The Existence of Positive Solutions for a New Coupled System of Multiterm Singular Fractional Integrodifferential Boundary Value Problems

Abstract

We discuss the existence of positive solutions for the coupled system of multiterm singular fractional integrodifferential boundary value problems 𝐷^𝛼₀⁺𝑢(𝑡)+𝑓₁(𝑡,𝑢(𝑡),𝑣(𝑡),(𝜙₁𝑢)(𝑡),(𝜓₁𝑣)(𝑡),𝐷^𝑝₀₊𝑢(𝑡),𝐷^𝜇1₀₊𝑣(𝑡),𝐷^𝜇2₀₊𝑣(𝑡),…,𝐷^𝜇𝑚₀₊𝑣(𝑡))=0,𝐷^𝛽₀₊𝑓𝑣(𝑡)+₂(𝑡,𝑢(𝑡),𝑣(𝑡),(𝜙₂𝑢)(𝑡),(𝜓₂𝑣)(𝑡),𝐷^𝑞₀₊𝑣(𝑡),𝐷^𝜈1₀₊𝑢(𝑡),𝐷^𝜈2₀₊𝑢(𝑡),…,𝐷^𝜈𝑚₀₊𝑢(𝑡))=0, 𝑢^(𝑖)(0)=0 and 𝑣^(𝑖)(0)=0 for all 0≤𝑖≤𝑛−2, [𝐷^𝛿1₀₊𝑢(𝑡)]_𝑡=1=0 for 2<𝛿₁<𝑛−1 and 𝛼−𝛿₁≥1, [𝐷^𝛿2₀₊𝑣(𝑡)]_𝑡=1=0 for 2<𝛿₂<𝑛−1 and 𝛽−𝛿₂≥1, where 𝑛≥4, 𝑛−1<𝛼,𝛽<𝑛, 0<𝑝,𝑞<1, 1<𝜇_𝑖,𝜈_𝑖<2(𝑖=1,2,…,𝑚), 𝛾_𝑗,𝜆_𝑗∶[0,1]×[0,1]→(0,∞) are continuous functions (𝑗=1,2) and (𝜙_𝑗𝑢)(𝑡)=∫^𝑡₀𝛾_𝑗(𝑡,𝑠)𝑢(𝑠)𝑑𝑠,(𝜓_𝑗𝑣)(𝑡)=∫^𝑡₀𝜆_𝑗(𝑡,𝑠)𝑣(𝑠)𝑑𝑠. Here 𝐷 is the standard Riemann-Liouville fractional derivative, 𝑓_𝑗(𝑗=1,2) is a Caratheodory function, and 𝑓_𝑗(𝑡,𝑥,𝑦,𝑧,𝑤,𝑣,𝑢₁,𝑢₂,…,𝑢_𝑚) is singular at the value 0 of its variables.

1. Introduction

During the last decade, there were a lot of manuscripts on fractional differential equations (see, e.g., [1–19] and the references therein). Fractional equations have been discussed extensively as valuable tools in the modeling of many phenomena in various fields of science and engineering. Indeed, we can find numerous applications in fluid mechanics, viscoelasticity, edge detection, porous media, and electromagnetism, as well as in various other areas. For more examples and details, see [3, 13–15] and references therein. On the other hand, there are many works about the existence of positive solutions of fractional differential equations (see, e.g., [1, 2, 6–8, 10, 17, 19] and the references therein).

2. The Problem

In this paper, we investigate the following coupled system of multiterm singular fractional integrodifferential boundary value problem: 𝐷^𝛼₀⁺𝑢(𝑡)+𝑓₁𝑡,𝑢(𝑡),𝑣(𝑡),𝜙₁𝑢(𝑡),𝜓₁𝑣(𝑡),𝐷^𝑝₀₊𝑢(𝑡),𝐷^𝜇1₀₊𝑣(𝑡),𝐷^𝜇2₀₊𝑣(𝑡),…,𝐷^𝜇𝑚₀₊𝑣(𝑡)𝐷=0,_𝛽0+𝑣(𝑡)+𝑓₂𝑡,𝑢(𝑡),𝑣(𝑡),𝜙₂𝑢(𝑡),𝜓₂𝑣(𝑡),𝐷^𝑞₀₊𝑣(𝑡),𝐷^𝜈1₀₊𝑢(𝑡),𝐷^𝜈2₀₊𝑢(𝑡),…,𝐷^𝜈𝑚₀₊𝑢(𝑡)𝑢=0,_(𝑖)(0)𝐷=0∀0≤𝑖≤𝑛−2,_𝛿10+𝑢(𝑡)_𝑡=1=0for2<𝛿₁<𝑛−1,𝛼−𝛿₁𝑣≥1,_(𝑖)(0)𝐷=0∀0≤𝑖≤𝑛−2,_𝛿20+𝑣(𝑡)_𝑡=1=0for2<𝛿₂<𝑛−1,𝛽−𝛿₂≥1, (1)(2)(3) where 𝑛≥4, 𝑛−1<𝛼, 𝛽<𝑛, 0<𝑝, 𝑞<1, 1<𝜇_𝑖,𝜈_𝑖<2 (𝑖=1,2,…,𝑚), 𝛾_𝑗 and 𝜆_𝑗 are positive-valued continuous functions on [0,1]×[0,1] (𝑗=1,2), 𝜙_𝑗𝑢(𝑡)=^𝑡₀𝛾_𝑗()𝑢(𝑠)𝜓𝑡,𝑠𝑑𝑠,_𝑗𝑣(𝑡)=^𝑡₀𝜆_𝑗(𝑡,𝑠)𝑣(𝑠)𝑑𝑠, (4)𝛾^𝑗₀=sup_𝑡∈[0,1]|∫^𝑡₀𝛾_𝑗(𝑡,𝑠)𝑑𝑠|<∞ and 𝜆^𝑗₀=sup_𝑡∈[0,1]|∫^𝑡₀𝜆_𝑗(𝑡,𝑠)𝑑𝑠|<∞. Also, 𝑓₁ and 𝑓₂ satisfy the local Caratheodory condition on [0,1]×𝔇(𝑓₁,𝑓₂∈Car([0,1]×𝔇)), where 𝔇 is a subset of ℝ^𝑚+5 and 𝑓_𝑗(𝑡,𝑥,𝑦,𝑧,𝑤,𝑣,𝑢₁,𝑢₂,…,𝑢_𝑚) may be singular at the value 0 of all its variables. Here, 𝐷 is the standard Riemann-Liouville fractional derivative. In fact, a function 𝑓 satisfies the local Caratheodory condition on [0,1]×𝔇 whenever(i)𝑓(⋅;𝑥,𝑦,𝑧,𝑤,𝑣,𝑢₁,𝑢₂,…,𝑢_𝑚)∶[0,1]→ℝ is measurable for all (𝑥,𝑦,𝑧,𝑤,𝑣,𝑢₁,𝑢₂,…,𝑢_𝑚)∈𝔇,(ii)𝑓(𝑡;⋅,⋅,…,⋅)∶𝔇→ℝ is continuous for almost all 𝑡∈[0,1],(iii)for each compact subset 𝜅⊂𝔇, there is a function 𝜑_𝜅∈𝐿¹[0,1] such that 𝑓𝑡;𝑥,𝑦,𝑧,𝑤,𝑣,𝑢₁,𝑢₂,…,𝑢_𝑚≤𝜑_𝜅(𝑡), (5)for almost all 𝑡∈[0,1] and all (𝑥,𝑦,𝑧,𝑤,𝑣,𝑢₁,𝑢₂,…,𝑢_𝑚)∈𝜅. The functions 𝑢,𝑣∈𝐶²[0,1] are called positive solutions of the problem (1), (2), and (3) whenever 𝑢>0 and 𝑣>0 on (0,1], 𝐷^𝛼₀⁺𝑢, 𝐷^𝛽₀₊𝑣∈𝐿¹[0,1], 𝑢, 𝑣 satisfy boundary conditions (2), (3), and (1) holds for almost all 𝑡∈[0,1]. In 2010, Agarwal et al. reviewed positive solutions of the singular Dirichlet problem 𝐷^𝛼₀⁺𝑢(𝑡)+𝑓𝑡,𝑢(𝑡),𝐷^𝜇₀₊𝑢(𝑡)=0, (6)𝑢(0)=𝑢(1)=0, where 1<𝛼<2, 0<𝜇≤𝛼−1 and 𝑓 satisfies the local Caratheodory condition on [0,1]×(0,∞)×ℝ [1]. In 2011, Stanek reviewed the singular problem 𝐷^𝛼₀⁺𝑢(𝑡)+𝑓𝑡,𝑢(𝑡),𝑢^′(𝑡),𝐷^𝜇₀₊𝑢(𝑡)=0, (7)𝑢(0)=0, 𝑢^′(0)=𝑢^′(1)=0, where 2<𝛼<3, 0<𝜇<1 and 𝑓 satisfies the local Caratheodory condition on [0,1]×𝔇 [19]. Here, 𝔇 is a subset of ℝ³. In 2012, Bai and Sun reviewed the singular problem 𝐷^𝛼₀⁺𝑢(𝑡)+𝑓𝑡,𝑢(𝑡),𝐷^𝜈₀⁺𝑢(𝑡),𝐷^𝜇₀₊𝑢(𝑡)=0, (8)𝑢(0)=𝑢^′(0)=𝑢^′′(0)=𝑢^′′(1)=0, where 3<𝛼≤4, 0<𝜈≤1, 1<𝜇≤2 and 𝑓 satisfies the local Caratheodory condition on [0,1]×𝔇 [8]. Again, 𝔇 is a subset of ℝ³. In this year, Agarwal et al. reviewed the singular problem ^𝑐𝐷^𝛼₀⁺𝑢(𝑡)+𝑓𝑡,𝑢(𝑡),𝑢^′(𝑡),^𝑐𝐷^𝜇₀₊𝑢(𝑡)=0, (9)𝑢^′(0)=0, 𝑢(1)=0, where 1<𝛼<2, 0<𝜇<1 and 𝑓 is a 𝐿^𝑞-Caratheodory function on [0,1]×𝔅 [2]. Here, 𝔅=(0,∞)×(−∞,0)×(−∞,0), 𝑞>1/(𝛼−1) and ^𝑐𝐷 is the Caputo fractional derivative. Many researchers have established the existence and uniqueness of solutions for some systems of nonlinear fractional differential equations, but there are few works about coupled system of multiterm nonlinear fractional differential equations (see, for example, [4, 5, 20, 21]). Also, there are a few papers discussing singular system of fractional differential equations (see, e.g., [6, 16]).

