Monotonic Positive Solutions of Nonlocal Boundary Value Problems for a Second-Order Functional Differential Equation

Abstract

We study the existence of at least one monotonic positive solution for the nonlocal boundary value problem of the second-order functional differential equation 𝑥^(𝑡)=𝑓(𝑡,𝑥(𝜙(𝑡))), 𝑡∈(0,1), with the nonlocal condition ∑^𝑚𝑘=1𝑎_𝑘𝑥(𝜏_𝑘)=𝑥₀, 𝑥^(0)+∑^𝑛𝑗=1𝑏_𝑗𝑥^(𝜂_𝑗)=𝑥₁, where 𝜏_𝑘∈(𝑎,𝑑)⊂(0,1), 𝜂_𝑗∈(𝑐,𝑒)⊂(0,1), and 𝑥₀,𝑥₁>0. As an application the integral and the nonlocal conditions ∫^𝑑𝑎𝑥(𝑡)𝑑𝑡=𝑥₀, 𝑥^(0)+𝑥(𝑒)−𝑥(𝑐)=𝑥₁ will be considered.

1. Introduction

The nonlocal boundary value problems of ordinary differential equations arise in a variety of different areas of applied mathematics and physics.

The study of nonlocal boundary value problems was initiated by Il’in and Moiseev [1, 2]. Since then, the non-local boundary value problems have been studied by several authors. The reader is referred to [3–22] and references therein.

In most of all these papers, the authors assume that the function 𝑓∶[0,1]×𝑅⁺→𝑅⁺ is continuous. They all assume thatlim_𝑥→∞𝑓(𝑥)lim_𝑥→0𝑥=0or∞,𝑓(𝑥)𝑥=0or∞.(1.1) These assumptions are restrictive, and there are many functions that do not satisfy these assumptions.

Here we assume that the function 𝑓∶[0,1]×𝑅⁺→𝑅⁺ is measurable in 𝑡∈[0,1] for all 𝑥∈𝑅⁺ and continuous in 𝑥∈𝑅⁺ for almost all 𝑡∈[0,1] is and there exists an integrable function 𝑎∈𝐿₁[0,1] and a constant 𝑏>0 such that||𝑓(𝑡,𝑥)||≤||𝑎(𝑡)||+𝑏|𝑥|,∀(𝑡,𝑥)∈[0,1]×𝐷.(1.2) Our aim here is to study the existence of at least one monotonic positive solution for the nonlocal problem of the second-order functional differential equation𝑥^(𝑡)=𝑓(𝑡,𝑥(𝜙(𝑡))),𝑡∈(0,1),(1.3) with the nonlocal condition^𝑚_𝑘=1𝑎_𝑘𝑥𝜏_𝑘=𝑥₀,𝑥^(0)+^𝑛_𝑗=1𝑏_𝑗𝑥^𝜂_𝑗=𝑥₁,(1.4) where 𝜏_𝑘∈(𝑎,𝑑)⊂(0,1),𝜂_𝑗∈(𝑐,𝑒)⊂(0,1),and𝑥₀,𝑥₁>0.

As an application, the problem with the integral and nonlocal conditions^𝑑𝑎𝑥(𝑡)𝑑𝑡=𝑥₀,𝑥^(0)+𝑥(𝑒)−𝑥(𝑐)=𝑥₁,(1.5) is studied.

It must be noticed that the nonlocal conditions𝑥(𝜏)=𝑥₀,𝜏∈(𝑎,𝑑),𝑥^(0)+𝑥^(𝜂)=𝑥₁,𝜂∈(𝑐,𝑒),_𝑚_𝑘=1𝑎_𝑘𝑥𝜏_𝑘=0,𝜏_𝑘∈(𝑎,𝑑),𝑥^(0)+^𝑛_𝑗=1𝑏_𝑗𝑥^𝜂_𝑗=0,𝜂_𝑗∈(𝑐,𝑒),_𝑑𝑎𝑥(𝑡)𝑑𝑡=0,𝑥^(0)+𝑥(𝑒)=𝑥(𝑐)(1.6) are special cases of our the nonlocal and integral conditions.

