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In this paper, we provide some appropriate conditions for the existence of solutions for a perturbed fractional neutral integro-differential system under the deformable derivative in a Banach space. Using the Banach contraction principle and Krasnoselskii's fixed point theorem, we establish some new existence theorems. Moreover, we provide two numerical examples to demonstrate the applicability of the theoretical results [ABSTRACT FROM AUTHOR]

The current paper introduces a novel generalization of cone metric spaces called type I and type II composed cone metric spaces. Therefore, examples of a type I and type II composed cone metric space, which is not a cone metric space, are given. We establish some results of fixed point precisely about Hardy–Rogers type contraction on C2CMS and provide examples. Finally, we present an application of our results and how our results solve the Fredholm integral equation of generalizing several existing and unique fixed point theorems. [ABSTRACT FROM AUTHOR]

In this paper, we establish a relation-theoretic set-valued version of the fixed point result of Ćirić and Prešić (Acta Math. Univ. Comen. LXXVI(2):143–147, 2007) on metric spaces endowed with an arbitrary binary relation. The results of this paper, generalize and unify the fixed point results of Ćirić and Prešić (Acta Math. Univ. Comen. LXXVI(2):143–147, 2007), Shukla and López (Quaest. Math. 45(3):1–16, 2019), and Shukla and Radenović (An. Ştiinţ. Univ. 'Al.I. Cuza' Iaşi, Mat. 63(2):339–350, 2017) in product spaces. Some examples are provided that justify and establish the importance of our results. As applications of our main result, we have established the existence of solutions to differential inclusion problems and the weak asymptotical stability and a global attractivity of the equilibrium point of a difference inclusion problem. The use of arbitrary binary relations in our results permits us to apply the results to the differential inclusion problems and difference inclusion problems with weaker assumptions than those used in the papers mentioned above. [ABSTRACT FROM AUTHOR]

In this paper, we consider fractional neutral differential equations with multipoint boundary value conditions involving Hadamard derivatives and integrals. We obtain the existence and uniqueness of the solution of the equation by using several fixed point theorems, and we also consider the Ulam–Hyers stability of the solution. In addition, we study the differential inclusion problem with multipoint boundary value conditions and prove the existence of the solution of the boundary value problem when the multivalued map has convex values. We also give several examples to illustrate the feasibility of the results. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we establish a relation-theoretic set-valued version of the fixed point result of Ćirić and Prešić (Acta Math. Univ. Comen. LXXVI(2):143–147, 2007) on metric spaces endowed with an arbitrary binary relation. The results of this paper, generalize and unify the fixed point results of Ćirić and Prešić (Acta Math. Univ. Comen. LXXVI(2):143–147, 2007), Shukla and López (Quaest. Math. 45(3):1–16, 2019), and Shukla and Radenović (An. Ştiinţ. Univ. ‘Al.I. Cuza’ Iaşi, Mat. 63(2):339–350, 2017) in product spaces. Some examples are provided that justify and establish the importance of our results. As applications of our main result, we have established the existence of solutions to differential inclusion problems and the weak asymptotical stability and a global attractivity of the equilibrium point of a difference inclusion problem. The use of arbitrary binary relations in our results permits us to apply the results to the differential inclusion problems and difference inclusion problems with weaker assumptions than those used in the papers mentioned above.

In the paper, we study a boundary value problem for a class of ψ-Hilfer fractional-order Langevin equations with multi-point integral boundary conditions. Existence and uniqueness results are established by using well-known fixed point theorems. Examples illustrating the main results are also included. [ABSTRACT FROM AUTHOR]

In this paper, several novel existence and multiplicity results are established for a coupled functional differential system with multi-parameters. The discussion is based upon fixed point theory, and our main findings enrich and complement those available in the literature. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we provide some appropriate conditions for the existence of solutions for a perturbed fractional neutral integro-differential system under the deformable derivative in a Banach space. Using the Banach contraction principle and Krasnoselskii’s fixed point theorem, we establish some new existence theorems. Moreover, we provide two numerical examples to demonstrate the applicability of the theoretical results

Abstract The current paper introduces a novel generalization of cone metric spaces called type I and type II composed cone metric spaces. Therefore, examples of a type I and type II composed cone metric space, which is not a cone metric space, are given. We establish some results of fixed point precisely about Hardy–Rogers type contraction on C2CMS and provide examples. Finally, we present an application of our results and how our results solve the Fredholm integral equation of generalizing several existing and unique fixed point theorems.