Let ‖𝑥‖₁=∫¹₀|𝑥(𝑡)|𝑑𝑡 be the norm of 𝐿¹[0,1], ‖𝑥‖=max{|𝑥(𝑡)|∶𝑡∈[0,1]} the norm of 𝐶[0,1], and ‖𝑢‖_∗=max{‖𝑢‖,‖𝑢^′‖,‖𝑢^′′‖} the norm of 𝑋=𝐶²[0,1]. As you know, (𝑋,‖⋅‖_∗) and (𝑋×𝑋,‖⋅‖_∗∗) are Banach spaces, where ‖(𝑢,𝑣)‖_∗∗=max{‖𝑢‖_∗,‖𝑣‖_∗}. Suppose that 𝐴𝐶[0,1] and 𝐴𝐶^𝑘[0,1] are the spaces of absolutely continuous functions and functions having absolutely continuous 𝑘th derivatives on [0,1], respectively. In this paper, we suppose that the functions 𝑓₁ and 𝑓₂ in (1) satisfy the following conditions:(H₁)𝑓₁,𝑓₂∈Car([0,1]×𝔇), where 𝔇=(0,∞)^𝑚+5. Also, there exists a positive constant 𝑚_𝑗 such that 𝑓_𝑗(𝑡,𝑥,𝑦,𝑧,𝑤,𝑣,𝑢₁,𝑢₂,…,𝑢_𝑚)≥𝑚_𝑗 for almost all 𝑡∈[0,1] and (𝑥,𝑦,𝑧,𝑤,𝑣,𝑢₁,𝑢₂,…,𝑢_𝑚)∈𝔇;(H₂) there exist positive mappings 𝑘₁,𝑘₂∈𝐿¹[0,1], 𝑃₁,𝑃₂∈𝐶(𝔇), and ℎ₁,ℎ₂∈𝐶([0,∞)^𝑚+5) such that 𝑃₁ and 𝑃₂ are nonincreasing and ℎ₁ and ℎ₂ are nondecreasing in all their variables, 𝑀₁=𝑚₁𝛼−𝛿₁Γ(𝛼+1),𝑀₂=𝑚₂𝛽−𝛿₂Γ,𝛽+1lim_𝑥→∞ℎ_𝑗(𝑥,𝑥,…,𝑥)𝑥=0,𝑗=1,2₁₀,𝑃₁𝑀₁𝑠^𝛼−1,𝑀₂𝑠^𝛽−1,𝑀₁^𝑠₀𝛾₁(𝑠,𝜏)𝜏^𝛼−1𝑑𝜏,𝑀₂^𝑠₀𝜆₁(𝑠,𝜏)𝜏^𝛽−1𝛿𝑑𝜏,₁−2𝑀₁Γ𝑠𝛼−𝑝^𝛼−𝑝,𝛿₂−2𝑀₂Γ𝛽−𝜇₁𝑠^𝛽−𝜇1,𝛿₂−2𝑀₂Γ𝛽−𝜇₂𝑠^𝛽−𝜇2𝛿,…,²−2𝑀₂Γ𝛽−𝜇_𝑚𝑠^{𝛽−𝜇𝑚}𝑑𝑠<∞,₁₀𝑃₂𝑀₁𝑠^𝛼−1,𝑀₂𝑠^𝛽−1,𝑀₁^𝑠₀𝛾₂(𝑠,𝜏)𝜏^𝛼−1𝑑𝜏,𝑀₂^𝑠₀𝜆₂(𝑠,𝜏)𝜏^𝛽−1𝛿𝑑𝜏,₂−2𝑀₂Γ𝑠𝛽−𝑞^𝛽−𝑞,𝛿₁−2𝑀₁Γ𝛼−𝜈₁𝑠^𝛼−𝜈1,𝛿₁−2𝑀₁Γ𝛼−𝜈₂𝑠^𝛼−𝜈2𝛿,…,¹−2𝑀₁Γ𝛼−𝜈_𝑚𝑠^{𝛼−𝜈𝑚}𝑓𝑑𝑠<∞,_𝑗𝑡,𝑥,𝑦,𝑧,𝑤,𝑣,𝑢₁,𝑢₂,…,𝑢_𝑚≤𝑃_𝑗𝑥,𝑦,𝑧,𝑤,𝑣,𝑢₁,𝑢₂,…,𝑢_𝑚+𝑘_𝑗(𝑡)ℎ_𝑗𝑥,𝑦,𝑧,𝑤,𝑣,𝑢₁,𝑢₂,…,𝑢_𝑚 (10)for almost all 𝑡∈[0,1] and (𝑥,𝑦,𝑧,𝑤,𝑣,𝑢₁,𝑢₂,…,𝑢_𝑚)∈𝔇. Since we suppose that the problem (1) is singular, we use regularization and sequential