Vol 7, No 10 (2016)

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Available online throug

ISSN 2229 – 5046

EXISTENCE RESULTS FOR IMPULSIVE FRACTIONAL DIFFERENTIAL EQUATIONS

WITH INTEGRAL BOUNDARY CONDITIONS

V. KRISHNAVENI*

Department of Mathematics,

Sri Ramakrishna Mission Vidyalaya, College of Arts and Science, Coimbatore, India.

K. SATHIYANATHAN

Department of Mathematics,

Sri Ramakrishna Mission Vidyalaya, College of Arts and Science, Coimbatore, India.

(Received On: 07-09-16; Revised & Accepted On: 06-10-16)

ABSTRACT

In this paper, we investigate the existence results for nonlinear impulsive fractional differential equations with integral

boundary conditions by using fixed point theorem and Green's function.

Keywords: Integral boundary; impulsive; fractional differential equation; Green's function; fixed point theorem.

1. INTRODUCTION

For the last decades, fractional calculus has received a great attention because fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various processes of science and engineering. The theory of impulsive differential equations describes processes which experience a sudden change of their state at certain moments. For an introduction of the basic theory of impulsive differential equations, see Lakshmikantham et al.[11], Bainov and Simeonov [4], and Samoilenko and Perestyuk [14] and the references therein. Integral boundary conditions have various applications in applied fields such as blood flow problems, chemical engineering, thermo-elasticity, underground water flow, population dynamics, and so forth. For a detailed description of the integral boundary conditions, we refer the reader to some recent papers [1, 2, 5, 6, 9, 10, 15-17, 19] and the references therein.

The Greens functions for boundary-value problems for ordinary and fractional differential equations have been investigated in detail in [8, 20]

In [3], the authors have studied the existence of solutions for fractional integro- differential equations with impulsive and integral conditions. In [7] the authors have studied a class of nonlinear fractional differential equations with impulsive and fractional integral boundary conditions.

In [18] the authors investigated the expression and properties of Green’s function for a second-order singular boundary value problem with integral boundary conditions and delayed argument.

In [12] the authors discussed nonlinear impulsive fractional differential equations involving the two-point and integral boundary conditions. The authors in [13] proved the existence and uniqueness of solutions of differential equations of fractional order with integral boundary conditions.

Inspired by the above works, we consider the existence of solutions for impulsive fractional differential equations with integral boundary conditions

⎩ ⎪ ⎨ ⎪ ⎧𝑐𝑐𝐷𝐷0+

𝑞𝑞 _𝑦𝑦₍_𝑡𝑡_{) =}_𝜔𝜔₍_𝑡𝑡₎_{𝑓𝑓�𝑡𝑡}_,_𝑦𝑦₍_𝑡𝑡_),_𝑦𝑦′₍_𝑡𝑡₎_�_{, 1 <}_𝑞𝑞_{< 2,} _𝑡𝑡 _{∈ 𝐽𝐽}_{= [0,1], t}_≠_t k,

∆𝑦𝑦|𝑡𝑡=𝑡𝑡𝑘𝑘 =𝐼𝐼𝑘𝑘�𝑦𝑦(𝑡𝑡𝑘𝑘)�, ∆𝑦𝑦′|𝑡𝑡=𝑡𝑡𝑘𝑘 =𝐽𝐽𝑘𝑘�𝑦𝑦(𝑡𝑡𝑘𝑘)�, 𝑡𝑡𝑘𝑘∈(0,1), 𝑘𝑘= 1,2, … ,𝑛𝑛,

𝑦𝑦(0) =𝑦𝑦(1) =∫ 𝑔𝑔₀1 (𝑠𝑠)𝑦𝑦(𝑠𝑠)𝑑𝑑𝑠𝑠,

(1.1)

(2)

where 𝐷𝐷₀ + 𝑞𝑞

𝐶𝐶 _{is the Caputo fractional derivative,} _𝜔𝜔₍_𝑡𝑡_):_𝐶𝐶₍_𝐽𝐽_,_𝑅𝑅

+ ), 𝐽𝐽= [0,1], 𝑓𝑓:𝐶𝐶(𝐽𝐽×𝑅𝑅×𝑅𝑅,𝑅𝑅 ), 𝐼𝐼𝑘𝑘,𝐽𝐽𝑘𝑘:𝐶𝐶(𝑅𝑅,𝑅𝑅),

∆𝑦𝑦|𝑡𝑡=𝑡𝑡𝑘𝑘 =𝑦𝑦(𝑡𝑡𝑘𝑘+)− 𝑦𝑦(𝑡𝑡𝑘𝑘− ), where 𝑦𝑦(𝑡𝑡𝑘𝑘+ ) and 𝑦𝑦(𝑡𝑡𝑘𝑘−) represent the right-hand limit of 𝑦𝑦(𝑡𝑡) at 𝑡𝑡=𝑡𝑡𝑘𝑘 respectively,

𝑘𝑘= 1,2,3, … ,𝑛𝑛, 0 =𝑡𝑡0<𝑡𝑡1<⋯<𝑡𝑡𝑛𝑛<𝑡𝑡𝑛𝑛+1= 1. ∆𝑦𝑦′|𝑡𝑡=𝑡𝑡𝑘𝑘 has a similar meaning for 𝑦𝑦′(𝑡𝑡) , and 𝑔𝑔 ∈ 𝐿𝐿1[0,1] is

nonnegative.

The rest of this paper is organized as follows. In section 2, we give some preliminaries about the fractional integral and derivative operator. In section 3, we present the expression and properties of Green's function associated with problem (1.1). In section 4, we get some existence results for problem (1.1) by means of standard fixed point theorem. Finally, in section 5, the example is also illustrated.

2. PRELIMINARIES

In this section, we give some preliminaries for discussing the solvability of problem (1.1).

Definition 2.1: The fractional (arbitrary) order integral of the function ℎ ∈ 𝐿𝐿1(𝐽𝐽, 𝑅𝑅₊) of order 𝑞𝑞 ∈ 𝑅𝑅₊ is defined by

𝐼𝐼₀𝑞𝑞+ℎ(𝑡𝑡) =_Γ1

(𝑞𝑞)� (𝑡𝑡 − 𝑠𝑠)𝑞𝑞−1ℎ(𝑠𝑠)𝑑𝑑𝑠𝑠,

1

0

where Γ(. ) is the Euler Gamma Function.

Definition 2.2: For a function ℎ given on the interval 𝐽𝐽, the Caputo-type fractional derivative of order 𝑞𝑞> 0 is defined by

𝐷𝐷

𝐶𝐶

0𝑞𝑞+ℎ(𝑡𝑡) =

1

Γ(𝑛𝑛 − 𝑞𝑞)� (𝑡𝑡 − 𝑠𝑠)𝑛𝑛−𝑞𝑞−1ℎ𝑛𝑛(𝑠𝑠)𝑑𝑑𝑠𝑠, 𝑛𝑛= [𝑞𝑞] + 1,

1

0

where the function ℎ(𝑡𝑡) has absolutely continuous derivatives up to order (𝑛𝑛 −1).

Lemma 2.1: Let > 0, then the differential equation 𝐶𝐶𝐷𝐷𝑞𝑞ℎ(𝑡𝑡) = 0 has the following solution:

ℎ(𝑡𝑡) =𝑐𝑐0+𝑐𝑐1𝑡𝑡+𝑐𝑐2𝑡𝑡2+⋯+𝑐𝑐(𝑛𝑛−1)𝑡𝑡𝑛𝑛−1, 𝑐𝑐𝑖𝑖 ∈ 𝑅𝑅,𝑖𝑖= 0,1,2, … ,𝑛𝑛 −1,𝑛𝑛= [𝑞𝑞] + 1.

Lemma 2.2: Let 𝑞𝑞> 0, then 𝐼𝐼𝑞𝑞𝐶𝐶𝐷𝐷₀𝑞𝑞₊ℎ(𝑡𝑡) =ℎ(𝑡𝑡) +𝑐𝑐₀+𝑐𝑐₁𝑡𝑡+𝑐𝑐₂𝑡𝑡2+⋯+𝑐𝑐₍_𝑛𝑛−₁₎𝑡𝑡𝑛𝑛−1, for some 𝑐𝑐_𝑖𝑖 ∈ 𝑅𝑅,

𝑖𝑖= 0,1,2, … ,𝑛𝑛 −1,𝑛𝑛= [𝑞𝑞] + 1.

We define the set of functions as follows

Let 𝐽𝐽′= [0,1]⧵{𝑡𝑡₁,𝑡𝑡₂, … . . ,𝑡𝑡_𝑛𝑛} and

𝑃𝑃𝐶𝐶[𝐽𝐽,𝑅𝑅] = {𝑦𝑦:𝐽𝐽 → 𝑅𝑅; 𝑦𝑦 ∈ 𝐶𝐶�(𝑡𝑡𝑘𝑘,𝑡𝑡𝑘𝑘+1),𝑅𝑅�,𝑦𝑦(𝑡𝑡𝑘𝑘+) 𝑎𝑎𝑛𝑛𝑑𝑑𝑦𝑦( 𝑡𝑡𝑘𝑘−) 𝑒𝑒𝑒𝑒𝑖𝑖𝑠𝑠𝑡𝑡 𝑤𝑤𝑖𝑖𝑡𝑡ℎ𝑦𝑦( 𝑡𝑡𝑘𝑘−) =𝑦𝑦(𝑡𝑡𝑘𝑘),𝑘𝑘= 1,2,3, … ,𝑛𝑛}

𝑃𝑃𝐶𝐶1_[_𝐽𝐽_,_𝑅𝑅_{] = {}_𝑦𝑦′_{∈ 𝑃𝑃𝐶𝐶}_[_𝐽𝐽_,_𝑅𝑅_];_𝑦𝑦′₍_𝑡𝑡

𝑘𝑘+) 𝑎𝑎𝑛𝑛𝑑𝑑𝑦𝑦′( 𝑡𝑡𝑘𝑘−) 𝑒𝑒𝑒𝑒𝑖𝑖𝑠𝑠𝑡𝑡 𝑎𝑎𝑛𝑛𝑑𝑑𝑦𝑦′𝑖𝑖𝑠𝑠𝑙𝑙𝑒𝑒𝑓𝑓𝑡𝑡𝑐𝑐𝑐𝑐𝑛𝑛𝑡𝑡𝑖𝑖𝑛𝑛𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑎𝑎𝑡𝑡𝑡𝑡𝑘𝑘,𝑘𝑘= 1,2,3, … ,𝑛𝑛}

Then 𝑃𝑃𝐶𝐶[𝐽𝐽,𝑅𝑅] is a Banach space with the norm

‖𝑦𝑦‖𝑃𝑃𝐶𝐶= sup𝑡𝑡∈𝐽𝐽|𝑦𝑦(𝑡𝑡)|, 𝑃𝑃𝐶𝐶1[𝐽𝐽,𝑅𝑅] is a Banach space with the norm

‖𝑦𝑦‖𝑃𝑃𝐶𝐶1= max{‖𝑦𝑦‖𝑃𝑃𝐶𝐶, �𝑦𝑦′�_{𝑃𝑃𝐶𝐶}}.