2. Integral Equation Representation

Consider the functional differential equation (1.3) with the nonlocal condition (1.4) with the following assumptions.(i)𝑓∶[0,1]×𝑅⁺→𝑅⁺ is measurable in 𝑡∈[0,1] for all 𝑥∈𝑅⁺ and continuous in 𝑥∈𝑅⁺ for almost all 𝑡∈[0,1] and there exists an integrable function 𝑎∈𝐿₁[0,1], and a constant 𝑏>0 such that ||𝑓(𝑡,𝑥)||≤||𝑎(𝑡)||+𝑏|𝑥|,∀(𝑡,𝑥)∈[0,1]×𝐷.(2.1)(ii)𝜙∶(0,1)→(0,1) is continuous.(iii)𝑏<1/(3−𝐵),𝐵=(∑^𝑛𝑗=1𝑏_𝑗+1)⁻¹.(iv)^𝑚_𝑘=1𝑎_𝑘>0,∀𝑘=1,2,…,𝑚,^𝑛_𝑗=1𝑏_𝑗>0,∀𝑗=1,2,…,𝑛.(2.2) Now, we have the following Lemma.

Lemma 2.1. The solution of the nonlocal problem (1.3)-(1.4) can be expressed by the integral equation 𝑥(𝑡)=𝐴𝑥₀−^𝑚_𝑘=1𝑎_𝑘^𝜏𝑘₀𝜏_𝑘+𝐵𝑡−𝐴_𝑚−𝑠𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠_𝑘=1𝑎_𝑘𝜏_𝑘𝑥¹−^𝑛_𝑗=1𝑏_𝑗^𝜂𝑗₀+𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠_𝑡0(𝑡−𝑠)𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠,(2.3) where 𝐴=(∑^𝑚𝑘=1𝑎_𝑘)⁻¹,𝐵=(∑^𝑛𝑗=1𝑏_𝑗+1)⁻¹.

Proof. Integrating (1.3), we get 𝑥^(𝑡)=𝑥^(0)+^𝑡0𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠.(2.4) Integrating (2.4), we obtain 𝑥(𝑡)=𝑥(0)+𝑥^(0)𝑡+^𝑡0(𝑡−𝑠)𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠.(2.5) Let 𝑡=𝜏_𝑘, in (2.5), we get ^𝑚_𝑘=1𝑎_𝑘𝑥𝜏_𝑘=^𝑛_𝑘=1𝑎_𝑘𝑥(0)+^𝑛_𝑘=1𝑎_𝑘𝜏_𝑘𝑥^(0)+^𝑚_𝑘=1𝑎_𝑘^𝜏𝑘₀𝜏_𝑘−𝑠𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠,(2.6) and we deduce that 𝑥(0)=𝐴𝑥₀−^𝑚_𝑘=1𝑎_𝑘𝜏_𝑘𝑥^(0)−^𝑚_𝑘=1𝑎_𝑘^𝜏𝑘₀𝜏_𝑘−𝑠𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠,𝐴=^𝑚_𝑘=1𝑎_𝑘⁻¹.(2.7) Substitute from (2.7) into (2.5), we obtain 𝑥(𝑡)=𝐴𝑥₀−^𝑚_𝑘=1𝑎_𝑘^𝜏𝑘₀𝜏_𝑘−𝑠𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠+𝑥^(0)𝑡−𝐴^𝑚_𝑘=1𝑎_𝑘𝜏_𝑘+_𝑡0(𝑡−𝑠)𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠.(2.8) Let 𝑡=𝜂_𝑗, in (2.4), we obtain ^𝑛_𝑗=1𝑏_𝑗𝑥^𝜂_𝑗=^𝑛_𝑗=1𝑏_𝑗𝑥^(0)+^𝑛_𝑗=1𝑏_𝑗^𝜂𝑗₀𝑥𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠,₁−𝑥^(0)=𝑥^(0)^𝑛_𝑗=1𝑏_𝑗+^𝑛_𝑗=1𝑏_𝑗^𝜂𝑗₀𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠,(2.9) and we deduce that 𝑥^(0)=𝐵𝑥₁−^𝑛_𝑗=1𝑏_𝑗^𝜂𝑗₀𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠,𝐵=^𝑛_𝑗=1𝑏_𝑗+1⁻¹.(2.10) Substitute from (2.10) into (2.8), we obtain 𝑥(𝑡)=𝐴𝑥₀−^𝑚_𝑘=1𝑎_𝑘^𝜏𝑘₀𝜏_𝑘+𝐵𝑡−𝐴_𝑚−𝑠𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠_𝑘=1𝑎_𝑘𝜏_𝑘𝑥¹−^𝑛_𝑗=1𝑏_𝑗^𝜂𝑗₀+𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠_𝑡0,(𝑡−𝑠)𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠,(2.11) which proves that the solution of the nonlocal problem (1.3)-(1.4) can be expressed by the integral equation (2.3).