techniques for the existence of positive solutions of the problem. In this way, for each natural number 𝑛 define the function 𝑓^𝑗_𝑛 (𝑗=1,2) by 𝑓^𝑗_𝑛𝑡,𝑥,𝑦,𝑧,𝑤,𝑣,𝑢₁,𝑢₂,…,𝑢_𝑚=𝑓_𝑗𝑡,𝜒_𝑛(𝑥),𝜒_𝑛𝑦,𝜒_𝑛(𝑧),𝜒_𝑛(𝑤),𝜒_𝑛(𝑣),𝜒_𝑛𝑢₁,𝜒_𝑛𝑢₂,…,𝜒_𝑛𝑢_𝑚 (11) for almost all 𝑡∈[0,1] and (𝑥,𝑦,𝑧,𝑤,𝑣,𝑢₁,𝑢₂,…,𝑢_𝑚)∈ℝ^𝑚+5, where 𝜒_𝑛(𝑥)=1𝑥if𝑥≥𝑛,1𝑛1if𝑥<𝑛. (12) Then condition (𝐻₁) implies that 𝑓^𝑗_𝑛∈Car([0,1]×ℝ^𝑚+5) and 𝑓^𝑗_𝑛(𝑡,𝑥,𝑦,𝑧,𝑤,𝑣,𝑢₁,𝑢₂,…,𝑢_𝑚)≥𝑚_𝑗 for almost all 𝑡∈[0,1] and (𝑥,𝑦,𝑧,𝑤,𝑣,𝑢₁,𝑢₂,…,𝑢_𝑚)∈ℝ^𝑚+5, (𝑗=1,2). Also, the condition (𝐻₂) implies 𝑓^𝑗_𝑛𝑡,𝑥,𝑦,𝑧,𝑤,𝑣,𝑢₁,𝑢₂,…,𝑢_𝑚≤𝑃_𝑗1𝑛,1𝑛1,…,𝑛+𝑘_𝑗(𝑡)ℎ_𝑗1+𝑥,1+𝑦,1+𝑧,1+𝑤,1+𝑣,1+𝑢₁,1+𝑢₂,…,1+𝑢_𝑚 (13) for almost all 𝑡∈[0,1] and (𝑥,𝑦,𝑧,𝑤,𝑣,𝑢₁,𝑢₂,…,𝑢_𝑚)∈[0,∞)^𝑚+5 and 𝑓^𝑗_𝑛𝑡,𝑥,𝑦,𝑧,𝑤,𝑣,𝑢₁,𝑢₂,…,𝑢_𝑚≤𝑃_𝑗𝑥,𝑦,𝑧,𝑤,𝑣,𝑢₁,𝑢₂,…,𝑢_𝑚+𝑘_𝑗(𝑡)ℎ_𝑗1+𝑥,1+𝑦,1+𝑧,1+𝑤,1+𝑣,1+𝑢₁,1+𝑢₂,…,1+𝑢_𝑚 (14) for almost all 𝑡∈[0,1] and (𝑥,𝑦,𝑧,𝑤,𝑣,𝑢₁,𝑢₂,…,𝑢_𝑚)∈𝔇. For obtaining solutions of the system (1), (2), and (3), for each natural number 𝑛, we will obtain solutions of the system 𝐷^𝛼₀⁺𝑢(𝑡)+𝑓¹_𝑛𝑡,𝑢(𝑡),𝑣(𝑡),𝜙₁𝑢(𝑡),𝜓₁𝑣(𝑡),𝐷^𝑝₀₊𝑢(𝑡),𝐷^𝜇1₀₊𝑣(𝑡),𝐷^𝜇2₀₊𝑣(𝑡),…,𝐷^𝜇𝑚₀₊𝑣(𝑡)𝐷=0,_𝛽0+𝑣(𝑡)+𝑓²_𝑛𝑡,𝑢(𝑡),𝑣(𝑡),𝜙₂𝑢(𝑡),𝜓₂𝑣(𝑡),𝐷^𝑞₀₊𝑣(𝑡),𝐷^𝜈1₀₊𝑢(𝑡),𝐷^𝜈2₀₊𝑢(𝑡),…,𝐷^𝜈𝑚₀₊𝑢(𝑡)=0, (15) via the boundary conditions (2) and (3) and by using solutions of this system, we will obtain solution of the system (1), (2), and (3). It has been proved that the fractional integral 𝐼^𝛼₀⁺ maps 𝐿¹[0,1] into 𝐿¹[0,1] whenever 𝛼∈(0,1) and maps 𝐿¹[0,1] into 𝐴𝐶^[𝛼]−1[0,1] whenever 