HTML.gif.

Lemma 2.1. ( [12]) A set B P C https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq36_HTML.gif is relatively compact in P C https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq37_HTML.gif if and only if the set B ̃ i https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq38_HTML.gif is relatively compact in the space C ( [ t i , t i + 1 ] ; X ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq39_HTML.gif, for every i = 0, 1,..., n.

The next step is to define the phase space. This will be done in the following way. The space https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq5_HTML.gif, will be formed by all measurable functions φ : ( - , 0 ] X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq40_HTML.gif with seminorm https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq41_HTML.gif. On the phase space https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq5_HTML.gif we assume the following condition. Let x : ( - , b ] X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq42_HTML.gif, b > 0 be a function such that x0 = φ, φ https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq43_HTML.gif and x [ 0 , b ] P C [ 0 , b ] ; X ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq44_HTML.gif. Then the following properties hold true.
  1. (i)

    x t is in https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq5_HTML.gif;

     
  2. (ii)

    x ( t ) X H x t https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq45_HTML.gif;

     
  3. (iii)

    x t K ( t - σ ) s u p { x ( s ) X : σ s t } + M ( t - σ ) x σ https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq46_HTML.gif, where H > 0 is a constant; K, M : [0, ∞) → [1, ∞), K(·) is continuous, M(·) is locally bounded and H, K, M are independent of x(·).

     

Remark 2.1. To treat retarded impulsive differential equation we suitable modified the axioms of the abstract phase space https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq5_HTML.gif. Actually, we drop the condition of continuity of the https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq5_HTML.gif-valued function tx t , since t x ( t ) P C https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq47_HTML.gif in not a continuous function.

Following the ideas of [15], we used the notations K ̄ https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq48_HTML.gif and M ̄ https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq49_HTML.gif which is defined by
K ̄ : = sup s [ 0 , b ] K ( s ) and  M ̄ : = sup s [ 0 , b ] M ( s ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equd_HTML.gif

In what following we give some examples of phase spaces whose the above axioms are satisfied.

Example 2.1. Consider the function g(θ) = e γθ , θ ≤ 0, γ ≥ 0, and let L2([0, π], ) be the space of square integrable Lebesgue measure functions endowed with the norm ξ L 2 = ( 0 π ξ ( x ) 2 d x ) 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq50_HTML.gif. Then we define the phase space norm https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq5_HTML.gif as being
B = φ : ( - , 0 ] L 2 ( [ 0 , π ] ; ) ; sup θ 0 e γ θ φ ( θ ) L 2 < ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Eque_HTML.gif

If https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq5_HTML.gif is endowed with the norm φ = sup θ 0 e γ θ φ ( θ ) L 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq51_HTML.gif, for all φ https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq52_HTML.gif then it is well known that ( , ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq53_HTML.gif is a phase space and the conditions (i)-(iii) are fulfilled. In these particular example it is possible to show that H = 1, K(t) = 1 and M(t) = e-γt, for all t ≥ 0.

Motivated by Pazy [16, 25] we adopt the following concept of mild solution to problem (1.1)-(1.3).

Definition 2.2. A function u : ( - , b ] X , b > 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq54_HTML.gif, is a local mild solution of problem (1.1)-(1.3) if the following conditions holds.
  1. (i)

    u0 = ϕ, φ https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq55_HTML.gif;

     
  2. (ii)

    the function u ( t ) P C ( [ 0 , b ] ; X ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq56_HTML.gif, x ( t i + ) = x ( t i ) + I i ( x t i ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq57_HTML.gif, for all i = 1,..., m;

     
  3. (iii)
    the integral equation below is satisfied,
    u ( t ) = U ( t , 0 ) ( φ ( 0 ) + g ( 0 , φ ) ) - g ( t , u t ) + 0 t U ( t , s ) f ( s , u s , 0 s e ( s , τ , u τ ) d τ ) d s + t i < t U ( t , t i ) I i ( u t i ) , t [ 0 , b ] , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equf_HTML.gif
     

is satisfied.

The tools used in this study are based on point fixed theory. For this reason, the next two theorems play important role in the development of our results.

Theorem 2.1. [[27], Leray-Schauder Alternative] Let C be a convex subset of a Banach space X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq2_HTML.gif, and assume 0 C. Let F : CC be a completely continuous operator, and let
( F ) : = x C ; x = λ F ( x ) , λ ( 0 , 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equg_HTML.gif

Then either ( F ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq58_HTML.gif is unbounded or F has a fixed point.

Theorem 2.2. [[28], Corollary 4.3.2] Suppose that D is a closed bounded convex subset of the Banach space X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq2_HTML.gif and that B and C are continuous function from D to X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq2_HTML.gif with

(ma 1) Bx + Cx D, for all x D.

(ma 2) C ( D ) ¯ https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq59_HTML.gifis compact set; and

(ma 3) there is a number 0 ≤ γ < 1 such that || Bx - By || ≤ γ || x - y ||, for all x, y D.

Then there is z D such that Bz + Cz = z.

Next, we stated some important conditions used in the proof of our results.

