A Coupled System of Nonlinear Fractional Differential Equations with Multipoint Fractional Boundary Conditions on an Unbounded Domain

Abstract

This paper investigates the existence of solutions for a coupled system of nonlinear fractional differential equations with m-point fractional boundary conditions on an unbounded domain. Some standard fixed point theorems are applied to obtain the main results. The paper concludes with two illustrative examples.

1. Introduction

In the last few decades, the subject of fractional calculus has gained considerable popularity and importance as it finds its applications in numerous fields of science and engineering. Some of the areas of recent applications of fractional models include fluid mechanics, solute transport or dynamical processes in porous media, material viscoelastic theory, dynamics of earthquakes, control theory of dynamical systems, and biomathematics. In the afore-mentioned areas, there are phenomena with estrange kinetics involving microscopic complex dynamical behaviour that cannot be characterized by classical derivative models. It has been learnt through experimentation that most of the processes associated with complex systems have nonlocal dynamics possessing long-memory in time, and the integral and derivative operators of fractional order do have some of these characteristics. Thus, due to the modeling capabilities of fractional integrals and derivatives for complex phenomena, the fractional modelling has emerged as a powerful tool and has accounted for the rapid development of the theory of fractional differential equations. Fractional differential equations also serve as an excellent tool for the description of hereditary properties of various materials and processes [1]. The presence of memory term in such models not only takes into account the history of the process involved but also carries its impact to present and future development of the process. For more details and applications, we refer the reader to the books [2–6]. For some recent work on the topic, see [7–27] and references therein.

The study of coupled systems involving fractional differential equations is also important as such systems occur in various problems of applied nature. For some recent results on systems of fractional differential equations, see [28–35].

Much of the work on fractional differential equations has been considered on finite domain and there are few papers dealing with infinite domain [36–43]. In this paper, we discuss the existence and uniqueness of the solutions of a coupled system of nonlinear fractional differential equations with 𝑚-point boundary conditions on an unbounded domain. Precisely, we consider the following problem:𝐷^𝑝𝑢(𝑡)((𝑡𝐷+𝑓𝑡,𝑣))=0,2<𝑝<3,_𝑞𝑣(𝑡)((𝑡𝑢(0)+𝑔𝑡,𝑢))=0,2<𝑞<3,=𝑢^′(0)=0,𝐷^𝑝−1𝑢(+∞)=^𝑚−2_𝑖=1𝛽_𝑖𝑢𝜉_𝑖,𝑣(0)=𝑣^′(0)=0,𝐷^𝑞−1𝑣(+∞)=^𝑚−2_𝑖=1𝛾_𝑖𝑣𝜉_𝑖, (1.1) where 𝑡∈𝐽=[0,+∞), 𝑓,𝑔∈𝐶(𝐽×ℝ,ℝ), 0<𝜉₁<𝜉₂<⋯<𝜉_𝑚−2<+∞, 𝐷^𝑝 and 𝐷^𝑞 denote Riemann-Liouville fractional derivatives of order 𝑝 and 𝑞, respectively, and 𝛽_𝑖>0,and𝛾_𝑖>0 are such that 0<∑^𝑚−2_𝑖=1𝛽_𝑖𝜉_𝑖^𝑝−1<Γ(𝑝) and 0<∑^𝑚−2_𝑖=1𝛾_𝑖𝜉_𝑖^𝑞−1<Γ(𝑞).

2. Preliminaries

For the convenience of the readers, in this section we first present some useful definitions and lemmas.

Definition 2.1 (see [5]). The Riemann-Liouville fractional derivative of order 𝛿 for a continuous function 𝑓 is defined by 𝐷^𝛿𝑓(𝑡)=1Γ()𝑑𝑛−𝛿𝑑𝑡^𝑛𝑡₀(𝑡−𝑠)^{𝑛−𝛿−1}𝑓(𝑠)𝑑𝑠,𝑛=[𝛿]+1, (2.1)
provided that the right-hand side is pointwise defined on (0,∞).

Definition 2.2 (see [5]). The Riemann-Liouville fractional integral of order 𝛿 for a function 𝑓 is defined as𝐼^𝛿𝑓(𝑡)=1Γ(𝛿)^𝑡₀(𝑡−𝑠)^𝛿−1𝑓(𝑠)𝑑𝑠,𝛿>0, (2.2)provided that such integral exists.

