Positive Solutions of an Initial Value Problem for Nonlinear Fractional Differential Equations

Abstract

We investigate the existence and multiplicity of positive solutions for the nonlinear fractional differential equation initial value problem 𝐷^𝛼₀₊𝑢(𝑡)+𝐷^𝛽₀₊𝑢(𝑡)=𝑓(𝑡,𝑢(𝑡)),𝑢(0)=0,0<𝑡<1, where 0<𝛽<𝛼<1,𝐷^𝛼₀₊ is the standard Riemann-Liouville differentiation and 𝑓∶[0,1]×[0,∞)→[0,∞) is continuous. By using some fixed-point results on cones, some existence and multiplicity results of positive solutions are obtained.

1. Introduction

Fractional differential equations have been subjected to an intense debate during the last few years (see, e.g., [1–5] and the references therein). This trend is due to the intensive development of the theory of fractional calculus itself and by the applications of such constructions in various sciences such as physics, mechanics, chemistry, and engineering [5–15]. The fractional differential equations started to be used extensively in studying the dynamical systems possessing memory effect. Comprehensive treatment of the fractional equations techniques such as Laplace and Fourier transform method, method of Green function, Mellin transform, and some numerical techniques are given in [5, 7, 9] and the references therein. In classical approach, linear initial fractional differential equations are solved by special functions [9, 16]. In some papers, for nonlinear problems, techniques of functional analysis such as fixed point theory, the Banach contraction principle, and Leray-Schauder theory are applied for solving such kind of the problems (see, e.g., [17–19] and the references therein). The existence of nonlinear fractional differential equations of one time fractional derivative is considered in [6, 7, 9, 20]. Also, the existence and multiplicity of positive solutions to nonlinear Dirichlet problem𝐷^𝛼₀₊𝑢(𝑡)+𝑓(𝑡,𝑢(𝑡))=0,0<𝑡<1,𝑢(0)=𝑢(1)=0,1<𝛼≤2,𝛼∈ℝ, (1.1) where 𝑓∶[0,1]×[0,∞)→[0,∞) is continuous and 𝐷^𝛼₀₊ is the Riemann-Liouville differentiation, have been reviewed by some authors (see e.g., [18–21] and the references therein).

In this paper, by using some fixed-point results, we investigate the existence and multiplicity of positive solutions for the nonlinear fractional differential equation initial value problem𝐷^𝛼₀₊𝑢(𝑡)+𝐷^𝛽₀₊𝑢(𝑡)=𝑓(𝑡,𝑢(𝑡)),𝑢(0)=0,0<𝑡<1, (1.2) where 0<𝛽<𝛼<1, 𝐷^𝛼₀₊ is the standard Riemann-Liouville differentiation, and 𝑓∶[0,1]×[0,∞)→[0,∞) is continuous. Now, we present some necessary notions. The Riemann-Liouville fractional integral of order 𝛼>0 is defined by 𝐼^𝛼𝑓(𝑡)∶=(1/Γ(𝛼))∫^𝑡₀(𝑡−𝜏)^𝛼−1𝑓(𝜏)𝑑𝜏 [20]. Also, the Riemann-Liouville fractional derivative of order 𝛼>0 is defined by 𝐷^𝛼𝑓(𝑡)∶=(1/Γ(𝑛−𝛼))(𝑑/𝑑𝑡)^𝑛∫^𝑡₀(𝑡−𝜏)^{𝑛−𝛼−1}𝑓(𝜏)𝑑𝜏, where 𝑛=[𝛼]+1 and the right side is pointwise defined on (0,∞) ([20]). The formula of Laplace transform for the Riemann-Liouville derivative is defined by𝐿𝐷^𝛼𝑓(𝑡);𝑠=𝑠^𝛼𝑓(𝑠)^𝑚−1_𝑘=0𝐷^𝑘𝐼^𝑚−𝛼𝑓0⁺𝑠^{𝑚−𝑘−1} (1.3) when the limiting values 𝑓^(𝑘)(0⁺) are finite and 𝑚−1<𝛼<𝑚. This formula simplifies to 𝐿{𝐷^𝛼𝑓(𝑡);𝑠}=𝑠^𝛼𝑓(𝑠) [21]. Also, two-parametric Mittag-Leffler function is defined by 𝐸_(𝛼,𝛽)(𝑧)=∑^∞_𝑘=0𝑧^𝑘/Γ(𝑘𝛼+𝛽) for 𝛼>0 and 𝛽>0 [21]. Analytic properties and asymptotical expansion of this function are given in [9]. For example, if 𝛼<2,𝜋𝛼/2<𝜇<min(𝜋,𝜋𝛼), 𝛽∈ℝand 𝑐₃ is a real constant, then |𝐸_𝛼,𝛽(𝑧)|≤𝑐₃/(1+|𝑧|), whenever |𝑧|≥0 and 𝜇≤|arg𝑧|≤𝜋. Also, by using the formula for integration of the Mittag-Leffler function term by term, we have (see [9]) ^𝑧₀𝑡^𝛽−1𝐸_𝛼,𝛽𝜆𝑡^𝛼𝑑𝑡=𝑧^𝛽𝐸_𝛼,𝛽+1𝜆𝑡^𝛼. (∗)