(H1) The function g : [ 0 , b ] × X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq60_HTML.gif satisfy the following condition

(H1.1) Let φ https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq61_HTML.gif and consider the extension y : ( - , b ] X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq62_HTML.gif of ϕ which is given by
y ( t ) = φ ( t ) , t 0 , U ( t , 0 ) φ ( 0 ) , t [ 0 , b ] . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equ6_HTML.gif
(2.2)

Then, each bounded set B of P C https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq63_HTML.gif the family of functions {g(t, y t + u t ), t [0, b], u B} is equi-continuous.

(H1.2) There are constants c1 and c2 such that g ( t , φ ) c 1 φ + c 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq64_HTML.gif, for all t ≥ 0 and φ . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq65_HTML.gif

(H2) The function f : [ 0 , b ] × × X X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq66_HTML.gif satisfies the following conditions.

(H2.1) The function (x, ϕ) → f(t, ϕ, x) is continuous for almost everywhere t [0, b].

(H2.2) The function tf(t, ϕ, x) is strong measurable for each ( φ , x ) × X . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq67_HTML.gif.

(H2.3) There is a positive continuous function m : [0, b] → [0, ∞) and a nondecreasing positive continuous function ψ: → [0, ∞) such that
f ( t , φ , x ) X m ( t ) ψ ( φ + x X ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equh_HTML.gif

for every ( t , φ , x ) [ 0 , b ] × × X . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq68_HTML.gif

(H3) The function e : [ 0 , b ] × [ 0 , b ] × X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq69_HTML.gif satisfy the following conditions.

(H3.1) The function ϕe(t, s, ϕ) is continuous almost everywhere for all t, s [0, b].

(H3.2) The function (t, s) → e(t, s, ϕ) is strong measurable for each φ . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq70_HTML.gif

(H3.3) There is a positive continuous function p : [0, b] → [0, ∞) and a nondecreasing integrable positive function Ω: → [0, ∞) such that
e ( t , s , φ ) X p ( s ) Ω ( φ ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equi_HTML.gif

for all ( t , s , φ ) [ 0 , b ] × [ 0 , b ] × . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq71_HTML.gif

(H4) For each function u : ( - , b ] X , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq72_HTML.gif with u 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq73_HTML.gif and u ( ) [ 0 , b ] P C ( [ 0 , b ] ; X ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq74_HTML.gif, τe(s, τ, u τ ), τ [0,b] and tf(t, u t , x), t [0, b] are measurable functions for almost everywhere s [0, b] and x X . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq75_HTML.gif

Now we are already to state and prove the main result of this article.

Theorem 2.3. Assume that the conditions (H1) - (H4) are satisfied. In addition, suppose that the following assumptions hold.

(t 1) The function g : [ 0 , b ] × X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq76_HTML.gif is completely continuous.

(t 2) The operators I i are completely continuous and there are positive constants, L i such that
I i ( x ) X L i φ , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equj_HTML.gif

for all φ https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq77_HTML.gif, x X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq78_HTML.gif, and i = 1,..., m.

(t3) For each bounded subsets B P C ( [ 0 , b ] ; X ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq79_HTML.gif and G X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq80_HTML.gif the set
U ( t , s ) f ( s , x ̄ s + y s , z ) , x B , z G , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equk_HTML.gif

is relatively compact for each ts, t, s [0, b], where x ̄ : ( - , b ] X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq81_HTML.gif is an extension of x in such manner that x ̄ ( t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq82_HTML.gif, t ≤ 0 and x ̄ ( t ) = x ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq83_HTML.gif, t [0, b].

If 1 - K ¯ ( M ̃ i = 1 m L i ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq84_HTML.gif and
0 t ξ ( s ) d s < C ̃ d s ψ ( s ) + Ω ( s ) d s , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equl_HTML.gif
where
C ̃ = ( K ¯ M ̃ + K ¯ M ̃ + M ¯ ) φ + c 2 ( M ̃ + 1 ) 1 - K ¯ ( M ̃ i = 1 m L i ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equm_HTML.gif
and
ξ ( t ) = max ( K ¯ M ̃ + K ¯ M ̃ + M ) φ 1 - K ̃ ( M ̃ i = 1 m L i ) K ̃ M ̃ m ( t ) , p ( t ) , t [ 0 , b ] , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equn_HTML.gif

then, the problems (1.1)-(1.3) have a mild solution.

Proof. Suppose that u : ( - , b ] X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq85_HTML.gif is a solution of (1.1)-(1.3) and let y : ( - , b ] X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq86_HTML.gif be a continuous extension of ϕ given in (H1). If we written the solution u(·) of the problem (1.1)-(1.3) as u(t) = x(t) + y(t), t (-∞, b], then we can see that x(t) = 0, t ≤ 0 and for t [0, b] the following integral equation hold true
x ( t ) = U ( t , 0 ) g ( 0 , φ ) - g ( t , x t + y t ) + 0 t U ( t , s ) f ( s , x s + y s , 0 s e ( s , τ , x τ + y τ ) d τ ) d s + t i < t U ( t , t i ) I i ( x t i + y t i ) , t [ 0 , b ] . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equo_HTML.gif
Motivated by this remark we consider the space
Λ = x : ( - , b ] X ; x ( θ ) = 0 , θ 0 , and x ( ) [ 0 , b ] P C ( [ 0 , b ] ; X ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equp_HTML.gif
endowed with the norm x Λ = sup t [ 0 , b ] x ( t ) X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq87_HTML.gif. Moreover, on Λ we define the operators Γ i : Λ → Λ, i = 1, 2, 3 given by
Γ 1 x ( t ) = U ( t , 0 ) f ( 0 , φ ) - f ( t , x t + y t ) Γ 2 x ( t ) = 0 t U ( t , s ) g ( s , x s + y s , 0 s e ( s , τ , x τ + y τ ) d τ ) d s Γ 3 x ( t ) = t i < t U ( t , t i ) I i ( x t i + y t i ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equq_HTML.gif