For the forthcoming analysis, we define the spaces𝑋=𝑢∈𝐶[0,+∞)∶sup_𝑡∈𝐽|𝑢(𝑡)|1+𝑡^𝑝−1,[)<+∞𝑌=𝑣∈𝐶0,+∞∶sup_𝑡∈𝐽|𝑣(𝑡)|1+𝑡^𝑞−1<+∞ (2.3)

equipped with the norms‖𝑢‖_𝑋=sup_𝑡∈𝐽|𝑢(𝑡)|1+𝑡^𝑝−1,‖𝑣‖_𝑌=sup_𝑡∈𝐽|𝑣(𝑡)|1+𝑡^𝑞−1. (2.4)(2.5)

Obviously 𝑋 and 𝑌 are Banach spaces.

Lemma 2.3 (see [38]). Let ℎ∈𝐶([0,+∞)). For 2<𝛼<3, the fractional boundary value problem 𝐷^𝛼𝑢(𝑡)(𝑡)𝑢(0)+ℎ=0,=𝑢^′(0)=0,𝐷^𝛼−1𝑢(+∞)=^𝑚−2_𝑖=1𝛽_𝑖𝑢𝜉_𝑖 (2.6)

has a unique solution𝑢(𝑡)=₀^+∞𝐺(𝑡,𝑠)ℎ(𝑠)𝑑𝑠, (2.7)

where𝐺(𝑡,𝑠)=𝐺^∗(𝑡,𝑠)+𝐺^∗∗(𝑡,𝑠), (2.8)

with𝐺^∗()=1𝑡,𝑠Γ(𝛼)𝑡^𝛼−1−(𝑡−𝑠)^𝛼−1𝑡,0≤𝑠≤𝑡<+∞,_𝛼−1𝐺,0≤𝑡≤𝑠<+∞._∗∗()=∑𝑡,𝑠^{𝑚−2𝑖=1}𝛽_𝑖𝑡^𝛼−1Γ(𝛼)−∑^𝑚−2_𝑖=1𝛽_𝑖𝜉_𝑖^𝛼−1𝐺^∗𝜉_𝑖,𝑠. (2.9)(2.10)

Lemma 2.4 (see [38]). For (𝑠,𝑡)∈[0,+∞)×[0,+∞), 𝐺(𝑡,𝑠)/1+𝑡^𝛼−1≤𝐿₁, where 𝐿₁=1Γ(𝛼)+∑^𝑚−2_𝑖=1𝛽_𝑖𝜉^𝛼−1_𝑚−2Γ(𝛼)Γ(𝛼)−∑^𝑚−2_𝑖=1𝛽_𝑖𝜉_𝑖^𝛼−1. (2.11)

3. Main Results

This section is devoted to some existence and uniqueness results for problem (1.1).

Define the space𝑍={(𝑢,𝑣)∣𝑢∈𝑋,𝑣∈𝑌} (3.1)

equipped with the norm‖(𝑢,𝑣)‖_𝑍=max‖𝑢‖_𝑋,‖𝑣‖_𝑌. (3.2)

Clearly 𝑍 is a Banach space.

Let an operator 𝑄∶𝑍→𝑍 be defined by𝑄(𝑢,𝑣)=𝑄₁(𝑣),𝑄₂(𝑢)=₀^+∞𝐺₁(𝑡,𝑠)𝑓(𝑡,𝑣(𝑠))𝑑𝑠,₀^+∞𝐺₂(𝑡,𝑠)𝑔(𝑡,𝑢(𝑠))𝑑𝑠, (3.3)

where 𝐺₁(𝑡,𝑠)=𝐺₁₁(𝑡,𝑠)+𝐺₁₂(𝑡,𝑠),𝐺₂(𝑡,𝑠)=𝐺₂₁(𝑡,𝑠)+𝐺₂₂(𝑡,𝑠), with𝐺₁₁()=1𝑡,𝑠Γ𝑝𝑡^𝑝−1−(𝑡−𝑠)^𝑝−1𝑡,0≤𝑠≤𝑡<+∞,_𝑝−1𝐺,0≤𝑡≤𝑠<+∞,₁₂()=∑𝑡,𝑠^{𝑚−2𝑖=1}𝛽_𝑖𝑡^𝑝−1Γ𝑝−∑^𝑚−2_𝑖=1𝛽_𝑖𝜉_𝑖^𝑝−1𝐺₁₁𝜉_𝑖,𝐺,𝑠₂₁()=1𝑡,𝑠Γ𝑞𝑡^𝑞−1−(𝑡−𝑠)^𝑞−1𝑡,0≤𝑠≤𝑡<+∞,_𝑞−1𝐺,0≤𝑡≤𝑠<+∞,₂₂()=∑𝑡,𝑠^{𝑚−2𝑖=1}𝛾_𝑖𝑡^𝑞−1Γ𝑞−∑^𝑚−2_𝑖=1𝛾_𝑖𝜉_𝑖^𝑞−1𝐺₂₁𝜉_𝑖,𝑠. (3.4)