Let 𝑃 be a cone in a Banach space 𝐸. The map 𝜃∶𝑃→[0,∞] is said to be a nonnegative continuous concave functional whenever 𝜃 is continuous and 𝜃(𝑡𝑥+(1−𝑡)𝑦)≥𝑡𝜃(𝑥)+(1−𝑡)𝜃(𝑦) for all 𝑥,𝑦∈𝑃 and 0≤𝑡≤1 [20]. We need the following fixed point theorems for obtaining our results.

Lemma 1.1 (see [22]). Let 𝐸 be a Banach space, 𝑃 a cone in 𝐸, and Ω₁,Ω₂ two bounded open balls of 𝐸 centered at the origin with Ω₁⊂Ω₂. Suppose that 𝐴∶𝑃∩(Ω₂⧵Ω₁)→𝑃 is a completely continuous operator such that either(i)‖𝐴𝑥‖≤‖𝑥‖,𝑥∈𝑃∩𝜕Ω₁ and ‖𝐴𝑥‖≥‖𝑥‖,𝑥∈𝑃∩𝜕Ω₂, or(ii)‖𝐴𝑥‖≥‖𝑥‖, 𝑥∈𝑃∩𝜕Ω₁ and ‖𝐴𝑥‖≤‖𝑥‖, 𝑥∈𝑃∩𝜕Ω₂holds. Then A has a fixed point in 𝑃∩(Ω₂⧵Ω₁).

Lemma 1.2 (see [23]). Let 𝑃 be a cone in a real Banach space 𝐸, 𝑐, 𝑏, and 𝑑 positive real numbers, 𝑃_𝑐={𝑥∈𝑃∶‖𝑥‖≤𝑐},𝜃 a nonnegative concave functional on 𝑃 such that 𝜃(𝑥)≤‖𝑥‖ for all 𝑥∈𝑃_𝑐 and 𝑃(𝜃,𝑏,𝑑)={𝑥∈𝑃∶𝑏≤𝜃(𝑥),‖𝑥‖≤𝑑}. (1.4) Suppose that 𝐴∶𝑃_𝑐→𝑃_𝑐 is completely continuous and there exist constants 0<𝑎<𝑏<𝑑≤𝑐 such that(𝑐₁){𝑥∈𝑃(𝜃,𝑏,𝑑)∶𝜃(𝑥)>𝑏}≠∅, and for some 𝑥∈𝑃(𝜃,𝑏,𝑑) we have 𝜃(𝐴𝑥)>𝑏,(𝑐₂)‖𝐴𝑥‖<𝑎 for all 𝑥 with ‖𝑥‖≤𝑎,(𝑐₃)𝜃(𝐴𝑥)>𝑏 for all 𝑥∈𝑃(𝜃,𝑏,𝑐) with ‖𝐴𝑥‖>𝑑.Then A has at least three fixed points 𝑥₁, 𝑥₂, and 𝑥₃ such that ‖𝑥₁‖<𝑎, 𝑏<𝜃(𝑥₂), 𝑎<‖𝑥₃‖ with 𝜃(𝑥₃)<𝑏.