𝛼≥1 [19]. Here, [𝛼] means the integral part of 𝛼 and 𝐴𝐶⁰[0,1]=𝐴𝐶[0,1]. Suppose that 𝛼>0 and 𝛼 is not a natural number. If 𝑥∈𝐶(0,1] and 𝐷^𝛼₀⁺𝑥∈𝐿¹[0,1], then 𝑥(𝑡)=𝐼^𝛼₀⁺𝐷^𝛼₀⁺𝑥(𝑡)+∑^𝑛_𝑘=1𝑐_𝑘𝑡^𝛼−𝑘 for all 𝑡∈(0,1], where 𝑛=[𝛼]+1 and 𝑐_𝑘∈ℝ for 𝑘=1,2,…,𝑛 [7]. Suppose that 𝜈∈(0,1), 𝜇∈(1,2), 𝑥∈𝐶²[0,1], and 𝑥(0)=𝑥^′(0)=0. Then, 𝐷^𝜇₀₊𝑥∈𝐶[0,1], 𝐷^𝜈₀⁺𝑥∈𝐶[0,1], 𝐷^𝜇₀₊𝑥(𝑡)=(1/Γ(2−𝜇))∫^𝑡₀(𝑡−𝑠)^1−𝜇𝑥^′′(𝑠)𝑑𝑠, and 𝐷^𝜈₀⁺𝑥(𝑡)=(1/Γ(2−𝜈))∫^𝑡₀(𝑡−𝑠)^1−𝜈𝑥^′′(𝑠)𝑑𝑠 [2].

3. Main Results

Now, we are ready to state and prove our main results. One can find main idea of next result in [17].

Lemma 1. Let 𝑛≥4 and 𝑛−1<𝛼<𝑛. For each 𝑔∈𝐿¹[0,1], 𝑢(𝑡)=∫¹₀𝐺_𝛼(𝑡,𝑠)𝑔(𝑠)𝑑𝑠 is the unique solution of the equation 𝐷^𝛼₀⁺𝑢(𝑡)+𝑔(𝑡)=0 in 𝐶²[0,1] which satisfies the boundary condition (2), where 𝐺_𝛼()=𝑡𝑡,𝑠_𝛼−1(1−𝑠)^{𝛼−𝛿1−1}−(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑡0≤𝑠≤𝑡≤1,_𝛼−1(1−𝑠)^{𝛼−𝛿1−1}Γ(𝛼)0≤𝑡≤𝑠≤1. (16)

Proof. It is easy to see that the functions 𝑢(𝑡)=−𝐼^𝛼₀⁺𝑔(𝑡)+𝑐₁𝑡^𝛼−1+𝑐₂𝑡^𝛼−2+⋯+𝑐_𝑛𝑡^𝛼−𝑛 are solutions of 𝐷^𝛼₀⁺𝑢(𝑡)+𝑔(𝑡)=0 in 𝐶(0,1] for all 𝑐_𝑖∈ℝ. Since [𝛼]≥3, 𝐼^𝛼₀⁺𝑔∈𝐴𝐶^[𝛼]−1[0,1] and so 𝑢(𝑡)=−𝐼^𝛼₀⁺𝑔(𝑡)+𝑐₁𝑡^𝛼−1 are solutions of 𝐷^𝛼₀⁺𝑢(𝑡)+𝑔(𝑡)=0 in 𝐶²[0,1], where 𝑐₁∈ℝ. The boundary condition (2) implies that 𝑐₂=⋯=𝑐_𝑛−1=0 and 0=𝑐₁Γ(𝛼)Γ𝛼−𝛿₁(1)^{𝛼−𝛿1−1}−1Γ𝛼−𝛿₁¹₀(1−𝑠)^{𝛼−𝛿1−1}𝑔(𝑠)𝑑𝑠. (17) Thus, 𝑢(𝑡)=(𝑡^𝛼−1/Γ(𝛼))∫¹₀(1−𝑠)^{𝛼−𝛿1−1}𝑔(𝑠)𝑑𝑠−(1/Γ(𝛼))∫^𝑡₀(𝑡−𝑠)^𝛼−1𝑔(𝑠)𝑑𝑠=∫¹₀𝐺_𝛼(𝑡,𝑠)𝑔(𝑠)𝑑𝑠 is the unique solution of the problem in 𝐶²[0,1].