for all t [0, b]. Using the fact that (U(t, s))tsis a evolution family of operators and assuming the conditions on f, g and the family of operator I i , i = 1,..., m, it is not difficult see that t → Γ i (t), t [0, b] is a normalized piecewise continuous function for all i = 1,..., m. This shows that Γ is well defined. In the next, we prove that the operator Γ = Γ1 + Γ2 + Γ3 satisfies all conditions of Theorem 2.1. As the proof is very long we split it into various steps.

Step 1. The operator 1 is completely continuous

Let x n Λ, n be a sequence of elements of Λ such that x n x as n → ∞ for some x Λ. From the boundedness of operators U(t, s) and the axioms of the phase space https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq5_HTML.gif it is easy to see that the set x t n , x t , n , t [ 0 , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq88_HTML.gif is bounded in https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq5_HTML.gif, which implies from condition (H1) the uniformity convergence of
g ( t , x t n + y t ) g ( t , x t + y t ) , as n https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equr_HTML.gif

on [0, b]. Thus, we have the continuity of Γ1. From condition (H 1), and the axiom (iii) follows immediately the Γ1 applies bounded sets of Λ into equi-continuous sets of Λ. On the other hand, again by (iii), and using the fact that f is a completely continuous function, soon as infers that, for each t [0, b], the set {g(t, x t + y t ), x B} is compact in X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq2_HTML.gif. The proof that Γ1 is a completely continuous operator is complete.

Step 2. The operator Γ2 is complete continuous

The condition (H3.1) permit us conclude that e ( t , s , x n s ) e ( t , s , x s ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq89_HTML.gif as n → ∞ almost everywhere for t, s [0, b]. By (H3.3), and the Lebesgue's dominated convergence theorem we conclude that
0 t e ( t , s , x n s ) d s 0 t e ( t , s , x s ) d s , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equs_HTML.gif
uniformly for t [0, b]. From the strong continuity of the operators (U(t, s))ts, we can conclude that
0 t U ( t , s ) f ( s , x s n + y s , 0 s e ( s , τ , x τ n + y τ ) d τ ) d s 0 t U ( t , s ) f ( s , x s + y s , 0 s e ( s , τ , x τ + y τ ) d τ ) d s , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equt_HTML.gif
as n → ∞, uniformly for t [0, b]. This fact and the properties of the evolution family U(t, s) lead us to the continuity of the operator Γ2. Next, we show that Γ2 takes bounded sets into equi-continuous sets. First, we observe from conditions (H2.3), (H3.3) and the axioms of phase space that
e ( t , s , x s + y s ) , t , s [ 0 , b ] , x B , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equu_HTML.gif
and
f ( t , x t + y t , 0 t e ( t , s , x s + y s ) d s ) ; t , s [ 0 , b ] , x B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equ7_HTML.gif
(2.3)

are bounded sets in X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq2_HTML.gif.

Let ε > 0 be the arbitrary positive real number and t1, t2 [0, b], t1 > t2. Thus, take into account the previous notes and using the assumption (iii) we see that the set
U ( t 2 - ε , s ) g ( s , x s + y s , 0 s e ( s , τ , x τ + y τ ) d τ ) d s , s [ 0 , t 2 - ε / 2 ] , x B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equv_HTML.gif
is relatively compact in X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq2_HTML.gif. Thus we have
Γ 2 u ( t 2 ) - Γ 2 u ( t 1 ) 0 t 2 - ε 2 [ U ( t 1 , t 2 - ε ) - U ( t 2 , t 2 - ε ) ] U ( t 2 - ε , s ) g ( s , x s + y s , 0 s e ( s , τ , x τ + y τ ) d τ ) d s + t 2 t 2 - ε 2 [ U ( t 1 , s ) - U ( t 2 , s ) ] g ( s , x s + y s , 0 s e ( s , τ , x τ + y τ ) d τ ) d s + t 2 t 1 U ( t 1 , s ) g ( s , x s + y s , 0 s e ( s , τ , x τ + y τ ) d τ ) d s , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equw_HTML.gif
thus, from the continuity of U(t, s) and the assumptions of compactness contained on the condition (t 3) we can infer the existence of 0 < δ < ε such that if |t1 - t2| < δ then
Γ 2 x ( t 1 ) - Γ 2 x ( t 2 ) ε ( t 2 - ε 2 ) + 2 ε 2 M ̃ sup s [ 0 , b ] f ( s , x s + y s , 0 s e ( s , τ , x τ + y τ ) d τ ) d s + ε M ̃ sup s [ 0 , b ] f ( s , x s + y s , 0 s e ( s , τ , x τ + y τ ) d τ ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equx_HTML.gif
This shows the equi-continuity of Γ2. In what follows, we show that for each t [0, b] the set
Θ ( t ) = 0 t U ( t , s ) f ( s , x s + y s , 0 s e ( s , τ , x τ + y τ ) d τ ) d s , x B , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equ8_HTML.gif
(2.4)
where B Λ, is pre-compact in Λ. To do that, we observe from (2.3) that for each s [0, t] the set
f ( s , x s + y s , 0 s e ( s , τ , x τ + y τ ) d τ ) , x B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equy_HTML.gif
is a bounded set. Then,
0 t U ( t , s ) f ( s , x s + y s , 0 s e ( s , τ , x τ + y τ ) d τ ) d s = 0 t - ε U ( t , s ) f ( s , x s + y s , 0 s e ( s , τ , x τ + y τ ) d τ ) d s + t - ε t U ( t , s ) f ( s , x s + y s , 0 s e ( s , τ , x τ + y τ ) d τ ) d s , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equz_HTML.gif
which implies by [[28], Lemma 1.3]
0 t U ( t , s ) f ( s , x s + y s , 0 s e ( s , τ , x τ + y τ ) d τ ) d s t c o ̃ U ( t , s ) f ( s , x s + y s , 0 s e ( s , τ , x τ + y τ ) d τ ) , s [ 0 , t ] + C ε , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equaa_HTML.gif