Note that the condition (𝑐₁) implies (𝑐₃) whenever 𝑑=𝑐.

2. Main Results

As we know, there is an integral form of the solution for the following equation: 𝐷^𝑎₀₊𝑢(𝑡)+𝐷^𝛽₀₊𝑢(𝑡)=𝑓(𝑡,𝑢(𝑡)),𝑢(0)=0,0<𝑡<1, (2.1) Suppose that the functions 𝑢 and 𝑓 are continuous on [0,1]. Then 𝑢(𝑡)=∫^𝑡₀𝐺(𝑡−𝜏)𝑓(𝜏,𝑢(𝜏))𝑑𝜏 is a solution for (2.1), where 𝐺(𝑡)=𝑡^𝛼−1𝐸_{𝛼−𝛽,𝛼}(−𝑡^𝛼−𝛽) and 𝐸_𝛼,𝛽 is the two-parameter function of the Mittag-Leffler type (see [9]). Now, we give an equivalent solution for (2.1). In fact, if we apply the Laplace transform to (2.1), then by using a calculation and finding the inverse Laplace transform we get that 𝑢(𝑡)=𝑡^𝛼−1𝐸_{𝛼−𝛽,𝛼}(−𝑡^𝛼−𝛽)∗𝑓(𝑡,𝑢(𝑡)) is an equivalent solution for (2.1). In this way, note that𝐷^𝛼𝑢(𝑡)+𝐷^𝛽𝑢(𝑡)=𝐷^𝛼𝐺(𝑡)+𝐷^𝛽𝐺(𝑡)∗𝑓(𝑡,𝑢(𝑡)), (2.2) where 𝐺(𝑡)=𝑡^𝛼−1𝐸_{𝛼−𝛽,𝛼}(−𝑡^𝛼−𝛽). But, we have𝐷^𝛼𝐺(𝑡)+𝐷^𝛽𝐺(𝑡)=𝑡⁻¹𝐸_{𝛼−𝛽,0}−𝑡^𝛼−𝛽+𝑡^{𝛼−𝛽−1}𝐸_{𝛼−𝛽,𝛼−𝛽}−𝑡^𝛼−𝛽=𝐸_{𝛼−𝛽,0}−𝑡^𝛼−𝛽−𝐸_{𝛼−𝛽,0}−𝑡^𝛼−𝛽−1𝑡1Γ.𝛼−𝛽 (2.3) Since lim_𝑡→0(1/𝑡)(1/Γ(𝛼−𝛽))=𝛿(𝑡), we get 𝐷^𝛼𝐺(𝑡)+𝐷^𝛽𝐺(𝑡)=𝛿(𝑡) and so𝐷^𝛼𝑢(𝑡)+𝐷^𝛽𝑢(𝑡)=𝛿(𝑡)∗𝑓(𝑡,𝑢(𝑡))=𝑓(𝑡,𝑢(𝑡)). (2.4) Now, we establish some results on existence and multiplicity of positive solutions for the problem (2.1). Let 𝐸=(𝐶[0,1],‖⋅‖_∞) be endowed via the order 𝑢≤𝑣 if and only if 𝑢(𝑡)≤𝑣(𝑡) for all 𝑡∈[0,1]. Consider the cone 𝑃={𝑢∈𝐸∣𝑢(𝑡)≥0} and the nonnegative continuous concave functional 𝜃(𝑢)=inf_1/2<𝑡<1|𝑢(𝑡)|. Now, we give our first result.