Observe that the problem (1.1) has a solution if and only if the operator 𝑄 defined by (3.3) has a fixed point.

Lemma 3.1. For (𝑠,𝑡)∈[0,+∞)×[0,+∞), one has 𝐺₁(𝑡,𝑠)1+𝑡^𝑝−1𝐺≤𝐿,²(𝑡,𝑠)1+𝑡^𝑞−1≤𝐿, (3.5) where 1𝐿=maxΓ𝑝+∑^𝑚−2_𝑖=1𝛽_𝑖𝜉^𝑝−1_𝑚−2Γ𝑝Γ𝑝−∑^𝑚−2_𝑖=1𝛽_𝑖𝜉_𝑖^𝑝−1,1Γ𝑞+∑^𝑚−2_𝑖=1𝛾_𝑖𝜉^𝑞−1_𝑚−2Γ𝑞Γ𝑞−∑^𝑚−2_𝑖=1𝛾_𝑖𝜉_𝑖^𝑞−1. (3.6)

Theorem 3.2. Assume that
(𝐻₁) there exist nonnegative functions 𝑎(𝑡),𝑏(𝑡)∈𝐶[0,+∞) such that 𝑓(𝑡,𝑥)≤𝑎(𝑡)|𝑥|+𝑏(𝑡),𝑡∈[0,+∞_0+∞),1+𝑡^𝑞−1𝑎(𝑡)1𝑑𝑡<𝐿,₀^+∞𝑏(𝑡)𝑑𝑡<+∞; (3.7)(𝐻₂) there exist nonnegative functions 𝑐(𝑡),𝑑(𝑡)∈𝐶[0,+∞) such that 𝑔𝑡,𝑦≤𝑐(𝑡)𝑦+𝑑(𝑡),𝑡∈[0,+∞_0+∞),1+𝑡^𝑝−1𝑐(𝑡)1𝑑𝑡<𝐿,₀^+∞𝑑(𝑡)𝑑𝑡<+∞. (3.8)
Then the system (1.1) has a solution.

Proof. Let us take 𝑅>max𝐿∫₀^+∞𝑏(𝑠)𝑑𝑠1−𝐿∫₀^+∞1+𝑠^𝑞−1𝑎(𝑠),𝑑𝑠𝐿∫₀^+∞𝑑(𝑠)𝑑𝑠1−𝐿∫₀^+∞1+𝑠^𝑝−1𝑐(𝑠)𝑑𝑠, (3.9) and define 𝐵_𝑅=(𝑢,𝑣)∈𝑍∣‖(𝑢,𝑣)‖_𝑍≤𝑅. (3.10) Obviously, 𝐵_𝑅 is a bounded closed and convex set of 𝑍.
As a first step, we show that the operator 𝑄 is 𝐵_𝑅→𝐵_𝑅.
For any (𝑢,𝑣)∈𝐵_𝑅, we have𝑄₁𝑣_𝑋=sup_𝑡∈𝐽11+𝑡^𝑝−1₀^+∞𝐺₁(𝑡,𝑠)𝑓(𝑠,𝑣(s))𝑑𝑠≤sup_𝑡∈𝐽11+𝑡^𝑝−1₀^+∞𝐺₁(𝑡,𝑠)(𝑎(𝑠)|𝑣(𝑠)|+𝑏(𝑠))𝑑𝑠≤𝐿₀^+∞1+𝑠^𝑞−1𝑎(𝑠)𝑑𝑠‖𝑣‖_𝑌+𝐿₀^+∞𝑏(𝑠)<𝑑𝑠𝐿∫₀^+∞𝑏(𝑠)𝑑𝑠1−𝐿∫₀^+∞1+𝑡^𝑞−1𝑎(𝑠)𝑑𝑠<𝑅. (3.11) Similarly, we can get 𝑄₂𝑢_𝑌=sup_𝑡∈𝐽11+𝑡^𝑞−1₀^+∞𝐺₂(𝑡,𝑠)𝑔(𝑠,𝑢(𝑠))𝑑𝑠≤𝐿₀^+∞1+𝑠^𝑝−1𝑐(𝑠)𝑑𝑠‖𝑢‖_𝑋+𝐿₀^+∞𝑑(𝑠)<𝑑𝑠𝐿∫₀^+∞𝑑(𝑠)𝑑𝑠1−𝐿∫₀^+∞1+𝑠^𝑝−1𝑐(𝑠)𝑑𝑠<𝑅. (3.12) That is, ‖𝑄(𝑢,𝑣)‖_𝑍≤𝑅. Thus, 𝑄𝐵_𝑅⊂𝐵_𝑅.
Next, we show that 𝑄 is completely continuous. By continuity of 𝑓,𝑔,𝐺₁, and 𝐺₂, it follows that 𝑄 is continuous. On the other hand, by a similar process used in [38], we can easily prove that the operators 𝑄₁and𝑄₂ are equicontinuous. Therefore it follows that 𝑄𝐵_𝑅 is an equicontinuous set. Also, it is uniformly bounded as 𝑄𝐵_𝑅⊂𝐵_𝑅. Thus, we conclude that 𝑄 is a completely continuous operator. Hence, by Schauder fixed point theorem, there exists a solution of (1.1). This completes the proof.