3. Existence of Solution

We study here the existence of at least one monotonic nondecreasing solution 𝑥∈𝐶[0,1] for the integral equation (2.3).

Theorem 3.1. Assume that (i)–(iv) are satisfied. Then the nonlocal problem (1.3)-(1.4) has at least one solution 𝑥∈𝐶[0,1].

Proof. Define the subset 𝑄_𝑟⊂𝐶(0,1) by 𝑄_𝑟={𝑥∈𝐶∶|𝑥(𝑡)|≤𝑟,𝑟=(𝐴𝑥₀+𝐵𝑥₁+(3−𝐵)‖𝑎‖)/(1−(3−𝐵)𝑏),𝑟>0}. Clear the set 𝑄_𝑟 which is nonempty, closed, and convex.
Let 𝐻 be an operator defined by(𝐻𝑥)(𝑡)=𝐴𝑥₀−^𝑚_𝑘=1𝑎_𝑘^𝜏𝑘₀𝜏_𝑘+𝐵𝑡−𝐴_𝑚−𝑠𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑s_𝑘=1𝑎_𝑘𝜏_𝑘𝑥¹−^𝑛_𝑗=1𝑏_𝑗^𝜂𝑗₀+𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠_𝑡0(𝑡−𝑠)𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠.(3.1)
Let 𝑥∈𝑄_𝑟, then||(𝐻𝑥)(𝑡)||≤𝐴𝑥₀+^𝑚_𝑘=1𝑎_𝑘^𝜏𝑘₀𝜏_𝑘+𝐵𝑡−𝐴_𝑚−𝑠||𝑓(𝑠,𝑥(𝜙(𝑠)))||𝑑𝑠_𝑘=1𝑎_𝑘𝜏_𝑘𝑥¹+^𝑛_𝑗=1𝑏_𝑗^𝜂𝑗₀+||𝑓(𝑠,𝑥(𝜙(𝑠)))||𝑑𝑠_𝑡0≤𝐴(𝑥𝑡₀−𝑠)||𝑓(𝑠,𝑥(𝜙(𝑠)))||𝑑𝑠+^𝑚_𝑘=1𝑎_𝑘+𝐵𝑥₁¹⁰||𝑎(𝑠)||+𝑏||𝑥(𝜙(𝑠))||𝑑𝑠+^𝑛_𝑗=1𝑏_𝑗+¹⁰||𝑎(𝑠)||+𝑏||𝑥(𝜙(𝑠))||𝑑𝑠₁₀≤𝐴𝑥₀||𝑎(𝑠)||+𝑏||𝑥(𝜙(𝑠))||𝑑𝑠+‖𝑎‖+𝑏sup_𝑡∈𝐼||𝑥(𝜙(𝑡))||+𝐵𝑥₁+𝐵^𝑛_𝑗=1+𝑏𝐵_𝑛𝑏_𝑗‖𝑎‖_𝑗=1≤𝐴𝑥₀𝑏_𝑗sup_𝑡∈𝐼||𝑥(𝜙(𝑡))||+‖𝑎‖+𝑏sup_𝑡∈𝐼||𝑥(𝜙(𝑡))||≤𝐴𝑥₀+𝐵𝑥₁+2‖𝑎‖+2𝑏‖𝑥‖+(1−𝐵)‖𝑎‖+𝑏(1−𝐵)‖𝑥‖+𝐵𝑥₁+(3−𝐵)‖𝑎‖+(3−𝐵)𝑏𝑟≤𝑟,(3.2) then 𝐻∶𝑄_𝑟→𝑄_𝑟 and {𝐻𝑥(𝑡)} is uniformly bounded in 𝑄_𝑟.
Also for 𝑡₁,𝑡₂∈[0,1] such that 𝑡₁<𝑡₂, we have(𝐻𝑥)𝑡₂−(𝐻𝑥)𝑡₁=𝐵𝑡₂−𝐴^𝑚_𝑘=1𝑎_𝑘𝜏_𝑘𝑥¹−^𝑛_𝑗=1𝑏_𝑗^𝜂𝑗₀+𝑓(𝑠,𝑥(𝜙(𝑡)))𝑑𝑠_𝑡20−𝐵𝑡𝑡₂₁−𝑠𝑓(𝑠,𝑥(𝜙(𝑡)))𝑑𝑠−𝐴^𝑚_𝑘=1𝑎_𝑘𝜏_𝑘𝑥¹−^𝑛_𝑗=1𝑏_𝑗^𝜂𝑗₀−𝑓(𝑠,𝑥(𝜙(𝑡)))𝑑𝑠_𝑡10=𝐵𝑡₂𝑡₁−𝑠𝑓(𝑠,𝑥(𝜙(𝑡)))𝑑𝑠−𝑡₁𝑥₁−^𝑛_𝑗=1𝑏_𝑗^𝜂𝑗₀+𝑓(s,𝑥(𝜙(𝑡)))𝑑𝑠_𝑡10+𝑡₂−𝑡₁𝑓(𝑠,𝑥(𝜙(𝑡)))𝑑𝑠_𝑡2𝑡1𝑡₂−𝑠𝑓(𝑠,𝑥(𝜙(𝑡)))𝑑𝑠.(3.3) Then ||(𝐻𝑥)𝑡₂−(𝐻𝑥)𝑡₁||≤𝐵||𝑡₂−𝑡₁||𝑥₁+^𝑛_𝑗=1𝑏_𝑗+||𝑡^𝜂𝑗₀||𝑎(𝑠)||+𝑏||𝑥(𝜙(𝑠))||𝑑𝑠₂+−𝑡₁||^𝑡1₀||𝑎(𝑠)||+𝑏||𝑥(𝜙(𝑠))||𝑑𝑠_𝑡2𝑡1≤𝐵||𝑡₂𝑡₂−𝑠||𝑎(𝑠)||+𝑏||𝑥(𝜙(𝑠))||𝑑𝑠−𝑡₁||𝑥₁+^𝑛_𝑗=1+||𝑡𝑏_𝑗[‖𝑎‖+𝑏𝑟₂]−𝑡₁||[‖𝑎‖+𝑏𝑟]+^𝑡2_𝑡1‖𝑎‖𝑑𝑠+𝑏𝑟𝑡₂−𝑡₁.(3.4) The above inequality shows that ||(𝐻𝑥)𝑡₂−(𝐻𝑥)𝑡₁||⟶0as𝑡₂⟶𝑡₁.(3.5) Therefore {𝐻𝑥(𝑡)} is equicontinuous. By the Arzelà-Ascoli theorem, {𝐻𝑥(𝑡)} is relatively compact.