Lemma 2.1. Define 𝑇∶𝑃→𝑃 by 𝑇𝑢(𝑡)∶=∫^𝑡₀𝐺(𝑡−𝜏)𝑓(𝜏,𝑢(𝜏))𝑑𝜏, where 𝐺(𝑡)=𝑡^𝛼−1𝐸_{𝛼−𝛽,𝛼}(−𝑡^𝛼−𝛽) and 𝐸_𝛼,𝛽(𝑧) is the two-parameter function of the Mittag-Leffler type. Then 𝑇 is completely continuous.

Proof. Since the mappings 𝐺 and 𝑓 are nonnegative and continuous, it is easy to see that 𝑇 is continuous. Now, we show that 𝑇 is a relatively compact operator. This implies that 𝑇 is completely continuous. Let Ω⊂𝑃 be a bounded subset. Then there exists a positive constant 𝑀>0 such that ‖𝑢‖≤𝑀 for all 𝑢∈Ω. Put 𝐿=sup_0≤𝑡≤1|𝑓(𝑡,𝑢(𝑡))|+1. Then, for each 𝑢∈Ω, we have |𝑇𝑢(𝑡)|=^𝑡₀(𝑡−𝜏)^𝛼−1𝐸_{𝛼−𝛽,𝛼}−(𝑡−𝜏)^𝛼−𝛽𝑓(𝜏,𝑢(𝜏))𝑑𝜏≤𝐿−𝑡^𝛼𝐸_{𝛼−𝛽,𝛼+1}−𝑡^𝛼−𝛽≤𝐿−𝑡^𝛼1+−𝑡^𝛼−𝛽≤𝐿𝑡^𝛼≤𝐿, (2.5) where 0<𝛼<1 and 𝑡∈[0,1]. Thus, 𝑇 is uniformly bounded. Now, we show that 𝑇 is equicontinuous. Let 𝑡,𝜏∈[0,1] and 𝑡₁≤𝑡₂. Thus, 𝑇𝑢𝑡₁−𝑇𝑢𝑡₂=^𝑡1₀𝐺𝑡₁−𝜏𝑓(𝜏,𝑢(𝜏))𝑑𝜏−^𝑡2₀𝐺𝑡₂𝑓((𝜏=−𝜏𝜏,𝑢))𝑑𝜏_𝑡10𝐺𝑡₁−𝜏𝑓(𝜏,𝑢(𝜏))−𝐺𝑡₂−𝜏𝑓(𝜏,𝑢(𝜏))𝑑𝜏+^𝑡1_𝑡2𝐺𝑡₂𝑓((𝜏≤−𝜏𝜏,𝑢))𝑑𝜏_𝑡10𝐺𝑡₁−𝜏𝑓(𝜏,𝑢(𝜏))−𝐺𝑡₂−𝜏𝑓(𝜏,𝑢(𝜏))𝑑𝜏+^𝑡1_𝑡2𝐺𝑡₂−𝜏𝑓(𝜏,𝑢(𝜏))𝑑𝜏. (2.6) Now, by using the formula for integration of the Mittag-Leffler function term by term given in (*), we obtain that 𝑇𝑢𝑡₁−𝑇𝑢𝑡₂≤𝑓𝑡^𝛼₁1+−𝑡₁^𝛼−𝛽−𝑡^𝛼₁1+−𝑡₁^𝛼−𝛽+𝑡₂−𝑡₁^𝛼1+−𝑡₂−𝑡₁^𝛼−𝛽+𝑡^𝛼₂1+−𝑡₂^𝛼−𝛽−𝑡^𝛼₁1+−𝑡₁^𝛼−𝛽−𝑡₂−𝑡₁^𝛼1+−𝑡₂−𝑡₁^𝛼−𝛽=𝑓𝑡^𝛼₂1+−𝑡₂^𝛼−𝛽−𝑡^𝛼₁1+−𝑡₁^𝛼−𝛽≤𝑓𝑡^𝛼₂−𝑡^𝛼₁−𝑡^𝛼₂𝑡₂^𝛼−𝛽−𝑡₁^𝛼−𝛽+𝑡₂^𝛼−𝛽𝑡^𝛼₂−𝑡^𝛼₁1+−𝑡₁^𝛼−𝛽1+−𝑡₂^𝛼−𝛽. (2.7) Thus, by using the formula 𝑡^𝑠₂−𝑡^𝑠₁=(𝑡₂−𝑡₁)/(𝑡₂^𝑠−1+⋯+𝑡₁^𝑠−1), we obtain a common factor (𝑡₁−𝑡₂). This implies that small changes of 𝑢 cause small changes of 𝑇𝑢. that is, 𝑇 is equicontinuous. Now by using the Arzela-Ascoli theorem, we get that 𝑇 is a relatively compact operator.