with diam(C ε ) < ε, where diam(·) denotes the diameter of the set C ε and co {·} the convex hull. Taking all this into account we see that for each fixed t [0, b], the set Θ(t) in (2.4) is relatively compact set in X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq2_HTML.gif. This completes the proof that the operator Γ2 is completely continuous.

Step 3. The operator Γ3 is completely continuous

To show that is Γ3 is a completely continuous, consider a bounded subset B of Λ and for each i = 1,..., m, define the set Π ̃ i C ( [ t i , t i + 1 ] ; X ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq90_HTML.gif as
Π ̃ i = j = 1 i - 1 U ( t , t j ) I j ( x t j ) , t [ t i , t i + 1 ] , x B ̃ i . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equab_HTML.gif
To prove that the sets Π ̃ i https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq91_HTML.gif, i = 1,..., m, are precompacts in C ( [ t i , t i + 1 ] ; X ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq92_HTML.gif, consider t1, t2 (t i , ti+1], t1 > t2. Using the continuity of (t, s) → U (t, s)x, and the compactness of sets I j (B), j = 1,..., m, given ε > 0 there is 0 < δ < ε such that if |t1 - t2| < δ we have
j = 1 i ( U ( t 1 , t j ) - U ( t 2 , t j ) ) I j ( x t j ) n ε , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equac_HTML.gif
uniformly for x B. On the other hand for t (t i , ti+1) fixed, from our hypothesis it is not difficult see that the set
Π ̃ i ( t ) = j = 1 i U ( t , t j ) I j ( x t j ) , x B , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equad_HTML.gif

is relatively compact in X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq2_HTML.gif

On the other hand, if t = t i , the set Π ̃ i ( t i ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq93_HTML.gif became
Π ̃ i ( t i ) = j = 1 i - 1 U ( t , t j ) I j ( x t j ) , x B . , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equae_HTML.gif
and proceeding as in the early case we infer that the set Π ̃ i ( t i ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq94_HTML.gif is relatively compact in X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq2_HTML.gif, The prove that the set
j = 1 i - 1 ( U ( t i , t j ) - U ( t 2 , t j ) ) I j ( x t j ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equaf_HTML.gif

is an equi-continuous set of functions is done in the same manner as at the beginning

of the section. The proof that Γ3 is completely continuous is finished.