Since all conditions of the Schauder theorem hold, then 𝐻 has a fixed point in 𝑄_𝑟 which proves the existence of at least one solution 𝑥∈𝐶[0,1] of the integral equation (2.3), wherelim_𝑡→0+𝑥(𝑡)=𝐴𝑥₀−^𝑚_𝑘=1𝑎_𝑘^𝜏𝑘₀𝜏_𝑘−𝐵𝐴_𝑚−𝑠𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠_𝑘=1𝑎_𝑘𝜏_𝑘𝑥¹−^𝑛_𝑗=1𝑏_𝑗^𝜂𝑗₀lim_𝑡→1−𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠=𝑥(0),𝑥(𝑡)=𝐴𝑥₀−^𝑚_𝑘=1𝑎_𝑘^𝜏𝑘₀𝜏_𝑘+𝐵1−𝐴_𝑚−𝑠𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠_𝑘=1𝑎_𝑘𝜏_𝑘𝑥¹−^𝑛_𝑗=1𝑏_𝑗^𝜂𝑗₀+𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠₁₀(1−𝑠)𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠=𝑥(1).(3.6) To complete the proof, we prove that the integral equation (2.3) satisfies nonlocal problem (1.3)-(1.4). Differentiating (2.3), we get 𝑥^(𝑡)=𝐵𝑥₁−^𝑛_𝑗=1𝑏_𝑗^𝜂𝑗₀𝑥𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠+^𝑡0𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠,(3.7)_(𝑡)=𝑓(𝑡,𝑥(𝜙(𝑡))).(3.8) Let 𝑡=𝜏_𝑘 in (2.3), we obtain 𝑥𝜏_𝑘=𝐴𝑥₀−^𝑚_𝑘=1𝑎_𝑘^𝜏𝑘₀𝜏_𝑘−𝑠𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠+^𝜏𝑘₀𝜏_𝑘−𝑠𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠,(3.9) which proves ^𝑚_𝑘=1𝑎_𝑘𝑥𝜏_𝑘=𝑥₀.(3.10) Also let 𝑡=𝜂_𝑗 in (3.7), we obtain 𝑥^𝜂_𝑗=𝐵𝑥₁−^𝑛_𝑗=1𝑏_𝑗^𝜂𝑗₀𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠+^𝜂𝑗₀𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠,(3.11) then ^𝑛_𝑗=1𝑏_𝑗𝑥^𝜂_𝑗=𝐵^𝑛_𝑗=1𝑏_𝑗𝑥¹−^𝑛_𝑗=1𝑏_𝑗^𝜂𝑗₀𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠+^𝑛_𝑗=1𝑏_𝑗^𝜂𝑗₀𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠.(3.12) Let 𝑡=0 in (3.7), we obtain 𝑥^(0)=𝐵𝑥₁−^𝑛_𝑗=1𝑏_𝑗^𝜂𝑗₀𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠.(3.13) Adding (3.12) and (3.13), we obtain 𝑥^(0)+^𝑛_𝑗=1𝑏_𝑗𝑥^𝜂_𝑗=𝑥₁.(3.14) This implies that there exists at least one solution 𝑥∈𝐶[0,1] of the nonlocal problem (1.3) and (1.4). This completes the proof.