Theorem 3.3. Assume that
(𝐻₃) there exist 0<𝜌₁<1 and nonnegative functions 𝑎₁(𝑡),𝑏₁(𝑡)∈𝐶[0,+∞) such that 𝑓(𝑡,𝑥)≤𝑎₁(𝑡)|𝑥|^𝜌1+𝑏₁(𝑡),𝑡∈[0,+∞_0+∞),1+𝑡^𝑞−1𝑎₁(𝑡)𝑑𝑡<+∞,₀^+∞𝑏₁(𝑡)𝑑𝑡<+∞. (3.13)(𝐻₄) there exist 0<𝜌₂<1 and nonnegative functions 𝑐₁(𝑡),𝑑₁(𝑡)∈𝐶[0,+∞) such that 𝑔𝑡,𝑦≤𝑐₁(𝑡)𝑦^𝜌2+𝑑₁(𝑡),𝑡∈[0,+∞_0+∞),1+𝑡^𝑝−1𝑐₁(𝑡)𝑑𝑡<+∞,₀^+∞𝑑₁(𝑡)𝑑𝑡<+∞. (3.14)
Then the system (1.1) has a solution.

Proof. In this case, we take 𝑅>max2𝐿₀^+∞𝑏₁(𝑠)𝑑𝑠,2𝐿₀^+∞1+𝑠^𝑞−1𝑎₁(𝑠)𝑑𝑠^{1/(1−𝜌1)},2𝐿₀^+∞𝑑₁(𝑠)𝑑𝑠,2𝐿₀^+∞1+𝑠^𝑝−1𝑐₁(𝑠)𝑑𝑠^{1/(1−𝜌2)}. (3.15) The rest of the proof is similar to that of Theorem 3.2. So we omit it.

Remark 3.4. By taking 𝜌₁,𝜌₂>1 (instead of 0<𝜌₁<1, 0<𝜌₂<1) in (𝐻₃) and (𝐻₄), one can show that (1.1) has a solution.

Theorem 3.5. Assume that
(𝐻₅) the functions 𝑓 and 𝑔 satisfy Lipschitz condition; that is, there exist nonnegative functions 𝐾₁(𝑡) and 𝐾₂(𝑡) such that𝑓(𝑡,𝑥)−𝑓𝑡,𝑦≤𝐾₁(𝑡[),𝑔())𝑥−𝑦,𝑡∈0,+∞𝑡,𝑥−𝑔𝑡,𝑦≤𝐾₂(𝑡)𝑥−𝑦,𝑡∈[0,+∞). (3.16) Then the problem (1.1) has a unique solution if 𝜇=𝐿₀^+∞𝐾₁(𝑠)1+𝑠^𝑞−1𝑑𝑠<1,𝜏=𝐿₀^+∞𝐾₂(𝑠)1+𝑠^𝑝−1𝑑𝑠<1. (3.17)