3. EXPRESSION AND PROPERTIES OF GREEN’S FUNCTION

Definition 3.1: A function 𝑦𝑦 ∈ 𝑃𝑃𝐶𝐶1[𝐽𝐽,𝑅𝑅]∩ 𝐶𝐶2[𝐽𝐽′,𝑅𝑅] with its Caputo derivative of order 𝑞𝑞 existing on 𝐽𝐽 is a solution of problem (1.1) if it satisfies (1.1). We give the following hypotheses:

(𝐻𝐻1) 𝜔𝜔 ∶ 𝐽𝐽 →[0, +∞) is a continuous function, and there exist 𝑡𝑡0∈ 𝐽𝐽 such that 𝜔𝜔(𝑡𝑡0) > 0;

(𝐻𝐻2) 𝑓𝑓 ∶ 𝐽𝐽 ×𝑅𝑅×𝑅𝑅 → 𝑅𝑅 is a continuous functions.

(𝐻𝐻3) 𝐼𝐼𝑘𝑘,𝐽𝐽𝑘𝑘 ∶ 𝑅𝑅 → 𝑅𝑅 are continuous functions.

(𝐻𝐻4) 𝑔𝑔 ∈ 𝐿𝐿1[0,1] is nonnegative and 𝜇𝜇 ∈[0,1),

where 𝜇𝜇=∫ 𝑔𝑔1 (𝑡𝑡)𝑑𝑑𝑡𝑡 (3.1) 0

Consider the following fractional impulsive boundary value problem :

�

𝐶𝐶_𝐷𝐷 0+

𝑞𝑞 _𝑦𝑦₍_𝑡𝑡_{) =}_𝜎𝜎₍_𝑡𝑡_{), 1 <}_𝑞𝑞_{< 2,} _{𝑡𝑡 ∈ 𝐽𝐽}

1=𝐽𝐽 ⧵{𝑡𝑡1,𝑡𝑡2, . . . ,𝑡𝑡𝑛𝑛},

∆𝑦𝑦|𝑡𝑡=𝑡𝑡𝑘𝑘 =𝐼𝐼𝑘𝑘�𝑦𝑦(𝑡𝑡𝑘𝑘)�, ∆𝑦𝑦′|𝑡𝑡=𝑡𝑡𝑘𝑘 =𝐽𝐽𝑘𝑘�𝑦𝑦(𝑡𝑡𝑘𝑘)�, 𝑡𝑡𝑘𝑘∈(0,1), 𝑘𝑘= 1,2, … ,𝑛𝑛,

𝑦𝑦(0) =𝑦𝑦(1) =∫ 𝑔𝑔₀1 (𝑠𝑠)𝑦𝑦(𝑠𝑠)𝑑𝑑𝑠𝑠,

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Proposition 3.1: The solution of problem (3.2) can be expressed by

𝑦𝑦(𝑡𝑡) =� 𝐻𝐻2(𝑡𝑡,𝑡𝑡𝑖𝑖) 𝑛𝑛+1

𝑖𝑖=1

�(𝑡𝑡𝑖𝑖_Γ− 𝑠𝑠₍_𝑞𝑞)₎𝑞𝑞−1𝜎𝜎(𝑠𝑠)𝑑𝑑𝑠𝑠

𝑡𝑡𝑖𝑖

𝑡𝑡𝑖𝑖−1

− � 𝐻𝐻1(𝑡𝑡,𝑡𝑡𝑖𝑖) 𝑛𝑛

𝑖𝑖=1

�(𝑡𝑡_Γ𝑖𝑖₍− 𝑠𝑠_{𝑞𝑞 −})𝑞𝑞−₁₎2𝜎𝜎(𝑠𝑠)𝑑𝑑𝑠𝑠

𝑡𝑡𝑖𝑖

+� 𝐻𝐻2(𝑡𝑡,𝑡𝑡𝑖𝑖)𝐼𝐼𝑖𝑖�𝑦𝑦(𝑡𝑡𝑖𝑖)� 𝑛𝑛

𝑖𝑖=1

𝐽𝐽𝑖𝑖�𝑦𝑦(𝑡𝑡𝑖𝑖)� , 𝑡𝑡𝑡𝑡 (𝑡𝑡𝑘𝑘,𝑡𝑡𝑘𝑘+1), (3.3)

𝑘𝑘= 0,1,2, …𝑛𝑛,𝑡𝑡0= 0,𝑡𝑡𝑛𝑛+1= 1. where

𝐻𝐻1(𝑡𝑡,𝑠𝑠) =𝐺𝐺1(𝑡𝑡,𝑠𝑠) +₁_{− 𝜇𝜇 � 𝑔𝑔}1 (𝑐𝑐)𝐺𝐺1 1

0

(𝑠𝑠,𝑐𝑐)𝑑𝑑𝑐𝑐 (3.4)

𝐻𝐻2(𝑡𝑡,𝑠𝑠) =𝐺𝐺1′(𝑡𝑡,𝑠𝑠) +₁_{− 𝜇𝜇 � 𝑔𝑔}1 (𝑐𝑐)𝐺𝐺1′ 1

0

(𝑠𝑠,𝑐𝑐)𝑑𝑑𝑐𝑐 (3.5)

𝐺𝐺1(𝑡𝑡,𝑠𝑠) =�𝑡𝑡_𝑠𝑠₍₁(1_{− 𝑡𝑡}− 𝑠𝑠_),), 𝑡𝑡 ≤ 𝑠𝑠_{𝑡𝑡 ≥ 𝑠𝑠},_{, (3.6)}

𝐺𝐺1′(𝑡𝑡,𝑠𝑠) =� −𝑡𝑡₍₁_{− 𝑡𝑡} 𝑐𝑐𝑜𝑜₎ (1_{𝑐𝑐𝑜𝑜}₍− 𝑠𝑠_−𝑠𝑠_),), 𝑡𝑡 ≤ 𝑠𝑠_{𝑡𝑡 ≥ 𝑠𝑠},_{, (3.7)}

Proof: suppose that 𝑦𝑦 is a solution of (3.2). Then, for some constants 𝑏𝑏₀,𝑏𝑏₁∈ 𝑅𝑅, we have

𝑦𝑦(𝑡𝑡) =𝐼𝐼₀𝑞𝑞+𝜎𝜎(𝑡𝑡)− 𝑏𝑏0− 𝑏𝑏1𝑡𝑡

=� (𝑡𝑡 − 𝑠𝑠_Γ₍_𝑞𝑞)₎𝑞𝑞−1𝜎𝜎(𝑠𝑠)𝑑𝑑𝑠𝑠 − 𝑏𝑏0− 𝑏𝑏1𝑡𝑡, 𝑡𝑡 ∈[0,1]. 𝑡𝑡

0

𝑦𝑦′₍_𝑡𝑡_{) =}_� (𝑡𝑡 − 𝑠𝑠)𝑞𝑞−2

Γ(𝑞𝑞 −1) 𝜎𝜎(𝑠𝑠)𝑑𝑑𝑠𝑠 − 𝑏𝑏1.

𝑡𝑡

0

If 𝑡𝑡 ∈(𝑡𝑡₁,𝑡𝑡₂], then for some constants 𝑐𝑐₀, 𝑐𝑐₁∈ 𝑅𝑅 we can write

𝑦𝑦(𝑡𝑡) =� (𝑡𝑡 − 𝑠𝑠_Γ₍)𝑞𝑞−1

𝑞𝑞) 𝜎𝜎(𝑠𝑠)𝑑𝑑𝑠𝑠 − 𝑐𝑐0− 𝑐𝑐1(𝑡𝑡 − 𝑡𝑡1), 𝑡𝑡

𝑡𝑡1

𝑦𝑦′₍_𝑡𝑡_{) =}_� (𝑡𝑡 − 𝑠𝑠)𝑞𝑞−1

Γ(𝑞𝑞) 𝜎𝜎(𝑠𝑠)𝑑𝑑𝑠𝑠 − 𝑐𝑐1. 𝑡𝑡

𝑡𝑡1

Using impulse conditions ∆𝑦𝑦|_𝑡𝑡₌_𝑡𝑡₁=𝐼𝐼₁�𝑦𝑦(𝑡𝑡₁)�, ∆𝑦𝑦′|_𝑡𝑡₌_𝑡𝑡₁=𝐽𝐽₁�𝑦𝑦(𝑡𝑡₁)�,

−𝑐𝑐0=� (𝑡𝑡1− 𝑠𝑠) 𝑞𝑞−1

Γ(𝑞𝑞) 𝜎𝜎(𝑠𝑠)𝑑𝑑𝑠𝑠 − 𝑏𝑏0− 𝑏𝑏1𝑡𝑡1+ 𝑡𝑡1

0 𝐼𝐼1�𝑦𝑦(𝑡𝑡1)�,

−𝑐𝑐1=� (𝑡𝑡1− 𝑠𝑠) 𝑞𝑞−2

Γ(𝑞𝑞 −1) 𝜎𝜎(𝑠𝑠)𝑑𝑑𝑠𝑠 − 𝑏𝑏1+𝐽𝐽1�𝑦𝑦(𝑡𝑡1)�.