In the next, we obtain a priori estimative of the solutions for the equation λ Γx λ = x λ , for λ (0, 1) and = Γ = Γ1 + Γ2 + Γ3. Let x be a solution of the equation λ Γ(x λ ) = x λ , in addition we use the notation m λ ( t ) = K ¯ sup s [ 0 , t ] x ( s ) + ( K ¯ M ̃ + M ¯ ) φ https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq95_HTML.gif, then we have
x ( t ) U ( t , 0 ) ( X ) g ( 0 , φ ) + g ( t , x t + y t ) + 0 t U ( t , s ) ( X ) f ( s , x s + y s , 0 s e ( s , τ , x τ + y τ ) d τ ) d s + t j < t U ( t , t j ) ( X ) I j ( x t j ) ( X ) M ̃ φ + c 1 m λ ( t ) + c 2 ( M ̃ + 1 ) + M ̃ 0 t m ( s ) ψ ( m λ ( s ) + 0 s p ( τ ) Ω ( m λ ( τ ) ) d τ ) d s + M ̃ t i < t L i m λ ( t ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equag_HTML.gif
this implies that
m λ ( t ) ( K ¯ M ̃ + K ¯ M ̃ + M ¯ ) φ + c 2 ( M ̃ + 1 ) 1 - K ¯ ( M ̃ i = 1 m L i ) + ( K ¯ M ̃ + K ¯ M ̃ + M ¯ ) φ 1 - K ¯ ( M ̃ i = 1 m L i ) M ̃ K ¯ 0 t m ( s ) ψ ( m λ ( s ) + 0 s p ( τ ) Ω ( m λ ( τ ) ) d τ ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equah_HTML.gif
If we take the right-hand side of the previous inequalities and call it of v(t) we have that m λ (t) ≤ v(t), for all t [0, b]. This leads us to the following inequality:
v ( t ) ( K ¯ M ̃ + K ¯ M ̃ + M ¯ ) φ + c 2 ( M ̃ + 1 ) 1 - K ¯ ( M ̃ i = 1 m L i ) + ( K ¯ M ̃ + K ¯ M ̃ + M ¯ ) φ 1 - K ¯ ( M ̃ i = 1 m L i ) K ¯ M ̃ 0 t m ( s ) ψ ( v ( s ) + 0 s p ( τ ) Ω ( v ( τ ) ) d τ ) d s , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equai_HTML.gif
this yields
v ( t ) ( K ¯ M ̃ + K ¯ M ̃ + M ¯ ) φ 1 - K ¯ ( M ̃ i = 1 m L i ) K ¯ M ̃ m ( t ) ψ ( v ( t ) + 0 t p ( τ ) Ω ( v ( τ ) ) d τ ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equaj_HTML.gif
Next, we considered the function ϖ ( t ) = v ( t ) + 0 t p ( s ) Ω ( v ( s ) ) d s https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq96_HTML.gif, thus we have that v(0) = ϖ(0) and v(t) ≤ ϖ(t), for all t [0, b], using this and the non-decreasingly properties of the function ψ(·), we get
ϖ ( t ) ( K ¯ M ̃ + K ¯ M ̃ + M ¯ ) φ 1 - K ¯ ( M ̃ i = 1 m L i ) K ¯ M ̃ m ( t ) ψ ( ϖ ( t ) ) + p ( t ) Ω ( ϖ ( t ) ) , t [ 0 , b ] . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equak_HTML.gif
Observe that if we define the function ξ ( t ) = max { ( K ̃ M ̃ + K ̃ M ̃ + M ) φ 1 - K ̃ ( M ̃ i = 1 m L i ) K ̃ M ̃ m ( t ) , p ( t ) } , t [ 0 , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq97_HTML.gif then we have
ϖ ( t ) ξ ( t ) ψ ( ϖ ( t ) ) + Ω ( ϖ ( t ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equal_HTML.gif
which implies that
ϖ ( t ) ψ ( ϖ ( t ) ) + Ω ( ϖ ( t ) ) ξ ( t ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equam_HTML.gif
for all t [0, b]. Integrating the early inequality from 0 to t we have
ϖ ( 0 ) ϖ ( t ) d s ψ ( s ) + Ω ( s ) 0 t ξ ( s ) d s < ϖ ( 0 ) d s ψ ( s ) + Ω ( s ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equan_HTML.gif

The early inequalities enable us to conclude that the set {x λ , x λ = Γx λ , λ (0, 1)} is bounded. From Theorem 2.1 the problem (1.1)-(1.3) has a mild solution. The proof of theorem is completed.   □

In the next result, the following conditions are used.

(G 1 ) There is a positive constant L f such that
g ( t , φ ) - g ( t , ψ ) X L g φ - ψ , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equao_HTML.gif

for every t [0, b] and φ , ψ . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq98_HTML.gif

(G 2 ) There are positive constants d i , i = 1,..., m, such that
I i ( x ) - I i ( y ) X d i x - y X , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equap_HTML.gif

for every x, y X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq99_HTML.gif..

Theorem 2.4. Assume that the condition (H 2)-(H 3) and (G1)-(G2) are satisfied. In addition, suppose that the assumption (iii) of Theorem (2.3) is satisfied. Then if
L g K ¯ + K ¯ M ̃ lim inf ξ ψ ( ξ + Ω ( ξ ) 0 b p ( τ ) d τ ) ξ 0 b m ( s ) d s + M ̃ K ¯ i = 1 n d i < 1 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equaq_HTML.gif
and
( L g + M ̃ i = 1 n d i ) < 1 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equar_HTML.gif

then the problem (1.1)-(1.3) has a mild solution.