Note that the Green function 𝐺_𝛼 in Lemma 1 has some properties. For example, 𝐺_𝛼, (𝜕/𝜕𝑡)𝐺_𝛼 and (𝜕²/𝜕𝑡²)𝐺_𝛼 are continuous functions on [0,1]×[0,1], 𝐺_𝛼(𝑡,𝑠)>0 on (0,1)×(0,1), 𝐺_𝛼(𝑡,𝑠)≤1/Γ(𝛼) for all (𝑡,𝑠)∈[0,1]×[0,1], ∫¹₀𝐺_𝛼(𝑡,𝑠)𝑑𝑠≥𝑡^𝛼−1/(𝛼−𝛿₁)Γ(𝛼+1) for all 𝑡∈[0,1], (𝜕/𝜕𝑡)𝐺_𝛼(𝑡,𝑠)>0 on (0,1)×(0,1), (𝜕/𝜕𝑡)𝐺_𝛼(𝑡,𝑠)≤1/Γ(𝛼−1) for all (𝑡,𝑠)∈[0,1]×[0,1], ∫¹₀(𝜕/𝜕𝑡)𝐺_𝛼(𝑡,𝑠)𝑑𝑠≥𝑡^𝛼−2/(𝛼−𝛿₁)Γ(𝛼) for all 𝑡∈[0,1], (𝜕²/𝜕𝑡²)𝐺_𝛼(𝑡,𝑠)>0 on (0,1)×(0,1), (𝜕²/𝜕𝑡²)𝐺_𝛼(𝑡,𝑠)≤1/Γ(𝛼−2) for all (𝑡,𝑠)∈[0,1]×[0,1], and ∫¹₀(𝜕²/𝜕𝑡²)𝐺_𝛼(𝑡,𝑠)𝑑𝑠≥(𝛿₁−2)𝑡^𝛼−3/(𝛼−𝛿₁)Γ(𝛼−1) for all 𝑡∈[0,1]. Let 𝑋=𝐶²[0,1]. Define the cone 𝑃 on 𝑋×𝑋 by 𝑥(0)𝑃=𝑥,𝑦∈𝑋×𝑋∶=𝑥^′(0)=𝑦(0)=𝑦^′(0)𝑥(𝑡)=0,,𝑥^′(𝑡),𝑥^′′(𝑡),𝑦(𝑡),𝑦^′(𝑡),𝑦^′′(𝑡)[.≥0,∀𝑡∈0,1] (18) It is easy to see that for each (𝑥,𝑦)∈𝑃, 𝑖=1,2,…,𝑚 and 𝑡∈[0,1] we have 𝐷^𝑝₀₊𝑥,𝐷^𝜈𝑖₀₊𝑥,𝐷^𝑞₀₊𝑦,𝐷^𝜇𝑖₀₊𝑦∈𝐶[0,1] and 𝐷^𝑝₀₊𝑥(𝑡)≥0, 𝐷^𝜈𝑖₀₊𝑥(𝑡)≥0, 𝐷^𝑞₀₊𝑦(𝑡)≥0, 𝐷^𝜇𝑖₀₊𝑦(𝑡)≥0. Now for each natural number 𝑛, define the operator 𝑇_𝑛 on 𝑃 by 𝑇_𝑛(𝑡)=𝑇𝑥,𝑦_1𝑛(𝑡)𝑇𝑥,𝑦_2𝑛(𝑡)=𝑥,𝑦₁₀𝐺_𝛼(𝑡,𝑠)𝑓¹_𝑛×𝑠,𝑥(𝑠),𝑦(𝑠),𝜙₁𝑥(𝑠),𝜓₁𝑦(𝑠),𝐷^𝑝₀₊𝑥(𝑠),𝐷^𝜇1₀₊𝑦(𝑠),𝐷^𝜇2₀₊𝑦(𝑠),…,𝐷^𝜇𝑚₀₊𝑦(𝑠)𝑑𝑠₁₀𝐺_𝛽(𝑡,𝑠)𝑓²_𝑛×𝑠,𝑥(𝑠),𝑦(𝑠),𝜙₂𝑥(𝑠),𝜓₂𝑦(𝑠),𝐷^𝑞₀₊𝑦(𝑠),𝐷^𝜈1₀₊𝑥(𝑠),𝐷^𝜈2₀₊𝑥(𝑠),…,𝐷^𝜈𝑚₀₊𝑥(𝑠)𝑑𝑠. (19)

Lemma 2. For each natural number 𝑛, 𝑇_𝑛 is a completely continuous operator on 𝑃.