𝑡𝑡1

0

Thus

𝑦𝑦(𝑡𝑡) =� (𝑡𝑡 − 𝑠𝑠_Γ₍_𝑞𝑞)₎𝑞𝑞−1𝜎𝜎(𝑠𝑠)𝑑𝑑𝑠𝑠+� �𝑡𝑡1 (𝑡𝑡1− 𝑠𝑠_Γ₍)_{𝑞𝑞 −}𝑞𝑞−2(₁₎𝑡𝑡 − 𝑡𝑡1)+(𝑡𝑡1_Γ− 𝑠𝑠₍_𝑞𝑞)₎𝑞𝑞−1�

0 𝜎𝜎(𝑠𝑠)𝑑𝑑𝑠𝑠 − 𝑏𝑏𝑐𝑐− 𝑏𝑏1𝑡𝑡 𝑡𝑡

𝑡𝑡1

+𝐽𝐽1�𝑦𝑦(𝑡𝑡1)�(𝑡𝑡 − 𝑡𝑡1) +𝐼𝐼1(𝑦𝑦(𝑡𝑡1))

𝑦𝑦′(𝑡𝑡) =� (_Γ𝑡𝑡 − 𝑠𝑠₍_{𝑞𝑞 −})𝑞𝑞−₁₎2𝜎𝜎(𝑠𝑠)𝑑𝑑𝑠𝑠 +� �(𝑡𝑡_Γ1₍_{𝑞𝑞 −}− 𝑠𝑠)₁₎𝑞𝑞−2 �𝜎𝜎(𝑠𝑠)𝑑𝑑𝑠𝑠 − 𝑏𝑏1+𝐽𝐽1�𝑦𝑦(𝑡𝑡1)� 𝑡𝑡1

0 . 𝑡𝑡

𝑡𝑡1

If ∈(𝑡𝑡_𝑘𝑘,𝑡𝑡_𝑘𝑘₊₁] , repeating the above procedure, we obtain

𝑦𝑦(𝑡𝑡) =� (𝑡𝑡 − 𝑠𝑠_Γ₍_𝑞𝑞)₎𝑞𝑞−1𝜎𝜎(𝑠𝑠)𝑑𝑑𝑠𝑠+� � �𝑡𝑡𝑖𝑖 (𝑡𝑡𝑖𝑖− 𝑠𝑠_Γ₍)_{𝑞𝑞 −}𝑞𝑞−2(₁₎𝑡𝑡 − 𝑡𝑡𝑖𝑖) + (𝑡𝑡𝑖𝑖_Γ− 𝑠𝑠₍_𝑞𝑞)₎𝑞𝑞−1� 𝜎𝜎(𝑠𝑠)𝑑𝑑𝑠𝑠

𝑡𝑡𝑖𝑖−1 𝑘𝑘

𝑖𝑖=1

− 𝑏𝑏0 𝑡𝑡

𝑡𝑡𝑘𝑘

−𝑏𝑏1𝑡𝑡+� 𝐽𝐽𝑖𝑖�𝑦𝑦(𝑡𝑡𝑖𝑖)�(𝑡𝑡 − 𝑡𝑡𝑖𝑖) + � 𝐼𝐼𝑖𝑖�𝑦𝑦(𝑡𝑡𝑖𝑖)� 𝑘𝑘

𝑖𝑖=1

(3.10)

𝑘𝑘

(4)

By the boundary conditions, we have

𝑏𝑏0=− � 𝑔𝑔(𝑡𝑡)𝑦𝑦(𝑡𝑡)𝑑𝑑𝑡𝑡 (3.11) 1

0

𝑏𝑏1=� (1− 𝑠𝑠) 𝑞𝑞−1

Γ(𝑞𝑞) 𝜎𝜎(𝑠𝑠)𝑑𝑑𝑠𝑠+� � �

(𝑡𝑡𝑖𝑖− 𝑠𝑠)𝑞𝑞−2(1− 𝑡𝑡𝑖𝑖)

Γ(𝑞𝑞 −1) +

(𝑡𝑡𝑖𝑖− 𝑠𝑠)𝑞𝑞−1

Γ(𝑞𝑞) � 𝜎𝜎(𝑠𝑠)𝑑𝑑𝑠𝑠

𝑡𝑡𝑖𝑖

𝑡𝑡𝑖𝑖−1 𝑘𝑘

𝑖𝑖=1 1

𝑡𝑡𝑘𝑘

+ � 𝐽𝐽𝑖𝑖�𝑦𝑦(𝑡𝑡𝑖𝑖)�(𝑡𝑡 − 𝑡𝑡𝑖𝑖) 𝑘𝑘

𝑖𝑖=1

+� 𝐼𝐼𝑖𝑖�𝑦𝑦(𝑡𝑡𝑖𝑖)� 𝑘𝑘

𝑖𝑖=1

(3.12)

Substituting (3.11) and (3.12) into (3.10), we have

𝑦𝑦(𝑡𝑡) =� 𝐺𝐺1′(𝑡𝑡,𝑡𝑡𝑖𝑖) 𝑛𝑛+1

𝑖𝑖=1

𝑡𝑡𝑖𝑖

− � 𝐺𝐺1(𝑡𝑡,𝑡𝑡𝑖𝑖) 𝑛𝑛

𝑖𝑖=1

�(_Γ𝑡𝑡𝑖𝑖₍− 𝑠𝑠_{𝑞𝑞 −})₁₎𝑞𝑞−2𝜎𝜎(𝑠𝑠)𝑑𝑑𝑠𝑠

𝑡𝑡𝑖𝑖

+� 𝐺𝐺1′(𝑡𝑡,𝑡𝑡𝑖𝑖)𝐼𝐼𝑖𝑖�𝑦𝑦(𝑡𝑡𝑖𝑖)� 𝑛𝑛

𝑖𝑖=1

𝐽𝐽𝑖𝑖�𝑦𝑦(𝑡𝑡𝑖𝑖)�+� 𝑔𝑔(𝑡𝑡)𝑦𝑦(𝑡𝑡)𝑑𝑑𝑡𝑡 1

0 where

� 𝑔𝑔(𝑡𝑡)

1

0

𝑦𝑦(𝑡𝑡)𝑑𝑑𝑡𝑡=� 𝑔𝑔(𝑡𝑡)

1

0

� 𝐺𝐺1′(𝑡𝑡,𝑡𝑡𝑖𝑖) 𝑛𝑛+1

𝑖𝑖=1

𝑡𝑡𝑖𝑖

𝑑𝑑𝑡𝑡 − � 𝑔𝑔(𝑡𝑡)

1

0

� 𝐺𝐺1(𝑡𝑡,𝑡𝑡𝑖𝑖) 𝑛𝑛

𝑖𝑖=1

�(_Γ𝑡𝑡𝑖𝑖₍− 𝑠𝑠_{𝑞𝑞 −})𝑞𝑞−₁₎2𝜎𝜎(𝑠𝑠)𝑑𝑑𝑠𝑠𝑑𝑑𝑡𝑡

𝑡𝑡𝑖𝑖

− � 𝑔𝑔(𝑡𝑡)

1

0

� 𝐺𝐺1(𝑡𝑡,𝑡𝑡𝑖𝑖) 𝑛𝑛

𝑖𝑖=1

𝐽𝐽𝑖𝑖�𝑦𝑦(𝑡𝑡𝑖𝑖)�𝑑𝑑𝑡𝑡+� 𝑔𝑔(𝑡𝑡) 1

0

� 𝐺𝐺1′(𝑡𝑡,𝑡𝑡𝑖𝑖)𝐼𝐼𝑖𝑖�𝑦𝑦(𝑡𝑡𝑖𝑖)� 𝑛𝑛

𝑖𝑖=1

𝑑𝑑𝑡𝑡+� 𝑔𝑔(𝑡𝑡)𝑑𝑑𝑡𝑡 ×

1

0

� 𝑔𝑔(𝑡𝑡)𝑦𝑦(𝑡𝑡)𝑑𝑑𝑡𝑡

1

0

Therefore, we have

� 𝑔𝑔(𝑠𝑠)

1

0

𝑦𝑦(𝑠𝑠)𝑑𝑑𝑠𝑠= 1

1− ∫ 𝑔𝑔₀1 (𝑠𝑠)𝑑𝑑𝑠𝑠�� 𝑔𝑔(𝑐𝑐)

1

0

� 𝐺𝐺1′(𝑠𝑠,𝑡𝑡𝑖𝑖) 𝑛𝑛+1

𝑖𝑖=1

�(𝑡𝑡𝑖𝑖_Γ− 𝑠𝑠₍_𝑞𝑞)₎𝑞𝑞−1𝜎𝜎(𝑠𝑠)𝑑𝑑𝑐𝑐

𝑡𝑡𝑖𝑖

𝑑𝑑𝑠𝑠

− � 𝑔𝑔(𝑐𝑐)

1

0

� 𝐺𝐺1(𝑠𝑠,𝑡𝑡𝑖𝑖) 𝑛𝑛

𝑖𝑖=1

�(𝑡𝑡_Γ𝑖𝑖₍− 𝑠𝑠_{𝑞𝑞 −})𝑞𝑞−₁₎2𝜎𝜎(𝑠𝑠)𝑑𝑑𝑐𝑐𝑑𝑑𝑠𝑠

𝑡𝑡𝑖𝑖

− � 𝑔𝑔(𝑠𝑠)

