Boundary value problems for systems of differential, difference and fractional equations. Positive solutions (Q2790392)

This well-written book is a collection of recent works by the authors who are pioneering researchers in the community of differential and difference equations. This particular book mainly focuses on existence and nonexistence of positive solutions of systems of second-order and higher-order differential, difference and fractional equations satisfying boundary value conditions including Riemann-Stieltjes integral boundary conditions. Many of the results heavily make use of the application of Guo-Krasnosel'skii's fixed point theorems. A few applications also resort to the Schauder fixed-point theorem, the Leray-Schauder fixed-point theorem, and fixed point index. The key tool, Greens' functions, are constructed and examples are given as illustrations of the main results for each problem discussed in the book.NEWLINENEWLINEThe book contains 5 chapters. Chapter 1 talks about the following system of second-order ordinary differential equations: NEWLINE\[NEWLINE (a(t)u'(t))'-b(t)u(t)+\lambda p(t) f(t,u(t),v(t))=0, 0<t<1,NEWLINE\]NEWLINENEWLINE\[NEWLINE (c(t)v'(t))'-d(t)v(t)+\mu q(t)g(t,u(t),v(t))=0, 0<t<1NEWLINE\]NEWLINE satisfying the Riemann-Stieltjes integral boundary conditions NEWLINE\[NEWLINE \alpha u(0)-\beta a(0) u'(0)=\int_0^1 u(s)d H_1(s), \gamma u(1)+\delta a(1) u'(1)=\int_0^1 u(s)d H_2(s),NEWLINE\]NEWLINENEWLINE\[NEWLINE \tilde{\alpha} v(0)-\tilde{\beta} c(0) v'(0)=\int_0^1 v(s)d K_1(s), \tilde{\gamma} v(1)+\tilde{\delta} c(1) v'(1)=\int_0^1 v(s)d K_2(s), NEWLINE\]NEWLINE Using conditions involving sub-linearity and super-linearity of nonnegative \(f\), \(g\) with respect to the second and third components, the application of Guo-Krasnosel'skii's fixed point theorems yields the existence of at least one positive solution and nonexistence of positive solutions of the above boundary value problem for both of the cases with parameters \(\lambda,\mu\) and the case without these parameters. The book also includes the problem when the nonlinear terms \(f\) and \(g\) are singular at \(t=0\) and \(t=1\). Several particular cases of the above problems are presented in Section 1.5 to demonstrate the main results of the general problems. At the end of the chapter, the authors study a similar problem but with an additional constant term in the integral boundary condition. The existence result of positive solutions is established by using the Schauder fixed point theorem. The nonexistence theorem and examples are also provided.NEWLINENEWLINEChapter 2 studies systems of higher-order ordinary differential equations NEWLINE\[NEWLINE u^{(n)}(t)+\lambda a(t) f(t,u(t),v(t)), 0<t<T,NEWLINE\]NEWLINENEWLINE\[NEWLINE v^{(m)}(t)+\mu b(t) g(t,u(t),v(t)), 0<t<TNEWLINE\]NEWLINE with multi-point boundary conditions NEWLINE\[NEWLINE u(0)=\sum\limits_{i=1}^p a_iu(\xi_i), u'(0)=\cdots=u^{(n-2)}(0)=0, \quad u(T) = \sum\limits_{i=1}^q b_i u(\eta_i),NEWLINE\]NEWLINENEWLINE\[NEWLINE v(0)=\sum\limits_{i=1}^r c_iv(\zeta_i), v'(0)=\cdots=v^{(m-2)}(0)=0, \quad v(T) = \sum\limits_{i=1}^l d_i v(\rho_i). NEWLINE\]NEWLINE Multi-point boundary conditions are a special case of integral boundary conditions. Similar to the first chapter, the Guo-Krasnosel'skii fixed point theorems and Schauder's fixed point theorem are used to established the existence and nonexistence of positive solutions of the above problem for cases with and without parameters \(\lambda,\mu\). Then the book specifies some particular problems and discusses the non-singularity case of \(f,g\). Different from Chapter 1, this chapter also studies the problem when \(f,g\) change