Proof. For any (𝑢₁,𝑣₁),(𝑢₂,𝑣₂)∈𝑍, we have 𝑄₁𝑣₂−𝑄₁𝑣_1𝑋=sup_𝑡∈𝐽11+𝑡^𝑝−1₀^+∞𝐺₁(𝑡,𝑠)𝑓𝑠,𝑣₂(𝑠)−𝑓𝑠,𝑣₁(𝑠)𝑑𝑠≤sup_𝑡∈𝐽0^+∞𝐺₁(𝑡,𝑠)1+𝑡^𝑝−1𝐾₁(𝑠)𝑣₂−𝑣₁(𝑠)𝑑𝑠≤𝐿₀^+∞𝐾₁(𝑠)1+𝑠^𝑞−1𝑑𝑠𝑣₂−𝑣_1𝑌=𝜇𝑣₂−𝑣_1𝑌. (3.18)
Similarly, it can be shown that𝑄₂𝑢₂−𝑄₂𝑢_1𝑌=sup_𝑡∈𝐽11+𝑡^𝑞−1₀^+∞𝐺₁(𝑡,𝑠)𝑔𝑠,𝑢₂(𝑠)−𝑓𝑠,𝑢₂(𝑠)𝑑𝑠≤𝐿₀^+∞𝐾₂(𝑠)1+𝑠^𝑝−1𝑑𝑠𝑢₂−𝑢_1𝑋=𝜏𝑢₂−𝑢_1𝑋. (3.19)
Thus, we get𝑄𝑢₂,𝑣₂−𝑄𝑢₁,𝑣_1𝑍≤max𝜇,𝜏𝑢₂,𝑣₂−𝑢₁,𝑣_1𝑍. (3.20) Obviously, 𝑄 is a contraction. Thus, the conclusion of the theorem follows by the contraction mapping principle. This completes the proof.

4. Example

Example 4.1. Consider the following multipoint boundary value problem on an unbounded domain: 𝐷^9/4𝑢(𝑡)+sin𝑡ln(1+|𝑣(𝑡)|)1+𝑡^7/4(2+𝑡)²+(1+cos2𝑡)𝑒^−𝑡𝐷=0,_11/4𝑣(𝑡)+𝑒^−5𝑡sin|𝑢(𝑡)|31+𝑡^5/4(1+𝑡)²+4(𝑡+4)²𝑢(0)=0,=𝑢^′(0)=0,𝐷^5/4𝑢()=2+∞5𝑢14+1𝑢(1),𝑣(0)10=𝑣^′(0)=0,𝐷^7/4𝑣()=3+∞𝑢1104+15𝑢(1). (4.1)

Here 𝑡∈[0,+∞), 𝑝=9/4, 𝑞=11/4, 𝜉₁=1/4, 𝜉₂=1, 𝛽₁=2/5, 𝛽₂=1/10, 𝛾₁=3/10, and 𝛾₂=1/5. One has𝑓((𝑡=(|𝑣(𝑡𝑡,𝑣))sin𝑡ln1+)|)1+𝑡^7/4(2+𝑡)²+(1+cos2𝑡)𝑒^−𝑡((𝑡=𝑒,𝑔𝑡,𝑢))^−5𝑡sin|𝑢(𝑡)|31+𝑡^5/4(1+𝑡)²+4(𝑡+4)². (4.2)

For 𝑎(𝑡)=1/(1+𝑡^7/4)(2+𝑡)², 𝑏(𝑡)=2𝑒^−𝑡, 𝑐(𝑡)=1/3(1+𝑡^5/4)(1+𝑡)², 𝑑(𝑡)=4/(𝑡+4)², by direct calculation we find that1𝐿=maxΓ𝑝+∑^𝑚−2_𝑖=1𝛽_𝑖𝜉^𝑝−1_𝑚−2Γ𝑝Γ𝑝−∑^𝑚−2_𝑖=1𝛽_𝑖𝜉_𝑖^𝑝−1,1Γ𝑞+∑^𝑚−2_𝑖=1𝛾_𝑖𝜉^𝑞−1_𝑚−2Γ𝑞Γ𝑞−∑^𝑚−2_𝑖=1𝛾_𝑖𝜉_𝑖^𝑞−11=maxΓ()+()+()9/42/51/10Γ(9/4)Γ(9/4)−2/5(1/4)^5/4−(),11/10Γ()+()+()11/43/101/5Γ(11/4)Γ(11/4)−3/10(1/4)^7/4−()𝑓((𝑡𝑥|(𝑡),𝑔(𝑡𝑦(𝑡)[),1/5=1.341213,𝑡,𝑥)≤𝑎)|+𝑏𝑡,𝑦≤𝑐)+𝑑,𝑡∈0,+∞_0+∞1+𝑡^𝑞−1𝑎(𝑡)1𝑑𝑡=2<1𝐿=0.745594,₀^+∞𝑏(𝑡)𝑑𝑡=2<+∞,_0+∞1+𝑡^𝑝−1𝑐(𝑡)1𝑑𝑡=3<1𝐿=0.745594,₀^+∞𝑑(𝑡)𝑑𝑡=1<+∞. (4.3)