Theorem 2.2. Suppose that in the problem (1.2) there exists a positive real number 𝑟>0 such that(𝐴₁)𝑓(𝑡,𝑢)≤𝛼𝑟 for all (𝑡,𝑢)∈[0,1]×[0,𝑟],(𝐴₂)𝑓(𝑡,𝑢)≥0 for all 𝑡∈[0,1] with 𝑢(𝑡)=0. Then the problem (1.2) has a positive solution 𝑢 such that 0≤|𝑢|≤𝑟.

Example 2.3. Consider the nonlinear fractional differential equation initial value problem 𝐷^3/2𝑢(𝑡)+𝐷^1/2𝑢(𝑡)+𝑢(𝑡)+sin𝑡=0,𝑢(0)=0,(0<𝑡<1). (2.8) Put 𝑟=2 and 𝛼=3/2. Since 𝑓(𝑡,𝑢)=𝑢(𝑡)+sin𝑡≤𝑢+1≤3=𝛼𝑟 for all (𝑡,𝑢)∈[0,1]×[0,2] and 𝑓(𝑡,𝑢)=𝑢+sin𝑡≥0 for all (𝑡,𝑢)∈[0,1]×{0}, by using Theorem 2.2 we get that this problem has a positive solution we get that this problem has a positive solution 𝑢 with 0≤‖𝑢‖≤2.

Proof. First, let us to consider the operator (𝑇𝑢)(𝑡)=∫^𝑡₀𝐺(𝑡−𝜏)𝑓(𝜏,𝑢(𝜏))𝑑𝜏, where 𝐺(𝑡)=𝑡^𝛼−1𝐸_{𝛼−𝛽,𝛼}(−𝑡^𝛼−𝛽)(0<𝛽<𝛼<1). By using Lemma 2.1, 𝑇 is completely continuous and note that 𝑢 is a solution of the problem (1.2) if and only if 𝑢=𝑇(𝑢). Let Ω₁={𝑢∈𝑃∶‖𝑢‖=0} and Ω₂={𝑢∈𝑃∶‖𝑢‖𝑢∈𝜕Ω₁} we have 𝑢(𝑡)=0 for all 𝑡∈[0,1]. By using the assumption (𝐴₂), we have (𝑇𝑢)(𝑡)=^𝑡₀𝐺(𝑡−𝜏)𝑓(𝜏,𝑢(𝜏))𝑑𝜏≥0=‖𝑢‖ (2.9) and so ‖𝑇𝑢‖≥‖𝑢‖. Also, for 𝑢∈𝜕Ω₂ we have 0≤𝑢(𝑡)≤𝑟 for all 𝑡∈[0,1]. By using the assumption (𝐴₁) we have ‖𝑇𝑢‖=max_0≤𝑡≤1^𝑡₀𝐺(𝑡−𝜏)𝑓(𝜏,𝑢(𝜏))𝑑𝜏≤𝛼𝑟^𝑡₀(𝑡−𝜏)^𝛼−1𝑑𝜏=𝑟𝑡^𝛼≤𝑟=‖𝑢‖. (2.10)
This completes the proof.