Proof. Let us consider the operator Γ: Λ → Λ defined as in Theorem 2.3. We claim that there is r > 0 such that Γ(B r ) B r . Suppose by contradiction that this assumption is false. Then for each r > 0 there are t r [0, b] and u r (·) B r such that Γ ( u r ) ( t r ) X r . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq100_HTML.gif This implies that
r M ̃ ( H φ + g ( 0 , φ ) X ) + M ̃ 0 t r f ( s , u s r + y s , 0 s e ( s , τ , u τ r + y τ ) d τ ) d s + t i < t r U ( t r , t i ) I i ( u t i r + y t i ) + g ( t r , u t r r + y t r ) X M ̃ ( H φ + g ( 0 , φ ) X ) + L g ( K ¯ r + ( K ¯ M ̃ H + M ¯ ) φ ) + sup s [ 0 , b ] g ( s , 0 ) + M ̃ ψ ( K ¯ r + ( K ¯ M ̃ H + M ¯ ) φ + Ω ( K ¯ r + ( K ¯ M ̃ H + M ¯ ) φ ) 0 b p ( τ ) d τ ) 0 b m ( s ) d s + M ̃ i = 1 n ( d i ( K ¯ r + ( K ¯ M ̃ H + M ¯ ) φ ) + I i ( 0 ) X ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equas_HTML.gif
take the lim inf in the previous inequality, we have
1 L g K ¯ + K ¯ M ̃ liminf ξ ψ ( ξ + Ω ( ξ ) 0 b p ( τ ) d τ ) ξ 0 b m ( s ) d s + M ̃ K ¯ i = 1 n d i , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equat_HTML.gif
which is contradictory with our assumptions. So let r > 0 be such a number and consider the restriction Γ B r https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq101_HTML.gif of the operator Γ on B r , that is, Γ B r : B r B r https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq102_HTML.gif Next, we split the operator Γ in the following way Γ = Γ1 + Γ2, where
Γ 1 ( u ) = U ( t , 0 ) ( φ ( 0 ) + g ( 0 , φ ) ) - g ( t , u t + y t ) + t i < t U ( t , t i ) I i ( u t i + y t i ) , t [ 0 , b ] and Γ 2 ( u ) = 0 t U ( t , s ) f ( s , u s + y s , 0 s e ( s , τ , u τ + y τ ) d τ ) d s , t [ 0 , b ] . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equau_HTML.gif
As shown in the proof of Theorem 2.3, it is not difficult to see that Γ2 is completely continuous and for u1, u2 Λ we have that
Γ 1 ( u 1 ) ( t ) - Γ 1 ( u 2 ) ( t ) X L f u t 1 - u t 2 + M ̃ i = 1 n d i u t i 1 - u t i 2 ( L f + M ̃ i = 1 n d i ) u 1 - u 2 Λ . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equav_HTML.gif

The previous inequality shows that Γ1 is contractive. Now, by Theorem 2.2, we can conclude that the problem (1.1)-(1.3) has a mild solution.   □

3. Applications

The main aim of this section is to apply our abstract results in concrete examples. To this end, we handle with a very special kind of operators. To be more specific, on the Banach space X = L 2 ( [ 0 , π ] , ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq103_HTML.gif we define the operator A : D ( A ) X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq104_HTML.gif given by Ax(ξ) = x"(ξ), ξ [0, π] with domain
D ( A ) = x X ; x X and x ( 0 ) = x ( π ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equaw_HTML.gif
It is well known that in this case A has a discrete spectrum which is given by -n2, n . Moreover, X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq2_HTML.gif has a completely orthonormal base formed by eigenfunctions of A associated with the eigenvalues -n2, which is given x n ( ξ ) = 2 π sin ( n ξ ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq105_HTML.gif, n . This implies that the following conditions are satisfied.
  1. (i)

    For each f X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq106_HTML.gif, n = 1 f , x n x n ( ξ ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq107_HTML.gif,

     
  2. (ii)

    For each f D(A), we have A f ( ξ ) = - n = 1 n 2 f , x n x n ( ξ ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq108_HTML.gif,

     
where 〈·,·〉 represents the inner product in X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq2_HTML.gif. Taking into account all these information, it is possible to prove that the operator A is the infinitesimal generator of a compact semigroup of bounded linear operators (T(t))t ≥ 0, which is given by
T ( t ) f ( ξ ) = n = 1 e - n 2 t f , x n x n ( ξ ) , t 0 , ξ [ 0 , π ] . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equax_HTML.gif
To guarantee the existence of an evolution family associated with the problem
u ( t ) = A ( t ) u ( t ) , t s , t , s [ 0 , b ] u ( s ) = x X , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equ9_HTML.gif
(3.1)

the following assumptions on the function a0 : [0, b] × [0, π] → are made

( a 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq109_HTML.gif There are constants c > 0 and α (0, 1) such that
a 0 ( t , ξ ) - a 0 ( s , ξ ) t - s α , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equay_HTML.gif

for all t, s [0, b] and almost everywhere ξ [0, π].

( b 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq110_HTML.gif there is a real number c0 such that
a 0 ( τ , ξ ) c 0 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equaz_HTML.gif

for all τ [0, ∞) and ξ [0, π].

Letting D(A(t)) = D(A) for all t ≥ 0 and A(t)x(ξ) = a0(t, ξ)x"(ξ), ξ [0, π], we have that the system (3.1) has an associated evolution family of operators (U(t, s))tswhich is given explicitly by the following formula:
( U ( t , s ) f ) ( ξ ) = n = 0 e - n 2 s t a 0 ( τ , ξ ) d τ f , x n x n ( ξ ) , ξ [ 0 , π ] , t [ 0 , b ] . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equba_HTML.gif
Using the properties of semigroup (T(t))t ≥ 0it is straightforward to show that U(t, s) satisfies the condition
U ( t , s ) ( X ) e - c 0 ( t - s ) , t s . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equbb_HTML.gif
Next, we consider the following partial differential equations
t u ( t , ξ ) + 0 ξ 0 t k 1 ( s , ξ 1 ) u ( s , ξ 1 ) sin ( ξ ) d s d ξ 1 = a 0 ( t , ξ ) 2 ξ 2 u ( t , ξ ) + 0 ξ 0 t k 1 ( s , ξ 1 ) u ( s , ξ 1 ) sin ( ξ ) d s d ξ 1 + - t k 2 ( - s ) P 2 ( s , u ( t , ξ ) ) d s + 0 t - s k 3 ( t - δ ) P 3 ( δ - t , u ( t , ξ ) d δ d s ) , u ( t , 0 ) = u ( t , π ) = 0 , s [ 0 , π ] , u ( s , ξ ) = φ ( s , ξ ) , s 0 , ξ [ 0 , π ] , u ( t i + , ξ ) = u ( t i , ξ ) + 0 ξ k 4 , i ( ξ ) u ( t i , ξ 1 ) d ξ 1 , i = 1 , , m . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equ10_HTML.gif
(3.2)