Proof. Let 𝑛 be a natural number, (𝑥,𝑦)∈𝑃, 𝜌¹(𝑡)=𝑓¹_𝑛𝑡,𝑥(𝑡),𝑦(𝑡),𝜙₁𝑥(𝑡),𝜓₁𝑦(𝑡),𝐷^𝑝₀₊𝑥(𝑡),𝐷^𝜇1₀₊𝑦(𝑡),𝐷^𝜇2₀₊𝑦(𝑡),…,𝐷^𝜇𝑚₀₊𝑦(𝑡) (20) and 𝜌²(𝑡)=𝑓²_𝑛(𝑡,𝑥(𝑡), 𝑦(𝑡), (𝜙₂𝑥)(𝑡), (𝜓₂𝑦)(𝑡), 𝐷^𝑞₀₊𝑦(𝑡), 𝐷^𝜈1₀₊𝑥(𝑡), 𝐷^𝜈2₀₊𝑥(𝑡), …, 𝐷^𝜈𝑚₀₊𝑥(𝑡)). Then, 𝜌¹,𝜌²∈𝐿¹[0,1] and there exist positive constants 𝑚₁ and 𝑚₂ such that 𝜌¹(𝑡)≥𝑚₁ and 𝜌²(𝑡)≥𝑚₂ for almost all 𝑡∈[0,1]. Since 𝐺_𝛼, 𝐺_𝛽, (𝜕/𝜕𝑡)𝐺_𝛼, (𝜕/𝜕𝑡)𝐺_𝛽, (𝜕²/𝜕𝑡²)𝐺_𝛼, and (𝜕²/𝜕𝑡²)𝐺_𝛽 are nonnegative and continuous functions on [0,1]×[0,1], 𝐺_𝛼(0,𝑠)=0, and 𝐺_𝛽(0,𝑠)=0 for all 𝑠∈[0,1], we get 𝑇¹_𝑛(𝑥,𝑦),𝑇²_𝑛(𝑥,𝑦)∈𝐶²[0,1], 𝑇¹_𝑛(𝑥,𝑦)(0)=𝑇²_𝑛(𝑥,𝑦)(0)=(𝑇¹_𝑛(𝑥,𝑦))^′(0)=(𝑇²_𝑛(𝑥,𝑦))^′(0)=0, 𝑇¹_𝑛(𝑥,𝑦)≥0, 𝑇²_𝑛(𝑥,𝑦)≥0, (𝑇¹_𝑛(𝑥,𝑦))^′≥0, (𝑇²_𝑛(𝑥,𝑦))^′≥0, (𝑇¹_𝑛(𝑥,𝑦))^′′≥0, and (𝑇²_𝑛(𝑥,𝑦))^′′≥0 on [0,1]. Thus, 𝑇_𝑛 maps 𝑃 into 𝑃. Suppose that {(𝑥_𝑘,𝑦_𝑘)}_𝑘≥1 is a convergent sequence in 𝑃 and lim_𝑘→∞(𝑥_𝑘,𝑦_𝑘)=(𝑥,𝑦). Then, lim_𝑘→∞𝑥_𝑘^(𝑗)(𝑡)=𝑥^(𝑗)(𝑡) and lim_𝑘→∞𝑦_𝑘^(𝑗)(𝑡)=𝑦^(𝑗)(𝑡) uniformly on [0,1] for 𝑗=0,1,2. Since for 𝑝,𝑞∈(0,1), 𝜇_𝑖,𝜈_𝑖∈(1,2) for all 𝑖=1,2,…,𝑚, 𝐷^𝑝₀₊𝑥_𝑘(𝑡)−𝐷^𝑝₀₊𝑥(𝑡≤𝑥_𝑘′′)−𝑥^′′Γ2−𝑝^𝑡₀(𝑡−𝑠)^1−𝑝𝑥𝑑𝑠≤^𝑘′′−𝑥^′′Γ,𝐷3−𝑝_𝑞0+𝑦_𝑘(𝑡)−𝐷^𝑞₀₊𝑦(𝑡≤𝑦_𝑘′′)−𝑦^′′Γ2−𝑞^𝑡₀(𝑡−𝑠)^1−𝑞𝑦𝑑𝑠≤^𝑘′′−𝑦^′′Γ,𝐷3−𝑞_𝜈𝑖0+𝑥_𝑘(𝑡)−𝐷^𝜈𝑖₀₊𝑥(𝑡≤𝑥_𝑘′′)−𝑥^′′Γ2−𝜈_𝑖^𝑡₀(𝑡−𝑠)^1−𝜈𝑖𝑥𝑑𝑠≤^𝑘′′−𝑥