1

0

𝑖𝑖=1

𝐽𝐽𝑖𝑖�𝑦𝑦(𝑡𝑡𝑖𝑖)�𝑑𝑑𝑠𝑠

+� 𝑔𝑔(𝑠𝑠)

1

0

� 𝐺𝐺1′(𝑠𝑠,𝑡𝑡𝑖𝑖)𝐼𝐼𝑖𝑖�𝑦𝑦(𝑡𝑡𝑖𝑖)� 𝑛𝑛

𝑖𝑖=1

𝑑𝑑𝑠𝑠�

and

𝑦𝑦(𝑡𝑡) =� 𝐺𝐺1′(𝑡𝑡,𝑡𝑡𝑖𝑖) 𝑛𝑛+1

𝑖𝑖=1

𝑡𝑡𝑖𝑖

𝑖𝑖=1

�(_Γ𝑡𝑡𝑖𝑖₍− 𝑠𝑠_{𝑞𝑞 −})₁₎𝑞𝑞−2𝜎𝜎(𝑠𝑠)𝑑𝑑𝑠𝑠

𝑡𝑡𝑖𝑖

+� 𝐺𝐺1′(𝑡𝑡,𝑡𝑡𝑖𝑖)𝐼𝐼𝑖𝑖�𝑦𝑦(𝑡𝑡𝑖𝑖)� 𝑛𝑛

𝑖𝑖=1

𝐽𝐽𝑖𝑖�𝑦𝑦(𝑡𝑡𝑖𝑖)�₁_{− 𝜇𝜇 �� 𝑔𝑔}1 (𝑐𝑐) 1

0

� 𝐺𝐺1′(𝑠𝑠,𝑡𝑡𝑖𝑖) 𝑛𝑛+1

𝑖𝑖=1

�(𝑡𝑡𝑖𝑖_Γ− 𝑠𝑠₍_𝑞𝑞)₎𝑞𝑞−1𝜎𝜎(𝑠𝑠)𝑑𝑑𝑐𝑐

𝑡𝑡𝑖𝑖

𝑑𝑑𝑠𝑠

− � 𝑔𝑔(𝑐𝑐)

1

0

𝑖𝑖=1

�(𝑡𝑡_Γ𝑖𝑖₍− 𝑠𝑠_{𝑞𝑞 −})𝑞𝑞−₁₎2𝜎𝜎(𝑠𝑠)𝑑𝑑𝑐𝑐𝑑𝑑𝑠𝑠

𝑡𝑡𝑖𝑖

− � 𝑔𝑔(𝑠𝑠)

1

0

𝑖𝑖=1

𝐽𝐽𝑖𝑖�𝑦𝑦(𝑡𝑡𝑖𝑖)�𝑑𝑑𝑠𝑠

+� 𝑔𝑔(𝑠𝑠)

1

0

� 𝐺𝐺1′(𝑠𝑠,𝑡𝑡𝑖𝑖)𝐼𝐼𝑖𝑖�𝑦𝑦(𝑡𝑡𝑖𝑖)� 𝑛𝑛

𝑖𝑖=1

𝑑𝑑𝑠𝑠�

𝐿𝐿𝑒𝑒𝑡𝑡 𝐻𝐻1(𝑡𝑡,𝑠𝑠) = 𝐺𝐺1(𝑡𝑡,𝑠𝑠) +₁_{− 𝜇𝜇 � 𝑔𝑔}1 (𝑐𝑐)𝐺𝐺1 1

0

(𝑠𝑠,𝑐𝑐)𝑑𝑑𝑐𝑐

𝐻𝐻2(𝑡𝑡,𝑠𝑠) = 𝐺𝐺1′(𝑡𝑡,𝑠𝑠) +₁_{− 𝜇𝜇 � 𝑔𝑔}1 (𝑐𝑐)𝐺𝐺1′ 1

0

(5)

Then,

𝑖𝑖=1

𝑡𝑡𝑖𝑖

𝑖𝑖=1

�(𝑡𝑡_Γ𝑖𝑖₍− 𝑠𝑠_{𝑞𝑞 −})𝑞𝑞−₁₎2𝜎𝜎(𝑠𝑠)𝑑𝑑𝑠𝑠

𝑡𝑡𝑖𝑖

+� 𝐻𝐻2(𝑡𝑡,𝑡𝑡𝑖𝑖).𝐼𝐼𝑖𝑖�𝑦𝑦(𝑡𝑡𝑖𝑖)� 𝑛𝑛

𝑖𝑖=1

.𝐽𝐽𝑖𝑖�𝑦𝑦(𝑡𝑡𝑖𝑖)�

Proposition 3.2: If (𝐻𝐻₄) holds, then we have

𝐻𝐻1(𝑡𝑡,𝑠𝑠) > 0,𝐺𝐺1(𝑡𝑡,𝑠𝑠) > 0, 𝑓𝑓𝑐𝑐𝑜𝑜 𝑡𝑡,𝑠𝑠 ∈(0,1) (3.13)

𝐻𝐻1(𝑡𝑡,𝑠𝑠)≥0,𝐺𝐺1(𝑡𝑡,𝑠𝑠)≥0, 𝑓𝑓𝑐𝑐𝑜𝑜 𝑡𝑡,𝑠𝑠 ∈(0,1) (3.14)

Proof: From the definition of 𝐻𝐻₁(𝑡𝑡,𝑠𝑠) and 𝐺𝐺₁(𝑡𝑡,𝑠𝑠), it is easy to obtain the result of (3.13) and (3.14).

Proposition 3.3: For 𝑡𝑡,𝑠𝑠 ∈[0,1], we have

𝑒𝑒(𝑡𝑡)𝑒𝑒(𝑠𝑠)≤ 𝐺𝐺1(𝑡𝑡,𝑠𝑠)≤ 𝐺𝐺1(𝑠𝑠,𝑠𝑠) =𝑠𝑠(1− 𝑠𝑠) =𝑒𝑒(𝑠𝑠)≤ 𝑒𝑒̅= max𝑡𝑡∈[0,1]𝑒𝑒(𝑠𝑠) =1₄ (3.15)

Proof: In fact, for 𝑡𝑡 ∈ 𝐽𝐽, 𝑠𝑠 ∈(0,1), we have

Case-1: If 0 <𝑡𝑡 ≤ 𝑠𝑠< 1, then

𝐺𝐺1(𝑡𝑡,𝑠𝑠)

𝐺𝐺1(𝑠𝑠,𝑠𝑠) =

𝑡𝑡(1− 𝑠𝑠) 𝑠𝑠(1− 𝑠𝑠)=

𝑡𝑡 𝑠𝑠≤1.

Case-2: If 0 <𝑠𝑠 ≤ 𝑡𝑡< 1, then

𝑠𝑠(1− 𝑡𝑡) 𝑠𝑠(1− 𝑠𝑠)=

1− 𝑡𝑡 1− 𝑠𝑠≤

1− 𝑠𝑠 1− 𝑠𝑠≤1.

In addition, by the definition of 𝐺𝐺₁(𝑡𝑡,𝑠𝑠), it is easy to obtain that

𝐺𝐺1(𝑡𝑡,𝑠𝑠)≤ 𝐺𝐺1(𝑡𝑡,𝑠𝑠), ∀𝑡𝑡 ∈ 𝐽𝐽, 𝑠𝑠 ∈(0,1).

Therefore

𝐺𝐺1(𝑡𝑡,𝑠𝑠)≤ 𝐺𝐺1(𝑡𝑡,𝑠𝑠) =𝑒𝑒(𝑠𝑠),∀𝑡𝑡,𝑠𝑠 ∈ 𝐽𝐽.

Similarly, we can prove that

𝐺𝐺1(𝑡𝑡,𝑠𝑠)≥ 𝑒𝑒(𝑡𝑡)𝑒𝑒(𝑠𝑠).

In fact, for all 𝑡𝑡,𝑠𝑠 we have that

Case-1: If 𝑡𝑡 ≤ 𝑠𝑠, then

𝑡𝑡(1− 𝑠𝑠) 𝑠𝑠(1− 𝑠𝑠) =

𝑡𝑡

𝑠𝑠≥ 𝑡𝑡 ≥ 𝑡𝑡(1− 𝑡𝑡).

Case-2: If 𝑡𝑡 ≤ 𝑠𝑠, then

𝑠𝑠(1− 𝑡𝑡) 𝑠𝑠(1− 𝑠𝑠) =

1− 𝑡𝑡

1− 𝑠𝑠 ≥(1− 𝑡𝑡) =𝑡𝑡(1− 𝑡𝑡).

So we have

𝐺𝐺1(𝑡𝑡,𝑠𝑠)≥ 𝑒𝑒(𝑡𝑡)𝑒𝑒(𝑠𝑠), ∀𝑡𝑡,𝑠𝑠 ∈ 𝐽𝐽.

Proposition 3.4: If (𝐻𝐻₄) holds, then for 𝑡𝑡,𝑠𝑠 ∈[0,1], we have

𝜌𝜌𝑒𝑒(𝑠𝑠)≤ 𝐻𝐻1(𝑡𝑡,𝑠𝑠)≤ 𝛾𝛾𝑠𝑠(1− 𝑠𝑠) =𝛾𝛾𝑒𝑒(𝑠𝑠)≤1₄𝛾𝛾 (3.16) where

𝛾𝛾=₁_{− 𝜇𝜇}1 , 𝜌𝜌=∫ 𝑒𝑒(𝜏𝜏)𝑔𝑔(𝜏𝜏)𝑑𝑑𝜏𝜏

1 0

1− 𝜇𝜇 (3.17)

Proof: By (3.4) and (3.14), we have

𝐻𝐻1(𝑡𝑡,𝑠𝑠) =𝐺𝐺1(𝑡𝑡,𝑠𝑠) +₁_{− 𝜇𝜇 � 𝐺𝐺}1 1(𝑠𝑠,𝜏𝜏)𝑔𝑔(𝜏𝜏)𝑑𝑑𝜏𝜏 1

(6)

≥ 1

1−𝜇𝜇∫ 𝐺𝐺1(𝑠𝑠,𝜏𝜏)𝑔𝑔(𝜏𝜏)𝑑𝑑𝜏𝜏 1

0

≥∫ 𝑒𝑒(𝜏𝜏)𝑔𝑔(𝜏𝜏)𝑑𝑑𝜏𝜏 1

0

1−𝜇𝜇 𝑠𝑠(1− 𝑠𝑠) =𝜌𝜌𝑒𝑒(𝑠𝑠),𝑡𝑡 ∈[0,1].