To model the problem (3.2) we choose as the phase space the set formed by all piecewise continuous functions φ : ( - , 0 ] X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq111_HTML.gif which sups ≤ 0h(θ) || φ(s) || < ∞, where h(θ) = e βθ , θ ≤ 0, and we denote this space by h https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq112_HTML.gifequipping it with the norm φ h = sup s 0 h ( s ) φ ( s ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq113_HTML.gif. In order to show that the conditions (H1)-(H4) are satisfied we needed to consider the following assumptions.

( p 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq114_HTML.gif The function P2 : (-∞, 0] × → [0, ∞) satisfies the following conditions:

( p 1.1 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq115_HTML.gif for each η , sP2(s, η ) is a measurable and bounded function,

( p 1.2 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq116_HTML.gif there is a positive constant L P 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq117_HTML.gif such that
P 2 ( s , η 1 ) - P 2 ( s , η 2 ) L P 2 η 1 - η 2 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equbc_HTML.gif

for all s ≤ 0, and η i , i = 1, 2.

( p 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq118_HTML.gif The functions s 0 π k 1 ( s , ξ 1 ) d ξ 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq119_HTML.gif and ξ - b 0 k 1 ( s , ξ ) d s https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq120_HTML.gif are bounded almost everywhere on [-b, 0] × [0, π], s 0 π k 1 ( s , ξ 1 ) 2 e - β s d ξ 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq121_HTML.gif is integrable on the interval [-b, 0].

( p 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq122_HTML.gif k2(·) L((-∞, π]) and sP2(s, η ) is measurable and bounded function for each η . In addition we assume the existence of positive constant L P 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq123_HTML.gif such that the following inequality hold true
P 2 ( s , η 2 ) - P 2 ( s , η 1 ) L P 2 η 2 - η 1 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equbd_HTML.gif

for almost everywhere s (-∞, 0] and η i [0, π], i = 1, 2.

( p 4 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq124_HTML.gif The function P3 : [-π, ∞) × satisfies the following conditions.

( p 4.1 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq125_HTML.gif for each η , sP3(s, η ), s [-∞, b), is a measurable and bounded function,

( p 4.2 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq126_HTML.gif there is a positive constant L P 3 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq127_HTML.gif such that
P 3 ( t , η 2 ) - P 3 ( t , η 1 ) L P 3 η 2 - η 1 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Eqube_HTML.gif

for all s [-π, ∞) and η i , i = 1, 2.

( p 4.3 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq128_HTML.gif The function k3(·) L([-π, ∞)).

To transform the problem (3.2) into the abstract system (1.1), we define the functions g : [ 0 , b ] × h X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq129_HTML.gif, f : [ 0 , b ] × × X X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq130_HTML.gif, e : [ 0 , T ] × [ 0 , T ] × h X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq131_HTML.gif and I i : X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq132_HTML.gif, i = 1, 2,..., n, respectively, given by,
g ( t , φ ) ( ξ ) = 0 ξ - t 0 φ ( s , ξ 1 ) sin ( ξ ) d s , t [ 0 , ) , ξ [ 0 , π ] , f ( t , φ , x ) ( ξ ) = - 0 k 2 ( - s ) P 2 ( s , φ ( 0 , ξ ) ) d s + x ( ξ ) , t [ 0 , b ] , ξ [ 0 , π ] e ( t , s , φ ) ( ξ ) = - s k 3 ( t - τ ) P 3 ( τ , φ ( 0 , ξ ) ) d τ , t , s [ 0 , T ] , ξ [ 0 , π ] , I i ( φ ) ( ξ ) = 0 ξ k 4 , i ( ξ ) φ ( 0 , ξ 1 ) d ξ 1 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equbf_HTML.gif
We shall show that the condition (H1) hold true. In fact, let x : (-∞, π] → L1(0, π) be a bounded function such that x [ 0 , π ] P C ( [ 0 , π ] ; L 1 ( 0 , π ) ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq133_HTML.gif we have
g ( t + h , x t + h ) ( ξ ) - g ( t , x t ) ( ξ ) 0 ξ t t + h x ( s , ξ 1 ) sin ( ξ ) d s d ξ t t + h 0 ξ x ( s , ξ 1 ) 2 d ξ 1 1 2 0 ξ k 1 ( s , ξ 1 ) 2 d ξ 1 1 2 sin ( ξ ) d s sup s [ - b , 0 ] 0 π x ( s , ξ 1 ) 2 d ξ 1 1 2 t t + h 0 π k 1 ( s , ξ 1 ) 2 d ξ 1 1 2 sin ( ξ ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equbg_HTML.gif
The previous inequalities jointly with the assumption ( p 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq118_HTML.gif show that the function tg(t, x t ) is uniformly continuous on bounded subsets of P C ( [ 0 , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq134_HTML.gif, L2(0, π)) which implies that the condition (H1.1) hold true. To prove that the condition (H1.2) is satisfied, we observe that
g ( t , φ ) ( ξ ) 0 ξ - t 0 k 1 ( s , ξ 1 ) φ ( s , ξ 1 ) d s d ξ 1 - t 0 0 ξ k 1 ( s , ξ 1 ) 2 e - β s d ξ 1 1 2 0 ξ φ ( s , ξ 1 ) 2 e β s d ξ 1 1 2 d s sup s 0 0 ξ φ ( s , ξ 1 ) 2 e β s d ξ 1 1 2 - t 0 0 π k 1 ( s , ξ 1 ) 2 e - β , s d ξ 1 1 2 d s , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equbh_HTML.gif

which implies the condition (H1.2).