In this paper, existence, localization and uniqueness results of solutions to elliptic Dirichlet boundary value problems are established. The approach is based on the nonlinear alternative of Leray-Schauder, the Brouwer fixed point theorem and the Galerkin method. MSC: 35J60, 47H10. [ABSTRACT FROM AUTHOR]

Abstract This paper aims to deal with the Cauchy problem of a fractional Lorentz force equation. By the methods of reducing and topological degree in cone, the existence and multiplicity of solutions to the problem were obtained, which extend and enrich some previous results.

This paper investigates some nonlinear third-order ordinary differential equations and inclusions with anti-periodic type integral boundary conditions and multi-strip boundary conditions. To establish the existence results for the given problems, we apply standard tools of fixed point theory for single-valued and multi-valued problems. The results are illustrated with the aid of examples. [ABSTRACT FROM AUTHOR]

We introduce a notion of nonlinear cyclic orbital (ξ − F) -contraction and prove related results. With these results, we address the existence and uniqueness results with periodic/anti-periodic boundary conditions for: 1. The nonlinear multi-order fractional differential equation L (D) θ (ς) = σ (ς , θ (ς)) , ς ∈ J = [ 0 , A ] , A > 0 , where L (D) = γ w c D δ w + γ w − 1 c D δ w − 1 + ⋯ + γ 1 c D δ 1 + γ 0 c D δ 0 , γ ♭ ∈ R (♭ = 0 , 1 , 2 , 3 , ... , w) , γ w ≠ 0 , 0 ≤ δ 0 < δ 1 < δ 2 < ⋯ < δ w − 1 < δ w < 1 ; 2. The nonlinear multi-term fractional delay differential equation L (D) θ (ς) = σ (ς , θ (ς) , θ (ς − τ)) , ς ∈ J = [ 0 , A ] , A > 0 ; θ (ς) = σ ¯ (ς) , ς ∈ [ − τ , 0 ] , where L (D) = γ w c D δ w + γ w − 1 c D δ w − 1 + ⋯ + γ 1 c D δ 1 + γ 0 c D δ 0 , γ ♭ ∈ R (♭ = 0 , 1 , 2 , 3 , ... , w) , γ w ≠ 0 , 0 ≤ δ 0 < δ 1 < δ 2 < ⋯ < δ w − 1 < δ w < 1 ; moreover, here D δ c is predominantly called Caputo fractional derivative of order δ. [ABSTRACT FROM AUTHOR]

This paper studies the existence of positive periodic solutions of the following delayed differential equation: u′+a(t)u=f(t,u(t−τ(t))), where a,τ∈C(R,R) are ω-periodic functions with ∫0ωa(t)dt=0, f:R×[0,∞)→R is continuous and ω-periodic with respect to t. By means of the fixed point theorem in cones, several new existence theorems are established. Our main results enrich and complement those available in the literature. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider fractional neutral differential equations with multipoint boundary value conditions involving Hadamard derivatives and integrals. We obtain the existence and uniqueness of the solution of the equation by using several fixed point theorems, and we also consider the Ulam–Hyers stability of the solution. In addition, we study the differential inclusion problem with multipoint boundary value conditions and prove the existence of the solution of the boundary value problem when the multivalued map has convex values. We also give several examples to illustrate the feasibility of the results.

In this article, we propose an iterative method, called the GA iterative method, to approximate the fixed points of generalized α-nonexpansive mappings in uniformly convex Banach spaces. Further, we obtain some convergence results of the new iterative method. Also, we provide a nontrivial example of a generalized α-nonexpansive mapping and with the example, we carry out a numeral experiment to show that our new iterative algorithm is more efficient than some existing iterative methods. Again, we present an interesting strategy based on the GA iterative method to solve nonlinear third-order boundary value problems (BVPs). For this, we derive a sequence named the GA–Green iterative method and show that the sequence converges strongly to the fixed point of an integral operator. Finally, the approximation of the solution for a nonlinear integrodifferential equation via our new iterative method is considered. We present some illustrative examples to validate our main results in the application sections of this article. Our results are a generalization and an extension of several prominent results of many well-known authors in the literature. [ABSTRACT FROM AUTHOR]

We discuss the solvability of an infinite system of first order ordinary differential equations on the half line, subject to nonlocal initial conditions. The main result states that if the nonlinearities possess a suitable 'sub-linear' growth then the system has at least one solution. The approach relies on the application, in a suitable Fréchet space, of the classical Schauder-Tychonoff fixed point theorem. We show that, as a special case, our approach covers the case of a system of a finite number of differential equations. An illustrative example of an application is also provided. [ABSTRACT FROM AUTHOR]

Abstract In the paper, we study a boundary value problem for a class of ψ-Hilfer fractional-order Langevin equations with multi-point integral boundary conditions. Existence and uniqueness results are established by using well-known fixed point theorems. Examples illustrating the main results are also included.