Thus all conditions of Theorem 3.2 are satisfied. Therefore, by Theorem 3.2, the couple system of nonlinear fractional differential (4.1) has at least one solution.

Example 4.2. Consider the following problem on an unbounded domain: 𝐷^𝑝𝑢(𝑡)+𝑀₁(𝑡)sin𝑣(𝑡)+𝑁₁(𝑡)𝐷=0,_𝑞𝑣(𝑡)+𝑀₂(𝑡)1+𝑢²(𝑡)+𝑁₂(𝑡)𝑢(0)=0,=𝑢^′(0)=0,𝐷^𝑝−1𝑢()=2+∞5𝑢14+1𝑢(1),𝑣(0)10=𝑣^′(0)=0,𝐷^𝑞−1𝑣()=3+∞𝑢1104+15𝑢(1). (4.4) Here 𝑡∈[0,+∞), 2<𝑝, 𝑞<3, 𝜉₁=1/4, 𝜉₂=1, 𝛽₁=2/5, 𝛽₂=1/10, 𝛾₁=3/10, and 𝛾₂=1/5, 𝑀₁(𝑡),𝑀₂(𝑡),𝑁₁(𝑡),𝑁₂(𝑡)∈𝐶([0,+∞),ℝ).
With𝑓(𝑡,𝑣(𝑡))=𝑀₁(𝑡)sin𝑣(𝑡)+𝑁₁(𝑡)((𝑡=𝑀,𝑔𝑡,𝑢))²(𝑡)1+𝑢²(𝑡)+𝑁₂(𝑡), (4.5) we have 𝑓(𝑡,𝑥)−𝑓𝑡,𝑦=𝑀₁(𝑡)sin𝑥−sin𝑦≤𝑀₁(𝑡[),𝑔()=𝑀)𝑥−𝑦,𝑡∈0,+∞𝑡,𝑥−𝑔𝑡,𝑦₂(𝑡1)1+𝑥²−11+𝑦²≤𝑀₂(𝑡)𝑥−𝑦,𝑡∈[0,+∞), (4.6)where 𝐾₁(𝑡)=|𝑀₁(𝑡)|, 𝐾₂(𝑡)=|𝑀₂(𝑡)|. So, the condition (𝐻₅) holds. Let us assume that 𝜇=𝐿₀^+∞𝑀₁(𝑠)1+𝑠^𝑞−1𝑑𝑠<1,𝜏=𝐿₀^+∞𝑀₂(𝑠)1+𝑠^𝑝−1𝑑𝑠<1. (4.7) For example, condition (4.7) holds if we take 9𝑝=4,𝑞=114,𝑀₁(𝑡)=11+𝑡^7/4(2+𝑡)²,𝑀₂(𝑡)=131+𝑡^5/4(1+𝑡)². (4.8) Thus all the conditions of Theorem 3.5 are satisfied. Therefore, by the conclusion of Theorem 3.5, the coupled system (4.4) has a unique solution.

5. Conclusion

We have shown the existence and uniqueness of solutions for a coupled system of nonlinear fractional differential equations with multipoint fractional boundary conditions on a semi-infinite domain. Our existence results are based on Schauder’s fixed point theorem, while the uniqueness result is obtained by applying Banach’s contraction mapping principle. The existence of solutions for (1.1) has been addressed for different kinds of growth conditions. Our approach is simple and can easily be applied to a variety of problems. This has been demonstrated by solving two examples.

Acknowledgment

The research of B. Ahmad was supported by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.