The next step is a proof that the function (x, ϕ) → f(t, ϕ, x) is continuous. However, with the help of condition ( p 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq118_HTML.gif we have
0 π f ( t , φ 2 , x 2 ) ( ξ ) - f ( t , φ 1 , x 1 ) ( ξ ) 2 4 L P 2 0 π φ 2 ( 0 , ξ ) - φ 1 ( 0 , ξ ) 2 d ξ + 4 0 π x 2 ( ξ ) - x 1 ( ξ ) 2 d ξ 4 L P 2 sup s 0 0 π e β s φ 2 ( s , ξ ) - φ 1 ( s , ξ ) 2 d ξ + 4 0 π x 2 ( ξ ) - x 1 ( ξ ) 2 d ξ , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equbi_HTML.gif
for all φ i https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq135_HTML.gif, x i X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq136_HTML.gif, i = 1, 2. Thus we have shown that the condition (H2.1) is fulfilled. In particular, as P2 is continuous in the second variable we have that for each ( φ , x ) × X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq137_HTML.gif fixed the function ξ - 0 k 2 ( - s ) P 2 ( s , φ ( 0 , ξ ) ) d s https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq138_HTML.gif is measurable. Thus from [[24], Theorem 1.2.1] we infer that tf(t, ϕ, x) is measurable for each ( φ , x ) × X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq139_HTML.gif. On the other hand, assuming that sP2(s, 0), s (-∞, 0] is bounded function, we have that
f ( t , φ , x ) ( ξ ) 2 4 L P 2 φ ( 0 , ξ ) - 0 k 2 ( - s ) d s 2 + 16 - 0 k 2 ( - s ) P 2 ( s , 0 ) d s 2 + 16 x ( ξ ) 2 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equbj_HTML.gif
which implies that
f ( t , φ , x ) X 4 L P 2 - 0 k 2 ( - s ) d s 2 , 4 π - 0 k 2 ( - s ) P 2 ( s , 0 ) d s 2 , 4 1 2 ( φ + x X ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equ11_HTML.gif
(3.3)

Thus the condition (H2.3) is fulfilled.

On the other hand, the same idea applied to prove that the previous functions is of Caratheádory type can be used to show that function e(·,·,·) satisfies the same property. Here it is mentioned that the functions that appear in the condition (H3.3) are given by
e ( t , s , φ ) X 4 max L P 2 - s k 2 ( - δ ) d δ 2 , 4 π - s k 2 ( - δ ) P 2 ( δ , 0 ) d δ 2 , 4 1 2 φ . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equ12_HTML.gif
(3.4)
Finally, it remains that the condition (H 4) is valid. However, we observe that
h L h ( ( - , 0 ] , L 2 ( ( 0 , π ) ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equbk_HTML.gif
where a function ϕ : (-∞, 0] → L2(0, π) is an element of L h ( ( - , 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq140_HTML.gif, L2(0, π)) if and only if
e s s sup s 0 0 π h ( s ) φ 2 ( s , ξ ) d ξ < , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equbl_HTML.gif
with the norm defined by φ L h = inf { a ; μ { θ ( - , 0 ] ; φ L 2 > a } = 0 } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_IEq141_HTML.gif, where μ = hdξ, and representing the Lebesgue measure on (-∞, 0]. Thus, following the ideas of [[29], Theorem 3.8] and using the fact that h(θ - t) ≤ G(-t)h(θ), θ ≤ 0, G(-t) = e-βt, t ≥ 0, we see that if u : (0, -∞] → L2(0, π) is admissible function in the sense of [29], then we derive the mensurability of tu t , t [0, b]. Thus, as e(·,·,·) and f(·,·,·) are measurable functions we infer that τe(t, τ, u τ ) and τf(t, u τ , x) for all t, τ [0, b], x L2(0, π). Now we will see that the conditions of the Theorem 2.3 hold. To see this, we observe that
g ( t , φ ) ( ξ + h ) - g ( t , φ ) ( ξ ) 0 ξ - t 0 φ ( s , ξ 1 ) ( sin ( ξ + h ) - sin ( ξ ) ) d s d ξ 1 + ξ ξ + h - t 0 φ ( s , ξ 1 ) sin ( ξ ) d s d ξ 1 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-135/MediaObjects/13660_2011_Article_125_Equbm_HTML.gif
taking the advantage of the previous inequality we have
g ( t , φ ) ( ξ + h ) - g ( t , φ ) ( ξ ) b sup s 0 0 π e β s φ ( s , ξ 1 ) 2 d ξ 1 0 π - h ( sin ( ξ + h ) - sin ( ξ ) ) d s + b h sup s 0