On the other hand, noticing 𝐺𝐺₁(𝑡𝑡,𝑠𝑠)≤ 𝑠𝑠(1− 𝑠𝑠), we obtain

𝐻𝐻1(𝑡𝑡,𝑠𝑠) =𝐺𝐺1(𝑡𝑡,𝑠𝑠) +₁_{− 𝜇𝜇 � 𝐺𝐺}1 1(𝑠𝑠,𝜏𝜏)𝑔𝑔(𝜏𝜏)𝑑𝑑𝜏𝜏 1

0

=𝑠𝑠(1− 𝑠𝑠) +₁_{− 𝜇𝜇 � 𝑠𝑠}1 (1− 𝑠𝑠)𝑔𝑔(𝜏𝜏)𝑑𝑑𝜏𝜏

1

0

=𝑠𝑠(1− 𝑠𝑠)�1 +₁_{− 𝜇𝜇}1 � 𝑔𝑔(𝜏𝜏)𝑑𝑑𝜏𝜏

1

0

�

=𝑠𝑠(1− 𝑠𝑠)�1 +₁_{− 𝜇𝜇}1 � =𝛾𝛾𝑒𝑒(𝑠𝑠), 𝑡𝑡 ∈[0,1].

The proof of Proposition 3.4 is complete.

Proposition 3.5: If (𝐻𝐻₄) holds, then for 𝑡𝑡,𝑠𝑠 ∈[0,1], we have

𝐻𝐻2(𝑡𝑡,𝑠𝑠)≤ 𝛾𝛾(1− 𝑡𝑡)≤ 𝛾𝛾 (3.18)

Proof:

𝐻𝐻2(𝑡𝑡,𝑠𝑠) =𝐺𝐺1′(𝑡𝑡,𝑠𝑠) +₁_{− 𝜇𝜇 � 𝐺𝐺}1 1′(𝑠𝑠,𝜏𝜏)𝑔𝑔(𝜏𝜏)𝑑𝑑𝜏𝜏 1

0

= 1 +₁_{− 𝜇𝜇 � 𝑔𝑔}1 (𝜏𝜏)𝑑𝑑𝜏𝜏

1

0

=₁_{− 𝜇𝜇}1 ≤ 𝛾𝛾

It follows from Proposition (3.1) that:

Lemma 3.1: If (𝐻𝐻₁)−(𝐻𝐻₂) hold, then a function 𝑦𝑦 ∈ 𝑃𝑃𝐶𝐶1[𝐽𝐽,𝑅𝑅]∩ 𝐶𝐶2[𝐽𝐽′,𝑅𝑅] is a solution of problem (1.1) iff

𝑦𝑦 ∈ 𝑃𝑃𝐶𝐶1_[_𝐽𝐽_,_𝑅𝑅_]_{is a solution of the impulsive fractional integral equation}

𝑖𝑖=1

�(𝑡𝑡𝑖𝑖_Γ− 𝑠𝑠₍_𝑞𝑞)₎𝑞𝑞−1[𝜔𝜔(𝑠𝑠)𝑓𝑓(𝑠𝑠,𝑦𝑦(𝑠𝑠),𝑦𝑦′₍_𝑠𝑠_))]_{𝑑𝑑𝑠𝑠} 𝑡𝑡𝑖𝑖

𝑖𝑖=1

�(𝑡𝑡_Γ𝑖𝑖₍− 𝑠𝑠_{𝑞𝑞 −})𝑞𝑞−₁₎2[𝜔𝜔(𝑠𝑠)𝑓𝑓(𝑠𝑠,𝑦𝑦(𝑠𝑠),𝑦𝑦′(𝑠𝑠))]𝑑𝑑𝑠𝑠

𝑡𝑡𝑖𝑖

𝑖𝑖=1

𝐽𝐽𝑖𝑖�𝑦𝑦(𝑡𝑡𝑖𝑖)�

𝑘𝑘= 0,1,2, … . . ,𝑛𝑛, 𝑡𝑡0= 0,𝑡𝑡𝑛𝑛+1= 1 (3.19)

Proof: Let 𝑦𝑦 ∈ 𝑃𝑃𝐶𝐶1[𝐽𝐽,𝑅𝑅]∩ 𝐶𝐶2[𝐽𝐽′,𝑅𝑅] be a solution of the boundary value problem (1.1), then by the same method as used in Proposition 3.1, we can prove that 𝑦𝑦 is a solution of the fractional integral equation (3.19).

Conversely, assume that 𝑦𝑦 satisfies the impulsive fractional integral equation (3.19).

If 𝑡𝑡 ∈[0,𝑡𝑡₁] then 𝑦𝑦(0) =∫ 𝑔𝑔₀1 (𝑠𝑠)𝑦𝑦(𝑠𝑠)𝑑𝑑𝑠𝑠, 𝑦𝑦(1) =∫ 𝑔𝑔₀1 (𝑠𝑠)𝑦𝑦(𝑠𝑠)𝑑𝑑𝑠𝑠 and using the fact that 𝑐𝑐𝐷𝐷𝛼𝛼 is the left inverse of 𝐼𝐼𝛼𝛼 we get

𝐷𝐷𝛼𝛼

(7)

If 𝑡𝑡 ∈[𝑡𝑡_𝑘𝑘,𝑡𝑡_𝑘𝑘₊₁),𝑘𝑘= 1,2, … ,𝑛𝑛 and using the fact that 𝐶𝐶𝐷𝐷𝛼𝛼𝐶𝐶= 0, where 𝐶𝐶 is a constant, we get

𝐷𝐷𝛼𝛼

𝐶𝐶 _𝑦𝑦₍_𝑡𝑡_{) =}_ℎ₍_𝑡𝑡₎_{for each}_{𝑡𝑡 ∈}_[_𝑡𝑡 𝑘𝑘,𝑡𝑡𝑘𝑘+1).

Also we can easily show that

∆𝑦𝑦|_𝑡𝑡₌_𝑡𝑡_𝑘𝑘 =𝐼𝐼_𝑘𝑘�𝑦𝑦(𝑡𝑡_𝑘𝑘)�, ∆𝑦𝑦′|_𝑡𝑡₌_𝑡𝑡_𝑘𝑘 =𝐼𝐼_𝑘𝑘�𝑦𝑦(𝑡𝑡_𝑘𝑘)�, 𝑘𝑘= 1,2, … … … … ,𝑛𝑛.

Hence the proof.

Define 𝑇𝑇 ∶ 𝑃𝑃𝐶𝐶1[𝐽𝐽,𝑅𝑅]→ 𝑃𝑃𝐶𝐶1[𝐽𝐽,𝑅𝑅] 𝑏𝑏𝑦𝑦

𝑇𝑇𝑦𝑦(𝑡𝑡) =� 𝐻𝐻2(𝑡𝑡,𝑡𝑡𝑖𝑖) 𝑛𝑛+1

𝑖𝑖=1

�(𝑡𝑡𝑖𝑖_Γ− 𝑠𝑠₍_𝑞𝑞)₎𝑞𝑞−1[𝜔𝜔(𝑠𝑠)𝑓𝑓(𝑠𝑠,𝑦𝑦(𝑠𝑠),𝑦𝑦′₍_𝑠𝑠_))]_{𝑑𝑑𝑠𝑠} 𝑡𝑡𝑖𝑖

𝑖𝑖=1

�(𝑡𝑡_Γ𝑖𝑖₍− 𝑠𝑠_{𝑞𝑞 −})𝑞𝑞−₁₎2[𝜔𝜔(𝑠𝑠)𝑓𝑓(𝑠𝑠,𝑦𝑦(𝑠𝑠),𝑦𝑦′(𝑠𝑠))]𝑑𝑑𝑠𝑠

𝑡𝑡𝑖𝑖

𝑖𝑖=1

𝐽𝐽𝑖𝑖�𝑦𝑦(𝑡𝑡𝑖𝑖)�, 𝑘𝑘= 0,1,2, …𝑛𝑛,𝑡𝑡0= 0,𝑡𝑡𝑛𝑛+1= 1 (3.20)

Using Lemma (3.1), Problem (1.1) reduces to a fixed point problem 𝑒𝑒=𝑇𝑇𝑒𝑒, where 𝑇𝑇is given by (3.20). Thus, The problem (1.1) has a solution iff operator T has a fixed point.

Lemma 3.2: Assume that (𝐻𝐻₁)−(𝐻𝐻₄) hold. Then 𝑇𝑇 ∶ 𝑃𝑃𝐶𝐶1[𝐽𝐽,𝑅𝑅]→ 𝑃𝑃𝐶𝐶1[𝐽𝐽,𝑅𝑅] is completely continuous.

Proof: Note that the continuity of 𝑓𝑓,𝑔𝑔,𝜔𝜔,𝐼𝐼_𝑘𝑘𝑎𝑎𝑛𝑛𝑑𝑑𝐽𝐽_𝑘𝑘together with 𝐻𝐻₁(𝑡𝑡,𝑠𝑠) 𝑎𝑎𝑛𝑛𝑑𝑑𝐻𝐻₂(𝑡𝑡,𝑠𝑠) ensures the continuity of 𝑇𝑇.