Theorem 2.4. Suppose that in the problem (2.1) there exist positive real numbers 0<𝑎<𝑏<𝑐 such that(𝐴₁)𝑓(𝑡,𝑢)<𝛼𝑎 for all (𝑡,𝑢)∈[0,1]×[0,𝑎],(𝐴₂)𝑓(𝑡,𝑢)>𝑁𝑏 for all (𝑡,𝑢)∈[1/2,1]×[𝑏,𝑐], where𝑁⁻¹=inf_1/2<𝑡<1^𝑡₀𝐺(𝑡−𝑠)𝑑𝑠, (2.11)(𝐴₃)𝑓(𝑡,𝑢)≤𝛼𝑐 for all (𝑡,𝑢)∈[0,1]×[0,𝑐].Then the problem (2.1) has at least there positive solutions 𝑢₁, 𝑢₂, and 𝑢₃ such that sup_0≤𝑡≤1|𝑢₁(𝑡)|<𝑎, 𝑏<inf_{1/2≤𝑡≤1}|𝑢₂(𝑡)|<sup_{1/2≤𝑡≤1}|𝑢₂(𝑡)|≤𝑐,𝑎<sup_0≤𝑡≤1|𝑢₃(𝑡)|≤𝑐 and inf_{1/2≤𝑡≤1}|𝑢₃(𝑡)|<𝑏.

Proof. Define 𝑃_𝑐={𝑥∈𝑃∶‖𝑥‖≤𝑐}. Then, ‖𝑢‖≤𝑐 for all 𝑢∈𝑃_𝑐. Note that, the assumption (𝐴₃) implies that 𝑓(𝑡,𝑢(𝑡))≤𝛼𝑐 for all 𝑡. Thus, ‖𝑇𝑢‖=sup_0≤𝑡≤1^𝑡₀𝐺(𝑡−𝜏)𝑓(𝜏,𝑢(𝜏))𝑑𝜏≤𝛼𝑐^𝑡₀(𝑡−𝜏)^𝛼−1𝑡𝑑𝜏=𝛼𝑐^𝛼𝛼=𝑐𝑡^𝛼≤𝑐. (2.12) Hence, 𝑇 is a operator on 𝑃_𝑐. Also, note that the assumption (𝐴₁) implies that 𝑓(𝑡,𝑢(𝑡))<𝛼𝑎 for all 0≤𝑡≤1. Thus, the condition (𝑐₂) in Lemma 1.2 holds. It is sufficient that we show that the condition (𝑐₁) in Lemma 1.2 holds. Put 𝑢(𝑡)=(𝑏+𝑐)/2 for all 0≤𝑡≤1. It is easy to see that 𝑢(𝑡)∈𝑃(𝜃,𝑏,𝑐) and 𝜃(𝑢)=𝜃((𝑏+𝑐)/2)>𝑏. Thus, {𝑢∈𝑃(𝜃,𝑏,𝑐)∶𝜃(𝑢)>𝑏}≠∅ and so 𝑏≤𝑢(𝑡)≤𝑐 for all 𝑢∈𝑃(𝜃,𝑏,𝑐) and 1/2≤𝑡≤1. But, the assumption (𝐴₂) implies that 𝑓(𝑡,𝑢(𝑡))≥𝑁𝑏 for all 1/2≤𝑡≤1 and so 𝜃(𝑇𝑢)=inf_{1/2≤𝑡≤1}|(𝑇𝑢)(𝑡)|=inf_{1/2≤𝑡≤1}^𝑡₀𝐺(𝑡−𝜏)𝑓(𝜏,𝑢(𝜏))𝑑𝜏>𝑁𝑏𝑁⁻¹=𝑏. (2.13) Thus, 𝜃(𝑇𝑢)>𝑏 for all 𝑢∈𝑃(𝜃,𝑏,𝑐). This shows that the condition (𝑐₁) in Lemma 1.2 holds. This completes the proof.

Acknowledgments

Research of the second and third authors was supported by Azarbaijan University of Shahid Madani. Also, the authors express their gratitude to the referees for their helpful suggestions which improved final version of this paper.