signs. The nonlinear alternative of Leray-Schauder type is used to establish existence theorems of positive solutions.NEWLINENEWLINEChapter 3 focuses on systems of second-order difference equations with multi-point boundary conditions: NEWLINE\[NEWLINE \Delta^2u_{n-1}+\lambda s_n f(n,u_n,v_n)=0, n=1,2,\ldots,N-1, NEWLINE\]NEWLINENEWLINE\[NEWLINE \Delta^2v_{n-1}+\mu t_n g(n,u_n,v_n)=0, n=1,2,\ldots,N-1. NEWLINE\]NEWLINE The problems with and without parameters and particular cases are studied. Existence and non-existence of positive solutions are obtained by Guo-Krasnosel'skii's fixed point theorems and fixed point index. Then the problem with additional positive constant terms in the boundary conditions is discussed in Section 3.4. Green's functions are constructed for each problem.NEWLINENEWLINEChapter 4 concentrates on systems of Riemann-Liouville fractional differential equations with uncoupled integral boundary conditions NEWLINE\[NEWLINE D_{0+}^\alpha u(t)+\lambda f(t,u(t),v(t))=0, 0<t<1,NEWLINE\]NEWLINENEWLINE\[NEWLINE D_{0+}^\beta v(t)+\mu g(t,u(t),v(t))=0, 0<t<1 NEWLINE\]NEWLINE with the uncoupled integral boundary conditions NEWLINE\[NEWLINE u(0)=u'(0)=\cdots=u^{(n-2)}(0)=0, u(1)=\int_0^1 u(s)dH(s),NEWLINE\]NEWLINENEWLINE\[NEWLINE v(0)=v'(0)=\cdots=v^{(m-2)}(0)=0, v(1)=\int_0^1 v(s)dK(s), NEWLINE\]NEWLINE where \(n-1<\alpha\leq n\), \(m-1<\beta\leq m\), \(n,m\geq 3\), and \(f,g\) are nonnegative. By using Guo-Krasnosel'skii's fixed point theorem, the existence and nonexistence of positive solutions of the above boundary value problem with and without parameters are investigated. The singular case is also discussed. In Section 4.3, the case of uncoupled boundary conditions with additional positive constants is presented by using the Schauder fixed point theorem. Section 4.4 discusses the sign-changing nonlinearity case. Main results are established by fixed-point index and Guo-Krasnosel'skii's fixed point theorem.NEWLINENEWLINEThe last Chapter 5 investigates the systems of Riemann-Liouville fractional differential equations with coupled integral boundary conditions: NEWLINE\[NEWLINE D_{0+}^\alpha u(t)+\lambda f(t,u(t),v(t))=0, 0<t<1,NEWLINE\]NEWLINENEWLINE\[NEWLINE D_{0+}^\beta v(t)+\mu g(t,u(t),v(t))=0, 0<t<1NEWLINE\]NEWLINE with the uncoupled integral boundary conditions NEWLINE\[NEWLINE u(0)=u'(0)=\cdots=u^{(n-2)}(0)=0, u(1)=\int_0^1 v(s)dH(s),NEWLINE\]NEWLINENEWLINE\[NEWLINE v(0)=v'(0)=\cdots=v^{(m-2)}(0)=0, v(1)=\int_0^1 u(s)dK(s), NEWLINE\]NEWLINE where \(n-1<\alpha\leq n\), \(m-1<\beta\leq m\), \(n,m\geq 3\), and \(f,g\) are nonnegative. The authors establish sufficient conditions for the existence and nonexistence of positive solutions by Guo-Krasnosel'skii's fixed point theorem. Then in Section 5.2, the problem without parameters is studied by fixed point index theory. The problem with \(f,g\) being singular is also discussed using the Guo-Krasnosel'skii fixed point theorem. When constants are added to the integral conditions, the Schauder fixed point theorem is used to for the existence and nonexistence of positive solutions in Section 5.3. The \(f,g\) sign-changing case is discussed in Section 5.4.NEWLINENEWLINEThe book provides a thorough analysis of the qualitative analysis of systems of differential, difference and Riemann-Liouville fractional differential equations with integral boundary conditions using tools from nonlinear functional analysis. This text is a great resource for graduate students and scholars to learn classic methods and latest development in this field. The thoughts in the book could encourage interest in further study in the area of differential equations or other fields.