Let Ω⊂ 𝑃𝑃𝐶𝐶1[𝐽𝐽,𝑅𝑅] be bounded. Then there exists positive constants 𝜆𝜆₁,𝜆𝜆₂ and 𝜆𝜆₃, such that

|𝑓𝑓(𝑡𝑡,𝑦𝑦(𝑡𝑡),𝑦𝑦′₍_𝑡𝑡_)|_{≤ 𝜆𝜆}

1 , |𝐼𝐼𝑘𝑘(𝑦𝑦)|≤ ,𝜆𝜆2𝑎𝑎𝑛𝑛𝑑𝑑 , |𝐽𝐽𝑘𝑘(𝑦𝑦)|≤ 𝜆𝜆3∀𝑦𝑦 ∈Ω. 𝑇𝑇ℎ𝑐𝑐𝑠𝑠 ∀𝑦𝑦 ∈ Ω , we have

|(𝑇𝑇𝑦𝑦)(𝑡𝑡)|≤ �|𝐻𝐻2(𝑡𝑡,𝑡𝑡𝑖𝑖)| 𝑛𝑛+1

𝑖𝑖=1

�(𝑡𝑡𝑖𝑖_Γ− 𝑠𝑠₍_𝑞𝑞)₎𝑞𝑞−1�𝜔𝜔(𝑠𝑠)𝑓𝑓�𝑠𝑠,𝑦𝑦(𝑠𝑠),𝑦𝑦′₍_𝑠𝑠₎_{�� 𝑑𝑑𝑠𝑠} 𝑡𝑡𝑖𝑖

+�|𝐻𝐻1(𝑡𝑡,𝑡𝑡𝑖𝑖)| 𝑛𝑛

𝑖𝑖=1

�(𝑡𝑡_Γ𝑖𝑖₍− 𝑠𝑠_{𝑞𝑞 −})𝑞𝑞−₁₎2�𝜔𝜔(𝑠𝑠)𝑓𝑓�𝑠𝑠,𝑦𝑦(𝑠𝑠),𝑦𝑦′₍_𝑠𝑠₎_{�� 𝑑𝑑𝑠𝑠} 𝑡𝑡𝑖𝑖

+�|𝐻𝐻2(𝑡𝑡,𝑡𝑡𝑖𝑖)|�𝐼𝐼𝑖𝑖�𝑦𝑦(𝑡𝑡𝑖𝑖)�� 𝑛𝑛

𝑖𝑖=1

�𝐽𝐽𝑖𝑖�𝑦𝑦(𝑡𝑡𝑖𝑖)��

≤ �|𝐻𝐻2(𝑡𝑡,𝑡𝑡𝑖𝑖)| 𝑛𝑛+1

𝑖𝑖=1

�(𝑡𝑡𝑖𝑖_Γ− 𝑠𝑠₍_𝑞𝑞)₎𝑞𝑞−1[𝑐𝑐𝜆𝜆1]𝑑𝑑𝑠𝑠 𝑡𝑡𝑖𝑖

𝑖𝑖=1

�(𝑡𝑡_Γ𝑖𝑖₍− 𝑠𝑠_{𝑞𝑞 −})𝑞𝑞−₁₎2[𝑐𝑐𝜆𝜆1]𝑑𝑑𝑠𝑠 𝑡𝑡𝑖𝑖

+𝜆𝜆2�|𝐻𝐻2(𝑡𝑡,𝑡𝑡𝑖𝑖)| 𝑛𝑛

𝑖𝑖=1

+𝜆𝜆3�|𝐻𝐻1(𝑡𝑡,𝑡𝑡𝑖𝑖)| 𝑛𝑛

𝑖𝑖=1

≤𝛾𝛾𝑐𝑐_Γ(₍𝑛𝑛_𝑞𝑞+ 1)_{+ 1) +}𝜆𝜆1 𝑛𝑛𝛾𝛾𝑐𝑐𝜆𝜆₄_Γ₍_𝑞𝑞1_{) +}𝑛𝑛𝛾𝛾𝜆𝜆2+𝑛𝑛𝛾𝛾𝜆𝜆_{4 =}3 𝜆𝜆 (3.21) where

𝑐𝑐= max

𝑡𝑡∈𝐽𝐽 𝜔𝜔(𝑡𝑡) (3.22)

Furthermore, for any 𝑡𝑡 ∈(𝑡𝑡_𝑘𝑘,𝑡𝑡_𝑘𝑘₊₁]

�(𝑇𝑇𝑦𝑦)′₍_𝑡𝑡₎_{� ≤ ��𝐺𝐺}

1′′𝑠𝑠𝑡𝑡(𝑡𝑡,𝑡𝑡𝑖𝑖)� 𝑛𝑛+1

𝑖𝑖=1

�𝑡𝑡𝑖𝑖 (𝑡𝑡𝑖𝑖_Γ− 𝑠𝑠₍_𝑞𝑞)₎𝑞𝑞−1��𝜔𝜔(𝑠𝑠)𝑓𝑓�𝑠𝑠,𝑦𝑦(𝑠𝑠),𝑦𝑦′₍_𝑠𝑠₎_{�� 𝑑𝑑𝑠𝑠} 𝑡𝑡𝑖𝑖−1

+�|𝐺𝐺′1𝑡𝑡(𝑡𝑡,𝑡𝑡𝑖𝑖)| 𝑛𝑛

𝑖𝑖=1

�𝑡𝑡𝑖𝑖 (𝑡𝑡_Γ𝑖𝑖₍− 𝑠𝑠_{𝑞𝑞 −})𝑞𝑞−₁₎2��𝜔𝜔(𝑠𝑠)𝑓𝑓�𝑠𝑠,𝑦𝑦(𝑠𝑠),𝑦𝑦′₍_𝑠𝑠₎_{�� 𝑑𝑑𝑠𝑠} 𝑡𝑡𝑖𝑖−1

+��𝐺𝐺1′′𝑠𝑠𝑡𝑡(𝑡𝑡,𝑡𝑡𝑖𝑖)��𝐼𝐼𝑖𝑖�𝑦𝑦(𝑡𝑡𝑖𝑖)�� 𝑛𝑛

𝑖𝑖=1

(8)

≤ ��𝐺𝐺1′′𝑠𝑠𝑡𝑡(𝑡𝑡,𝑡𝑡𝑖𝑖)� 𝑛𝑛+1

𝑖𝑖=1

�(𝑡𝑡𝑖𝑖_Γ− 𝑠𝑠₍_𝑞𝑞)₎𝑞𝑞−1[𝑐𝑐𝜆𝜆1]𝑑𝑑𝑠𝑠 𝑡𝑡𝑖𝑖

𝑖𝑖=1

�(_Γ𝑡𝑡𝑖𝑖₍− 𝑠𝑠_{𝑞𝑞 −})𝑞𝑞−₁₎2[𝑐𝑐𝜆𝜆1]𝑑𝑑𝑠𝑠 𝑡𝑡𝑖𝑖

+𝜆𝜆2��𝐺𝐺1′′𝑠𝑠𝑡𝑡(𝑡𝑡,𝑡𝑡𝑖𝑖)� 𝑛𝑛

𝑖𝑖=1

+𝜆𝜆3�|𝐺𝐺′1𝑡𝑡(𝑡𝑡,𝑡𝑡𝑖𝑖)| 𝑛𝑛

𝑖𝑖=1

≤𝑐𝑐_Γ(𝑛𝑛₍_𝑞𝑞+ 1)_{+ 1) +}𝜆𝜆1 𝑛𝑛𝑐𝑐𝜆𝜆_Γ₍_𝑞𝑞1_{) +}𝑛𝑛𝜆𝜆2+𝑛𝑛𝜆𝜆3=𝜆𝜆̅ (3.23)

On the other hand, From (3.23) for 𝑡𝑡₁, 𝑡𝑡₂𝑡𝑡 (𝑡𝑡_𝑘𝑘, 𝑡𝑡_𝑘𝑘₊₁] with 𝑡𝑡₁<𝑡𝑡₂, we have

�(𝑇𝑇𝑦𝑦)′₍_𝑡𝑡

2)−(𝑇𝑇𝑦𝑦)′(𝑡𝑡1)� ≤ � |(𝑇𝑇𝑦𝑦)′(𝑠𝑠)| 𝑑𝑑𝑠𝑠 𝑡𝑡2

𝑡𝑡1

≤ 𝜆𝜆̅(𝑡𝑡2− 𝑡𝑡1) (3.24)

It follows from (3.21), (3.23) 𝑎𝑎𝑛𝑛𝑑𝑑 (3.24) that 𝑇𝑇is equicontinuous on all subintervals (𝑡𝑡_𝑘𝑘,𝑡𝑡_𝑘𝑘₊₁],𝑘𝑘= 1,2 …𝑛𝑛. Thus by Arzela-Ascoli Theorem, the operator 𝑇𝑇 ∶ 𝑃𝑃𝐶𝐶1[𝐽𝐽,𝑅𝑅]→ 𝑃𝑃𝐶𝐶1[𝐽𝐽,𝑅𝑅] is completely continuous.

To prove our main result, we also need the following lemma.

Lemma 3.3 (Schauder Fixed point Theorem): Let 𝐷𝐷be non empty, closed, bounded, convex subset of a banach space X, and suppose that 𝑇𝑇:𝐷𝐷 → 𝐷𝐷 is completely continuous operator. Then 𝑇𝑇has fixed point 𝑒𝑒 ∈ 𝐷𝐷.