^′′Γ3−𝜈_𝑖,𝐷^𝜇𝑖₀₊𝑦_𝑘(𝑡)−𝐷^𝜇𝑖₀₊𝑦(𝑡≤𝑦_𝑘′′)−𝑦^′′Γ2−𝜇_𝑖^𝑡₀(𝑡−𝑠)^1−𝜇𝑖𝑦𝑑𝑠≤^𝑘′′−𝑦^′′Γ3−𝜇_𝑖,𝜙_𝑗𝑥_𝑘(𝑡)−𝜙_𝑗𝑥(𝑡)≤sup_𝑡∈[0,1]^𝑡₀𝛾_𝑗(𝑡,𝑠)𝑑𝑠𝑥_𝑘−𝑥=𝛾^𝑗₀𝑥_𝑘,𝜓−𝑥_𝑗𝑦_𝑘(𝑡)−𝜓_𝑗𝑦(𝑡)≤sup_𝑡∈[0,1]^𝑡₀𝜆_𝑗(𝑡,𝑠)𝑑𝑠𝑦_𝑘−𝑦=𝜆^𝑗₀𝑦_𝑘−𝑦, (21) for 𝑗=1,2 and 𝑘≥1, we have lim_𝑘→∞𝐷^𝑝₀₊𝑥_𝑘(𝑡)=𝐷^𝑝₀₊𝑥(𝑡), lim_𝑘→∞𝐷^𝑞₀₊𝑦_𝑘(𝑡)=𝐷^𝑞₀₊𝑦(𝑡), lim_𝑘→∞𝐷^𝜈𝑖₀₊𝑥_𝑘(𝑡)=𝐷^𝜈𝑖₀₊𝑥(𝑡), lim_𝑘→∞𝐷^𝜇𝑖₀₊𝑦_𝑘(𝑡)=𝐷^𝜇𝑖₀₊𝑦(𝑡) for 𝑖=1,2,…,𝑚, and lim_𝑘→∞(𝜙_𝑗𝑥_𝑘)(𝑡)=(𝜙_𝑗𝑥)(𝑡), lim_𝑘→∞(𝜓_𝑗𝑦_𝑘)(𝑡)=(𝜓_𝑗𝑦)(𝑡) uniformly on [0,1] for 𝑗=1,2. Also, we have 𝐷^𝑝₀₊𝑥_𝑘≤𝑥_𝑘^′′Γ2−𝑝^𝑡₀(𝑡−𝑠)^1−𝑝𝑥𝑑𝑠≤^𝑘′′Γ,𝐷3−𝑝_𝑞0+𝑦_𝑘≤𝑦_𝑘^′′Γ2−𝑞^𝑡₀(𝑡−𝑠)^1−𝑞𝑦𝑑𝑠≤^𝑘′′Γ,𝐷3−𝑞_𝜈𝑖0+𝑥_𝑘≤𝑥_𝑘^′′Γ2−𝜈_𝑖^𝑡₀(𝑡−𝑠)^1−𝜈𝑖𝑥𝑑𝑠≤^𝑘′′Γ3−𝜈_𝑖,𝐷^𝜇𝑖₀₊𝑦_𝑘≤𝑦_𝑘^′′Γ2−𝜇_𝑖^𝑡₀(𝑡−𝑠)^1−𝜇𝑖𝑦𝑑𝑠≤^𝑘′′Γ3−𝜇_𝑖, (22) and also ‖𝜙_𝑗𝑥_𝑘‖≤𝛾^𝑗₀‖𝑥_𝑘‖ and ‖𝜓_𝑗𝑦_𝑘‖≤𝜆^𝑗₀‖𝑦_𝑘‖ for 𝑗=1,2 and 𝑘≥1. Now, put 𝜌¹_𝑘(𝑡)=𝑓¹_𝑛𝑡,𝑥_𝑘(𝑡),𝑦_𝑘(𝑡),𝜙₁𝑥_𝑘(𝑡),𝜓₁𝑦_𝑘(𝑡),𝐷^𝑝₀₊𝑥_𝑘(𝑡),𝐷^𝜇1₀₊𝑦_𝑘(𝑡),𝐷^𝜇2₀₊𝑦_𝑘(𝑡),…,𝐷^𝜇𝑚₀₊𝑦_𝑘(𝑡) (23) and𝜌²_𝑘(𝑡)=𝑓²_𝑛(𝑡,𝑥_𝑘(𝑡),𝑦_𝑘(𝑡),(𝜙₂𝑥_𝑘)(𝑡),(𝜓₂𝑦_𝑘)(𝑡),𝐷^𝑞₀₊𝑦_𝑘(𝑡), 𝐷^𝜈1₀₊𝑥_𝑘(𝑡), 𝐷^𝜈2₀₊𝑥_𝑘(𝑡),…,𝐷^𝜈𝑚₀₊𝑥_𝑘(𝑡)) for all 𝑘≥1. Then, lim_𝑘→∞𝜌^𝑗_𝑘(𝑡)=𝜌^𝑗(𝑡) for almost all 𝑡∈[0,1] and 𝑗=1,2. Since 𝑓^𝑗_𝑛∈Car([0,1]×ℝ^𝑚+5) and {𝑥_𝑘}_𝑘≥1, {𝑦_𝑘}_𝑘≥1 are bounded sequences in 𝐶²[0,1] and also by using the above inequalities, {𝐷^𝑝₀₊𝑥_𝑘}_𝑘≥1,