Abstract In this paper, several novel existence and multiplicity results are established for a coupled functional differential system with multi-parameters. The discussion is based upon fixed point theory, and our main findings enrich and complement those available in the literature.

Abstract This paper is concerned with the impact of the parameter on the existence of different types of solutions for a class of nonlinear fractional integral boundary value problems with a parameter that causes the sign of Green’s function associated with the BVP to change. By using the Guo–Krasnoselskii fixed point theorem, the Leray–Schauder nonlinear alternative, and the analytic technique, we give the range of the parameter for the existence of strong positive solutions, strong negative solutions, negative solutions, and sign-changing solutions for the boundary value problem. Some examples are given to illustrate our main results.

Abstract This paper studies the existence of positive periodic solutions of the following delayed differential equation: u′+a(t)u=f(t,u(t−τ(t))), $$ u'+a(t)u=f\bigl(t,u\bigl(t-\tau (t)\bigr)\bigr), $$ where a,τ∈C(R,R) $a, \tau \in C(\mathbb{R},\mathbb{R})$ are ω-periodic functions with ∫0ωa(t)dt=0 $\int_{0}^{\omega }a(t)\,dt=0$, f:R×[0,∞)→R $f:\mathbb{R}\times [0, \infty)\to \mathbb{R}$ is continuous and ω-periodic with respect to t. By means of the fixed point theorem in cones, several new existence theorems are established. Our main results enrich and complement those available in the literature.

Abstract In this paper, we introduce a new concept of coupled non-separated boundary conditions and solve a coupled system of fractional differential equations supplemented with these conditions. The existence results obtained in the given configuration are not only new but also yield some new special results corresponding to particular values of the parameters involved in the problem. For the illustration of the existence and uniqueness result, an example is constructed.

Our concern is the existence of positive solutions of the discrete Neumann boundary value problem where $f: [1,T]_{\mathbb{Z}}\times\mathbb{R}^{+}\to\mathbb{R}$ is a sign-changing function. By using the Guo-Krasnosel'skiĭ fixed point theorem, the existence and multiplicity of positive solutions are established. The nonlinear term $f(t,z)$ may be unbounded below or nonpositive for all $(t,z)\in[1,T]_{\mathbb{Z}}\times\mathbb{R}^{+}$. [ABSTRACT FROM AUTHOR]

We discuss the existence and uniqueness of solutions for a fractional differential equation of order q ∊ (n - 1, n] with slit-strips type boundary conditions. The slit-strips type boundary condition states that the sum of the influences due to finite strips of arbitrary lengths is related to the value of the unknown function at an arbitrary position (nonlocal point) in the slit (a part of the boundary off the two strips). The desired results are obtained by applying standard tools of the fixed point theory and are well illustrated with the aid of examples. We also extend our discussion to the cases of arbitrary number of nonlocal points in the slit, the nonlocal multi-substrips conditions and Riemann-Liouville type slit-strips boundary conditions. [ABSTRACT FROM AUTHOR]

A class of weak Ψ-contractions satisfying the C-condition is defined on metric spaces. The existence and uniqueness of fixed points of such maps are discussed both on metric spaces and on partially ordered metric spaces. The results are applied to a first order periodic boundary value problem. [ABSTRACT FROM AUTHOR]

Using the Kuratowski measure of noncompactness and progressive estimation method, we obtain the existence results of mild solutions for impulsive partial neutral second-order functional integro-differential equations with infinite delay in Banach spaces. The compactness condition of the impulsive term, some restrictive conditions on a priori estimation and noncompactness measure estimation have been deleted. Our conditions are simple and our results essentially improve and extend some known results. As applications, some examples are provided to illustrate the obtained results. MSC: 34K30, 34K40, 35R10, 47D09. [ABSTRACT FROM AUTHOR]

Using the Mönch fixed point theorem, this article proves the existence of mild solutions for nonlinear mixed type integro-differential functional evolution equations with nonlocal conditions in Banach spaces. Some restricted conditions on a priori estimation and measure of noncompactness estimation have been deleted, and compactness conditions of evolution operators or compactness conditions on a nonlinear term [InlineEquation not available: see fulltext.] have been weakened. Our results extend and improve many known results. MSC: 34G20, 34K30. [ABSTRACT FROM AUTHOR]