4. EXISTENCE OF SOLUTIONS

In this section we apply Lemma 3.3 to establish the existence of solutions to problem (1.1). Let us define

𝜃𝜃 = lim_|_𝑦𝑦_|+|_𝑧𝑧_|_→_∞�max_{𝑡𝑡∈𝐽𝐽} |𝑓𝑓_|_𝑦𝑦(_{| + |}𝑡𝑡,𝑦𝑦,𝑧𝑧_𝑧𝑧)|_|�,

𝜃𝜃1= lim_|_𝑦𝑦_|_→_∞|𝐼𝐼𝑘𝑘_|(_𝑦𝑦𝑦𝑦_{| ,})| 𝜃𝜃2= lim_|_𝑦𝑦_|_→_∞|𝐽𝐽𝑘𝑘_|_𝑦𝑦(𝑦𝑦_{| ,})| 𝑘𝑘= 1, 2, … …𝑛𝑛

Theorem 4.1: Assume that (𝐻𝐻₁)−(𝐻𝐻₄) hold. Suppose further that

𝛿𝛿= max{𝛿𝛿1 ,𝛿𝛿2 } < 1 (4.1) where

𝛿𝛿1=2𝑐𝑐_Γ(₍𝑛𝑛_𝑞𝑞+ 1)_{+ 1)}𝛾𝛾𝜃𝜃+2₄𝑐𝑐𝑛𝑛𝛾𝛾𝜃𝜃_Γ₍_𝑞𝑞_{) +}𝑛𝑛𝛾𝛾𝜃𝜃1+𝑛𝑛𝛾𝛾𝜃𝜃_{4 ,}2 and

𝛿𝛿2=2_Γ𝑐𝑐(₍𝑛𝑛_𝑞𝑞+ 1)_{+ 1)}𝜃𝜃+2_Γ𝑐𝑐𝑛𝑛𝜃𝜃₍_𝑞𝑞_{) +}𝑛𝑛𝜃𝜃1+𝑛𝑛𝜃𝜃2

where 𝑐𝑐is defined in (3.22) . Then problem (1.1) has atleast one solution 𝑒𝑒 ∈ 𝑃𝑃𝐶𝐶1[𝐽𝐽,𝑅𝑅]⋂𝐶𝐶2[𝐽𝐽′,𝑅𝑅].

Proof: We shall use Schauder’s fixed point theorem to prove that T has a fixed point. First, recall that the operator

𝑇𝑇 ∶ 𝑃𝑃𝐶𝐶1_[_𝐽𝐽_,_𝑅𝑅_]_{→ 𝑃𝑃𝐶𝐶}1_[_𝐽𝐽_,_𝑅𝑅_]_{is completely continuous (see the proof of lemma (3.2)).}

On account of (4.1), we can choose 𝜃𝜃′>𝜃𝜃, 𝜃𝜃₁′ >𝜃𝜃₁𝑎𝑎𝑛𝑛𝑑𝑑𝜃𝜃₂′ >𝜃𝜃₂ such that

𝛿𝛿′₁=2𝑐𝑐_Γ(₍𝑛𝑛_𝑞𝑞+ 1)_{+ 1) +}𝛾𝛾𝜃𝜃′ 2₄𝑐𝑐𝑛𝑛𝛾𝛾𝜃𝜃_Γ₍_𝑞𝑞_{) +}′ 𝑛𝑛𝛾𝛾𝜃𝜃1′ +𝑛𝑛𝛾𝛾𝜃𝜃2 ′

4 (4.2)

and

𝛿𝛿′2=2𝑐𝑐(𝑛𝑛+ 1)𝜃𝜃 ′ Γ(𝑞𝑞+ 1) +

2𝑐𝑐𝑛𝑛𝜃𝜃′

Γ(𝑞𝑞) +𝑛𝑛𝜃𝜃1′ +𝑛𝑛𝜃𝜃2′ (4.3)

By the definition of 𝜃𝜃, there exists 𝑙𝑙> 0 such that

|𝑓𝑓(𝑡𝑡,𝑦𝑦,𝑧𝑧)| <𝜃𝜃′_(|_𝑦𝑦_{| + |}_𝑧𝑧_|),_∀_{𝑡𝑡 ∈ 𝐽𝐽}_{, |}_𝑦𝑦_{| + |}_𝑧𝑧_{| >}_𝑙𝑙_,

|𝑓𝑓(𝑡𝑡,𝑦𝑦,𝑧𝑧)| <𝜃𝜃′_(|_𝑦𝑦_{| + |}_𝑧𝑧_{|) +}_𝑀𝑀

1, ∀𝑡𝑡 ∈ 𝐽𝐽, 𝑦𝑦,𝑧𝑧 ∈ 𝑅𝑅, (4.4) where

𝑀𝑀1=_{𝑡𝑡∈𝐽𝐽}_,|max_𝑦𝑦_|+|_𝑧𝑧_|_≤𝑙𝑙|𝑓𝑓(𝑡𝑡,𝑦𝑦,𝑧𝑧)| < +∞

Similarly, we have

�|𝐼𝐼𝑘𝑘(𝑦𝑦)| <𝜃𝜃1′|𝑒𝑒| +𝑀𝑀𝑘𝑘, ∀𝑦𝑦 ∈ 𝑅𝑅, 𝑘𝑘= 1,2 …𝑛𝑛

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It follows from (3.20), (4.4) and (4.5) that

|(𝑇𝑇𝑦𝑦)(𝑡𝑡)| ≤ �|𝐻𝐻2(𝑡𝑡,𝑡𝑡𝑖𝑖)| 𝑛𝑛+1

𝑖𝑖=1

�𝑡𝑡𝑖𝑖 (𝑡𝑡𝑖𝑖− 𝑠𝑠_Γ₍_𝑞𝑞)₎𝑞𝑞−1��𝜔𝜔(𝑠𝑠)𝑓𝑓�𝑠𝑠,𝑦𝑦(𝑠𝑠),𝑧𝑧(𝑠𝑠)��𝑑𝑑𝑠𝑠

𝑖𝑖=1

�𝑡𝑡𝑖𝑖 (_Γ𝑡𝑡𝑖𝑖₍− 𝑠𝑠_{𝑞𝑞 −})𝑞𝑞−₁₎2��𝜔𝜔(𝑠𝑠)𝑓𝑓�𝑠𝑠,𝑦𝑦(𝑠𝑠),𝑧𝑧(𝑠𝑠)��𝑑𝑑𝑠𝑠

+�|𝐻𝐻2(𝑡𝑡,𝑡𝑡𝑖𝑖)|�𝐼𝐼𝑖𝑖�𝑦𝑦(𝑡𝑡𝑖𝑖)�� 𝑛𝑛

𝑖𝑖=1

≤ �|𝐻𝐻2(𝑡𝑡,𝑡𝑡𝑖𝑖)| 𝑛𝑛+1

𝑖𝑖=1

�𝑡𝑡𝑖𝑖 (𝑡𝑡𝑖𝑖_Γ− 𝑠𝑠₍_𝑞𝑞)₎𝑞𝑞−1[𝛾𝛾(𝜃𝜃′_(|_𝑦𝑦_{| + |}_𝑧𝑧_{|) +}_𝑀𝑀_)]_{𝑑𝑑𝑠𝑠} 𝑡𝑡𝑖𝑖−1

𝑖𝑖=1

�𝑡𝑡𝑖𝑖 (_Γ𝑡𝑡𝑖𝑖₍− 𝑠𝑠_{𝑞𝑞 −})𝑞𝑞−₁₎2[𝛾𝛾(𝜃𝜃′_(|_𝑦𝑦_{| + |}_𝑧𝑧_{|) +}_𝑀𝑀_)]_{𝑑𝑑𝑠𝑠} 𝑡𝑡𝑖𝑖−1

+�|𝐻𝐻2(𝑡𝑡,𝑡𝑡𝑖𝑖)|(𝜃𝜃1′|𝑦𝑦| +𝑀𝑀𝑘𝑘) 𝑛𝑛

𝑖𝑖=1

(𝜃𝜃2′|𝑦𝑦| +𝑀𝑀�𝑘𝑘)

≤(_Γ𝑛𝑛₍_𝑞𝑞+ 1)_{+ 1)}𝑐𝑐𝛾𝛾��2𝜃𝜃′��|𝑦𝑦|�𝑃𝑃𝐶𝐶1�+𝑀𝑀�+ 𝑛𝑛𝑐𝑐𝛾𝛾

4Γ(𝑞𝑞)��2𝜃𝜃′��|𝑦𝑦|�𝑃𝑃𝐶𝐶1�+𝑀𝑀��

+𝑛𝑛𝛾𝛾�𝜃𝜃1′��|𝑦𝑦|�𝑃𝑃𝐶𝐶1�+𝑀𝑀_𝑘𝑘�+𝑛𝑛𝛾𝛾

4�𝜃𝜃2′��|𝑦𝑦|�𝑃𝑃𝐶𝐶1�+𝑀𝑀�_𝑘𝑘�

≤ 𝛿𝛿₁′‖𝑦𝑦‖_{𝑃𝑃𝐶𝐶}1+𝑀𝑀(1) (4.6) Where 𝛿𝛿₁′ is defined by (4.2) and 𝑀𝑀(1)is defined by