Corollary 3.2. The solution of the problem (1.3)-(1.4) is monotonic nondecreasing.

Proof. Let𝑡₁<𝑡₂, we deduce from (2.3) that 𝑥𝑡₁=𝐴𝑥₀−^𝑚_𝑘=1𝑎_𝑘^𝜏𝑘₀𝜏_𝑘+𝐵𝑡₁−𝑠𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠−𝐴^𝑚_𝑘=1𝑎_𝑘𝜏_𝑘𝑥¹−^𝑛_𝑗=1𝑏_𝑗^𝜂𝑗₀+𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠_𝑡10<𝐴𝑥𝑡₁₀−𝑠𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠−^𝑚_𝑘=1𝑎_𝑘^𝜏𝑘₀𝜏_𝑘+𝐵𝑡₂−𝑠𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠−𝐴^𝑚_𝑘=1𝑎_𝑘𝜏_𝑘𝑥¹−^𝑛_𝑗=1𝑏_𝑗^𝜂𝑗₀+𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠_𝑡20𝑡₂−𝑠𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠=𝑥𝑡₂,(3.15) which proves that the solution 𝑥 of the problem (1.3)-(1.4) is monotonic nondecreasing.

3.1. Positive Solution

Let 𝑏_𝑗=0,𝑗=1,2,…𝑛 and 𝑥₁=0, then the nonlocal problem condition (1.4) will be^𝑚_𝑘=1𝑎_𝑘𝑥𝜏_𝑘=𝑥₀,𝑥^(0)=0.(3.16)

Theorem 3.3. Let the assumptions (i)–(iv) of Theorem 3.1 be satisfied. Then the solution of the nonlocal problem (1.3)–(3.16) is positive 𝑡∈[𝑑,1].

Proof. Let 𝑏_𝑗=0,𝑗=1,2,…𝑛 and 𝑥₁=0 in the integral equation (2.3) and the nonlocal condition (1.4), then the solution of the nonlocal problem (1.3)–(3.16) will be given by the integral equation 𝑥(𝑡)=𝐴𝑥₀−^𝑚_𝑘=1𝑎_𝑘^𝜏𝑘₀𝜏_𝑘−𝑠𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠+^𝑡0(𝑡−𝑠)𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠,(3.17) where 𝐴=(∑^𝑚𝑘=1𝑎_𝑘)⁻¹.
Let 𝑡∈[𝑑,1], then^𝜏𝑘₀𝜏_𝑘−𝑠𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠≤^𝑡0(𝑡−𝑠)𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠,𝜏_𝑘≤𝑡,_𝑚_𝑘=1𝑎_𝑘^𝜏𝑘₀𝜏_𝑘−𝑠𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠≤^𝑚_𝑘=1𝑎_𝑘^𝑡0(𝑡−𝑠)𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠.(3.18) Multiplying by 𝐴=(∑^𝑚𝑘=1𝑎_𝑘)⁻¹, we obtain 𝐴^𝑚_𝑘=1𝑎_𝑘^𝜏𝑘₀𝜏_𝑘−𝑠𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠≤𝐴^𝑚_𝑘=1𝑎_𝑘^𝑡0=(𝑡−𝑠)𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠_𝑡0(𝑡−𝑠)𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠,(3.19) and the solution 𝑥 of the nonlocal problem (1.3) and (3.16), given by the integral equation (3.17), is positive for 𝑡∈[𝑑,1]. This complete the proof.