𝑀𝑀(1)₌𝑀𝑀(𝑛𝑛+ 1)𝑐𝑐𝛾𝛾 Γ(𝑞𝑞+ 1) +

𝑀𝑀𝑛𝑛𝑐𝑐𝛾𝛾

4Γ(𝑞𝑞) +𝑛𝑛𝛾𝛾𝑀𝑀𝑘𝑘+𝑛𝑛𝛾𝛾₄𝑀𝑀�𝑘𝑘

Similarly, from (3.20), (4.4) and (4.5), we get

�(𝑇𝑇𝑦𝑦) ′₍_𝑡𝑡₎_{� ≤ ��𝐺𝐺}

1′′𝑠𝑠𝑡𝑡(𝑡𝑡,𝑡𝑡𝑖𝑖)� 𝑛𝑛+1

𝑖𝑖=1

�𝑡𝑡𝑖𝑖 (𝑡𝑡𝑖𝑖− 𝑠𝑠_Γ₍_𝑞𝑞)₎𝑞𝑞−1��𝜔𝜔(𝑠𝑠)𝑓𝑓�𝑠𝑠,𝑦𝑦(𝑠𝑠),𝑧𝑧(𝑠𝑠)��𝑑𝑑𝑠𝑠

+��𝐺𝐺1′𝑡𝑡(𝑡𝑡,𝑡𝑡𝑖𝑖)� 𝑛𝑛

𝑖𝑖=1

�𝑡𝑡𝑖𝑖 (_Γ𝑡𝑡𝑖𝑖₍− 𝑠𝑠_{𝑞𝑞 −})𝑞𝑞−₁₎2��𝜔𝜔(𝑠𝑠)𝑓𝑓�𝑠𝑠,𝑦𝑦(𝑠𝑠),𝑧𝑧(𝑠𝑠)��𝑑𝑑𝑠𝑠

+��𝐺𝐺1′′𝑠𝑠𝑡𝑡(𝑡𝑡,𝑡𝑡𝑖𝑖)��𝐼𝐼𝑖𝑖�𝑦𝑦(𝑡𝑡𝑖𝑖)�� 𝑛𝑛

𝑖𝑖=1

≤ ��𝐺𝐺1′′𝑠𝑠𝑡𝑡(𝑡𝑡,𝑡𝑡𝑖𝑖)� 𝑛𝑛+1

𝑖𝑖=1

�𝑡𝑡𝑖𝑖 (𝑡𝑡𝑖𝑖− 𝑠𝑠_Γ₍_𝑞𝑞)₎𝑞𝑞−1[𝛾𝛾(𝜃𝜃′_(|_𝑦𝑦_{| + |}_𝑧𝑧_{|) +}_𝑀𝑀_)]_{𝑑𝑑𝑠𝑠} 𝑡𝑡𝑖𝑖−1

𝑖𝑖=1

�𝑡𝑡𝑖𝑖 (_Γ𝑡𝑡𝑖𝑖₍− 𝑠𝑠_{𝑞𝑞 −})𝑞𝑞−₁₎2[𝛾𝛾(𝜃𝜃′_(|_𝑦𝑦_{| + |}_𝑧𝑧_{|) +}_𝑀𝑀_)]_{𝑑𝑑𝑠𝑠} 𝑡𝑡𝑖𝑖−1

+��𝐺𝐺1′′𝑠𝑠𝑡𝑡(𝑡𝑡,𝑡𝑡𝑖𝑖)�(𝜃𝜃1′|𝑦𝑦| +𝑀𝑀𝑘𝑘) 𝑛𝑛

𝑖𝑖=1

(𝜃𝜃2′|𝑦𝑦| +𝑀𝑀�𝑘𝑘)

≤(_Γ𝑛𝑛₍_𝑞𝑞+ 1)_{+ 1)}𝑐𝑐��2𝜃𝜃′_��_|_𝑦𝑦_|_�

𝑃𝑃𝐶𝐶1�+𝑀𝑀�+_Γ𝑛𝑛𝑐𝑐₍_𝑞𝑞₎�(2𝜃𝜃′(�|𝑦𝑦|�𝑃𝑃𝐶𝐶1) +𝑀𝑀)��+𝑛𝑛�𝜃𝜃1′��|𝑦𝑦|�𝑃𝑃𝐶𝐶1�+𝑀𝑀𝑘𝑘�

+𝑛𝑛�𝜃𝜃2′��|𝑦𝑦|�𝑃𝑃𝐶𝐶1�+𝑀𝑀�𝑘𝑘�

≤ 𝛿𝛿2′‖𝑦𝑦‖𝑃𝑃𝐶𝐶1+𝑀𝑀(2) (4.7) Where 𝛿𝛿₂′ is defined by (4.3) and 𝑀𝑀(2)is defined by

𝑀𝑀(2)₌𝑀𝑀(𝑛𝑛+ 1)𝑐𝑐 Γ(𝑞𝑞+ 1) +

𝑀𝑀𝑛𝑛𝑐𝑐

Γ(𝑞𝑞) +𝑛𝑛𝑀𝑀𝑘𝑘+𝑛𝑛𝑀𝑀�𝑘𝑘

It follows from (4.6) and (4.7) that

‖𝑇𝑇𝑦𝑦‖𝑃𝑃𝐶𝐶1 ≤ 𝛿𝛿′‖𝑇𝑇𝑦𝑦‖_{𝑃𝑃𝐶𝐶}1+𝑀𝑀′,∀𝑦𝑦 ∈ 𝑃𝑃𝐶𝐶1[𝐽𝐽,𝑅𝑅] where

𝛿𝛿′_{= max{}_𝛿𝛿

1′,𝛿𝛿2′} < 1, 𝑀𝑀′ = max�𝑀𝑀(1),𝑀𝑀(2)� .

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Consequently, Lemma 3.3 implies that 𝑇𝑇has a fixed point in 𝐵𝐵_𝑜𝑜, and the proof is complete.

Remark 4.1: Condition (4.1) is certainly satisfied if |𝑓𝑓(𝑡𝑡,𝑦𝑦,𝑧𝑧)|

|𝑦𝑦|+|𝑧𝑧| →0 uniformly in 𝑡𝑡 ∈ 𝐽𝐽as |𝑦𝑦| + |𝑧𝑧|→+∞, |𝐼𝐼𝑘𝑘(𝑦𝑦)|

|𝑦𝑦| →

0 as |𝑦𝑦|→+∞ ,|𝐽𝐽𝑘𝑘(𝑦𝑦)|

|𝑦𝑦| →0 as |𝑦𝑦|→+∞ , ( 𝑘𝑘= 1,2 …𝑛𝑛) .

5. EXAMPLE

We consider the following boundary value problem:

⎩ ⎪ ⎪ ⎨ ⎪ ⎪

⎧ 𝐷𝐷₀₊32 _𝑒𝑒₍_𝑡𝑡_{) =}_𝑏𝑏

0𝑡𝑡

1

6 �𝑏𝑏₁5�𝑏𝑏₂𝑡𝑡 − 𝑒𝑒+𝑒𝑒′− 1

20𝑒𝑒′− 𝑏𝑏3𝐼𝐼𝑛𝑛(1 +𝑒𝑒2)�, 𝑡𝑡 ∈ 𝐽𝐽, 𝑡𝑡 ≠ 1 2 ,

𝐶𝐶

∆𝑒𝑒|_𝑡𝑡

1=1₂=

1 10 𝑒𝑒 �

1

2�, ∆𝑒𝑒′|𝑡𝑡1=1₂ =

1 6 𝑒𝑒 �

1

2�,

𝑒𝑒(0) =𝑒𝑒(1) =�₂₅𝑡𝑡2

1

0

𝑒𝑒(𝑡𝑡)𝑑𝑑,

(5.1)

Here 𝑞𝑞=3

2,𝑛𝑛= 1,𝑏𝑏0,𝑏𝑏1,𝑏𝑏2,𝑏𝑏3 are positive real numbers.

This problem has at least one solution in

𝑃𝑃𝐶𝐶1_[_𝐽𝐽_,_𝑅𝑅_]_⋂_𝐶𝐶_{^2 [}_𝐽𝐽′_,_𝑅𝑅_]_Where_𝐽𝐽′₌_�_0,1 2� ∪ �

1 2, 1�.

Proof: It follows from (5.1) that

𝜔𝜔(𝑡𝑡) =𝑏𝑏0𝑡𝑡

1 6,

𝑓𝑓(𝑡𝑡,𝑒𝑒,𝑒𝑒′_{) =}_𝑏𝑏

15�𝑏𝑏2𝑡𝑡 − 𝑒𝑒+𝑒𝑒′−₂₀1 𝑒𝑒′− 𝑏𝑏3𝐼𝐼𝑛𝑛(1 +𝑒𝑒2) (5.2)

𝑡𝑡1=1_{2 ,} 𝐼𝐼1(𝑒𝑒) =₁₀1 𝑒𝑒, 𝐽𝐽1(𝑒𝑒) =1₆𝑒𝑒 (5.3)

From the definition of 𝜔𝜔 ,𝑓𝑓,𝐼𝐼₁𝑎𝑎𝑛𝑛𝑑𝑑𝐽𝐽₁ , it is easy to see that (𝐻𝐻₁)−(𝐻𝐻₄) hold. So

𝜃𝜃 ≤₂₀1, 𝜃𝜃1≤₁₀1 ,𝜃𝜃2≤1₆, 𝜇𝜇=∫ 𝑡𝑡

2

25 1

0 𝑑𝑑𝑡𝑡. We have

𝐺𝐺1(𝑡𝑡,𝑠𝑠) =− �𝑠𝑠_𝑡𝑡₍₁(1− 𝑡𝑡_{− 𝑠𝑠}_{), 0}), 0_{≤ 𝑡𝑡 ≤ 𝑠𝑠 ≤}≤ 𝑠𝑠 ≤ 𝑡𝑡 ≤1,_1,

𝐺𝐺𝑡𝑡′(𝑡𝑡,𝑠𝑠) =� 𝑠𝑠₍_{𝑠𝑠 −} , 0_{1), 0}≤ 𝑠𝑠 ≤ 𝑡𝑡 ≤_{≤ 𝑡𝑡 ≤ 𝑠𝑠 ≤}1,_1,

𝐺𝐺1′𝑠𝑠(𝑡𝑡,𝑠𝑠) =�(_𝑡𝑡𝑡𝑡 −_,01) , 0_{≤ 𝑡𝑡 ≤ 𝑠𝑠 ≤}≤ 𝑠𝑠 ≤ 𝑡𝑡 ≤_1,1,

𝐺𝐺1′′𝑠𝑠𝑡𝑡(𝑡𝑡,𝑠𝑠) = 1

𝜇𝜇 ≤_25,1 𝐻𝐻1(𝑡𝑡,𝑠𝑠)≤25_96,𝐻𝐻2(𝑡𝑡,𝑠𝑠)≤25_24,

𝛿𝛿1≤0.25, 𝛿𝛿2≤0.4.

Thus, our conclusion follows from theorem 4.1.

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