Example 3.4. Consider the nonlocal problem of the second-order functional differential equation (1.3) with two-point boundary condition 𝑥^(0)=0,𝑥(𝜂)=𝑥₀,𝜂∈(𝑎,𝑑)⊂(0,1).(3.20) Applying our results here, we deduce that the two-point boundary value problem (1.3)–(3.20) has at least one monotonic nondecreasing solution 𝑥∈𝐶[0,1] represented by the integral equation 𝑥(𝑡)=𝑥₀−^𝜂0(𝜂−𝑠)𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠+^𝑡0(𝑡−𝑠)𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠.(3.21) This the solution is positive with 𝑡>𝜂.

4. Nonlocal Integral Condition

Let 𝑥∈𝐶[0,1] be the solution of the nonlocal problem (1.3) and (1.4).

Let 𝑎_𝑘=𝑡_𝑘−𝑡_𝑘−1,𝜏_𝑘∈(𝑡_𝑘−1,𝑡_𝑘)⊂(𝑎,𝑑)⊂(0,1) and let 𝑏_𝑗=𝜉_𝑗−𝜉_𝑗−1,𝜂_𝑗∈(𝜉_𝑗−1,𝜉_𝑗)⊂(𝑐,𝑒)⊂(0,1), then^𝑚_𝑘=1𝑡_𝑘−𝑡_𝑘−1𝑥𝜏_𝑘=𝑥₀,𝑥^(0)+^𝑛_𝑗=1𝜉_𝑗−𝜉_𝑗−1𝑥^𝜂_𝑗=𝑥₁.(4.1) From the continuity of the solution 𝑥 of the nonlocal problem (1.3) and (1.4), we obtainlim_{𝑚𝑚→∞}_𝑘=1𝑡_𝑘𝑥−𝑡_𝑘−1𝑥𝜏_𝑘=^𝑑𝑎𝑥(𝑠)𝑑𝑠,_(0)+lim_{𝑛𝑛→∞}_𝑗=1𝜉_𝑗−𝜉_𝑗−1𝑥^𝜂_𝑗=𝑥^(0)+^𝑒𝑐𝑥^(𝑠)𝑑𝑠,(4.2) and the nonlocal condition (1.4) transformed to the integral condition^𝑑𝑎𝑥(𝑠)𝑑𝑠=𝑥₀,𝑥^(0)+𝑥(𝑒)−𝑥(𝑐)=𝑥₁,(4.3) and the solution of the integral equation (2.3) will be𝑥(𝑡)=(𝑑−𝑎)⁻¹𝑥⁰−^𝑑𝑎^𝑡0+((𝑏−𝑐)+1)₋₁(𝑡−𝑠)𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠𝑑𝑡(𝑡−1)𝑥₁−^𝑒𝑐^𝑡0+𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠𝑑𝑡_𝑡0𝑓(𝑠,𝑥(𝜙(𝑠)))𝑑𝑠.(4.4) Now, we have the following theorem.

Theorem 4.1. Let the assumptions (i)–(iv) of Theorem 3.1 be satisfied. Then the nonlocal problem 𝑥^(𝑡)=𝑓(𝑡,𝑥(𝜙(𝑡))),𝑡∈(0,1)_𝑑𝑎,𝑥(𝑠)𝑑𝑠=𝑥₀,𝑥^(0)+𝑥(𝑒)−𝑥(𝑐)=𝑥₁(4.5) has at least one monotonic nondecreasing solution 𝑥∈𝐶[0,1] represented by (4.4).